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Annual Meeting Optical Society of America

Decay of quantum coherence using quantum trajectories

H. J. Carmichael, L. Tian, P. Kochan

Published Jan 1, 1992• 0 citations
Publisher

Abstract

In the quantum trajectory approach an open quantum system is represented by a stochastic pure-state wave function. The mixed-state density operator that satisfies the usual master equation is recovered from the quantum trajectories by performing either an ensemble average or, for stationary systems, an average over time. This unraveling of the master equation dynamics into pure-state trajectories provides new insight into the decay of quantum coherence in systems that are open to the environment. Under some conditions macroscopic superposition states are preserved as macroscopic superposition states along individual trajectories, but are reduced to mixtures by the trajectory average. Under other conditions one of the states in a macroscopic superposition becomes dominant (in amplitude) over the other along each quantum trajectory. We discuss the mechanisms that produce these dynamics and assess the implications for the observation of Schrodinger cat states in optics experiments. We present results for some specific examples of macroscopic superposition states generated by a cavity QED system. The system involves a small collection of atoms in an optical cavity driven by a coherent laser field. Under strong-coupling conditions this system produces a variety of Schrodinger cat states whose precise form depends on the number of atoms, the method of excitation (through a cavity mirror or from the side), the initial state of the atoms, and the position of the atoms in the cavity.

DOI
10.1364/oam.1992.mff4

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In the quantum trajectory approach an open quantum system is represented by a stochastic pure-state wave function. The mixed-state density operator that satisfies the usual master equation is recovered from the quantum trajectories by performing either an ensemble average or, for stationary systems, an average over time. This unraveling of the master equation dynamics into pure-state trajectories provides new insight into the decay of quantum coherence in systems that are open to the environment. Under some conditions macroscopic superposition states are preserved as macroscopic superposition states along individual trajectories, but are reduced to mixtures by the trajectory average. Under other conditions one of the states in a macroscopic superposition becomes dominant (in amplitude) over the other along each quantum trajectory. We discuss the mechanisms that produce these dynamics and assess the implications for the observation of Schrodinger cat states in optics experiments. We present results for some specific examples of macroscopic superposition states generated by a cavity QED system. The system involves a small collection of atoms in an optical cavity driven by a coherent laser field. Under strong-coupling conditions this system produces a variety of Schrodinger cat states whose precise form depends on the number of atoms, the method of excitation (through a cavity mirror or from the side), the initial state of the atoms, and the position of the atoms in the cavity.
The decay of coherence when a quantum system interacts with a much larger environment is usually described by a master equation for the system reduced density matrix and emphasizes the evolution of an entire ensemble. We consider two methods that have been developed recently to simulate the evolution of single realizations. Quantum-state diffusion involves both diffusion, where the individual quantum trajectory fluctuates through a Wiener process deriving from the environment, and localization to a coherent state, an eigenstate of the relevant Lindblad operator describing the coupling of the system to the environment. We demonstrate the localization process for different initial states and utilize the Wigner function to depict this localization in phase space. We concentrate on quantum states that can be expressed as a superposition of appropriate coherent states. For an initial superposition of two coherent states (a Schr\"odinger ``cat''), one of the two components will dominate the evolution. For initial Fock states, which can be described as a continuous superposition of coherent states on a ring, localization takes place when one coherent state is selected from that ring where each component has nearly the same energy as the original Fock state. We also consider the localization from a nonclassical squeezed ground state, which can be expressed as a superposition of coherent states along a line in phase space. The second simulation method considered is the state vector Monte Carlo, or ``quantum jump,'' approach, which relates to the direct counting of decay quanta. In the case of an initial Schr\"odinger ``cat,'' we find that when no quantum is detected the ``cat'' shrinks, but when a quantum is detected, the Schr\"odinger ``cat'' ``jumps'' from one type of ``cat'' to another with different internal phase. For an initial squeezed state we show how quantum jumps lead to individual realizations which are superpositions of two squeezed states.
We challenge the standard picture of decohering Schrödinger cat states as an ensemble average obeying a Lindblad master equation, brought about locally from an irreversible interaction with an environment. We generate self-consistent collections of pure system states correlated with specific environmental records, corresponding to the function of the wave-particle correlator first introduced in Carmichael et al. [ Phys. Rev. Lett. 85 (2000)]. In the spirit of Carmichael et al. [in Coherent States: Past, Present and Future , World Scientific, 1994], we find that the complementary unravelings evince a pronounced disparity when the “position” and “momentum” of the damped cavity mode — an explicitly open quantum system — are measured. Intensity-field correlations may largely deviate from a monotonic decay, while Wigner functions of the cavity state display contrasting manifestations of quantum interference when conditioned on photon counts sampling a continuous photocurrent. In turn, the conditional photodetection events mark the contextual diffusion of both the net charge generated at the homodyne detector, and the electromagnetic field amplitude in the resonator.
By definition, an ensemble of quantum trajectories reproduces the density matrix of an open system after ensemble averaging. We present a general class of stochastic Schrodinger equations for quantum trajectories which unifies and extends existing methods. Various quantum jump and state-diffusion methods are recognized as opposite limiting cases. The inherent freedom of choice can be used to significantly reduce the calculation time in specific cases. Examples of this are methods with adaptive noise, for which a specific observable becomes noise free to first order. Linear stochastic equations can also be constructed, which allow analytical solutions. As an illustration, the decoherence of Schrodinger cat states of a cavity mode is discussed.
A quantum trajectory represents a single history of the state vector of an open quantum system. The stochastic nature of its evolution can be viewed as resulting from the quantum probabilities for the outcomes of measurements performed on the bath. We derive stochastic Schrödinger equations for various trajectory models, starting from the full Schrödinger equation for the system and the bath combined. The state reduction of the bath at a measurement also projects the state of the system on a pure state, thereby destroying the entanglement. The master equation for the system's density matrix is obtained as a result of the discussion, and it is not taken as its starting point. The general concepts are illustrated in the specific case of an atom coupled to the radiation field.
A reason is discussed (may be not the only one) for why we do not see any superposition of macroscopic states in the real world. Under the general assumption that quantum macrostates are statistical ensembles of microstates, it is shown that any superposition of macrostates is reduced in a very short time by the unitary dynamics of the ordinary Schrödinger equation, deducing the Born rule without having to postulate it. In more detail, the macroscopic and microscopic degrees of freedom are decoupled in the Schrödinger equation, yielding an effective stochastic equation for the macroscopic variables, with the ensemble average of the microscopic amplitudes that acts as a self-generated internal white noise. The stochastic equation is shown to be a reducing Itô equation if some general causality conditions are met, predicting a very quick collapse of any macroscopic superposition upon formation, with probabilities which satisfy the Born rule. In the context of the von Neumann measurement scheme, the relevance of the result is discussed as a simple dynamical solution of the measurement problem.
A Schrödinger-cat state is a coherent superposition of macroscopically distinguishable quantum states, in quantum optics usually realized as superposition of coherent states.Protocols to prepare photonic cats have been presented for atomic systems.Here we investigate in what manner and how well the preparation protocols can be transferred to a solid-state platform, namely, a semiconductor quantum-dot-cavity system.In quantum-dotcavity systems there are many disruptive influences like cavity losses, the radiative decay of the quantum dot, and the coupling to longitudinal acoustic phonons.We show that for one of the protocols these influences kill the quantum coherence between the states forming the cat, while for a second protocol a parameter regime can be identified where the essential characteristics of Schrödinger-cat states survive the environmental influences under conditions that can be realized with current equipment.