By definition, an ensemble of quantum trajectories reproduces the density matrix of an open system after ensemble averaging. We present a general class of stochastic Schrodinger equations for quantum trajectories which unifies and extends existing methods. Various quantum jump and state-diffusion methods are recognized as opposite limiting cases. The inherent freedom of choice can be used to significantly reduce the calculation time in specific cases. Examples of this are methods with adaptive noise, for which a specific observable becomes noise free to first order. Linear stochastic equations can also be constructed, which allow analytical solutions. As an illustration, the decoherence of Schrodinger cat states of a cavity mode is discussed.
By definition, an ensemble of quantum trajectories reproduces the density matrix of an open system after ensemble averaging. We present a general class of stochastic Schrodinger equations for quantum trajectories which unifies and extends existing methods. Various quantum jump and state-diffusion methods are recognized as opposite limiting cases. The inherent freedom of choice can be used to significantly reduce the calculation time in specific cases. Examples of this are methods with adaptive noise, for which a specific observable becomes noise free to first order. Linear stochastic equations can also be constructed, which allow analytical solutions. As an illustration, the decoherence of Schrodinger cat states of a cavity mode is discussed.
We introduce jump-time unraveling as a distinct description of open quantum systems. As our starting point, we consider quantum jump trajectories, which emerge, physically, from continuous quantum measurements, or, formally, from the unraveling of Markovian quantum master equations. If the stochastically evolving quantum trajectories are ensemble-averaged at specific times, the resulting quantum states are solutions to the associated quantum master equation. We demonstrate that quantum trajectories can also be ensemble-averaged at specific jump counts. The resulting jump-time-averaged quantum states are then solutions to a discrete, deterministic evolution equation, with time replaced by the jump count. This jump-time evolution represents a trace-preserving quantum dynamical map if and only if the associated open system does not exhibit dark states. In the presence of dark states, on the other hand, jump-time-averaged states may decay into the dark states and the jump-time evolution may eventually terminate. Jump-time-averaged quantum states and the associated jump-time evolution are operationally accessible in continuous measurement schemes, when quantum jumps are detected and used to trigger the readout measurements. We illustrate the jump-time evolution with the examples of a two-level system undergoing relaxation or dephasing, a damped harmonic oscillator, and a free particle exposed to collisional decoherence.
The ensemble-averaged dynamics of open quantum systems are typically irreversible. We show that this irreversibility need not hold at the level of individually monitored quantum trajectories. Our main results are analytical stochastic Schrödinger equations for quantum reverse diffusion, along with corresponding stochastic master equations. These equations describe the exact and approximate stochastic reverse processes for continuously monitored Pauli channels, including time-dependent depolarizing noise. We show that the reverse processes generalize the forward dynamics by combining the noise effects of the forward processes with an additional stochastic drift that dynamically steers a quantum state back to its initial configuration. Consequently, the exact reverse stochastic Schrödinger equations admit closed-form solutions that can be implemented in real-time without the need for variational techniques. Our findings establish an analytical framework for quantum state recovery, noise-resilient quantum gates, quantum generative modelling, quantum tomography via forward-reverse cycles, and potential paradigms for quantum error correction based on reverse diffusion.
A quantum trajectory represents a single history of the state vector of an open quantum system. The stochastic nature of its evolution can be viewed as resulting from the quantum probabilities for the outcomes of measurements performed on the bath. We derive stochastic Schrödinger equations for various trajectory models, starting from the full Schrödinger equation for the system and the bath combined. The state reduction of the bath at a measurement also projects the state of the system on a pure state, thereby destroying the entanglement. The master equation for the system's density matrix is obtained as a result of the discussion, and it is not taken as its starting point. The general concepts are illustrated in the specific case of an atom coupled to the radiation field.
Abstract The dynamics of Gaussian states for open quantum systems described by Lindblad equations can be solved analytically for systems with quadratic Hamiltonians and linear Lindbladians, showing the familiar phenomena of dissipation and decoherence. It is well known that the Lindblad dynamics can be expressed as an ensemble average over stochastic pure-state dynamics, which can be interpreted as individual experimental implementations, where the form of the stochastic dynamics depends on the measurement setup. Here we consider quantum-jump and stochastic Schrödinger dynamics for initially Gaussian states. While both unravellings converge to the same Lindblad dynamics when averaged, the individual dynamics can differ qualitatively. For the stochastic Schrödinger equation, Gaussian states remain Gaussian during the evolution, with stochastic differential equations governing the evolution of the phase-space centre and a deterministic evolution of the covariance matrix. In contrast to this, individual pure-state dynamics arising from the quantum-jump evolution do not remain Gaussian in general. Applying results developed in the non-Hermitian context for Hagedorn wavepackets, we formulate a method to generate quantum-jump trajectories that is described entirely in terms of the evolution of an underlying Gaussian state. To illustrate the behaviours of the different unravellings in comparison to the Lindblad dynamics, we consider two examples in detail, which can be largely treated analytically, a harmonic oscillator subject to position measurement and a damped harmonic oscillator. In both cases, we highlight the differences as well as the similarities of the stochastic Schrödinger and the quantum-jump dynamics.
In the quantum trajectory approach an open quantum system is represented by a stochastic pure-state wave function. The mixed-state density operator that satisfies the usual master equation is recovered from the quantum trajectories by performing either an ensemble average or, for stationary systems, an average over time. This unraveling of the master equation dynamics into pure-state trajectories provides new insight into the decay of quantum coherence in systems that are open to the environment. Under some conditions macroscopic superposition states are preserved as macroscopic superposition states along individual trajectories, but are reduced to mixtures by the trajectory average. Under other conditions one of the states in a macroscopic superposition becomes dominant (in amplitude) over the other along each quantum trajectory. We discuss the mechanisms that produce these dynamics and assess the implications for the observation of Schrodinger cat states in optics experiments. We present results for some specific examples of macroscopic superposition states generated by a cavity QED system. The system involves a small collection of atoms in an optical cavity driven by a coherent laser field. Under strong-coupling conditions this system produces a variety of Schrodinger cat states whose precise form depends on the number of atoms, the method of excitation (through a cavity mirror or from the side), the initial state of the atoms, and the position of the atoms in the cavity.
Summary form only given. Quantum trajectories of state vectors can be used to describe open systems that otherwise can only be described using a density matrix. Besides the numerical advantage, quantum trajectories provide insight in the actual evolution of a system under continuous observation. Two distinct methods are usually considered, the quantum-jump method, associated with direct photodetection, and the quantum state diffusion (QSD) method, associated with homodyne or heterodyne mixing. We derive a general stochastic equation for quantum trajectories.