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Register a new userChaos in Continuously Monitored Quantum Systems: An Optimal Path Approach
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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. 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 non-commuting 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.
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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 systems characteristic function. By generalizing the Chantasri-Dressel-Jordan (CDJ) formalism (Chantasri et al.~2013 and 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/refrigerator experiments.
We put forth a unifying formalism for the description of the thermodynamics of continuously monitored systems, where measurements are only performed on the environment connected to a system. We show, in particular, that the conditional and unconditional entropy production, which quantify the degree of irreversibility of the open systems dynamics, are related to each other by the Holevo quantity. This, in turn, can be further split into an information gain rate and loss rate, which provide conditions for the existence of informational steady-states (ISSs), i.e. stationary states of a conditional dynamics that are maintained owing to the unbroken acquisition of information. We illustrate the applicability of our framework through several examples.
Gaussian states of quantum oscillators are fully characterized by the mean values and the covariance matrix of their quadrature observables. We consider the dynamics of a system of oscillators subject to interactions, damping, and continuous probing which maintain their Gaussian state property. Such dynamics is found in many physical systems that can therefore be efficiently described by the ensuing effective representation of the density matrix $rho(t)$. Our probabilistic knowledge about the outcome of measurements on a quantum system at time $t$ is not only governed by $rho(t)$ conditioned on the evolution and measurement outcomes obtained until time $t$, but is also modified by any information acquired after $t$. It was shown in [Phys. Rev. Lett. 111, 160401 (2013)] that this information is represented by a supplementary matrix, $E(t)$. We show here that the restriction of the dynamics of $rho(t)$ to Gaussian states implies that the matrix $E(t)$ is also fully characterized by a vector of mean values and a covariance matrix. We derive the dynamical equations for these quantities and we illustrate their use in the retrodiction of measurements on Gaussian systems.
Monitoring a quantum observable continuously in time produces a stochastic measurement record that noisily tracks the observable. For a classical process such noise may be reduced to recover an average signal by minimizing the mean squared error between the noisy record and a smooth dynamical estimate. We show that for a monitored qubit this usual procedure returns unusual results. While the record seems centered on the expectation value of the observable during causal generation, examining the collected past record reveals that it better approximates a moving-mean Gaussian stochastic process centered at a distinct (smoothed) observable estimate. We show that this shifted mean converges to the real part of a generalized weak value in the time-continuous limit without additional postselection. We verify that this smoothed estimate minimizes the mean squared error even for individual measurement realizations. We go on to show that if a second observable is weakly monitored concurrently, then that second record is consistent with the smoothed estimate of the second observable based solely on the information contained in the first observable record. Moreover, we show that such a smoothed estimate made from incomplete information can still outperform estimates made using full knowledge of the causal quantum state.
We discuss the problem of estimating a frequency via N-qubit probes undergoing independent dephasing channels that can be continuously monitored via homodyne or photo-detection. We derive the corresponding analytical solutions for the conditional states, for generic initial states and for arbitrary efficiency of the continuous monitoring. For the detection strategies considered, we show that: i) in the case of perfect continuous detection, the quantum Fisher information (QFI) of the conditional states is equal to the one obtained in the noiseless dynamics; ii) for smaller detection efficiencies, the QFI of the conditional state is equal to the QFI of a state undergoing the (unconditional) dephasing dynamics, but with an effectively reduced noise parameter.
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