Violation of self-similarity in the expansion of a one-dimensional Bose gas
P. Pedri,1,2L. Santos,1 P. O¨ hberg,3 and S. Stringari2 1
Institut fu¨r Theoretische Physik, Universita¨t Hannover, D-30167 Hannover, Germany 2Dipartimento di Fisica, Universita` di Trento and BEC-INFM, I-38050 Povo, Italy
3Department of Physics, University of Strathclyde, Glasgow G4 0NG, Scotland 共Received 29 April 2003; published 1 October 2003兲
The expansion of a one-dimensional Bose gas after releasing its initial harmonic confinement is investigated employing the Lieb-Liniger equation of state within the local-density approximation. We show that during the expansion the density profile of the gas does not follow a self-similar solution, as one would expect from a simple scaling ansatz. We carry out a variational calculation, which recovers the numerical results for the expansion, the equilibrium properties of the density profile, and the frequency of the lowest compressional mode. The variational approach allows for the analysis of the expansion in all interaction regimes between the mean-field and the Tonks-Girardeau limits, and in particular shows the range of parameters for which the expansion violates self-similarity.
DOI: 10.1103/PhysRevA.68.043601 PACS number共s兲: 03.75.Kk, 05.30.Jp
I. INTRODUCTION
The experimental achievement of Bose-Einstein conden-sation共BEC兲 关1兴has aroused a large interest in the physics of ultracold atomic gases. Among the topics related to this field, the physics of low-dimensional atomic gases has recently attracted significant attention. The development of the trap-ping techniques has allowed for the realization of very an-isotropic geometries, where the confinement is so strong in one or two dimensions that at low temperatures the transver-sal motion is ‘‘frozen,’’ and does not contribute to the dy-namics of the system. In this way two- 关2–5兴 and one-dimensional 关2,6,7兴systems have been accomplished. Low-dimensional gases present significantly different properties compared to the three-dimensional ones. A remarkable ex-ample is provided by the existence of quasicondensation 关8 –11兴, whose effects have been recently observed experi-mentally 关12兴.
During the last years, the one-dimensional 共1D兲 Bose gases have been the subject of growing interest, in particular, the limit of impenetrable bosons 关13兴, which behave to a large extent as a noninteracting Fermi system, acquiring some remarkable properties. The conditions for the experi-mental realization of strongly correlated 1D gases are rather restrictive 关8,14兴, since a large radial compression, a suffi-ciently small density, and eventually a large scattering length are needed. Fortunately, recent experimental developments have opened perspectives in this sense. Especially interesting is the possibility to modify at will the interatomic interac-tions by means of Feshbach resonances关15兴and the capabil-ity of loading an atomic gas in an optical lattice关16兴.
From the theoretical side, the physics of 1D Bose gases was first investigated by Girardeau关13兴, who considered the limit of impenetrable bosons, also called Tonks-Girardeau 共TG兲gas, pointing out a nontrivial relation with the physics of ideal Fermi gases. This analysis was later extended by Lieb and Liniger 关21兴, who solved analytically the problem for any regime of interactions, using Bethe ansatz. Yang and Yang关22兴extended the analysis including finite temperature effects. Recently, the experimental accessibility of trapped
gases have encouraged the investigation of the harmonically trapped case. The Bose-Fermi 共BF兲 mapping has been em-ployed to the case of an inhomogeneous gas in the TG limit 关23兴. However, there is unfortunately, to the best of our knowledge, no exact solution for arbitrary interaction strength in the case of trapped gases. The problem of the equilibrium of a trapped gas can be analyzed using a local-density approximation and employing the Lieb-Liniger共LL兲 equation of state locally to evaluate the equilibrium density profiles 关17兴. A similar formalism has been recently em-ployed to analyze the collective oscillations in the presence of harmonic trapping 关18兴. Both Refs. 关17兴 and 关18兴 have shown the occurrence of a continuous transition from the mean-field共MF兲regime to the TG one as the intensity of the interaction is varied. Recently, Gangardt and Shlyapnikov 关19兴have discussed the stability and phase coherence of 1D trapped Bose gases. These authors have analyzed the local correlation properties and found that inelastic decay pro-cesses, such as three-body recombination, are suppressed in the TG regime, and intermediate regimes between MF and TG. This fact opens promising perspectives towards the ac-complishment of strongly interacting 1D Bose gases with large number of particles. This analysis have been very re-cently extended to the case of finite temperatures关24兴.
The expansion of a one-dimensional Bose gas in a guide was analyzed in Ref. 关20兴by means of a hydrodynamic ap-proach based on the local LL model. The expansion dynam-ics was shown to be different for different interaction strengths, and its analysis could be employed to discern be-tween the TG and MF regimes. In particular, the self-similar solution is violated.
In this paper, we extend the analysis of Ref. 关20兴by in-troducing a variational approach, which permits us to study the asymptotic regime at large expansion times. This method is shown to be in excellent agreement with previous direct numerical simulations, and additionally permits us to recover the results of Refs.关17,18兴. More importantly, our variational approach allows us to determine the regime of parameters for which the self-similarity of the expansion is violated.
the local LL model which we employ to analyze the expan-sion dynamics. In Sec. III we briefly discuss the numerical results obtained in Ref.关20兴. In Sec. IV we present a varia-tional approach which allows us to investigate in detail the expansion dynamics for arbitrary regimes of parameters. Fi-nally we conclude in Sec. V.
II. LOCAL LIEB-LINIGER MODEL
We analyze in the following a dilute gas of N bosons confined in a very elongated harmonic trap with radial and axial frequencies andz (Ⰷz). We assume that the transversal confinement is strong enough so that the interac-tion energy per particle is smaller than the zero-point energy ប of the transversal trap. In this way, the transversal
dy-namics is effectively frozen and the system can be consid-ered as dynamically 1D. In this section we briefly review the formalism introduced in Ref. 关17兴.
We assume that the interparticle interaction can be ap-proximated by a ␦ function pseudopotential. Therefore the Hamiltonian that describes the physics of the 1D gas becomes
Hˆ1D⫽Hˆ1D 0 ⫹
兺
j⫽i Nmz2zi2
2 , 共1兲
where
Hˆ1D0 ⫽⫺ ប 2
2m j
兺
⫽1N
2
zj2⫹g1Di
兺
⫽1N⫺1
兺
j⫽i⫹1N
␦共zi⫺zj兲 共2兲
is the homogeneous Hamiltonian in the absence of the har-monic trap, m is the atomic mass, and g1D⫽⫺2ប2/ma
1D. The scattering problem under one-dimensional constraints was analyzed in detail by Olshanii关14兴, and it is character-ized by the one-dimensional scattering length a1D ⫽(⫺a2/2a)关1⫺C(a/a)兴, with a the three-dimensional scattering length, a⫽
冑
2ប/m is the oscillator length in the radial direction, and C⫽1.4603 . . . . As shown by Lieb and Liniger关21兴, the homogeneous Hamiltonian Hˆ1D0 can be diagonalized exactly by means of Bethe ansatz 关25兴. In the thermodynamic limit, a 1D gas at zero temperature with a given linear density n is characterized by the energy per particle⑀共n兲⫽ ប 2
2mn
2e„␥共n兲…, 共3兲
where␥⫽2/n兩a1D兩. The function e(␥) fulfills
e共␥兲⫽ ␥ 3
3共␥兲
冕
⫺1 1g共x兩␥兲x2dx, 共4兲
where g(x兩␥) and(␥) are the solutions of the LL system of equations 关21兴
g共x兩␥兲⫽ 1 2⫹
1 2
冕
⫺11 2共␥兲
2共␥兲⫹共y⫺x兲2g共y兩␥兲d y , 共5兲
共␥兲⫽␥
冕
⫺11
g共x兩␥兲dx. 共6兲
We assume next that at each point z the gas is in local equi-librium, with local energy per particle provided by Eq. 共3兲. Then, one can obtain the corresponding hydrodynamic equa-tions for the density and the atomic velocity
tn⫹
z共nv兲⫽0, 共7兲
tv⫹v
zv⫽⫺
1 m
z
冋
le共n兲⫹ 1 2mz2z2
册
, 共8兲where
le共n兲⫽
冉
1⫹n
n
冊
⑀共n兲 共9兲is the Gibbs free energy per particle. The hydrodynamic de-scription holds if the local chemical potential, related to the interaction energy in the homogeneous system, is much larger than the kinetic energy associated to the density modu-lations. It assumes, in particular, that the density of the gas varies smoothly within the typical distance fixed by the heal-ing length 共local-density approximation兲. Note that for the case of n兩a1D兩→⬁, one obtains le(n)⫽g1Dn, retrieving the 1D Gross-Pitaevskii equation 关26兴, whereas for the case n兩a1D兩→0, one getsle(n)⫽2ប2n2/2m, and the equation
of Ref. 关27兴is recovered. The system has only one control parameter 关8,17,18兴, namely, A⫽N兩a1D兩2/a
z
2, where a
z
⫽
冑
ប/mz is the harmonic-oscillator length in the z direc-tion. The regime AⰇ1 corresponds to the MF limit, in which the stationary-state density profile has a parabolic form. On the other hand, the regime AⰆ1 corresponds to the TG re-gime, which is characterized by a stationary-state density profile with the form of a square root of a parabola.III. NUMERICAL RESULTS
In Ref.关20兴, Eqs.共3兲,共5兲,共6兲, and共9兲were employed to simulate numerically the expansion of a 1D gas in the frame-work of the hydrodynamic formalism. The expansion follows the sudden removal of the axial confinement, while the radial one is kept fixed. In particular, it was observed that during the expansion the density profile is well described by the expression
n共z,t兲⫽nm共t兲
冋
1⫺冉
z
R共t兲
冊
2册
s(t), 共10兲
where nm(t) provides the appropriate normalization, R(t) is
[image:2.612.315.561.152.275.2]At this point we discuss the physics behind this violation of the self-similarity. If the local chemical potential presents a fixed power-law dependence on the density, le⬀n, it is easy to show from the hydrodynamic equations 共7兲 and共8兲 that there exists a self-similar solution of the form n ⫽(n0/b)关1⫺(z/bR)2兴1/, where b¨⫽z
2
/b⫹1. For the par-ticular case of the TG gas, the scaling law can be also ob-tained from the exact BF mapping关20兴. However, sincele
is obtained from the LL equations, the dependence ofle on
n is quadratic for a low density and linear for a large one. Therefore,ledoes not fulfill a fixed power-law dependence
during the expansion, and the self-similarity is violated. In particular, as the expansion proceeds the whole system ap-proaches the low-density regime, and consequently, the ex-ponent s decreases monotonically. In the following section, we analyze in more detail this effect.
IV. VARIATIONAL CALCULATION
In this section, we complete our understanding of the ex-pansion of a one-dimensional Bose gas in a guide by means of a variational ansatz using a Lagrangian formalism. The Lagrangian density for the system is of the form
L⫽⫺mn t ⫺
1 2mn
冉
z
冊
2
⫺12mz2z2n⫺共n兲n, 共11兲
where the velocity field is defined as v⫽/z. The equa-tions of motion are obtained from the functional derivation of the actionA⫽兰Ldtdz:␦A/␦⫽0 共continuity equation兲, ␦A/␦n⫽0 共which after partial derivation with respect to z provides the Euler equation兲. From the numerical results we have observed that the density is at any time well described by Eq. 共10兲. Therefore, we assume the following ansatz for the density:
n⫽C共s兲 b
冉
1⫺z2
R2b2
冊
s
, 共12兲
where b and s are time-dependent variables, R is the initial Thomas-Fermi radius, and C(s) is related to the normaliza-tion to the total number of particles. For the field we consider the following form:
⫽1 2␣z
2⫹1 4z
4, 共13兲
where␣ and are time-dependent parameters. We stress at this point that in the analysis of the self-similar expansion of a BEC 关28 –30兴 a quadratic ansatz 共in z) for the field provides the exact solution. However, for the problem under consideration in this paper, it is necessary to include higher-order terms to account for the violation of the self-similarity. We have checked that terms of higher order than z4introduce only small corrections, and therefore we reduce to the form of Eq.共13兲.
We are interested in the dynamics of the parameters b and s, related to the size and the shape of the cloud, respectively. Integrating the Lagrangian density in z, L⫽兰Ldz, one finds a Lagrangian for the above-mentioned parameters:
L共␣˙ ,␣,˙ ,,b,s兲⫽mNR 2
2
再
⫺␣˙ b2 2s⫹3⫺
3 2
˙ b4 共2s⫹5兲共2s⫹3兲
⫺ ␣
2b2
2s⫹3⫺2
␣b4 共2s⫹5兲共2s⫹3兲
⫺ 
2b6
共2s⫹7兲共2s⫹5兲共2s⫹3兲⫺ b2z2 2s⫹3
冎
⫺
冕
dzn⑀共n兲. 共14兲We perform a gauge transformation 关31兴
L共t,q,q˙兲→L共t,q,q˙兲⫹d
dtg共t,q兲, 共15兲
where
g共t,q兲⫽mNR 2
2
再
⫺␣b2 2s⫹3⫺
3 2
b4
共2s⫹5兲共2s⫹3兲
冎
. 共16兲The resulting Lagrangian is of the form L ⫽L(␣,,b˙,b,s˙,s). Imposing the conservation laws L/␣ ⫽L/⫽0, we obtain a Lagrangian depending only on the two relevant parameters s and b, of the form L⫽K⫺V, where
V
E1D⫽
冉
B共b,s兲f0共s兲 ⫹
A2b2
关0f0共s0兲兴2 1
2s⫹3
冊
, 共17兲K
E1D⫽
A2共M11b˙2⫹2 M12b˙s˙⫹M22s˙2兲 关0f0共s0兲兴
[image:3.612.56.298.55.258.2]2共2s⫹3兲 , 共18兲 FIG. 1. Time evolution of the exponent s(t) for A⫽0.43,
⫽2(20) kHz, and N⫽200 atoms 关z⫽2⫻(1.8) Hz at t⫽0].
where E1D⫽ប2/2m兩a1D兩2 is the typical energy associated with the interatomic interactions. In Eq. 共17兲, we use the function B(b,s)⫽兰d y (1⫺y2)s⑀„n(y )…/E1D, where we inte-grate over the rescaled axial coordinate y⫽z/Rb. In Eqs. 共17兲 and 共18兲, we define the dimensionless central density 0⫽n0兩a1D兩, where n0 is the initial central density and the parameter s0⫽s(t⫽0). We have additionally employed the auxiliary functions fn(s)⫽兰yn(1⫺y2)sd y , and the
coefficients
M11⫽1, 共19兲
M12⫽ ⫺b
2s⫹3, 共20兲
M22⫽
b2共121⫹186s⫹96s2⫹16s3兲
4共s⫹1兲共2s⫹3兲2共2s⫹5兲2 . 共21兲
From the Lagrangian L, one obtains the corresponding Euler-Lagrange equations for the parameters b and s. In or-der to find the initial conditions0and s0 for the expansion, we have numerically minimized the potential V in the pres-ence of the harmonic trap for different values of A, assuming b⫽1共see Fig. 2兲. When AⰇ1, s0tends to 1, as expected for the MF case. On the contrary, when AⰆ1, s0 tends to 1/2 共TG profile兲. As expected from Ref. 关17兴, for AⰆ1, 0 ⬀A1/2, whereas for AⰇ1, the MF dependence 0⬀A2/3 is recovered.
Our variational approach allows us to calculate the lowest compressional mode, offering an alternative method as the one discussed in Ref.关18兴. Expanding the potential V around the equilibrium solution up to second order in b共see Fig. 3兲, and neglecting for small oscillations the time dependence of s, we obtain
m
2
z
2⫽1⫹ 1 2A2
关0
2f0共s0兲兴2
f2共s0兲 2B
b2兩b⫽1. 共22兲
Our results show a continuous transition from the MF value, m⫽
冑
3z, to the TG one, m⫽2z, in excellentagree-ment with the results obtained by means of a sum rule formalism关18兴.
From the corresponding Euler-Lagrange equations, we have obtained the dynamics of b(t) and s(t). We have checked in all our calculations that the energy and number of particles remain a constant of motion. We have compared the variational results with our simulations based on the exact resolution of the hydrodynamic equations关20兴, obtaining an excellent agreement 共see Fig. 1兲.
We have analyzed the asymptotic value b˙⬁ for different values of A. It is easy to obtain, that for a power-law depen-dence of the local chemical potentialle⬀n, the derivative
of the scaling parameter asymptotically approaches a value b˙⬁⫽
冑
2/z. Therefore a continuous transition from b˙⬁⫽z 共TG兲 to b˙⬁⫽
冑
2z 共MF兲 is expected. We recover this [image:4.612.320.559.55.253.2]dependence from our variational calculations 共see Fig. 4兲. We have analyzed the behavior of s during the expansion dynamics for different values of A. In particular, we have defined the asymptotic ratio ⫽s⬁/s0, with s⬁⫽s(t→⬁) 共see Fig. 5兲. Deeply in the TG regime (AⰆ1) or in the MF one (AⰇ1),⯝1, i.e., in those extreme regimes, the expan-sion is well described by a self-similar solution. However, for intermediate values,⬍1, i.e., the expansion is not self-similar. The self-similarity is maximally violated in the vi-cinity of A⫽1, althoughdeparts significantly from 1 for a range 0.01ⱗAⱗ100. The behavior at large A can be under-stood as follows. If the gas is at t⫽0 deeply in the MF regime, the change of the functional dependence ofle with the density occurs at very long expansion times, when the initial interaction energy of the gas has been fully transferred into kinetic energy. Therefore, for large values of A the self-similarity is recovered.
FIG. 2. Equilibrium values共at t⫽0 before opening the trap兲of the exponent s, as a function of the parameter A. The dashed lines denote the MF limit, s0⫽1, and the TG one, s0⫽1/2.
V. CONCLUSIONS
In this paper, we have extended the analysis of Ref. 关20兴 on the expansion dynamics of a one-dimensional Bose gas in a guide. We have shown that the expansion violates under certain conditions the self-similarity, and in this sense differs significantly from the expansion dynamics of a BEC. We have shown that the problem can be solved by employing the hydrodynamic approach and the local Lieb-Liniger model. We have developed a variational approach based on a La-grangian formalism to study the expansion for any regime of parameters. We have identified the possible physical situa-tions at which self-similarity is violated. This should occur in a rather wide range of parameters (0.01ⱗAⱗ100). The par-ticular properties of the expansion of a gas in the strongly interacting regime could therefore be employed to discern between mean-field and strongly interacting regimes. In ad-dition, the asymptotic behavior of the expanded cloud could be employed to discriminate between different initial inter-action regimes of the system.
Our discussion has been restricted to the analysis of the density properties. In fact the present formalism cannot de-scribe the dynamics of the coherence in the system, i.e., we are limited to the diagonal terms of the corresponding single-particle density matrix. The description of the nondiagonal terms lies beyond the scope of this paper, and requires other techniques of analysis关19,32兴.
ACKNOWLEDGMENTS
We acknowledge support from Deutsche Forschungsge-meinschaft 共SFB 407兲, the RTN Cold Quantum gases, ESF PESC BEC2000⫹, Royal Society of Edinburgh, and the Ministero dell’Istruzione, dell’Universita` e della Ricerca 共MIUR兲. L.S. and P.P. acknowledge the Alexander von Hum-boldt Foundation, the Federal Ministry of Education and Re-search, and the ZIP Programme of the German Government. Discussions with M.D. Girardeau, M. Lewenstein, D.S. Petrov, and G.V. Shlyapnikov are acknowledged. We espe-cially thank C. Menotti for providing us with the data of Ref. 关18兴.
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[image:5.612.55.298.57.254.2]FIG. 4. Expansion velocity. Asymptotic value of b˙ as a function of the parameter A.
FIG. 5. Violation of the self-similarity. Value of ⫽s⬁/s0as a
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关31兴Lagrangian共14兲does not depend on b˙ and s˙, and consequently one obtains the dynamics only for ␣ and  imposingL/b
⫽L/s⫽0, and later on from these two conditions one can obtain the dynamics of b and s. The gauge transformation共15兲 turns out to simplify significantly the calculations.