Complex Chemical Potential: Signature of Decay in a Bose-Einstein Condensate

Abstract

We explore the zero-temperature statics of an atomic Bose-Einstein condensate in which a Feshbach resonance creates a coupling to a second condensate component of quasi-bound molecules. Using a variational procedure to find the equation of state, the appearance of this binding is manifest in a collapsing ground state, where only the molecular condensate is present up to some critical density. Further, an excited state is seen to reproduce the usual low-density atomic condensate behavior in this system, but the molecular component is found to produce an underlying decay, quantified by the imaginary part of the chemical potential. Most importantly, the unique decay rate dependencies on density (∼ ρ 3/2 ) and on scattering length (∼ a 5/2 ) can be measured in experimental tests of this theory.Since the production of atomic Bose-Einstein condensates (BEC) in the laboratory [1, 2], many schemes have been proposed whereby the experimentalist may control the interatomic interactions governing the behavior of these gases [3]. One such proposal involves a Feshbach resonance in which two atoms combine to form a quasibound molecule [4,5]. This molecule is described as the intermediate state or closed channel of the scattering reaction as the constituent atoms in general have different spin configurations in the bound state than in the scattering state. Due to its dependence on the internal spin states, the energy difference, or detuning between the scattered and bound states can thus be tuned using the Zeeman effect in an external magnetic field. As the binding energy of the molecular state is brought close to the energy of the colliding atoms, the appearance of these loosely bound molecules increases. Consequently, the coupling between atoms and molecules acts to modify the effective interatomic interactions. Near zero energy, these interactions are described by the s-wave scattering length, which in turn can be tuned by varying the external magnetic field. This degree of control suggests that the negative scattering length of initially unstable condensates may be tuned to positive values thereby rendering the condensate stable. We provide a many-body variational description of such a scenario, specifically using the case of 85 Rb to compare our results with experiment [6,7]. For a uniform system our findings reveal a collapsing ground state and a decaying excited state, where the latter fits the behavior seen from experiment. In the excited state, a complex chemical potential is obtained in which the imaginary part determines the inverse decay time.Before embarking on the many-body analysis of the coupled atom-molecule BEC, it is first necessary to review the relevant two-body physics underlying the interatomic interactions. It can be seen that the two-body formalism provides a means by which the many-body equations can be renormalized by replacing the interaction strength by the s-wave scattering length as the relevant parameter. In two-atom scattering, there are a number of different channels or outcomes of the scatter...

D-dimensional Bose gases and the Lambert W function

“…These ideas have become important in statistical physics in recent years. 7,26 For the four-dimensional ideal Bose gases near condensation the complex analysis of the chemical potential and pressure is incomplete. The branch cuts and singularities of these thermodynamic functions in the complex plane need further study.…”

On the Quantum Instability of Attractive Bose Systems

“…This suggests that despite being innately attractive, the ensemble can remain stable against collapse if there is a nonnegative scattering length [1]. Nevertheless, we find a ground state that tends toward collapse, with the expected, positive-pressure case realized only at the cost of including an inherent quantum instability [2,3,4]. Elucidation of this result is provided by a series of derivations presented in the following sections.…”

“…2 Obtaining the asymptotic behavior for large |x| requires the usual expansion |x − x ′ | = x(1 − x · x ′ /x 2 + · · · ), where the leading term is kept in the denominator, but the first two terms are retained in the exponential in (5). Substitution of this expanded Green's function into (4) leads to the asymptotic scattering wave function,…”

“…Hence, the energy eigenvalue is denoted as 2k 2 . 2 We use the notation R , as the wave function is "repelled" by the potential. The point where the zero energy line intersects the radial axis is the scattering length, a, which must be non-negative for repulsion.…”

Coherent Decay of Bose-Einstein Condensates

“…(B.1) gives the energy due to the coupling of atoms to both of the molecular states. In this energy component, there is a ξ * D term with a second order part given by k,k ,k δ(k − k + k )v 1 2 k + 1 2 k ξ * (0) (k , t)D (2) (k, k , t) + ξ * (1) (k , t)D (1) (k, k , t) + H. c. 11) which can be more compactly written as q,q ,P δG * (q, P, t)D G 2 (q, q , P)δG(q, P, t) + δΣ * (q, P, t)D Σ 2 (q, q , P)δΣ(q, P, t) + q,P δφ * (P, t)D Gφ (q, P)δG(q, P, t) + δχ * (P, t)D Gχ (q, P, t)δG(q, P, t) +δω * (P, t)D Σω (q, P)δΣ(q, P, t) + δν * (P, t)D Σν (q, P)δΣ(q, P, t) , In this compact notation we have used the definitions of q and P given by Eqs. (A.10) as well as the shorthand of (A.12).…”