Quantum simulation of low-dimensional systems Quantum simulation of low-dimensional systems using interference experiments.using interference experiments.

Anatoli Polkovnikov,Anatoli Polkovnikov,Boston UniversityBoston University

Collaboration:

Ehud AltmanEhud Altman - - The Weizmann Institute of ScienceThe Weizmann Institute of ScienceEugene Demler Eugene Demler - - Harvard UniversityHarvard UniversityVladimir Gritsev Vladimir Gritsev - - Harvard UniversityHarvard University

Quantum SimulationsQuantum Simulations

1.1. Universal: can simulate any unitary evolutionUniversal: can simulate any unitary evolution

2.2. Simulating specific (interacting) HamiltoniansSimulating specific (interacting) Hamiltonians

a)a) direct: simulate the system we realizedirect: simulate the system we realize

b)b) Indirect: simulate one system realizing another Indirect: simulate one system realizing another one.one.

This talk: simulating a quantum impurity model in a 1D This talk: simulating a quantum impurity model in a 1D interacting Fermi gas using interference between interacting Fermi gas using interference between homogeneous 1D bosons.homogeneous 1D bosons.

This talk:This talk:

Interference between two systems of interacting bosons: Interference between two systems of interacting bosons:

• measurements and interferencemeasurements and interference

• shot noiseshot noise

• noise due to phase fluctuationsnoise due to phase fluctuations

• full distribution function and quantum simulationsfull distribution function and quantum simulations

• outlook.outlook.

Interference between two condensates.Interference between two condensates.

dx

TOFTOF

† †1 2 1 2

int

( , ) ( , ) ( , ) ( , ) ( , )

( , ) ( , )

x t a x t a x t a x t a x t

x t x t

† †int 1 2 2 1( , ) ( , ) ( , ) ( , ) ( , )x t a x t a x t a x t a x t

Free expansion:Free expansion:

11 1 1 1

22 2 2 2

( / 2) ( , ) ~ exp ,

( / 2) ( , ) ~ exp ,

mv m x dt a x t a iQ x Q

tmv m x d

a x t a iQ x Qt

† †

int 1 2 2 1( , ) exp( ) exp( ), md

x t a a iQx a a iQx Qt

1,2

1,2 int ( ) cosia Ne x N Qx Andrews Andrews et. al. 1997et. al. 1997

What do we observe?What do we observe?

b) Uncorrelated, but well defined phases b) Uncorrelated, but well defined phases intint(x)(x)=0=0

2 2int int( ) ( ) ~ cos cos ~ cos ( ) 0x y N Qx Qy N Q x y

Hanbury Brown-Twiss EffectHanbury Brown-Twiss Effect

x

TOFTOF

c) Initial number state. c) Initial number state.

† † 2int int 1 1 2 2( ) ( ) ~ cos ( ) ~ cos ( )x y a a a a Q x y N Q x y

Work with original bosonic fields:Work with original bosonic fields:† †

int 1 2 2 1( ) ~ exp( ) exp( ) =0x a a iQx a a iQx

int ( ) cosx N Qx

a)a) Correlated phases Correlated phases (( = 0) = 0)

int ( ) cosx N Qx

Y. Castin and J. Dalibard, 1997Y. Castin and J. Dalibard, 1997

P. Anderson, 1984, Do two superfluids which have never seen one another possess a definitive P. Anderson, 1984, Do two superfluids which have never seen one another possess a definitive phase?phase?

Y. Castin and J. Dalibard, 1997 for a specific gedanken experiment: yes but this phase is Y. Castin and J. Dalibard, 1997 for a specific gedanken experiment: yes but this phase is spontaneously generated by measurement!!!spontaneously generated by measurement!!!

A.P., E. Altman and E. Demler, 2005: yes for ToF experiments.A.P., E. Altman and E. Demler, 2005: yes for ToF experiments.

How do we analyze the image?How do we analyze the image?

( ) ~ c.c.i iQxx Ae e 2 ( )( ) ( ) ~ c.c.iQ x yx y A e

2 ( )~ ( ) ( ) iQ x yA dxdy x y e This analysis does not take into account quantum This analysis does not take into account quantum effects, i.e. Need to correct for this.effects, i.e. Need to correct for this.† ( ), ( ) 0.a x a y

Define an observable (Define an observable (interference amplitude squared interference amplitude squared ):):

2 ( , ) ( , ) exp ( ) ( , )A dxdy x t y t iQ x y dx x t

24 2

2

2

2 11 two coherent states

2 1 11 two Fock states

2

A A NNA

N N N

The interference amplitude The interference amplitude does notdoes not fluctuate at large N! fluctuate at large N!

† 2 2 † 2 2 † 2 2 † † 2 2 † †4 1 1 2 2 1 1 2 2 1 1 1 1 2 24 4 2A a a a a a a a a a a a a a a

2 † † † †2 1 1 2 2 1 1 2 2 A A a a a a a a a a depends only on Ndepends only on N

† †1 2 2 1( , ) ~ exp( ) exp( )x t a a iQx a a iQx

Decrease the effect of shot noise.Decrease the effect of shot noise.

Extended condensates.Extended condensates.

Reduce the interference contrast.Reduce the interference contrast.

This talk: how to analyze this reduction of the contrast.This talk: how to analyze this reduction of the contrast.

x

z

z1

z2

AQ

†int 1 20

( ) exp( ) ( ) ( ) c.c.L

x iQx a z a z dz

† †2 1 1 1 2 2 1 2 2 1 20 0

( ) ( ) ( ) ( )L L

A a z a z a z a z dz dz

Identical homogeneous condensates:Identical homogeneous condensates:

2†

2 1 10( ) (0)

LA L a z a dz

Interference amplitude contains information about fluctuations Interference amplitude contains information about fluctuations within each condensate.within each condensate.

2int int

2 † †1 1 2 1 2 2 1 2 1 20 0

( ) ( ) cos ( )

( ) ( ) ( ) ( )L L

x y A Q x y

A a z a z a z a z dz dz

Fluctuating Condensates.Fluctuating Condensates.

Scaling with L: two limiting casesScaling with L: two limiting cases

†int 1 2( ) ( ) ( ) exp( ) . . exp( ) . .z zz z

x a z a z iQx c c N iQx i c c

A L

Ideal condensates:Ideal condensates:L x

z

Interference contrast Interference contrast does not depend on L.does not depend on L.

L x

z

Dephased condensates:Dephased condensates:

A L

Contrast scalesContrast scales as L as L-1/2-1/2..

Formal derivation:Formal derivation:

2†

2 1 10( ) (0)

LA L a z a dz

Ideal condensate: Ideal condensate: †1 1( ) (0) ca z a

22 cA L

L

Thermal gas:Thermal gas:

†1 1( ) (0) ~ exp( / )a z a z

2A L

L

Intermediate case (quasi long-range order).Intermediate case (quasi long-range order).2

†2 1 10

( ) (0)L

A L a z a dz

z

1D condensates (Luttinger liquids):1D condensates (Luttinger liquids):

1/ 2†1 1( ) (0) /

K

ha z a z

L

1/ 22 1/ 1/2 , Interference contrast /

KK Kh hA L L

Repulsive bosons with short range interactions: Repulsive bosons with short range interactions: 2

2

2

Weak interactions 1

Strong interactions (Fermionized regime) 1

K A L

K A L

Finite temperature:Finite temperature:

1 1/22

2

1K

hh

A Lm T

Angular Dependence.Angular Dependence.

† ( tan )int 1 20

†1 20

( ) ( ) ( ) c.c.

exp( ) ( ) ( ) +c.c., tan

L iQ x z

L iqz

x a z a z e dz

iQx a z a z e dz q Q

2 † †1 1 1 2 2 1 2 2 2 1 1 20 0

( ) ( ) ( ) ( ) ( ) cos ( )L L

A q a z a z a z a z q z z dz dz

q is equivalent to the relative momentum of the two q is equivalent to the relative momentum of the two condensates (always present e.g. if there are dipolar condensates (always present e.g. if there are dipolar oscillations).oscillations).

z

x(z 1)

x(z 2)

(for the imaging beam (for the imaging beam orthogonal to the orthogonal to the page, page, is the angle of is the angle of the integration axis the integration axis with respect to z.)with respect to z.)

Two-dimensional condensates at finite temperatureTwo-dimensional condensates at finite temperature

CCDcamera

x

z

Time of

flight

z

xy

imaginglaser

(picture by Z. Hadzibabic)(picture by Z. Hadzibabic)

22 †

0 0( ) ( ', ) (0,0)

yX L

Q yA X X L dx a x y a

Elongated condensates: Elongated condensates: LLxx>>>>LLy y ..

Observing the Kosterlitz-Thouless transition

Above KT transition Ly

Lx

2 ( )QA X X

Below KT transition

2 2 2QA X

Universal jump of at TKT

KTT T 1/ 4

KTT T 1/ 2

2 2 2( )QA X X

Always algebraic scaling, easy to detect.Always algebraic scaling, easy to detect.

Zoran Hadzibabic, Zoran Hadzibabic, Peter Kruger, Marc Cheneau, Baptiste Battelier, Peter Kruger, Marc Cheneau, Baptiste Battelier, Sabine Sabine Stock,Stock, and Jean Dalibard (2006).and Jean Dalibard (2006).

integration

over x axis

X

z

z

integration

over x axisz

x

integration distance X

(pixels)

Contrast after integration0.4

0.2

00 10 20 30

middle Tlow T

high T

2 2

2212

0

( ) 1 1( ) ~ ~ (0, ) ~

XQA X

C X g x dxX X X

Interference contrast:Interference contrast:

Exp

onen

t

central contrast

0.5

0 0.1 0.2 0.3

0.4

0.3 high T low T

T (K)1.0 1.1 1.2

1.0

0

“universal jump in the superfluid density”

c.f. Bishop and Reppy

Z. Hadzibabic et. al.Z. Hadzibabic et. al.

Vortex proliferationVortex proliferation

Fraction of images showing at least one dislocation:

0

10%

20%

30%

central contrast0 0.1 0.2 0.3 0.4

high T low T

Higher Moments.Higher Moments.

2 † †1 1 1 2 2 1 2 2 1 20 0( ) ( ) ( ) ( )

L LA a z a z a z a z dz dz is an observable is an observable

quantum operatorquantum operator

2† 2 1/

2 1 2 1 1 1 20 0( ) ( )

L L KA dz dz a z a z L

Identical condensates. Mean:Identical condensates. Mean:

Similarly higher momentsSimilarly higher moments2

† † (2 1/ )2 1 1 1 1 1 1 10 0

.. ... ( )... ( ) ( )... ( )L L n K

n n n nA dz dz a z a z a z a z L

Probe of the higher order correlation functions. Probe of the higher order correlation functions.

Universal (size independent) Universal (size independent) distribution function:distribution function:

2 2 2 2 2( ) : ( )n nW A A A W A dA

Shot noise contribution: Shot noise contribution: AA2n2n~ A~ A2n2n / L/ L1-1/K1-1/K Shot Shot

noise is subdominant for K>1 at T=0.noise is subdominant for K>1 at T=0.

Sketch of the derivationSketch of the derivation

Action:Action:

1/ 2 1 1/ 22 2~

ni K Kc n c h na e A C L Z

With periodic boundary conditions we find:With periodic boundary conditions we find:

These integrals can be evaluated using Jack polynomials These integrals can be evaluated using Jack polynomials ((Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995))

1

21

2 2 2 20 1

( 1/ 2 )1 (1 1/ )

(1/ 2 ) (1 ) (1 1/ 2 )

K KZ

K K

2 1

2

1 24 4

0 1 2

1/ 1/ 24

(1/ 2 ) 1/ 2 1 1

K KZ

K K

Two simple limits:Two simple limits:

2 221: !, ( ) exp( )nK Z n W A C CA

Central limit theorem! Also at finite T.

x

z

z1

z2

A

Strongly interacting Tonks-Girardeau regimeStrongly interacting Tonks-Girardeau regime

2 2 22 0

224 2

22

: 1, ( ) ,

6

nK Z W A A A

Z ZA

ZA K

Weakly interacting BEC like regime.Weakly interacting BEC like regime.

Connection to the impurity in a Luttinger liquid problem.Connection to the impurity in a Luttinger liquid problem.

Boundary Sine-Gordon theory:Boundary Sine-Gordon theory:

2 2

0 0

exp ,

2 cos 2 (0, )2 x

Z D S

KS dx d g d

21/ 2

22( ) , 2 ,!

nK

nn

xZ x Z x g

n

Same integrals as in the expressions for (we rely on Euclidean Same integrals as in the expressions for (we rely on Euclidean invariance).invariance).

2nA

2 20 00

( ) ( ) (2 / ) ,Z x W A I Ax A dA

1/ 2 1 1/ 20

K Kc hA C L

P. Fendley, F. Lesage, H. Saleur (1995).

20 02 0

0

2( ) ( ) (2 / ) ,W A Z ix J Ax A xdx

A

Experimental simulation of the quantum impurity problemExperimental simulation of the quantum impurity problem

1.1. Do a series of experiments and determine the distribution function.Do a series of experiments and determine the distribution function.

T. Schumm, et. al., Nature Phys. 1, 57 (2005).

Distribution of interference phases (and amplitudes) from two 1D condensates.Distribution of interference phases (and amplitudes) from two 1D condensates.

2.2. Evaluate the integral.Evaluate the integral.2 2

0 00( ) ( ) (2 / ) ,Z x W A I Ax A dA

3.3. Read the result. Read the result.

Relevance of the boundary SG model to other problems.Relevance of the boundary SG model to other problems.

An isolated impurity in a 1D Fermi gas.An isolated impurity in a 1D Fermi gas.

2 2

0 02 cos 2 (0, )

2 xS dx d g dK

K<1, attractive interactions – impurity is relevantK<1, attractive interactions – impurity is relevant

interacting electron gasinteracting electron gas scattering on impurityscattering on impurity

K>1, repulsive interactions – impurity is irrelevantK>1, repulsive interactions – impurity is irrelevantKane and Fisher, 1992Kane and Fisher, 1992

We can directly simulate the partition function for this We can directly simulate the partition function for this problem in interference experiments.problem in interference experiments.

2 22 cos 2 (0, )

2 xS dxd u u h u g d uK

hh plays the role of the relative momentum plays the role of the relative momentum qq

I. Affleck, W. Hofstetter, D. R. Nelson, U. Schollwock, J.Stat.Mech. 0410 P003 (2004)

Interacting flux lines in 2D superconductors.Interacting flux lines in 2D superconductors.

( )Z x can be found using Bethe ansatz methods for half integer K.can be found using Bethe ansatz methods for half integer K.

In principle we can find In principle we can find WW::

20 02 0

0

2( ) ( ) (2 / ) ,W A Z ix J Ax A xdx

A

Difficulties: need to do analytic continuation. Difficulties: need to do analytic continuation. The problem becomes increasingly harder as The problem becomes increasingly harder as K K increases.increases.

Use a different approach based on spectral determinant:Use a different approach based on spectral determinant:

Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999);

Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001)

2

0

( ) 1n n

bxZ ix

E

2 1/ 2 sin / 2

4 (1 1/ 2 )K K

b K K

0 1 2 3 4

Pro

babi

lity

W(

)

K=1 K=1.5 K=3 K=5

Evolution of the distribution function.Evolution of the distribution function.

22A A

Universal Gumbel distribution at large K Universal Gumbel distribution at large K

((-1)/-1)/

exp( / )e

/e

1

( 1)( 1) exp[ ( 1)]

exp( ) 1( ) exp( )

e 1K K

K x K x

x KW x x x

K

Generalized extreme value distribution:Generalized extreme value distribution:

Emergence of extreme value statistics on other instances: Emergence of extreme value statistics on other instances:

1max { }n n nz z z E. Bretin, Phys. Rev. Lett. E. Bretin, Phys. Rev. Lett. 9595, 170601 (2005) , 170601 (2005)

From independent From independent random variables to random variables to correlated intervalscorrelated intervals

Also Also 1/f1/f noise noise2

[0, ]( ) ( )

t Tw T h t h

Other examples of extreme value statistics.Other examples of extreme value statistics.

Extension: direct probing of fermionic superfluidityExtension: direct probing of fermionic superfluidity

†int 1 20

( ) exp( ) ( ) ( ) c.c.L

x iQx a z a z dz

2 2int int( ) ( ) cos ( ) cos ( )x y A Q x y B Q x y

† †2 1 1 1 2 2 1 2 2 1 20 0

† †2 1 1 1 2 2 2 2 1 1 20 0

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

L L

L L

A a z a z a z a z dz dz

B a z a z a z a z dz dz

BCS modelBCS model

2 22

2 22, 2 1

4 4f

LB A L

E

Conclusions.Conclusions.Analysis of interference between independent condensates reveals a Analysis of interference between independent condensates reveals a wealth of information about their internal structure.wealth of information about their internal structure.

a)a) Shot noise and phase fluctuations are responsible for decrease of the Shot noise and phase fluctuations are responsible for decrease of the interference contrast. Shot noise is subdominant in large systems with interference contrast. Shot noise is subdominant in large systems with (quasi) long range order.(quasi) long range order.

b)b) Scaling of interference amplitudes with Scaling of interference amplitudes with LL or or revealsreveals correlation correlation function exponents. function exponents.

c)c) Probability distribution of amplitudes gives the information about Probability distribution of amplitudes gives the information about higher order correlation functions.higher order correlation functions.

d)d) Interference of two Luttinger liquids allows one to obtain partition Interference of two Luttinger liquids allows one to obtain partition function of a 1D quantum impurity problem (also related to variety of function of a 1D quantum impurity problem (also related to variety of other problems) and thus to simulate it.other problems) and thus to simulate it.

Extensions to other cases: fermions, out of equilibrium systems, spin Extensions to other cases: fermions, out of equilibrium systems, spin systems, etc.systems, etc.

Quench experiments in 1D and 2D systems:Quench experiments in 1D and 2D systems:

T. Schumm . et. al., Nature Physics 1, 57 - 62 (01 Oct 2005)

Study dephasing as a function of time. What sort of Study dephasing as a function of time. What sort of information can we get?information can we get?

Analyze dynamics of phase coherence:Analyze dynamics of phase coherence:

1 2( ( ) ( )) ( )( ) e ei t t i tf t

Idea: extract energies of excited states Idea: extract energies of excited states and thus go beyond static probes.and thus go beyond static probes.

Relevant model:Relevant model:

Excitations: solitons and breathers.Excitations: solitons and breathers.

Can create solitons Can create solitons only in pairs. Expect only in pairs. Expect damped oscillations :damped oscillations :

2( ) sin

ei t sm t

t

solitonssolitons

breathers (bound solitons) breathers (bound solitons) Can create isolated Can create isolated breathers. Expect breathers. Expect undamped oscillations:undamped oscillations:

( )e sini tbm t

Analogy with a Josephson junction.Analogy with a Josephson junction.

2

2

22 cosH

K

2

0 1/ 4 22

1 1exp ,

42 h

EEnn

breathersbreatherssoliton pairs soliton pairs (only with q(only with q0)0)

Numerical simulationsNumerical simulations

0 10 20 30 40 500.80

0.85

0.90

0.95

1.00N=1, U=0.5, J=1

Pha

se C

oher

ence

Time

Hubbard model, Hubbard model, 2 chains, 6 sites each2 chains, 6 sites each

0.0 0.2 0.4 0.60.00

0.01

0.02

0.03

0.04

0.05

0.06N=1, U=0.5, J=1, J=0.1, M=6

Po

we

r S

pe

ctru

m

Frequency

bb0202

bb2424

bb4646 bb0404bb26262s2s0101 2b2b0202

Fourier analysis of the oscillations is a way to perform Fourier analysis of the oscillations is a way to perform spectroscopy.spectroscopy.

Quench in 2D condensatesQuench in 2D condensates

Expect a very sharp change in TExpect a very sharp change in TKTKT as as

a function of the layer separation. a function of the layer separation.

A simple entropic argument:A simple entropic argument:

rrEnergy ~ JEnergy ~ Jrr2 2 expect confinement expect confinement

KT argumentKT argument2 2

2 2 ln 02

c cKT KT

RT T

m m

JJ=0:=0:

2 2* *2 2 ln 0c c

KT KT

RT T

m m

JJ>0:>0:

Sudden change in T/TSudden change in T/TKTKT can result in the Kibble- can result in the Kibble-

Zurek mechanism of the topological defect formation.Zurek mechanism of the topological defect formation.

Start at T>TStart at T>TKTKT quench to T<T quench to T<TKTKT..

If quench is fast If quench is fast we expect that vortices do not thermalize. we expect that vortices do not thermalize. Have nonequilibrium vortex population.Have nonequilibrium vortex population.

RG calculation for various RG calculation for various values of vortex fugacity. values of vortex fugacity.

Neglect dependence Neglect dependence cc(T).(T).

22 †

0( ) ( ) (0) cos( )

L

QA q L a z a qz dz

Angular (momentum) Dependence.Angular (momentum) Dependence.

1qL

2

22 2

21 1/

( ) , ideal condensates ( 1);

1( ) , finite T (short range correlations);

1

1( ) , quasi-condensates finite K.

Q

Q

Q K

A q q K

A qq

A qq

has a cusp singularity for K<1, relevant for fermions.2 ( )QA q