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Applications of the ML-MCTDHB to theNonequilibrium Quantum Dynamics of
Ultracold SystemsPeter Schmelcher
Zentrum fur Optische QuantentechnologienUniversitat Hamburg
624. WE-HERAEUS-SEMINAR: SIMULATING QUANTUM PROCESSES AND DEVICES
BAD HONNEF, SEPTEMBER 19-22, 2016
The Centre for Optical Quantum Technologies
Experiment and Theory -Workshop and Guest Pro-gram
Cold Atoms – p.1/44
in collaboration with
L. Cao, S. Krönke, O. Vendrell (ML-MCTDHB)L. Cao, S. Krönke, R. Schmitz, J. Knörzer and S.Mistakidis (Applications)
funded by the SFB 925 ’Light induced dynamics andcontrol of correlated quantum systems’ of the GermanScience Foundation (DFG).
Cold Atoms – p.2/44
Contents
1. Introduction and motivation2. Methodology: The ML-MCTDHB approach3. Tunneling mechanisms in double and triple wells4. Multi-mode quench dynamics in optical lattices5. Many-body processes in black and grey matter-wave
solitons6. Correlated dynamics of a single atom coupling to an
ensemble7. Concluding remarks
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1. Introduction and Motivation
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Introduction and Motivation
An exquisite control over the external and internaldegrees of freedom of atoms developed over decadeslead to the realization of Bose-Einstein Condensation indilute alkali gases at nK temperatures.Key tools available:
Laser and evaporative coolingMagnetic, electric and optical dipole trapsOptical lattices and atom chipsFeshbach resonances (mag-opt-conf) for tuning ofinteraction
Cold Atoms – p.5/44
Introduction and MotivationEnormous degree of control concerning preparation,processing and detection of ultracold atoms !Weak to strongly correlated many-body systems:
BEC nonlinear mean-field physics (solitons, vortices,collective modes,...)Strongly correlated many-body physics (quantumphases, Kondo- and impurity physics, disorder,Hubbard model physics, high Tc superconductors,...)
Few-body regime:Novel mechanisms of transport and tunnelingAtomtronics (Switches, diodes, transistors, ....)Quantum information processing
Cold Atoms – p.6/44
Introduction: Some facts
Hamiltonian: H =∑
i
(p2i
2mi+ V (ri)
)+ 1
2
∑i,j,i !=j W (ri − rj)
V is the trap potential: harmonic, optical lattice, etc.
W describes interactions: contact gδ(ri − rj), dipolar, etc.
Dynamics is governed by TDSE: i!∂tΨ(r1, ..., rN , t) = HΨ(r1, ..., rN , t)
Ideal Bose-Einstein condensate: no interaction g = 0 ⇒ Macroscopic matter wave.
Φ(r1, ..., rN ) =N∏
i=1
φ(ri)
Hartree product: bosonic exchange symmetry.
Interaction g #= 0: Mean-field description leads to Gross-Pitaevskii equation with cubicnonlinearity, exact for N → ∞, g → 0.
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Introduction: Some facts
Finite, and in particular ’stronger’ interactions:Correlations are ubiquitous
A multiconfigurational ansatz is necessary
Ψ(r1, ..., rN , t) =∑
i
ciΦi(r1, ..., rN , t)
⇒ Ideal laboratory for exploring the dynamics ofcorrelations (beyond mean-field):
Preparation of correlated initial statesSpreading of localized/delocalized correlations ?Time-dependent ’management’ and control of correlations ?Is there universality in correlation dynamics ?
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Introduction and Motivation
Calls for a versatile tool to explore the (nonequilibrium)quantum dynamics of ultracold bosons: Wish list
Take account of all correlations (numerically exact)Applies to different dimensionalityTime-dependent Hamiltonian: DrivingWeak to strong interactions (short and long-range)Few- to many-body systemsMixed systems: different species, mixeddimensionalityEfficient and fast
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Introduction and Motivation
Multi-Layer Multi-Configuration Time-Dependent Hartreefor Bosons (ML-MCTDHB) is a significant step in thisdirection !
In the following: A brief account of the methodology andthen some selected diverse applications to ultracoldbosonic systems.
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2. Methodology: The ML-MCTDHBApproach
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The ML-MCTDHBMethod
aim: numerically exact solution of the time-dependent Schrödinger equationfor a quite general class of interacting many-body systems
history: [H-D Meyer.WIREs Comp. Mol. Sci. 2, 351 (2012).]MCTDH (1990): few distinguishable DOFs, quantum molecular dynamicsML-MCTDH (2003): more distinguishable DOFs, distinct subsystemsMCTDHF (2003): indistinguishable fermionsMCTDHB (2007): indistinguishable bosons
idea:use a time-dependent, optimally moving basis in the many-body Hilbert space
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Hierarchy within ML-MCTDHBWe make an ansatz for the state of the total system |Ψt〉 with time-dependencies ondifferent layers:
top layer |Ψt〉 =∑M1
i1=1 ...∑MS
iS=1 Ai1,...,iS (t)S⊗
σ=1|ψ(σ)
iσ(t)〉
species layer |ψ(σ)k (t)〉 =
∑"n|Nσ
Cσk;"n(t) |%n〉(t)
particle layer |φ(σ)k (t)〉 =
∑nσi=1 B
σk;i(t)|ui〉
.
xC yC zCxA xB
NA NB NC
MA MB
MC
mA mB mC
MA MB MC,x MC,y MC,z
Mc Lachlan variational principle: Propagate the ansatz |Ψt〉 ≡ |Ψ({λit})〉, λit ∈ Caccording to i∂t|Ψt〉 = |Θt〉 with |Θt〉 ∈ span{ ∂
∂λkt|Ψ({λit})〉} minimizing the
error functional |||Θt〉 − H|Ψt〉||2
[AD McLachlan. Mol. Phys. 8, 39 (1963).]In this sense, we obtain a variationally optimally moving basis!Dynamical truncation of Hilbert space on all layersSingle species, single orbital on particle layer → Gross-Pitaevskii equation !(Nonlinear excitations: Solitons, vortices,...)
Cold Atoms – p.13/44
The ML-MCTDHB equations of motion
top layer EOM:
i∂tAi1,...,iS =M1∑
j1=1
...
MS∑
jS=1
〈ψ(1)i1
...ψ(S)iS
| H |ψ(1)j1
...ψ(S)jS
〉Aj1,...,jS
with |ψ(1)j1
...ψ(S)jS
〉 ≡ |ψ(1)j1
〉 ⊗ ...⊗ |ψ(S)jS
〉
⇒ system of coupled linear ODEs with time-dependent coefficients due to thetime-dependence in |ψ(σ)
j (t)〉 and |φ(σ)j (t)〉
⇒ reminiscent of the Schrödinger equation in matrix representation
species layer EOM:
i∂tCσi;"n = 〈%n|(1− P spec
σ )Mσ∑
j,k=1
∑
"m|Nσ
[(ρspecσ )−1]ij〈H〉σ,specjk |%m〉 Cσk;"m
⇒ system of coupled non-linear ODEs with time-dependent coefficients due to thetime-dependence of the |φ(σ)
j (t)〉 and of the top layer coefficients
Cold Atoms – p.14/44
The ML-MCTDHB equations of motion
particle layer EOM:
i∂t|φ(σ)i 〉 = (1− P part
σ )mσ∑
j,k=1
[(ρpartσ )−1]ij〈H〉σ,partjk |φ(σ)k 〉
⇒ system of coupled non-linear partial integro-differential equations(ODEs, if projected on |u(σ)
k 〉, respectively) with time-dependentcoefficients due to time-dependence of the Cσ
i;"n and Ai1,...,iS
Lowest layer representations:
Discrete Variable Representation (DVR):implemented DVRs: harmonic, sine (hardwall b.c.), exponential (periodic b.c.),
radial harmonic, LaguerreFast Fourier Transform
Stationary states via improved relaxation involving imaginary time propagation !
S Kronke, L Cao, O Vendrell, P S, New J. Phys. 15, 063018 (2013).L Cao, S Kronke, O Vendrell, P S, J. Chem. Phys. 139, 134103 (2013).
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3. Tunneling mechanisms in thedouble and triple well
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Few-boson systems - Perspectives
Extensive experimental control of few-boson systemspossible: Loading, processing and detection[I. Bloch et al, Nature 448, 1029 (2007)]
Bottom-up understanding of tunneling processes andmechanismsAtomtronics perspective providing us withcontrollable atom transport on individual atom level:
Diodes, transistors, capacitors, sources anddrains
Double well, triple well, waveguides, etc.
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Few-boson systems: Double Well
No interactions: Rabi oscillations.Weak interactions: Delayed tunneling.Intermediate interactions:
Tunneling comes almost to a hold in spite ofrepulsive interactions.Pair tunneling takes over !
Very strong interactions: Fragmented pair tunneling.
! N = 2 atomsK. Winkler et al., Nature 441, 853 (2006); S. Fölling et al., Nature 448, 1029 (2007)S. ZOLLNER, H.D. MEYER AND P.S., PRL 100, 040401 (2008); PRA 78, 013621 (2008)
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200 250 300 350 400
pair
prob
abilit
y p 2
time t
g=0g=0.2g=25
Cold Atoms – p.18/44
Interband Tunneling: Motivation
Here: Bottom-up approach of understanding thetunneling mechanisms !
Triple well is minimal system analog of asource-gate-drain junction for atomtronicsTriple well shows novel tunneling scenarios ⇔ Impacton transportStrong correlation effects beyond single bandapproximation !Beyond the well-known suppression of tunneling:Multiple windows of enhanced tunneling i.e. revivalsof tunneling: Interband tunneling involving higherbands !
Cold Atoms – p.19/44
Interband Tunneling: Analysis Tool
Methodology: Multi-Layer Multi-ConfigurationTime-Dependent Hartree for Bosons
Novel number-state representation includinginteraction effects for analysis
Three bosons: Single, pair and triple modes.
Cold Atoms – p.20/44
Interband Tunneling: Single boson tunnelingThree bosons initially in the left well: Ψ ≈ |3, 0, 0〉0(a) g = 0.1 and (b) g = 3.26
Single boson tunneling to middle andright well via |3, 0, 0〉0 ⇔ |2, 1, 0〉1 ⇔|2, 0, 1〉1 i.e. via first-excited states !
Cold Atoms – p.21/44
Interband Tunneling: Single boson tunnelingThree bosons initially in the middle well: Ψ ≈ |0, 3, 0〉0(a) g = 9.85
Single boson tunneling to left and rightwell via |0, 3, 0〉0 ⇔ |1, 2, 0〉3 ⇔|0, 2, 1〉3 i.e. via second-excited states!
Cold Atoms – p.22/44
Interband Tunneling: Two boson tunnelingThree bosons initially in the middle well: Ψ ≈ |0, 3, 0〉0(a) g = 5.8
Two boson tunneling to the left and rightwell via |0, 3, 0〉0 ⇔ |1, 1, 1〉6 i.e. twofirst-excited states !
Cao et al, NJP 13, 033032 (2011)
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4. Multi-mode quench dynamics inoptical lattices
Cold Atoms – p.24/44
Main features
Focus: Correlated non-equilibrium dynamics of inone-dimensional finite lattices following a suddeninteraction quench from weak (SF) to strong interactions!
Phenomenology: Emergence of density-wave tunneling,breathing and cradle-like processes.Mechanisms: Interplay of intrawell and interwell dynamicsinvolving higher excited bands.Resonance phenomena: Coupling of density-wave andcradle modes leads to a corresponding beatingphenomenon !⇒ Effective Hamiltonian description and tunability.
Incommensurate filling factor ν > 1(ν < 1)
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Post quench dynamics....
Fluctuations δρ(x, t) of the one-body density for weaker(a) and stronger (b) quench: Spatiotemporal oscillations.
20 40 60 80 100 120 140 160
−0.04
−0.02
0
0.02
0.04
cradle
breathingover-barrier
20 403010
a
b
c
Densitywaves andtheir tunnel-ing.
s.f.
m.f.
Cold Atoms – p.26/44
Mode analysis
Density tunneling mode: Global ’envelope’ breathingIdentification of relevant tunneling branches(number state analysis)Fidelity analysis shows 3 relevant frequencies:pair and triple mode processesTransport of correlations and dynamical bunchingantibunching transitions
On-site breathing and craddle mode: Similar analysispossible involving now higher excitations
Cold Atoms – p.27/44
Craddle and tunneling mode interaction
0.5 1 1.5 2 2.5 3 3.5 4 4.5
10
20
30
40
50
60
70
80
g
/
0 20 40 60 80 1000.04
0.02
0
0.02
0.04
0.06
t1(t)
2(t)
1
2
3
4
5
6
7
8
9
x 10 3
(a) (b)
Fourier spectrum of the intrawell-asymmetry ∆ρL(ω):
Avoided crossing of tunneling and craddle mode !
⇒ Beating of the craddle mode - resonant enhancement.S.I. Mistakidis, L. Cao and P. S., JPB 47, 225303 (2014), PRA 91, 033611 (2015)
Cold Atoms – p.28/44
5. Many-body processes in black andgrey matter-wave solitons
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Setup and preparation
N weakly interacting bosons in a one-dimensionalboxInitial many-body state: Little depletion, density andphase as close as possible to dark soliton in thedominant natural orbitalPreparation: Robust phase and density engineeringscheme.CARR ET AL, PRL 103, 140403 (2009); PRA 80, 053612(2009); PRA 63, 051601 (2001); RUOSTEKOSKI ET AL, PRL104, 194192 (2010)
Cold Atoms – p.30/44
Density dynamics
-10
-5
0
5
10
x(u
nitso
fξ)
(a)
0 1 2 3 4 5 6t (units of τ )
-10
-5
0
5
10
x(u
nitso
fξ)
(b)
0 1 2 3 4 5 6-10
-5
0
5
10mean-field
0.00
0.01
0.02
0.03
0 1 2 3 4 5 6-10
-5
0
5
10mean-field
Reduced one-body densityρ1(x, t)
N = 100, γ = 0.04
Black (top) and grey(bottom) solitonM = 4 optimized orbitalsInset: Mean-field theory(GPE)Slower filling process ofdensity dip for moving soli-ton
Cold Atoms – p.31/44
Evolution of contrast and depletion
0 1 2 3 4 5 6t (units of τ )
0.2
0.4
0.6
0.8
1.0
c(t)/c(0)
β = 0.0
β = 0.1
β = 0.2
β = 0.3
β = 0.4
β = 0.5
β = 0.6
β = 0.70.0 0.35 0.7
β
2
8
14
τc
(un
itso
fτ
)
0 1 2 3 4 5 6t (units of τ )
0.02
0.05
0.10
0.15
0.20
0.25
d(t)
(a)β = 0.0
β = 0.1
β = 0.2
β = 0.3
β = 0.4
β = 0.5
β = 0.6
β = 0.7
0 1 2 3 4 5 6t (units of τ )
0.0
0.2
0.4
0.6
0.8
1.0
λ i(t)
(b)
0.0 1.0 2.0 3.0 4.0 5.0 6.0t (units of τ )
0.0
0.005
0.01
0.015
0.02
λ i(t)
Relative contrast c(t)/c(0) of darksolitons for various β = u
s
(c(t) = max ρ1(x,0)−ρ1(xst ,t)
max ρ1(x,0)+ρ1(xst ,t)
)
Dynamics of quantum depletiond(t) = 1−maxi λi(t) ∈ [0, 1] andevolution of the natural populationsλi(t) for β = 0.0 (solid black lines)and β = 0.5 (dashed dotted red lines).ρ1(t) =
∑Mi=1 λi(t) |ϕi(t)〉〈ϕi(t)|
Cold Atoms – p.32/44
Natural orbital dynamics
-10
-5
0
5
10x
(uni
tsofξ
)
(a) (c)
0 1 2 3 4 5 6t (units of τ )
-10
-5
0
5
10
x(u
nits
ofξ
)
(b)
0 1 2 3 4 5 6t (units of τ )
(d)0 1 2 3 4 5 6
-10
-5
0
5
10
0.000
0.012
0.024
0.036
0 1 2 3 4 5 6-10
-5
0
5
10
0 1 2 3 4 5 6-10
-5
0
5
10
0.000
0.030
0.060
0.090
0 1 2 3 4 5 6-10
-5
0
5
10
0 π 2πphase
Density and phase (inset) evolution of the dominant and second domi-nant natural orbital. (a,b) black soliton (c,d) grey soliton β = 0.5.
Cold Atoms – p.33/44
Localized two-body correlationsTwo-body correlation function g2(x1, x2; t) for a black soliton(first row) and a grey soliton β = 0.5 (second) at timest = 0.0 (first column), t = 2.5τ (second) and t = 5τ (third).S. Krönke and P.S., PRA 91, 053614 (2015)
-10
-5
0
5
10
x2
(uni
tsof
ξ)
(a) 1.014
1.000
0.986
0.972
0.966
-10 -5 0 5 10x1 (units of ξ)
-10
-5
0
5
10
x2
(uni
tsof
ξ)
(d) 1.014
1.000
0.986
0.972
0.964
(b) 2.50
2.00
1.50
1.00
0.59
-10 -5 0 5 10x1 (units of ξ)
(e)1.08
1.04
1.00
0.96
0.92
(c)1.36
1.20
1.00
0.80
0.65
-10 -5 0 5 10x1 (units of ξ)
(f)1.20
1.10
1.00
0.90
0.84
Cold Atoms – p.34/44
6. Correlated dynamics of a singleatom coupling to an ensemble
Cold Atoms – p.35/44
Setup and preparation
Bipartite system: impurity atom plus ensemble of e.g. bosons of differentmF = ±1 trapped in optical dipole trapApplication of external magnetic field gradient separates speciesInitialization in a displaced ground ie. coherent state via RF pulse to mF = 0 forimpurity atom.⇒ Single atom collisionally coupled to an atomic reservoir: Energy and correlationtransfer - entanglement evolution.
J. KNORZER, S. KRONKE AND P.S., NJP 17, 053001 (2015)Cold Atoms – p.36/44
Energy transfer
Spatiotemporally localized inter-species coupling:Focus on long-time behaviour over many cycles.Energy transfer cycles with varying particle numberof the ensemble
Cold Atoms – p.37/44
One-body densities
Time-evolution of densities for the two species (ensemble-top, impurity-bottom) forfirst eight impurity oscillations.Impurity atom initiates oscillatory density modulations in ensemble atoms.Backaction on impurity atom.
Cold Atoms – p.38/44
Coherence analysis
Time-evolution of normalized excess energy ∆Bt with
Husimi distribution QBt (z, z
∗) = 1π 〈z|ρBt |z〉 , z ∈ C of
reduced density ρBt at certain time instants.Cold Atoms – p.39/44
Coherence measure
Distance (operator norm) to closest coherent state,as a function of time for different atom numbers inthe ensemble.
Cold Atoms – p.40/44
Correlation analysis
(a) Short-time evolution of the von Neumann entanglement entropy SvN(t) andinter-species interaction energy EAB
int (t) = 〈HAB〉t.(b) Long-time evolution of SvN(t). for NA = 2 (blue solid line), NA = 4 (red,dashed) and NA = 10 (black, dotted).
J. KNORZER, S. KRONKE AND P.S., NJP 17, 053001 (2015)
Cold Atoms – p.41/44
7. Concluding remarks
Cold Atoms – p.42/44
Conclusions
ML-MCTDHB is a versatile efficient tool for thenonequilibrium dynamics of ultracold bosons.Few- to many-body systems can be covered: Shownhere for the emergence of collective behaviour.Tunneling mechanismsMany-mode correlation dynamics: From quench todriving.Beyond mean-field effects in nonlinear excitations.Open systems dynamics, impurity and polarondynamics, etc.Mixtures !
Cold Atoms – p.43/44
Thank you for your attention !
Cold Atoms – p.44/44
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