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arXiv:1807.10398

Cavity QED with tunneling: an application of adiabatic elimination in quantum trajectory theory

Charles H. Baldwin, Perry R. Rice

Published Jul 26, 2018• 0 citations
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Abstract

We model the evolution of a single atom in a cavity interacting with two lasers, one far off resonance which creates an optical potential lattice and one near resonance, which can interact with the atom. The atom may tunnel between sites in the lattice. We find the photon counting statistic $g^{(2)}(τ)$ depends on the tunneling rate. Additionally we derive methods to work with a Bose-Einstein Condensate in the same situation, in ther Josephson junction regime. It is shown that oscillations in $g^{(2)}(τ)$ are directly proportional to the tunneling coefficient. We work in the bad-cavity limit and give an example of the use of adiabatic elimination in quantum trajectory theory

DOI
10.48550/arxiv.1807.10398

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We model the evolution of a single atom in a cavity interacting with two lasers, one far off resonance which creates an optical potential lattice and one near resonance, which can interact with the atom. The atom may tunnel between sites in the lattice. We find the photon counting statistic g(2)(τ)g^{(2)}(τ)g(2)(τ) depends on the tunneling rate. Additionally we derive methods to work with a Bose-Einstein Condensate in the same situation, in ther Josephson junction regime. It is shown that oscillations in g(2)(τ)g^{(2)}(τ)g(2)(τ) are directly proportional to the tunneling coefficient. We work in the bad-cavity limit and give an example of the use of adiabatic elimination in quantum trajectory theory
We model the evolution of a single atom in a cavity interacting with two lasers, one far off resonance which creates an optical potential lattice and one near resonance, which can interact with the atom.The atom may tunnel between sites in the lattice.We find that the photon counting statistic g (2) (τ ) depends on the tunneling rate.Additionally we derive methods to work with a Bose-Einstein Condensate in the same situation.It is shown that oscillations in g (2) (τ ) are directly proportional to the tunneling coefficient given in our models.
We study the interplay between the dynamics of a Bose-Einstein condensate in a double-well potential and that of an optical cavity mode. The cavity field is superimposed to the double-well potential and affects the atomic tunneling processes. The cavity field is driven by a laser red detuned from the bare cavity resonance; the dynamically changing spatial distribution of the atoms can shift the cavity in and out of resonance. At resonance the photon number is hugely enhanced and the atomic tunneling becomes amplified. The Josephson-junction equations are revisited and the phase diagram is calculated. We find solutions with finite imbalance and at the same time find a lack of self-trapping solutions due to the emergence of a new separatrix resulting from enhanced tunneling.
The method of adiabatic elimination has been widely adopted in quantum optics in the past several decades. In the study of cavity-based light-matter interactions, the bad-cavity limit is often encountered, where the damping rate of the cavity is much larger than the interaction strength. The fast-damped cavity will quickly relax to a quasi-stationary state, and one can eliminate the cavity from the equation of motion by setting its time derivative to zero. Elimination of the cavity in the bad-cavity limit can reduce the dimensionality of the equations of motion of the system. However, we find that the adiabatic elimination procedure performed in the quantum Langevin equation leads to an incorrect commutation relation, which was rarely discussed in the former studies, as far as we know. Here, we show the incorrect commutation relation arises from the fact that the high frequency of the vacuum noise should be cut off to perform adiabatic elimination, but the noise with high frequency cutoff is still treated as white noise with infinite bandwidth and delta commutation relation. We also study the correlation function and show that the high frequency part of noise contributes very little when averaged over the bath. Therefore, the adiabatic elimination method can reduce the complexity of the calculations while maintaining physical reliability.
We have developed a theory of a Josephson junction formed by two tunnel-coupled Bose-Einstein condensates in a double-well potential in the regime of strong atom-atom interaction for an arbitrary total number NNN of bosons in the condensates. The tunnel resonances in the junction are shown to be periodically spaced by the interaction energy, forming a single-atom staircase sensitive to the parity of NNN even for large NNN. One of the manifestations of the staircase structure is the periodic modulation with the bias energy of the visibility of the interference pattern in lattices of junctions.
We review our theoretical advances in quantum tunneling of Bose-Einstein condensates in optical traps and in microcavities. By employing a real physical system, the frequencies of the pseudo Goldstone modes in different phases between two optical traps are studied respectively, which are the crucial feature of the non-Abelian Josephson effect. When the optical lattices are under gravity, we investigate the quantum tunneling in the “Wannier-Stark localization” regime and “Landau-Zener tunneling” regime. We finally get the total decay rate and the rate is valid over the entire range of temperatures. At high temperatures, we show how the decay rate reduces to the appropriate results for the classical thermal activation. At intermediate temperatures, the results of the total decay rate are consistent with the thermally assisted tunneling. At low temperatures, we obtain the pure quantum tunneling ultimately. And we study the alternating-current and direct-current (ac and dc) photonic Josephson effects in two weakly linked microcavities containing ultracold two-level atoms, which allows for direct observation of the effects. This enables new investigations of the effect of many-body physics in strongly coupled atom-cavity systems and provides a strategy for constructing novel interference devices of coherent photons. In addition, we propose the experimental protocols to observe these quantum tunneling of Bose-Einstein condensates.