The stationary solution ρ of a quantum master equation can be represented as an ensemble of pure states in a continuous infinity of ways. Here we suggest a definition of the the “most natural” ensemble. The first requirement is that the ensemble be physically realizable through monitoring the system’s environment. We call this an ’unraveling’ of the master equation. The second requirement is that it be the most robust of such ensembles. Robustness of an ensemble is defined in terms of the survival probability S(t) which is the average probability for one of its elements to remain unchanged after a time t . The most natural ensemble can thus be said to be the one produced by the maximally robust unraveling. An application of interest is the atom laser. We show that the nature of the maximally robust unraveling is closely related to the question of whether the output beam satisfies the conditions required to be a true laser beam.
The stationary solution ρ of a quantum master equation can be represented as an ensemble of pure states in a continuous infinity of ways. Here we suggest a definition of the the “most natural” ensemble. The first requirement is that the ensemble be physically realizable through monitoring the system’s environment. We call this an ’unraveling’ of the master equation. The second requirement is that it be the most robust of such ensembles. Robustness of an ensemble is defined in terms of the survival probability S(t) which is the average probability for one of its elements to remain unchanged after a time t . The most natural ensemble can thus be said to be the one produced by the maximally robust unraveling. An application of interest is the atom laser. We show that the nature of the maximally robust unraveling is closely related to the question of whether the output beam satisfies the conditions required to be a true laser beam.
Monitoring the fluorescent radiation of an atom unravels the master equation evolution by collapsing the atomic state into a pure state which evolves stochastically. A robust unraveling is one that gives pure states that, on average, are relatively unaffected by the master-equation evolution (which applies once the monitoring ceases). The ensemble of pure states arising from the maximally robust unraveling has been suggested to be the most natural way of representing the system [H.M. Wiseman and J.A. Vaccaro, Phys. Lett. A 250, 241 (1998)]. We find that the maximally robust unraveling of a resonantly driven atom requires an adaptive interferometric measurement proposed by Wiseman and Toombes [Phys. Rev. A 60, 2474 (1999)]. The resultant ensemble consists of just two pure states which, in the high driving limit, are close to the eigenstates of the driving Hamiltonian $\ensuremath{\Omega}{\ensuremath{\sigma}}_{x}/2.$ This ensemble is the closest thing to a classical limit for a strongly driven atom. We also find that it is possible to reasonably approximate this ensemble using just homodyne detection, an example of a continuous Markovian unraveling. This has implications for other systems, for which it may be necessary in practice to consider only continuous Markovian unravelings.
As discussed in the preceding paper [Wiseman and Vaccaro, preceding paper, Phys. Rev. A 65, 043605 (2002)], the stationary state of an optical or atom laser far above threshold is a mixture of coherent field states with random phase, or, equivalently, a Poissonian mixture of number states. We are interested in which, if either, of these descriptions of ${\ensuremath{\rho}}_{\mathrm{ss}}$ as a stationary ensemble of pure states, is more natural. In the preceding paper we concentrated upon the question of whether descriptions such as these are physically realizable (PR). In this paper we investigate another relevant aspect of these ensembles, their robustness. A robust ensemble is one for which the pure states that comprise it survive relatively unchanged for a long time under the system evolution. We determine numerically the most robust ensembles as a function of the parameters in the laser model: the self-energy $\ensuremath{\chi}$ of the bosons in the laser mode, and the excess phase noise $\ensuremath{\nu}.$ We find that these most robust ensembles are PR ensembles, or similar to PR ensembles, for all values of these parameters. In the ideal laser limit $(\ensuremath{\nu}=\ensuremath{\chi}=0),$ the most robust states are coherent states. As the phase noise or phase dispersion is increased through $\ensuremath{\nu}$ or the self-interaction of the bosons $\ensuremath{\chi},$ respectively, the most robust states become more and more amplitude squeezed. We find scaling laws for these states, and give analytical derivations for them. As the phase diffusion or dispersion becomes so large that the laser output is no longer quantum coherent, the most robust states become so squeezed that they cease to have a well-defined coherent amplitude. That is, the quantum coherence of the laser output is manifest in the most robust PR ensemble being an ensemble of states with a well-defined coherent amplitude. This lends support to our approach of regarding robust PR ensembles as the most natural description of the state of the laser mode. It also has interesting implications for atom lasers in particular, for which phase dispersion due to self-interactions is expected to be large.
A laser, be it an optical laser or an atom laser, is an open quantum system that produces a coherent beam of bosons (photons or atoms, respectively). Far above threshold, the stationary state ${\ensuremath{\rho}}_{\mathrm{ss}}$ of the laser mode is a mixture of coherent-field states with random phase, or, equivalently, a Poissonian mixture of number states. This paper answers the question: can descriptions such as these, of ${\ensuremath{\rho}}_{\mathrm{ss}}$ as a stationary ensemble of pure states, be physically realized? Here physical realization is as defined previously by us [H. M. Wiseman and J. A. Vaccaro, Phys. Lett. A 250, 241 (1998)]: an ensemble of pure states for a particular system can be physically realized if, without changing the dynamics of the system, an experimenter can (in principle) know at any time that the system is in one of the pure-state members of the ensemble. Such knowledge can be obtained by monitoring the baths to which the system is coupled, provided that coupling is describable by a Markovian master equation. Using a family of master equations for the (atom) laser, we solve for the physically realizable (PR) ensembles. We find that for any finite self-energy $\ensuremath{\chi}$ of the bosons in the laser mode, the coherent-state ensemble is not PR; the closest one can come to it is an ensemble of squeezed states. This is particularly relevant for atom lasers, where the self-energy arising from elastic collisions is expected to be large. By contrast, the number-state ensemble is always PR. As the self-energy $\ensuremath{\chi}$ increases, the states in the PR ensemble closest to the coherent-state ensemble become increasingly squeezed. Nevertheless, there are values of $\ensuremath{\chi}$ for which states with well-defined coherent amplitudes are PR, even though the atom laser is not coherent (in the sense of having a Bose-degenerate output). We discuss the physical significance of this anomaly in terms of conditional coherence (and hence conditional Bose degeneracy).
It is well known that the state operator of an open quantum system can be generically represented as the solution of a time-local equation — a quantum master equation. Unraveling in quantum trajectories offers a picture of open system dynamics dual to solving master equations. In the unraveling picture, physical indicators are computed as Monte Carlo averages over a stochastic process valued in the Hilbert space of the system. This approach is particularly adapted to simulate systems in large Hilbert spaces. We show that the dynamics of an open quantum system generically admits an unraveling in the Hilbert space of the system described by a Markov process generated by ordinary stochastic differential equations for which rigorous concentration estimates are available. The unraveling can be equivalently formulated in terms of norm-preserving state vectors or in terms of linear “ostensible” processes trace preserving only on average. We illustrate the results in the case of a two level system in a simple boson environment. Next, we derive the state-of-the-art form of the Diósi-Gisin-Strunz Gaussian random ostensible state equation in the context of a model problem. This equation provides an exact unraveling of open systems in Gaussian environments. We compare and contrast the two unravelings and their potential for applications to quantum error mitigation.
Stochastic unravelings provide a useful way to represent open quantum system dynamics in terms of pure state realizations, and have been widely studied both from a fundamental and from a computational point of view. They were initially formulated for Markovian dynamics described by the Gorini-Kossakowski-Sudarshan-Lindblad master equation. However, due to recent technological and experimental development, most physical relevant dynamics present temporal correlations beyond the Markov approximation. Such correlations cause decay rates to turn temporarily negative, thus requiring the generalization of stochastic unravelings from Markovian to non-Markovian scenarios. Indeed, many unraveling techniques have been introduced in this regime, and a comprehensive review of the different jump methods is currently missing. In this work, we provide an overview of widely used quantum jump unraveling techniques for non-Markovian systems and also discuss them in terms of their numerical efficiency, divisibility requirements, Hilbert space extension, and measurement interpretation.
An open quantum system in steady state ${\stackrel{^}{\ensuremath{\rho}}}_{\mathrm{ss}}$ can be represented by a weighted ensemble of pure states ${\stackrel{^}{\ensuremath{\rho}}}_{\mathrm{ss}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k}^{}{\ensuremath{\wp}}_{k}|{\ensuremath{\psi}}_{k}〉〈{\ensuremath{\psi}}_{k}|$ in infinitely many ways. A physically realizable (PR) ensemble is one for which some continuous measurement of the environment will collapse the system into a pure state $|\ensuremath{\psi}(t)〉$, stochastically evolving such that the proportion of time for which $|\ensuremath{\psi}(t)〉\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}|{\ensuremath{\psi}}_{k}〉$ equals ${\ensuremath{\wp}}_{k}$. Some, but not all, ensembles are PR. This constitutes the preferred ensemble fact. We present the necessary and sufficient conditions for a given ensemble to be PR, and illustrate the method by showing that the coherent state ensemble is not PR for an atom laser.