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Physical Review A

Quantum optical tunneling: A representation-free theory valid near the state-equation turning points

P. D. Drummond

Published Jun 1, 1986• 31 citations
Publisher

Abstract

The problem of representation dependence in the treatment of quantum tunneling in absorptive optical bistability is considered. Disregarding this problem can lead to order-of-magnitude errors in calculated tunneling rates. A new description is presented, valid near the state-equation turning points, in which regime the discrepancies between different representations are shown to disappear. This description is also useful for treating critical fluctuations and dispersive cases.

DOI
10.1103/physreva.33.4462

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The problem of representation dependence in the treatment of quantum tunneling in absorptive optical bistability is considered. Disregarding this problem can lead to order-of-magnitude errors in calculated tunneling rates. A new description is presented, valid near the state-equation turning points, in which regime the discrepancies between different representations are shown to disappear. This description is also useful for treating critical fluctuations and dispersive cases.
We specialize the quantum-statistical theory of optical bistability developed in "Optical Bistability. II" to a low-QQQ cavity. From a linearized Fokker-Planck equation derived in the positive PPP representation, the steady-state correlation matrix and spectrum of fluctuations are calculated. Expressions which normally assume a positive-definite diffusion matrix are shown to be valid for nonpositive-definite diffusion. For absorptive bistability we find agreement with results derived previously by quite different methods. More generally we extend previously published results by including atomic and cavity detunings. We compare the transmitted spectrum for dispersive bistability with that for absorptive bistability, and photon antibunching in dispersive bistability is discussed.
The problem of the global stability of steady states in optical bistability is considered with emphasis toward the distinction between quantum switching (characterized by a classical diffusive process) and quantum tunneling (characterized by a nonclassical process with a singular and nonpositive Glauber-Sudarshan distribution). Relative to treatments of quantum switching in optical bistability for which nonpositive definite diffusion is excluded, we address the issue of global stability when the atom-field interaction is purely radiative and nonclassical processes (such as squeezing and antibunching) are involved. In the 1-D approximation, we point out certain anomalous features in the eigenvalue structure of the switching problem associated with a change in sign of the diffusion coefficient in the Glauber-Sudarshan representation. To substantiate the validity of the 1-D approximation, we explore the topology of the 2-D generalized potential. Finally, we compare our result with those previously obtained to emphasize in qualitative terms the distinction between switching and tunneling for bistable optical systems. Our analysis is motivated by an attempt to identify explicit experimental signatures of quantum tunneling.
For the model of Drummond and Walls describing dispersive optical bistability, quantum tunneling rates are calculated in the low-cavity-damping limit. Without damping the system can be described by an appropriate Hamilton operator. By expanding the density operator into eigenstates of this Hamilton operator the equation of motion of the density operator is cast into an equation of motion for the density matrix. In this equation we take into account only diagonal elements and those nondiagonal elements for which the corresponding eigenstates of the Hamilton operator are nearly degenerate. The tunneling rates are then determined by the eigenvalues of the truncated density-matrix equation.
A quantum treatment of a coherently driven cavity with a medium of two-level atoms is given. The system displays both absorptive and dispersive bistability. Quantum fluctuations are included via a Fokker-Planck equation in a generalized PPP representation, after adiabatic elimination of the atomic variables. The transmitted spectrum and correlation function is calculated in a linearized (Gaussian) approximation. In the limit of large detuning and radiative damping, the quantum fluctuations are identical to those of a simple nonlinear polarizability theory. An analysis is given of the nonclassical statistical properties of the intensity-correlation spectrum.
For the model of Drummond and Walls describing dispersive optical bistability, stationary solutions as well as quantum-tunneling rates are calculated by solving a pseudo-Fokker-Planck equation (PFPE), i.e., a Fokker-Planck-like equation with a non-positive-definite diffusion matrix, in terms of matrix continued fractions. Due to the quantum nature of the model, an interesting oscillating variation of the decay rates of the PFPE as a function of the scaling factor for the photon number is found for small cavity damping.
A quantum treatment of a coherently driven dispersive cavity is given based on a cubic nonlinearity in the polarisability of the internal medium. This system displays bistability and hysteresis in the semiclassical solutions. Quantum fluctuations are included via a Fokker-Planck equation in a generalised P representation. The transmitted light shows a transition from a single-peaked spectrum to a double-peaked spectrum above the threshold of the lower branch. Fluctuations in the field are reduced on the upper branch and both photon bunching and photon antibunching are predicted, for different operating points. An exact solution obtained for the steady-state generalised P function shows decidedly non-equilibrium behaviour, e.g. the lack of a Maxwell construction.