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.
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 ∣\ensuremathψ(t)〉, stochastically evolving such that the proportion of time for which ∣\ensuremathψ(t)〉=∣\ensuremathψk〉 equals \ensuremath℘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.
The density matrix $\ensuremath{\rho}$ for an n-level system is decomposed into the minimum number of pure states necessary to calculate physical observables. The corresponding physical system is first represented by a set B of n pure states $|{\ensuremath{\beta}}_{i}〉,$ together with their statistical weights. The time evolution of the system is therefore calculated as B(t)=UB(t0), with the propagator U operating on each member of the set, in contrast to the more laborious $\ensuremath{\rho}(t)=U\ensuremath{\rho}{(t}_{0}{)U}^{\ifmmode\dagger\else\textdagger\fi{}}.$ At least one of the states can be eliminated from the set by reducing its weight to zero via a simple transformation of the density matrix. When there are degenerate weights, the transformation is applied using the weight with the largest degeneracy. Thus, even if none of the original statistical weights are equal to zero, so that $\mathrm{rank}(\ensuremath{\rho})=n,$ the system can be described by a set of m states with m<n. The resulting minimal set provides a concommitant increase in calculational efficiency that depends on the fractional degeneracy. The independent evolution of the states comprising the set also provides an inherent parallelism that can be readily utilized in numerical computations. In addition to efficiency and conceptual simplification, this decomposition can provide physical insight, since the explicit contribution of each pure state to the dynamics of the system is readily apparent.
In this chapter we shall deal with the quantum dynamics of an open system. By open quantum system we mean here a system which interacts with an environment (see Sec. 9.4): since we are not interested in the dynamics of the environment, we shall have to describe the evolution of the system in some “effective” way. In particular, if we consider only the evolution of the system, it will be non-unitary, and this will represent the subject of this chapter. As we know, Hamiltonian quantum dynamics is unitary and changes pure states into pure states. On the other hand, non-unitary dynamics changes an initially pure state into a mixture, which must be described by a density matrix (see Ch. 5). In the case of macroscopic systems, the coupling with the environment may be arbitrarily reduced and therefore its influence can be made correspondingly small (see Sec. 1.1). Microscopic systems, however, always couple to the environment and this coupling cannot be considered negligible. This is the reason why the quantum theory of open systems is one of the most important and fundamental chapters of quantum mechanics that, though born in quantum optics, has many implications in almost all fields of physics. The present chapter can be seen as a further development of the measurement theory (see Ch. 9), as open systems manifest a decoherent dynamics (see in particular Sec. 9.4).
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).
Introduction As discussed in Chapter 1, to understand the general evolution, conditioned and unconditioned, of a quantum system, it is necessary to consider coupling it to a second quantum system. In the case in which the second system is much larger than the first, it is often referred to as a bath, reservoir or environment , and the first system is called an open system . The study of open quantum systems is important to quantum measurement for two reasons. First, all real systems are open to some extent, and the larger a system is, the more important its coupling to its environment will be. For a macroscopic system, such coupling leads to very rapid decoherence . Roughly, this term means the irreversible loss of quantum coherence, that is the conversion of a quantum superposition into a classical mixture. This process is central to understanding the emergence of classical behaviour and ameliorating, if not solving, the so-called quantum measurement problem . The second reason why open quantum systems are important is in the context of generalized quantum measurement theory as introduced in Chapter 1. Recall from there that, by coupling a quantum system to an ‘apparatus’ (a second quantum system) and then measuring the apparatus, a generalized measurement on the system is realized. For an open quantum system, the coupling to the environment is typically continuous (present at all times).
A Markovian open quantum system which relaxes to a unique steady state ${\ensuremath{\rho}}_{\mathrm{ss}}$ of finite rank can be decomposed into a finite physically realizable ensemble (PRE) of pure states. That is, as shown by R. I. Karasik and H. M. Wiseman [Phys. Rev. Lett. 106, 020406 (2011)], in principle there is a way to monitor the environment so that in the long-time limit the conditional state jumps between a finite number of possible pure states. In this paper we show how to apply this idea to the dynamics of a double quantum dot arising from the feedback control of quantum transport, as previously considered by C. P\"oltl, C. Emary, and T. Brandes [Phys. Rev. B 84, 085302 (2011)]. Specifically, we consider the limit where the system can be described as a qubit, and show that while the control scheme can always realize a two-state PRE, in the incoherent-tunneling regime there are infinitely many PREs compatible with the dynamics that cannot be so realized. For the two-state PREs that are realized, we calculate the counting statistics and see a clear distinction between the coherent and incoherent regimes.
Introduction In our treatment of quantum mechanics we discussed the notion of the state of a given system. The state was specified by giving the eigenvalues of a maximal set of mutually commuting observables. This represented the most information we could have about a given system in principle. In practice, of course, we do not really have all of the necessary information about the eigenvalues available. No experimentalist sets out with the primary intention of obtaining results which correspond to a maximal set of mutually compatible observables for any macroscopic system, such as a gas. Even a very small volume of gas contains an enormous number of particles. The quantum mechanical state would require us to know a great deal of information about the internal states of the electrons and nuclei of the gas molecules. The experimentalist may only be concerned with gross properties such as the pressure, internal energy, or specific heat for example. Most of the detailed information about the detailed state of the system is neither observed, nor of direct interest. Instead we must content ourselves with a description of the average, or most likely, behaviour of the system in the probabilistic sense. The usual technique for doing this consists of the introduction of an ensemble, meaning a very large collection of systems whose measured macroscopic properties are the same as those for the system we are interested in, but whose unobserved properties can vary.