Tunable Landau-Zener transitions in a spin-orbit coupled Bose-Einstein condensate

Abraham J. Olson,1, ∗ Su-Ju Wang,1 Robert J. Niffenegger,1

Chuan-Hsun Li,2 Chris H. Greene,1 and Yong P. Chen1, 2, 3, †

1Department of Physics, Purdue University, West Lafayette IN 479072School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907

3Birck Nanotechnology Center, Purdue University, West Lafayette IN 47907(Dated: October 8, 2013)

The Landau-Zener (LZ) transition is one of the most fundamental phenomena in quantum dy-namics. It describes nonadiabatic transitions between quantum states near an avoided crossing thatcan occur in diverse physical systems. Here we report experimental measurements and tuning of LZtransitions between the dressed eigenlevels of a synthetically spin-orbit (SO) coupled Bose-Einsteincondensate (BEC). We measure the transition probability as the BEC is accelerated through theSO avoided crossing, and study its dependence on the coupling between the diabatic (bare) states,eigenlevel slope, and eigenstate velocity—the three parameters of the LZ model that are indepen-dently controlled in our experiments. Furthermore, we performed time-resolved measurements ofthe LZ transitions to determine the diabatic switching time. Our observations show excellent quan-titative agreement with the LZ model and numerical simulations of the quantum dynamics in thequasimomentum space. The tunable LZ transition may be exploited to enable a spin-dependentatomtronic transistor, and open new possibilities to realize novel quantum states based on nonadi-abatic synthetic gauge fields.

PACS numbers: 03.75.Lm, 67.85.De

Controllable “synthetic” gauge fields can be createdusing laser-dressed adiabatic states in ultracold atomicgases [1, 2]. Rapid experimental progress has, amongmany other developments, realized measurements of bothbosonic and fermionic ultracold atoms in synthetic spin-orbit (SO) gauge fields [3–5]. Such developments havemotivated many recent proposals for using more elabo-rate laser-dressed synthetic gauge fields to create quan-tum simulators using ultracold atoms [6] to realize novelquantum states such as topological insulators [7, 8] andMajorana fermions [9, 10].

For the laser-dressed synthetic gauge fields realized inexperiments and proposed in theories, it is typically as-sumed that the system adiabatically follows the dressedeigenlevels [1, 2]. Naturally, the hitherto unexploredregimes in which this adiabatic assumption no longerholds are also of interest, as more complex studies andcoupling schemes proceed. For example, some propos-als for more exotic synthetic gauge fields rely on non-adiabatic (diabatic) schemes [1, 11]. In addition, thebehavior of quantum states near avoided crossings areof interest, because topological phase changes can occurthere (e.g. in topological semi-metals and topologicalinsulators [12, 13]). Motivated by this potential, we in-vestigate the interband transitions as a synthetically SOcoupled Bose-Einstein condensate is accelerated throughthe SO band crossing. We find that the Landau-Zener(LZ) theory provides an excellent quantitative model forunderstanding such transitions.

The Landau-Zener model [14] describes the transitionof a quantum state between two adiabatic eigenlevels

when some parameter that controls the eigenstate of thesystem is linearly varied in time [15]. The LZ model as-sumes the time-dependent Schrodinger equation

i~∂

∂t

(φ1φ2

)=

(E1(x) Ω/2Ω/2 E2(x)

)(φ1φ2

), (1)

where the diabatic (“bare”) states, φ1,2, have energies,E1,2 which linearly vary with some adiabatic parameter,x, and cross at xc (E1(xc) = E2(xc)). The difference inslopes between the energy levels at the crossing is definedas β = [∂E1(x)/∂x− ∂E2(x)/∂x]x=xc

. Ω is the couplingbetween the two diabatic energy levels. This coupling“dresses” the bare energy levels and forms new adiabaticeigenlevels separated by Ω at the avoided crossing.

If a quantum state begins in one adiabatic eigenlevelfar from the avoided crossing and is given some eigen-state velocity, v = dx/dt, of the adiabatic parameterin the direction toward the avoided crossing, it acquiressome probability, PLZ , to make a diabatic transition tothe other adiabatic eigenlevel as it moves past xc. Thisdiabatic transition probability is determined to be

PLZ = exp[−2π(Ω/2)2/(~vβ)

]. (2)

With small velocities or strong coupling, the adiabatictheorem holds and negligible transfer occurs. However,with high velocities or weak coupling, the diabatic tran-sition probability can be significant.

Previous experimental measurements of LZ transi-tions have been performed with diverse physical sys-

arX

iv:1

310.

1818

v1 [

cond

-mat

.qua

nt-g

as]

7 O

ct 2

013

2

qi

qi

qc

(a) Experiment setup

BBias

g

E2

E1

BEC

Ω1 (bottom)

Ω2 (top)

yx

z

v

v

v

(c) Acceleration methods

(i)

(ii)ΔωL

ΔωL

Ω1 (bottom) (top) Ω2

Δ

(b) Raman Laser Energies

|+1>|0>

|-1>q/kr

FIG. 1. (a,b) Counter-propagating, linearly polarized laser beams couple the mF = −1, 0 states of the 87Rb BEC via a Ramantransition. A bias field Zeeman splits the mF states, and it is controlled by an external magnetic field coil aligned along thez-direction. Gravity acts in the −y-direction. (c) Two acceleration methods used to study Landau-Zener transitions at thisavoided crossing of the SO eigenlevels: (i) the acceleration induced by the force of the trapping potential drives transitions fromthe upper to lower dressed eigenlevel, (ii) the acceleration induced by the gravitational force drives transitions from the lowerto upper dressed eigenlevel. The dashed black curves indicate two ”bare” spin state energy levels, the solid lines indicate thetwo SO coupled eigenlevels [with the color indicating the bare state spin component, red (blue) for spin mF = −1 (0)]. Solidand dotted arrows depict adiabatic (intraband) motion and diabatic (interband) LZ transition, respectively.

tems. Some examples include ultracold atoms in acceler-ated optical lattices [16–18], Feshbach associated ultra-cold molecules [19], Rydberg atoms [20], as well as insolid-state qubits [21, 22] and spin-transistors [23]. Inthis paper, we measure the LZ transition probability ofa BEC with synthetic 1D SO coupling of equal Rashbaand Dresselhaus types [3]. The SO coupling is the resultof adiabatic “dressed” states formed by Raman couplingthe “bare” quadratic dispersion curves of two mF spinstates [24, 25]. The coupling of the spin states by theRaman field leads to a SO coupling of the form:

HSO =

(~2

2m (q + kr)2 − δ/2 ΩR/2

ΩR/2~2

2m (q − kr)2 + δ/2

)(3)

where ΩR is the Raman coupling strength, ~kr is thesingle-photon recoil momentum from the coupling lasers,m is atomic mass, δ is the Raman detuning, ~ is thereduced Planck’s constant, and ~q is the quasimomen-tum. Applying the Landau-Zener model to a SO cou-pled BEC, the adiabatic parameter is q, the velocity isv = dq/dt, the coupling strength is ΩR, and β is definedas the diabatic curve slope difference by making a linearapproximation of HSO near the diabatic crossing point,qc.

For our experiment, we produce nearly-pure 3D BECsof 2×104 87Rb atoms in an optical dipole trap [26], withtrapping frequencies tuned in the range of ωz,y/2π ≈ 180-450 Hz and ωx/2π ≈ 50-90 Hz. To create synthetic spin-orbit coupling, we employ counter-propagating Ramanbeams along the y-axis which couple the |mF 〉 states ofthe F = 1 ground state manifold of 87Rb (see Fig. 1),similar to that of Lin et al. [3, 27]. The two Ra-man beams are generated from the same laser source(ωL = 2π×383 240 GHz), have a frequency difference of∆ωL = 2π×3.5MHz, and have perpendicular linear po-

larizations when incident on the BEC [Fig. 1(a)]. Thedetuning provided by the quadratic Zeeman shift on the|mF = +1〉 state (εq ≈ 2π×3.4 kHz) allows for the sys-tem to be approximated by the two-state descriptionof Eq. (3). The natural energy scale for the SO cou-pled system is the recoil energy from the coupling laser

fields, ER =~2k2r2m = ~ × 3.75 kHz, where kr = 2π/λ

and λ = 782.26 nm. Neglecting atom-atom interactions,the dynamics of the BEC relevant for the experimentshere can be described by a 1D Schrodinger equation,where H = HSO + Htrap. Recalling the relation of po-sition and momentum operators y = i∂/∂q, it is elu-cidating to express the trapping term in the y-axis as

Htrap = −mω2y

2d2

dq2 , which shows how the trapping poten-tial acts as a “kinetic energy” in quasimomentum space[28, 29].

As schematically shown in Fig. 1(c), measurements ofthe Landau-Zener transition probability, PLZ , were per-formed by first preparing the BEC in either the upper orlower SO eigenlevel, with initial quasimomentum ~qi farfrom the SO diabatic crossing at ~qc. The BEC was thenaccelerated [30] through the diabatic crossing, either bythe optical trapping force or by gravity. Depending onthe accelerating force, the BEC acquired different eigen-state velocities, dq/dt, as it passed qc (detailed discussionof preparing different dq/dt is presented in Appendix A).After the crossing and when the BEC was sufficientlyfar from the diabatic crossing such that the diabatic andadiabatic eigenstates matched to better than 97%, theRaman beams and dipole trap were instantly turned offto map the adiabatic dressed eigenstates to the bare spinstates. A Stern-Gerlach field was then applied to sepa-rate the bare mF spin states in time-of-flight (TOF), andabsorption images measured the population of each spinstate to determine PLZ .

Fig. 2 (a,b) shows the measurement of PLZ for increas-

3

ing coupling strengths, ΩR, and different eigenstate ve-locities, dq/dt, with the theoretically calculated PLZ fromEq. (2) shown by the solid curves. In agreement with theLZ model, the transition probability increases for smallercoupling strengths or larger dq/dt. The LZ model and ex-perimental results are in good quantitative agreement tothe level of our experimental resolution of PLZ , which islimited by technical noise in the experimental imaging.Shown in Fig. 2 (c), we also measured PLZ over a range ofδ. The results further validate the expected LZ behavior,where PLZ does not depend on δ (since β is independentof δ for this SO coupled system).

To measure the effect of changing β, it is necessaryto measure PLZ at a different diabatic crossing. Shownin Fig. 3(a), the third-spin state in the system allowsprobing of the diabatic crossing of the |mF = ±1〉 stateswhere β = 8ER/kr, twice the value of the ground statecrossing [31]. The coupling between the |mF = −1〉and |mF = +1〉 states is a four-photon process, andthe strength of the coupling is numerically found to beΩ4p/ER ≈ 0.12(ΩR/ER)1.75 [32]. To compare the twocrossing in the LZ model, we present in Fig. 3(c) the PLZvalue over a range of diabatic state coupling strengths,Ω, which defines the energy gap at the avoided crossingand is ΩR for the lower diabatic crossing and Ω4p for theexcited state diabatic crossing. The measurements of theexcited state crossing is again in good agreement withthe theoretically calculated PLZ from the LZ model.

This tunable LZ transition in the SO coupled BECcould be used to create a unique spin-dependent “atom-tronic” device [33, 34], an analog of a transistor in whichΩ acts as the gate voltage, PLZ (the output in one of thespin components) as the current, and the “drift velocity”dq/dt induced by the force that acts as the source-drainvoltage (note the qualitative similarity of Fig. 2 (b) totransistor characteristic curves). An important charac-teristic in such devices is the “switching time” of thedevice to transition between two states. To that end,we also made time-dependent measurements of the LZtransition process. We performed such measurementsby instantaneously turning off the Raman coupling dur-ing the LZ transition process (at time t since the BECstarts from qi at t = 0) and thus mapping the BECdressed states on to their bare-spin component basis.These are then separated by a Stern-Gerlach pulse andimaged after TOF to determine PLZ,DIA(t), where thediabatic subscript indicates the measurements are madein the bare, not adiabatic, basis [35]. For the experi-ments here where the atoms start in the |mf = 0〉 state,PLZ,DIA(t) = N|mF=0〉/Ntotal.

Fig. 4 (a) shows the resulting measurements ofPLZ,DIA(t) as time t is varied for a fixed v at the crossing(dq/dt = 5.0kr/ms) and four different Raman coupling

1.7 kr/ms

2.3 kr/ms

5.0 kr/ms

8.2 kr/ms

9.6 kr/ms

0.0

0dq/dt (kr/ms)

ΩR

0.28 ΕR

ΩR

v= dq/dt

FIG. 2. (a-b) Measurement of the LZ transition probability,PLZ , over a range of Raman coupling strengths, ΩR, and withdifferent eigenstate velocities (∂q/∂t) at the diabatic cross-ing. The data in (b) is from the same experiments as (a), butplotted to show the effect of ∂q/∂t on PLZ . All solid lines arecalculated from the LZ model using Eq. (2) and the exper-imental values of ΩR, ∂q/∂t, and β = 4ER/kr with no freeparameters. The data with ∂q/∂t = 1.7 kR/ms correspondsto the case of acceleration due to gravity (Fig. 1c.ii); the other∂q/∂t data are measured by applying an optical dipole trap-ping force of different magnitudes (controlled by the opticaltrap laser power) to accelerate the BEC (Fig. 1c.i). All ex-periments were performed with ∼ 1× 104 atoms in the BEC.Each data point is the average of 3-5 measurements, and er-ror bars indicate an average 10% uncertainty in atom numberdue to technical noise. (c) Measurement of the LZ transi-tion probability over a range of Raman detuning, δ, fromresonance. No discernible change of PLZ was observed, inagreement with the theoretical model. Measurement was per-formed with ∂q/∂t = 1.7 kR/ms (supplied by gravity) withtwo values of ΩR: 0.28ER (closed squares) and 0.43ER (opencircles). The values calculated from Eq. (2) are shown asdashed lines.

strengths. The solid lines are results of the numericalsimulation of the time-dependent, 1D Schrodinger equa-tion (see Appendix B for description of numerical simu-lation). By fitting the experimental measurements to asigmoid function (similar to [17]), we extract the time ittakes to transition between the diabatic states (“switch-ing time”, tswitchdia ) in Fig. 4(b). We rescale the extracted

4

(c)

Ω (ER)

(a) (b)E

(q)/

ER

q/kr q/kr

Lower Crossing (β=4 ER/kr)

Upper Crossing (β=8 ER/kr)theoryexp

theoryexp

FIG. 3. Measurement of the LZ transition probability at theupper crossing of the diabatic dispersion relations for the mF

spin states. (a) The full eigenlevels of the three-state sys-tem where the dashed lines indicate the bare state energylevels and the solid color lines indicate SO coupled adiabaticeigenlevels with color representing the mF components (redfor |mF = −1〉, blue for |mF = 0〉, and green for |mF = +1〉).The dashed box, magnified in (b), indicates the upper cross-ing. (c) The lower crossing data (black squares, same datashown in Fig. 2(a)) are measurements of PLZ for the crossingof the |mF = −1〉 and |mF = 0〉 states where β = 4 ER/kr,and the upper crossing data (grey circles) are measurementsfor the crossing of the |mF = −1〉 and |mF = +1〉 states whereβ = 8 ER/kr. The eigenstate velocity in both cases wasdq/dt = 1.7 kR/ms supplied by gravity.

switching times to τswitchdia = tswitchdia (vβ/2~)1/2 and findagreement with Ref. [36]’s predicted value of τswitchdia =2.5 in the diabatic limit [(ΩR/2)/(~vβ/2)1/2 << 1]. Gen-eral agreement is found with the numerical simulation inboth the final PLZ,DIA value and the timescale of thetransition. The oscillatory behavior predicted by the nu-merical simulations was not observed in our data, wherethe experimental resolution is limited by technical noise.

In summary, we have measured the interband Landau-Zener transition probability in a spin-orbit coupled BEC.The coupling strength, diabatic slopes, and eigenstatevelocity at the avoided crossing were each varied inde-pendently and shown to agree with the LZ prediction ofEq. (2). This work can pave the way for utilizing non-adiabatic transitions in synthetic gauge fields to realizenovel physical phenomena. We have also demonstrated

2kr

|-1> |0>

(a)

0.45 ER0.59 ER0.81 ER0.93 ER

ΩR

(ΩR/2)/(ħvβ/2)1/2τ

=

(v

β/2

ħ)1/

2sw

itch

dia (b)

FIG. 4. (a) Measurement of the time-dependent LZ transi-tion in the diabatic basis. The bare-state spin componentsof the BEC for different points in time as it passes throughthe diabatic crossing are measured by instantly turning offthe Raman beams to map the dressed states onto the barediabatic states. The inset shows experimental absorption im-ages (at three representative points in time indicated by thearrows at t = 0, 180, 380 µs respectively for ΩR = 0.59ER)of the time dependent transition (false color added to distin-guish the spin components). The change in BEC aspect ratiois due to a quadrupole mode excited by the state preparationprocess. Solid curves in (a) are from direct solution of thetime dependent Schrodinger equation (TDSE). (b) Measure-ments of the diabatic switching time, tswitchdia , for different ΩR,with dq/dt = 5.0 kR/ms, β = 4 ER/kr, and ωy/2π = 338 Hz.Here tswitchdia is found by fitting each set of data to a sigmoidfunction [17]. The switching time is scaled by (vβ/2~)1/2 tocompare with the theory of Ref. [36]. The black dots are theresults from direct solutions of the TDSE.

the versatility of using both gravitational and opticaltrapping forces to prepare and drive quantum states insynthetic gauge fields. Finally, the dynamics of the tran-sition process was directly measured and was in excellentagreement with our numerical calculations. The inter-band transitions studied in our work are entirely due tothe breakdown of adiabaticity in the system, in contrastto the transitions due to 2-body collisions studied in pre-vious experiments [29, 37]. In future work, our approachcould be used to probe more complex synthetic gaugefields and to observe Stueckelberg interference [18]. An-other natural continuation of this work would be to study

5

the effect of interactions on this LZ transition process[38].

ACKNOWLEDGMENTS

The research was supported in part by DURIP-AROGrant No. W911NF-08-1-0265 and the Miller Family En-dowment. A.J.O. acknowledges support of the U.S. Na-tional Science Foundation Graduate Research FellowshipProgram. S-J.W. and C.G. acknowledge support fromthe U.S. National Science Foundation. We thank HuiZhai for helpful comments.

Appendix A: Variable eigenstate velocity

To achieve the variable eigenstate velocities at the di-abatic crossing point, the BEC was first prepared in anoptical dipole trap in the |mF = 0〉 state at q = 1kr. Thetrapping potential was instantly removed and the BECwould fall under gravity in the |mF = 0〉 diabatic statefor 1.2 ms, at which point it would reach q ≈ −1.0kr. TheRaman coupling was then instantly turned on along withthe trapping potential, which caused the BEC to be “ac-celerated back” (decelerated in real space) through thediabatic crossing with an acceleration dependent on the

trap frequency ωy. After passing the crossing, the popu-lation of the BEC in each spin component was measuredto determine PLZ . The eigenstate velocities dq/dt weredetermined from time resolved measurements of the BECas it crossed qc, see Table I. For the numerical simulationsof the time-dependent LZ transition shown in Fig. 4, thewidth (σw) of the initial condensate momentum distribu-tion was set to match that of the experimentally recordedvalues, also shown in Table I.

ωy/2π (Hz) dq/dt (kr/ms) σw (kr)264 2.3 0.31338 5.0 0.40397 8.2 0.44449 9.6 0.47

TABLE I. The corresponding optical dipole trap frequency ωyfor each of the measured dq/dt at q = qc, as well as the initialmomentum width of the BEC, where the momentum distri-bution was fitted by p(q) = 1√

2πσwexp

[−(q − qi)2/(2σ2

w)].

Appendix B: Time-dependent Schrodinger equationsimulation

To solve the one-dimensional time-dependentSchrodinger equation for the spin-orbit coupled BEC ina harmonic trap, we apply the Chebychev propagationmethod [39].

i~∂tΨ(q, t) = HΨ(q, t) =

[− 1

2mω2 ∂

2

∂q2+

(~2

2m (q + kr)2 − δ/2 ΩR/2

ΩR/2~2

2m (q − kr)2 + δ/2

)]Ψ(q, t), (A4)

where Ψ(q, t) = Ψ↑(q, t),Ψ↓(q, t)T is a two-componentcolumn vector written in the bare state basis (|mF =−1〉, |mF = 0〉). Expanding the evolution operator in

terms of the Chebychev polynomials with a renormaliza-tion of the Hamiltonian Hnorm whose eigenvalue rangesfrom [λmin, λmax], we arrive at

U(dt) = e−iHdt/~ =

∞∑n=0

anφn(−iHnorm) =

∞∑n=0

anφn

(−iH + I(λmax + λmin)/2

(λmax − λmin)/2

), (A5)

where φn(x) is the complex Chebychev polynomial of or-der n. The expansion coefficients are

6

an = ei(λmax+λmin)dt/2~(2− δn,0)Jn

((λmax − λmin)dt

2~

). (A6)

òò

ò

ò

ò

ò

ò

ò

ò

ò

ò

ò

ì

ì

ì

ì

ì

ì

ì

ì

ì

ì

ì

ì

àà

à

à

à

à

à

à

à

à

à

à

ô

ô

ô

ô

ô

ô

ô

ô

ôô ô ô

ð

ð

ð

ð

ð

ð

ð

ðð ð ð ð

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

1.0

WRHERL

PL

Z

1.7 kRms2.3 kRms5.0 kRms8.2 kRms9.6 kRmsdqdt

FIG. 5. Comparison of the exact numerical solution (shapedsymbols) and the Landau-Zener formula (solid curves) for thenon-adiabatic transition probability as a function of the Ra-man coupling ΩR . Different colors correspond to differenteigenstate velocities at the crossing point.

Jn(x) is the Bessel function of order n. The wave func-tion at any time is obtained by applying the evolutionoperator to an given initial wave function: Ψ(q, t+ dt) =U(dt)Ψ(q, t). To perform the Hamiltonian operation, werepresent our wave functions and operators in the Fourierdiscrete variable representation (Fourier-DVR) [40]. Thegrid points are qi = qmin + i(qmax − qmin)/(N + 1) fori = 1, 2, ..., N . We take qmin, qmax = −6kr, 6kr, andN = 500. To converge the series expansion, the de-gree of the expansion in Eq.(A5) must be larger thanR = (λmax − λmin)dt/2~. In our simulation, we choosethe degree to be the least integer greater than or equal to1.5R. Since the parameter R depends on dt, we increasethe efficiency of our codes by appropriately choosing asuitable time step (dt = 0.01~/ER) for each time propa-gation.

In our simulation, a Gaussian wave packet in one ofthe adiabatic states serves as the initial wave function.Note that the adiabatic states (|+〉, |−〉) are related tothe bare states by a unitary transformation. With themethod mentioned in the first paragraph, we can evolveour system to any later time to study non-adiabatic inter-band transitions. Defining the probability for an atomto stay in |±〉 as P±(t) =

∑Ni=1 |Ψ±(qi, t)|2, we extract

the asymptotic values of the probability for the atomto be in the other adiabatic state right after the wavepacket passes the avoided crossing. This probability isthe familiar Landau-Zener transition probability if the

energy band is linear in the bare state basis. Near theavoided crossing region, the energy bands in the spin-orbit coupled system are well described by two linearlines, and Fig. 5 shows that the simple LZ formula gives avery good approximation to the non-adiabatic inter-bandtransition probability in the spin-orbit coupled BEC.

∗ olsonaj@purdue.edu† yongchen@purdue.edu

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[30] The acceleration in the real space can be either positiveor negative.

[31] The excited state crossing studied here is the same di-abatic crossing where a “Zitterbewegung” was recentlyobserved [41].

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