Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Frontiers in Optics

Multiple Most-Likely Path Solutions for Continuously Monitored, Driven Qubits

Philippe Lewalle, Areeya Chantasri, Andrew N. Jordan

Published Jan 1, 2016• 0 citations
Publisher

Abstract

We examine most-likely paths for continuously monitored pure-state qubits, obtained as an extremum of a stochastic path integral. By considering the evolution of the Lagrange Manifold, we locate multiple-path solutions, analogous to optical caustics.
DOI
10.1364/fio.2016.ftu2g.4

Related papers

Prediction and characterization of multiple extremal paths in continuously monitored qubits

Philippe Lewalle, Areeya Chantasri, Andrew N. Jordan

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.

arXiv:1612.07861•Physical Review A•Apr 21, 2017•24 citations

Chaos in continuously monitored quantum systems: An optimal-path approach

Philippe Lewalle, John Steinmetz, Andrew N. Jordan

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.

arXiv:1803.07615•Physical Review A•Jul 31, 2018•17 citations

Quantum caustics in resonance-fluorescence trajectories

Mahdi Naghiloo, Desmond Kai Teck Tan, P. M. Harrington +3 more

We employ phase-sensitive amplification to perform homodyne detection of the resonance fluorescence from a driven superconducting artificial atom. Entanglement between the emitter and its fluorescence allows us to track the individual quantum state trajectories of the emitter conditioned on the outcomes of the field measurements. We analyze the ensemble properties of these trajectories by considering trajectories that connect specific initial and final states. By applying the stochastic path-integral formalism, we calculate equations of motion for the most-likely path between two quantum states and compare these predicted paths to experimental data. Drawing on the mathematical similarity between the action formalism of the most-likely quantum paths and ray optics, we study the emergence of caustics in quantum trajectories: places where multiple extrema in the stochastic action occur. We observe such multiple most-likely paths in experimental data and find these paths to be in reasonable quantitative agreement with theoretical calculations.

arXiv:1612.03189•Physical Review A•Nov 3, 2017•30 citations

Purification of a monitored qubit: exact path-integral solution

Matheus M. R. Poltronieri Martins, Henrique Santos Lima

We investigate the purification dynamics of a single qubit under continuous in time monitoring. By employing a collisional model framework where the system interacts sequentially with ancillary qubits, we describe the conditioned evolution of the density matrix through a stochastic master equation. We show that for initial mixed states, the dynamics reduce to a multiplicative Langevin equation for a single scalar parameter representing the state's purity. This stochastic process is solved exactly using the Onsager-Machlup path integral formalism, allowing us to derive the full probability distribution for the qubit's trajectories. Our analytical results reveal that purification is characterized by a dynamical crossover from a diffusion dominated regime to a measurement dominated regime, visible in the emergence of a bimodal state distribution. The analytical solutions are in strong agreement with numerical simulations, providing a robust theoretical benchmark for the study of information extraction in monitored quantum systems.

quant-pharXiv:2605.12783•May 14, 2026•0 citations

Stochastic path-integral formalism for continuous quantum measurement

Areeya Chantasri, Andrew N. Jordan

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.

arXiv:1507.07016•Physical Review A•Sep 21, 2015•60 citations

Multiple Most Likely Paths in Diffusive Quantum Trajectories of Pure-State Qubits

Philippe Lewalle, Areeya Chantasri, Andrew N. Jordan

Bulletin of the American Physical Society•Mar 14, 2017•0 citations
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 employ phase-sensitive amplification to perform homodyne detection of the resonance fluorescence from a driven superconducting artificial atom. Entanglement between the emitter and its fluorescence allows us to track the individual quantum state trajectories of the emitter conditioned on the outcomes of the field measurements. We analyze the ensemble properties of these trajectories by considering trajectories that connect specific initial and final states. By applying the stochastic path-integral formalism, we calculate equations of motion for the most-likely path between two quantum states and compare these predicted paths to experimental data. Drawing on the mathematical similarity between the action formalism of the most-likely quantum paths and ray optics, we study the emergence of caustics in quantum trajectories: places where multiple extrema in the stochastic action occur. We observe such multiple most-likely paths in experimental data and find these paths to be in reasonable quantitative agreement with theoretical calculations.
We investigate the purification dynamics of a single qubit under continuous in time monitoring. By employing a collisional model framework where the system interacts sequentially with ancillary qubits, we describe the conditioned evolution of the density matrix through a stochastic master equation. We show that for initial mixed states, the dynamics reduce to a multiplicative Langevin equation for a single scalar parameter representing the state's purity. This stochastic process is solved exactly using the Onsager-Machlup path integral formalism, allowing us to derive the full probability distribution for the qubit's trajectories. Our analytical results reveal that purification is characterized by a dynamical crossover from a diffusion dominated regime to a measurement dominated regime, visible in the emergence of a bimodal state distribution. The analytical solutions are in strong agreement with numerical simulations, providing a robust theoretical benchmark for the study of information extraction in monitored quantum systems.
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.