Radial action-phase quantization in Bose–Einstein condensates
Introduction
The amplitude-phase formulation of the 1D stationary (time-independent) Schroedinger equation defined by its energy eigenvalue E and its potential V is known since Milne showed long ago [1] its link with the Ermakov–Milne–Pinney (EMP) nonlinear differential equation [2], [3] The dot overscript defines the derivative along the chosen (usually radial) spatial degree of freedom while and m is the particle mass. The constant Ermakov coefficient A in Eq. (1) is that additional degree of freedom with respect to the standard Schroedinger quantum description which yields the amplitude-phase holographic representation of a quantum eigenstate. Indeed it provides the following action quantization rule along a closed return circuit Γ in the configuration space, in terms of the usual quantum hydrodynamical particle momentum where S is the quantum-phase action which is defined by the Madelung ansatz of the Schroedinger wave function while is solution of Eq. (1). In Eq. (2), the positive integer n is the total number of nodes of the eigenstate [4], [5], [6]. The action-phase quantization Eq. (2) has been successfully illustrated in the case of the (an)harmonic oscillator [4], [5], the central-field Kepler system [6] and the particle in the box [5]. It yields an interesting classical interpretation of the probabilistic Born postulate in terms of the time spent by the quantum system about any point of the configuration space, as already suggested by White in the early thirties [7]. Note the ground state quantum-phase action which is equal to h and not to like in Sommerfeld's semiclassical action quantization rule. This remark clearly emphasizes the basic difference between the exact EMP theory and the Wentzel–Kramers–Brillouin (WKB) semiclassical approximation, due to the alternative definition of the particle momentum as the gradient of the quantum-phase action S. The EMP transformation between the Schroedinger wave function Ψ and the general solution a of Eq. (1) may involve a great variety of integration constants and/or invariants [8], [9]. It was used, for instance, in quantum defect theory [10] and its link with the WKB approximation for solving one-dimensional wave equations was emphasized [11]. Then the extension of the EMP theory to a particular class of time-dependent wave-packet quantum systems displayed the central role devoted to the Ermakov invariant in order to connect the time-dependent Schroedinger equation, the Feynman-kernel time propagator and the Wigner-function methods [12], [13]. Its further generalization to the time-dependent Gross–Pitaevskii (GP) nonlinear Schroedinger equation that describes the mean-field approximation of a dilute gaseous Bose–Einstein condensate (BEC) was achieved by use of an exact analytical moment method [14].
The present Letter considers a stationary vortex state of such a 2D axisymmetrical BEC and shows that the action quantization rule Eq. (2) along the radial degree of freedom still survives in the presence of the cubic GP nonlinearity for particular discrete values of the Ermakov constant. This result is reminiscent of Einstein's pioneering attempt to quantize classical trajectories on a 2D torus [15]. Indeed, once the radial component of the particle momentum is allowed in addition to its azimuthal vortex component (where m is the vortex circulation integer), the question of the quantum counterpart of classical quasiperiodic orbits on such a 2D torus arises. Einstein's conclusion was that these later were not quantizable, in contrast with periodic orbits. It seems that the present work provides an illustration of this conjecture: the discrete series of nonlinear EMP quantum trajectories defined by the 's and Eq. (2) resemble Einstein's discrete quantizable periodic orbits on the torus.
The Letter is organized in seven sections as follows: Section 2 provides, for each chemical potential μ and Ermakov constant A, the accumulated—or path-integral—quantum phase through the corresponding EMP transformation of the nonlinear stationary GP equation by use of the Madelung ansatz . This yields in Section 3 the discrete series of values that define regular radial BEC order parameters (i.e. radial GP nonlinear eigenstates). In Section 4 we use the strong Thomas–Fermi particle–particle interaction description of the dilute gaseous BEC in order to display analytical solutions and, therefore (Section 5), the corresponding Ermakov splitting of the radial energy spectrum of the condensate that lifts the degeneracy of the linear case [4], [5], [6]. Typical values of this splitting are obtained in Section 6 when considering the three main vortex-nucleation series of BEC experiments: they range from a few percents to 11% of the vortex azimuthal kinetic energy per particle. We conclude in Section 7 by pointing out interesting experimental perspectives and/or theoretical developments of the present work.
Section snippets
EMP description of a 2D GP nonlinear quantum system
The mean-field zero-temperature stationary GP equation of a BEC reads [16], [17], [18]: where is the external (usually harmonic, like in the present work) confining potential, the coupling constant is defined by the scattering length or, equivalently, by the healing length related to the peak density n of the condensate. Eq. (3) defines the discrete “nonlinear eigenvalue” (or chemical potential) μ that is related to the
The radial action-phase Ermakov quantization
There is another generic property of all EMP systems: the Ermakov amplitude defined by Eq. (1) diverges at the boundaries of the system [1], [2], [3], [4], [5], [6], [9]. Although the location of this divergence can always be repelled outside of any compact manifold in the configuration space for small enough values of the Ermakov constant A in the linear case (and therefore outside of the physical space that includes the quantum system, making this divergence more mathematical than physical in
The Thomas–Fermi approximation
The Thomas–Fermi approximation of Eq. (7) consists in neglecting the quantum kinetic energy, defined by the Laplacian term in Eqs. (4), (7) (namely the two first terms in the l.h.s. of this later equation) in comparison with the total energy of the particle. It is quite acceptable in a dilute gaseous alkaline BEC, when the condensate particle number , i.e. when the interatomic repulsion is strong [16], [17], [18]. Then Eq. (7) reduces to the algebraic equation where
The Ermakov splitting of the radial energy spectrum
The Thomas–Fermi approximation of the nonlinear GP Schroedinger Eq. (4), which yields , leads to the following convenient perturbation scheme concerning the nonlinear eigenvalues (or chemical potential) : where (respectively ) is the chemical potential of the first excited vortex state (respectively the vortexless ground state) and where the brackets label the quantum expectation value to be performed with the integrable nonlinear
BEC vortex experiments
Let us now summarize and illustrate the above EMP Thomas–Fermi description of vortex radial energy quantization by use of the MIT [21], [22], JILA [23] and ENS [24], [25], [26], [27] experimental results about vortex nucleation. Eqs. (22), (23) yield as the ratio of radial kinetic energy per particle over azimuthal kinetic energy per particle for the BEC vortex state. The dimensionless energy parameter is given by the value of the ground state chemical
Conclusion
The above results show that the radial quantized energy per particle in BEC-vortex nucleation experiments is expected to range over some percents of the azimuthal one. Although this property explains why a pure azimuthal vortex velocity field is quite a good approximation for the description of standard BEC-vortex experiments, it suggests a new experimental challenge (using phase-imprinting vortex-nucleation techniques such as JILA's [29] or MIT's [30] ones?) that would consist in detecting the
Acknowledgements
The author is deeply grateful to Y. Castin, E. Cornell, J. Dalibard, J. de Freitas-Pacheco, G. Kavoulakis, D. Schuch and P. Valiron for several very useful comments. He acknowledges the hospitality of the Science Institute of the University of Iceland (Reykjavik) and warmly thanks G. Björnsson and V. Gudmundsson for numerous fruitful discussions there.
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