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The Rochester Conferences on Coherence and Quantum Optics and the Quantum Information and Measurement meeting

Stochastic Path Integral for Pre-/Post-selected Continuous Quantum Measurement

Areeya Chantasri, Justin Dressel, Andrew N. Jordan

Published Jan 1, 2013• 0 citations
Publisher

Abstract

We propose a stochastic-path-integral method to study a continuously measured qubit. Results for the most-likely qubit trajectories in Hilbert space contrained by pre- and post-selection as boundary conditions are presented.
DOI
10.1364/qim.2013.w6.43

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We present a stochastic path integral formalism for continuous quantum measurement that enables the analysis of rare events using action methods. By doubling the quantum state space to a canonical phase space, we can write the joint probability density function of measurement outcomes and quantum state trajectories as a phase space path integral. Extremizing this action produces the most likely paths with boundary conditions defined by preselected and postselected states as solutions to a set of ordinary differential equations. As an application, we analyze continuous qubit measurement in detail and examine the structure of a quantum jump in the Zeno measurement regime.
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We examine optimal paths between initial and final states for diffusive quantum trajectories in continuously monitored pure-state qubits, obtained as extrema of a stochastic path integral. We demonstrate the possibility of ``multipaths'' in the dynamics of continuously monitored qubit systems, wherein multiple optimal paths travel between the same pre- and postselected states over the same time interval. Optimal paths are expressed as solutions to a Hamiltonian dynamical system. The onset of multipaths may be determined by analyzing the evolution of a Lagrangian manifold in this phase space and is mathematically analogous to the formation of caustics in ray optics or semiclassical physics. Additionally, we develop methods for finding optimal traversal times between states or optimal final states given an initial state and evolution time; both give insight into the measurement dynamics of continuously monitored quantum states. We apply our methods in two systems: a qubit with two noncommuting observables measured simultaneously and a qubit measured in one observable while subject to a Rabi drive. In the two-observable case we find multipaths due to caustics, bounded by a diverging Van Vleck determinant, and their onset time. We also find multipaths generated by paths with different winding numbers around the Bloch sphere in both systems.
We predict that continuously monitored quantum dynamics can be chaotic. The optimal paths between past and future boundary conditions can diverge exponentially in time when there is time-dependent evolution and continuous weak monitoring. Optimal paths are defined by extremizing the global probability density to move between two boundary conditions, and are then expressed as solutions to a Hamiltonian dynamical system. We investigate the onset of chaos in pure-state qubit systems with optimal paths generated by a periodic Hamiltonian. Specifically, chaotic quantum dynamics are demonstrated in a scheme where two noncommuting observables of a qubit are continuously monitored, and one measurement strength is periodically modulated. The optimal quantum paths in this example bear similarities to the trajectories of the kicked rotor, or standard map, which is a paradigmatic example of classical chaos. We emphasize connections with the concept of resonance between integrable optimal paths and weak periodic perturbations, as well as our previous work on ``multipaths,'' and connect the optimal path chaos to instabilities in the underlying quantum trajectories.
Introduction A very general concept of a quantum trajectory would be the path taken by the state of a quantum system over time. This state could be conditioned upon measurement results, as we considered in Chapter 1. This is the sort of quantum trajectory we are most interested in, and it is generally stochastic in nature. In ordinary use, the word trajectory usually implies a path that is also continuous in time. This idea is not always applicable to quantum systems, but we can maintain its essence by defining a quantum trajectory as the path taken by the conditional state of a quantum system for which the unconditioned system state evolves continuously. As explained in Chapter 1, the unconditioned state is that obtained by averaging over the random measurement results which condition the system. With this motivation, we begin in Section 4.2 by deriving the simplest sort of quantum trajectory, which involves jumps (that is, discontinuous conditioned evolution). In the process we will reproduce Lindblad's general form for continuous Markovian quantum evolution as presented in Section 3.6. In Section 4.3 we relate these quantum jumps to photon-counting measurements on the bath for the model introduced in Section 3.11, and also derive correlation functions for these measurement records. In Section 4.4 we consider the addition of a coherent field (the ‘local oscillator’) to the output before detection. In the limit of a strong local oscillator this is called homodyne detection, and is described by a continuous (diffusive) quantum trajectory.