parametric instability rates in periodically driven...

Parametric Instability Rates in Periodically Driven Band Systems

S. Lellouch,1,* M. Bukov,2 E. Demler,3 and N. Goldman1,†1Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles,

CP 231, Campus Plaine, B-1050 Brussels, Belgium2Department of Physics, Boston University,

590 Commonwealth Avenue, Boston, Massachusetts 02215, USA3Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

(Received 20 October 2016; revised manuscript received 18 February 2017; published 5 May 2017)

In this work, we analyze the dynamical properties of periodically driven bandmodels. Focusing on the caseof Bose-Einstein condensates, and using a mean-field approach to treat interparticle collisions, we identify theorigin of dynamical instabilities arising from the interplay between the external drive and interactions. Wepresent a widely applicable generic numerical method to extract instability rates and link parametricinstabilities to uncontrolled energy absorption at short times. Based on the existence of parametric resonances,we then develop an analytical approach within Bogoliubov theory, which quantitatively captures theinstability rates of the system and provides an intuitive picture of the relevant physical processes, including anunderstanding of how transverse modes affect the formation of parametric instabilities. Importantly, ourcalculations demonstrate an agreement between the instability rates determined from numerical simulationsand those predicted by theory. To determine the validity regime of the mean-field analysis, we compare thelatter to the weakly coupled conserving approximation. The tools developed and the results obtained in thiswork are directly relevant to present-day ultracold-atom experiments based on shaken optical lattices and areexpected to provide an insightful guidance in the quest for Floquet engineering.

DOI: 10.1103/PhysRevX.7.021015 Subject Areas: Atomic and Molecular Physics,Quantum Physics, Superfluidity

I. INTRODUCTION

Periodically driven systems have been widely studiedover the past decades, revealing rich and exotic phenomenain a large class of physical systems. Some of the originalapplications include examples from classical mechanics,e.g., the periodically kicked rotor [1] and the Kapitzapendulum [2], which display fascinating effects, such asdynamical stabilization [3] or integrability to chaos tran-sitions. More recently, studies of their quantum counter-parts have led to remarkable applications in a wide range ofphysical platforms, such as ion traps [4], photonic crystals[5], irradiated graphene [6–8] and ultracold gases in opticallattices [9,10].A major reason for the renewed interest in periodically

driven quantum systems is the fact that they constitute theessential ingredient for Floquet engineering [6,9–12]. Tosee this, recall that periodically driven quantum systemsobey Floquet’s theorem [13], which implies that the time-evolution operator Uðt; 0Þ of a system, whose dynamics is

generated by the Hamiltonian Hðtþ TÞ ¼ HðtÞ, can bewritten as [11,14]

Uðt; 0Þ ¼ T exp

�−iZ

t

0

dt0Hðt0Þ�

¼ e−iKkickðtÞe−itHeff eiKkickð0Þ; ð1Þ

where T denotes time ordering. Here, we introduced thetime-independent effective Hamiltonian Heff , as well asthe time-periodic “kick” operator Kkickðtþ TÞ ¼ KkickðtÞ,which has zero average over one driving period [11]. It thenfollows that the stroboscopic time evolution (t ¼ NT,N ∈ Z) of such systems is governed by

UðNT; 0Þ ¼ e−iKkickð0Þe−iNTHeff eiKkickð0Þ ¼ e−iNTHF ;

HF ¼ e−iKkickð0ÞHeffeiKkickð0Þ:

Hence, up to a change of basis [15], the stroboscopictime evolution is completely described by the effectiveHamiltonian Heff , which can be designed by suitablytailoring the driving protocol [11,12,16–18]. Moreover,we note that the dynamics taking place within each periodof the drive, the so-called micromotion, is entirely capturedby the kick operator KkickðtÞ. On the theoretical side,perturbative methods to compute both the effectiveHamiltonian and the micromotion operators have been

*[email protected]†[email protected]

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.

PHYSICAL REVIEW X 7, 021015 (2017)

2160-3308=17=7(2)=021015(32) 021015-1 Published by the American Physical Society

https://doi.org/10.1103/PhysRevX.7.021015




https://creativecommons.org/licenses/by/4.0/

https://creativecommons.org/licenses/by/4.0/

developed in the form of an inverse-frequency expansion[11,14,16,19], variants of which include the Floquet-Magnus [12,20], the van Vleck [17], and the Brillouin-Wigner [18] expansions.On the experimental side, Floquet engineering has been

used to explore a wide variety of physical phenomena,ranging from dynamical trapping of ions in Paul traps [4] toone-way propagating states in photonic crystals [5]. In thecontext of ultracold quantum gases, this concept soonresonated with the idea of quantum simulation [21], as itbecame clear that Floquet engineering could offer powerfulschemes to reach novel models and properties, typicallyinaccessible in conventional condensed-matter materials[9,10]. For instance, shaken optical lattices have been usedto explore dynamical localization [22–26], photon-assistedtunneling [27,28], and driven-induced superfluid-to-Mott-insulator transitions [29–32]. In the field of disorderedquantum systems, the periodically kicked rotor has beenexploited as a flexible and successful simulator forAnderson localization [33–35]. More recently, a series ofexperiments implemented shaken or time-modulated opti-cal-lattice potentials in view of designing artificial gaugefields for neutral atoms [36–42], some of which led tonontrivial topological band structures [9] or frustratedquantum models [43,44]. Moreover, an experiment dem-onstrated the control over the population of targetedFloquet states [45]. Other applications of Floquet engineer-ing in cold atoms include the experimental realization ofstate-dependent lattices [46] and dynamical topologicalphase transitions [47], as well as proposals to dynamicallystimulate quantum coherence [48] or to create subwave-length optical lattices [49], synthetic dimensions [50], orpure Dirac-type dispersions [11,51]. In an even broadercontext, Floquet engineering is at the core of photo-inducedsuperconductivity [52–54] and topological insulators [55],and time crystals [56–58].Today, the theory of periodically driven quantum sys-

tems (including the resulting artificial gauge fields andtopological band structures [9,10]) is well established forsingle particles, and there is a strong need for incorporatinginterparticle interactions in the description of such systems.Indeed, interparticle interactions constitute an essentialingredient in simulating condensed-matter systems [21],and the relevance of many theoretical proposals stronglyrelies on the extent to which the systems of interest remainstable away from the noninteracting limit. Moreover, intheir quest for realizing new topological states of matter,experiments based on driven quantum systems are morethan ever eager to consider interacting systems [59];indeed, building on the successful optical-lattice imple-mentation of flat energy bands with nontrivial Chernnumbers [39], mastering interactions in this context wouldallow for the engineering of intriguing strongly correlatedstates, such as fractional Chern insulators [60]. In thisframework, we point out that a useful tool to derive the

effective low-energy physics of strongly correlated sys-tems, namely, the Schrieffer-Wolff transformation, wasgeneralized to periodically driven systems [61].However, until now, the presence of interparticle inter-

actions has led to complications. Recent experiments ontime-modulated (“shaken”) optical lattices [36–39,62] havereported severe heating, particle loss, and dissipativeprocesses, whose origin presumably stems from a richinterplay between the external drive and interparticlecollisions. In this sense, understanding the role of inter-actions in driven systems, and their relation to heatingprocesses and instabilities, is both of fundamental andpractical importance. On the theory side, several comple-mentary approaches have been considered to tackle theproblem. On the one hand, a perturbative scattering theory,leading to a so-called “Floquet Fermi golden rule” [63,64],has been developed to estimate heating and loss rates, and itshowed good agreement with the band-population dynam-ics reported in Ref. [39]. Extensions to Bose-Einsteincondensates (BEC) have been proposed [65,66]; however,to the best of our knowledge, the resulting loss rates havenot yet been confirmed by experiments or by numericalsimulations. On the other hand, various studies analyzedthe dynamical instabilities that occur in BEC trappedin moving, shaken, or time-modulated optical lattices[67–73]. For instance, Ref. [73] analyzed such dynamicalinstabilities through a numerical analysis of theBogoliubov–de Gennes (BdG) equations; however, no linkwas made with heating processes, and the approach lackeda conclusive comparison with analytics. Importantly, sev-eral studies pointed out the parametric nature of thoseinstabilities [70,71,74,75]. Interestingly, we note that para-metric instabilities, and their impact on energy absorptionand thermalization processes, have also been studiedtheoretically in a wider context, ranging from Luttingerliquids [76] to cosmology [77–79]. In the context of drivenoptical lattices, the parametric amplification of scatteredatom pairs, through phase-matching conditions involvingthe band structure and the drive, has also been identified[80–83]. Altogether, there is a strong need for a combinedanalytical and numerical study of driven optical-latticemodels, which would provide analytical estimates ofinstability and heating rates supported by numericalsimulations.

A. Scope of the paper

The object of the present paper is to explore the physicsof periodically driven bosonic band models, where inter-particle interactions are treated within mean-field(Bogoliubov) theory. Indeed, as in Refs. [65,66,73,75],we assume that the driven atomic gas forms a BEC, and weanalyze the properties of the corresponding Bogoliubovexcitations in response to the external drive. To do so, wedevelop a generic and widely applicable method to inves-tigate the short-time dynamics of periodically driven

LELLOUCH, BUKOV, DEMLER, and GOLDMAN PHYS. REV. X 7, 021015 (2017)

021015-2

bosonic systems and to extract the corresponding instabilityrates, both numerically and analytically.We first focus on a shaken one-dimensional (1D) lattice

model and extensively explore the occurrence of instabil-ities and heating in this system, before considering moreelaborate models. In a previous study on a similar model[73], a dynamical instability of the condensate was found tooccur above a critical interaction strength, which wasnumerically calculated as a function of the drive amplitudeK and frequency ω. This allowed the author to numericallymap out the stability boundaries between the stable andunstable phases. More recently, Refs. [74,75] suggestedthat such boundaries could be understood in terms of asimple energy-conservation criterion, which could be for-mulated within a parametric-instability analysis. In thiswork, we offer a significant advance in the description andunderstanding of the instabilities that affect bosonic peri-odically driven systems, both qualitatively and quantita-tively, as we summarize below:(1) We show how to obtain the instability rates as a

function of the model parameters, both numericallyand analytically, and we establish their relation to thegrowth of physical quantities in the system, such asthe energy, thus providing a link between dynamicalinstability and heating. Moreover, we identify differ-ent instability regimes in the system, which areassociated with different time scales and character-ized by a different behavior in the growth of physicalquantities. We introduce several observables thatreveal clear signatures of these instabilities, includ-ing the noncondensed (depleted) fraction and themomentum distribution of quasiparticle modes (seeFig. 1); we discuss how they could be observed incurrent ultracold-atom experiments.

(2) We present a full numerical solution of the mean-field problem, determining the stability diagram ofthe model from first principles (see Fig. 2). We stressthat the quantitative character and versatility of ourprocedure make it general enough to be applicable toa wide class of periodically driven systems, includ-ing those involving resonant modulations, higher-dimensional lattices, various geometries (e.g., squareor honeycomb lattices, continuous space), and spin-dependent lattices. Moreover, it can also be enrichedto incorporate a full experimental sequence, e.g.,including adiabatic state preparation [62]. Our re-sults are expected to determine experimentallyfavorable regimes by providing ab initio numerical(and analytical) estimates for the instability rates in avariety of experimental configurations.

(3) We perform an analytical treatment of the problembased on the existence of parametric resonances inthe Bogoliubov–de Gennes equations. In Ref. [74],following a weak-coupling conserving approxima-tion, it was argued that a driven-lattice model was

stable against parametric resonance provided thatthe drive frequency satisfies ω > 2Weff , with Weff

the bandwidth associated with Heff in the Bogoliu-bov approximation. Intuitively, this can be under-stood by recalling that the elementary excitationsof the system (i.e., the Bogoliubov phonons) arealways created in pairs, and thus, whenever the drivefrequency exceeds twice the maximum possibleexcitation energy, stability is ensured by energyconservation. However, our rigorous analytical der-ivation reveals that this simple criterion needs to berevised: As we demonstrate, an accurate qualitativeexplanation requires taking into account bothhigher-order photon-absorption processes and thedetuning from resonance within the parametric-resonance treatment [84]. This analysis allows us,for the first time, to derive the functional dependenceof the instability growth rate on the model param-eters and, ultimately, to understand all the features ofthe stability diagram (see Fig. 2).

(4) We extend the stability analysis to two-dimensional(2D) models and study the effects due to transversedirections, considering both the case of a transverselattice (i.e., a discrete transverse degree of freedom)and that of transverse tubes (i.e., a continuoustransverse degree of freedom). In the case ofcontinuous degrees of freedom, a theoretical sim-plification in the equations allows one to obtain

-3

-2

-1

0

1

2

34

20

-2-4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

nq/Vol

axq

x

qy

FIG. 1. The quasiparticle momentum distribution nq of a 2Dshaken BEC trapped in a one-dimensional lattice (along x), andallowed to move along a continuous (tube) transverse direction(y), develops unique resonance structures in the presence of aperiodic drive. Dominating the dynamics in the early stage ofevolution, they signal the onset of instabilities and providea clear experimental signature of the parametric resonancephenomenon explained in this work. The momentum distri-bution is shown after 60 driving periods (t ¼ 60T). Here, thedistribution nq is divided by the system’s volume Vol, ξdenotes the BEC healing length, and ax is the lattice spacingconstant. The precise model and the corresponding parametersare the same as in Fig. 17.

PARAMETRIC INSTABILITY RATES IN PERIODICALLY … PHYS. REV. X 7, 021015 (2017)

021015-3

very simple analytical formulas for instability rates,which are in perfect agreement with our full numeri-cal simulations. Moreover, this study provides aclear physical picture of how instabilities are en-hanced by the presence of transverse modes in thesystem, as already anticipated in Refs. [64,66]. Moregenerally, we argue that our theory should provideguidance for experiments; It illustrates, for instance,the advantage of working at high frequency and thenecessity of using a strong transverse confinement toreduce parametric instabilities. We also include adiscussion of finite-size effects, making a linkbetween the physics of double wells and that ofoptical lattices.

B. Outline

This paper is organized as follows. In Sec. II, we derivethe general mean-field equations that will constitute thebasis of our analysis. Section III is devoted to the numericalsolution of those equations, including details on the generalprocedure used and a presentation of the results, such as thestability diagram of the model under consideration, theidentification of several time scales and instability regimesin the problem, and the dynamics of various physicalobservables. In Sec. IV, we perform an analytical treatmentof the problem: Mapping the Bogoliubov equations on aparametric oscillator (Sec. IVA), we build an effectivemodel from which analytical instability rates can beinferred (Sec. IV B). The analytical results are presentedin Sec. IV C, including a discussion of the validity regimesof the approach. The case of finite-size systems is presentedin Sec. V. In Sec. VI, we discuss the case where a transversedirection is present (a lattice or a continuous one); thisincludes simple analytical formulas for instability rates aswell as a physical understanding of the enhancement ofinstabilities by transverse modes. Finally, in Sec. VII,we discuss the application of our results to the weaklyinteracting Bose-Hubbard model in the mean-field regime;in order to understand the role of nonlinear processes andstudy the regime of validity of our linearized analysis, weemploy a weak-coupling conserving approximation tostudy the leading-order (in the interaction strength) featuresof particle-conserving dynamics; we identify clear signa-tures of the instabilities (including the noncondensedfraction and the momentum distribution of quasiparticles)that could be directly probed by current ultracold-atomexperiments in modulated optical lattices.

II. MEAN-FIELD EQUATIONS ANDSTABILITY ANALYSIS

Consider a Bose gas in a shaken 1D lattice, with mean-field interactions, governed by the time-dependent Gross-Pitaevskii equation (GPE) [10,73]

i∂tan ¼ −Jðanþ1 þ an−1Þ þ K cosðωtÞnan þUjanj2an;ð2Þ

where n labels the lattice sites, J > 0 denotes the tunnelingamplitude for nearest-neighbor processes, U > 0 is the on-site interaction strength, and the time-periodic modulationhas an amplitude K and a frequency ω ¼ 2π=T. Note thatwe set ℏ ¼ 1 throughout this work.Equation (2) describes a wide variety of physical

systems. On the one hand, it is expected to provide agood description for the shaken 1D Bose-Hubbard model[10], when treated in the weakly interacting regime: In thiscase, the time-modulated system in Eq. (2) can be realizedby mechanically modulating an optical lattice filled withweakly interacting bosonic atoms [10]; see also Sec. VII for

FIG. 2. Numerical (top panel) and analytical (bottom panel)stability diagram for the model described by Eq. (2). Theinstability rate Γ (expressed in units of the hopping amplitudeJ, see text) is plotted as a function of the modulationamplitude K=ω [defined in Eq. (2)] and interaction strengthg, for a driving frequency ω ¼ 5J. The vertical lines and letterssummarize the agreement and disagreement regions, as dis-cussed in Sec. IV C. A: Away from the zeros of J 0ðK=ωÞ andJ 2ðK=ωÞ, the analytics successfully capture the instabilityrates extracted from numerics. B: Agreement zone near the(close) zeros of J 0ðK=ωÞ and J 2ðK=ωÞ, where both analyticsand numerics predict a stable behavior. C/D: Close to a zero ofJ 0ðK=ωÞ, the analytical perturbative approach breaks down.E: Potential quantitative disagreement when the contributionof the second harmonic has a weak amplitude; the instability isthen partly due to higher harmonics (see Sec. IV B).


021015-4

further discussion. On the other hand, some physical systemsare exactly described by Eq. (2): for instance, nonlinearoptical systems [85], including helical photonic crystal [5].In all cases, Eq. (2) defines a close self-consistent problem,which constitutes the core of the present study.Similar to the analysis of Ref. [73], we study the

dynamical instabilities of a BEC described by Eq. (2).To do so, we proceed along the following steps:

(i) First, we determine the time evolution of the con-

densate wave function að0Þn ðtÞ by solving the fulltime-dependent GPE and assuming that the initial

state að0Þn ðt ¼ 0Þ forms a BEC. More precisely, ourchoice for the initial state corresponds to the solutionof the static (effective) GPE:

−Jeffðað0Þnþ1 þ að0Þn−1Þ þUjað0Þn j2að0Þn ¼ μað0Þn ;

Jeff ¼ JJ 0ðK=ωÞ;ð3Þ

where we introduced the effective tunneling ampli-tude Jeff renormalized by the drive [10] and whereJ 0 denotes the zeroth-order Bessel function. For the

model under consideration, we find that að0Þn ðt ¼ 0Þis the Bloch state eip0n of momentum p0 ¼ 0 ifJ 0ðK=ωÞ > 0 (homogeneous condensate) and p0 ¼π for J 0ðK=ωÞ < 0. Note that such a choice takesinto account the initial kick due to the launchingof the modulation [11] [86]. Altogether, this choicefor the initial state features both the effects oftunneling renormalization and initial launching ofthe drive, which are both present in our model (2).We emphasize that this prescription for the initialstate is the only step of our calculations that relies onthe existence of a well-defined high-frequency limit,as provided by the inverse-frequency expansion[11,14,16]; indeed, all subsequent results are basedon the full time-dependent equations. Note that it isnot the purpose of this paper to discuss how toprepare the system in this initial state; one possibilityis to use Floquet adiabatic perturbation theory (seeRefs. [62,87]), which is also the experimentallypreferred strategy.

(ii) Given the time-dependent solution for the conden-

sate wave function að0Þn ðtÞ, we analyze its stability byconsidering a small perturbation

anðtÞ ¼ að0Þn ðtÞ½1þ δanðtÞ� ð4Þ

and linearizing the Gross-Pitaevskii equation (2) inδan; we refer to Appendix A for a discussion onthe specific parametrization chosen in Eq. (4). Thisyields the time-dependent Bogoliubov–de Gennesequations, which take the general form

i

� _δan_δa�n

�¼

F nðtÞ Ujað0Þn j2−Ujað0Þn j2 −F nðtÞ�

!�δanδa�n

�; ð5Þ

where we introduced the operator F nðtÞ, whoseaction on δan is defined by

F nðtÞδan ≡ −JLðað0ÞδanÞn

að0Þn

þ 2Ujað0Þn j2δan

þ K cosðωtÞnδan − i_að0Þn

að0Þn

δan; ð6Þ

and where the discrete operator L is defined byLð·Þn ≡ ð·Þnþ1 þ ð·Þn−1.

At this stage, let us emphasize that the Bogoliubovequations (5) contain the complete time dependence ofthe problem, which includes the effects related to the

micromotion [note that the BEC wave function að0Þn ðtÞ iscomputed exactly, not stroboscopically]. Importantly, andas will become apparent below in Sec. IV, the full-timedependence of the system plays a crucial role in its stability.The Bogoliubov equations of motion (5) can be time

evolved over one driving period T, which allows one todetermine the associated “time-evolution” (propagator)matrix ΦðTÞ. From this, we extract the “Lyapunov”exponents ϵq, which are related to the eigenvalues λq ofΦðTÞ by λq ¼ e−iϵqT. The appearance of Lyapunov expo-nents with positive imaginary parts thus indicates adynamical instability [71,73], i.e., an exponential growthof the associated modes at the rate sq ¼ Imϵq, where sqdenotes the growth rate of the momentum mode q. As weshall see later, a quantitative indicator of the instability isthe maximum growth rate of the spectrum,

ΓðJ;U;K;ωÞ≡maxq

sq; ð7Þ

which, in the following, will be referred to as the instabilityrate Γ. This choice will be justified in Sec. III B. We pointout that the so-defined instability rate is independent of thereference frame (or gauge) and that it is associated with thestroboscopic dynamics (t ¼ T × integer), by construction.For the model under investigation, one can considerably

simplify the problem by working in a rotating frame, inwhich translational invariance is manifest. This is achievedby applying the following gauge transformation [88]:

anðtÞ → e−inðK=ωÞ sinðωtÞanðtÞ: ð8Þ

In this frame, and going to momentum space, the systemof coupled Bogoliubov–de Gennes equations reduces to a2 × 2 matrix [73], and the dynamics of the mode withmomentum q ∈ BZ (first Brillouin zone) is describedby [89]


021015-5

i∂t

�uqvq

�¼�εðq; tÞ þ g g

−g −εð−q; tÞ − g

��uqvq

�; ð9Þ

where

εðq; tÞ ¼ 4J sin

�q2

�sin

�q2þ p0 −

KωsinðωtÞ

�; ð10Þ

with p0 denoting the momentum of the initial BEC wavefunction. Throughout this paper, we set the lattice constantto unity (ax ¼ 1). Here, we introduced the parameterg≡ ρU, where ρ ¼ jað0Þð0Þj2 denotes the condensatedensity; the latter enters the normalization of the solution

að0Þn ðtÞ; the amplitudes uq, vq of the Bogoliubov modes aredefined by the expansion of the fluctuation term δan interms of Bloch waves

δanðtÞ ¼Xq

uqðtÞeiqn þ v�qðtÞe−iqn: ð11Þ

Equation (9) is numerically easier to integrate and will alsoconstitute the starting point of the analytical analysis(see Sec. IV).

III. NUMERICAL SIMULATION

The general procedure outlined above can be imple-mented and solved numerically, regardless of the preciseform of the physical band model. In more involved models,

the initial state að0Þn ð0Þ can generically be determined byfinding the ground state of the effective Hamiltonian Heff ,i.e., by solving the equivalent of Eq. (3) using imaginary-

time propagation. The condensate wave function að0Þn ðtÞ isthen determined by solving the time-dependent GPE withthis initial condition, through real-time propagation (e.g.,using a Crank-Nicolson integration scheme). Finally, theBogoliubov equations are also solved over one drivingperiod by real-time propagation, yielding the operatorΦðTÞ, which is then exactly diagonalized (e.g., using aLanczos algorithm). In the present case, this procedure canbe shortcut by directly solving Eqs. (9), and we havechecked that this produces the same results for all physicalquantities. We now present the numerical results obtainedfor our model.

A. Stability diagrams

The stability diagram of the model in Eq. (2), whichdisplays the behavior of the instability rate Γ as a functionof the interaction strength g ¼ Uρ and modulation ampli-tude K=ω, is shown in Fig. 2, for a reasonably large drivingfrequency ω ¼ 5J. The stability boundary is similar tothe one previously reported in Refs. [73,90]; in particular,the system is found to be stable in regions whereJ 0ðK=ωÞ ¼ 0, which can be attributed to the fact that

the dynamics is frozen by the cancellation of the effectivetunneling [73]. At the transition to instability, the instabilityrate builds up continuously from zero and then increaseswhen going further in the unstable regime.Figure 3 shows similar stability diagrams for two other

values of the driving frequency, ω ¼ 10J and ω ¼ J. In thefirst case, we observe that the diagram is mostly unaffectedby a change of ω, provided g is rescaled as g ∝ ω2. Moregenerally, except in what we refer to as the “low-frequencyregime” (defined by the condition ω < 4jJeff j for thepresent model; see the next paragraph), transitions toinstability always occur at some finite g, and the corre-sponding rate is found to mostly depend on the quantitiesK=ω and g=ω2 [91]. As we shall see in Sec. IV, this can beunderstood from the fact that the instability rate dependson a competition between ω and the Bogoliubov dispersionassociated with the linearized effective GPE [i.e., theBogoliubov dispersion stemming from the linear analysisof Eq. (3)],

EavðqÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4jJeff jsin2ðq=2Þ(4jJeff jsin2ðq=2Þþ2g)

q; ð12Þ

ω = 10J(a)

0 2 4 6 8 10

K/ω

0

8

16

24

32

40

g/J

0

1

2

3

4

5

6

7 Γ

Γω = J(b)

0 2 4 6 8 10

K/ω

0

0.1

0.2

0.3

0.4

g/J

0

0.05

0.1

0.15

0.2

0.25

FIG. 3. Numerical instability rate Γ as a function of interactionstrength g ¼ Uρ and modulation amplitude K=ω, for two valuesof the driving frequency (ω ¼ 10J and ω ¼ J). At large ω (i.e.,outside the “low-frequency” regime, see text), the instability ratesdepend mostly only on K=ω and g=ω2, while in the low-frequency regime, instabilities can occur at infinitesimal inter-action strength.


021015-6

where the two cases p0 ¼ 0, π are taken into accountthrough the absolute value jJeff j. As soon as the transitionoccurs at sufficiently large g, the term 4jJeff j sin2 q=2 in thisdispersion becomes negligible compared to g, which resultsin Eav ∝

ffiffiffig

p, explaining the scaling indicated above.

This is no longer true in the “low-frequency regime” (seeFig. 3 for ω ¼ J), which we generically define through thecriterion according to which ω is smaller than the effectivefree-particle bandwidth (i.e., ω < 4jJeff j for the modelunder consideration). In that case, we observe that insta-bilities may occur at any finite interaction strength g, whichis related to the fact that ω is smaller than the bandwidth ofthe effective Bogoliubov dispersion (12) at g ¼ 0; see alsothe analytical analysis in Sec. IV for more details.Finally, in the singular case where Jeff ¼ 0, the effective

Bogoliubov dispersion EavðqÞ in Eq. (12) vanishes, andthus, the system is necessarily stable.

B. Dynamics of physical observables

Instead of solving the Bogoliubov equations [Eq. (5)]over one driving period only, we can also use these tocompute the full time evolution of physical quantities,hence revealing both their long-time and micromotiondynamics. To do so, one has to choose an initial conditionfor the fluctuation term, δanðt ¼ 0Þ. In this section, weconsider a generic (small) perturbation δanðt ¼ 0Þ, whichhas Fourier components over all Bogoliubov modes, i.e., ofthe generic form δanð0Þ ¼ α

PqAqeiqn þ Bqe−iqn, with Aq,

Bq being complex numbers and α a small amplitude. InSec. VII, we discuss a more physical initial condition, in theframework of the periodically driven Bose-Hubbard model.Altogether, given such an initial condition, one can numeri-cally evolve Eqs. (5) using real-time propagation, which

yields δanðtÞ [and thus also anðtÞ ¼ að0Þn ðtÞ½1þ δanðtÞ� inthe linearized approximation], hence revealing the behaviorof physical quantities.

1. Energy growth and heating

As an illustration, we first evaluate the energy in therotating frame, which is given at time t by

EðtÞ ¼ −JXn

½anþ1ðtÞeinðK=ωÞ sinðωtÞ

þ an−1ðtÞe−inðK=ωÞ sinðωtÞ�a�nðtÞ þU2

Xn

janðtÞj4:

ð13Þ

In the linear approximation, one can explicitly recastthis expression as a function of δan and only keep termsof lowest order in δan. This yields EðtÞ ¼ E½að0Þ�ðtÞ þEfluctðtÞ, where the energy of the fluctuations EfluctðtÞ isgiven to lowest order by

EfluctðtÞ ≈ −JXn

½að0Þnþ1δanþ1einðK=ωÞ sinðωtÞ

þ að0Þn−1δan−1e−inðK=ωÞ sinðωtÞ�a�ð0Þn δa�n

þ U2

Xn

jað0Þn j4½δa2n þ δa�2n þ 4jδanj2�: ð14Þ

The behavior of the energy of the fluctuations in Eq. (14)over several driving periods is shown in Fig. 4, in the stableand the unstable regimes (red curves), for an initialcondition corresponding to Aq ¼ 1 and Bq ¼ 0. In thestable regime, it displays modulations due to micromotionbut no long-term growth. Conversely, as anticipated for aparametric instability, the energy displays an exponentialgrowth in the unstable regime. By fitting this growth, wefind that its rate is given by 2Γ [the factor 2 stems from the

5

10

20

50

0 T 2T 4T 6T 8T

Ene

rgy

dens

ity o

f the

fluc

tuat

ions

(in

uni

ts o

f J )

time t

ω =5J,g=4J,K/ω =7.5From linearized equationFrom nonlinear equation

10

1000

100000

1e+07

1e+09

0 T 2T 4T 6T 8T

Ene

rgy

dens

ity o

f the

fluc

tuat

ions

(in

uni

ts o

f J )

time t

ω=5J,g=7J,K/ω=7.5From linearized equationFrom nonlinear equation

FIG. 4. Time evolution of the energy density of the fluctuations(in units of J) over eight driving periods for two values of g. In thestable regime (top panel, g ¼ 4), the energy displays variationsdue to micromotion but no long-term growth, while in theunstable regime (bottom panel, g ¼ 7), the energy curve featuresexponential growth, defining a heating rate compatible with themaximal Lyapunov exponent Γ (see Fig. 5). The parameters areAq ¼ 1 and Bq ¼ 0 (for the initial condition, see text), ω ¼ 5J,and K=ω ¼ 7.5.


021015-7

square moduli (jδanj2;…) in Eq. (14); see also below,Eq. (17)], which expresses the fact that the maximallyunstable mode dominates the growth of the energy in thesystem. This unstable behavior, and the validity range of therelated regime, will be further addressed in the next para-graphs. Figure 5 shows the diagram obtained by extractingthe heating rates from energy curves and is found to be inexcellent agreement with the stability diagram of Fig. 2.Therefore, we conclude that the instability rate Γ can be usedas a quantitative estimator of heating in such systems.

2. Validity regimes of the calculation

The computation of the full dynamics also highlights inwhich regimes our previous calculation of Γ is expectedto hold.On the one hand, the calculation of the rate Γ is based on

a Floquet treatment of the Bogoliubov equations (5) andthus describes the dynamics at times

t ≫ T:

In turn, the dynamics within one driving period, apparent inFig. 4, cannot be captured by our analysis.On the other hand, the analysis presented so far is based

on the linear approximation of the GPE. To investigatenonlinear effects, we compared the time evolution obtainedfrom the linearized Eq. (5) (red curves in Fig. 4, aspreviously discussed) with the time evolution of theoriginal nonlinear GPE in Eq. (2) (grey dashed lines inFig. 4), for the same initial condition. While the two graphsagree reasonably well in the stable regime, we find asignificant deviation at longer times, within the unstableregime, because of the growth of the fluctuation term. Thisillustrates the intuitive fact that our linear-approximation-based treatment only holds at short times, imposing asecond validity condition on our theory:

t ≪ tlin:

At longer times, nonlinear corrections to the evolutiondamp the exponential growth of the energy, which leads toa slowdown (saturation) of the heating dynamics.Altogether, we find two natural time scales in the

problem, and our approach thus requires the conditionT ≪ tlin to be relevant. Such a condition is fulfilled in awide window of realistic system parameters.

3. Instability regimes

A third natural time scale arises when studying thegrowth of physical observables. In the linear treatment ofthe GPE, a generic physical observable can be expressed as

OðtÞ ¼Xq

O½uqðtÞ; vqðtÞ�; ð15Þ

where O½uqðtÞ; vqðtÞ� is a functional, which is quadratic inthe Bogoliubov modes uqðtÞ and vqðtÞ; for instance, this isstraightforward for the energy when substituting Eq. (11)into Eq. (14). This also applies to the noncondensedfraction when considering a weakly interacting Bose-Hubbard model in the mean-field interacting regime [seeEq. (45) in Sec. VII]. Using the fact that in our Floquettreatment of the Bogoliubov equation, each mode qstroboscopically evolves according to the rate sq, onecan rewrite Eq. (15) as

OðtNÞ ¼Xq

O½uqð0Þ; vqð0Þ�e2sqtN ; ð16Þ

where the time tN ≡ NT denotes an integer multiple of thedriving period T. By introducing Γ ¼ maxq sq, as consid-ered above, two cases may arise:(a) If ΓtN ≫ 1, the growth of the maximally unstable

mode dominates in Eq. (16), and we find that theobservables stroboscopically grow exponentially withthe rate 2Γ,

OðtNÞ ∝ e2ΓtN ; ð17Þas already observed for the energy in Sec. III B 1.Remarkably, this rate only involves the most unstablemode, and it is the same for all physical quantities. Asannounced above, this justifies the use of Γ as a globalinstability rate in our theoretical analysis.

(b) If ΓtN ≪ 1, all exponentials in Eq. (16) can belinearized, yielding

OðtNÞ ¼ Oð0Þ þ tNXq

2sqO½uqð0Þ; vqð0Þ� þOðt2NÞ:

ð18ÞIn this regime, the energy increases linearly in time,with a rate given by the slope 2

PqsqO½uqð0Þ; vqð0Þ�.

The latter now involves a summation over all modessince no single mode contribution can be singled out at

ω=5J Heating rate

0 2 4 6 8 10

K/ω

0

2

4

6

8

10g/

J

0

1

2

3

4

5

6

FIG. 5. Heating rate, obtained by fitting the exponential growthof the energy in Fig. 4. The resulting diagram is very similar tothe instability diagram (Fig. 2), up to an overall numerical factor≈2 (see text).


021015-8

these early times (tN ≪ 1=Γ). Note also that, in thisregime, the rate depends on the physical observableconsidered (through the functional O).

For most values of the system parameters [such asthose used in Fig. 4], ΓT ≫ 1, so the linear regime[Eq. (18)] is hidden in the first oscillation (which, asstated above, is not accurately captured by our Floquetapproach). Yet, very close to the stability boundary,where Γ is small, this marginal regime can becomeapparent in a very narrow range of parameters, as weillustrate in Fig. 6.

Here, three regimes clearly show up: At very short timest ≪ T, the initial growth is not described by our Floquetapproach, as previously explained; at intermediate timesT ≪ t ≪ Γ−1, the growth is linear and accurately describedby Eq. (18); at longer times t ≫ Γ−1, the growth isexponential and well described by Eq. (17). Note thatthe linear regime is expected to be hard to probe inexperiments since it appears at very short times, typicallya few milliseconds (see also discussion in Sec. VIII).

IV. ANALYTICAL APPROACH

A. Linearized Gross-Pitaevskii equationas a parametric oscillator

In momentum space, the Bogoliubov–de Gennes equa-tions (9) constitute a set of uncoupled equations, whichindependently govern the time evolution of excitations witha given momentum q: The mean-field analysis effectivelyreduces to a single-particle problem, and one can studyhow instabilities occur independently in each of thosemodes. The general mechanism underlying the appearanceof parametric instabilities was suggested in Ref. [74],where a weakly interacting Bose-Hubbard model wasinvestigated through a weak-coupling particle-number-conserving approximation; in that study, parametric reso-nances were shown to appear in the Bogoliubov equations,whenever the drive frequency was reduced below twice thesingle-particle bandwidth. A similar approach was consid-ered in Ref. [92], which also provides an intuitive picture ofthe underlying mechanisms, in terms of the so-called Kreinsignature associated with the Bogoliubov modes.The same phenomenon occurs in the present framework,

as can be seen by performing a change of basis that recastsEq. (9) into the following form (see Appendix C fordetails):

i∂t

�~u0q~v0q

�¼�EavðqÞ1þ WqðtÞ þ

gEavðqÞ

�0 hqðtÞe−2iEavðqÞt

−hqðtÞe2iEavðqÞt 0

��~u0q~v0q

�: ð19Þ

Here, EavðqÞ is the Bogoliubov dispersion associated withthe effective (time-averaged) GPE, within the Bogoliubovapproximation [see Eq. (12)];WqðtÞ is a diagonal matrix ofzero average over one driving period, which will play norole in the following (see Appendix C for its exactexpression); and hqðtÞ is a (real-valued) function thatcan be Fourier expanded as

hqðtÞ ¼J24sin2ðq=2Þ

X∞l¼−∞

½J lðK=ωÞ þ J −lðK=ωÞ�eilωt

¼ 4Jsin2ðq=2ÞX∞l¼−∞

J 2lðK=ωÞei2lωt; ð20Þ

with J lðzÞ the lth Bessel function of the first kind.Importantly, the specific form of Eq. (19) allows one toclearly distinguish between the contribution due to thetime-averaged dynamics, as captured by the effectivedispersion EavðqÞ, and the contribution due to the full-time evolution [WqðtÞ, hqðtÞ]. As will be shown below, thefull-time dependence of the system is crucial for under-standing the origin of its instabilities, through the propertiesof the real-valued function hqðtÞ.Casting the equations of motion in the form (19) allows

one to directly identify any parametric resonance effects. Tosee this, we recall the reasoning of Ref. [74], which wasbased on applying a rotating-wave approximation (RWA)

5

10

100

1000

10000

0 T 2T 4T 6T 8T

Ene

rgy

dens

ity o

f the

fluc

tuat

ions

(in

uni

ts o

f J )

time t

ω=5J,g=5.7J,K/ω=7.5

linear regime exponential regime

From linearised equationFrom non-linear equation

Rate from Eq.(17)Rate from Eq.(16)

FIG. 6. Time evolution of the energy density of the fluctuations(in units of J) over eight driving periods for the initial conditionsAq ¼ 1 and Bq ¼ 0 (see text), for ω ¼ 5J, K=ω ¼ 7.5, andg ¼ 5.7. In this regime, where Γ is small, three regimes clearlyshow up: At very short times t ≪ T, the initial growth is notdescribed by our Floquet approach, as previously explained; atintermediate timesT ≪ t ≪ Γ−1, the growth is linear and describedby Eq. (18) (red dotted line); at large times t ≫ Γ−1, the growth isexponential and described by Eq. (17) (red short-dotted line).


021015-9

to Eq. (19). This approach amounts to neglecting all fast-oscillating terms in Eq. (19), which is equivalent toperforming a time average over a drive period T; we notethat this treatment captures the stroboscopic dynamics(t ¼ T × integer), which is consistent with the definitionof the instability rates, as extracted from Lyapunov expo-nents in Sec. II. If one disregards the expression in Eq. (20)for now, and if one naively assumes that the lowestfrequency appearing in hqðtÞ in Eq. (19) is ω, then adominant contribution to the dynamics is expected whenthe resonance condition ω ¼ 2EavðqÞ is fulfilled, resultingin the time-independent nondiagonal term in the matrixdisplayed in Eq. (19). Keeping this resonant term onlyyields a time-independent 2 × 2 matrix that can be dia-gonalized and whose eigenvalues (i.e., the Lyapunovexponents) are found to exhibit an imaginary part, yieldinga nonzero instability rate. This argument was invokedin Ref. [74] to justify the intuitive stability criterionω > 2Weff [with Weff ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4jJeff jð4jJeff j þ 2gÞp

the band-width of the effective Bogoliubov dispersion], which statesthat instability arises from the absorption of the energy ω tocreate a pair of Bogoliubov excitations on top of thecondensate. However, a more careful examination showsthat the instability rates inferred from this idea are notconsistent with our numerical simulations. Moreover, toget a sense of the additional dilemma one is faced with, wenote that the expression for hqðtÞ in Eq. (20) actually onlycontains even harmonics of the modulation, and therefore,following the reasoning above, the resonance conditionshould then read 2ω ¼ 2EavðqÞ. Yet, such a criterionprovides a wrong estimate of the stability-instabilityboundaries. Hence, it appears that this simple explanationneeds to be thoughtfully revised.In the following, we present a more rigorous analytical

evaluation of the instability rates for our system. Similarto Ref. [74], our derivation relies on the fact that theBogoliubov equations (19) precisely take the form of a so-called parametric oscillator. In Appendix B, we brieflyrecall some useful results about this paradigmatic model[3], which describes a harmonic oscillator of eigenfre-quency ω0 driven by a weak sinusoidal perturbation offrequency ω and amplitude α. In brief, such a modeldisplays a parametric instability for ω ≈ 2ω0. Importantly,the instability is maximal when this resonance condition isfulfilled, but it occurs in a whole range of parametersaround this point. As detailed in Landau and Lifshits [3],the width of the resonance domain and the instability ratesin the vicinity of the resonance can be calculated perturba-tively by introducing the detuning δ ¼ ω − 2ω0 and solv-ing the equations perturbatively in δ and amplitude α (seeAppendix B).More specifically, the Bogoliubov equations (19) exactly

take the form of the parametric oscillator [Eq. (B2)], withthe frequency ω0 of the unperturbed oscillator beingidentified with the dispersion EavðqÞ. Thus, it immediately

follows that the model is equivalent to a set of independentparametric oscillators, one for each mode q. Two pointsshould be emphasized here: First, all of those parametricoscillators depend on q in a different way and will thusexhibit different resonance conditions; second, the functionhqðtÞ in Eq. (19) is not a pure sinusoid as in Eq. (B2), butrather, it contains all (even) harmonics of the drivingfrequency ω. Altogether, given those two remarks, reso-nances are expected as soon as one of the harmonics of themodulation, of energy lω, is close to twice the energy of

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14

g/J

ω=5J,K/ω=7.5Full numerics

Numerics up to harmonic 2Numerics up to harmonic 4Numerics up to harmonic 6

Numerics with all odd harmonics

0

0.2

0.4

0.6

0.8

1

1.2

Γ Γ

1.4

1.6

1.8

2

0 1 2 3 4 5 6 7 8 9 10

g/J

ω=5J,K/ω=7.5 NumericsNumerics on second harmonic alone

Analytics up to order 2Analytics up to order 1

FIG. 7. Top panel: Instability rate Γ as a function of theinteraction strength g for ω ¼ 5J and K=ω ¼ 7.5, numericallycomputed from the Bogoliubov equations (cf. Sec. II), taking intoaccount only a few harmonics of the modulation in the functionhqðtÞ. As theoretically predicted from the expression of hqðtÞ[Eq. (20)], the odd harmonics do not contribute, while the firsteven ones successfully describe the full numerical calculation.Bottom panel: Comparing the analytical and numerical resultsfor the instability rate ΓðgÞ, using the same parameters as above.The red crosses correspond to the full numerical solution, whilethe blue dashed lines are numerically obtained by only taking thesecond harmonic of the drive into account. As genericallyobserved in the A zones of the stability diagram (Fig. 2), thenumerical instability rates are well captured by the perturbativeanalytical approach, implemented here up to order 1 (red dotted-dashed line) and 2 (blue dotted line).


021015-10

any of the (effective, time-averaged) Bogoliubov modes,2EavðqÞ.Before digging into more technical details, let us

acquire some intuition about the general mechanismsbehind this phenomenon. To do so, consider the stabilitydiagram in Fig. 2 (i.e., outside of the low-frequencyregime [93]). In this case, in the lowest part of the stabilitydiagram (small g), ω is generically large compared to thebandwidth of the effective dispersion EavðqÞ, so noresonance can occur. When increasing g at fixed K=ω,the first mode to exhibit a resonance will be that ofmaximal EavðqÞ, i.e., q ¼ π. Naively, the first resonancewould be due to the first harmonic l ¼ 1 and would occurfor 2Eavðq ¼ πÞ ¼ ω, which was also the argument inRef. [74]. However, as already discussed, the functionhqðtÞ only contains even harmonics, and the first reso-nance to occur is, therefore, the one corresponding tol ¼ 2. At first sight, this might seem to be in contradictionwith the intuitive stability criterion 2EavðπÞ ¼ ω since thel ¼ 2 resonance is centered around EavðπÞ ¼ ω; however,as we shall discuss in more detail below, the key point is

that the resonance domain has a finite width: Althoughcentered around EavðπÞ ¼ ω, its boundaries are in factclose to the point where 2EavðπÞ ¼ ω. Resonances due tohigher harmonics (fourth, sixth, etc.) become importantonly for higher values of g, and we can, to a firstapproximation, restrict the analysis to the second har-monic only.To confirm this intuition, we have verified that restricting

our analysis to the second harmonic only is generallysufficient to estimate the stability boundary and to recoverthe instability rate in its vicinity [94]. Deeper in the unstableregion, the fourth and, eventually, the sixth harmonics areprogressively required to recover the agreement with thenumerical results (see Fig. 7).

B. Effective model

To simplify the calculations, we restrict our analysisto the second harmonic (see discussion above); we stressthat higher-order resonances can be treated along the samelines. The equations of motion [Eq. (19)] then read

i∂t

�~u0q~v0q

�¼�EavðqÞ1þ WqðtÞ þ

αqEavðqÞ2

�0 cosð2ωtÞe−2iEavðqÞt

− cosð2ωtÞe2iEavðqÞt 0

��~u0q~v0q

�; ð21Þ

with

αq ¼ 16JJ 2ðK=ωÞ sin2ðq=2Þg

½EavðqÞ�2:

Therefore, for a fixed q, these equations are nowstrictly equivalent to the equations of motion for theparametric oscillator of Eq. (B2), with the followingidentifications:

ω0 → EavðqÞ;ω → 2ω;

α → αq: ð22Þ

As a result, each mode q exhibits a resonance domaincentered around EavðqÞ ¼ ω. The instability rate is generi-cally maximal around the resonance point (defining themaximal rate smax

q ), and it decreases until it cancels at theedges of the resonance domain. In order to obtain analyticalestimates of the instability rate and the width of theresonance domain, one can apply the procedure detailedin Appendix B, which amounts to solving the problemperturbatively in the detuning δ ¼ 2ω − 2EavðqÞ andamplitude αq. To zeroth order, the instability only ariseswhen the resonance condition is precisely fulfilled, whichdefines the rate “on resonance,” s�q, which is given by

s�q ¼ g4JJ 2ðK=ωÞ sin2ðq=2Þ

EavðqÞ: ð23Þ

To first order, the instability rate reads [see Eq. (B3) withthe substitutions (22)]

sq ¼ s�q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

� ½ω − EavðqÞ�EavðqÞ4JJ 2ðK=ωÞ sin2ðq=2Þg

�2

s; ð24Þ

whenever the argument in the square root is positive, andsq ¼ 0 otherwise. The associated instability rate sq is thusmaximal on resonance (hence, smax

q ¼ s�q), and it decreaseswhen going towards the boundaries of the resonancedomain. At second order, sq is given by the solution ofthe implicit equation (B4) with the substitutions (22),which can be found numerically; this can be performedto improve the accuracy of the analytical rate’s value. Inparticular, at this order, we note that the actual maximalinstability point is then slightly shifted from the resonancepoint (i.e., smax

q ≠ s�q). Altogether, the total instability rate isgiven by [Eq. (7)]

Γ ¼ maxq

sq: ð25Þ

As a function of g [which enters EavðqÞ; see Eq. (12)], theinstability rate of a given mode q forms a “bumped curve”


021015-11

(see Fig. 8), which to a first approximation simply followsfrom Eq. (24). Since the resonance domain for each mode iscentered around a different energy (and therefore a differentg), the curves corresponding to various modes are slightlyshifted, and the total instability rate is given by the envelopeof all those curves (see Fig. 8).In particular, the curve centered around the smallest

values of g (i.e., the leftmost curve in Fig. 8, near thetransition point) is the one associated with the mode ofhighest Eav, namely, q ¼ π. Let us denote by gmax the valueof g where this curve is maximal: At first order, thiscorresponds to the solution of the resonance conditionω ¼ EavðπÞ ¼


. Then, we identifytwo cases:(a) For g > gmax, there always exists, in the thermody-

namic limit, a single mode q that is maximallyunstable, and the total instability rate, Eq. (25), isthus given by the rate of this particular mode. At firstorder (see discussion above), this mode is the onefulfilling the resonance condition ω ¼ EavðqÞ, and theinstability rate is given by the rate on resonance of thisparticular mode qres, so

Γ ¼ s�qres ¼��J 2ðK=ωÞJ 0ðK=ωÞ

�� gω� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

g2 þ ω2

q− g; ð26Þ

where we have explicitly reexpressed qres using theexpression ω ¼ EavðqresÞ.

(b) Conversely, for g < gmax, the instability rate is onlygoverned by the mode q ¼ π (see Fig. 8), and it cannotbe estimated from the knowledge of the associated rateon resonance. Therefore, to capture the behavior of theinstability rate near the transition point, as well as the

stability boundary itself, one can indeed restrict ouranalysis to the mode q ¼ π, but one in turn has toresort to the perturbative expansion around the reso-nance point discussed in Appendix B. In other words,in that case, one has

Γ ¼ sπ; ð27Þ

where sπ is given at lowest order by Eq. (24) withq ¼ π. The stability boundary can also be computed atany required order following the procedure indicatedin Appendix B, which yields, up to second order,

ω¼EavðπÞþ4JJ 2ðK=ωÞg

EavðπÞ−4J2J 2

2ðK=ωÞg2EavðπÞ3

þOðg3Þ:

ð28Þ

Interestingly, in the high-frequency regime, sincethe transition occurs at sufficiently large g, so thatEav ∝

ffiffiffig

p, one recovers that the boundary is a function

of the combination g=ω2, as previously observednumerically in Sec. III.

When decreasing the frequency, the transition occurs atlower and lower values of g: Since all corrections of anyorder tend to vanish at small g in Eq. (28), the transitionpoint at vanishing g is simply obtained for ω ¼ 4Jeff,recovering the criterion for entering the low-frequencyregime previously discussed. In this regime, which in somesense corresponds to a negative gmax, one is always in case(a) and the rate Γ is given by Eq. (26).Altogether, to capture the behavior of the instability rate

in the whole range of parameters, one should use Eqs. (24)and (25), which include all modes as well as the firstcorrection due to the finite detuning. Note that the pertur-bative expansion yielding those expressions is expected tohold, provided αq is not too large, i.e., EavðqÞ is not toosmall (see below for a discussion of the breaking points ofthe approach).

C. Results based on the analytical approach

We now show the results obtained from this analyticalapproach for the instability rate, Eq. (25), with sq beinggenerically computed to second order in the detuning(unless explicitly specified). The stability diagram is verysimilar to the one obtained numerically (see Fig. 2). Let usinvestigate in more detail this agreement and comment onthe small differences between the numerical and analyticaldiagrams.

1. Agreement regions

For values of K=ω that are not too close to the zeros ofthe Bessel functions J 0 and J 2 [95] (i.e., not too close tothe edges of the lobes in the stability diagram, zones A inFig. 2), the analytical calculation gives a very good estimate

0

0.5

1

Γ 1.5

2

2.5

3

0 15 30 45 60 75 90

g/J

ω = 5J,K/ω = 7.5

gmax/J

All modesq = π

q = 7π/8q = 3π/4

q = π /2

FIG. 8. Total instability rate Γ (red crosses) and instability ratessq of a few individual modes (dashed lines), numericallycomputed as a function of interaction strength g for ω ¼ 5Jand K=ω ¼ 7.5. The instability rate associated with an individualmode q is maximal for EavðqÞ ¼ ω and decreases around it. Thetotal instability rate is given by the envelope of all those curves.


021015-12

of both the transition point and the instability rate, up tomoderate interactions g (Fig. 7). At higher g, moreharmonics become important (presumably, in a complexand “coupled” way; we verified that building independenteffective models for separate harmonics does not reproducethe numerics accurately). Zones where J 4 vanishes are“favorable” since the first correction to the effective model[Eq. (21)] is absent, and the agreement between analyticsand numerics survives for larger values of g.In the vicinity of the “common” zeros of J 2ðK=ωÞ and

J 0ðK=ωÞ (i.e., “between” the lobes of the stability dia-gram, zones B in Fig. 2), both the analytics and numericsagree and predict a stable regime.

2. Disagreement regions

A detailed description of the small disagreementregions is provided in Appendix D. In brief, the analyticsbreak down into two main regions: (i) Around the firstzero of J 0 (i.e., the only zero of J 0 where J 2 issignificant; zones C on Fig. 2), the perturbative approachbreaks down because of the vanishing of the effectivedispersion Eav, and the analytical approach fails. (ii) Nearthe common zeros of J 0 and J 2 (i.e., “between” the lobesof the stability diagram), we find that the numerics andthe analytics disagree in the way the stable zones close atlarge g. We identified two main causes for this effect: thebreakdown of the perturbative approach due to thevanishing of Eav, and the vanishing of the secondharmonics near the zeros of J 2. The consequences ofthese factors taken together are very involved since theycan both compensate for each other and compete. In brief,at the left border of each lobe (zones D in Fig. 2), thenumerics displays a transition to instability (althoughshowing up only at large g), which is not captured bythe analytics. At the right border of the lobes (zones E),the instability predicted by the full numerics is most likelydue to a joint effect of the second and higher-orderharmonics and is therefore not accurately captured bythe analytical approach [see Fig. 21(b)].Nevertheless, the disagreement regions constitute very

narrow zones in the stability diagram. Figure 2 summarizesthe zones where the analytical and numerical calculationsagree (in green) and the ones where they disagree quali-tatively (in red) or just quantitatively (in orange).

3. Scaling in limiting cases

The analytical solutions provide a general scalingbehavior for the instability rates as a function of the modelparameters, which we analyze for three limiting cases.(a) Weak interactions: If ω is larger than the effective

free-particle bandwidth of the model (high-frequencyregime), the system is stable at g ≈ 0, and it features atransition to an unstable phase at some finite gc, with ascaling [see Eq. (24)]

Γ ∝ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðg − gcÞ

p:

Conversely, if ω is smaller than the effective free-particle bandwidth (low-frequency regime), the sys-tem is typically unstable at very small g, and [seeEq. (26)]

Γ ∝ g:

(b) Weak driving amplitudes: The analysis of the tran-sition to the unstable phase from K ¼ 0 (where thesystem is stable) to finite K is subtle since this limitdoes not commute with the thermodynamic limit.Indeed, one has to note that the resonance domainof each mode has a vanishing width when K → 0.Therefore, for any finite-size system, instabilities atvanishing K occur only for discrete values of theparameters (fulfilling one of the resonance conditionsassociated with a specific mode); it is only whenincreasing K that those instability regions grow inparameter space, eventually merging and forming thefirst lobe of the stability diagram. In the thermody-namic limit, one nevertheless finds the general scaling

Γ ∝ K2;

which stems from the second-order Bessel function,which dominates at low K in Eq. (26).

(c) Low driving frequency: In this limit, one finds [seeEq. (26)]

Γ ∝ ω × fðK=ωÞ;

where f is a generic function of K=ω only.

V. FINITE-SIZE SYSTEMS: FROM A DOUBLEWELL TO THE ENTIRE LATTICE

The analytical approach, which treats different modesindependently, allows one to readily predict the behavior offinite-size systems under periodic modulation, close to aparametric resonance. In this case, the “continuous”dispersion EavðqÞ is replaced by discrete energy modes.For a small number of lattice sites (and thus modes), thetotal instability rate Γ ¼ maxqsq, which is given by theenvelope of the curves associated with individual modes(see Fig. 8), may not be a monotonic curve as a function ofg. Instead, it is rather composed of disjointed bumps if theresonance domains of two consecutive modes are discon-nected. Figure 9 shows how individual modes contributeto the total growth rate when increasing the number ofsites, both inside and outside the low-frequency regime.Interestingly, the situation is slightly different in these twocases. Outside the low-frequency regime, i.e., for ω≳ Jeff,the onset of instability is governed by the mode q ¼ π


021015-13

(see Sec. IV), and therefore, the critical frequency, belowwhich the system becomes unstable, happens to be thesame for both an infinite and a two-site system. Since themode q ¼ π remains the “most unstable” one, all the wayup to large values of g (see Fig. 8), the stability diagram of atwo-site system is in fact very similar to the one associatedwith an infinite system (whereas a one-site system istrivially stable) [96]. Conversely, in the low-frequencyregime (ω≲ Jeff ), instability occurs already at vanishinglysmall g, induced by a certain resonant mode qres fulfilling

the resonance condition (see Sec. IV). Increasing thenumber of sites therefore lowers the instability boundaryas a function of g since increasing the number of points inmomentum space allows for the states to get closer to themaximally unstable mode. Moreover, if the number of sitesN is too small, the discretization in momentum space is sorough that the resonance domains of consecutive modesmay be disconnected, resulting in “islands” of instability inthe phase diagram. These islands begin to merge withincreasing N, as the resonance domains of adjacent modesoverlap, only gradually approaching the phase diagram ofan infinite system. Both our analytical and numericalmethods capture this behavior and agree for any numberof lattice sites. Interestingly, since the instability rate isdetermined by its maximal value over all modes (i.e., theenvelope of all curves in Fig. 9), its estimate obtained forfinite-size systems is already very decent, and furtherincreasing the number of sites just smoothens the curveof the instability rate as a function of g.We point out that for very small systems, there might be

minor corrections due to edge states: Indeed, the previoustreatment, formulated in momentum space, tacitly impliesperiodic boundary conditions. However, in the lab frame,the drive actually breaks translational invariance, and theboundary conditions are intrinsically open, possibly yield-ing different selection rules for states near the edges.Although such a difference is expected to play a negligiblerole in large systems, it may lead to minor but noticeabledeviations in small systems.

VI. EFFECT OF TRANSVERSE DIRECTIONS:2D LATTICES VS TUBES

So far, our study focused on the origin of parametricinstabilities that occur in a driven system, which satisfiesthe 1D GPE on a lattice (2). As further discussed inSec. VII, this model can be used to describe the physics ofweakly interacting bosonic atoms, trapped in a 1D opticallattice. However, from an experimental point of view, it isintriguing to determine the effects of transverse directionson the stability diagram and instability rates. Indeed,optical-lattice experiments involving weakly interactingbosons [39,41] typically feature continuous transversedegrees of freedom, commonly referred to as “tubes” or“pancakes.” In addition, experiments realizing artificialmagnetic fields [97,98] involve two-dimensional opticallattices, and hence, it is relevant to study the fate ofinstabilities as one transforms a 1D optical lattice into afull 2D lattice, by adding sites (and allowing for hoppingprocesses) along a transverse direction.In this section, we extend our previous analysis to study

the effects of (i) a secondary tight-binding-lattice direction(resulting in a ladder or a full 2D lattice) and (ii) anadditional continuous (“tube’”) degree of freedom. In thefollowing, we assume that the periodic modulation remains

0

0.01

0.02

0.03ΓΓ

0.04

0.05

0.06

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

g/J

g/J

ω=J,K/ω=7.5(a)

N=2N=4N=8

N=64q=π

q=π/2q=3π/4q=π/4

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50 60 70 80 90 100

ω=5J,K/ω=7.5(b)

N=2N=4N=8

N=64q=π

q=π/2q=3π/4q=7π/8

FIG. 9. Total instability rate and contributions of a few modesas a function of the interaction strength g, for ω ¼ J (low-frequency regime) and ω ¼ 5J (outside the low-frequencyregime), and for an increasing number of lattice sites N ¼ 2,4, 6, 64. The total instability rate is the envelope of the curvesassociated with “available” modes in the system (for N ¼ 2,only q ¼ 0, π; for N ¼ 4, only q ¼ 0; π=2; π; 3π=2...). The curvegets smoother when increasing the number of sites. Outside thelow-frequency regime, the transition to instability is governedby the mode π and already captured by a two-site system. In thelow-frequency regime, the instability rate at low g, which isgoverned by a certain mode qres, is probed with more and moreaccuracy when gradually increasing N: Here, the two-sitesystem is always stable since qres > π, the four-site systemdisplays a transition since qres > π=2, and then this transitionlowers when probing qres.


021015-14

exclusively aligned along the original tight-binding latticedimension.

A. Two-dimensional lattice geometry

In this section, we consider the addition of a transverselattice direction, aligned along the y axis; by doing so, wekeep the same time-dependent modulation as before, whichis hence exclusively aligned along the x axis. Our aim is tostudy how an increase in the number of sites along thetransverse direction affects the instability rates, previouslyevaluated for the 1D configuration. Theoretical studies[64,66] have reported that heating and collisional processesare enhanced by the presence of a transverse direction,which provides crucial information for current experimen-tal studies.For this extended model, the time-dependent GPE reads

i∂tam;n ¼ −Jðam;nþ1 þ am;n−1 þ amþ1;n þ am−1;nÞþ K cosðωtÞmam;n þ Ujam;nj2am;n; ð29Þ

where each site of the underlying (2D) square lattice isnow labeled by two integers ðm; nÞ. As for the 1D case(Sec. II), the condensate wave function at time t ¼ 0 isagain given by a Bloch state eiðpxmþpynÞ, with momentumpx ¼ 0 if J 0ðK=ωÞ > 0 and px ¼ π if J 0ðK=ωÞ < 0; forthis choice of the time modulation, the momentum alongthe y direction is necessarily py ¼ 0. The time-dependentBogoliubov–de Gennes equations, which govern the timeevolution of the mode q ¼ ðqx; qyÞ, take the form

i∂t

�uqvq



��uqvq

�;

ð30Þ

where

εðq; tÞ ¼ 4J sinqx2sin

�qx2þ px −

Kωsinωt

�þ 4J sin2

qy2;

which is a straightforward generalization of Eq. (9).Therefore, both the numerical and the analytical methodspresented in the previous sections can be readily appliedto Eq. (30).Figures 10 and 11 show the stability diagrams, obtained

numerically by increasing the number Ny of transverselattice sites (from 4 to 16); they correspond to the drivefrequency ω ¼ 10J (i.e., high-frequency regime, where ωis greater than the effective bandwidth) and ω ¼ 5J (i.e.,low-frequency regime), respectively. Since the motionaldegrees of freedom along the x and y directions areindependent of each other, the problem being separable,the situation is analogous to the one previously discussed inthe case of finite-size systems.

Let us summarize the results: Away from the low-frequency regime (Fig. 10), namely, for ω > 4ðJ þ JeffÞ[the drive frequency is again compared to the modified(effective) free-particle bandwidth], the system is stablefor small g, and the onset of instability is governed bythe mode of maximal energy (i.e., the first mode to exhibit aresonance), which now corresponds to q ¼ ðπ; πÞ.Therefore, as soon as two sites are present in the transversedirection, the transition point to the unstable regime is wellcaptured. Moreover, both the instability boundary and theinstability rates in its vicinity are expected to remainunaffected when increasing Ny since they are dominatedby the mode ðπ; πÞ [see Eq. (27)]. Conversely, in the low-frequency regime, there is an instability at vanishingly smallg, which is driven by a certain mode ðqx; qyÞ: In this case,adding the number of transverse sites lowers the instabilityboundary, as increasing the resolution in momentum spaceallows one to get closer to this maximally unstable mode.Using Eq. (30), one can readily extend the analytical

approach of Sec. IV. Indeed, all the results of Sec. IVremain applicable in the present case, provided that thedispersion EavðqÞ is now replaced by

ω=10J

(a)Γ

0 2 4 6 8 10

K/ω

0

2

4

6

8

10

g/J

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

ω=10J

(b)Γ

0 2 4 6 8 10

K/ω

0

2

4

6

8

10

g/J

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

FIG. 10. Numerical instability rate as a function of interactionstrength g ¼ Uρ and modulation amplitude K=ω, for ω ¼ 10J(i.e., outside the low-frequency regime) and for two differentnumbers of lattice sites in the transverse direction: Ny ¼ 4 (toppanel) and Ny ¼ 16 (bottom panel).


021015-15

EavðqÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4jJeff jsin2qx=2þ 4Jsin2qy=2Þ

q×

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4jJeff jsin2qx=2þ 4Jsin2qy=2þ 2gÞ

q: ð31Þ

The resulting stability diagrams are shown in Fig. 12 forthe same parameters as in Fig. 10, and they reveal anexcellent agreement with the latter. Interestingly, the agree-ment between analytics and numerics is even better thanfor the 1D configuration since the presence of transversemodes in the effective dispersion Eq. (31) prevents EavðqÞfrom vanishing when J 0ðK=ωÞ ¼ 0 [99].

B. Continuous transverse degrees of freedom:The case of tubes

We now consider the case where the transverse degreeof freedom corresponds to free motion along a continuum,in one or more transverse directions. In this case, theBogoliubov–de Gennes equations still take the form ofEq. (30), now with

εðq; tÞ ¼ 4J sinqx2sin

�qx2þ px −

Kωsinωt

�þ q⊥2

2m;

so the previous analysis still holds, provided that the time-averaged Bogoliubov dispersion EavðqÞ is now replaced by

EavðqÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4jJeff jsin2qx=2þ q⊥2=2mÞ

q×

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4jJeff jsin2qx=2þ q⊥2=2mþ 2gÞ

q: ð32Þ

Interestingly, since this dispersion is unbounded, ω willalways be smaller than the total bandwidth, and hence,there will always be (at least) one mode that is precisely seton resonance. In other words, the system necessarily fallsinto the low-frequency regime previously introduced,where it is unstable at any finite g. As explained inSec. IV B, the instability rate can be evaluated, at lowestorder, by calculating the rate on resonance associated withthe mode(s) satisfying the condition ω ¼ EavðqresÞ [seeEq. (26)]:

ω=5J

(a)

0 2 4 6 8 10

K/ω

0

2

4

6

8

10g/

J

0

0.5

1

1.5

2

2.5

3Γ

Γω=5J

(b)

0 2 4 6 8 10

K/ω

0

2

4

6

8

10

g/J

0

0.5

1

1.5

2

2.5

3

3.5

FIG. 11. Numerical instability rate as a function of interactionstrength g ¼ Uρ and modulation amplitude K=ω, for ω ¼ 5J(i.e., in the low-frequency regime) and for two different numbersof lattice sites in the transverse direction: Ny ¼ 4 (top panel) andNy ¼ 16 (bottom panel).

ω=10J

(a)

0 2 4 6 8 10

K/ω

0

2

4

6

8

10

g/J

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8Γ

Γω=10J

(b)

0 2 4 6 8 10

K/ω

0

2

4

6

8

10

g/J

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

FIG. 12. Analytical instability rate as a function of interactionstrength g ¼ Uρ and modulation amplitude K=ω, for ω ¼ 10J(i.e., outside the low-frequency regime) and for two differentnumbers of lattice sites in the transverse direction: Ny ¼ 4 (toppanel) and Ny ¼ 16 (bottom panel). This figure is to be comparedwith Fig. 10.


021015-16

s�qres ¼ 4gJ 2ðK=ωÞ sin2ðqresx =2ÞJ=EavðqresÞ;¼ 4gJ 2ðK=ωÞ sin2ðqresx =2ÞJ=ω: ð33Þ

Importantly, although there are generically several resonantmodes qres (corresponding to different qx and q⊥), they donot have the same instability rate s�qres : The total instabilityrate Γ is then attributed to the most unstable mode, namely,

Γ ¼ maxqres

ðs�qresÞ ¼ s�qmum ; ð34Þ

where we introduced the notation qmum to denote themomentum of the most unstable mode.We then identify two cases:(i) If ω >


, which is often thecase in realistic configurations, there exists a reso-nant mode at qmum

x ¼ π, which is thus the mostunstable one; qmum⊥ is then adjusted so as to respectthe resonance condition. The maximally unstablemode thus reads

qmumx ¼ π;

ðqmum⊥ Þ2=2m ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2 þ ω2

q− g − 4jJeff j: ð35Þ

In this case, the total instability rate is given by thesimple analytical formula

Γ ¼ 4JjJ 2ðK=ωÞjgω: ð36Þ

(ii) If ω <ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4jJeff jð4jJeff j þ 2gÞp

, the most unstablemode, which obeys the resonance conditionω ¼ EavðqÞ, is necessarily reached at qmum⊥ ¼ 0,yielding

qmumx ¼ 2 arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2 þ ω2

p− g

4jJeff j

s; qmum⊥ ¼ 0:

ð37Þ

In this case, the total instability rate is given by

Γ ¼� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

g2 þ ω2

q− g��J 2ðK=ωÞ

J 0ðK=ωÞ�� gω : ð38Þ

Interestingly, these results do not depend on the numberof transverse dimensions since all that matters is theexistence of an unbounded continuum. Figure 13 comparesthe numerical stability diagram to the analytical formulaEq. (36), for realistic (experimental) parameters. As visiblein Fig. 14, which shows cuts through the diagrams ofFig. 13, the agreement is excellent at low interaction ormodulation amplitude, and the scaling of the instabilityrate is remarkably simple: Γ increases linearly with g

and quadratically with K. Such simple results couldpromisingly be compared with experiments. Note that inexperiments, one typically has J=ℏ ≈ 100 Hz − 1 kHz, sothe inverse instability rates are typically of the order of1–10 ms.At this stage, it is worth comparing those results with the

more traditional Fermi Golden Rule (FGR) approach: First,in the present case, and contrary to what is expected for aFGR regime [63], the final rate can be inferred from thecontribution of a single (well-identified) mode and not froma sum over all modes (which would then involve the densityof states in the description). Second, the form of the finalformula in Eq. (36) is significantly different from thatresulting from a FGR argument, where the rate wouldtypically scale as g2 (as dictated by the squared matrixelement of the perturbation [63]). When considering adriven weakly interacting degenerate Bose gas, one expectsthe short-time dynamics to be dominated by the parametricinstability identified in this work; at longer times, as thecondensate significantly depletes, heating rates areexpected to be dominated by a FGR behavior [63,64],not captured in the present analysis. We believe that thosetwo distinct mechanisms and instability regimes could beprobed by current ultracold-atom experiments.

ω=32J

0 2 4 6 8 10

K/ω

0

8

16

24

32

40

g/J

0

0.5

1

1.5

2

2.5

3

3.5

4Γ

Γω=32J

0 2 4 6 8 10

K/ω

0

8

16

24

32

40

g/J

0

0.5

1

1.5

2

2.5

3

3.5

4

FIG. 13. Numerical (top panel) and analytical (bottom panel)instability rates Γ as a function of the modulation amplitude K=ωand interaction strength g, for a high drive frequency ω ¼ 32J.


021015-17

As a practical remark, we emphasize that the resultspresented in this section suggest that working with a latticeseems to be more favorable than with tubes or pancakes, inview of reducing parametric instabilities in periodicallydriven Bose systems.

VII. PERIODICALLY DRIVENBOSE-HUBBARD MODEL

In this section, we discuss the relation between theGross-Pitaevskii equation introduced in Eq. (2) and the (fullquantum) driven Bose-Hubbard model [10]. Our goal is toclarify the validity of our analysis in view of describingparametric instabilities and heating in the context of thisquantum model.

A. Effective Hamiltonian and micromotion operator

For the sake of concreteness, here we focus on the 1Dmodel of Eq. (2), but the discussion applies to all modelsinvestigated in the paper (in particular, the two-dimensionalmodels of Sec. VI). Consider a system of weakly interact-ing bosons, trapped in a shaken 1D lattice, as described bythe periodically driven Bose-Hubbard Hamiltonian [10]

HðtÞ ¼ −JXn

ða†nþ1an þ H:c:Þ þ K cosðωtÞXn

na†nan

þU2

Xn

a†na†nanan; ð39Þ

where J > 0 denotes the tunneling amplitude of nearest-neighbor hopping, and U > 0 is the on-site interactionstrength. The on-site potential term describes a time-periodic modulation of amplitude K and frequencyω ¼ 2π=T.As we already stated in the Introduction, the dynamics is

stroboscopically governed by the effective FloquetHamiltonian [see Eq. (1)]. In the absence of interactions,it is exactly [100] given by [10–12]

HeffðU ¼ 0Þ ¼ −JeffXn

ða†nþ1an þ H:c:Þ;

Jeff ¼ JJ 0ðK=ωÞ: ð40Þ

In addition, the kick operator [Eq. (1)] reads [16]

KkickðtÞ ¼ i log ½RðtÞe−iKrotkickðtÞ�; ð41Þ

where RðtÞ ¼ e−iK=ω sinðωtÞX is the unitary transformation tothe rotating frame (with X ≡ Σnna

†nan the position operator

on the lattice). Here, Krotkick denotes the kick operator in the

rotating frame [16], which is explicitly given by

KrotkickðtÞ ¼ −2J

Xq∈BZ

�Zt

0

cosðq − K=ω sinωt0Þdt0

− J 0ðK=ωÞ cosðqÞ�a†kak: ð42Þ

Whenever interactions are present, Eq. (40) is no longerexact, and the effective Hamiltonian is a much morecomplicated (possibly nonlocal) object [59,101]. In thehigh-frequency regime, it can be approximated using theinverse-frequency expansion [11], which to lowest orderyields

Heff ¼ −JeffXn

ða†nþ1an þ H:c:Þ

þ U2

Xn

a†na†nanan þOðω−1Þ: ð43Þ

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 5 10 15 20 25 30 35 40

/JΓΓ

g/J

Numerics, K/ω = 1Analytics, K/ω = 1Numerics, K/ω = 2Analytics, K/ω = 2

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

/J

K/ω

Numerics, g = 17JAnalytics, g = 17JNumerics, g = 38JAnalytics, g = 38J

FIG. 14. Top panel: Instability rate as a function of theinteraction strength g for ω ¼ 32J and for two values of themodulation amplitude. As captured by the analytical expressionEq. (36), the instability rate increases linearly with g. Bottompanel: Instability rate as a function of the modulation amplitudeK=ω for ω ¼ 32J and for two values of the interactionparameter g. As captured by the analytical expressionEq. (36), the instability rate increases quadratically withK=ω at low amplitudes.


021015-18

Hence, in the infinite-frequency limit, the periodic drivemerely renormalizes the hopping matrix element J → Jeff .Since the Bessel function takes both positive and negativevalues, it is possible to tune the amplitude-to-frequencyratio such that J 0ðK=ωÞ ¼ 0, in which case tunnelingis completely suppressed [22,23,29]. Therefore, even aweakly interacting lattice system can effectively behave asa strongly interacting one under periodic driving, in thesense that the interaction strength U potentially dominatesover the tunneling when Jeff ≈ 0. We recall that the latticedispersion flips sign when J 0ðK=ωÞ < 0, in which case thestable minimum appears at quasimomentum q ¼ π:Namely, at equilibrium, bosons are expected to condensein a finite-momentum state in this regime. We point outthat corrections to the Hamiltonian (43) become non-negligible away from the high-frequency limit and thatthese are expected to manifest themselves over longertime scales.

B. Relation to the mean-field approachand observables

The mean-field approach presented in this paper [basedon Eq. (2)] is expected to provide a good description forthe shaken 1D Bose-Hubbard model [Eq. (39)] at shorttimes, as far as the weakly interacting regime is concerned.More specifically, if one assumes that the initial state is thatof fully condensed bosons, the system in Eq. (39) can betreated within mean-field theory [102]. In this approxima-tion, the annihilation and creation operators an, a

†n are

merely replaced by classical fields an, a�n, and theHeisenberg equations of motion for an, a†n lead to thetime-dependent GPE in Eq. (2).Throughout this work, our choice for the initial state

corresponds to the Bogoliubov ground state of the effectiveHamiltonian (43) [see Sec. II]. In order to compute heatingrates for such a model, following the procedure detailed inSec. III B, we also had to fix the initial condition for thefluctuation term δa, which, in our GPE-based approach,could be taken to be an arbitrary small perturbation on topof the condensate.However, our choice for the initial condensate wave

function (i.e., ground state of Heff ) suggests a naturalchoice for the initial fluctuation term δaðt ¼ 0Þ. Indeed,one could assume that at t ¼ 0, this fluctuation term isprecisely given by the Bogoliubov wave function of acondensate in the ground state of Heff , as obtained from anumerical diagonalization of the corresponding (effective)Bogoliubov equations, namely,

δanðt ¼ 0Þ ¼Xq

uqeiqn þ v�qe−iqn;

where uq, vq correspond to the solution of

�εavðqÞ þ g g

−g −εavð−qÞ − g

��uqvq

�¼ EavðqÞ

�uqvq

�;

ð44Þ

where εavðqÞ and EavðqÞ depend on the model underconsideration. Here, εavðqÞ denotes the time average ofthe instantaneous dispersion εðq; tÞ [see Eq. (10) for the 1Dmodel, and Eq. (30) for the 2D case], and EavðqÞ is thetime-averaged Bogoliubov dispersion [see Eq. (12) for the1D model, and Eq. (31) for the 2D case].A relevant observable to quantify heating and losses,which

can be probed in experiments, is the noncondensed density,which, in the Bogoliubov approximation, reads [103]

ρncðtÞ ¼1

Vol

Xq

jvqðtÞj2; ð45Þ

with Vol the volume of the system. Given the above initialcondition, it is straightforward to apply the numerical toolsdeveloped in Sec. III B [104], so as to describe the timeevolution of the noncondensed density Eq. (45) in the system.For instance, Fig. 15 shows the exponential growth rate of thenoncondensed fraction (referred to as the loss rate) obtainedfor the 2D lattice model of Sec. VI A, which is thereforeexpected to describe particle losses in the associated2D-driven Bose-Hubbard model. Note that, unsurprisingly,this is very similar to the stability diagram, Fig. 11(b), whichcorresponds to the same parameters (see Sec. III B).The aforementioned aspects are intrinsic to the mean-

field Bogoliubov approximation in the frame of themodel under consideration. More generally, there are afew important points one has to keep in mind, whenadopting a mean-field approach to study out-of-equilibrium

ω=5J Loss rate

0 2 4 6 8 10

K/ω

0

2

4

6

8

10

g/J

0

1

2

3

4

5

6

FIG. 15. Loss rate (exponential growth rate of the noncon-densed density) for the 2D lattice model of Sec. VI A, for ω ¼ 5Jand Ny ¼ 16. This rate is obtained by fitting the exponentialgrowth of the noncondensed fraction, similarly to what was donein Fig. 5 for the energy.


021015-19

ergodic (nonintegrable) bosonic systems subject toperiodic driving:

(i) As already alluded to above, the Floquet Hamilto-nian becomes an increasingly nonlocal operator, thefurther the drive frequency deviates from the infin-ite-frequency limit. An intriguing and interestingpart of this intrinsic nonlocality is due to Floquetmany-body resonances [105] appearing because ofthe hybridization of many-body states induced bythe drive. These resonances appear as a result ofenergy absorption and provide shortcuts betweenstates in energy space separated by multiples of thedrive frequency. Hence, in the unstable phase, theyare expected to affect the heating rates at timest ∼ U−1, when interaction effects between quasipar-ticles become important.

(ii) In fact, one can anticipate (using the inverse-frequency expansion in the rotating frame) thatthe Floquet Hamiltonian of the driven Bose-Hubbard model contains complicated three- andhigher-body interaction terms, starting from orderω−3. It is currently unknown how these terms modifyand limit the application of Bogoliubov’s mean-fieldtheory associated with the effective (Floquet)Hamiltonian.

Despite the open character of these potential issues relatedto interacting bosonic systems, we expect that our GPE-based analysis captures the dominant contribution to theshort-time evolution of the periodically driven Bose-Hubbard model in Eq. (39), in particular, the onset ofinstability.

C. Time evolution beyond the linearized regime:The conserving approximation

Since the mean-field Bogoliubov approximationassumes a macroscopic occupation of the condensatemode, it remains valid provided the number of noncon-densed atoms remains small compared to the total numberof atoms throughout the entire subsequent evolution. Asexpected, this assumption fails in the unstable regions,where the depletion of the condensate grows exponentially

and the mean-field approach typically holds at short times.Thus, the time scale for reaching a sizable dynamicaldepletion sets a natural upper bound on the validity ofmean-field approaches.One way to avoid the problems associated with particle

conservation is to apply the weak-coupling conservingapproximation (WCCA) [74]. Based on a Keldysh fieldtheory formalism, the WCCA is the minimal extension ofBogoliubov theory, which includes the proper effectiveinteractions between the Bogoliubov quasiparticles andthe condensate, order by order in the original interactionstrength U, while ensuring particle conservation at alltimes. Truncated to linear order in U, the WCCA isequivalent to the Bogoliubov-Hartree-Fock approximation[106], and in the following, we restrict ourselves to thiscase. The unknown variables in the WCCA formalismare [107]

ϕðtÞ ¼ e−iR

t

0μðt0Þdt0 haq¼0i;

F11ðt; qÞ ¼1

2hfaqðtÞ; a†qðtÞgic;

F12ðt; qÞ ¼e−2i

Rt

0μðt0Þdt0

2hfaqðtÞ; a−qðtÞgic; ð46Þ

where ϕðtÞ ¼ a0nðtÞe−iR

t

0μðt0Þdt0 ffiffiffiffiffiffiffiffi

Volp

is the q ¼ 0 mode ofthe spatially homogeneous (i.e., n-independent) condensatewave function, and where μðtÞ ¼ −zJ cosðK=ω sinωtÞ þ gis the chemical potential, which is irrelevant for U(1)-conserving dynamics. The subscript c stands for connectedcorrelators: hAðtÞBðtÞic ¼ hAðtÞBðtÞi − hAðtÞihBðtÞi. Theequal-time correlator F11ðt; qÞ is, apart from an additiveconstant, the phonon density. In momentum space, it isrelated to the momentum distribution function of thequasiparticles nqðtÞ by nqðtÞ ¼ F11ðt; qÞ − 1=2. Note thatnqðtÞ does not include the condensate delta function peak atq ¼ 0. The WCCA EOM for the equal-time correlatorsrepresents a simple system of nonlinear and nonlocal inspace equations

i∂tϕðtÞ ¼ εq¼0ðtÞϕðtÞ þUVol

�½ϕðtÞ��½ϕðtÞ�2 þ 2ϕðtÞ

Zq0F11ðt; q0Þ þ ½ϕðtÞ��

Zq0F12ðt; q0Þ

�;

∂tF11ðt; qÞ ¼ 2Im

UVol

�½ϕðtÞ�2 þ

Zq0F12ðt; q0Þ

�½F12ðt; qÞ��

�;

i∂tF12ðt; qÞ ¼½εqðtÞ þ ε−qðtÞ�F12ðt; qÞ þ 2

UVol

�2

�jϕðtÞj2 þ

Zq0F11ðt; q0Þ

�F12ðt; qÞ

þ�½ϕðtÞ�2 þ

Zq0F12ðt;q0Þ

�F11ðt;qÞ

��; ð47Þ

where � denotes complex conjugation.


021015-20

The conserved quantity itself is the total number ofparticles (condensed and excited):

N ¼ jϕðtÞj2 þZqnqðtÞ

¼ jϕðtÞj2 þZq

�F11ðt; qÞ −

1

2

�¼ const: ð48Þ

One recognizes the GPE equation for the spatially homo-geneous condensate density ϕðtÞ and identifies the addi-tional phonon feedback terms. Quite generally, if oneeliminates all terms containing integrals over q, theWCCA EOM decouples into the familiar GPE for thecondensate [see Eq. (2)], while the second two equationsare equivalent to the BdG equations, Eq. (44) [108].Opening up a channel between the quasiparticles and the

condensate, we find that the phonon buildup rate decreases(and, consequently, the condensate depletion slows down)compared to the linearized BdG equations, in agreementwith intuitive expectations. In other words, the WCCAdeviates from the exponentially growing linearized solu-tion, which sets a natural scale on the validity of the mean-field approximation.To justify the applicability of the mean-field approach at

short times, we compare the dynamical buildup of quasi-particle excitations in the parametrically unstable regime,predicted by the linearized EOM [see Eq. (9)], and theWCCA (to order U). Here, we initialize the system in theBogoliubov ground state corresponding to the nondrivenHamiltonian HðK ¼ 0Þ. Figure 16 shows the time evolu-tion of the quasiparticle excitations (top panel) and thecondensate density (bottom panel) for the 2D model ofSec. VI B. One can clearly identify two stages of evolution:In stage I, which continues for about t≲ 220T drivingperiods, the dynamics is governed by the single-particlemean-field physics. The dotted vertical line at t� ≈ 60Tmarks the time up to which the dynamics predicted by thelinearized mean-field equations agrees with the WCCA.Thus, for t ≤ t�, this initial regime features an exponentialbuildup of phonons at the momenta satisfying the resonantcondition ω ¼ EavðqÞ. For 60T ≲ t≲ 220T, the time evo-lution is characterized by a different growth rate than theone predicted by the mean-field equations for this systemconfiguration. By the time t ≈ t�, the backaction effect ofthe quasiparticles onto the condensate becomes sizable,and the dynamics no longer follows the initial linearizedexponential growth. Based on the available data, it is notpossible to determine whether the growth remains expo-nential or follows another law. In stage II, t≳ 220T, thepopulation of the resonant q modes slows down signifi-cantly, although it never really saturates, as can be seenfrom the condensate depletion at longer times [see Fig. 16(bottom panel)]. Because of the unbounded character of thedispersion along the qy direction, the presence of resonantmodes set by the condition ω ¼ EavðqÞ does not allow forthe formation of a Floquet steady state. Nevertheless, the

growth rate diminishes significantly, and the dynamicsslows down, as expected for a prethermal regime.Interestingly, the condensate evolution curve [Fig. 16(bottom panel)] changes curvature in stage II, comparedto stage I. Curiously, we note that a very similar change ofbehavior was reported in a cosmological context [77].Important features of the quasiparticle dynamics are

conveniently reflected in their momentum distributionfunction nqðtÞ [109]. Figure 17 shows a comparisonbetween the momentum distributions, as obtained by timeevolving the linearized and WCCA EOM over a longduration (t ¼ 1000T), for the same 2D model. The visiblelines, which materialize largely populated modes,

Jt0 50 100 150

1/V

ol Σ

qnq(t

)

0

1

2

3

4

5

6

7

8

9

10WCCAlinearised

ΙΙΙ

Jt0 500 1000 1500

|φ(t

)|2 /V

ol

0

2

4

6

8

10

12

14

16

18

WCCAlinearised

FIG. 16. Time dependence of the quasiparticle (phonon)density (top panel) and the condensate density (bottom panel)using the linearized (red line) and the conserving (blue line)approximation, for the 2D band model from Sec. VI B (with alattice along the x direction and the continuum along the ydirection). There are two discernible regimes: I and II (see text),delineated by a vertical dashed line. The vertical dotted line marksthe time up to which the linearized mean-field dynamics and theWCCA agree well. The model parameters are ρ=Vol ¼ 17.34,U=J ¼ 1.011 (g=J ¼ 17.53), m ¼ 1=ð20JÞ, and healing lengthto x-lattice constant ratio ξ=ax ¼ 1.47, ω=J ¼ 32.37, andK=ω ¼ 1.0. We use a total of N ¼ NxNy ¼ 5 × 104 momentumpoints with Nx ¼ 50 and Ny ¼ 1000 to reach the thermodynamiclimit. As an initial state, we choose the Bogoliubov ground state.


021015-21

correspond to on-resonance modes [EavðqÞ ¼ ω], whichhave indeed been shown to be unstable. In the linearizeddynamics, the largest peaks are visible around the max-imally unstable modes, as predicted by our theory [seeEqs. (35) and (37)]; for the considered parameters, the mostunstable mode indeed corresponds to qmum

x ¼ �π, andqmumy is adjusted so that the resonance condition ω ¼

EavðqÞ is fulfilled (see Sec. VI B). We stress that the largewidth of the peaks around qx ¼ �π, in the top panel ofFig. 17, is an artifact due to the significant population ofmomentum modes at long times, which yields saturation onthis plot; we refer to Fig. 1 for a more striking illustrationof these peaks, as calculated at shorter times (for the samemodel parameters). In contrast, the WCCA couples allmomenta through the q0 integrals in Eq. (47), allowing the

population to further spread along the x axis. All modesfulfilling the resonance condition now exhibit peaks ofsimilar magnitudes, precisely materializing the curve ofequation ω ¼ EavðqÞ; note, though, that the populationsof the modes along this curve are not completely homo-geneous, presumably due to the nonlinear character of theWCCA EOM and/or the differences in the instability ratesof resonant modes. Moreover, as we discussed in Sec. IV C,modes that are not precisely on resonance, but sufficientlyclose to it, are also affected by the instability.A summary of the various behavior types of the

momentum distribution, which are expected for the differ-ent parameter regimes, is presented in Sec. VIII (seeFigs. 19 and Table II).We conclude this section by noticing that while the

WCCA to leading order in U features a prethermal Floquetregime at high driving frequencies [74], the absence ofquasiparticle collisions prevents the onset of thermalizingdynamics at later times. Including the next-to-leading orderin U has been done for the periodically driven OðNÞ model[110,111], where the crossover from a prethermal steadystate to a thermal regime becomes clearly visible.

VIII. SUMMARY OF THE MAIN RESULTSAND CONCLUDING REMARKS

The analysis presented in this work reveals the crucialrole played by parametric instabilities in the short-timedynamics of periodically driven band models. It provides aquantitative description and a qualitative understanding ofthe stability diagrams of these periodically driven systems,capturing both the stability boundaries and the instabilityrates in its vicinity. This work also identifies clear sig-natures of these early-stage instabilities, which could bedetected in current cold-atom experiments.On the one hand, the quantitative description of the

instability rates will prove helpful to guide experimentalistsin finding stable regimes of operation. A first indicator isthe instability rates themselves. Table I summarizes thescaling behavior found for instability rates in the thermo-dynamic limit as a function of the model parameters, asderived in Sec. IV C. Remarkably, these scaling laws holdin all cases investigated in this work, i.e., 1D and 2D latticesand high- and low-frequency regimes.Other important information concerns the identification

of the most unstable mode qmum. Table II recalls thecharacteristics of this most unstable mode for the various

FIG. 17. Quasiparticle momentum distribution function nq aftera duration of t ¼ 1000T, for the 2D band model from Sec. VI B(with a lattice along the x direction and continuum along the ydirection). The top panel shows the linearized mean-field model,while the bottom panel shows the conserving approximation(WCCA) to order U. The model parameters are the same as inFig. 16. Here, we set the lattice constant ax ¼ 1 to unity, whileξ ¼ 1.47ax denotes the BEC healing length.

TABLE I. Limiting scalings of instability rates in the thermodynamic limit, in terms of the interaction strength g,the modulation amplitude K, and frequency ω. The high- and low-frequency regimes are defined through acomparison between ω and the free-particle effective bandwidth (see Sec. III A).

Low g Low K Low ω

High-frequency regime Low-frequency regime

Γ ∝ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðg − gcÞ

pΓ ∝ g Γ ∝ K2 Γ ∝ ω × fðK=ωÞ


021015-22

relevant regimes discussed in this work. In the high-frequency regime, where ω is larger than the free-particleeffective bandwidth, the onset of instability is necessarilydriven by the mode q ¼ π (or qx;y;z ¼ π if several latticedegrees of freedom are present). In the low-frequencyregime, the most unstable mode qmum obeys the resonancecondition ω ¼ EavðqÞ. Yet, in dimensions higher thanone, we found that this condition is not sufficient tounambiguously determine qmum since a whole set ofresonant modes (associated with different rates) typicallysatisfy it. Our analysis showed that two situations canoccur and that these could be distinguished by comparingthe frequency ω to the effective Bogoliubov bandwidthassociated with the shaken-lattice direction alone: If ω >ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4jJeff jð4jJeff j þ 2gÞp

, then qmumx ¼ π and qmum⊥ ≠ 0 [see

Eq. (35)]; conversely, if ω <ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4jJeff jð4jJeff j þ 2gÞp

, wefind qmum

x < π and qmum⊥ ¼ 0 [see Eq. (37)]. Let us empha-size that, in principle, the so-called low-frequency regimecan be reached at arbitrarily large frequencies; for instance,this typically occurs whenever the free-particle effectivedispersion is unbounded, as in the case where a continu-ous direction is present. Figures 18 and 19 show themomentum distribution as calculated for the hybrid-2Dband model of Sec. VI B (i.e., a shaken lattice along the xdirection and a continuum along the y direction); byexploring the different parameter regimes, these resultsillustrate all the behaviors described above (Table II).Those predictions on the instability rates can readily be

tested in present-day experiments. For typical experimentalparameters, the ratio ω=J is generally set in the range10–100; therefore, in the case of pure-lattice systems, experi-mentswould typically lie in thehigh-frequency regimedefinedin Sec. III A, and themost unstablemode is thus expected to beqmumx;y;z ¼ π. In turn, experiments with a continuous degree offreedom fall into the second column of Table II, and we thusexpect themaximal instability to occur through qmumx ¼ π andfinite qmum⊥ . We point out that experiments could also inves-tigate the low-frequency regime of driven-lattice systems (firstcolumn of Table II), by working with lower drive frequenciesω ∼ J (see, e.g., Ref. [28]).The different observables that we introduced are acces-

sible in current experiments, and probing them should

allow clear experimental signatures of the predicted insta-bility behavior. In particular, measuring the momentumdistribution of the gas, which is routinely performed throughtime-of-flight techniques, and identifying the positions of thepeaks would give us access to the momentum of the mostunstable modes, which is one of the most recognizablecharacteristics of parametric instabilities.On the other hand, several important conceptual points

arise from the detailed analysis presented in this paper: Inparticular, the intuitive stability criterion ω > 2Weff is notsufficient to properly estimate the instability boundary andrates in the system, as becomes clear from the rigorousderivation we provided. More precisely, the parametricresonance condition for the instability is fulfilled forω ¼ EavðqÞ, but in addition, one has to take the (possibly)large width of the resonance window into account in order

TABLE II. Characterization of the most unstable mode (mum) for various relevant regimes. The low-frequencyregime is generically defined through the criterion according to which the drive frequency ω is smaller than theeffective free-particle bandwidth (e.g., ω < 4jJeff j for the 1D shaken-lattice model of Sec. III A). We stress that theeffective free-particle dispersion is unbounded for models involving a continuous spatial dimension, in which casethe system necessarily corresponds to the low-frequency regime (for all ω).

Low-frequency regime High-frequency regime

ω <ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4jJeff jð4jJeff j þ 2gÞp

ω >ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4jJeff jð4jJeff j þ 2gÞp

qmumx < π qmum

x ¼ π qmumx;y;z ¼ π

qmum⊥ ¼ 0 qmum⊥ > 0 (at the onset of instability)See Eq. (37) See Eq. (35)

(a) (b)

(d)(c)

FIG. 18. Quasiparticle momentum distribution function nq aftera duration of t ¼ 10T, for the 2D band model from Sec. VI B(with a lattice along the x direction and a continuum along the ydirection), computed using the linearized dynamics, for differentvalues of the relevant parameter


=ω; seeTable II. The suggested ellipses correspond to on-resonancemodes [EavðqÞ ¼ π], which are unstable, while the most unstablemodes, which are revealed by the main peaks in the momentumdistribution, are accurately described by our theoretical predic-tions (see Table II).


021015-23

to properly estimate the position of the boundary; we notethat similar effects were found in the context of the many-body Kapitza pendulum [75].As a result of our study, the following general guidelines

deserve to be highlighted: In lattice systems, working athigh frequency (much larger than the effective bandwidthof Bogoliubov excitations) is favorable since no parametricresonance can then occur, ensuring a stable evolution. Wealso analyzed the effect of transverse directions, be it a deeplattice or tubes; although the instability is genericallyincreased in both cases by the fact that more and moremodes are available in the system (offering more possibil-ities to have resonances), we showed that working with alattice should be much more favorable to find stableregimes and have less heating. We stress that our analysisresults from a tight-binding approximation, namely, therestriction of the study to a single Bloch band. Thissimplification is reasonable for situations where the drivefrequency is set away from any interband resonances,which is indeed possible in experiment. Interestingly, wenote that such interband excitations have been generated

and studied experimentally in Ref. [112]. The interplaybetween interband transitions, as generated by setting thedrive frequency close to resonance with interband spacing,and the existence of parametric instabilities should showrich behaviors, which could be explored through thetheoretical approach presented in this work.Finally, we emphasize that the instability rates derived

from parametric resonances (see, e.g., the scalings inTable I) differ quantitatively from the ones computedwithin a FGR approach [64], which generically scale asg2 for low interactions. In particular, we note that contraryto the case of parametric resonances, the rates emanatingfrom the FGR approach strongly depend on the dimension-ality of the system, through its density of states, whichindicates that the two phenomena are indeed very distinctin essence. Since the system is bosonic, we expect that theparametric instability should dominate the short-timedynamics because of its exponential growth rate. It is onlyat later times that Fermi’s golden rule comes to rule thebehavior of the system, when the dynamics will bedominated by collision processes leading to thermalization

(a)

(b) (d)

(c) (e)

(f )

FIG. 19. Quasiparticle momentum distribution function nq for the 2D band model from Sec. VI B (with a lattice along the x directionand a continuum along the y direction), as a function of the parameter


=ω and of qx (upper line) [resp. q⊥ (lowerline)], after integration over q⊥ (resp. qx), and computed as follows: (left panels) after a time t ¼ 10T using the linearized dynamics,(middle panels) same time but using the WCCA, and (right panels) after a longer time t ¼ 101T using the WCCA. The dotted blueline indicates our theoretical prediction for the most unstable mode (see Table II), which is verified in all regimes: Forffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4jJeff jð4jJeff j þ 2gÞp=ω > 1, we find qmum⊥ ¼ 0 and qmum

x < π, as predicted by Eq. (37); forffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4jJeff jð4jJeff j þ 2gÞp

=ω < 1, we findqmumx ¼ π and qmum⊥ > 0, as given by Eq. (35). Note that, in the latter regime, the peaks are only apparent after a long time (here 101T,

right panel) because of the small values of the corresponding instability rates; see also Fig. 18(a), where these peaks are made visible. Asalready observed in Fig. 16, both linearized and WCCA agree very well on the position of the peaks, although the WCCA predicts abroadening of the instability zone due to nonlinear couplings between the modes. Note that at longer times (right panel), higher-orderparametric resonances are also visible. The apparent “quantized” character of the resonant peaks for t ¼ 101T is a numerical artifact dueto the discretization of the variable


=ω used in the numerical simulation.


021015-24

at a higher temperature, a feature that is generically knownfor quantum Bose fluids [77]. Hence, both phenomena takeplace in different regimes of the time evolution. Moreprecisely, for typical experimental parameters, the insta-bility rates derived from parametric resonances are found tobe of the order of 1–10 ms, while the Fermi golden ruledecay usually spreads over times of the order of 100 ms[64]. It is within the scope of present-day experiments toobserve those two distinct regimes.

ACKNOWLEDGMENTS

The authors are pleased to acknowledge J. Berges, I.Bloch, J. C. Budich, J. Dalibard, M. Dalmonte, P. Gaspard,M. Lacki, C. Laflamme, J. Lutsko, J. Nager, S.Nascimbene, H. Pichler, A. Polkovnikov, M. Reitter, U.Schneider, K. Wintersperger, and P. Zoller for insightfuldiscussions. The work in Brussels is supported by the ERCStarting Grant TopoCold, the FRS-FNRS (Belgium), theUniversité Libre de Bruxelles (ULB) and the BelgianScience Policy Office (PAI Project No. P7/18). M. B.acknowledges support by NSF Grant No. DMR-1506340and Grant No. ARO W911NF1410540. E. D. was sup-ported by Harvard-MIT CUA, NSF Grant No. DMR-1308435, AFOSR Quantum Simulation MURI, andAFOSR MURI Photonic Quantum Matter. Part of thecomputational work reported in this paper was performedon the Shared Computing Cluster, which is administered byBoston University’s Research Computing Services. Theauthors also acknowledge the Research ComputingServices group for providing consulting support, whichhas contributed to the results reported within this paper.

APPENDIX A: DIFFERENTPARAMETRIZATIONS OF THE BOGOLIUBOVEXPANSION IN THE TIME-DEPENDENT CASE

In this appendix, we recall the different parametrizationsthat are commonly used for the Bogoliubov expansion intime-dependent systems. Starting from the time-dependentGPE on the lattice,

i∂tan ¼ −Jðanþ1 þ an−1Þ þ K cosðωtÞnan þUjanj2an;ðA1Þ

and given the time-dependent solution for the condensate

wave function að0Þn ðtÞ, the basic idea of the Bogoliubovexpansion is to consider a small perturbation on top

of að0Þn ðtÞ.A natural approach consists in writing anðtÞ ¼ að0Þn ðtÞ þ

δanðtÞ, and to linearize the Gross-Pitaevskii equation interms of δan. This yields the time-dependent BdG equa-tions, which take the general form

i

� _δan_δa�n

�¼�

MnðtÞ Uað0Þn2

−Uað0Þn�2 −MnðtÞ�

��δanδa�n

�; ðA2Þ

where we introduced the notation

MnðtÞδan ≡ −Jðδanþ1 þ δan−1Þ þ 2Ujað0Þn j2þ K cosðωtÞnδan: ðA3Þ

The resulting form of the Bogoliubov equations inEq. (A2), which features spurious complex time-dependentoff-diagonal terms, is quite unusual and, in fact, not veryconvenient. First of all, we find that those terms can lead tonumerical instabilities, when solving this equation usinga finite-difference scheme. Then, we note that the form(A2) does not allow for a simple comparison with thestandard static case: Indeed, setting K ¼ 0 and writing

að0Þn ðtÞ ¼ ffiffiffiρ

pe−iμt, with ρ the condensate density and μ ¼

g − 2J the chemical potential (and g≡Uρ), we find thatEq. (A2) yields the nonstandard form

i

� _δan_δa�n

�¼�−JLþ 2g ge−2iμt

ge2iμt JL − 2g

��δanδa�n

�; ðA4Þ

where the discrete operator L is defined by Lð·Þn ≡ð·Þnþ1 þ ð·Þn−1. The equations of motion (A4) are stillexplicitly time dependent, and hence, they do not allow fora direct identification of the quasiparticle spectrum.In the static case, a common way of solving this issue is

to use a slightly different parametrization, namely, anðtÞ ¼e−iμt½að0Þn ðtÞ þ δanðtÞ�, where we have factored out thedynamical phase μ associated with the free evolution.Using this parametrization, we obtain the Bogoliubovequations in the form

i

� _δan_δa�n

�¼�−JΔþ g g

g JΔ − g

��δanδa�n

�; ðA5Þ

where Δδan ≡ δanþ1 − 2δan þ δan−1 denotes the discreteLaplacian. The resulting equations of motion correspond tothe standard form of the BdG equations, which are indeedtime independent (the spurious complex off-diagonal termshave disappeared), allowing for a direct extraction of theBogoliubov spectrum.In the driven-system situation, the time-dependent phase

of the BECwave function að0Þn ðtÞ does not reduce to a trivialphase μ, as it also contains an additional dynamical phaseinduced by the periodic driving. Therefore, factorizing e−iμt

alone in the Bogoliubov parametrization would not besufficient to get rid of all the spurious terms. In order tosolve that issue, we generalize the method discussed above

by factorizing the entire time-dependent phase of að0Þn ðtÞ:This is achieved by writing


021015-25

anðtÞ ¼ eiϕnðtÞ½ρnðtÞ þ δanðtÞ�; ðA6Þ

where we have introduced the modulus and phase of the

BEC wave function, að0Þn ðtÞ ¼ ρnðtÞeiϕnðtÞ.For the model considered in this work, translational

invariance is recovered in the rotating frame, which allowsone to further simplify the parametrization: Indeed, takinginto account the fact that ρnðtÞ is uniform (it does notdepend on n), we can equivalently write Eq. (A6) in theform [Eq. (4)]

anðtÞ ¼ að0Þn ðtÞ½1þ δanðtÞ�; ðA7Þ

where we have now factorized the full condensate wavefunction. Linearizing the Gross-Pitaevskii equation in δanthen yields the time-dependent Bogoliubov–de Gennesequations, which take the general form considered in themain text,

i

_δan_δa�n

!¼

F nðtÞ Ujað0Þn j2−Ujað0Þn j2 −F nðtÞ�

!�δanδa�n

�; ðA8Þ

where we introduced the operator F nðtÞ, whose action onδan is defined by

F nðtÞδan ≡ −JLðað0ÞδanÞn

að0Þn

þ 2Ujað0Þn j2δan

þ K cosðωtÞnδan − i_að0Þn

að0Þn

δan: ðA9Þ

Compared to Eq. (A2), the differences are twofold. Firstof all, the off-diagonal terms are real in this approach,

which is numerically more practical. Then, we note that thelast term in Eq. (A9), which involves the logarithmic

derivative of the BEC wave function að0Þn ðtÞ, features atime-dependent chemical potential, which takes intoaccount the entire dynamical phase induced by the periodicdriving; in particular, we recover the chemical potentialterm −μ ¼ −gþ 2J in the static limit K → 0.For the reasons detailed above, we have chosen to

consider the convenient parametrization defined inEq. (A7), both in our numerical and in our analyticalextraction of the instability rates; see also Eqs. (47) for theextension to WCCA. We stress that the parametrization inEq. (A7) should be revised for more elaborate models(multiband, multicell,...), where ρnðtÞ is not uniform, usingthe more general form (A6). Finally, we emphasize that allparametrizations lead to the same physical results and thatchoosing one or the other is just a matter of technicalconvenience.

APPENDIX B: USEFUL RESULTS ON THEPARAMETRIC OSCILLATOR

The simplest parametric oscillator is the systemdescribed by the equation of motion

xðtÞ þ ω20½1þ α cosðωtÞ�xðtÞ ¼ 0; ðB1Þ

describing a harmonic oscillator of pulsation ω0 perturbedby a sinusoidal modulation of pulsation ω and amplitudeα ≪ 1. Equivalently, one can introduce the standard oper-ators γ, γ† defined by x ¼ ðγ þ γ†Þ= ffiffiffiffiffiffiffiffi

2ω0

pand p ¼

iffiffiffiffiffiffiffiffiffiffiω0=2

p ðγ† − γÞ, and use the parametrization γðtÞ ¼~u0ðtÞγðt ¼ 0Þ − ~v0�ðtÞγ†ðt ¼ 0Þ to rewrite the equation ofmotion (B1) in the following way (see Ref. [74] for details):

i∂t

�~u0ðtÞ~v0ðtÞ

�¼�ω01þWðtÞσz þ

αω0

2

�0 cosðωtÞe−2iω0t

− cosðωtÞe2iω0t 0

��~u0ðtÞ~v0ðtÞ

�; ðB2Þ

where WðtÞ ¼ αω0=2 cosðωtÞ. As mentioned in Sec. IVA,the simplest way to identify the appearance of parametricresonance in this model is to perform a RWA treatment ofEq. (B2). Since the latter consists in taking the time averageover a drive period in Eq. (B2), the zero-average termWðtÞσz does not contribute, and a pronounced contributionto the dynamics is found when ω ¼ 2ω0. Keeping thecorresponding resonant terms only yields a time-independent 2 × 2 matrix in Eq. (B2), which can straight-forwardly be diagonalized. Its eigenvalues then exhibit animaginary part responsible for an exponential growth of thesolution. The instability rate (here, on resonance) sðRÞ isthus given by the imaginary part of the eigenvalues of this2 × 2 matrix, yielding sðRÞ ¼ αω0=4. This corresponds tothe rate on resonance, i.e., precisely when ω ¼ 2ω0 [113].

However, a key feature that will have crucial importancein the following is that instability in fact arises not onlywhen the resonance condition is precisely satisfied but in afull range of parameters around the resonance point. Toinvestigate more precisely how the rate behaves in thevicinity of the resonance, and at which point it willeventually drop to zero (which will give the limits of theinstability domain), one can follow the general procedureindicated in Landau and Lifshits [3]. The idea is to writeω ¼ 2ω0 þ δ and to solve the problem perturbatively in thedetuning δ and α [114]. At lowest order, we seek a solutionof Eq. (B1) of the form xðtÞ ¼ AðtÞ cos½ðω0 þ δ=2Þt� þBðtÞ sin½ðω0 þ δ=2Þt�, where AðtÞ and BðtÞ vary slowlycompared to the oscillating terms. Inserting this intoEq. (B1) and solving it to first order in δ eventually


021015-26

yields that AðtÞ and BðtÞ behave as est, with a rate sgiven by

s ¼ sðRÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ð2δ=αω0Þ2

q; sðRÞ ¼ αω0=4: ðB3Þ

At first order, the instability rate is thus maximal onresonance and decreases around it, and instability arisesfor all values

ω ∈ ½2ω0 − αω0=2; 2ω0 þ αω0=2�:

This calculation can be extended to higher orders byadding in the ansatz for the solution harmonics that will

be of higher order in the detuning. For instance, at secondorder, the idea is to seek a solution of the form

xðtÞ¼A0ðtÞcos½ðω0þδ=2Þt�þB0ðtÞsin½ðω0þδ=2Þt�þA1ðtÞcos½3ðω0þδ=2Þt�þB1ðtÞsin½3ðω0þδ=2Þt�

and to solve Eq. (B1) to second order in δ and α. We findan unstable regime for

ω ∈ ½2ω0 − αω0=2 − α2ω0=32; 2ω0 þ αω0=2 − α2ω0=32�;

while the instability rate s is the solution of the implicitequation

��

s2 þ ω20 − ω2=4þ αω2

0=2 ωs αω20=2 0

−ωs s2 þ ω20 − ω2=4 − αω2

0=2 0 αω20=2

αω20=2 0 s2 − 8ω2

0 3ωs

0 αω20=2 −3ωs s2 − 8ω2

0

��¼ 0; ðB4Þ

which can be numerically solved. Similarly to the first-order result, the instability rate continuously decreases fromits maximal value to zero at the edges of the instabilitydomain; yet, the latter is no longer centered on theresonance point but is slightly shifted; therefore, themaximal instability rate, which is reached at the centerof the instability domain, does not coincide with theresonance point (although it is not far from it). We concludethis appendix by pointing out that the rates are independentwith respect to the time-dependent term WðtÞσz, at allorders of this perturbative treatment.

APPENDIX C: CHANGE OF BASISLEADING TO EQ. (19)

In this appendix, we describe the change of basisintroduced in Eq. (19). Starting from the Bogoliubovequations written in Fourier space [Eq. (9)],

i∂t

�uqvq



��uqvq

�; ðC1Þ

where

εðq; tÞ ¼ 4J sin

�q2

�sin

�q2þ p0 −

KωsinðωtÞ

�;

one first introduces the change of basis that diagonalizesthe time-averaged part of the equation,

�uqvq

�¼�coshðθqÞ sinhðθqÞsinhðθqÞ coshðθqÞ

��u0qv0q

�; ðC2Þ

where coshð2θqÞ≡ ð4jJeff jsin2ðq=2Þ þ gÞ=EavðqÞ andsinhð2θqÞ≡ g=EavðqÞ, with

EavðqÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4jJeff jsin2ðq=2Þð4jJeff jsin2ðq=2Þ þ 2gÞ

qthe time-averaged Bogoliubov dispersion. Then, defining asecond transformation

�~u0q~v0q

�¼�e2iEavðqÞt 0

0 1

��u0qv0q

�; ðC3Þ

the Bogoliubov equations take the final form

i∂t

�~u0q~v0q

�¼�EavðqÞ1þ WqðtÞ þ sinhð2θqÞ

�0 hqðtÞe−2iEavðqÞt

−hqðtÞe2iEavðqÞt 0

��~u0q~v0q

�; ðC4Þ

where sinhð2θqÞ ¼ g=EavðqÞ, EavðqÞ is the Bogoliubov dispersion associated with the effective GPE within the Bogoliubovapproximation [see Eq. (12)], WqðtÞ is a diagonal matrix of zero average over one driving period,


021015-27

WqðtÞ ¼ 4J sinðq=2ÞXl≠0

J lðK=ωÞ

×

�cosh2θq sinðq2 − lωtÞ þ sinh2θq sinðq2 þ lωtÞ 0

0 −sinh2θq sinðq2 − lωtÞ þ cosh2θq sinðq2 þ lωtÞ

�;

and hqðtÞ is a (real-valued) function that can be Fourierexpanded as

hqðtÞ ¼J24 sin2ðq=2Þ

X∞l¼−∞

½J lðK=ωÞ þ J −lðK=ωÞ�eilωt

¼ 4J sin2ðq=2ÞX∞l¼−∞

J 2lðK=ωÞei2lωt; ðC5Þ

with J lðzÞ the lth Bessel function of the first kind.Because of the formal similarity with Eq. (B2), the

instability rates associated with the equations of motion(C4) can be directly evaluated, following the treatmentdescribed in Appendix B. In particular, it is thereforeapparent that the time-dependent term WqðtÞ does notcontribute to these extracted rates, highlighting the con-venience of the specific frame (gauge) considered in thisanalysis. We recall that, while the short-time behavior(micromotion) of physical observables generally dependson the chosen frame, the instability rates defined in Sec. II,which are associated with the long-time (stroboscopic)dynamics, are independent of this choice (see Figs. 4 and 6for an illustration of the distinction between the micro-motion and long-time behaviors).

APPENDIX D: DISAGREEMENT REGIONS

The analytical approach is based on a perturbativesolution in the detuning and the amplitude of the pertur-bation, which is performed on an effective model restrictedto the second harmonic (see Sec. IV B). We observe that itbreaks down into two main regions:

(i) The main discrepancy with the numerical resultsoccurs around the first zero of J 0 (the only onewhere J 2 is not small as well, zones C in Fig. 2), theanalytics fails to capture the numerically observedinstability, and we also notice a completely differentbehavior between the first-order and second-orderanalytics (see Fig. 20). This comes from the fact that,in this case, the effective dispersion Eav is flat, andthe perturbative approach breaks down. More pre-cisely, the first-order analytical solution gives aresonance domain centered on ω ¼ EavðqÞ ≈ 0 butof infinite width [∝ EavðqÞ−1], thus displaying astrong instability. The second-order correction dis-places the center of the resonance domain as∝ EavðqÞ−3, sweeping away all instability in theconsidered parameter range. More generally, the

perturbative approach does not converge and cannotcapture the observed behavior (which seems to bedue to the second harmonic alone).

(ii) Slight discrepancies also occur “between” the lobesof the stability diagram, as the numerics and theanalytics disagree on the way those “black zones”close (or not) at large g. First, near the zeros of J 0

(i.e., at the left border of each lobe, zones D inFig. 2), the numerics displays a transition to insta-bility which is not captured by the analytics (sim-ilarly to what was observed in case C). However, thisfailure of the analytics is less marked than in case C(in the sense that it appears for much larger g) fortwo main reasons: On the one hand, the fact thatJ 2ðK=ωÞ is also small can partly kill the divergen-ces due to the vanishing of J 0ðK=ωÞ; on the otherhand, the stable regime extends here to much largerg, coherently with the fact that the second harmonichas a very small amplitude [indeed, as visible inFig. 21(a), the numerical instability seems rather tobe from higher harmonics). More precisely, at pointswhere J 2ðK=ωÞ ¼ 0, the analytics, which is essen-tially built on a second-harmonic-based model, isalways stable and cannot reproduce the numericalinstabilities due to higher harmonics (which never-theless show up only at large g). Near such points

0

5

10

15Γ

20

25

30

0 1 2 3 4 5 6 7 8 9 10

g/J

ω=5J,K/ω=2.4

NumericsAnalytics up to order 2Analytics up to order 1

Numerics on second harmonic alone

FIG. 20. Comparison between the numerically and analyticallycomputed instability rate Γ in the case K=ω ¼ 2.4 [zone C on thestability diagram of Fig. 2, where J 0ðK=ωÞ ≈ 0 and J 2ðK=ωÞ islarge]. Since the effective Bogoliubov dispersion is flat, theanalytical perturbative approach breaks down.


021015-28

(zones E), the numerical instability seems to be dueto a joined effect of the second and next harmonics,and it is therefore not accurately captured by theanalytical approach [see Fig. 21(b)].

[1] G. Casati, B. V. Chirikov, F. M. Izrailev, and J. Ford,Stochastic Behavior of a Quantum Pendulum under aPeriodic Perturbation, Lecture Notes Phys. 93, 334 (1979).

[2] P. Kapitza, Dynamical Stability of a Pendulum When itsPoint of Suspension Vibrates, Sov. Phys. JETP 21, 588(1951).

[3] L. Landau and E. Lifshits, Theoretical Physics (PergamonPress, Oxford, 1969), Vol. 1.

[4] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland,Quantum Dynamics of Single Trapped Ions, Rev. Mod.Phys. 75, 281 (2003).

[5] L. Lu, J. D. Joannopoulos, and M. Soljačić, TopologicalPhotonics, Nat. Photon 8, 821 (2014).

[6] J. Cayssol, B. Dora, F. Simon, and R. Moessner, FloquetTopological Insulators, J. Phys. Status Solidi Rapid Res.Lett. 7, 101 (2013).

[7] T. Oka and H. Aoki, Photovoltaic Hall Effect in Graphene,Phys. Rev. B 79, 081406 (2009).

[8] T. Kitagawa, T. Oka, A. Brataas, L. Fu, and E. Demler,Transport Properties of Nonequilibrium Systems under theApplication of Light: Photoinduced Quantum Hall Insula-tors without Landau Levels, Phys. Rev. B 84, 235108(2011).

[9] N. Goldman, J. C. Budich, and P. Zoller, TopologicalQuantum Matter with Ultracold Gases in Optical Lattices,Nat. Phys. 12, 639 (2016).

[10] A. Eckardt, Atomic Quantum Gases in Periodically DrivenOptical Lattices, arXiv:1606.08041v1 [Rev. Mod. Phys.(to be published)].

[11] N. Goldman and J. Dalibard, Periodically-Driven QuantumSystems: Effective Hamiltonians and Engineered GaugeFields, Phys. Rev. X 4, 031027 (2014).

[12] M. Bukov, L. D’Alessio, and A. Polkovnikov, UniversalHigh-Frequency Behaviour of Periodically-Driven Quan-tum Systems: From Dynamical Stabilization to FloquetEngineering, Adv. Phys. 64, 139 (2015).

[13] J. H. Shirley, Solution of the Schrodinger Equation with aHamiltonian Periodic in Time, Phys. Rev. 138B, B979(1965).

[14] S. Rahav, I. Gilary, and S. Fishman, Effective Hamiltoniansfor Periodically Driven Systems, Phys. Rev. A 68, 013820(2003).

[15] This change of basis is important to keep in mind whenpreparing the system in a given initial state.

[16] N. Goldman, J. Dalibard, M. Aidelsburger, and N. Cooper,Periodically-Driven Quantum Matter: The Case of Reso-nant Modulations, Phys. Rev. A 91, 033632 (2015).

[17] A. Eckardt and E. Anisimovas, High-Frequency Approxi-mation for Periodically-Driven Quantum Systems from aFloquet-Space Perspective, New J. Phys. 17, 093039 (2015).

[18] T. Mikami, S. Kitamura, K. Yasuda, N. Tsuji, T. Oka, andH. Aoki, Brillouin-Wigner Theory for High-FrequencyExpansion in Periodically Driven Systems: Application toFloquet Topological Insulators, Phys. Rev. B 93, 144307(2016).

[19] S. Rahav, I. Gilary, and S. Fishman, Time IndependentDescription of Rapidly Oscillating Potentials, Phys. Rev.Lett. 91, 110404 (2003).

[20] S. Blanes, F. Casas, J. A. Oteo, and J. Ros, The MagnusExpansion and Some of Its Applications, Phys. Rep. 470,151 (2009).

[21] I. M. Georgescu, S. Ashhab, and F. Nori, QuantumSimulation, Rev. Mod. Phys. 86, 153 (2014).

[22] D. H. Dunlap and V. M. Kenkre, Dynamic Localization ofa Charged Particle Moving under the Influence of anElectric Field, Phys. Rev. B 34, 3625 (1986).

0

1

2

3

Γ Γ

4

5

6

7

8

9

0 5 10 15 20 25 30 35 40 45 50

g /J

ω=5J,K/ω=5.52NumericsNumerics on second harmonic

Numerics on second and fourth harmonicsAnalytics up to order 2Analytics up to order 1

(a)

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40 45 50

g /J

ω=5J,K/ω=5NumericsNumerics on second harmonic

Numerics on second and fourth harmonicsAnalytics up to order 2Analytics up to order 1

(b)

FIG. 21. Comparison between the numerically and analytically computed instability rate Γ in the cases K=ω ¼ 5.52 [(left panel), zoneD of the stability diagram of Fig. 2, where J 0ðK=ωÞ ≈ 0] and K=ω ¼ 5 [(right panel), zone E of the stability diagram of Fig. 2, whereJ 2ðK=ωÞ ≈ 0]. The disagreement in those cases can be due to a breakdown of the perturbative approach [when J 0ðK=ωÞ vanishes, (leftpanel)] and/or a vanishing of the second harmonic [when J 2ðK=ωÞ vanishes, (right panel)].


021015-29

https://doi.org/10.1007/BFb0021757

https://doi.org/10.1103/RevModPhys.75.281


https://doi.org/10.1038/nphoton.2014.248

https://doi.org/10.1002/pssr.201206451

https://doi.org/10.1002/pssr.201206451

https://doi.org/10.1103/PhysRevB.79.081406



https://doi.org/10.1038/nphys3803

http://arXiv.org/abs/1606.08041v1

http://arXiv.org/abs/1606.08041v1


https://doi.org/10.1080/00018732.2015.1055918

https://doi.org/10.1103/PhysRev.138.B979

https://doi.org/10.1103/PhysRev.138.B979

https://doi.org/10.1103/PhysRevA.68.013820



https://doi.org/10.1088/1367-2630/17/9/093039



https://doi.org/10.1103/PhysRevLett.91.110404


https://doi.org/10.1016/j.physrep.2008.11.001

https://doi.org/10.1016/j.physrep.2008.11.001



[23] D. H. Dunlap and V. M. Kenkre, Effect of Scattering on theDynamic Localization of a Particle in a Time-DependentElectric Field, Phys. Rev. B 37, 6622 (1988).

[24] M. Holthaus, Collapse of Minibands in Far-InfraredIrradiated Superlattices, Phys. Rev. Lett. 69, 351 (1992).

[25] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O.Morsch, and E. Arimondo, Dynamical Control of Matter-Wave Tunneling in Periodic Potentials, Phys. Rev. Lett.99, 220403 (2007).

[26] A. Eckardt, M. Holthaus, H. Lignier, A. Zenesini, D.Ciampini, O. Morsch, and E. Arimondo, Exploring Dy-namic Localization with a Bose-Einstein Condensate,Phys. Rev. A 79, 013611 (2009).

[27] A. Eckardt and M. Holthaus, AC-Induced Superfluidity,Europhys. Lett. 80, 50004 (2007).

[28] C. Sias, H. Lignier, Y. Singh, A. Zenesini, D. Ciampini,O. Morsch, and E. Arimondo, Observation of Photon-Assisted Tunneling in Optical Lattices, Phys. Rev. Lett.100, 040404 (2008).

[29] A. Eckardt, C. Weiss, and M. Holthaus, Superfluid-Insulator Transition in a Periodically-Driven OpticalLattice, Phys. Rev. Lett. 95, 260404 (2005).

[30] D. Ciampini, O. Morsch, and E. Arimondo, QuantumControl in Strongly Driven Optical Lattices, Int. J.Quantum. Inform. 09, 139 (2011).

[31] A. Zenesini, H. Lignier, D. Ciampini, O. Morsch, and E.Arimondo, Coherent Control of Dressed Matter Waves,Phys. Rev. Lett. 102, 100403 (2009).

[32] A. Zenesini, H. Lignier, O. Morsch, D. Ciampini, and E.Arimondo, Tunneling Control and Localization for Bose-Einstein Condensates in a Frequency Modulated OpticalLattice, Laser Phys. 20, 1182 (2010).

[33] S. Fishman, D. R. Grempel, and R. E. Prange, Chaos,Quantum Recurrences and Anderson Localization, Phys.Rev. Lett. 49, 509 (1982).

[34] G. Casati, I. Guarneri, and D. L. Shepelyansky, AndersonTransition in a One-Dimensional System with ThreeIncommensurate Frequencies, Phys. Rev. Lett. 62, 345(1989).

[35] M. Lopez, J. F. Clément, P. Szriftgiser, J. C. Garreau, andD. Delande, Experimental Test of Universality of theAnderson Transition, Phys. Rev. Lett. 108, 095701 (2012).

[36] M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y.Chen, and I. Bloch, Experimental Realization of StrongEffective Magnetic Fields in an Optical Lattice, Phys. Rev.Lett. 107, 255301 (2011).

[37] M. Aidelsburger, M. Atala, M. Lohse, J. Barreiro, B.Paredes, and I. Bloch, Realization of the HofstadterHamiltonian with Ultracold Atoms in Optical Lattices,Phys. Rev. Lett. 111, 185301 (2013).

[38] H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton,and W. Ketterle, Realizing the Harper Hamiltonian withLaser-Assisted Tunneling in Optical Lattices, Phys. Rev.Lett. 111, 185302 (2013).

[39] M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J.Barreiro, S. Nascimbene, N. R. Cooper, I. Bloch, and N.Goldman, Measuring the Chern Number of HofstadterBands with Ultracold Bosonic Atoms, Nat. Phys. 11, 162(2015).

[40] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T.Uehlinger, D. Greif, and T. Esslinger, ExperimentalRealisation of the Topological Haldane Model, Nature(London) 515, 237 (2014).

[41] C. J. Kennedy, W. C. Burton, W. C. Chung, and W.Ketterle, Observation of Bose-Einstein Condensation ina Strong Synthetic Magnetic Field, Nat. Phys. 11, 859(2015).

[42] C. E. Creffield, G. Pieplow, F. Sols, and N. Goldman,Realization of Uniform Synthetic Magnetic Fields byPeriodically Shaking an Optical Square Lattice, New J.Phys. 18, 093013 (2016).

[43] A. Eckardt, P. Hauke, P. Soltan-Panahi, C. Becker, K.Sengtock, and M. Lewenstein, Frustrated Quantum Anti-ferromagnetism with Ultracold Bosons in a TriangularLattice, Europhys. Lett. 89, 10010 (2010).

[44] J. Struck, C. Olschlager, R. Le Targat, P. Soltan-Panahi,A. Eckardt, M. Lewenstein, P. Windpassinger, and K.Sengstock, Quantum Simulation of Frustrated ClassicalMagnetism in Triangular Optical Lattices, Science 333,996 (2011).

[45] R. Desbuquois, M. Messer, F. Görg, K. Sandholzer,G. Jotzu, and T. Esslinger, Controlling the Floquet StatePopulation and Observing Micromotion in a PeriodicallyDriven Two-Body Quantum System, arXiv:1703.07767[Phys. Rev X (to be published)].

[46] G. Jotzu, M. Messer, F. Görg, D. Greif, R. Desbuquois,and T. Esslinger, Creating State-Dependent Lattices forUltracold Fermions by Magnetic Gradient Modulation,Phys. Rev. Lett. 115, 073002 (2015).

[47] N. Fläschner, D. Vogel, M. Tarnowski, B. S. Rem,D. S. Lühmann, M. Heyl, J. C. Budich, L. Mathey, K.Sengstock, and C. Weitenberg, Observation of a Dynami-cal Topological Phase Transition, arXiv:1608:05616.

[48] A. Robertson, V. M. Galitski, and G. Refael, DynamicStimulation of Quantum Coherence in Systems of LatticeBosons, Phys. Rev. Lett. 106, 165701 (2011).

[49] S. Nascimbène, N. Goldman, N. R. Cooper, and J.Dalibard, Dynamic Optical Lattices of Sub-wavelengthSpacing for Ultracold Atoms, Phys. Rev. Lett. 115, 140401(2015).

[50] H. M. Price, T. Ozawa, and N. Goldman, SyntheticDimensions for Cold Atoms from Shaking a HarmonicTrap, Phys. Rev. A 95, 023607 (2017).

[51] J. C. Budich, Y. Hu, and P. Zoller,Helical Floquet Channelsin 1D Lattices, Phys. Rev. Lett. 118, 105302 (2017).

[52] G. Goldstein, C. Aron, and C. Chamon, Photo-InducedSuperconductivity in Semiconductors, Phys. Rev. B 91,054517 (2015).

[53] D. Fausti, R. I. Tobey, N. Dean, S. Kaiser, A. Dienst, M. C.Hoffmann, S. Pyon, T. Takayama, H. Takagi, and A.Cavalleri, Light-Induced Superconductivity in a Stripe-Ordered Cuprate, Science 331, 189 (2011).

[54] M. Knap, M. Babadi, I. Martin, and E. Demler, DynamicalCooper Pairing in Non-equilibrium Electron-PhononSystems, Phys. Rev. B 94, 214504 (2016).

[55] N. H. Lindner, G. Refael, and V. Galitski, Floquet Topo-logical Insulator in Semiconductor Quantum Wells, Nat.Phys. 7, 490 (2011).


021015-30






https://doi.org/10.1209/0295-5075/80/50004




https://doi.org/10.1142/S0219749911007150

https://doi.org/10.1142/S0219749911007150


https://doi.org/10.1134/S1054660X10090410













https://doi.org/10.1038/nature13915

https://doi.org/10.1038/nature13915



https://doi.org/10.1088/1367-2630/18/9/093013

https://doi.org/10.1088/1367-2630/18/9/093013

https://doi.org/10.1209/0295-5075/89/10010

https://doi.org/10.1126/science.1207239


http://arXiv.org/abs/1703.07767



http://arXiv.org/abs/1608:05616












[56] K. Sacha, Modeling Spontaneous Breaking of Time-Translation Symmetry, Phys. Rev. A 91, 033617 (2015).

[57] D. V. Else, B. Bauer, and C. Nayak, Floquet Time Crystals,Phys. Rev. Lett. 117, 090402 (2016).

[58] J. Zhang, P. W. Hess, A. Kyprianidis, P. Becker, A. Lee,J. Smith, G. Pagano, I. D. Potirniche, A. C. Potter, A.Vishwanath, N. Y. Yao, and C. Monroe, Observation of aDiscrete Time Crystal, arXiv:1609.08684.

[59] E. Anisimovas, G. Zlabys, B. M. Anderson, G. Juzeliunas,and A. Eckardt, The Role of Real-Space Micromotion forBosonic and Fermionic Floquet Fractional Chern Insula-tors, Phys. Rev. B 91, 245135 (2015).

[60] A. G. Grushin, A. Gómez-León, and T. Neupert, FloquetFractional Chern Insulators, Phys. Rev. Lett. 112, 156801(2014).

[61] M. Bukov, M. Kolodrubetz, and A. Polkovnikov,Schrieffer-Wolff Transformation for Periodically DrivenSystems: Strongly Correlated Systems with ArtificialGauge Fields, Phys. Rev. Lett. 116, 125301 (2016).

[62] P. Weinberg, M. Bukov, S. Vajna, L. D’Alessio, A.Polkovnikov, and M. Kolodrubetz, Adiabatic PerturbationTheory and Geometry of Periodically-Driven Systems,arXiv:1606.02229.

[63] T. Bilitewski and N. R. Cooper, Population Dynamics in aFloquet Realization of the Harper-Hofstadter Hamilto-nian, Phys. Rev. A 91, 063611 (2015).

[64] T. Bilitewski and N. R. Cooper, Scattering Theory forFloquet-Bloch States, Phys. Rev. A 91, 033601 (2015).

[65] S. Choudhury and E. Mueller, Stability of a Floquet Bose-Einstein Condensate in a One-Dimensional Optical Lat-tice, Phys. Rev. A 90, 013621 (2014).

[66] S. Choudhury and E. Mueller, Transverse CollisionalInstabilities of a Bose-Einstein Condensate in a DrivenOne-Dimensional Lattice, Phys. Rev. A 91, 023624 (2015).

[67] L. Fallani, L. De Sarlo, J. E. Lye, M. Modugno, R. Saers,C. Fort, and M. Inguscio, Observation of DynamicalInstability for a Bose-Einstein Condensate in a Moving1D Optical Lattice, Phys. Rev. Lett. 93, 140406 (2004).

[68] M. Modugno, C. Tozzo, and F. Dalfovo, Role of Trans-verse Excitations in the Instability of Bose-EinsteinCondensates Moving in Optical Lattices, Phys. Rev. A70, 043625 (2004).

[69] Y. Zheng, M. Kostrun, and J. Javanainen, Low-Acceler-ation Instability of a Bose-Einstein Condensate in anOptical Lattice, Phys. Rev. Lett. 93, 230401 (2004).

[70] M. Kramer, C. Tozzo, and F. Dalfovo,Parametric Excitationof a Bose-Einstein Condensate in a One-DimensionalOptical Lattice, Phys. Rev. A 71, 061602 (2005).

[71] C. Tozzo, M. Kramer, and F. Dalfovo, Stability Diagramand Growth Rate of Parametric Resonances in Bose-Einstein Condensates in One-Dimensional Optical Latti-ces, Phys. Rev. A 72, 023613 (2005).

[72] L. De Sarlo, L. Fallani, J. E. Lye, M. Modugno, R. Saers,C. Fort, and M. Inguscio, Unstable Regimes for a Bose-Einstein Condensate in an Optical Lattice, Phys. Rev. A72, 013603 (2005).

[73] C. E. Creffield, Instability and Control of a Periodically-Driven Bose-Einstein Condensate, Phys. Rev. A 79,063612 (2009).

[74] M. Bukov, S. Gopalakrishnan, M. Knap, and E. Demler,Prethermal Floquet Steady States and Instabilities in thePeriodically Driven, Weakly Interacting Bose-HubbardModel, Phys. Rev. Lett. 115, 205301 (2015).

[75] R. Citro, E. G. Dalla Torre, L. D’Alessio, A. Polkovnikov,M. Babadi, T. Oka, and E. Demler, Dynamical Stability ofa Many-Body Kapitza Pendulum, Ann. Phys. (Amsterdam)360, 694 (2015).

[76] M. Bukov and M. Heyl, Parametric Instability in Peri-odically Driven Luttinger Liquids, Phys. Rev. B 86,054304 (2012).

[77] J. Berges, K. Boguslavski, S. Schlichting, and R.Venugopalan, Basin of Attraction for Turbulent Thermal-ization and the Range of Validity of Classical-StatisticalSimulations, J. High Energy Phys. 05 (2014) 054.

[78] J. Berges, K. Boguslavski, S. Schlichting, and R.Venugopalan, Universality Far from Equilibrium: FromSuperfluid Bose Gases to Heavy-Ion Collisions, Phys. Rev.Lett. 114, 061601 (2015).

[79] A. Pineiro Orioli, K. Boguslavski, and J. Berges,UniversalSelf-Similar Dynamics of Relativistic and NonrelativisticField Theories Near Nonthermal Fixed Points, Phys. Rev.D 92, 025041 (2015).

[80] N. Gemelke, E. Sarajlic, Y. Bidel, S. Hon, and S. Chu,Parametric Amplification of Matter Waves in PeriodicallyTranslated Optical Lattices, Phys. Rev. Lett. 95, 170404(2005).

[81] G. K. Campbell, J. Mun, M. Boyd, E. W. Streed, W.Ketterle, and D. E. Pritchard, Parametric Amplificationof Atoms, Phys. Rev. Lett. 96, 020406 (2006).

[82] B. M. Anderson, L. W. Clark, J. Crawford, A. Glatz,I. S. Aronson, P. Scherpelz, L. Feng, C. Chin, and K.Levin, Floquet-Band Engineering of Shaken BosonicCondensates, arXiv:1612.06908 [Phys. Rev. Lett. (to bepublished)].

[83] S. I. Mistakidis, T. Wulf, A. Negretti, and P. Schmelcher,Resonant Quantum Dynamics of Few Ultracold Bosons inPeriodically Driven Finite Lattices, J. Phys. B 48, 244004(2015).

[84] L. D. Landau and E. M. Lifshitz, Mechanics (Elsevier,Amsterdam, 2008).

[85] I. Carusotto and C. Ciuti, Quantum Fluids of Light, Rev.Mod. Phys. 85, 299 (2013).

[86] This is generically done by applying the kick operator [11]at time t ¼ 0. However, for this model, the action of thekick operator Kkickð0Þ reduces to a multiplication by atrivial phase when acting on a Bloch state, so there is, infact, no need to “rotate.”

[87] V. Novicenko, E. Anisimovas, and G. Juzeliunas, FloquetAnalysis of a Quantum System with Modulated PeriodicDriving, Phys. Rev. A 95, 023615 (2017).

[88] This transformation is formally reminiscent of a standardgauge transformation in electromagnetism, i.e., whengoing from a gauge where the electric field is written asE ¼ −∇V to a gauge where E ¼ −dA=dt, where V and Adenote the scalar and vector potentials.

[89] The quantity q represents the momentum of the corre-sponding Bogoliubov quasiparticle with respect to theground state [as becomes clear from the equation


021015-31
























https://doi.org/10.1016/j.aop.2015.03.027

https://doi.org/10.1016/j.aop.2015.03.027



https://doi.org/10.1007/JHEP05(2014)054



https://doi.org/10.1103/PhysRevD.92.025041

https://doi.org/10.1103/PhysRevD.92.025041






https://doi.org/10.1088/0953-4075/48/24/244004

https://doi.org/10.1088/0953-4075/48/24/244004




anðtÞ ¼ að0Þn ðtÞ½1þ δanðtÞ�]. This will always be tacitlyimplied in the following, when referring to as mode q.

[90] At least for sufficiently high frequency, as consideredin the calculations of Ref. [73]; see discussion belowconcerning the behavior associated with the low-frequencyregime.

[91] Slight deviations from this scaling appear only for smallvalues of K=ω.

[92] G. Salerno, T. Ozawa, H. M. Price, and I. Carusotto,Floquet Topological System Based on Frequency-Modulated Classical Coupled Harmonic Oscillators,Phys. Rev. B 93, 085105 (2016).

[93] The same analysis holds in the low-frequency regimeunless the system is already unstable at low g, where aresonance can show up [since 2EavðqÞ ¼ lω may alreadyhave a solution for some q0, l0]. The instability rate is thenno longer governed by the mode q ¼ π and the lowestharmonic, but by the mode q0 and the harmonic l0.

[94] See below for a precise discussion about the validity of thisapproximation.

[95] We recall that the zeros of J 2 are very close to the zeros ofJ 0, but slightly smaller. Moreover, J 0ðxÞ has an addi-tional zero around x ¼ 2.4, where J 2ðxÞ is not smallcompared to unity.

[96] Only very small systems with an odd number of sites (thuswithout featuring a mode q ¼ π) are expected to displaydifferences on the stability boundary.

[97] J. Dalibard, F. Gerbier, G. Juzeliūnas, and P. Öhberg,Colloquium: Artificial Gauge Potentials for NeutralAtoms, Rev. Mod. Phys. 83, 1523 (2011).

[98] N. Goldman, G. Juzeliunas, P. Ohberg, and I. B. Spielman,Light-Induced Gauge Fields for Ultracold Atoms, Rep.Prog. Phys. 77, 12 (2014).

[99] We recall that this singular vanishing of the Bessel functionis responsible for a breakdown of the perturbative approachin the 1D case, which resulted in small disagreement zones(C and D) between the numerical and analytical stabilitydiagrams (Fig. 2).

[100] Equation (40) was derived in the thermodynamic limit.For finite-size systems with open boundary conditions,there are additional corrections that become relevant forω≲ 2Jeff .

[101] M. Bukov, L. D’Alessio, and A. Polkovnikov, Adv. Phys.64, 139 (2015).

[102] V. M. Popov, Functional Integrals in Quantum FieldTheory and Statistical Physics (Reidel, Dordrecht, 1983).

[103] Y. Castin, Coherent Atomic Matter Waves, edited by R.Kaiser, C. Westbrook, and F. David, Proceedings of the

Les Houches Summer School of Theoretical Physics(EDP Sciences/Springer-Verlag, Berlin, 2001), Vol. LXXII,Chap. “Bose-Einstein condensates in atomic gases: Simpletheoretical results.”

[104] At least in dimension d > 1, since the 1D case isproblematic because of the divergence of the quantumdepletion in the thermodynamic limit (which is indepen-dent of the drive and, hence, exists even at time t ¼ 0).

[105] M. Bukov, M. Heyl, D. A. Huse, and A. Polkovnikov,Heating and Many-Body Resonances in a PeriodicallyDriven Two-Band System, Phys. Rev. B 93, 155132 (2016).

[106] A. Griffin, Conserving and Gapless Approximations for anInhomogeneous Bose Gas at Finite Temperatures, Phys.Rev. B 53, 9341 (1996).

[107] We implicitly assume here that the system is translationalinvariant and condensation occurs in the q ¼ 0 mode. Forthe general case, please consult the Supplemental Materialto Ref. [74].

[108] The Bogoliubov equations [see Eq. (8)] can indeed beexactly rewritten in the form

∂tF11ðt; qÞ ¼ 2gImð½F12ðt; qÞ��Þ;i∂tF12ðt; qÞ ¼ ½εqðtÞ þ ε−qðtÞ − 2μþ 4g�F12ðt; qÞ

þ 2gF11ðt; qÞ;

where we have defined the correlators in Eq. (46).[109] The condensate delta peak at q ¼ 0 is not part of nqðtÞ

since this mode is treated additionally via the field ϕðtÞ.[110] S. A. Weidinger and M. Knap, Floquet Prethermalization

and Regimes of Heating in a Periodically Driven, Inter-acting Quantum System, arXiv:1609.09089 [Phys. Rev.Lett. (to be published)].

[111] A. Chandran and S. L. Sondhi, Interaction-StabilizedSteady States in the Driven OðNÞ Model, Phys. Rev. B93, 174305 (2016).

[112] M. Weinberg, C. Ölschläger, C. Sträter, S. Prelle, A.Eckardt, K. Sengstock, and J. Simonet, MultiphotonInterband Excitations of Quantum Gases in Driven OpticalLattices, Phys. Rev. A 92, 043621 (2015).

[113] There are also other resonances for ω ¼ 2ω0=n, but theyare of higher order, in the sense that both the magnitude ofthe instability rates and the width of the resonance domainscale as αn.

[114] The idea is indeed that the nth order correction in δ willscale as αn.


021015-32



https://doi.org/10.1088/0034-4885/77/12/126401

https://doi.org/10.1088/0034-4885/77/12/126401

https://doi.org/10.1080/00018732.2015.1055918

https://doi.org/10.1080/00018732.2015.1055918









parametric instability rates in periodically driven...

Documents