Density ripples in expanding low-dimensional gases as a probe of correlations

Density ripples in expanding low-dimensional gases as a probe of correlations

A. Imambekov,^1,2I. E. Mazets,^3,4D. S. Petrov,^5,6V. Gritsev,⁷ S. Manz,³ S. Hofferberth,⁸T. Schumm,^3,9 E. Demler,⁸and J. Schmiedmayer³

1Department of Physics, Yale University, New Haven, Connecticut 06520, USA

2Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA

3Atominstitut, Fakultät für Physik, TU-Wien, Stadionallee 2, 1020 Vienna, Austria

4Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia

5Laboratoire Physique Théorique et Modéles Statistique, Université Paris Sud, CNRS, 91405 Orsay, France

6Russian Research Center Kurchatov Institute, Kurchatov Square, 123182 Moscow, Russia

7Physics Department, University of Fribourg, Chemin du Musee 3, 1700 Fribourg, Switzerland

8Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

9Wolfgang Pauli Institute, University of Vienna, 1090 Vienna, Austria

We investigate theoretically the evolution of the two-point density correlation function of a low-dimensional ultracold Bose gas after release from a tight transverse confinement. In the course of expansion thermal and quantum fluctuations present in the trapped systems transform into density fluctuations. For the case of free ballistic expansion relevant to current experiments, we present simple analytical relations between the spec- trum of “density ripples” and the correlation functions of the original confined systems. We analyze several physical regimes, including weakly and strongly interacting one-dimensional 共1D兲 Bose gases and two- dimensional共2D兲 Bose gases below the Berezinskii-Kosterlitz-Thouless共BKT兲 transition. For weakly inter- acting 1D Bose gases, we obtain an explicit analytical expression for the spectrum of density ripples which can be used for thermometry. For 2D Bose gases below the BKT transition, we show that for sufficiently long expansion times the spectrum of the density ripples has a self-similar shape controlled only by the exponent of the first-order correlation function. This exponent can be extracted by analyzing the evolution of the spectrum of density ripples as a function of the expansion time.

I. INTRODUCTION

A. Quantum noise studies of ultracold atoms

Quantum correlations can be used to identify and study interesting quantum phases and regimes in ultracold atomic systems. Recent experimental advances include detection of the Mott insulator phase of bosonic 关1兴 and fermionic 关2兴 atoms in optical lattices, production of correlated atom pairs in spontaneous four-wave mixing of two colliding Bose- Einstein condensates 关3兴, studies of dephasing关4兴and inter- ference distribution functions 关5兴 in coherently split one- dimensional 共1D兲 atomic quasicondensates 共QC兲, observation of the Berezinskii-Kosterlitz-Thouless 共BKT兲 transition 关6,7兴 in two-dimensional 共2D兲 quasicondensates 关8兴, and Hanbury-Brown-Twiss correlation measurements for nondegenerate共ND兲metastable⁴He关9兴and³He atoms关10兴, bosonic关11兴and fermionic关12兴atoms in optical lattices, and in atom lasers 关13兴. In one-dimensional atomic gases 关14–20兴,in situmeasurements of correlations have been at- tained by means of photoassociation spectroscopy关21兴or by measuring the three-body inelastic decay关22兴using the pro- portionality of the corresponding rates to the zero-distance two-particle and three-particle correlation functions, respec- tively关23兴.

Recently it was demonstrated that one can detect single neutral atoms in a tight trap or guide 关24–29兴. However, direct共not inferred from any kind of atomic loss rate关21,22兴兲 observation of interatomic correlations at short distances in trapped ultracold atomic gases is hindered in many cases by

either the finite spatial resolution of the optical detection technique or the very low detection efficiency of the scan- ning electron microscope 关27兴. Therefore one needs to re- lease ultra cold atoms from the trap, diluting the atomic cloud in the course of expansion.

In this paper we address the question of how the correla- tions in the low-dimensional system evolve during the time- of-flight expansion and discuss how the density variations in the time-of-flight images relate to the properties of the origi- nal trapped quantum gas. These “density ripples” in the ex- panding gas reflect the original thermal or quantum phase fluctuations existing in the cloud under confinement. Such phase fluctuations are already present in three-dimensional 共3D兲Bose-condensed clouds under an external confinement with large aspect ratio关30兴. Their effect on density ripples of expanding clouds has been observed 关31–33兴, but quantita- tive analysis of such experiments was complicated since one had to take into account interactions in the course of expan- sion. However, for sufficiently strong transverse confinement reached in current experiments with low-dimensional gases 共chemical potential of the order of the transverse confine- ment frequency兲, the gas expands rapidly in the transverse direction so interactions during the expansion stage can be safely neglected. Then one can develop a simple analytical theory, which directly relates the spectrum of the density ripples after the expansion to the correlation functions of the original fluctuating condensates. Similar question has been considered for 3D clouds expanding in the gravitational field but only for noninteracting atoms 关34兴. We also note the density ripples we discuss are different from the density

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Published in "Physical Review A 80: 033604, 2009"

which should be cited to refer to this work.

modulations which appear due to interactions during expan- sion and have been studied in Refs.关35,36兴.

B. Density ripples in expanding condensates: Preview We consider one- or two-dimensional atomic gases re- leased from a tight trap formed by a scalar potential as real- ized on atom chips or in optical lattice experiments. We con- sider the situation when free expansion takes place in all three dimensions. This should be contrasted to the expansion of such a gas inside a waveguide关16,37–45兴, with the trans- verse confinement being permanently maintained. In the lat- ter case, the nonlinear atomic coupling constant

g_1D= 2ប␻_⬜^as, 共1兲 where␻_⬜is the transverse trapping frequency and asis the atomics-wave scattering length, remains the same. While a bosonic gas rarifies during such expansion, collisions remain important. For example, in the 1D case dynamics asymptoti- cally reaches the limiting Tonks-Girardeau共TG兲 关46兴regime of impenetrable bosons. In our case, if the fundamental fre- quency of the potential of the transverse confinement is much larger than the initial chemical potential of the atoms, the expansion in the transverse directions is determined mainly by the kinetic energy stored in the initial localized state of the transverse motion. Interatomic collisions play almost no role in the expansion. Moreover tight transverse confinement decouples the motion of trapped atoms in the longitudinal and transverse directions. Thus when analyzing density ripples we can reduce the problem to the same num- ber of dimensions as the initial trap共see discussion below in Sec.II兲. For a 1D trap we consider a one-dimensional spec- trum of density ripples, and for atoms which were originally confined in a pancake trap we analyze two-dimensional den- sity ripples.

Before we consider a general formalism, it is useful to present the analysis for the simplest situation. Let us assume that the initial state can be described using the mean-field Bogoliubov approach关47–49兴. Let⌿ˆ

kជ

†be the creation opera- tor of atoms at momentumkជright before the expansion. After free expansion during time t, in the Heisenberg representa- tion we have ⌿ˆ

kជ

†共t兲=⌿ˆ

kជ

†e^iប2k2t/2m, where m is the atomic mass. Then the density operator at timet is given by

␳共rជ^,t兲=1 L

兺

kជ₁,kជ₂⌿ˆ

kជ₁

†⌿ˆ

kជ₂e^{−i共kជ1−kជ2兲·rជ}e^{共itប2/2m兲共k12−k22兲}, 共2兲 and for the density correlation function we obtain

具␳共rជ1,t兲␳共rជ2,t兲典= 1 L²

兺

kជ₁,kជ₂,kជ₃,kជ₄具⌿ˆ

kជ₁

†⌿ˆ_kជ

2⌿ˆ

kជ₃

†⌿ˆ_kជ

4典

⫻e^{−i共kជ1−kជ2兲·rជ1}e^{−i共kជ3−kជ4兲·rជ2}

⫻e^{共itប2/2m兲共k12−k22兲}e^{共itប2/2m兲共k32−k42兲}. 共3兲 The expectation value 具⌿ˆ

kជ₁

†⌿ˆ

kជ 2⌿ˆ

kជ₃

†⌿ˆ

kជ

4典 should be taken in the original condensate before the expansion. Within the mean-field Bogoliubov theory only a state withk= 0 is mac-

roscopically occupied. Thus in Eq.共3兲we take two operators to be

冑

N=

冑

n_1DL, wheren_1Dis the atomic density before the expansion. Thus Eq. 共3兲can be written as

具␳共rជ1,t兲␳共rជ2,t兲典=n_1D² +n_1D L _q

兺

ជ⫽0

e^{iqជ·共rជ1−rជ2兲}

⫻

冋

^{1 + 2具⌿ˆq†⌿ˆq典+共具⌿ˆ−q⌿ˆq典}

+具⌿ˆ

−q

† ⌿ˆ

q

†典兲cosប²q²t

m

册

^{. 共4兲}

The Bogoliubov theory predicts expectation values of 具⌿ˆ

−q⌿ˆ

q典,具⌿ˆ

−q

† ⌿ˆ

q

†典and 1 + 2具⌿ˆ

q

†⌿ˆ

q典as 具⌿ˆ

−q⌿ˆ

q典=具⌿ˆ

−q

† ⌿ˆ

q

†典= − ␮

2Eq

冋

^{1 + 2nB}

冉

^kEB^qT

冊 册

^{, 共5兲}

1 + 2具⌿ˆ

q

†⌿ˆ

q典=⑀q+␮

E_q

冋

^{1 + 2nB}

冉

^kEB^qT

冊 册

^{, 共6兲}

where ⑀q=ប²q²/共2m兲, ␮=g_1Dn_1D is the chemical potential, Eq=

冑

_⑀q共2␮⁺⑀q兲is the Bogoliubov excitation spectrum, and n_B is the Bose occupation number.

From these equations we can easily find the mean-field spectrum of density ripples具兩␳^MF共q兲兩²典 关see Eq.共21兲and the discussion nearby for the precise mathematical definition of the spectrum兴

具兩␳^MF共q兲兩²典=n_1D

冋

^{1 + 2nB}

^冉

^kEBqT

^冊 册

⫻

冋

^E⑀qq

+ ␮

E_q

冉

^{1 − cosប2mq2t}

冊 册

^{. 共7兲}

The general character of the spectrum is clear from Eq.共7兲. As a function of momentum it is not monotonic. We find minima near បq²t/m= 2␲ⁿ and maxima close to បq²t/m

=␲共2n− 1兲, wheren is a positive integer number. Note that while the positions of maxima and minima are essentially universal, the amplitude of individual maxima depends on both temperature and interaction strength.

The mean-field analysis leading to Eq.共7兲is conceptually simple but has limited applicability. It is applicable for weakly interacting 3D Bose condensates if the interactions during expansion are switched off using Feshbach reso- nances 关50兴. In lower dimensions, thermal and quantum phase fluctuations are expected to suppress true long-range order in 2D Bose condensates at finite temperature关51兴and in 1D Bose condensates even at zero temperature 关52兴. In this paper we show how the analysis of the density ripples can be extended to more complicated but experimentally rel- evant situations when the mean-field approach breaks down.

We will find a similar structure to Eq. 共7兲: positions of maxima and minima of the spectrum are given by the same approximate universal conditions on the momenta. However explicit expressions for the strength of individual maxima will be very different. They will contain rich information about fluctuations of low-dimensional condensates.

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C. Relation to other work

Conceptually, the question we consider in this paper is somewhat similar to the interpretation of cosmological ob- servations. In the latter case, quantum fluctuations present in the early universe after its expansion result in observable anisotropies of the cosmic microwave background radiation 关53,54兴and in the density ripples of matter which eventually evolve into galaxies 关55兴. In our case, density ripples of the expanding clouds contain important information about corre- lations present in the trapped state. Analogies between prop- erties of condensates and cosmology have attracted signifi- cant attention recently 关56–63兴.

In addition to the mentioned approaches, several other techniques have been used to experimentally study correla- tions of low-dimensional gases. Some of them rely on cre- ation of two copies of the same cloud关64–68兴, while others require analysis of noise correlations关69兴orin situdensity- fluctuation statistics 关70兴. Interference experiments between two low-dimensional clouds关4,5,8兴can also be used to char- acterize two-point and multipoint correlation functions 关71–74兴. Analysis of density ripples is a much simpler ex- periment and, as we discuss in this paper, can be used for thermometry. This is particularly important for weakly inter- acting 1D Bose quasicondensates关75,76兴, for which the stan- dard approach to measuring temperature by fitting density profiles cannot be extended to temperatures of the order of the chemical potential. In this regime, the chemical potential is very weakly dependent on the temperature关77兴, thus finite temperature leads only to small corrections to the “inverted parabola” density profile 关78兴. An improved thermometry method based on comparison of in situ measured density profiles with solutions of Yang and Yang equations 关79兴 in the local-density approximation has been developed in Ref.

关19兴.

There has been significant theoretical interest in correla- tion functions of the 1D Bose gas. At distances much larger than the healing length, correlation functions are described by Luttinger liquid theory关80–82兴. In the weakly interacting quasicondensate regime, correlation functions can be de- scribed by extension of Bogoliubov theory to low- dimensional gases关76,77,83–85兴. In the strongly interacting regime, one can use “fermionization”关46兴of a 1D Bose gas to evaluate correlation functions at all distances as certain determinants 关86兴. The Lieb-Liniger model 关87兴 which de- scribes the 1D Bose gas is exactly solvable, and one can also analytically obtain zero-distance two-point关88,89兴and three- point 关90兴 density correlations for any interaction strength and extract certain dynamical correlation functions 关91–95兴 from the exact solution. Various numerical techniques have been used as well关78,96–98兴and recent results including the decoherent quantum regime关89,99兴are summarized in Refs.

关100,101兴.

Two-dimensional systems have also been a subject of considerable experimental关8,15,68,102–105兴and theoretical work关75,85,106–114兴.

D. Structure of the paper

This paper is organized as follows. In Sec. II we derive simple analytical relations between the density ripples after

the expansion and the correlation functions of the original system before the expansion. In Sec. III A we analyze the case of weakly interacting 1D Bose gases and obtain explicit expression for the spectrum of density ripples. In Sec. III B we consider the case of a strongly interacting 1D Bose gas.

In Sec. III C we review general features of the density- density correlation function in expanding 1D Bose clouds. In Sec. IVwe discuss 2D Bose systems below the BKT transi- tion 关6,7兴. We summarize our results and make concluding remarks in Sec. V.

II. FREE EXPANSION

In this section we focus on the atoms expanding from a one-dimensional trap. The atom field operator evolution dur- ing the free expansion is given by 关115兴

⌿ˆ共r,t兲=

冕

^{d3r⬘G3共r−r⬘,t兲⌿ˆ共r⬘,0兲, 共8兲}

where the Green’s function of free motion is

G₃共r−r⬘^,t兲=G₁共x−x⬘^,t兲G₁共y−y⬘^,t兲G₁共z−z⬘^,t兲, 共9兲

G₁共␰,t兲=

冑

2␲^miបtexp

冉

^im2បt^␰2

冊

^{, 共10兲}

withmbeing the atomic mass. Tight transverse confinement decouples the motion of trapped atoms in the共y,z兲plane and along the waveguide axis x so that the transverse motion is confined to its ground state f_⬜共y,z兲 and ⌿ˆ共r, 0兲

=f_⬜共y,z兲␺^ˆ共x, 0兲. This, alongside with Eq. 共9兲, allows for a separation of motion in the longitudinal and transverse direc- tions, effectively reducing the problem to 1D.

We introduce the two-particle density matrix for the lon- gitudinal motion as

␳共x1,x₂;x₁⬘^,x2⬘;t兲=具␺^ˆ†共x₁⬘,t兲␺^ˆ†共x₂⬘,t兲␺^ˆ共x2,t兲␺^ˆ共x1,t兲典.

共11兲 Then we define the two-point density correlation function

g₂共x1,x₂;t兲=␳共x1,x₂;x₁,x₂;t兲

n共x1,t兲n共x2,t兲 , 共12兲 wheren共x,t兲=具␺^ˆ†共x,t兲␺^ˆ共x,t兲典. The free evolution of the two- particle density matrix is given by the convolution of the two-particle density matrix at t= 0 with four respective Green’s functions, one for each spatial argument, two of the Green’s functions being complex conjugate. We are inter- ested in the casex₁=x₁⬘^andx2=x₂⬘in the final state. Then we obtain

␳共x1,x₂;x₁,x₂;t兲=

冕

^dx3

冕

^dx3⬘

冕

^dx4

冕

^{dx4⬘G1共x1−x3,t兲}

⫻G₁共x₂−x₄,t兲G₁^ⴱ共x₁−x₃⬘^,t兲G₁^ⴱ共x₂−x₄⬘^,t兲

⫻␳共x3,x₄;x₃⬘^,x4⬘;0兲. 共13兲 We assume that the product of the typical velocity of the atoms in the x direction and the expansion time is much

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smaller than the size of the trapped atomic cloud. Then we are allowed to consider a uniform sample with length L

→⬁, with the 1D number densityn_1D=N/L being kept con- stant in the thermodynamic limit 共Nbeing the total number of atoms兲. Note that this limit is opposite to the conventional limit of infinitely large expansion times, in which density in real space reflects the initial momentum distribution共in that regime, it was recently proposed关116兴that noise correlations in density profiles can be used to probe properties of low- dimensional gases兲.

In our limitn共x,t兲=n_1Dis constant in time, and the two- particle density matrix is translationally invariant共it does not change if all four of its spatial arguments are shifted by the same amount兲 at any time. The density correlation function then depends on the coordinate difference only so we use the notation

g₂共x1−x₂;t兲 ⬅g₂共x1,x₂;t兲. 共14兲 Using the translational invariance of the two-particle density matrix and the identity

␦共x兲= 1 2␲

冕

−⬁

⬁

dyexp共iyx兲, 共15兲

we arrive at

共x1−x₂;x₁−x₂;t兲= m

4␲បt

冕

_−⬁^{⬁ dx}

冕

_−⬁^{⬁ dx⬘exp}

再

^i4បtm

⫻关共x1−x₂−x兲²−共x1−x₂−x⬘兲²兴

冎

⫻共x;x⬘^;0兲, 共16兲 where

共x;x⬘;t兲 ⬅␳

冉

^x2,− x 2;x⬘

2,−x⬘

2;t

冊

^{. 共17兲}

Obviously, ␳共x₁,x₂;x₁,x₂;t兲=共x₁−x₂;x₁−x₂;t兲=n_1D² g₂共x₁

−x₂;t兲. The physical meaning of Eq.共16兲is that the motion of the center of mass of an atomic pair plays no role in the dynamics of establishing g₂共x1−x₂;t兲, which is fully deter- mined by the relative motion. The relative-motion degree of freedom is characterized by the reduced massm/2关117兴.

Let us now consider some properties of the two-particle density matrix 共x1;x₂;t兲 for bosons. Changing the sign of x₁ or x₂ is equivalent to a permutation of two bosons and, hence, does not change the two-particle density matrix, i.e.,

共x₁;x₂;t兲=共兩x₁兩;兩x₂兩;t兲. 共18兲 For the regimes we consider the density matrix of neutral bosons can be assumed to be real. This, together with the Hermicity property, results in

共x1;x₂;t兲=共x2;x₁;t兲. 共19兲 Using Eqs. 共18兲 and 共19兲 and Fourier transforming the Green’s functions, we can reduce Eq. 共16兲to

共x;x;t兲= 2

␲

冕

0

⬁

dq

冕

0

⬁

dXcosqxcosqX

⫻

冉 冏

^X−បqtm

冏

^;

冏

^X+បqtm

冏

^;0

冊

^{. 共20兲}

Alternatively, this equation can be written as 具兩␳共q兲兩²典=

冕

−⬁

⬁

dXcosqX

冓

^␺ˆ†

^冉

^{បqtm ,0}

^冊

⫻␺^ˆ†共X,0兲␺^ˆ

冉

^{X+បqtm ,0}

冊

^{␺ˆ共0,0兲}

冔

^{. 共21兲}

Here 具兩␳共q兲兩²典 is the spectrum of density ripples at time t, which in experiment can be obtained by Fourier transform- ing absorption images after expansion 关118兴. It is related to two-point density correlation function as关119兴

具兩␳共q兲兩²典=n_1D²

冕

_−⬁^{⬁ dxexp共iqx兲关g2共x;t兲− 1兴. 共22兲}

Equations共20兲and共21兲provide a simple analytical relation between the properties of the density ripples after the expan- sion and the correlation functions before the expansion.

It is straightforward to generalize the above analysis to the 2D case. In particular, the analog of Eq.共21兲for the time evolution of the two-point density correlation function has the same form, withXsubstituted byrandq treated as a 2D vector. Namely, we obtain

具兩␳共q兲兩²典=

冕

R²

d²rcosq·r

⫻

冓

^␺ˆ†

^冉

^បmqt,0

^冊

^{␺ˆ†共r,0兲␺ˆ}

^冉

^r+បmqt,0

^冊

^{␺ˆ共0,0兲}

冔

^.

共23兲 III. 1D BOSE GASES

A. Weakly interacting 1D Bose gases

In this subsection we will consider the spectrum of den- sity ripples of weakly interacting 1D quasicondensates. Be- fore we proceed to full analytical theory, let us return to the simple mean-field Bogoliubov approach, which we discussed in the introduction. Readers may be skeptical about the ap- plicability of the mean-field approach to 1D. Indeed, there is no long-range order for 1D gases even at zero temperature 关52兴, and mean-field approach is generally not applicable.

However, in the regime of weak interactions and under cer- tain conditions on the expansion timet, the spectrum of den- sity ripples is captured correctly by the mean-field approach as we will verify later in a rigorous calculation.

We consider Eq.共7兲in the limit⑀qⰆ␮^,EqⰆkBT. In this case one can neglect the first term in the second parentheses, use an approximation E_q⬀q, and expand the Bose occupa- tion number, leading to

具兩␳^MF共q兲兩²典

n_1D² ⬇2mkBT共1 − cosប²q²t/m兲

ប²n_1Dq² . 共24兲

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Let us now present a full calculation, which does not make a mean-field approximation. For weak interactions Bo- goliubov theory has been extended to low-dimensional qua- sicondensates 关77兴, and can be used to calculate correlation functions at all distances. For quasicondensates, the fluctua- tions of the phase are described by the Gaussian action. For Gaussian actions, higher order correlation functions are sim- ply related to two-point correlation functions共see, e.g., Refs.

关120,121兴兲, and the four-point correlation function in Eq.

共21兲 factorizes into products of two-point correlation func- tions of bosonic fields as关74兴

具␺^ˆ†共x₁⬘,0兲␺^ˆ†共x₂⬘,0兲␺^ˆ共x1,0兲␺^ˆ共x2,0兲典

= 兿_i,j=1² 具␺^ˆ†共xi⬘,0兲␺^ˆ共xj,0兲典

具␺^ˆ†共x₁⬘^,0兲␺^ˆ共x₂⬘^,0兲典具␺^ˆ†共x₁,0兲␺^ˆ共x₂,0兲典. 共25兲 This equation gives correct result for all values of x₁,x₂,x₁⬘ and x₂⬘ in the leading order over 1/K expansion 共see the definition of KⰇ1 below兲 and can be used to evaluate the spectrum of the density ripples in weakly interacting conden- sates for all times. The two-point correlation function 具␺^ˆ†共x₁⬘, 0兲␺^ˆ共x1, 0兲典=n_1Dg₁共x₁⬘^−x1兲is translationally invariant and is simply related to predictions of the Bogoliubov theory.

For 1D quasicondensates, one has关77兴 g₁共x兲=具␺^ˆ†共x,0兲␺^ˆ共0,0兲典

n_1D = exp

冋

^{−2K1 f}

^冉

␰^xh

冊 册

^{, 共26兲}

whereK=␲បn1D/共mc兲Ⰷ1 is the Luttinger liquid parameter, cis the speed of sound,␰h=ប/

冑

m␮is the healing length, and

␮ is the chemical potential. When only lowest transverse mode is occupied 共␮Ⰶប␻_⬜兲, speed of sound is given by c

=

冑

2ប␻⬜n_1Das/m. The dimensionless function f共s兲 depends on the temperature and equals

f共s兲= 2

冕

0

⬁dk共1 − cosks兲兵关uk

2+v_k²兴nk+v_k²其, 共27兲

where

u_k=1

2

冋 冉

^{k2k+ 42}

冊

^1/4+

冉

^{k2k+ 42}

冊

^1/4

册

^{, 共28兲}

vk=1

2

冋 ^冉

^{k2k+ 42}

^冊

^1/4−

^冉

^{k2k+ 42}

^冊

^1/4

册

^{, 共29兲}

nk= 1

exp共␮

冑

k²共k²+ 4兲/2kBT兲− 1. 共30兲 For finite temperatures, the function f共s兲 has the following asymptotic behavior:

f共s兲 ⬇␲兩s兩kBT

␮ ^{+C for} ␲兩s兩kBT

␮ ^{Ⰷ1, 共31兲} whereC⬅C共kBT/␮兲is of orderO共1兲for kBT⬃␮^.

Quasicondensate theory is valid关77,100,101兴for tempera- tures,

kBT/␮ⰆK/␲^, 共32兲 significantly beyond the regime of validity of Luttinger liq- uid theory, which is restricted tok_BT/␮Ⰶ1. The longitudinal density profile of a quasicondensate in external harmonic confinement follows the inverted parabola shape under con- dition共32兲, see, e.g., Ref.关78兴. Due to the low fraction of the thermally populated excited states, it is problematic to ex- tract the temperature of the gas from fitting bimodal distri- butions to the observed density profiles. Below we show that the spectrum of density ripples can be used as a convenient tool to characterize the temperature and is sensitive to tem- peratures of the order of the chemical potential.

To be specific, let us consider the case of ⁸⁷Rb atoms 共scattering length as= 5.2 nm兲 with density n_1D= 40 ␮^m−1 and transverse confinement frequency ␻_⬜^{= 2}␲⫻2 kHz, re- sulting in Luttinger liquid parameter K⬇47 and healing length ␰h⬇0.37 ␮m. We can use Eqs.共25兲–共30兲to numeri- cally evaluate in-trap correlation functions. By performing then a numerical integration of Eq.共21兲for various tempera- tures and expansion times, we can evaluate the spectrum of density ripples under condition 共32兲, and the results are shown in Figs.1 and2. In the inset to Fig.2 we also show g₂共x;t兲evaluated using the inverse of Eq.共22兲. In the quasi- condensate regime the behavior ofg₂共x;t兲follows the quali- tative discussion of Sec. III C.

There are several qualitative features that should be noted.

The spectrum of density ripples is not a monotonic function and can also have several maxima. The positions of the maxima only weakly depend on the temperature, and are mostly determined by the expansion time. The amplitude of the ripples, on the other hand, significantly depends both on the expansion time and temperature.

0.4

FIG. 1. Normalized spectrum of density ripples 具兩␳共q兲兩²典/共n_1D² ␰h兲 for weakly interacting 1D quasicondensate of

87Rb atoms with density n_1D= 40 ␮m⁻¹, transverse confinement frequency␻_⬜= 2␲⫻2 kHz, Luttinger liquid parameterK⬇47, and healing length␰h⬇0.37 ␮m. Expansion time is fixed att= 27 ms 共with 1/␰h

冑

_បt/m⬇11.8兲, and temperatures equal 共top to bottom兲 T= 40 nK共k_BT/␮= 1兲, T= 27 nK共k_BT/␮= 0.67兲, T= 12 nK共k_BT/␮

= 0.3兲, andT= 0. Values on the axes of this and subsequent plots are dimensionless. Dots are obtained by numerical integration of Eq.

共21兲 in the weakly interacting limit making use of Eqs.共25兲–共30兲. Solid lines correspond to analytical results 关Eq. 共37兲兴, which are derived under condition共33兲.

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Let us now derive a simple analytical expression for the spectrum of density ripples, which is valid in the regime 关justified below after Eq.共39兲兴

␲

␰h

冑

^បtm kBT

␮ ^{Ⰷ1. 共33兲}

Under this condition one can use Eq.共31兲 and approximate the two-point correlation function by

g₁共x兲 ⬇exp共−兩x兩/␭T兲 for 兩x兩Ⰷ␰h

␮

kBT, 共34兲 where␭Tis defined by

␭T=2K␰h␮

␲^kBT =2ប²n_1D

mk_BT , 共35兲 and does not depend on interaction strength, as long as Eq.

共32兲is satisfied.

Using Eqs.共25兲and共34兲, the second line of Eq.共21兲can be written as

g₁共បqt/m兲²g₁共X兲²

g₁共X−បqt/m兲g1共X+បqt/m兲⬇exp−X

␭T

for Xⱕបqt

m and exp−បqt m␭T

otherwise.

The constant term exp−បqt/m␭T is responsible for g₂共x→⬁,t兲= 1. Since according to Eq.共21兲we need to take a Fourier transform of the above expression, subtracting the constant on the whole interval共0 ,⬁兲does not affect具兩␳共q兲兩²典 for q⫽0, and we obtain

具兩␳共q兲兩²典 n_1D² ⬇2

冕

0

បqt/m

dxcosqx

冉

^exp−␭T^x

− exp−បqt m␭T

冊

^.

共36兲 This integral can be evaluated in a closed form and leads to an analytical answer

具兩␳共q兲兩²典 n_1D² ␰h

⬇␭Tq−e^{−2បqt/m␭T}共␭Tqcosបq²t/m+ 2 sinបq²t/m兲 q␰h共1 +␭T2q²兲 .

共37兲 Note that the last equation reduces to Eq. 共24兲 when ␭Tq Ⰷ1 and បqt/m␭TⰆ1. Figures 1 and 2 show an excellent agreement between the analytical result and numerical inte- gration described earlier after Eq. 共32兲. The analytical result shows the same nonmonotonic behavior as the numerical calculations. The parameter␭Tdefines a time scale

tc⬇6.5m␭T2

ប , 共38兲

after which only a single maximum persists. When several maxima and minima are present, their positions can be esti- mated by

បq²t

m ⬇␲共2n− 1/2⫿1/2兲, 共39兲 where the upper 共lower兲 sign corresponds to the nth maxi- mum 共minimum兲. These conditions can be understood as a

“standing wave” conditions in Eq. 共36兲 and become more precise at lower temperatures.

The appearance of minima and maxima in the spectrum of density ripples can be understood in terms of matter-wave

near-field diffraction. The analogous effect for light waves 共in the spatial domain兲is known as the Talbot effect关122兴.

Its matter-wave counterpart has been also observed in dif- fraction of atoms on a grating关123兴. In our case, we observe near-field diffraction in the time domain. For each expansion time, a certain momentum contribution will be “imaged”

onto itself, leading to a minimum in the spectrum of density ripples for a given momentumq. As compared to diffraction on a regular grating with a fixed period, the typical fluctua- tion length in the trapped cloud is not constant but distrib- uted around the thermal length␭T. Therefore, minima in the spectrum appear for any sufficiently small expansion time, according to condition共39兲.

Condition 共33兲can now be justified in the regime where 具兩␳共q兲兩²典 is near its largest values. In such case most of the

FIG. 2. 共Color online兲 Normalized spectrum of density ripples 具兩␳共q兲兩²典/共n_1D² ␰h兲 with the same parameters as in Fig.1 but for a fixed temperature T= 27 nK共k_BT/␮= 0.67兲, and various times of flight: t= 49 ms 共red, solid兲, t= 27 ms 共green, dashed兲, and t

= 9.5 ms共blue, dotted兲. Dots are obtained by numerical integration of Eq. 共21兲 in the weakly interacting limit making use of Eqs.

共25兲–共30兲. Lines correspond to analytical results关Eq.共37兲兴, which are derived under condition 共33兲. Inset shows g₂共x;t兲, obtained from具兩␳共q兲兩²典using the inverse of Eq.共22兲.

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contributions to Eq. 共21兲 come from distances of the order

冑

បt/m, and Eq.共33兲follows from Eq. 共32兲.

So far we have been assuming that the quasicondensate is deep in the 1D regime,␮^,kBTⰆប␻_⬜. While Eqs.共25兲–共30兲 are valid only under such assumption, Eqs.共34兲and共35兲also work in the weakly interacting quasi-1D regime,

␮,kBT⬃ប␻⬜. 共40兲 Indeed, they rely only on the 1D nature of long-range corre- lations, weakness of interactions, and the property cK

=␲ⁿ1D/m, which is a consequence of the Galilean invariance 关81兴. In Eqs.共32兲and共33兲, the Luttinger liquid parameterK can be obtained asK=ប␲ⁿ1D/共mc兲, where the square of the sound velocity ccan be determined from compressibility as c²=n_1D共⳵␮/⳵ⁿ1D兲/m. For chemical potential␮, one can use an approximate relation关124兴␮⁼ប␻_⬜共

冑

1 + 4a_sn_1D− 1兲.

Let us now briefly review the conditions under which one can neglect interactions in expanding 1D clouds and the ef- fects of finite condensate length L. Transverse expansion takes place at the times of the order of inverse transverse confinement ␻_⬜⁻¹. Up to the times of this order, one cannot neglect interactions during the expansion. Correlation func- tions which enter Eq. 共21兲 will be smeared up to the dis- tances of the order ␦^x⬃c/␻_⬜⁼␰h␮/␻_⬜, and smearing will only weakly affect the final result for 具兩␳共q兲兩²典 if q␦xⰆ1.

Thus to observe an oscillating spectrum of density ripples, one needs to satisfy the condition

␰h

冑

បt^m

␮

ប␻⬜Ⰶ1, 共41兲 which easily holds for the parameters shown in Figs.1and2.

In addition, one can use Feshbach resonances 关50兴to com- pletely switch off interactions during the expansion.

Locally, corrections due to finite L can be neglected if finite limits of integration in Eq. 共16兲 lead to smearing of delta functions up to the distances at which the correlation functions change considerably. This change can occur either because of the variations of the density in external confine- ment at distances⬃Lor because of the decay of correlations for finite temperatures at distances of the order ⬃K␰h/a.

Thus for finite temperatures these conditions read as mL

បtmin共L,K␰h/a兲Ⰷ1 共42兲 and are easily satisfied for parameters considered earlier and, e.g., longitudinal frequency␻x= 2␲⫻5 Hz. Under condition 共42兲 one can take the inhomogeneity of the density profile into account within the local-density approximation by aver- aging the prediction of Eq.共37兲.

B. Strongly interacting 1D Bose gases

Let us now describe the evolution of the two-point density correlation functiong₂共x;t兲of a strongly interacting 1D Bose gas. A dimensionless parameter which controls the strength of interactions at zero temperature can be written as

␥⁼_ប^mg₂ ^1D

n_1D=2m␻_⬜^as

បn1D

Ⰷ1. 共43兲

Under such conditions, the bosonic wave function takes on fermion properties, and the density correlation function in the trapg₂共x; 0兲is the same as for noninteracting fermions of the same density and temperature. In particular, it vanishes at x= 0, and one hasg₂共0 ; 0兲= 0. However, the correlation func- tions that contain creation and annihilation operators at dif- ferent points, such as 共x1;x₂; 0兲 in Eq. 共20兲, are not the same as for noninteracting fermions. This happens because bosonic operators when written in terms of fermionic opera- tors, contain a “string” which ensures proper commutation relations.

In the Appendix we derive a representation of共x1;x₂; 0兲 as a Fredholm-type determinant, which can be easily evalu- ated numerically. Combining this representation with Eq.

共20兲, we evaluate g₂共x;t兲 after various expansion times nu- merically. The results for zero temperature are shown in Fig.

3, while the results for finite temperature kBT=␮

⬇1.2共␲បn_1D兲²/2mare shown in Fig.4. In spite of a consid- erable change in the temperature, there is no qualitative change in the behavior ofg₂共x;t兲. The qualitative behavior of FIG. 3. Two-point density correlation functiong₂共x;t兲of a zero- temperature strongly interacting 1D Bose gas 共Tonks-Girardeau limit兲for different timestafter the release of the gas from the trap.

Different curves correspond to t= 0 共solid兲, t= 0.25m/共បn_1D² 兲 共dashed兲, andt=m/共បn_1D² 兲 共dotted兲.

FIG. 4. Two-point density correlation function g₂共x;t兲 for the same parameters as in Fig.3but for a finite temperature, k_BT=␮

⬇1.2共␲បn_1D兲²/2m.

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g₂共x;t兲in Figs.3and4is in agreement with Eq.共44兲below and␭C⬃n_1D⁻¹ for the Tonks-Girardeau gas.

C. General remarks about 1D case

Before concluding this section we would like to provide a qualitative analysis of the evolution of the density correlation functiong₂共x;t兲as a function of the expansion timet.

The general structure of the two-particle density matrix 共x₁;x₂; 0兲of a 1D Bose gas is shown schematically in Fig.

5. Because of the Bose symmetry,共x1;x₂; 0兲is represented in the共x1,x₂兲plane by two infinite perpendicular “bands” of a typical transverse size ␭C共correlation length兲. Asymptoti- cally, asx₁→⫾⬁ andx₂=⫾x₁, →n_1D² . There are several possible cases of atomic correlations near the point x₁=x₂

= 0 in a trapped 1D gas. In general, at t= 0, we have

␳共0 ; 0 ; 0兲=n_1D² g₂共0 ; 0兲. In the case of the Tonks-Girardeau gas of impenetrable bosons 关46,86兴 g₂共0 ; 0兲⬅g₂^TG共0兲= 0 (at zero temperature g₂^TG共x兲= 1 −关sin共␲ⁿ1Dx兲/共␲ⁿ1Dx兲兴² 关46兴).

Another possibility is a weakly interacting degenerate gas 共quasicondensate兲, where g₂共0 ; 0兲⬅g₂^QC共0兲⬇1 关77,88,96,100,101兴. Finally, the 1D Bose gas can be nonde- generate 共thermal兲, in which caseg₂共0 ; 0兲⬅g₂^ND= 2. As the interparticle distance grows, the density correlation function quite rapidly approaches its asymptotic value g₂共x→⬁; 0兲

= 1 at the distances of the order of␭C.

One can show that time-dependent density correlation function can be written as

g₂共x;t兲= 1 +␬共␭C,x,t兲+关g2共0;0兲− 1兴h共␭C,x,t兲. 共44兲 The first term共unity兲stems from the band of nonzero values of aligned along the linex₁=x₂ 共see Fig.5兲. It represents the density correlation function of an ideal gas of distin- guishable particles at equilibrium. The second term,

␬共␭C,x,t兲, reflects the Bose-Einstein statistics of the atoms and appears due to the second “band” along x₂= −x₁. Its maximum value,␬共␭C, 0 ,t兲, increases from 0 to 1 on a typi- cal time scale ⬃m␭C2/ប. As 兩x兩 grows, this term asymptoti- cally approaches 0 on a length scale given by␭C. The third term describes washing out of initial short-range 共micro- scopic兲 correlations. The maximum value of h共␭C,x,t兲 is reached at x= 0, it decreases from 1 to 0 on a time scale

⬃m␭C2/ប, andh共␭C,x,t兲⬇0 if兩x兩Ⰷ␭C. In the course of free evolution, the density correlation properties of an expanding Bose gas become similar to that of an ideal Bose gas at temperature kBT⬃ប²/共m␭C2兲.

IV. 2D BOSE GASES BELOW THE BEREZINSKII- KOSTERLITZ-THOULESS TEMPERATURE Let us now discuss the properties of density ripplesj in expanding 2D clouds. Recently 2D condensates have been realized experimentally in several groups 关8,15,68,102,103,105兴. Reduced dimensionality has dramatic effect on thermal fluctuations. In the case of 2D Bose gases there is no true long-range order for any finite temperature 关51兴. For uniform 2D Bose clouds at sufficiently low tem- peratures, the two-point correlation function behaves at large distances as 关75,85,106,107兴

具␺^ˆ†共r,0兲␺^ˆ共0,0兲典 ⬇n_2D

冉

^␭r2D

冊

^{␩ for rⰇ ␭2D. 共45兲}

For weakly interacting 2D Bose gas at small temperatures, one can evaluate parameters of Eq. 共45兲 from microscopic theory. The dimensionless parameter characterizing weak- ness of interactions is written as关85,107兴

˜g=as

冑

^8␲m_ប^{␻⬜Ⰶ1. 共46兲}

The exponent␩ ^{in Eq.}共45兲equals关85,107兴

␩^{= T}

T_dⰆ1 for k_BTⰆk_BT_d=2␲ប²n_2D

m , 共47兲 and ␭2D equals the de Broglie wavelength of thermal phononsបc/共kBT兲atkBTⰆ␮, and the two-dimensional heal- ing length␰2D=ប/

冑

m␮at high temperatureskBTⰇ␮.

Equation共45兲remains valid for␩smaller than

␩c= 1/4, 共48兲

at which point the BKT关6,7,120兴transition takes place due to proliferation of vortices, and correlation functions start to decay exponentially with distance.

Such a transition for ultra cold 2D Bose gases has been observed recently 关8,68,105兴, and its microscopic origin has been elucidated. Experiments of Ref.关8兴studied interference of two independent 2D Bose clouds, which requires imaging along the “in-plane” direction and inevitably leads to aver- aging over inhomogeneous densities. Study of the spectrum of density ripples in expanding clouds with imaging in trans- verse direction 共as done in Ref. 关68兴兲 avoids this problem altogether and can provide access to properties of correla-

ΛC

TG QC

ND 4 2 0 2 4 4

2 0 2 4

x1arb.units x2arb.units

0 1 2

FIG. 5. Density plot of the two-particle density matrix 共x₁;x₂; 0兲of a 1D Bose gas, see Eqs.共11兲and共17兲 共the density bar represents thescale in units ofn_1D² 兲. Initially共att= 0兲the bosonic system can be a Tonks-Girardeau gas共TG兲or a weakly interacting quasicondensate 共QC兲 or a nondegenerate 共ND兲 thermal gas. The central共x₁⬇x₂⬇0兲part of the two-particle density matrix in these cases is shown in three respective insets. The bar shows the typical correlation scale␭C.

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tions at fixed density. The interplay between the BKT transi- tion and the effects of the external confinement is a rather complicated question even for weakly interacting Bose gas 关103,104,109,108兴, and we will only discuss the uniform case here.

Even for weak interactions, one cannot use quasiconden- sate theory to analytically describe correlations as functions of microscopic parameters in the vicinity of the BKT transi- tion or to predict the transition temperature and has to resort to fully numerical methods 关110兴. Nevertheless, the factor- ization property 关Eq. 共25兲兴 remains valid for large-distance behavior of correlation functions for all␩below the critical value 1/4 since large-distance fluctuations of the phase are still described by the Gaussian theory. Using that together with Eq. 共45兲, we will now obtain the prediction for the spectrum of density ripples which is valid as long as only points with relative distances much larger than ␭2Dcontrib- ute significantly to the integral in Eq. 共23兲. We will show below that this regime is realized if

冑

^បtm Ⰷ ␭2D. 共49兲 We introduce a dimensionless variable

y=បq²t

m , 共50兲

and use expression Eq. 共45兲 for allr. Using symmetries of the resulting integral, the expression for具兩␳共q兲兩²典is written as 具兩␳共q兲兩²典 ⬇n_2D² ␭_2D²

冉

^m␭បt2D2

冊

^{1−␩F共␩,y兲, 共51兲}

whereF共␩^,y兲is a dimensionless function defined by 关125兴 F共␩,y兲= 4

y^1+␩

冕

0

⬁

drxcosrx

冕

0

⬁

dry

⫻

再 ^{冋 冑}

^{共rx+y兲2+rx2r2y+}

^冑

^{r共ry2x−y兲2+ry2}

^册

^{␩− 1}

冎

^.

共52兲 We find that the spectrum of density ripples remains self- similar in the course of expansion and the shape of the spec- trum is a function of ␩ only. Plots ofF共␩,y兲 for three dif- ferent values of ␩ are shown in Fig. 6 and have a similar structure. Positions of maxima and minima are very well described by Eq. 共39兲, where the upper 共lower兲 sign corre- sponds to thenth maximum共minimum兲. In Eq.共23兲typical distances which contribute to 具兩␳共q兲兩²典 near its maximum at y⬇␲can be estimated as⬃

冑

_បt/m, which leads to condition 共49兲. Note however, that self-similarity starts breaking down for sufficiently largeyeven when condition 共49兲is satisfied.

Scaling of the magnitude of 具兩␳共q兲兩²典 with time in the self-similar regime can be used to extract ␩. For example, the integral of具兩␳共q兲兩²典from zero to its first minimum scales with time as

冕

0 冑2␲m/បt

dq具兩␳共q兲兩²典⬀t^1/2−␩, 共53兲 and the exponent changes considerably as␩changes from 0 to the critical value 1/4

For small ␩, one can derive an expansion ofF共␩^,y兲as F共␩^,y兲= 4

y^1+␩关␩^f1共y兲+␩^2f2共y兲+␩^3f3共y兲+¯兴, 共54兲 where f₁共y兲can be evaluated analytically as

f₁共y兲= 2␲^sin2y

2. 共55兲

The term f₂共y兲leads to a finite value of F共␩^,y兲 at the first minimum. By including effects of f₂共y兲 and f₃共y兲, one can derive

F共␩^,2␲兲 F共␩^,␲兲 ⬇ 1

2^␩共1.19␩^{+ 0.38}␩²兲 for ␩Ⰶ1, 共56兲 which coincides with the direct numerical evaluation up to 2.5% for␩^{= 0.25.}

For weakly interacting uniform 2D Bose gases at low temperatures, one can also obtain predictions which are not limited by Eq.共49兲. Under condition

n_2D␰2D

2 Ⰷ1 共57兲

an extension of Bogoliubov theory to 2D quasicondensates describes correlations at all distances 关77兴. Such a theory is valid up to temperatures of the order

kBT

␮ ^log kBT

␮ ^⬃n2D␰2D2 Ⰷ1 共58兲 and predicts the exponent 共47兲. The correlation function is written as

0 5 10 15 20

0.0 0.5 1.0 1.5 2.0

F(,y)

y

FIG. 6. Dependence of universal functions F共␩^,y兲 on y

=បq²t/mplotted for three different values of correlation exponents

␩. Under condition共49兲functionsF共␩^,y兲determine the self-similar shape of the spectrum of density ripples according to Eq. 共51兲. Curves from top to bottom correspond to ␩^{= 0.25} 共solid, the Berezinskii-Kosterlitz-Thouless point兲, ␩^{= 0.15} 共dashed兲, and ␩

= 0.10共dotted兲.

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