# Single-particle density matrix for a time-dependent strongly interacting one-dimensional Bose gas

**ABSTRACT** We derive a $1/c$-expansion for the single-particle density matrix of a strongly interacting time-dependent one-dimensional Bose gas, described by the Lieb-Liniger model ($c$ denotes the strength of the interaction). The formalism is derived by expanding Gaudin's Fermi-Bose mapping operator up to $1/c$-terms. We derive an efficient numerical algorithm for calculating the density matrix for time-dependent states in the strong coupling limit, which evolve from a family of initial conditions in the absence of an external potential. We have applied the formalism to study contraction dynamics of a localized wave packet upon which a parabolic phase is imprinted initially.

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**ABSTRACT:**Nonequilibrium dynamics of a Lieb-Liniger system in the presence of the hard-wall potential is studied. We demonstrate that a time-dependent wave function, which describes quantum dynamics of a Lieb-Liniger wave packet comprised of N particles, can be found by solving an $N$-dimensional Fourier transform; this follows from the symmetry properties of the many-body eigenstates in the presence of the hard-wall potential. The presented formalism is employed to numerically calculate reflection of a few-body wave packet from the hard wall for various interaction strengths and incident momenta. Comment: revised version, improved notation, Fig. 5 addedNew Journal of Physics 11/2009; · 4.06 Impact Factor - SourceAvailable from: Robert Pezer[Show abstract] [Hide abstract]

**ABSTRACT:**We use Gaudin's Fermi-Bose mapping operator to calculate exact solutions for the Lieb-Liniger model in a linear (constant force) potential (the constructed exact stationary solutions are referred to as the Lieb-Liniger-Airy wave functions). The ground state properties of the gas in the wedge-like trapping potential are calculated in the strongly interacting regime by using Girardeau's Fermi-Bose mapping and the pseudopotential approach in the $1/c$-approximation ($c$ denotes the strength of the interaction). We point out that quantum dynamics of Lieb-Liniger wave packets in the linear potential can be calculated by employing an $N$-dimensional Fourier transform as in the case of free expansion.Physical Review A 05/2010; · 3.04 Impact Factor

Page 1

arXiv:0907.4608v1 [cond-mat.quant-gas] 27 Jul 2009

Single-particle density matrix for a time-dependent strongly interacting

one-dimensional Bose gas

R. Pezer∗

Faculty of Metallurgy, University of Zagreb, 44103 Sisak, Croatia

T. Gasenzer

Institut f¨ ur Theoretische Physik, Universit¨ at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

H. Buljan

Department of Physics, University of Zagreb, PP 332, Zagreb, Croatia

(Dated: July 27, 2009)

We derive a 1/c-expansion for the single-particle density matrix of a strongly interacting time-

dependent one-dimensional Bose gas, described by the Lieb-Liniger model (c denotes the strength

of the interaction). The formalism is derived by expanding Gaudin’s Fermi-Bose mapping operator

up to 1/c-terms. We derive an efficient numerical algorithm for calculating the density matrix for

time-dependent states in the strong coupling limit, which evolve from a family of initial conditions in

the absence of an external potential. We have applied the formalism to study contraction dynamics

of a localized wave packet upon which a parabolic phase is imprinted initially.

PACS numbers: 05.30.-d,03.75.Kk

I.INTRODUCTION

One of the most attractive many-body quantum sys-

tems nowadays has been introduced by Lieb and Lin-

iger in their landmark paper more than forty years ago

[1]. The system is composed of N identical δ-interacting

bosons in one spatial dimension and is referred to as a

Lieb-Liniger (LL) gas. They have presented an explicit

form of the many-body wave function for a homogeneous

gas with periodic boundary conditions [1], including

equations describing the ground state and the excitation

spectrum. In the strongly interacting ”impenetrable-

core” regime [2], such a one dimensional (1D) system

is referred to as the Tonks-Girardeau (TG) gas; exact so-

lutions in this limit are obtained by Girardeau’s Fermi-

Bose mapping [2]. Following Ref. [1], Yang and Yang

[3] have eliminated a possible existence of phase tran-

sitions in the LL system by proving analyticity of the

partition function. After many recent experimental suc-

cesses [4, 5, 6, 7, 8, 9] in realization of effectively one

dimensional (1D) interacting gases, from the weak up to

the strongly interacting TG regime [5], the LL model has

attracted considerable attention of the physics commu-

nity. There is a clear reason for this; nontrivial quantum

many-body systems are notoriously oblique to a quan-

titative analysis, and therefore possibility of an exact

treatment in particular cases, together with experimental

realization, is of great value. Moreover, exact solutions

can be useful as a benchmark for approximate treatments

aiming to describe a broader range of physical systems.

Even though exact LL many-body wave functions can

∗Electronic address: rpezer@phy.hr

be constructed in some cases (e.g., stationary [1, 10, 11,

12, 13, 14, 15, 16] or time-dependent wave functions

[17, 18, 19, 20]), the calculation of observables (correla-

tion functions) from such solutions usually poses a major

difficulty in practice [21, 22, 23, 24, 25, 26, 27, 28, 29,

30, 31, 32]. Various methods have been employed to over-

come this difficulty including, for example, the quantum

inverse scattering method (e.g., see [21, 29]) and quan-

tum Monte Carlo integration [25]. A recent discussion of

several exact methods for the calculation of correlation

functions of a nonequilibrium 1D Bose gas can be found

in Ref. [32]. In the TG limit, the momentum distribution

can be analytically studied for a ring geometry, and also

for harmonic confinement (e.g., see [33, 34]). Numerical

methods for the calculation of the reduced single-particle

density matrix (RSPDM) can be performed efficiently

for various TG states (ground state, excited and time-

dependent states, see Ref. [35] for hard-core bosons on

the lattice, and Ref. [36] for the continuous TG model

[2]).

Ultracold atoms in 1D atomic wave guides enter the

strongly interacting regime at low temperatures, in tight

transverse confinement, and with strong effective interac-

tions [37, 38, 39]. The correlations functions of a LL gas

can in this limit be calculated by using 1/c expansions

(e.g., see [22, 24, 27]) from the TG (c → ∞) regime.

These calculations in the strongly interacting limit ex-

ploit the fact that a bosonic LL gas is dual to a fermionic

system [40], such that weakly interacting fermions cor-

respond to strongly interacting bosons and vice versa

[40, 41]. A strongly interacting 1D Bose gas was stud-

ied in Ref. [42] by using perturbation theory for the dual

fermionic system. In Ref. [27], the dynamic structure

factor was calculated for zero and finite temperatures.

Here we calculate the 1/c correction for the RSPDM of

a Lieb-Liniger gas, which adds upon a recently obtained

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2

formula for the RSPDM of a TG gas [36]. The method

is derived by using the 1/c term of the so-called Fermi-

Bose (FB) mapping operator introduced by Gaudin [17].

The FB operator method provides us with exact time-

dependent solutions of a LL model [17, 19] in the ab-

sence of external potentials and other boundary condi-

tions; it was recently used to study free expansion of a

LL gas [20]. We derive an efficient numerical algorithm

for calculating the RSPDM for time-dependent states in

the strong coupling limit, which evolve from a family of

initial conditions in the absence of an external potential.

We employ it to study the evolution of a many-body wave

packet with a parabolic phase imprinted at t = 0, which

corresponds to focusing with a lens in optics, a technique

experimentally feasible in 1D atomic gases [9].

This work complements the studies of nonequilibrium

dynamics of interacting Bose gases which have been ad-

dressed by use of the LL model away from [18, 19, 20, 32]

and in the TG limit [35, 36, 43, 44, 45, 46, 47, 48, 49].

The phenomenological relevance of these studies is un-

derlined by the fact that nonequilibrium dynamics is ac-

cessible experimentally [7, 8].

The paper is organized as follows: Section II gives a

detailed account of the formalism, where we outline the

procedure for the calculation of the RSPDM. In section

III we present the example of a Bose gas initially in the

ground state within a harmonic trap, effectively in the

TG regime; at t = 0 a parabolic phase is applied on the

initial state and the harmonic potential is turned off. The

subsequent dynamics leads to focusing of the cloud and

local increase of the 1/c correction. Details of the deriva-

tion and relevant mathematical identities are collected in

the Appendices.

II.

STRONGLY-INTERACTING LIEB-LINIGER GAS

FORMALISM FOR THE TIME-DEPENDENT

The Lieb-Liniger model describes a system of bosons

interacting via pointlike interactions; the many-body

Schr¨ odinger equation for the LL gas of N such bosons

reads [1]

i∂ψLL

∂t

= −

N

?

i=1

∂2ψLL

∂x2

i

+

?

1≤i<j≤N

2cδ(xi−xj)ψLL. (1)

Here, ψLL(x1,...,xN,t) is the time-dependent bosonic

wave function, c is the strength of the interaction. For

now, we assume the absence of any external potential and

boundary conditions.Under these circumstances, the

time-dependent LL model (1) can be solved by employing

the Fermi-Bose transformation [17, 19]. This method can

be applied, e.g., to exactly study free expansion from an

initially localized state [20]. If ψF(x1,...,xN,t) is an an-

tisymmetric (fermionic) wave function, which obeys the

Schr¨ odinger equation for a noninteracting Fermi gas,

i∂ψF

∂t

= −

N

?

i=1

∂2ψF

∂x2

i

, (2)

then the wave function

ψLL(x1,...,xN,t) =

?

N(c)

?

1≤r<l≤N

??

?

sgn(xl− xr)+

1

c

?

∂

∂xl

−

∂

∂xr

ψF(x1,...,xN,t) (3)

obeys Eq. (1) as pointed out in Ref. [17] (see also

Refs. [19, 20]). Here, N(c) is a normalization constant,

and the differential operator in the square brackets de-

notes the Fermi-Bose mapping operator [19]. The deriva-

tives do not act on any of the sign functions (the sign

functions can be avoided by working in only one sector of

the configuration space, e.g., for x1< x2... < xN, e.g.,

see Refs. [19, 20]). We assume that ψF(x1,...,xN,t)

is normalized to unity. Equation (3) can be reorganized

in a finite power series with terms of order 1/cm, where

m = 0,1,...,N(N−1)/2. By using the Fermi-Bose map-

ping operator in this form, one obtains a systematic ex-

pansion of the exact many-body wave function ψLL in

the inverse interaction strength 1/c.

The expansion of the wave function (3) in the inverse

coupling strength 1/c is particularly useful in the strong

coupling limit, as it allows an approximate calculation

of one-body observables contained within the reduced

single-particle density matrix (RSPDM). In the Tonks-

Girardeau limit, where c = ∞, a formula for efficient cal-

culation of the RSPDM has recently been derived [36].

Here we generalize that result to include the 1/c term

in the expansion. By keeping only the 1/c terms in the

expansion, Eq. (3) reduces to

ψLL(x1,...,xN,t) ≃

?

?∂ψF

∂xl

N(c)

?

1≤i<j≤N

−∂ψF

∂xr

sgn(xj−xi)

?

ψF+

1

c

?

1≤r<l≤N

sgn(xl− xr)

??

(x1,...,xN,t)

(4)

The first term is simply the TG gas wave function [2],

while the second term gives the 1/c correction to the TG

wave function when the coupling constant c is finite. This

expression is the starting point for all results that will be

derived in this paper. The RSPDM is defined as

ρLL(x,y,t) =N

?

dx2···dxNψLL(x,x2,...,xN,t)∗

× ψLL(y,x2,...,xN,t). (5)

On inserting Eq. (4) into Eq. (5)) we obtain a formal

Page 3

3

expression for the O(1/c) correction of the RSPDM:

ρLL(x,y,t) = N(c)ˆI(X)ψ∗

+

cN(c)

1≤r<l≤N

F(x,X,t)ψF(y,X,t)

1

?

ˆI(X)

?

sgn(xl− xr)

?∂ψF

∂xl

+ sgn(xl− xr)ψ∗

?∂ψF

∂xl

−∂ψF

∂xr

?∗

(x,X,t)

ψF(y,X,t)

F(x,X,t)

−∂ψF

∂xr

?

(y,X,t)

?

+ O(1/c2).

(6)

Here, X = (x2,...,xN) and the integral operatorˆI(X)

is defined as:

ˆI(X) = N

N

?

n=2

?∞

−∞

dxnsgn(x − xn)sgn(y − xn). (7)

The first term on the right hand side of Eq. (6) is the

TG gas RSPDM [36]. It can be proven (as is done in Ap-

pendix A) that only the partial derivatives with respect

to the first coordinate x1in Eq. (6) give a nonvanishing

contribution. After eliminating the vanishing terms from

Eq. (6) we are left with

ρLL(x,y,t) = N(c)ρTG(x,y,t) +1

N

?

l=2

cN(c)

?∗

×

ˆI(X)

?

sgn(x − xl)

?∂ψF

∂x1

(x,X,t)

ψF(y,X,t)+

sgn(y − xl)ψ∗

F(x,X,t)

?∂ψF

∂x1

?

(y,X,t)

?

+ O(1/c2)

= ρTG(x,y,t) +1

c[η(x,y,t) + η(y,x,t)∗] + O(1/c2).

(8)

In Eq.

η(x,y,t):

(8) we have implicitly defined the quantity

η(x,y,t) =

N

?

l=2

ˆI(X)sgn(x − xl)

?∂ψF

∂x1

?∗

(x,X,t)

?∗

(x,X,t)

ψF(y,X,t)

= (N − 1)ˆI(X)sgn(x − x2)

?∂ψF

∂x1

ψF(y,X,t).

(9)

By using the calculation presented in Appendix A it fol-

lows that

?

dx[η(x,x,t) + η(x,x,t)∗] = 0,

that is, the 1/c correction to the single-particle density

ρLL(x,x,t) increases the density in some regions of space,

but also lowers it in others such that the integral over

the terms of order 1/c is zero. By using this result we

see that in the leading 1/c approximation we can take

N(c) = 1; this fact has already been utilized in the last

line of Eq. (8). It is straightforward to verify that the

integralsˆI(X)sgn(x−xl)

pendent of l (2 ≤ l ≤ N), which yields the second identity

in Eq. (9). The last line of Eq. (8) verifies that to or-

der 1/c the RSPDM possesses as required the symmetry

ρLL(x,y,t) = ρLL(y,x,t)∗.

Up to this point we did not make any assumptions

on the structure of ψF (except that it is antisymmet-

ric and normalized), that is, the derivation was general,

valid even if the wave functions should describe the gas

in an external potential etc. Quite generally, ψFcan also

be considered as a function of c, and one could expand

it in powers of 1/c. ¿From this point on, ψF will be

represented as a Slater determinant formed from single

particle wave functions,

?

∂ψF

∂x1

?∗

(x,X)ψF(y,X) are inde-

ψF =

1

√N!

?

P

(−)PφP1(x1,t)···φPN(xN,t);(10)

such wave functions can be used to study dynamics on an

infinite line, which arises from initial conditions given by

Eqs. (4) and (10). In Eq. (10), φj(x,t) (j = 1,...,N) are

orthonormal single-particle wave functions which obey

the Schr¨ odinger equation i∂φj/∂t = −∂2φj/∂x2. P de-

notes a permutation P(1,...,N) = (P1,...,PN) of the

particle number indices, and (−)Pis its signature. In

this form ψF enables us to rewrite Eq. (8) such that it

involves 1D integrals only, resulting in certain algebraic

cofactors suitable for numerical calculation.

A.Algorithm for RSPDM calculation

For the sake of clarity of the presentation, we first de-

scribe the algorithm for calculation of the RSPDM, and

only afterwards provide its derivation. Without loss of

generality we consider the case x < y. The first step is

to calculate the integrals

Ik,l(x,y,t) = δkl− 2

?y

x

dx′φ∗

k(x′,t)φl(x′,t)(11)

and

Ik,l(y,t) = −δkl+ 2

?y

−∞

dx′φ∗

k(x′,t)φl(x′,t);(12)

for k,l = 1,...,N.

Page 4

4

These integrals are arranged in the following matrices:

P(x,y,t) =

I1,1(x,y,t) I1,2(x,y,t) ... I1,N(x,y,t)

I2,1(x,y,t) I2,2(x,y,t) ... I2,N(x,y,t)

...

IN,1(x,y,t) IN,2(x,y,t) ... IN,N(x,y,t)

...

...

...

, (13)

and

P(l)(x,y,t) =

I1,1(x,y,t) I1,2(x,y,t) ... I1,l(y,t) ... I1,N(x,y,t)

I2,1(x,y,t) I2,2(x,y,t) ... I2,l(y,t) ... I2,N(x,y,t)

...

IN,1(x,y,t) IN,2(x,y,t) ... IN,l(y,t) ... IN,N(x,y,t)

...

...

...

.(14)

Let us define the column vector

Ψ(x,t) =

φ1(x,t)

...

φN(x,t)

,(15)

and its first spatial derivative

Ψ′(x,t) =

φ′

1(x,t)

...

φ′

N(x,t)

. (16)

The TG reduced single-particle density matrix is given

by [36]

ρTG(x,y,t) = det[P(x,y,t)]

Ψ†(x,t)?P(x,y,t)−1?TΨ(y,t). (17)

The quantity η(x,y,t) can also be written in a convenient matrix form (suitable for efficient numerical implemen-

tation):

η(x,y,t) =

N

?

l=1

?

det

?

P(l)(x,y,t)

?

Ψ′†(x,t)

?

P(l)(x,y,t)−1?T

Ψ(y,t)

?

− det[P(x,y,t)]Ψ′†(x,t)?P(x,y,t)−1?TΨ(y,t) (18)

If any of the matrices P(l)happen to be singular, we can

resort to a direct calculation via algebraic cofactors (see

the proof of the algorithm in Appendix B). This hap-

pens rarely and only for some particular high-symmetry

points. Equations (13)–(18) provide the grounds for an

efficient numerical method for calculating the RSPDM,

which is a generalization of the previously introduced

method for the TG gas [36].

It is convenient to calculate the diagonal correction

to the RSPDM. In this case, the matrices appearing in

Eq. (18) are very simple and it is straightforward to ob-

tain

c

?ρLL(x,x,t)

N(c)

− ρTG(x,x,t)

?Tr(ρ′)Tr(I) − Tr(ρ′· I)?

?

≃

(x,t)

(19)

where on the right hand side ρ is N ×N matrix given by

ρk,l(x,t) = φ∗

k(x,t)φl(x,t)

and primes denote the first spatial derivative. We can

also omit the normalisation constant from Eq. (19) since

N(c) = 1 within the 1/c approximation considered here.

III. CONTRACTION DYNAMICS OF

STRONGLY INTERACTING 1D BOSE GAS

In this section we apply the presented formalism to

investigate time-evolution of a many-body wave packet

which contracts in space after a parabolic phase was im-

printed on it. More specifically, we consider N strongly

interacting bosons which are initially in the ground state

in the presence of a harmonic potential. We assume that

this initial state is well described by the Tonks-Girardeau

ground state wave function, a symmetrized Slater deter-

minant. As described in more detail below, at time t = 0,

a parabolic phase is suddenly imprinted onto this state,

and the harmonic potential is turned off. Imprinting the

Page 5

5

parabolic phase is experimentally feasible [9] and is equiv-

alent to the action of a focusing lens on the travelling-

wave state of a light beam in optics. In the context of

atomic gases this can be achieved by applying, over a

short period of time, a tight longitudinal harmonic po-

tential to the one-dimensional gas. This results in pro-

viding different initial momenta to different parts of the

wave packet, with the momenta being proportional to the

distance from the center of the wave packet and directed

towards the center.

In order to connect the model written in Eq. (1) to

physical units we first write the 1D Schr¨ odinger equation

for this system:

i?∂ψLL

∂τ

=

N

?

i=1

?

−?2

2m

∂2

∂ξ2

i

+mω2

HO

2

ξ2

i

?

ψLL

+

?

1≤i<j≤N

2gδ(ξi− ξj)ψLL,

where g is the 1D scattering length [37], ξ is the spa-

tial coordinate, and τ denotes time. Here we denote the

axial trapping frequency by ωHO. By multiplying the

above expression with 2/?ωHOwe obtain a dimensionless

equation suitable for numerical calculation and a physical

interpretation of the relevant scales:

i∂ψLL

∂t

=

N

?

i=1

?

−∂2

∂x2

i

+ x2

i

?

ψLL

+

?

1≤i<j≤N

2

2g

aHOEHO

δ(xi− xj)ψLL. (20)

where we have denoted zero point energy of the har-

monic trap with EHO =

2/ωHO, while space is in units aHO =

other words xi = ξi/aHO, and t = ωHOτ/2.

set c = 2g/aHOEHO we easily verify that it is exactly

the interaction strength of the LL model c, see Eq. (1).

For concreteness, let us assume that the harmonic trap

has frequency ωHO= 60 Hz, which is loaded with87Rb

atoms, N = 12. These values yield aHO = 1.38µm. As

we have already stated, we assume that the effective in-

teraction strength c is sufficiently large such that the sys-

tem is initially in the Tonks-Girardeau regime, and that

the 1/c correction is negligible at t = 0. Under these

conditions the ground state is given by

?ωHO

2

. The unit of time is

??/mωHO; in

If we

ψg.s.=

?

1≤i<j≤N

sgn(xj− xi)

N

det

j,m=1[φj(xm)]/

√

N!,

(21)

where φj(x) is the jth eigenstate of the SP harmonic

oscillator −φ′′

parabolic phase to the system. Technically, each of the

SP wave functions in the determinant (21) is multiplied

by

j+ x2φj = Ejφj. At t = 0 we imprint a

P(x) = e−i(x/b)2,

where we have chosen b =

function which starts to evolve in the absence of a trap-

ping potential is

√2. Thus, the initial wave

ψLL(x1,...,xN,0) ≈

?

1≤i<j≤N

sgn(xj− xi)

N

det

j,m=1[φj(xm)P(xm)]/

√

N!. (22)

After this initial excitation the wave packet starts to con-

tract. Figure 1 displays the evolution of the SP x-space

density and the momentum distribution (SP k-space den-

sity) for c = 50; the momentum distribution is defined as

[33]

nLL(k,t) = (2π)−1

?∞

−∞

dx

?∞

−∞

dx′e−ik(x−x′)ρLL(x,x′,t).

It should be emphasized that the results are scalable, that

is, the same results can be obtained for smaller values of

c provided that the initial state is broader (this corre-

sponds to the fact that for the LL gas in equilibrium

the regime is determined by the interaction strength c

divided by the linear density [1]). As the gas becomes

more dense during the contraction, the 1/c correction

to the TG density becomes larger and clearly visible in

the x-space density. Note that the gas compresses more

strongly for the finite c in comparison to the TG results.

After the wave packet reaches the maximally dense con-

figuration, it starts to freely expand and the 1/c correc-

tion starts to become less visible.

note that while the changes in the x-space density are

nonnegligible in contraction, the 1/c correction in the k-

space density is practically negligible at all times. ¿From

our simulations we find quite generally that the 1/c cor-

rection to the k-space density is much less sensitive to

changes in the coupling strength than the x-space density.

Note that the total x-space density in the presented form

of the 1/c approximation can become negative in some

parts of the space, if the approximation is used beyond

its range of validity (e.g., if c is not sufficiently large).

This gives a clear indication that higher order terms, of

order O(1/c2), become relevant, at least in those parts

of the space where the density is negative.

We emphasize again that the results presented here,

for an evolution in one dimension without further bound-

ary conditions, are correct up to 1/c for all times of the

evolution, i.e., there is no accumulation of the error in

time due to the approximation and we can extract the

asymptotic behaviour of the LL gas.

It is interesting to

IV.CONCLUSION

We have studied the 1/c expansion in the strong cou-

pling limit of the Lieb-Liniger model, exploiting the for-

malism based on a mapping between free Fermi and inter-

acting Bose gas. More specifically we have developed an

Page 6

6

ρLL

x

2.7 ms

1.8 ms

1.3 ms

0.8 ms

0 ms

−6

−4

−20246

0

0.1

0.2

0.3

0.4

nLL

k

2.7 ms

1.8 ms

1.3 ms

0.8 ms

0 ms

−3

−2

−1

0123

0

1

2

3

4

5

FIG. 1:

gas with N=12 bosons and c = 50 at different times of the propagation. At the time of 1.3 ms the cloud attains the maximal

compression. The corresponding shapes for the TG gas are also shown as black dot-dashed lines. The difference between

red solid and black dot-dashed lines corresponds to the 1/c correction. The curves are shifted on the ordinate axis for better

visibility. See text for details.

(color online) The x-space density (left, red solid line), and momentum distribution (right, red solid line) of the LL

efficient algorithm to calculate the reduced one body den-

sity matrix of the strongly interacting LL gas, which is

based on the method previously developed for the Tonks-

Girardeau gas [36], and the Fermi-Bose transformation

[17, 19, 20]. The method is suitable to describe dynamics

on an infinite line, in the absence of external potentials,

and for initial states which can be described by Equations

(4) and (10). An attractive feature is stable accuracy for

all times since the error is not being accumulated dur-

ing the evolution. For the x-space density we provide

a very compact formula Eq. (19) that can be employed

for quite large numbers of particles enabling fast calcu-

lation of the nonequilibrium x-space density dynamics.

In addition, the efficiency of the method is demonstrated

in an example discussed in Sec. III; we have examined

nonequilibrium dynamics where a parabolic phase is im-

printed onto a localized wave packet after which the wave

packet contracts for a while. Our simulation studies the

setup which was experimentally realized in Ref. [9] for a

weakly interacting Bose gas at finite temperature.

Acknowledgments

R.P. and H.B. acknowledge support from the Croat-

ian Ministry of Science (Grant No. 119-0000000-1015).

T.G. acknowledges support by the Deutsche Forschungs-

gemeinschaft. This work is also supported in part by

the Croatian-German scientific collaboration funded by

DAAD and MZOˇS, and the Croatian National Founda-

tion for Science.

APPENDIX A: FORMULAS

Integrals in Eq. (6) involving derivatives in integration

coordinates all vanish due to following expression for k ?=

1 and for any l ?= k:

?∂ψF

∂xk

ˆI(X)sgn(xk− xl)

?∗

(x,X)

ψF(y,X)

+ˆI(X)sgn(xk− xl)ψ∗

F(x,X)

?∂ψF

∂xk

?

(y,X)

=ˆI(X)sgn(xk− xl)

∂

∂xk

[ψ∗

F(x,X)ψF(y,X)]

This is obtained by collecting the corresponding terms

from the first and the second part of the 1/c correction

to the RSPDM in Eq. (6). Let us now look at the kth

integration only. Integrating by parts we have:

[sgn(xk− xl)ψ∗

F(x,X)ψF(y,X)]

???

∞

−∞−

?∞

−∞

dxk2δ(xk− xl)ψ∗

F(x,X)ψF(y,X)

The first term is zero due to boundary condition at infin-

ity and the second term is zero because of the antisym-

metry of ψF. By using the method presented above it is

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7

straightforward to see that

?∞

−∞

dx[η(x,x,t) + η(x,x,t)∗] = 0

where η is defined in Eq.

is that the normalization constant N(c) = 1 in the 1/c

approximation cosidered here.

(9).A direct consequence

APPENDIX B: DETERMINANT EXPANSION

In this appendix we derive Eq. (18). The derivation

is similar to that for the RSPDM of a TG gas [36]. To

simplify the notation, we will suppress the time variable

in all formulas. By inserting Eq. (10) for ψF in Eq. (9)

for η we obtain

η(x,y) =N(N − 1)

N!

?

ij

(−)i+j[φ

′

i(x)]∗φj(y)

?

P,Q

(−)P(−)QIp2,q2Ip3,q3...IpN,qN. (B1)

Here, P ∈ SN−1 is a permutation of the N − 1 indices

not including i: (p2,...,pN) = P(1...i − 1 i + 1...N).

Similarly, Q ∈ SN−1 permutes the N − 1 indices not

including j: (q2,...,qN) = Q(1...j − 1 j + 1...N).

We considerthe sum

Let (q′′

N) be the ordered series of integers

{1,...,N}\{l,j}, where l ?= j, and Q′∈ SN−2 a

permutation of these N − 2 numbers, (q′

over thepermutations.

3,...,q′′

3,...,q′

N) =

Q′(q′′

in Eq. (B1) can be written as

3,...,q′′

N). With these, the sum over permutations

1

(N − 2)!

?

P,Q

(−)P+QIp2,q2Ip3,q3···IpN,qN

=

1

(N − 2)!

N

?

l=1

(l?=j)

(−)l+1?

P,Q′

(−)P+Q′Ip2,lIp3,q′

3···IpN,q′

N

=

N

?

l=1

(l?=j)

?

P

(−)l+1(−)PIp2,lIp3,q′′

3···IpN,q′′

N

=

N

?

l=1

(l?=j)

detP(l)

i,j(x,y). (B2)

Here, P(l)

inating the ith row and the jth column from the matrix

P(l)defined in Eq.(13). This yields the following result

for η(x,y):

i,jis a N − 1 × N − 1 matrix obtained by elim-

η(x,y) =

?

ij

[φ

′

i(x)]∗φj(y)

?

l=1

(l?=j)

(−)i+jdetP(l)

i,j(x,y)

(B3)

The final result, Eq. (18), is obtained from (B3) with the

use of standard matrix algebra connecting minors, the

cofactor matrix, and the matrix inverse.

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