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Interacting bosons in a nearly-resonant potential well

We establish that the ability of a localized trapping potential to bind weakly-interacting bosons is dramatically enhanced in the vicinity of the threshold of formation of the single-particle bound-state of the trap. Specifically, for repulsive particles and a super-threshold trapping potential the equilibrium number of bound bosons and the size of the ground state diverge upon approaching the single-particle threshold from above. For attractive interactions and a sub-threshold trap a collective bound state always forms for a sufficiently large number of bosons despite the inability of interparticle attraction alone to form a two-body bound state.

We establish that the ability of a localized trapping potential to bind weakly-interacting bosons is dramatically enhanced in the vicinity of the threshold of formation of the single-particle boundstate of the trap. Specifically, for repulsive particles and a super-threshold trapping potential the equilibrium number of bound bosons and the size of the ground state diverge upon approaching the single-particle threshold from above. For attractive interactions and a sub-threshold trap a collective bound state always forms for a sufficiently large number of bosons despite the inability of interparticle attraction alone to form a two-body bound state.

I. RESONANT BINDING
It is empirically known that a surprisingly large number of short-range interactions operating in Nature have resonant character: the two-particle attraction is either barely sufficient to form a two-body bound state or misses the binding threshold by a small amount. In nuclear physics the prominent examples include the neutronnucleon and neutron-alpha particle attractions which are nearly binding [1]. In atomic physics the same is true for interactions between the atoms of 3 He [2] and of those of the spin-polarized Hydrogen family [3]. At the same time two 4 He atoms form a weakly-bound dimer whose binding energy is four orders of magnitude smaller than the depth of the 4 He-4 He potential well [4].
The proximity to the two-body binding threshold can be quantified by the s-wave scattering length whose magnitude significantly exceeds the range of interparticle forces, thus implying an effectively long-range interaction [1]. This regime is fundamentally interesting because systems consisting of resonantly-interacting particles may exhibit a series of effects which are independent of the microscopic details of the interaction between the constituents. One such phenomenon is the well-understood Efimov effect [5], which occurs in a system of three bosons where resonant two-body forces trigger a formation of an arbitrarily large number of loosely bound levels in a three-particle system. Conclusive experimental observation of the Efimov physics became possible only recently in an ultra-cold gas of Cs atoms [6] because, in contrast to other nearly-resonant systems found in Nature, the two-body scattering length and thus interparticle interactions in a cold gas can be precisely controlled by using Feshbach resonances [7]. This allowed the measurement of the dependence of the three-body recombination rate on the two-body scattering length which contains signatures specific to the Efimov effect [8].
The goal of this paper is to establish the existence of a novel resonant effect which may also be observed in a cold bosonic gas: we will demonstrate that the number of particles bound by a nearly-resonant potential well increases as the well is made more shallow, diverging exactly at the single-particle threshold. The fact that such an effect might exists is implied by variational analysis given in our previous work [9]. Similar to the Efimov effect the ultimate origin of this counterintuitive behavior is an effectively large attraction range of the potential well. Whereas the Efimov effect requires an attractive part of the physical potential to bind three particles, the phenomenon discussed below takes place in the presence of interparticle repulsion.
In the laboratory attractive wells have been realized using the optical dipole force [10], in which atoms are attracted to the intense region of a focused laser beam. Strong confinement in three dimensions can be obtained at the intersection of two beams, each typically having diameter of about 15 µm [11]. The binding properties of such traps can be tuned by adjusting the laser power. Even tighter traps can be achieved by using holographic techniques [13], optical superlattices [14], near-field [15], and white-light techniques [16].
Additionally, tuning the parameters of such localized potential placed next to the classical edge of a Bose-Einstein condensate, might allow manipulation of a welldefined number of particles [9]; it also provides a new context for studying the quantum many-body problem, as this involves the states of an interacting boson system. The experimental feasibility of such a setup was discussed in Ref. [9] as well as reasons why tighly focused potential is needed.

II. THE MODEL
We consider n interacting bosons of mass m in the presence of a well-localized attractive potential U (r). We assume a system of sufficiently low density so that the range of interactions between particles need not be included as a parameter. Therefore we study the Gross-Pitaevskii (GP) energy functional [17] (1) subject to the normalization condition Here we have assumed that the many-body wavefunction can be written as a product of the single-particle functions ψ(r). The parameter g represents the short-ranged interaction between the particles. The external potential U (r) will be chosen in the form of an attractive "shell" of radius a: because this is a simple form that will bind particles without having singular behavior in the wavefunction at the origin. The case of more realistic potentials will be discussed in Section VI. For the case of one particle, there is a single bound state for V > V c = 2 /2ma 2 , and this condition will continue to be relevant for the many-body problem.

III. RESULTS FROM AN APPROXIMATE WAVEFUNCTION
The ground-state wavefunction minimizes the energy functional (1). We can deduce its general form from inspection. If the ground state is to be bound, it will have to be relatively large in the well-localized region where U (r) is nonzero. At sufficiently large distances it is decreasing more rapidly than 1/r, to ensure normalizability. It will pass from large to small in a way that keeps the gradient term small; this suggests ψ ≈ 1/r at intermediate distances. The effect of the interparticle repulsion (the last term in Eq. (1)) will be to suppress large values of ψ.
In a previous paper [9] these considerations led us to study the two-parameter trial wavefunction such that ψ = C for r < a, and ψ(r) = Ca exp(−αr + αa)/r for larger r. We will briefly recapitulate the results. The form of the wavefunction for larger r is the exact wavefunction for a bound state of noninteracting particles, and will remain a good approximation to the interacting wavefunction at large r, where ψ 4 will be negligibly small. The wavefunction extends to a distance of order 1/α, which will be large for large n and also for V close to V c . Thus we are particularly interested in the case of small α. Integration of Eq. (2) gives We note that for small α this reduces to C 2 ≈ α/a 2 , rather than α 3 (i.e. the reciprocal of the "volume" of the wavefunction). This implies that the two length scales a and 1/α both play roles; the limit a → 0 cannot be taken. Eq. (1) leads to where E 1 (x) is the exponential integral [18]. Using Eq. (4) to eliminate mention of α, the leading terms of Eq. (5) are Minimizing this expression with respect to C gives an estimate for the energy. Assuming g > 0 (i.e. the particles repel each other), For large n this becomes independent of n, and the corresponding value of α is small. The energy at n → ∞ is greater or less than the energy at n = 1, depending on whether g is greater or less than g c = 3πa 3 V c /2. For g less than g c , this would seem to indicate that the potential can bind an infinite number of particles (with a very swollen wavefunction); however, including the last term of (5) gives a shallow minimum for g < g c , and then for large n the state is unstable against the unbinding of particles (i.e. it is within the continuum of a state with smaller n). Our main goal is to establish whether this minimum is real, and if this is the case, what is its behavior as the single-particle threshold is approached from above, V → V c + 0. The two regimes for the interparticle strength (the cases g > g c and g < g c ) are in essence a small-n effect. The ground-state energy for two particles is greater or lesser than the energy for one, depending on whether the interaction parameter is large or small. This shows that for V > V c there is a n-body bound state; however, the total energy for large n becomes independent of n, so that the chemical potential decreases to zero. This occurs because for small α the single-particle wavefunctions extend to large distances from the attractive center, making it correspondingly unlikely that two particles will be at the same place, so that the interparticle interactions can be effectively made as small as one likes (by tuning α). The effect depends on the way the size of the ground state scales with the number of bound bosons, and does not occur in lower dimensions [9].

IV. GROSS-PITAEVSKII EQUATION
The approach just outlined has the advantage that the mathematics is transparent, and yields results for the energy and wavefunction in which the role of the various parameters in the problem is explicit. It is a robust method that gives an upper bound on the exact groundstate energy. However, the results are only as good as the trial wavefunction. Therefore we will find the true minimum of Eq. (1) by numerical means. This gives the best possible wavefunction that can be written as a simple product of single particle wavefunctions. In the process it verifies that the simple variational function is qualitatively correct, and puts the theory of resonant binding on solid footing. Additionally we will show that non-trivial cooperative bound states also form for a sub-critical well in the presence of a weak interparticle attraction.
In what follows we will choose a = 1 and V c = 2 /2ma 2 = 1, which sets the length and energy scales for the problem.
Taking a variational derivative of Eq.(1) gives the GP equation [17] − 1 r where γ = g(n − 1). Unlike the case of the linear Schrödinger equation, the eigenvalue ǫ for which the GP equation (8) has a normalizable solution is not directly proportional to the minimum value of Eq. (1); however, they are related by We solved Eq. (8) by integrating from r = 0 with initial conditions ψ = 1, dψ/dr = 0 and chosen values for V , γ, and ǫ. The integration was in the form of a power-law representation of ψ(r) for r < 1, which avoided the problems coming from the singular form of the GP equation at small r and gave accurate values for ψ and its derivative at r = 1; thereafter, we used second-order Runge-Kutta integration with equal-sized steps in the variable y = √ r. In general this gave a function that diverged to plus or minus infinity at large r. By tuning ǫ we could find a case for which ψ remained small to chosen large values of r. This condition defines the eigenvalue ǫ to high accuracy. Once a normalizable solution had been found, it was rescaled to satisfy the normalization condition (2). This also changes the value of γ, but not V or ǫ; only at the end of the calculation do we find what value of the particle number n the solution corresponds to. Figure 1 shows how the ground-state energy varies with the number of bosons n, and compares the numerical solution of the GP equation, Eqs. (8)- (9), to the variational result (the minimum value of Eq. (5)). The parameter V = 1.5, and g = 2, 4, 6, ..., 20. The GP solution is always below the variational prediction, because it uses a better wavefunction.
Although the variational results differ quantitatively from the solution to the GP theory, qualitatively they are similar. We offer the following observations: *The energy is negative for all n, so that the attractive potential has a bound state with an infinite number of particles.
*For g < g c , the energy has a very shallow minimum as a function of the number of particles. This implies that it will be difficult to attach a definite number of particles to a well by placing it in tunnelling contact with a source (such as a condensate droplet): there is no equilibrium at all, or the number of particles bound will be large and very susceptible to small perturbations. The minimum in E(n) is even less pronounced in the GP solution than it is in the variational result. We have indicated the apparent position of the minima, using the "+" graph marker.
*For g > g c , the energy increases with the number of particles. Then it is possible to bind a well-defined number of particles by letting the system equilibrate with a source. However, this scheme will only work well for a small number of particles: the plateau at large n will make the system insensitive to the number of particles. In the variational study, we found a counterintuitive behavior for the value n x at which the minimum of the total energy occurs: when g is below the critical value g c , n x becomes large, diverging as V approaches the binding threshold. This means that an arbitrarily large number of particles can be stably bound by making the potential weak! We have verified that this behavior is also exhibited by the solution to the GP theory. Here it is useful to observe that we are trying to find the minimum of the total energy both with respect to the parameter n and the normalized wavefunction ψ. Imposing the condition that the derivative of Eq.(1) with respect to n (with ψ fixed) is zero, and combining the outcome with Eq. (9), we obtain a relationship Imposing this condition proved to be easier than trying to minimize E over n. We constructed Figure 2 by finding normalized solutions to the GP equation (8) for various values of γ, and iterating until Eq. (11) holds. Although our work demonstrates the existence of the effect of resonant binding only within the GP approximation, we expect that it is also present in the exact theory. As a nearly-resonant well is made more shallow approaching the single-particle binding threshold, the effective range of the central attraction increases, and the well binds an increasingly larger number of bosons. Therefore the microscopic details of the well are not expected to be of qualitative significance. The large range is responsible for the survival of the effect in the presence of weak short-range repulsion between the bosons because it allows the particles to take advantage of the central attraction while minimizing mutual contact interactions [19]. The interactions become more important as the well is made deeper, thus explaining the existence of the minimum in Fig. 2. For a sufficiently deep well the number of bound particles n x increases as the well deepens which is the normally expected behavior.
Additionally, we argue that our results are immune to the effects of correlations neglected in the GP approximation. Indeed, the GP theory can be viewed as a variational method. Therefore the exact energy will be lower than our calculations indicate. What they do show is that near resonance the wavefunction can accommodate an unexpectedly large number of particles, because each particle manages to sample the attractive well without interacting with the other particles too strongly. The effect of correlations would be that the particles evade each other more effectively, and thus enhance the effect we predict.
The issue of validity of the GP theory was also addressed in Ref. [9] where the same problem was studied in general dimension. Specifically, in one dimension a version of the problem in question was solved exactly by using the Bethe ansatz methods and the results were compared with the GP solution. Although the exact ground-state wavefunction is indeed not of the product (Hartree) type, in the limit of large particle number the exact ground-state energy turns out to be nearly identical to its GP counterpart. This example represents "the worst case scenario" as the effect of correlations is expected to be strongest in one dimension. Therefore we conclude that it is very likely that the GP theory is fully adequate in three spatial dimensions.

V. WEAKLY ATTRACTING PARTICLES
The variational analysis also predicts an interesting phenomenon for the case of weakly-attracting particles in the presence of a potential that is too weak to bind a single particle (g < 0 and V ≤ 1). The second term of Eq. (6) is negative for sufficiently large n, pointing to the existence of a many-body bound state. For the particular case V = 1, the critical value of the number of particles needed is given by n c = 3π/2|g|. When V < 1, the minimum energy can be positive (indicating a metastable state), but becomes negative for sufficiently large n. The possibility of a cooperatively bound state of this sort was first pointed out by Migdal [20]. Figure 3 shows the results of numerical solution of the GP theory for the case of weakly attracting particles in the presence of a V ≤ 1 potential. In agreement with the variational argument, collective bound states are seen to form for a sufficiently large number of bosons. Specifically, the farther away is the binding threshold, the larger number of bosons is required to form a bound state.

VI. BEYOND THE MODEL
The delta-shell potential cannot be achieved experimentally. Then one may ask how our results are relevant to the real world.
The variational results are quite general for a localized external potential. Since the trial wavefunction is constant for r < a, the external potential only enters through its average value The solution to the GP equation will be affected slightly. We can anticipate the effects by comparing the delta shell problem to that of the attractive square well U (r) = −W for r < a. Outside the well the GP equation is the same for the square well and the delta shell, and so the solutions (for given γ and ǫ) are the same; the only question is how the value for W corresponds to that for V . The condition that relates them is that the values of the logarithmic derivative of the wavefunctions should be the same at r = a. For the delta shell, the solution to Eq.(8) (for γ = 0) is ψ = C sinh( |ǫ|r)/r, and is very nearly constant for the cases of interest; however for the square well the solution inside (again for noninteracting particles) has the form ψ = C sin( W − |ǫ|r)/r. The logarithmic derivative of this function takes on all values and there is an infinite set of values of W for which the match can be achieved. The effect of the particle interactions will be to somewhat suppress the accumulation of particles (places where ψ is large) and to push the consecutive bound states to larger values of W . However, the feature of interest in the resonantly bound system is the number of particles that can be bound, and this is a property of the wavefunction at large distances. Thus we can expect that the the differences between different external potentials can be absorbed into the definitions of V c and g.

VII. CONCLUSIONS
To summarize, our work demonstrates that a superthreshold localized trapping potential can bind an unexpectedly large number of weakly-repulsive bosons while a sub-threshold trap can trigger a formation of a cooperative bound state of weakly-attractive bosons even when two-body attraction is insufficient to form a two-body state. These effects are insensitive to the microscopic details of the trapping potential and have their origin in the large effective range of nearly-resonant trapping potential. We hope that these results will be experimentally tested in the near future.

VIII. ACKNOWLEDGMENTS
This work was supported by the Thomas F. Jeffress and Kate Miller Jeffress Memorial Trust.