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ULTRACOLD ATOMIC GASES 19
the phase angle rotating from q1 on one wall to q2 on the other (cf. Fig. 1.1). This would not be possible for systems with a scalar order parameter (as the Ising model), which would rather be separated into two domains. By imposing these boundary conditions, the free energy F of a certain system (for given temperature and system size) acquires a dependence on the pinning angles: F = F(q1, q2). We consider the free-energy dif-ference DF(q) = F( q, q) F(q, q), for a small twisting angle 2q. This difference is an extensive quantity, proportional to the system volume L3. Moreover, it is proportional to the square of hhrfii, the average gradient1 of the order-parameter phase [20]. hhrfii is equal to 2q/L: The total angle difference (2q) divided by the length over which the change takes place (L). The expansion of DF for small q leads to the definition of the helicity modulus, U, through U encodes the response of the system to an imposed phase twist: A large value of U corresponds to a rigid systems, where the phase twist has a large free-energy cost. This definition of U through the small-q expansion of DF(q) is general, and it holds for in-stance for the XY and Heisenberg spin models. In the specific case of the Bose gas, the helicity modulus corresponds to the superfluid density rs, via which follows from the connection between (h¯/m)rf(x) and the superfluid velocity vs (cf. Eq. 1.18). This definition of rs through Eqs. 1.19 and 1.20 (or analogous expressions related to twisted boundary conditions), is only based on equilibrium properties, and it is of fundamental importance for several applications: It has been applied to the ideal Bose gas, leading to the exact expression [20, 21].
for a three-dimensional system in the thermodynamic limit, it is a key ingredient in the formulation of the finite-size scaling theory (cf. Section 3.4.1 and Ref. [22]), and it is at the basis of the measure of the superfluid density in quantum Monte Carlo (cf. Section 3.3.2 and Ref. [23]).
Ultracold atomic gases
The first atomic system used to probe low-temperature physics and the effects of quan-tum mechanics has been liquid helium [24], for which the l transition (cf. Section 1.1) takes place at Tc ’ 2.17 K. The critical temperature for Bose-Einstein condensation of non-interacting bosons with the mass and density of 4He (r ’ 2 1028 m 3) is Tc0 ’ 3.13 K. While being quantitatively different from Tc, the non-interacting theory captures the correct order of magnitude for Bose-Einstein condensation, which motivated the first connection between the phenomena of BEC and superfluidity [19]. Nevertheless, liquid helium is a strongly interacting system: The main features of the interaction potential (the repulsive core for distances below 2.5 Å and the minimum at 2.8 Å) take place at distances comparable to the typical interparticle separation (of the order of 3–4 Å). This modifies the zero-temperature condensate fraction, which is reduced to below 10% (as compared to the non-interacting case, where N0/N = 1 at zero temperature).
Liquid helium has to be compared with the currently available ultracold atomic gases, a different class of systems where Bose-Einstein condensation takes place [25]. The progress in cooling techniques allowed to reach temperatures as low as tens of nK. The transition to the solid phase, which would exist at such low temperature, is avoided by using extremely dilute systems (with typical densities of the order of 1019– 1020 m 3, leading to interparticle distances of approximately 100 nm). These systems are therefore metastable. The major instability mechanism is given by inelastic three-body losses, which are scattering processes resulting into a bound dimer and a third atom carrying away the binding energy. The rate of these collisions sets a finite life-time, typ-ically larger than the other time scales of the system2. Considering the example of 87Rb, the temperature at which the phase-space density rl3th becomes of order one is 102 nK. It is in this temperature regime that the first direct observation of Bose-Einstein condensation has been realized, with clouds of Rb, Li and Na atoms [1, 2, 3]. Alkali atomic species are the most commonly used for ultracold experiments. Other species are chosen for some specific features, as the long-range interactions caused by the large magnetic dipole moment of Er atoms [26].
Differently from liquid helium, ultracold atomic systems offer an unprecedented de-gree of control, and their application spectrum largely exceeds the sole observation of Bose-Einstein condensation. Some of the main tunability directions include the interac-tion strength tuning through Feshbach resonances, the confinement of the atomic clouds on the sites of an optical lattice, the mimicking of artificial magnetic fields for neutral atoms, and the use of anisotropic confinement to obtain low-dimensional systems (see Ref. [4] for a review). In this work, we concentrate on the low-temperature properties of spinless bosonic atoms, in the regime of strong interactions.
The main observable to detect the onset of Bose-Einstein condensation is the mo-mentum distribution, n (k), which develops a narrow peak at zero momentum in the condensed phase (cf. Section 3.3.4). In the majority of experiments, an atomic cloud of less than 107 atoms is confined by an external trap, well approximated by an harmonic potential. In this case the BEC transition is also visible in the non-uniform density pro-file of the gas. More recently, the same transition has been realized for a gas confined in a uniform-box trap [27, 28]. This novel set-up has an advantage in the study of critical behavior with diverging correlation lengths, since there is no inhomogeneity caused by the external trapping potential.
A useful criterion to classify experimental and theoretical studies of ultracold atomic systems is the strength of interactions. The theory for non-interacting bosons is only qualitatively valid (apart from cases where interactions are artificially tuned to zero strength), despite the correct order-of-magnitude estimate of the critical temperature in the strongly-interacting case of liquid helium. The ideal-gas model fails in capturing several properties of interacting systems, including the shape of the condensate wave function in a harmonic trap. Moreover, the BEC transition is qualitatively modified by the presence of any finite interaction strength, and the critical behavior of the interact-ing system belongs to a distinct universality class (cf. Section 3.4.1). This motivates why several contradictory predictions have been proposed for the lowest-order effect of weak interactions on the Bose-gas critical temperature, taking more than 40 years before consensus was reached (cf. Ref. [14] for a detailed review).
The next level of approximation is a mean-field approximation, as Gross-Pitaevskii theory [29]. This approach is valid in the regime of low temperature and weak inter-actions, where it correctly reproduces several observables, including for instance the condensate wave function for a trapped gas. However, it cannot describe systems with strong thermal or quantum fluctuations. A recent example of this limitation is the ob-servation of stable quantum droplets of dipolar quantum gases in regimes where mean-field theory predicts a mechanical collapse [30, 31]. Beyond-mean-field corrections have been predicted [32], and later identified in the equation of state of an interacting BEC [33].
This classification in terms of the interaction strength naturally continues towards the regime of strong interactions and the unitary limit. This is the main subject of this work, and the status of experiments in this direction is reviewed in Section 2.3.
Density matrix and path integrals
The density matrix is the basic tool to treat a statistical mixture of quantum states. In particular, it is at the basis of the path-integral formulation for equilibrium thermody-namics, extensively used in this work.
The partition function Z of a quantum system at inverse temperature b is the trace of the density-matrix operator exp( bH). The simplest way to represent this trace is through a basis where the Hamiltonian H operator is diagonal, so that where En are the eigenvalues of the Hamiltonian H. The formulation at the basis of the work described here is, however, is in the position representation.
This expression includes the full set of single-particle eigenfunctions Yn and eigenval-ues En of H, and the summation has to be replaced by an integration in case of a con-tinuous spectrum. An analogous definition holds for the density matrix of an N-body system, where Yn and En are the eigenfunctions and eigenvalues of the N-body Schrödinger equation, and where we use the shorthands X = fx1, . . . , xN g and X0 = fx10, . . . , x0N g. Despite encoding the full thermodynamics of a many-body system, the expression in Eq. 1.25 is rarely of practical use for realistic many-body problem (that is, for a large number of interacting quantum particles), since it requires the full knowledge of the spectrum and eigenfunctions of H. Nevertheless, it constitutes the basis for powerful approximation schemes, leading to unbiased solutions for some challenging problems in many-body physics.
The density matrix is conveniently treated within the path-integral formalism. In its original formulation, this formalism consists in a rewriting of the real-time propagator of a quantum system [34]. The central quantity is the amplitude for the system being in state jAi at time t0 and in state jBi at time t1, given by B exp i H(t1h¯ t0) A . (1.26).
This propagator, generally unknown, can be rewritten as a weighted sum over all the space-time paths connecting jA, t = t0i with jB, t = t1i. Through the formal replace-ment of the time interval t1 t0 with an imaginary-time variable, the real-time prop-agator in Eq. 1.26 is related to the density matrix in Eq. 1.25, for which the same idea of summing over paths is applicable [35, 36]. The analogy goes even further, since the main approximation scheme used for imaginary-time path integrals (that is, the dis-cretization into small imaginary-time intervals) was already used to treat the original time-evolution problem. This is discussed more in detail in Section 3.2.1. Furthermore, the path-integral formulation provides a mapping between the thermodynamics of a quantum system in d dimensions with a classical system in d + 1 dimensions, which can be treated by Monte Carlo algorithms.
DENSITY MATRIX AND PATH INTEGRALS 23
For bosons, the eigenstates Yn in Eq. 1.25 are symmetric under the exchange of any pair of coordinates, xi $ xj. In this case, the bosonic density matrix rbosN (X, X0; b) re-mains unchanged when X0 is replaced by PX0 x0P1 , . . . , x0PN , where P = P1, . . . , PN is a permutation of the indices 1, . . . , N. Starting from the N-body density matrix for distin-guishable particles, rN (X, X0; b), its bosonic counterpart is obtained as an average over all permutations P, rbosN X, X0; b = N! åP rN X, PX0; b . (1.27).
For a given permutation P, a cycle is a subset of f1, . . . , Ng such that its elements only exchange place with one another, upon repeated application of P (see Fig. 1.2). The cy-cle length is the number of elements in one such subset. From the bosonic density ma-trix, it is simple to state a rough condition for the appearance of long cycles at thermal equilibrium. The typical interparticle distance is r 1/3, and the distinguishable-particle density matrix rN is exponentially suppressed if any pair xj, x0j is at distance larger than lth (cf. Section 2.1.2). A permutation P different from the identity has a relevant weight in Eq. 1.27 only if the distance between any pair xj, x0Pj is comparable to lth. If this is not the case, the contribution of P is strongly suppressed. Therefore, the regime where cycles longer than one become important is when rl3th is of the order of unity, corre-sponding to the BEC criterion in Section 1.1. High temperature or low density decrease the statistical weight of permutations which include long cycles, while low temperature or high density have the opposite effect. In the next section, the concept of permutation.
Figure 1.2: Schematic view of some of the 4! = 24 permutations for N = 4, together with their decomposition into cycles (in parentheses).
cycles is quantitatively related to the appearance of a BEC phase, for non-interacting bosons. The general connection of permutation cycles with BEC and superfluidity is given in Sections 3.3.2 and 3.3.3, within the path-integral framework.
Recursion method for non-interacting bosons
For the non-interacting Bose gas, the study of permutation cycles leads to a power-ful method to exactly compute observables as the energy and the condensate fraction. This method provides insight into the connection between permutation cycles and Bose-Einstein condensation. Moreover, it is valuable to benchmark our QMC algorithm (cf. Fig. 3.15 and Table 3.1).
This shows that the two terms in Eq. 1.30 correspond to the two available permutations for two particles, P = 1, 2 and P = 2, 1 (with the notation of Fig. 1.2). The recursion relation in Eq. 1.29 provides a compact way to obtain ZN (b), which would naively re-quire a sum over all N! permutations. ZN (b) gives access to several thermodynamic observables. The average energy, for instance, is obtained through its definition, and a recursion relation for ¶ZN (b)/¶b which is derived from Eq. 1.29.
The recursive expression for ZN (b) is connected with the statistics of permutation cycles. The k-th term in Eq. 1.29 is the product of z(kb) and ZN k(b), which corre-sponds to assigning a given particle to a cycle of length k and weighting this choice with the partition function of the other N k particles, ZN k(b). Therefore, the normal-ized probability for a given particle to be on a cycle of length k reads.
As particles are identical, the average number of particles on cycles of length k is k pk.
In Chapter 3, this expression is used to benchmark our QMC technique (cf. Fig. 3.15).
for cases in which the ground state energy is equal to zero (cf. Eq. 1.3). The conden-sate fraction hN0i is å N0 N0 p(N0). Moreover, the integer derivative of the average permutation-cycle occupation is proportional to p(N0) [40, 39, 41].
for k 1. This establishes a connection between the condensate fraction and the ap-pearance of long permutation cycles. At large N and T < Tc0, for instance, kpk drops to zero for k ’ hN0 i, which allows to identify the condensate fraction with the length of the longest cycle with a finite occupation number.
The recursion scheme for ZN (b) also gives access to other observables, like the one-body reduced off-diagonal density matrix [40, 42] and the superfluid fraction (cf. Sec-tion 3.3.2), but it is restricted to ideal particles. The generalization to interacting systems can only be obtained approximately, for instance through corrections to Eq. 1.29 based on the high-temperature virial coefficients (see for instance Ref. [43], where this is used to study the unitary Bose gas).
Few-body physics with strong interactions
In this chapter, we consider systems of two or three strongly-interacting quantum par-ticles. Two limits are of particular interest, the scaling and the unitary limit. In the scal-ing limit, the range of the interparticle potential vanishes. This is the relevant regime for experiments with dilute ultracold atomic gases, where the interaction range is typi-cally much smaller than the average interparticle distance between atoms. The unitary limit corresponds to interactions which realize a diverging scattering length. This is the extreme strongly-interacting limit, since the corresponding cross section for two-body collisions saturates a theoretical upper bound. Unitary interactions can be induced in ultracold atomic samples, through the Feshbach-resonance technique (see Section 2.3).
We first review the general basics of scattering theory for two quantum particles (see Section 2.1). Analytical results for the case of zero-range unitary interactions allow us to compute thermal correlation functions, including the momentum distribution and the pair-correlation function. These are of particular interest, since they share some relevant features with the case of a unitary-interacting many-body system, described in Chapter 4. For the case of three strongly-interacting particles, the Efimov effect appears, consisting in a scale-invariant sequence of three-body bound states. This phenomenon is connected to universal properties, which do not depend on microscopic details of the interparticle interactions. In Section 2.2, we review the description of the Efimov effect and verify that our theoretical model (based on a three-body cutoff) is consistent with the universal theory. The verification is based on numerical results for the ground-state trimer, obtained through the quantum Monte Carlo technique (cf. Chapter 3).
Throughout Sections 2.1 and 2.2, we combine two different approaches to the few-body problem. The study of two-body scattering and of the Efimov effect is based on wave functions for single quantum states, while the density-matrix approach is used to obtain finite-temperature results. For N = 2, the latter provides analytical, approx-imate expressions for the relevant correlation functions. More generally, numerically exact results can be obtained at any temperature, by combining the density-matrix ap-proach with the quantum Monte Carlo technique. This powerful method is also used to describe the many-body case (cf. Chapter 4).
Two-body physics
In the present section, we describe the two-body quantum problem for strong, short-ranged interactions. We first review the basics of scattering theory for two particles, providing the specific examples of the square-well and zero-range model potentials (see Section 2.1.1). From the study of wave functions, we then move to the finite-temperature density-matrix formalism, and report the exact solution for the density matrix of two particles with zero-range, unitary interactions (see Section 2.1.2.1). This is useful for two applications. On one hand, it gives direct access to thermal correlation functions for two particles, like the one-particle-reduced density matrix g(1) (r) and the pair-correlation function g(2) (r) (see Section 2.1.3). These show the small-distance features typical of systems with strong, short-ranged interactions, like the cusp singularity of g(1) (r) and the divergence of the pair-correlation function. On the other hand, the exact knowledge of the two-body density matrix constitutes the essential ingredient for our quantum Monte Carlo scheme (see Section 3.2.1). This technique is used to benchmark other calculations for N = 2 (e.g. for correlation functions), and to obtain new results for N 3, concerning Efimov trimers (cf. Section 2.2.2) and the unitary Bose gas (cf. Chapter 4).
Scattering theory
where P = (p1 + p2), p = (p1 p2)/2, r = x1 x2, and r = jrj. H corresponds to the Hamiltonian of a free particle with mass 2m, while the presence of two-body interactions only affects Hrel. We consider the relative-motion problem, and assume that the V2(r) decays to zero at large r. We look for scattering eigenstates Yrelk(r) of Hrel, that solve the time-independent Schrödinger equation, 2 Yrelk(r) = EkYrelk(r), pm + V2(r) (2.2).
TWO-BODY PHYSICS 29
with a positive energy Ek = h¯2k2/m. For large distance r, a scattering eigenstate is the superposition of the incoming plane wave (with wave number k) and a scattered spherical wave:
Yrelk(r) eikr cos q + f (k, q) eikr r , (2.3).
where q is the angle between k and r. The scattering amplitude f (k, q) encodes the ef-fects of interactions on a two-body collision process. The total scattering cross section s(k), for instance, is the solid-angle integral of j f (k, q)j2. As the system is character-ized by axial symmetry (for rotations with axis k), a scattering state can be written as a partial-wave expansion where Al are constant coefficients and Pl are the Legendre polynomials. The radial wave function Rk,l (r) solves the one-dimensional Schrödinger equation.
For k going to zero, the total cross section s(k) tends to 4pa2 (cf. Eq. 2.9), which is the area of a sphere of radius a. Thus the scattering length can be interpreted as the length scale over which the incoming plane wave is substantially modified by interactions. In the case of a hard-sphere potential (where V2(r) is infinite for distances smaller than the particle diameter, and zero otherwise), this interpretation becomes exact, as the scatter-ing length a is equal to the particle diameter [45].
For ultracold atomic gases, the typical interaction potential V2(r) is strongly repul-sive at small r, has a minimum at intermediate distance, and falls off as a power law. The attractive large-r tail is Vtail(r).
For 87Rb, for instance, lvdW 4 nm [5]. This potential supports several molecular bound states with large binding energy. This indicates that ultracold gases are only metastable, while the equilibrium state would correspond to a solid phase. Therefore, this real-istic potential cannot be used directly within thermal-equilibrium theories (including the path-integral QMC technique) [46]. However, ultracold atomic systems are exper-imentally realized in the dilute regime, where the range of the two-body potential is significantly smaller than the typical interparticle distance. Theoretical predictions in this regime only depend on the scattering properties of the potential, rather than on the specific form of V2(r). At low temperature, in particular, they depend only on the 1If V2(r) decays at large r as r n, with n > 3, Eq. 2.10 is only valid for the partial waves such that l < (n 3)/2. For larger n, the small-k limit of dl (k) is kn 2 modulo p [45].
TWO-BODY PHYSICS 31
s-wave scattering length a. Thus an arbitrary model can be chosen, provided it repro-duces the required value of a. The choice is dictated by simplicity in the calculations, and by the fact the model potential should be amenable to a thermal-equilibrium treat-ment (that is, it should not have deeply-bound molecular states, which may lead to a instability towards a solid phase). An example of this choice is the use of the hard-sphere potential to study bosonic gases with small scattering length (see for instance [47]). In Sections 2.1.1.1 and 2.1.1.2, we describe two possible models: The attractive square-well potential, and its zero-range limit. The latter is then used throughout the rest of this chapter and in Chapter 4.
Table of contents :
1 Bosons at low temperature
1.1 Bose-Einstein condensation
1.2 Ultracold atomic gases
1.3 Density matrix and path integrals
1.4 Recursion method for non-interacting bosons
2 Few-body physics with strong interactions
2.1 Two-body physics
2.1.1 Scattering theory
2.1.2 Two-body density matrix
2.1.3 Correlation functions
2.2 Three-body physics and Efimov effect
2.2.1 Efimov trimers
2.2.2 Three-body-cutoff model
2.3 Ultracold atoms with strong interactions
2.3.1 Feshbach resonances
2.3.2 Signatures of the Efimov effect
3 Path-Integral quantum Monte Carlo
3.1 Monte Carlo sampling
3.1.1 Direct sampling
3.1.2 Markov-chain Monte Carlo
3.1.3 Variable-dimensionality sampling
3.2 Sampling quantum paths
3.2.1 Path integrals and Monte Carlo
3.2.2 Closed-sector moves
3.2.3 Open-sector moves
3.2.4 Sector-changing moves
3.3 Observables
3.3.1 Spatial correlation functions
3.3.2 Permutation cycles and superfluid densits
3.3.3 One-body-reduced density matrix
3.3.4 Momentum distribution and condensate fraction
3.4 Additional aspects related to PIMC
3.4.1 Superfluid transition and finite-size scaling
3.4.2 Large-momentum tail of n (k)
4 Many-body unitary bosons
4.1 Imaginary-time discretization revisited
4.2 Efimov liquid
4.2.1 Liquid/gas phase coexistence
4.2.2 QMC observation of the Efimov liquid
4.3 Normal-gas phase
4.3.1 Momentum distribution
4.3.2 Contact density
4.4 BEC phase
4.4.1 Critical temperature
4.4.2 Momentum distribution
4.4.3 Contact density
5 Classical three-body hard-core model
5.1 Sampling configurations at fixed pressure
5.1.1 Box rescaling
5.1.2 Insertion/removal move
5.2 Packing problem
5.2.1 Variational ansätze
5.2.2 Simulated annealing
5.3 Finite-pressure phase diagrams
5.3.1 Two dimensions
5.3.2 Three dimensions
