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Quantum and classical Josephson oscillations for a Bose-Einstein condensate

Juha Javanainen, Janne Ruostekoski

Published Apr 5, 2011• 0 citations
OpenAlex

Abstract

We study a Bose-Einstein condensate in a double-well trap when the atom number in each potential well is monitored continuously via light scattering. The quantum mechanical solution and an ab-initio classical stochastic description that incorporates the back-action of the atom number measurements as phase diffusion are compared for the Josephson oscillations. The conjecture emerges that if the continuous measurements can resolve the classical behavior, that is what they will report.

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We study a Bose-Einstein condensate in a double-well trap when the atom number in each potential well is monitored continuously via light scattering. The quantum mechanical solution and an ab-initio classical stochastic description that incorporates the back-action of the atom number measurements as phase diffusion are compared for the Josephson oscillations. The conjecture emerges that if the continuous measurements can resolve the classical behavior, that is what they will report.
We develop a classical theoretical description for nonlinear many-body dynamics that incorporates the back-action of a continuous measurement process. The classical approach is compared with the exact quantum solution in an example with an atomic Bose–Einstein condensate in a double-well potential where the atom numbers in both potential wells are monitored by light scattering. In the classical description, the back-action of the measurements appears as diffusion of the relative phase of the condensates on each side of the trap. When the measurements are frequent enough to resolve the system dynamics, the system behaves classically. This happens even deep in the quantum regime, and demonstrates how classical physics emerges from quantum mechanics as a result of measurement back-action.
We use quantum trajectories to simulate Josephson oscillations of atomic condensates between the two sides of a double-well potential. In the simulations the atoms in both wells are monitored using off-resonant light scattering, and the ultimate outcome of our thought experiment is a sequence of photon counts probing the numbers of the atoms in each potential well. We show how to reconstruct the Josephson oscillations from the observed photon counts using Bayesian inference, and study the oscillations quantitatively by averaging the inferred time-dependent oscillation amplitude over a large number of realizations. Scaling behaviors that characterize the oscillations are uncovered and related to physics principles such as measurement back-action. It turns out that the scalings hold true for quite small atom numbers, so that in this sense four atoms in a potential well may already make a Bose-Einstein condensate.
The properties of an atomic Bose-Einstein condensate in a double-well potential are investigated through a two-mode analysis. An analytic solution for the semiclassical tunnelling and self-trapping dynamics is compared with numerical simulations of the quantum dynamics, which exhibit collapses and revivals for a closed system. A continuous non-destructive measurement technique to monitor the Josephson tunnelling oscillations is presented, in which the condensate in one well dispersively shifts the phase of a coherent probe beam in proportion to atom-number. The evolution of the resulting homodyne photocurrent and Bloch Q distributions shows that oscillations develop even when the initial state possesses phase symmetry. The conditional dynamics of the condensate which result from measurement back-action also appear in certain semiclassical formulations. The homodyne measurement technique is incorporated into a proposed weak-force detector. A maximally entangled initial state, which is the ground state for a double condensate with strong attractive atomic interactions, enables a high-precision measurement. The dynamics of quantum many-body multimode systems of interacting bosons are simulated using phase-space methods. The use of the Wigner technique predicts novel noise effects in fibre solitons. The positive-P representation is used to simulate the formation of mesoscopic Bose-Einstein condensates via evaporative cooling in three dimensional atom traps. The results indicate highly non-classical behaviour near the critical point, and provide evidence for the spontaneous formation of vortices. Comparisons with corresponding mean-field calculations reveal large differences between the semiclassical and fully quantum results. Finally, the possibility of future progress with alternative phase-space methods is considered.
We study a Bose-Einstein condensate (BEC) in a double-well potential subject to an unsharp continuous measurement of the atom number in one of the two wells. We investigate the backaction of the measurement on the quantum dynamics and the viability to monitor the ensuing time evolution. For vanishing interatomic interactions, mainly the expectation values of the measured local observable can be inferred from the measurement record. Conversely, in the presence of moderate interatomic interactions, the entire many-body state---modified by the measurement---is monitored with unit fidelity and, at the same time, the measurement effects a transition from quantum to mean-field (classical) behavior of the BEC. We show that this perfect state estimation is possible because the interatomic interactions enhance the information gained via the measurement.
Using the mean-field Gross-Pitaevskii (GP) equation we study the evolution of Josephson current in a repulsive Bose-Einstein condensate in an optical-lattice potential initiated by a sudden displacement of the trapping potential. Similarly, we also study the evolution of quantum tunneling in a double-well potential in one and three dimensions initiated by a trap displacement leading to a periodic oscillation of the condensate across the barrier which could be observed experimentally. The frequencies of Josephson current found directly from the solution of the GP equation are in agreement with the experiment by Cataliotti et al. [2001 Science 293 843]. The breakdown of the Josephson current for a large trap displacement is also in agreement with another experiment by Cataliotti et al. [2003 New J. Phys. 5 71]. PACS numbers: 03.75.Lm, 03.75.KkQuantum tunneling and Josephson current in Bose-Einstein condensates 2 1.