Effective-range description of a Bose gas under strong one- or two-dimensional confinement

We point out that theories describing s-wave collisions of bosonic atoms confined in one-dimensional (1D) or two-dimensional (2D) geometries can be extended to much tighter confinements than previously thought. This is achieved by replacing the scattering length by an energy-dependent scattering length which was already introduced for the calculation of energy levels under 3D confinement. This replacement accurately predicts the position of confinement-induced resonances in strongly confined geometries.


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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT In this paper, we present an accurate analytical description for scattering in 1D and 2D geometries based on the findings of [13,14].
We consider a gas of bosonic atoms in an optical lattice and assume that there is little tunnelling between the lattice cells, so that each cell is independent. The atoms in a cell are confined by a trapping potential which will be assumed harmonic (which is true near the centre of the cell). Let us consider a pair of atoms in such a cell. For a harmonic potential, the centre-of-mass motion decouples from the relative motion and the stationary Schrödinger equation for the relative motion wave function ψ( r) reads Here, r = (x, y, z) is the relative coordinate with separation r, µ the reduced mass, U(r) the isotropic atom-atom interaction potential, V( r) the trapping potential, and E the relative energy. For 2D confinement (tube or wave guide geometry), the atoms are strongly confined in the xy-directions and (almost) free to move in the z-direction, therefore we set where ρ = ρ and ρ is the projection of r on the xy-plane. For 1D confinement ( pancake geometry), the atoms are strongly confined in the z-direction and (almost) free to move in the xy-directions Here, ω is the trapping frequency at the centre of the cell and we define σ = √h /(2µω) as the typical length scale associated to the trap in the confined directions.
Any scattering solution ψ( r) of equation (1) is composed of an incident wave and a scattered wave. A plane wave basis can be used for the incident wave, and the scattered wave can be expressed with the noninteracting Green's function G( r, r ) of the system for U(r) = 0. Namely, for 2D confinement, one has where φ nm ( ρ) denotes the unit-normalized 2D isotropic harmonic oscillator eigenstate of principal quantum number n and angular quantum number m, and q nm = (2µ/h 2 )E − (2n + 1 + |m|)/σ 2 is the wavenumber of the incident plane wave. The Green's function reads For 1D confinement, one has 3 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT where ϕ n (z) denotes the unit-normalized 1D harmonic oscillator eigenstate of vibrational index n, and q n is the wavevector of the incident plane wave with norm q n = (2µ/h 2 )E − (n + (1/2))/σ 2 . The Green's function reads where H (1) α is the first Hankel function. We first assume that the interaction potential U(r) has a finite range r b . This means that there is a separation r 0 r b beyond which the wavefunction is essentially a solution of equation (1) with U(r) = 0, that is to say a solution of the noninteracting problem. This is indeed the case for typical atomic interactions which drop off as −C 6 /r 6 van der Waals potentials. Because of this fast drop-off, the wavefunction (at sufficiently low energy) reaches its noninteracting form for r 4 (1/20)β 4 6 , where β 6 = (2µC 6 /h 2 ) 1/4 [16]. The separation r 0 is therefore on the order of β 6 , which ranges typically from 2 to 5 nm.
We further assume that the interaction potential U scatters only s-waves. The scattering of partial waves of arbitrary order under cylindrical confinement was treated in detail in [17]. Retaining only s-wave scattering is valid in the absence of shape resonances and if the coldcollision condition kr 0 < 1 is satisfied, wherehk = √ 2µE/h is the collisional momentum (see appendix).
Under these two assumptions, the noninteracting form of equations (2) and (3) is obtained by first taking the limit r r of the Green's function (since r > r 0 r b ∼ r ) and then approximating the remaining integral over r by φ ν,µ (0) (or ϕ ν (0)) times a quantity 4πA which does not depend on the indices ν or µ appearing inside the Green's function (taking into account the dependence on ν, µ would introduce higher-order partial wave scattering [17] -see appendix). These two steps are formally equivalent to taking the Green's function out of the integral in equations (2) and (3) and evaluate it at r = 0. Note that this can be achieved at any r by replacing the potential U(r) by a regularized contact interaction [18]. For 2D confinement, one finds [19] and for 1D confinement [7], where the factors A mn and A n are to be determined. The wavefunction ψ can also be expanded in spherical partial waves where Y ,m are the spherical harmonics and θ the angle between r and the z-axis, ϕ the angle between ρ and the x-axis. At short separations, the confining potential V( r) is negligible, so one can expect that within a certain range r 1 related to the confinement length σ (see appendix), the wavefunction is close to a solution of equation (1) with V( r) = 0, that is to say, a solution of the free-space scattering problem. Beyond r 0 , the partial waves of this solution reach their noninteracting form which is known to be a combination of regular and irregular spherical Bessel functions. Consistent with the s-wave approximation, there is no irregular Bessel function for = 0, i.e. no scattered partial wave. For the s-wave ( = 0), we have where η is a normalization factor to be determined and a(k) = − tan δ k /k is the energy-dependent s-wave scattering length introduced in [13,15] (δ k is the usual s-wave phase shift, related to the s-wave component of the reactance matrix K s (k) = − tan δ k ). This energy-dependent scattering length contains all the effects of the interaction on the wavefunction in the region r 0 < r < r 1 , and for any collisional energy E. For moderately tight traps σ r 0 leading to small collisional energies, there is a range of r for which equation (7) simplifies to where a = lim k→0 a(k) is the scattering length of the potential. However, for very tight lattices, σ may be close to r 0 and only equation (7) holds. The essence of the method used in [7,19] is to assume that σ r 0 and match the noninteracting expressions (4) and (5), respectively, with the free-space expression (8) in the region r 0 < r < r 1 where they both are valid. (In [7], this is implicitly achieved by use of a 3D regularized contact interaction.) By performing the matching procedure up to zeroth order in the asymptotic expansion in r/σ near r = 0, they obtain two relations yielding the two unknown factors η and A nm (or A n ). From that knowledge, they deduce the effective 1D and 2D interaction strengths in the quasi-1D or 2D regime (q n , q nm k) with B ≈ 0.915 and ζ(1/2)/ √ 2 ≈ −1.033, where ζ is the Riemann zeta function. The singularity in these expressions as a function a, σ, or q 0 corresponds to the confinement-induced resonance. Note, however, that these analytical formulae are valid only when σ is large with respect to r 0 .
We stress here that the method can be extended by matching the s-wave component of the noninteracting expressions (4) or (5) with the more general free-space expression (7). It is straightforward to see that a matching to zeroth order in r/σ results in the same conditions as those of [7,19], but with the scattering length a being replaced by the energy-dependent a(k) in all the formulae. In particular, a is replaced by a(k) in the expressions for η, A nm , A n and equations (9) and (10). Surprisingly, we find that once the two unknown factors η and A nm (or A n ) are determined this way, the expression (4) or (5) match the expression (7) up to second order in r/σ, i.e. the next two orders automatically match without the need for extra parameters. This was checked numerically both in 2D and 1D confinement, and we give in the appendix In both panels, the s-wave component of the solution to the free-space scattering problem at energy E =h 2 k 2 /2µ is represented as a thin black line. For r > β 6 it has the asymptotic form corresponding to equation (7). The thick black line represents the 1D or 2D wavefunction in the noninteracting region obtained from equation (4) or (5). It is determined by a matching procedure with the free-space scattering wavefunction, as explained in the text. The previous theories [7,11] were based on a matching with the solution to free-space scattering at zero energy (dotted line), which has the asymptotic form corresponding to equation (8) for r > β 6 . The resulting noninteracting 1D and 2D wave functions are represented by thick grey lines. They do not connect to the zero-energy wave function for the considered confinements. The lengths β 6 , a and 1/k are indicated by arrows.
an analytical derivation of this result for 2D confinement, based on mathematical assumptions which we checked numerically. As a result, the expressions (4) or (5) and (7) can be matched within less than 1% for σ r 0 . In figure 1, we illustrate the matching of the wavefunctions for a confinement σ close to r 0 , in the case of a van der Waals interaction U(r) = − C 6 r 6 , for which r 0 ≈ β 6 , as stated earlier. The figure shows both the s-wave component of the solution to the 3D free-space problem (which is highly oscillatory for r < r 0 , and has the asymptotic form (7) for r > r 0 ) and the noninteracting wavefunction along either the z-or ρ-directions. For better comparison, we should plot only the s-wave component of the noninteracting wavefunction, but this happens to be very slowly converging numerically. The inclusion of higher-order partial waves thus induces a second-order difference (because ψ 2,m (r) ∝ r 2 for small r), but one can still appreciate the very good matching of the two functions even at these extreme confinements. In contrast, the two functions calculated from the original theory do not match.
The fact that we can gain two orders in the matching procedure by using the free-space expression (7) implies that the replacement a → a(k) greatly improves the accuracy and the range of validity of the original theory. Although we do not have an estimation of the accuracy on g 1 and g 2 , we expect the extended theory to give reasonably good predictions for σ r 0 . This improvement brought by the simple replacement a → a(k) is not so surprising in view of the results reported in [13,15,[21][22][23], which have shown the relevance of energy-dependent scattering lengths for the accurate calculation of energy levels in 3D confined geometries. It has been long known in other contexts that contact interaction models are significantly improved when the coupling constants are proportional to the energy-dependent scattering length (or reactance matrix) [24]. Similar extensions of (9) and (10) were considered in [25]- [27] in order to take into account the energy dependence due to a scattering resonance at low energy. In [27], a renormalized contact interaction was used, leading to the replacement of a by the quantity (2µ/4πh 2 )T(k), where T(k) = (4πh 2 /2µ)a(k)(1 + ika(k)) −1 is the T -matrix. This complex quantity is equivalent to the real a(k) at low energy. Here, we focus on the energy dependence for strong confinement even in the absence of any resonance.
In the quasi-1D (q nm k) or quasi-2D (q n k) regimes, the collisional energy is set by the confinement length σ and can be relatively high, although the effective collisions in the weak directions are cold. An interesting consequence of the extended theory is that if the confinement is strong enough, the collisional energy can probe the energy-dependence of a(k). For a standard contact potential [18], a(k) is constant and equal to a. However, a more realistic a(k) has some energy dependence. For instance, in the effective-range approximation, a(k) has the resonant form: where r e is the effective range of the potential [28]. This approximation works well for shortrange interactions with a large scattering length a r e . In the case of van der Waals interactions, the effective range r e is a simple function of a and β 6 [29,30]  More elaborate analytical expressions of a(k) valid for any a have been derived for van der Waals interactions [30]. The interest of equations (10) and (11) is that they give a simple two-parameter description of the collisions for a wide range of energies.
To illustrate these ideas, we calculated the 1D interaction strength g 1D for a van der Waals interaction consistent with the Lennard-Jones parameters of the numerical calculation reported in [14]. The authors observed a difference between their numerical calculation and the analytic formula (9) where a is taken as the zero-energy scattering length. They suggested that this difference comes the fact that the confinement-induced resonance in g 1D results from a Feshbach resonance with a trap bound state, whose binding energy is not predicted accurately by a pseudopotential based only on the scattering length. As a result, the formula (9) does not predict the resonance at the right location. However, we show in figure 2 that the same formula used in conjunction with the replacement a → a(k) in the effective range approximation reproduces the numerical calculations very well. This is because the effective range approximation is able to reproduce the binding energy of the last bound state accurately. The only region where the effective range approximation fails is for small scattering lengths a β 6 , where it predicts a spurious resonance, as visible in figure 2.  [14] and confirmed by us, while we calculated the dots in right panel (pancake geometry) using the adaptive grid refinement method of [20]. The dashed curves correspond to the formulae (9) and (10) with the zero-energy scattering length a, and the solid curves corresponds to the same formulae using the energy-dependent scattering length a(k) instead of a.
Here, a(k) has been calculated in the effective-range approximation, equation (11), which is valid almost everywhere except for small positive scattering lengths where the approximation causes a spurious resonance.
We also calculated the 2D interaction strength and checked that a similar situation occurs in the pancake configuration. Using the adaptive grid refinement method of [20], we solved the Schrödinger equation (1) for a Lennard-Jones interaction U(r) = (C 12 /r 12 ) − (C 6 /r 6 ) and a cylindrical harmonic trap. The tight pancake limit is obtained by setting the ratio of axial and radial frequencies to 400 (thus leading to a spatial aspect ratio of 1/20), and the tight confinement scale is set to σ = 1.18 β 6 . The parameter C 12 is adjusted to set the number of bound states supported by the interaction and the scattering length. From this calculation, we obtained the eigenenergies and then used equation (21) of [31] to extract the 2D scattering length. We found that it shows very little dependence on the number of bound states, which can be as low as 2, saving computational efforts. Using equation (7) of [19] (or equation (15) of [32]), we could then relate the 2D scattering length to the interaction strength g 2D for any q 0 -we chose a q 0 given by the zero-point momentum in the weak direction. Figure (2) compares this numerical g 2D with the analytical formula (9) for the same q 0 . Again, the position of the confinement-induced resonance for negative scattering lengths [11] is correctly predicted by (9) provided the energydependent scattering length is used. As previously, the effective range approximation works well, except for small scattering lengths. These results also suggest that the observation of the resonance may provide useful information about the effective range of the interaction.
In summary, we have shown that the effective 1D or 2D interactions of ultracold bosons in strongly confined systems are governed by 3D collisions at a relatively high energy determined 8 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT by the confinement. The effect of these high-energy collisions can be well described by a single quantity, the energy-dependent scattering length, up to extremely tight confinements. For van der Waals interactions, this quantity itself can be expressed in the effective range approximation in terms of the zero-energy scattering length and the van der Waals length. This parametrized energy-dependent scattering length leads to an accurate analytic prediction of the confinementinduced resonance both in 1D and 2D confinements.

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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT where the L ν is the Laguerre polynomial. For equation (A.5) a partial wave expansion appears not feasible, so we resort to evaluating the expressions along the ρ and z-axes. Along the z-axis (ρ = 0), we find the following expansion Along the ρ-axis (z = 0), we find the following expansion In these expansions, ζ refers to the Hurwitz zeta function which we define as follows for −1 < s < 1, generalizing the definition given in [14,33]. The first expansion (A.6) was found using the counter-term method explained in these references with the refined counter term  For instance, for α = 1%, one gets σ > 1.43r 0 . This indicates that the method works even for a confinement length σ on the order of the range r 0 .
We can also check that this is consistent with our assumption that only s-waves are scattered. Higher-order partial wave scattering arises if we take into account the dependence of A n0 on ν. The first correction is A n0 → A n0 + (ν + 1/2)B n0 , and leads to the term B n0 σ 2 /z 3 along the z-axis. This term is to be matched with the leading-order term of the scattered d-wave term (−K 2 )η 2 φ n0 (0)y 2 (kz)(−5)P 2 (cos 0), where K 2 is the d-wave reactance matrix element. In the absence of shape resonance, K 2 is purely determined by the long-range van der Waals behaviour of the interaction, K 2 ≈ − 1 100 (kβ 6 ) 4 -see equation (8) of [30]. This leads to d-wave scattering is negligible if |B n0 | |A n0 |. In the quasi-1D regime (q n0 k and k ∼ 1/σ), and using the value of η 2 (A.19), one finds σ 4 1 40 |1 − σ/a(k)| β 4 6 , which for a wide range of scattering lengths is consistent with the condition (A.20) and the cold collision condition kr 0 < 1 given in the text.