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Conference on Lasers and Electro-Optics

A Hamiltonian for cavity decay

S. M. Dutra, Gerard Nienhuis

Published Aug 29, 2005• 0 citations
Publisher

Abstract

Summary form only given. The conventional way to treat cavity dissipation in quantum optics is by applying the system-reservoir approach. The small system consists of the quantized discrete modes of a fictitious perfect cavity. The reservoir is the external field described by continuous annihilation and creation operators for the modes of free-space. This conventional approach has been used since the early days of the laser. The Hamiltonian that describes this approach became known as the Gardiner-Collett Hamiltonian. Despite being useful in describing several phenomena in cavities with a high Q factor, it is well known that this approach breaks down when the Q decreases. We have derived the Gardiner-Collett Hamiltonian from a rigorous modes of the universe expansion, as an approximation for high-Q, and obtained the coupling strength between the discrete cavity annihilation and creation operators (small system) and the continuous annihilation and creation operators of free-space (reservoir). We have found that the coupling strength depends on the frequency even when the reflectivity is frequency independent. The physical consequence of this frequency dependence is that for short times compared to one cavity round-trip time, the reservoir (free-space) is no longer Markovian. Our explicit expression for this frequency dependent coupling strength unlocks this fast time non-Markovian regime that had been previously inaccessible within the Gardiner-Collett Hamiltonian. Finally, we also show how to generalize the Gardiner-Collett Hamiltonian to arbitrary quality factors. This generalized Hamiltonian can be applied to develop a full quantum theory of excess noise in lasers.

DOI
10.1109/iqec.2000.908131

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Summary form only given. The conventional way to treat cavity dissipation in quantum optics is by applying the system-reservoir approach. The small system consists of the quantized discrete modes of a fictitious perfect cavity. The reservoir is the external field described by continuous annihilation and creation operators for the modes of free-space. This conventional approach has been used since the early days of the laser. The Hamiltonian that describes this approach became known as the Gardiner-Collett Hamiltonian. Despite being useful in describing several phenomena in cavities with a high Q factor, it is well known that this approach breaks down when the Q decreases. We have derived the Gardiner-Collett Hamiltonian from a rigorous modes of the universe expansion, as an approximation for high-Q, and obtained the coupling strength between the discrete cavity annihilation and creation operators (small system) and the continuous annihilation and creation operators of free-space (reservoir). We have found that the coupling strength depends on the frequency even when the reflectivity is frequency independent. The physical consequence of this frequency dependence is that for short times compared to one cavity round-trip time, the reservoir (free-space) is no longer Markovian. Our explicit expression for this frequency dependent coupling strength unlocks this fast time non-Markovian regime that had been previously inaccessible within the Gardiner-Collett Hamiltonian. Finally, we also show how to generalize the Gardiner-Collett Hamiltonian to arbitrary quality factors. This generalized Hamiltonian can be applied to develop a full quantum theory of excess noise in lasers.
We use Thomson's classical concept of mode of a leaky cavity to develop a quantum theory of cavity damping. This theory generalizes the conventional system-reservoir theory of high-QQQ cavity damping to arbitrary Q. The small system now consists of damped oscillators corresponding to the natural modes of the leaky cavity rather than undamped oscillators associated with the normal modes of a fictitious perfect cavity. The formalism unifies semiclassical Fox-Li modes and the normal modes traditionally used for quantization. It also lays the foundations for a full quantum description of excess noise. The connection with Siegman's semiclassical work is straightforward. In a wider context, this theory constitutes a radical departure from present models of dissipation in quantum mechanics: unlike conventional models, system and reservoir operators no longer commute with each other. This noncommutability is an unavoidable consequence of having to use natural cavity modes rather than normal modes of a fictitious perfect cavity.
We present a rigorous first principles derivation of the heuristic approach behind most of cavity QED: the quantization of the field in a high-Q cavity as if the mirrors were perfect with losses modelled by a coupling with the outside. We derive the Hamiltonian describing this coupling, thereby allowing us to avoid the coarse-graining approximation implicit in the usual Langevin or master equation treatments of cavity damping. We obtain an explicit expression for the coupling constant that describes the non-Markovian nature of the reservoir in the regime beyond the validity of the coarse-graining approximation.
The exact non-Markovian dynamics of a microcavity strongly coupled to a general reservoir at arbitrary temperature are studied. With the exact master equation for the reduced density operator of the cavity system, we analytically solve the time evolution of the cavity state and the associated physical observables. We show that the non-Markovian dynamics are completely determined by the propagating (retarded) and correlation Green functions. Comparing the non-Markovian behavior at finite temperature with those at the zero-temperature limit or Born-Markov limit, we find that the non-Markovian memory effect can dramatically change the coherent and thermal dynamics of the cavity. We also numerically study the dissipation dynamics of the cavity through the mean mode amplitude decay and the average photon number decay in the microwave regime. It is shown that the strong coupling between the cavity and the reservoir results in a long-time dissipationless evolution to the cavity field amplitude, and its noise dynamics undergo a critical transition from the weak to the strong coupling due to the non-Markovian memory effect.
The equations of motion for the resonant modes in a lossy cavity, which couples them through the nonresonant thermal loss mechanism described by Senitzky, are examined. The second-order perturbation treatment of dissipation is justified for long time intervals by reference to an intermittent similarity transformation. Self-consistent expressions for commutators such as 〈[Pn(t1), Pm(t2)]〉d, Pn being the operator-amplitude of the electric field in the nth mode, as averaged over the loss mechanism d with respect to its diagonal density matrix, are obtained from Laplace transform expressions for these operators. These averaged commutators describe correlations of fluctuations in different modes, in a way compatible with the ordinary commutation relations for lossless modes. The self-consistent anticommutator expressions, averaged over the loss mechanism, imply the Bose-Einstein formula for thermal energy in each mode at time t2 = t1. This is shown from direct examination of anticommutators and also from an integration of the correlation function. Commutators for the total electric and magnetic fields at two space-time points in the cavity have the proper causal properties. The dissipation terms in the field equations of motion cannot be derived from an equivalent Hamiltonian unless a modified analysis such as that of Kemeny is undertaken. Finally, the approximate equations of motion for the mode operator-amplitudes of a parametric amplifier are derived in the creation-annihilation operator framework and the spectral density of noise obtained at the signal frequency. The equilibrium noise present due to coupling with the idle modes is proportional to the number of photons in the idle frequency mode instead of that number plus one, in contrast to the conclusion of Wagner and Hellwarth. In this system the loss-averaged operators evolve from initial thermal equilibrium values to parametric equilibrium values according to nonlinear equations of motion containing time dependent commutators.
The quantum theory is developed for an arbitrarily stratified one-dimensional optical cavity which transmits light at one of its end The field is expanded in terms of the modes of the universe and each universal mode is quantized separately by means of the usual method. A class of coherent states which yields oscillation wave shapes of the cavity quasimodes as the field eigenvalue is studied. A state of this class gives an eigenvalue for the annihilation operator of a universal mode which is an analytic function of the mode frequency except for a mode-dependent phase factor. The mathematical meaning of the phase factor is studied. A new scheme of field quantization is developed which absorbs the above phase factor and yields simpler forms of the eigenvalue equations for the above class of states. For a state of the class, the local intensity, its fluctuation, and the various correlations of the field decay with time. For the local field inside or outside of an optical structure which is coupled to the free space, the almost definite phase relationship among the universal modes plays a decisive role.
Abstract In this paper, we study the exact entanglement dynamics of two two-level atoms in a dissipative cavity. We use the Gardiner–Collett Hamiltonian to model the dissipative cavity, in which we assume that the two atoms resonantly interact with the cavity field and the cavity field itself interacts with the surrounding medium. Then, with the help of the Fano’s technique we show that this system can be regarded as two atoms interacting with a heat bath. In such a case, we find that there exists a decoherence-free state that does not evolve in time. At this time there exists a so-called super-radiant state which decays in time due to dissipation. At last, we use the quantum Zeno effect to preserve the entanglement which already has been stored in the system.