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arXiv:1612.07861 · Physical Review A

Prediction and characterization of multiple extremal paths in continuously monitored qubits

Philippe Lewalle, Areeya Chantasri, Andrew N. Jordan

Published Apr 21, 2017• 24 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.95.042126

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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.
We generalize and extend the stochastic path-integral formalism and action principle for continuous quantum measurement introduced in Chantasri, Dressel, and Jordan [Phys. Rev. A 88, 042110 (2013)], where the optimal dynamics, such as the most likely paths, is obtained by extremizing the action of the stochastic path integral. In this work, we apply exact functional methods as well as develop a perturbative approach to investigate the statistical behavior of continuous quantum measurement. Examples are given for the qubit case. For qubit measurement with zero-qubit Hamiltonian, we find analytic solutions for average trajectories and their variances while conditioning on fixed initial and final states. For qubit measurement with unitary evolution, we use the perturbation method to compute expectation values, variances, and multi-time-correlation functions of qubit trajectories in the short-time regime. Moreover, we consider continuous qubit measurement with feedback control, using the action principle to investigate the global dynamics of its most likely paths, and finding that in an ideal case, qubit state stabilization at any desired pure state is possible with linear feedback. We also illustrate the power of the functional method by computing correlation functions for the qubit trajectories with a feedback loop to stabilize the qubit Rabi frequency.
Abstract In the geometry of quantum evolutions, a geodesic path is viewed as a path of minimal statistical length connecting two pure quantum states along which the maximal number of statistically distinguishable states is minimum. In this paper, we present an explicit geodesic analysis of the dynamical trajectories that emerge from the quantum evolution of a single-qubit quantum state. The evolution is governed by an Hermitian Hamiltonian operator that achieves the fastest possible unitary evolution between given initial and final pure states. Furthermore, in addition to viewing geodesics in ray space as paths of minimal length, we also verify the geodesicity of paths in terms of unit geometric efficiency and vanishing geometric phase. Finally, based on our analysis, we briefly address the main hurdles in moving to the geometry of quantum evolutions for open quantum systems in mixed quantum states.
We consider the evolution of a quantum simple harmonic oscillator in a general Gaussian state under simultaneous time-continuous weak position and momentum measurements.We deduce the stochastic evolution equations for position and momentum expectation values and the covariance-matrix elements from the characteristic function of the system.By generalizing the Chantasri-Dressel-Jordan (CDJ) formalism (Chantasri et al., 2013(Chantasri et al., , 2015) ) to this continuous-variable system, we construct its stochastic Hamiltonian and action.Action extremization gives us the equations for the most likely readout paths and quantum trajectories.For steady states of the covariance-matrix elements, the analytical solutions for these most likely paths are obtained.Using the CDJ formalism, we calculate final-state probability densities exactly starting from any initial state.We also demonstrate the agreement between the optimal-path solutions and the averages of simulated clustered stochastic trajectories.Our results provide insights into the time dependence of the mechanical energy of the system during the measurement process, motivating their importance for quantum measurement-engine or refrigerator experiments.Although quantum mechanics fundamentally limits our ability to extract information via simultaneous measurements of noncommuting observables, insights into the dynamics of the system during such a process can lead to a deeper understanding of the physics of measurement.
We consider the temporal correlations of the quantum state of a qubit subject to simultaneous continuous measurement of two noncommuting qubit observables. Such qubit state correlators are defined for an ensemble of qubit trajectories, which has the same fixed initial state and can also be optionally constrained by a fixed final state. We develop a stochastic path integral description for the continuous quantum measurement and use it to calculate the considered correlators. Exact analytic results are possible in the case of ideal measurements of equal strength and are also shown to agree with solutions obtained using the Fokker-Planck equation. For a more general case with decoherence effects and inefficiency, we use a diagrammatic approach to find the correlators perturbatively in the quantum efficiency. We also calculate the state correlators for the quantum trajectories which are extracted from readout signals measured in a transmon qubit experiment, by means of the quantum Bayesian state update. We find an excellent agreement between the correlators based on the experimental data and those obtained from our analytical and numerical results.