dynamical quasicondensation of hard-core bosons at finite

Lev Vidmar

Penn state University

Dynamical quasicondensation of hard-core bosons at finite momenta

arXiv:1505:05150

Lev Vidmar

Collaboration (U Munich and MPQ Garching):

S. Langer S.S. Hodgman, M. Schreiber, S. Braun

J.P. Ronzheimer

Dynamical quasicondensation of hard-core bosons at finite momenta

U. SchneiderI. BlochF. Heidrich-Meisner

Design far from equilibrium systems to

uncover conserved quantities that govern the dynamics (e.g., that give rise to ballistic current)

induce ordered states that do not exist in the equilibrium phase diagram

Goal:

Bosons on a lattice

Emergence of (quasi) long-range phase coherence when cooling an ideal Bose gas into a Bose-Einstein q = 0 (quasi) condensate

Spontaneous emergence of quasi-long-range order far from equilibrium: quasicondensation at a finite momentum

Here:

Standard picture:

Ɛ(q)

n(q)

vg(q)Ɛ,vg

q(d-1)-π/2 π/2-π π0

Ɛ

ρ(ε)

2J

-2J

(a) (b)

(c)

Ultracold atoms on optical lattices may realize such novel states, provided that (among others):

Bosons on a lattice

Ɛ(q)

n(q)

vg(q)Ɛ,vg

q(d-1)-π/2 π/2-π π0

Ɛ

ρ(ε)

2J

-2J

(a) (b)

(c)

The initial state can be prepared with high fidelity

The condensation at finite momenta should be resolved from time-of-flight measurement

?

Finite momentum condensationCondensation at q = 0Greiner et al (2002)

(a)

(b)

Load 39K atoms get BEC

Adiabatically ramp up deep optical lattice

Experiments with ultracold bosons

Dephasing: One boson per site

Ronzheimer et al, PRL (2013)

Initial state preparation

| 0i =Y

i2L0

a†i |;i

Bose-Hubbard modelTime evolution:

H = �JX

hi,ji

a†iaj +U

2

X

i

ni(ni � 1)

U/J ⇡ 20 … very close to the hard-core regime

H = �JX

hi,ji

a†iaj +U

2

X

i

ni(ni � 1)

Integrability and Bose-Hubbard model

Bose-Hubbard model, U/J = 1 Hard-core bosons

a†iai + aia†i = 1

Hard-core bosons map to noninteracting spinless fermions

Cazalilla et al, Rev. Mod. Phys. (2011)

ni = nfiLocal density:

Integrals of motion: nfk

Mapping to spin-1/2: XX chain

H = �JX

hi,ji

a†iaj = �JX

hi,ji

f†i fj =

X

k

✏knfk

Initial state and the quench protocol

t = 0: Inhomogeneous initial state

View 1: Bosons in a trap

……

View 2: Domain wall

| 0i =Y

i2L0

a†i |;i

Quantum quench: removal of the confining potential

View 1: Sudden expansion of hard-core bosons on an empty lattice

View 2: Domain wall meltingRigol and Muramatsu (PRL, 2004)

Antal et al (PRE, 1999), Gobert et al (PRE, 2005), Lancaster and Mitra (PRE, 2010), …

Recent experiments

Experiment

DMRGBallistic dynamics of a strongly interacting 1D system due to integrability

Hard-core bosons in 1D

Fermi-Hubbard model in 2D

Initial state: band insulatorU/J = 0 U/J = 12

Schneider et al, Nature Physics (2012)

Ronzheimer et al, PRL (2013)

Xia et al, Nature Physics (2015)Quantum distillation: upon removal of the confining trap, region of doubly occupied sites in the center get purified

Initial state:

Density distribution

Bose-Hubbard model in 1D and 2D

| 0i =Y

i2L0

a†i |;i

Bose-Hubbard model in 1D

Theoretical predictions

Dynamical quasicondensation at finite momenta:

Rigol and Muramatsu (PRL, 2004)

(Also: Daley et al, PRA, 2005)

0 50 100-50-100j

t E (τ

·N)

0.080.120.160.200.24

-π/2 π/2-π π0

t E (τ

·N)

0

1/4

1/2

3/4

1

0

1/4

1/2

3/4

1

0

1

0.8

0.6

0.4

0.2

-1.5 -1 -0.5 0 0.5 1 1.5

10

0.1

|⟨aj a

j+r⟩|

✝

rq (d-1)1

0 1

(a)

(c)

(b)

(d)

nj, n(q) (a.u.)n(

x) (a

.u.)

~

x~

10π

0

(r)

r10 20

tE (τ·N)

^^

0 50 100-50-100j

t E (τ

·N)

0.080.120.160.200.24

-π/2 π/2-π π0

t E (τ

·N)

0

1/4

1/2

3/4

1

0

1/4

1/2

3/4

1

0

1

0.8

0.6

0.4

0.2

-1.5 -1 -0.5 0 0.5 1 1.5

10

0.1

|⟨aj a

j+r⟩|

✝

rq (d-1)1

0 1

(a)

(c)

(b)

(d)

nj, n(q) (a.u.)

n(x)

(a.u

.)~

x~

10π

0

(r)

r10 20

tE (τ·N)

^^

Two time regimes in dynamics

Times tE < (N/4)(~/J)

Bosons: build-up of the quasicondensatesDynamical emergence of long-range correlations

Times Dynamical fermionization

Rigol and Muramatsu (PRL, 2005)

LV et al (PRB, 2013)

tE � N(~/J)

nHCBk (t ! 1) ! nf

k

Asymptotic MDF of HCBs given by integrals of motion

Minguzzi and Gangardt, (PRL, 2005)

Hard-core bosons

Bose-Hubbard model

Tonks-Girardeau gas

Long-lived!

Density distribution

0 50 100-50-100j

t E (τ

·N)

0.080.120.160.200.24

-π/2 π/2-π π0

t E (τ

·N)

0

1/4

1/2

3/4

1

0

1/4

1/2

3/4

1

0

1

0.8

0.6

0.4

0.2

-1.5 -1 -0.5 0 0.5 1 1.5

10

0.1

|⟨aj a

j+r⟩|

✝

rq (d-1)1

0 1

(a)

(c)

(b)

(d)

nj, n(q) (a.u.)

n(x)

(a.u

.)~

x~

10π

0

(r)

r10 20

tE (τ·N)

^^

0 50 100-50-100j

t E (τ

·N)

0.080.120.160.200.24

-π/2 π/2-π π0

t E (τ

·N)

0

1/4

1/2

3/4

1

0

1/4

1/2

3/4

1

0

1

0.8

0.6

0.4

0.2

-1.5 -1 -0.5 0 0.5 1 1.5

10

0.1

|⟨aj a

j+r⟩|

✝

rq (d-1)1

0 1

(a)

(c)

(b)

(d)

nj, n(q) (a.u.)

n(x)

(a.u

.)~

x~

10π

0

(r)

r10 20

tE (τ·N)

^^

Dynamical emergence of long-range correlations

……

0 50 100-50-100j

t E (τ

·N)

0.080.120.160.200.24

-π/2 π/2-π π0

t E (τ

·N)

0

1/4

1/2

3/4

1

0

1/4

1/2

3/4

1

0

1

0.8

0.6

0.4

0.2

-1.5 -1 -0.5 0 0.5 1 1.5

10

0.1

|⟨aj a

j+r⟩|

✝

rq (d-1)1

0 1

(a)

(c)

(b)

(d)

nj, n(q) (a.u.)

n(x)

(a.u

.)~

x~

10π

0

(r)

r10 20

tE (τ·N)

^^

x = (j � j0)/(2tE)

Each half of the chain: Current carrying state

Emergence of power-law correlations

A(r) ⇠ r�1/2

�(r) = ±⇡

2r

ha†jaj+ri = A(r)ei�(r)

0 50 100-50-100j

t E (τ

·N)

0.080.120.160.200.24

-π/2 π/2-π π0

t E (τ

·N)

0

1/4

1/2

3/4

1

0

1/4

1/2

3/4

1

0

1

0.8

0.6

0.4

0.2

-1.5 -1 -0.5 0 0.5 1 1.5

10

0.1

|⟨aj a

j+r⟩|

✝

rq (d-1)1

0 1

(a)

(c)

(b)

(d)

nj, n(q) (a.u.)

n(x)

(a.u

.)~

x~

10π

0

(r)

r10 20

tE (τ·N)

^^

… ground-state exponent

Lancaster and Mitra (PRE, 2010)

Each half of the chain: Current carrying state


Emergence of power-law correlations

A(r) ⇠ r�1/2

�(r) = ±⇡

2r

ha†jaj+ri = A(r)ei�(r)

0 50 100-50-100j

t E (τ

·N)

0.080.120.160.200.24

-π/2 π/2-π π0

t E (τ

·N)

0

1/4

1/2

3/4

1

0

1/4

1/2

3/4

1

0

1

0.8

0.6

0.4

0.2

-1.5 -1 -0.5 0 0.5 1 1.5

10

0.1

|⟨aj a

j+r⟩|

✝

rq (d-1)1

0 1

(a)

(c)

(b)

(d)

nj, n(q) (a.u.)

n(x)

(a.u

.)~

x~

10π

0

(r)

r10 20

tE (τ·N)

^^

… ground-state exponent

Lancaster and Mitra (PRE, 2010)

0 50 100-50-100j

t E (τ

·N)

0.080.120.160.200.24

-π/2 π/2-π π0

t E (τ

·N)

0

1/4

1/2

3/4

1

0

1/4

1/2

3/4

1

0

1

0.8

0.6

0.4

0.2

-1.5 -1 -0.5 0 0.5 1 1.5

10

0.1

|⟨aj a

j+r⟩|

✝

rq (d-1)1

0 1

(a)

(c)

(b)

(d)

nj, n(q) (a.u.)

n(x)

(a.u

.)~

x~

10π

0

(r)

r10 20

tE (τ·N)

^^

Singularities in the quasimomentum distribution at finite quasimomenta

n(q) =1

L

X

j,l

e�iq(j�l)dha†jali

n(q)


Power-law correlations emerge spontaneously

0 50 100-50-100j

t E (τ

·N)

0.080.120.160.200.24

-π/2 π/2-π π0

t E (τ

·N)

0

1/4

1/2

3/4

1

0

1/4

1/2

3/4

1

0

1

0.8

0.6

0.4

0.2

-1.5 -1 -0.5 0 0.5 1 1.5

10

0.1

|⟨aj a

j+r⟩|

✝

rq (d-1)1

0 1

(a)

(c)

(b)

(d)

nj, n(q) (a.u.)

n(x)

(a.u

.)~

x~

10π

0

(r)

r10 20

tE (τ·N)

^^

To remember:

Singularities in the quasimomentum distribution at finite quasimomenta

n(q) =1

L

X

j,l

e�iq(j�l)dha†jali

n(q)

(there is no coherence in the initial state!)

There are two independent sources of coherence

Experimental realization

Dynamical quasicondensation at finite momenta:

arXiv:1505:05150

norm

aliz

ed d

ensi

ty (a

.u.)

tE = 0τ tE = 7τ tE = 36τ

expa

nsio

ntim

e-of

-flig

ht

tE

tTOF

tE = 14τ(a)

time-of-flight position (µm)-200 0 200 -200 0 200 -200 0 200 -200 0 200

(f) (g) (h) (i)

-100 0 100 -100 0 100 -100 0 100 -100 0 100in situ position (d)

(b) (c) (d) (e) experimenttheory

t = 0

t = tE

t = tE+tTOF norm

aliz

ed d

ensi

ty (a

.u.)

Expansion in a lattice (with interactions!)

Expansion in a free space (without interactions)

Time-of-flight (TOF) measurement

In-situ absorption imaging

| (t)i = e�iHt| 0i

Measure density distribution

Measure momentum distribution

nj

nk

Experimental sequence

norm

aliz

ed d

ensi

ty (a

.u.)

tE = 0τ tE = 7τ tE = 36τ

expa

nsio

ntim

e-of

-flig

ht

tE

tTOF

tE = 14τ(a)


(f) (g) (h) (i)



t = 0

t = tE

t = tE+tTOF norm

aliz

ed d

ensi

ty (a

.u.)

Results: density distribution in the lattice

Theoretical modeling: two input parameters only | 0i =Y

i2L0

a†i |;i

Total number of particles in the 3D cloud of isolated 1D chains

Finite entropy in the system, which results in holes in the initial state

Si = �kB(ni log ni + (1� ni) log(1� ni))

S/N3D =X

i=(x,y,z)

Si

/N3D ⇡ 1.2kB

probability to find a particle at site i

norm

aliz

ed d

ensi

ty (a

.u.)

tE = 0τ tE = 7τ tE = 36τ

expa

nsio

ntim

e-of

-flig

ht

tE

tTOF

tE = 14τ(a)


(f) (g) (h) (i)



t = 0

t = tE

t = tE+tTOF norm

aliz

ed d

ensi

ty (a

.u.)

Results: density distribution in the lattice

0 10 20 30 40 50tE (τ)

50

100

Rc (

d)

Density dynamics consistent with ballistic expansionRonzheimer et al, PRL (2013)

norm

aliz

ed d

ensi

ty (a

.u.)

tE = 0τ tE = 7τ tE = 36τ

expa

nsio

ntim

e-of

-flig

ht

tE

tTOF

tE = 14τ(a)


(f) (g) (h) (i)



t = 0

t = tE

t = tE+tTOF norm

aliz

ed d

ensi

ty (a

.u.)

Results: TOF distribution

norm

aliz

ed d

ensi

ty (a

.u.)

tE = 0τ tE = 7τ tE = 36τ

expa

nsio

ntim

e-of

-flig

ht

tE

tTOF

tE = 14τ(a)


(f) (g) (h) (i)



t = 0

t = tE

t = tE+tTOF norm

aliz

ed d

ensi

ty (a

.u.)

norm

aliz

ed d

ensi

ty (a

.u.)

tE = 0τ tE = 7τ tE = 36τ

expa

nsio

ntim

e-of

-flig

ht

tE

tTOF

tE = 14τ(a)


(f) (g) (h) (i)



t = 0

t = tE

t = tE+tTOF norm

aliz

ed d

ensi

ty (a

.u.)

Comparison: theory - experiment

Very good agreement for both the number of peaks as well as the position of peaks:

Main peaks: very close to

The number of side peaks consistent with emergence of two independent quasicondensates

~q = ±(⇡/2)(~/d)

tTOF = 6ms tTOF = 10ms tTOF = 12ms tTOF = 15ms

02468

101214

corrected time-of-flight position (µm)

corrected time-of-flight position (µm)t T

OF (m

s)lin

e de

nsity

(a.u

.)

(a) (b) (c) (d)

0-600 600 0-600 600 0-600 600 0-600 600

-600 -400 -200 0 200 400 600

(e)

Fit the experimental TOF distribution with six gaussians

Very similar results: the two main peaks very close to

norm

aliz

ed d

ensi

ty (a

.u.)

tE = 0τ tE = 7τ tE = 36τ

expa

nsio

ntim

e-of

-flig

ht

tE

tTOF

tE = 14τ(a)


(f) (g) (h) (i)



t = 0

t = tE

t = tE+tTOF norm

aliz

ed d

ensi

ty (a

.u.)

Results: TOF distribution (fitting)

~q = ±(⇡/2)(~/d)

More about theoretical modeling

Typical initial states zero entropy

finite entropy

TOF distribution nTOF(r) =X

rj ,rl

w⇤0(r � rj , tTOF)w0(r � rl, tTOF)ha†rj arli

nTOF(x) =1

d

r↵

⇡

e

�x

2↵

X

j,l

e

x(l+j)↵e

� 12 (l

2+j

2)↵e

�ix(l�j)�e

i(l2�j

2) �2 ha†

j

a

l

iGerbier et al, PRL (2008)

One-particle density matrix

tE = 10⌧ tE = 14⌧

norm

aliz

ed d

ensi

ty (a

.u.)

tE = 0τ tE = 7τ tE = 36τ

expa

nsio

ntim

e-of

-flig

ht

tE

tTOF

tE = 14τ(a)


(f) (g) (h) (i)



t = 0

t = tE

t = tE+tTOF norm

aliz

ed d

ensi

ty (a

.u.)

Instead of the generic thermalization dynamics, the atoms bunch at one distinct momentum

The quasi-long-range order emerges spontaneously far from equilibrium, where it is very surprising to find coherence at all

The role of integrability and dimensionality?

Emergence of long-range order in steady states of generic systems?

Conclusions

Open questions

Thank you

0 20 40 60 80 100N

0

0.1

0.2

0.3

0.4P

(N

)

S = 0S > 0

Distribution of probabilities P(N) to find particles in chains with particle number N

One-particle density matrix

tE = 10⌧ tE = 14⌧

Dimensionality

Hard-core bosons on two-leg ladder: fermionic momenta not conserved

J?/J = 0 J?/J = 0.5 J?/J = 1

Density profiles

t=0 t>0

LV, Langer, McCulloch, Schneider, Schollwöck, Heidrich-Meisner, PRB (2013)

BHM at finite interactions

U=4 U=10

Expansion from Mott insulator

Expansion from superfluid

U/J(U/J)c ⇡ 3.4

hard-core U=0

Density profiles are identical for all bosonic and fermionic Mott insulators

Virtually identical to the ones of non-interacting fermions

Initial state at finite U/J

1D Mott insulators (bosons and fermions)

Asymptotic expansion dynamics governed by non-interacting fermions

Dynamical fermionization for Bose-Hubbard model

Particle-hole symmetric MDF

Virtually identical density profiles

at asymptotic times …

Flat velocity distribution n(vk)

dynamical quasicondensation of hard-core bosons at finite

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