sophia walls - Academia.edu (original) (raw)

Papers by sophia walls

Research paper thumbnail of Memory in a sequence of weak and short duration measurements of non-commuting observables

arXiv (Cornell University), Feb 13, 2024

We use continuous, stochastic quantum trajectories within a framework of quantum state diffusion ... more We use continuous, stochastic quantum trajectories within a framework of quantum state diffusion (QSD) to describe alternating measurements of two non-commuting observables. Projective measurement of an observable completely destroys memory of the outcome of a previous measurement of the conjugate observable. In contrast, measurement under QSD is not projective and it is possible to vary the rate at which information about previous measurement outcomes is lost by changing the strength of measurement. We apply our methods to a spin 1/2 system and a spin 1 system undergoing alternating measurements of the Sz and Sx spin observables. Performing strong Sz measurements and weak Sx measurements on the spin 1 system, we demonstrate return to the same eigenstate of Sz to a degree beyond that expected from projective measurements and the Born rule. Such a memory effect appears to be greater for return to the ±1 eigenstates than the 0 eigenstate. Furthermore, the spin 1 system follows a measurement cascade process where an initial superposition of the three eigenstates of the observable evolves into a superposition of just two, before finally collapsing into a single eigenstate, giving rise to a distinctive pattern of evolution of the spin components.

Research paper thumbnail of Stochastic quantum trajectories demonstrate the Quantum Zeno Effect in open spin 1/2, spin 1 and spin 3/2 systems

Journal of physics. A, Mathematical and theoretical, Apr 4, 2024

Research paper thumbnail of Stochastic quantum trajectories demonstrate the Quantum Zeno Effect in an open spin system

arXiv (Cornell University), Sep 21, 2022

We investigate the Quantum Zeno Effect in spin 1/2, spin 1 and spin 3/2 open quantum systems unde... more We investigate the Quantum Zeno Effect in spin 1/2, spin 1 and spin 3/2 open quantum systems undergoing Rabi oscillations. The systems interact with an environment designed to perform continuous measurements of an observable, driving the systems stochastically towards one of the eigenstates of the corresponding operator. The system-environment coupling constant represents the strength of the measurement. Stochastic quantum trajectories are generated by unravelling a Markovian Lindblad master equation using the quantum state diffusion formalism. This is regarded as a better representation of system behaviour than consideration of the averaged evolution since the latter can mask the effect of measurement. Complete positivity is maintained and thus the trajectories can be considered as physically meaningful. Increasing the measurement strength leads to greater dwell by the system in the vicinity of the eigenstates of the measured observable and lengthens the time taken by the system to return to that eigenstate, thus demonstrating the Quantum Zeno Effect. For very strong measurement, the Rabi oscillations develop into randomly occurring near-instantaneous jumps between eigenstates. The stochastic measurement dynamics compete with the intrinsic, deterministic quantum dynamics of the system, each attempting to drive the system in the Hilbert space in different ways. As such, the trajectories followed by the quantum system are heavily dependent on the measurement strength which other than slowing down and adding noise to the Rabi oscillations, changes the paths taken in spin phase space from a circular precession into elaborate figures-of-eight.

Research paper thumbnail of Toy model illustrating the effect of measurement dependence on a Bell inequality

arXiv (Cornell University), Jul 14, 2023

Bell's inequalities rely on the assumption of measurement independence, namely that the probabili... more Bell's inequalities rely on the assumption of measurement independence, namely that the probabilities of adopting configurations of hidden variables describing a system prior to measurement are independent of the choice of physical property that will be measured. Weakening this assumption can change the inequalities to accommodate experimental data. We illustrate this by considering quantum measurement to be the dynamical evolution of hidden variables to attractors in their phase space that correspond to eigenstates of system observables. The probabilities of adopting configurations of these variables prior to measurement then depend on the choice of physical property measured by virtue of the boundary conditions acting on the dynamics. Allowing for such measurement dependence raises the upper limit of the CHSH parameter in Bell's analysis of an entangled pair of spin half particles subjected to measurement of spin components along various axes, whilst maintaining local interactions. We demonstrate how this can emerge and illustrate the relaxed upper limit using a simple toy model of dynamical quantum measurement. The conditioning of the hidden variable probability distribution on the chosen measurement settings can persist far back in time in certain situations, a memory that could explain the correlations exhibited in an entangled quantum system.

Research paper thumbnail of Stochastic quantum trajectories demonstrate the Quantum Zeno Effect in an open spin system

We investigate the Quantum Zeno Effect in spin 1/2, spin 1 and spin 3/2 open quantum systems unde... more We investigate the Quantum Zeno Effect in spin 1/2, spin 1 and spin 3/2 open quantum systems undergoing Rabi oscillations. The systems interact with an environment designed to perform continuous measurements of an observable, driving the systems stochastically towards one of the eigenstates of the corresponding operator. The system-environment coupling constant represents the strength of the measurement. Stochastic quantum trajectories are generated by unravelling a Markovian Lindblad master equation using the quantum state diffusion formalism. This is regarded as a better representation of system behaviour than consideration of the averaged evolution since the latter can mask the effect of measurement. Complete positivity is maintained and thus the trajectories can be considered as physically meaningful. Increasing the measurement strength leads to greater dwell by the system in the vicinity of the eigenstates of the measured observable and lengthens the time taken by the system to return to that eigenstate, thus demonstrating the Quantum Zeno Effect. For very strong measurement, the Rabi oscillations develop into randomly occurring near-instantaneous jumps between eigenstates. The stochastic measurement dynamics compete with the intrinsic, deterministic quantum dynamics of the system, each attempting to drive the system in the Hilbert space in different ways. As such, the trajectories followed by the quantum system are heavily dependent on the measurement strength which other than slowing down and adding noise to the Rabi oscillations, changes the paths taken in spin phase space from a circular precession into elaborate figures-of-eight.

Research paper thumbnail of Memory in a sequence of weak and short duration measurements of non-commuting observables

arXiv (Cornell University), Feb 13, 2024

We use continuous, stochastic quantum trajectories within a framework of quantum state diffusion ... more We use continuous, stochastic quantum trajectories within a framework of quantum state diffusion (QSD) to describe alternating measurements of two non-commuting observables. Projective measurement of an observable completely destroys memory of the outcome of a previous measurement of the conjugate observable. In contrast, measurement under QSD is not projective and it is possible to vary the rate at which information about previous measurement outcomes is lost by changing the strength of measurement. We apply our methods to a spin 1/2 system and a spin 1 system undergoing alternating measurements of the Sz and Sx spin observables. Performing strong Sz measurements and weak Sx measurements on the spin 1 system, we demonstrate return to the same eigenstate of Sz to a degree beyond that expected from projective measurements and the Born rule. Such a memory effect appears to be greater for return to the ±1 eigenstates than the 0 eigenstate. Furthermore, the spin 1 system follows a measurement cascade process where an initial superposition of the three eigenstates of the observable evolves into a superposition of just two, before finally collapsing into a single eigenstate, giving rise to a distinctive pattern of evolution of the spin components.

Research paper thumbnail of Stochastic quantum trajectories demonstrate the Quantum Zeno Effect in open spin 1/2, spin 1 and spin 3/2 systems

Journal of physics. A, Mathematical and theoretical, Apr 4, 2024

Research paper thumbnail of Stochastic quantum trajectories demonstrate the Quantum Zeno Effect in an open spin system

arXiv (Cornell University), Sep 21, 2022

We investigate the Quantum Zeno Effect in spin 1/2, spin 1 and spin 3/2 open quantum systems unde... more We investigate the Quantum Zeno Effect in spin 1/2, spin 1 and spin 3/2 open quantum systems undergoing Rabi oscillations. The systems interact with an environment designed to perform continuous measurements of an observable, driving the systems stochastically towards one of the eigenstates of the corresponding operator. The system-environment coupling constant represents the strength of the measurement. Stochastic quantum trajectories are generated by unravelling a Markovian Lindblad master equation using the quantum state diffusion formalism. This is regarded as a better representation of system behaviour than consideration of the averaged evolution since the latter can mask the effect of measurement. Complete positivity is maintained and thus the trajectories can be considered as physically meaningful. Increasing the measurement strength leads to greater dwell by the system in the vicinity of the eigenstates of the measured observable and lengthens the time taken by the system to return to that eigenstate, thus demonstrating the Quantum Zeno Effect. For very strong measurement, the Rabi oscillations develop into randomly occurring near-instantaneous jumps between eigenstates. The stochastic measurement dynamics compete with the intrinsic, deterministic quantum dynamics of the system, each attempting to drive the system in the Hilbert space in different ways. As such, the trajectories followed by the quantum system are heavily dependent on the measurement strength which other than slowing down and adding noise to the Rabi oscillations, changes the paths taken in spin phase space from a circular precession into elaborate figures-of-eight.

Research paper thumbnail of Toy model illustrating the effect of measurement dependence on a Bell inequality

arXiv (Cornell University), Jul 14, 2023

Bell's inequalities rely on the assumption of measurement independence, namely that the probabili... more Bell's inequalities rely on the assumption of measurement independence, namely that the probabilities of adopting configurations of hidden variables describing a system prior to measurement are independent of the choice of physical property that will be measured. Weakening this assumption can change the inequalities to accommodate experimental data. We illustrate this by considering quantum measurement to be the dynamical evolution of hidden variables to attractors in their phase space that correspond to eigenstates of system observables. The probabilities of adopting configurations of these variables prior to measurement then depend on the choice of physical property measured by virtue of the boundary conditions acting on the dynamics. Allowing for such measurement dependence raises the upper limit of the CHSH parameter in Bell's analysis of an entangled pair of spin half particles subjected to measurement of spin components along various axes, whilst maintaining local interactions. We demonstrate how this can emerge and illustrate the relaxed upper limit using a simple toy model of dynamical quantum measurement. The conditioning of the hidden variable probability distribution on the chosen measurement settings can persist far back in time in certain situations, a memory that could explain the correlations exhibited in an entangled quantum system.

Research paper thumbnail of Stochastic quantum trajectories demonstrate the Quantum Zeno Effect in an open spin system

We investigate the Quantum Zeno Effect in spin 1/2, spin 1 and spin 3/2 open quantum systems unde... more We investigate the Quantum Zeno Effect in spin 1/2, spin 1 and spin 3/2 open quantum systems undergoing Rabi oscillations. The systems interact with an environment designed to perform continuous measurements of an observable, driving the systems stochastically towards one of the eigenstates of the corresponding operator. The system-environment coupling constant represents the strength of the measurement. Stochastic quantum trajectories are generated by unravelling a Markovian Lindblad master equation using the quantum state diffusion formalism. This is regarded as a better representation of system behaviour than consideration of the averaged evolution since the latter can mask the effect of measurement. Complete positivity is maintained and thus the trajectories can be considered as physically meaningful. Increasing the measurement strength leads to greater dwell by the system in the vicinity of the eigenstates of the measured observable and lengthens the time taken by the system to return to that eigenstate, thus demonstrating the Quantum Zeno Effect. For very strong measurement, the Rabi oscillations develop into randomly occurring near-instantaneous jumps between eigenstates. The stochastic measurement dynamics compete with the intrinsic, deterministic quantum dynamics of the system, each attempting to drive the system in the Hilbert space in different ways. As such, the trajectories followed by the quantum system are heavily dependent on the measurement strength which other than slowing down and adding noise to the Rabi oscillations, changes the paths taken in spin phase space from a circular precession into elaborate figures-of-eight.