Reentrant behavior of the breathing-mode-oscillation frequency in a one-dimensional Bose gas
Exciting temporal oscillations of the density distribution is a high-precision method for probing ultracold trapped atomic gases. Interaction effects in their many-body dynamics are particularly puzzling and counter-intuitive in one spatial dimension (1D) due to enhanced quantum correlations. We consider 1D quantum Bose gas in a parabolic trap at zero temperature and explain, analytically and numerically, how oscillation frequency depends on the number of particles, their repulsion and the trap strength. We identify the frequency with the energy difference between the ground state and a particular excited state. This way we avoided resolving the dynamical evolution of the system, simplifying the problem immensely. We find an excellent quantitative agreement of our results with the data from the Innsbruck experiment [Science 325, 1224 (2009)].
pacs:03.75.Kk 03.75.Hh, 67.85.-d,
All existing ultracold-gas experiments are carried out with systems which are spatially inhomogeneous due to the presence of an external confining potential Pitaevskii and Stringari (2003); Pethick and Smith (2008). Exciting temporal oscillations of the gas density distribution in such a confined geometry is a basic tool for investigating the spectrum of collective excitations and phase diagram Mewes et al. (1996); Jin et al. (1996); Chevy et al. (2002); Kinast et al. (2004, 2005); Altmeyer et al. (2007); Riedl et al. (2008). One-dimensional (1D) gases have their own specifics: Enhanced quantum correlations affect their collective excitations spectrum drastically, masking out signatures of the Bose and Fermi statistics of the constituent particles Gogolin et al. (1999); Giamarchi (2004). The exactly solvable homogeneous Lieb-Liniger gas model Lieb and Liniger (1963) is a paradigmatic demonstration of that statement. There bosons interact through a -function potential of strength Increasing suppresses spatial overlap between any two bosons. This leads to a many-body excitation spectrum identical to that of a free Fermi gas in the limiting case of infinite repulsion, , known as the Tonks-Girardeau (TG) gas Girardeau (1960). The presence of an external parabolic potential makes the low-lying part of excitation spectrum to be discrete. The first excited state of the gas, a dipole mode, is interaction independent. It is associated with the center-of-mass oscillations at a trap frequency . The second excited state is doubly degenerate for and One mode with the interaction-independent frequency comes from center-of-mass oscillations. Another mode is called the breathing (or compressional) mode. Being excited by a small instantaneous change of the trapping frequency , this mode has the frequency which depends on the number of particles in the trap, and the gas temperature
Experimental investigations of the breathing mode oscillations in 1D ultracold-gas experiments have been reported by several groups Moritz et al. (2003); Haller et al. (2009); Fang et al. (2014). It was found that the frequency ratio as a function of the interaction strength, goes through two crossovers: from the value down to and then back to (see, e.g., Fig. 2), as the system goes from non-interacting to weakly interacting, and then from weakly interacting to strongly interacting regime Haller et al. (2009). The latter crossover has been described theoretically for going to infinity, by the approach based on the local density approximation (LDA) Menotti and Stringari (2002). A description of the former crossover has been done only numerically for the few particles: by using the multilayer multiconfiguration time-dependent Hartree method Schmitz et al. (2013) and using numerical diagonalization Tschischik et al. (2013). Experiments Moritz et al. (2003); Fang et al. (2014) were done in the regime of weak coupling, for which is expected as goes to infinity at zero temperature. To what extent are the observed deviations from the value due to finite and is an open question. Answering it paves a way towards understanding interaction effects in dynamics and thermalization of 1D quantum gases.
In this Rapid Communication we present the analytic and numerical results for the breathing-mode-oscillation frequency in the repulsive Lieb-Liniger gas in a parabolic trap of frequency . Using the Hartree approximation we explain how the decrease of from the value down to as the interparticle repulsion increases is linked to a transition from the Gaussian Bose–Einstein condensate (BEC) to the Thomas–Fermi (TF) BEC regime. By further increasing the repulsion strength, goes back to the value This return is associated with the transition from the TF BEC to the Tonks-Girardeau regime and is described within local density approximation. We perform extensive diffusion Monte Carlo simulations for a gas containing up to particles. As the number of particles increases, predictions from the simulations converge to the ones from the Hartree and LDA in their respective regimes. This makes our results for applicable for arbitrary number of particles and value of the repulsion strength. We find an excellent quantitative agreement with the data from the Innsbruck experiment Haller et al. (2009). We also estimate relevant temperature scales for the Palaiseau experiment Fang et al. (2014).
Model and sum rules. — The model we consider is the Lieb-Liniger gas of repulsive bosons in a parabolic trap. The Hamiltonian for particles is
Here is the particle mass, and is the particle potential energy in a parabolic trap. The length scales in the model are set by the -wave scattering length , related to the coupling constant and by the harmonic oscillator length . Three zero-temperature quantum regimes shown schematically in Fig. 1 are identified for model (1) based on its thermodynamic and local correlation properties Petrov et al. (2004).
We employ a sum rule approximation, which makes it possible to get for arbitrary , , and from ground-state properties of Hamiltonian (1) solely Abraham and Bonitz (2014). More specifically, is obtained by calculating the response of the gas to a change of the trap frequency:
where , and is the center-of-mass coordinate. The average is taken with respect to the ground-state wave function of Hamiltonian (1). Neglecting in amounts to replacing with , the latter operator being used in Ref. Menotti and Stringari (2002).
By changing one excites many modes rather than a single breathing mode. These modes cause given by Eq. (2) to be different from the breathing mode frequency. Their contribution could be diminished by a proper choice of How good is our choice, for that purpose is seen by comparing the exact spectrum of model (1) for with given by Eq. (2) for arbitrary value of . We found that given by Eq. (2) with misses up to of the deviation from value, while with it misses at most.
Gaussian BEC to TF BEC crossover. — We approximate (normalized to ) with the Hartree variational wave function for This function is found by minimizing the functional with respect to The procedure amounts to solving the Hartree eigenvalue equation (same as the Gross-Pitaevskii equation)
for the minimal possible . Here and and are dimensionless length and energy given in units of and respectively. The Hartree parameter reads
The ground-state density distribution found with respect to the Hartree state is and the average of the operators and is . Substituting this expression into Eq. (2) and taking into account that we find that depends on , , and through a single parameter within the Hartree approximation.
We explore the dependence of on Deep in the Gaussian BEC regime, , we use a series expansion in the harmonic oscillator wave functions for and solve Eq. (3) perturbatively. We get
where . Perturbation theory for the many-body wave functions of Hamiltonian (1) extends the validity range of Eq. (5) to arbitrary Note that the Hartree approximation is only valid in the large limit. Indeed, Eq. (3) in which is replaced with minimizes the energy functional for any . This implies Being substituted into Eq. (2) both and lead to Eq. (5) with replaced by that is, to the result which is correct in the large limit only. Note also that the ground-state wave function of Hamiltonian (1) obtained with perturbation theory and used for the sum rule (2) with gives Eq. (5) correctly. This supports our approach to DMC simulations (detailed later in the Rapid Communication), in which we rely on Eq. (2) and .
In the case Eq. (3) results in an inverted parabola density profile, characteristic of the TF BEC regime
In the case of arbitrary we solve Eq. (3) numerically. The plot of as a function of is shown in Fig. 2. We observe a smooth crossover between the (TF BEC) and (Gaussian BEC) regimes. We find that at defined as a reference point separating these regimes (see Fig. 1).
TF BEC to TG crossover. — This crossover is associated to an interplay of the parameters and ; see Fig. 1. It may not be captured within the Hartree approximation, which does not contain as a parameter independent of . Instead, we may use LDA Menotti and Stringari (2002). It is only valid in the large limit, and is based on the assumption that the local chemical potential at a point is equal to the chemical potential in a homogeneous system that has the same density Therefore for and vanishes for in model (1). Here is the Thomas-Fermi radius of the gas cloud, whose value is set by the normalization condition The dependence of on in the homogeneous Lieb-Liniger model (Eq. (1) with ) was found in Ref. Lieb and Liniger (1963). Using Eq. (2) with we get readily. The result depends on and through a single parameter within LDA Petrov et al. (2000); Dunjko et al. (2001); Menotti and Stringari (2002).
In the limiting case of impenetrable bosons, , the local chemical potential is equal to the Fermi energy, This leads to the semicircular LDA density profile , characteristic of the TG regime, see Fig. 1. Excitation spectrum of model (1) deep in the TG regime, can be found perturbatively in . For that we use a mapping from the gas of strongly repulsive bosons to that of weakly attractive fermions Cheon and Shigehara (1999). A perturbative solution for the ground state energy is given in Ref. Paraan and Korepin (2010). Analyzing the excited states results in the expansion Zhang et al. (2014)
where is calculated for all
In the case , local chemical potential is of the Gross–Pitaevskii form, and the shape of the density profile is given by Eq. (6), characteristic of the TF BEC regime. This implies
We find that at defined as a reference point separating these regimes (see Fig. 1).
DMC simulations. — How does depend on model parameters for small , and how good are the Hartree approximation/LDA in that case? To answer these questions quantitatively we perform large-scale numerical simulations based on the diffusion Monte Carlo (DMC) algorithm Boronat and Casulleras (1994). This algorithm amounts to solving many-body Schrödinger equation in imaginary time and makes it possible to calculate ground-state energy to arbitrarily high precision. The convergence rate of the simulations can be enhanced greatly by doing an importance sampling with a guiding wave function . We use with the parameter minimizing the variational energy. This function is known to work very well in a number of 1D systems Astrakharchik and Giorgini (2003, 2006); Astrakharchik and Brouzos (2013); Garcia-March et al. (2013).
We use the sum rule approximation (2), which only requires the knowledge of the ground state properties of the model. For the number of particles ranging from to we pushed DMC to its limits to perform high-accuracy simulations. Specifically, up to CPU hours were used to get each data point for . The results obtained are shown as a function of in Fig. 2 and of in Fig. 3. Dashed lines interpolating the data points are obtained by using a Padé approximation and Eqs. (5) and (7) for the asymptotic values of We see in Fig. 2 that the Hartree and DMC curves are indistinguishable from each other for at any . The minimal value of at which these two curves are close to each other decreases with increasing It reaches the value and the minimal value of reaches at We may thus locate the TF BEC regime of the model from Fig. 2 by setting where . Figure 3 shows the same data points as in Fig. 2, as a function of the LDA parameter . Evidently, LDA and DMC curves coincide for at any .
Comparison with experiments. — The Innsbruck group loaded three-dimensional (3D) BEC of atoms into an array of 1D tubes formed by retro-reflected laser beams. The frequency of the external parabolic potential along the tube direction is and the maximal number of atoms per tube is about The Innsbruck group data shown in Figs. 2 and 3 of the present Rapid Communication are taken from Figs. 2 and 3(a) of Ref. Haller et al. (2009) for . We see that DMC simulations for and are compatible with the experimental data points. This match suggests that the temperature effects play little role in the experiment. The temperature of the 1D gas can be estimated by assuming that it is inherited from the 3D BEC, whose temperature is between and Note1 (). The degeneracy temperature of an ideal Bose gas, defined as ( is the Boltzmann constant) is about We see that is at least twice as low as
The ETH experiment examined what happens with the breathing oscillations if the temperature of the 3D BEC prepared to be loaded into an array of 1D tubes gets higher Moritz et al. (2003). The parameters and correspond to the TF BEC regime of the 1D gas. It was found that the breathing mode persists and grows from the value to (with the uncertainty about ). These findings could be interpreted as the increase of due to the increase of the temperature of the 1D gas, assuming that it is in thermal equilibrium.
The Palaiseau group prepared a single tube with atoms using atom-chip setup Fang et al. (2014). The number of atoms in the tube is given in Table 1, and Data points from the Palaiseau group shown in Figs. 2 and 3 of the present Rapid Communication are taken from Fig. 3(a) of Ref. Fang et al. (2014). The parameters and correspond to the TF BEC regime for all data points. We see that the frequencies for the first five of them match our theoretical predictions within the error bars. The frequencies for the last two of them are higher than the theory predicts.
We get the gas temperature by comparing the height of the density profile in the tube center, calculated theoretically Note1 () with the one measured in experiment Fang et al. (2014). The values of and are given in Table. 1. According to Ref. Bouchoule et al. (2007), finite-temperature effects are relevant above for the range of parameters chosen in the experiment. We see from Table 1 that increases monotonously from the value for the first data point to for the last one. Note that is nearly three times larger than (and, therefore, than ) for all data points. Thus, may not define the crossover temperature in the experiment Fang et al. (2014).
Summary. — We investigated the breathing mode frequency in model (1) at zero temperature by identifying the energy difference between a particular excited state and the ground state. This way we avoided dealing with the dynamical evolution of the initial state of the system. Our theory predicts the reentrant behavior of and fully explains the recent experiment Haller et al. (2009) for the repulsive interparticle interaction. The extension of the present theory to the finite-temperature case requires a separate study. The existing phenomenological approaches Fang et al. (2014); Hu et al. (2014) are yet to be tested against the predictions from the exact dynamical evolution of the system.
Acknowledgements.We thank the Palaiseau group Fang et al. (2014) for providing access to their experimental data and the Innsbruck group Haller et al. (2009) for numerous enlightening discussions. The Barcelona Supercomputing Center (The Spanish National Supercomputing Center – Centro Nacional de Supercomputación) is acknowledged for the provided computational facilities. The work of A.Iu.G. was supported by grant from Region Ile-de-France DIM NANO-K. G.E.A. acknowledges partial financial support from the DGI (Spain) Grant No. FIS2011-25275 and Generalitat de Catalunya Grant No. 2009SGR-1003.
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