the of systems - uni ulm

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).


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


1. Introduction and Motivation


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


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


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.


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 ?



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



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.


2. Methodology: The ML-MCTDHBApproach


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


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,...)


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


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).


3. Tunneling mechanisms in thedouble and triple well


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.


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


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 !


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.


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 !


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!


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)


4. Multi-mode quench dynamics inoptical lattices


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)


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.


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


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)


5. Many-body processes in black andgrey matter-wave solitons


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)


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


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)|


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.


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


6. Correlated dynamics of a singleatom coupling to an ensemble


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


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.


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.


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)


7. Concluding remarks


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 !


Thank you for your attention !


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