Classical versus quantum dynamics of the atomic Josephson junction

Abstract

We compare the classical (mean-field) dynamics with the quantum dynamics of atomic Bose-Einstein condensates in double-well potentials. The quantum dynamics are computed using a simple scheme based upon the Raman-Nath equations. Two different methods for exciting a non-equilbrium state are considered: an asymmetry between the wells which is suddenly removed, and a periodic time oscillating asymmetry. The first method generates wave packets that lead to collapses and revivals of the expectation values of the macroscopic variables, and we calculate the time scale for these revivals. The second method permits the excitation of a single energy eigenstate of the many-particle system, including Schrödinger cat states. We also discuss a band theory interpretation of the energy level structure of an asymmetric double-well, thereby identifying analogies to Bloch oscillations and Bragg resonances. Both the Bloch and Bragg dynamics are purely quantum and are not contained in the mean-field treatment.

Impurity in a Bose-Einstein condensate in a double well

“…As explained in Ref. [27], in this latter picture, the tilt plays the role of the quasimomentum of the particle, and the eigenvalue structure seen in Fig. 1 as a function of tilt can then be viewed as a band structure plotted as a function of quasimomentum.…”

“…II and the mean-field theory presented in this section is analogous to that between quantum and classical mechanics for a single particle: the N → ∞ limit of the many-body theory gives the mean-field theory, and is equivalent to theh → 0 limit of single-particle quantum theory [26]. In fact, the precise relationship is NJ /U = (S/h) 2 , where S is the classical action [27].…”

“…The energy level structure shown in Fig. 1 does not have the usual periodic form we expect of a band structure, but it can be made periodic by applying a simple transformation [27].…”

“…Furthermore, the use of a Feshbach resonance would allow the interaction between the quantum system and the measurement device to be tuned between being weak and strong. In this context, we note that the interesting question of the classical (mean-field) limit of bosons in a double-well potential has been addressed in a number of papers [26][27][28] with the general conclusion that this occurs when N → ∞, although with the caveat that quantum effects remain important close to the separatrix [27,28], which is the boundary in phase space outside of which self-trapping occurs.…”

Analytical results for Josephson dynamics of ultracold bosons

“…For a semiclassical evaluation of j(t) according to equations (18) and (19) we need semiclassical expressions for E n − E n±k and the overlap coefficients c n . While the spectrum was discussed in section II, we start here with the latter.…”