We study the quantum properties of the electromagnetic field in optical cavities coupled to an arbitrary number of escape channels. We consider both inhomogeneous dielectric resonators with a scalar dielectric constant ǫ(r) and cavities defined by mirrors of arbitrary shape. Using the Feshbach projector technique we quantize the field in terms of a set of resonator and bath modes. We rigorously show that the field Hamiltonian reduces to the system-and-bath Hamiltonian of quantum optics. The field dynamics is investigated using the input-output theory of Gardiner and Collet. In the case of strong coupling to the external radiation field we find spectrally overlapping resonator modes. The mode dynamics is coupled due to the damping and noise inflicted by the external field. For wave chaotic resonators the mode dynamics is determined by a non-Hermitean random matrix. Upon including an amplifying medium, our dynamics of open-resonator modes may serve as a starting point for a quantum theory of random lasing.
Propagation of the Wigner function is studied on two levels of semiclassical propagation: one based on the Van Vleck propagator, the other on phase-space path integration. Leading quantum corrections to the classical Liouville propagator take the form of a time-dependent quantum spot. Its oscillatory structure depends on whether the underlying classical flow is elliptic or hyperbolic. It can be interpreted as the result of interference of a pair of classical trajectories, indicating how quantum coherences are to be propagated semiclassically in phase space. The phase-space path-integral approach allows for a finer resolution of the quantum spot in terms of Airy functions.
Feshbach's projector technique is employed to quantize the electromagnetic field in optical resonators with an arbitrary number of escape channels. We find spectrally overlapping resonator modes coupled due to the damping and noise inflicted by the external radiation field. For wave chaotic resonators the mode dynamics is determined by a non-Hermitean random matrix. Upon including an amplifying medium, our dynamics of open-resonator modes may serve as a starting point for a quantum theory of random lasing. Theoretically, much is known [2] about the subthreshold radiation from random lasers but only a few results exist in the non-linear lasing regime. Progress in this regime has been hampered by the unusual properties of the random laser modes. First, the mode amplitudes and mode frequencies in random lasers depend on the statistical properties of the underlying random medium. Random laser modes therefore must be analyzed in a statistical fashion, quite in contrast to traditional laser resonators. Moreover, the character of the modes depends on the amount of disorder. For strong disorder, localization of light may set in and give rise to well separated modes centered in different regions of space. By contrast, weak disorder leads to a poor confinement of light and to strongly overlapping modes.Standard laser theory [3,4] only applies to quasidiscrete modes and cannot account for lasing in the presence of overlapping modes. Various quantization schemes have been proposed [5][6][7] to replace the quasidiscrete modes of standard theory by quasimodes or Fox-Li modes of "bad" resonators. Unfortunately, these schemes are not well suited for a quantum statistical description required for random lasers. Statistics naturally enters the random-scattering theory pioneered by Beenakker [2], but that approach is restricted to linear media and cannot describe lasers above the lasing threshold. So far, to our knowledge, there is no satisfactory scheme for the field quantization in random media.In the present paper we develop such a quantization scheme for optical resonators with overlapping modes. The resonator may have an irregular shape or may contain weak random scatterers to ensure chaotic scattering of light inside the cavity. We employ a technique previously applied to condensed matter physics, the Feshbach projector formalism. Using that method we show that the electromagnetic field Hamiltonian of open resonators reduces to the well-known system-and-bath Hamiltonian of quantum optics. Chaotic scattering enters that Hamiltonian in two ways. First, the frequency spectrum of the resonator modes shows the correlations and "level repulsion" typical for wave chaos. Second, the resonator mode amplitudes are not damped separately but coupled by dissipation. Both effects are related to the spectral properties of non-Hermitean random matrices and must eventually be included in a quantum theory of random lasing.FIG. 1. Sketch of a chaotic resonator that is connected to the external radiation field via a number of openings.We start ...
We show that the time evolution of entanglement under incoherent environment coupling can be faithfully recovered by monitoring the system according to a suitable measurement scheme.PACS numbers: 03.67.Mn,03.65.Yz,42.50.Lc Quantum information processing requires the ability to produce entangled states and coherently perform operations on them. Under realistic laboratory conditions, however, entanglement is degraded through uncontrolled coupling to the environment. It is of crucial practical importance to quantify this degradation process [1][2][3], though also extremely difficult in general, due to the intricate mathematical notions upon which our understanding of entanglement relies [4][5][6]. Up to now, no general observable is known which would complement such essentially formal concepts with a specific experimental measurement setup.In the present Letter, we come up with a dynamical characterization of entanglement, through the continuous observation of a quantum system which evolves under incoherent coupling to an environment. We show that, at least for small, yet experimentally relevant systems, there is an optimal measurement strategy to monitor the entanglement of the time evolved, mixed system state. Mixed state entanglement is then given as the average entanglement of the pure states generated by single realisations of the optimal measurement-induced, stochastic time evolution.Consider a bipartite quantum system composed of subsystems A and B, interacting with its environment. Due to this coupling, an initially pure state |Ψ 0 of the composite system will evolve into a mixed state ρ(t), in a way governed by the master equatioṅwhere the Hamiltonian H generates the unitary system dynamics. The superoperators L k describe the effects of the environment on the system, and, for a Markovian bath, have the standard form [7]where the operators J k depend on the specific physical situation under study.To extract the time evolution of entanglement under this incoherent dynamics, one solution is to evaluate a given entanglement measure M (ρ) for the solution ρ(t), at all times t. One starts from one of the known pure state measures M (Ψ) [5,6,8], together with a pure state decomposition of ρ,where the p i are the positive, normalized weights of each pure state |Ψ i . The most naive generalization for a mixed state would then be to consider the averagewhich, however, is not suitable, since the decomposition (3) is not unique: M would thus give rise to different values of entanglement for different valid decompositions of ρ [9], inconsistently with the general requirements for a bona fide entanglement measure [5,6]. The proper definition of M (ρ) therefore is the infimum of all possible averages M [10], but holds two main drawbacks: (i) it turns into a hard numerical problem for higher dimensional or multipartite systems, and, (ii) even for bipartite qubits, where analytical solutions for some measures M (ρ) are known [8], there is no obvious interpretation of this optimal decomposition, in physical terms. Our a...
We use quantum diffusive trajectories to prove that the time evolution of two-qubit entanglement under spontaneous emission can be fully characterized by optimal continuous monitoring. We analytically determine this optimal unraveling and derive a deterministic evolution equation for the system's concurrence. Furthermore, we propose an experiment to monitor the entanglement dynamics in bipartite two-level systems and to determine the disentanglement time from a single trajectory.
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