It is known that a quantum master equation that describes dissipative relaxation (such as spontaneous decay) can be separated in a part that describes spontaneous jumps and a jump-free part. The jump-free part of the evolution is determined by a non-Hermitian Hamiltonian H, which acts on state vectors, rather than on density operators. We identify a class of systems for which the complete evolution is entirely determined by the jump-free evolution, which is governed by H. Although this class is restricted, it contains a sizable number of systems of physical significance. Among these are two-, three-, and four-state atoms. The method is used to find explicit expressions of time-dependent expectation values in terms of the complex eigenvalues of H. As a first application we evaluate exactly the photon statistics in the driven \ensuremath{\Lambda} system. As a second example of the method, we treat the configuration of a V system driven by two standing waves. This system constitutes an effective beam splitter.
It is known that a quantum master equation that describes dissipative relaxation (such as spontaneous decay) can be separated in a part that describes spontaneous jumps and a jump-free part. The jump-free part of the evolution is determined by a non-Hermitian Hamiltonian H, which acts on state vectors, rather than on density operators. We identify a class of systems for which the complete evolution is entirely determined by the jump-free evolution, which is governed by H. Although this class is restricted, it contains a sizable number of systems of physical significance. Among these are two-, three-, and four-state atoms. The method is used to find explicit expressions of time-dependent expectation values in terms of the complex eigenvalues of H. As a first application we evaluate exactly the photon statistics in the driven \ensuremath{\Lambda} system. As a second example of the method, we treat the configuration of a V system driven by two standing waves. This system constitutes an effective beam splitter.
Within the framework of the quantum-statistical approach, utilizing both non-Hermitian Hamiltonian and Lindblad’s jump operators, one can derive various generalizations of the von Neumann equation for reduced density operators, also known as hybrid master equations. If one considers the evolution of pure states only, i.e., disregarding the coherence between states and spontaneous transitions from pure to mixed states, then one can resort to quantum-mechanical equations of the Schrödinger type. We derive them from the hybrid master equations and study their main properties, which indicate that our equations have a larger range of applicability compared to other generalized Schrödinger equations proposed hitherto. Among other features, they can describe not only systems which remain in the stationary eigenstates of the Hamiltonian as time passes, but also those which evolve from those eigenstates. As an example, we consider a simple but important model, a quantum harmonic oscillator driven by both Hamiltonian and non-Hamiltonian terms, and derive its classical limit, which turns out to be the damped harmonic oscillator. Using this model, we demonstrate that the effects of dissipative environments of different types can cancel each other, thus resulting in an effectively dissipation-free classical system. Another discussed phenomenon is whether a non-trivial quantum system can reduce to a classical system in free motion, i.e., without experiencing any classical Newtonian forces. This uncovers a large class of quantum-mechanical non-Hamiltonian systems whose dynamics are not determined by conventional mechanics’ potentials and forces, but rather come about through quantum statistical effects caused by the system’s environment.
A unified quantum-statistical framework is proposed for open three-level atomic systems when two lower-energy states are coupled to a single upper state. We suggest a non-Hermitian generalization of the closed model’s Hamiltonian by the analytical continuation of real-valued parameters related to the energies into a complex domain, as well as by considering some terms previously neglected by the procedure of Hermitian conjugation. We show that such systems can have two types of density operator’s evolution, depending on whether they are expected to be sustainable, or in a process of irreversible decay. This approach not only explains previously heuristically introduced terms in equations of motion for the density matrix but also suggests new terms which predict new dissipative effects. We then consider a special case, namely the evolution of pure states described by a generalized Schrödinger equation, which turns out to be nonlinear with respect to the atomic wavefunctions. As a final step toward unification, in addition to the non-Hermitian Hamiltonian generalization, we also include Markovian quantum jumps by adding the Lindblad dissipator. The resulting equations not only expand the range of applicability of open three-level models, but also suggest that dissipative phenomena in quantum-optical systems in vivo are more diverse and significant than previously believed.
One possible realization of non-Hermitian systems is based on open quantum systems by omitting quantum jumping terms in the master equation. This is a good approximation at short times where the effects of quantum jumps can be ignored. However, the jumps can affect the long-time dynamics of the system, motivating us to take the jumps into account in these studies. In this paper, by treating the quantum jumps as perturbations, we examine the effect of the quantum jumps on the non-Hermitian system. For this purpose, we first derive an effective Hamiltonian to describe the dynamics of the open quantum system based on the master equation, then expand the eigenstates and eigenenergies up to the first and second order in the quantum jumps. Finally, we apply our theory to a dissipative two-level system and dissipative fermionic superfluids. The effect of the quantum jump on the dynamics and the nonequilibrium phase transition is demonstrated and discussed.
We present the complete analytical solution of the cavity QED of a two-level atom and a field mode at zero temperature. It includes both dissipation of the field due to a finite Q value of the cavity and incoherent decay mechanisms for the atom. This analytical solution is provided by a powerful method for treating general master equations that appear in quantum optics. As distinct from the usual approaches we first deal with that part of the master equation which describes the dissipative coupling of the field and the atom to their reservoirs. Rather than using number-state or dressed-state bases we expand the density operator into the eigenstates of the nonunitary parts of the master equation which model the dissipative part of the dynamics. The set of these eigenstates is the damping basis. With the aid of this expansion we find the eigenvalues and eigenstates of the total Liouville operator. The evolution of an arbitrary initial state is then known. We employ these results to give an exact solution of the dynamics of the photon field in realistic experiments with one-atom masers at very low temperatures. It includes detuning, cavity leakage effects, spontaneous decay mechanisms for the atoms, a Fizeau-type velocity distribution for the atomic beam, and a statistical parameter for the probability of the excitation of incoming atoms, covering the limits of Poissonian pumping and of regular pumping. On the same grounds one can treat the one-atom laser, consisting of a single atom which stays in permanent interaction with the field mode and which is continuously pumped by external heat baths.
We present a closed-form solution to the eigenvalue problem of a class of master equations that describe open quantum systems with loss and dephasing but without gain. The method relies on the existence of a conserved number of excitations in the Hamiltonian part and the fact that none of the Lindblad operators describe an excitation of the system. In the absence of dephasing Lindblad operators, the eigensystem of the Liouville operator can be constructed from the eigenvalues and eigenvectors of the effective non-Hermitian Hamiltonian used in the quantum jump approach. Open versions of spin chains, the Tavis-Cummings model, and coupled Harmonic oscillators without gain can be solved using this technique.
Every open-system dynamics can be associated to infinitely many stochastic pictures, called unravelings, which have proved to be extremely useful in several contexts, both from the conceptual and the practical point of view. Here, focusing on quantum-jump unravelings, we demonstrate that there exists inherent freedom in how to assign the terms of the underlying master equation to the deterministic and jump parts of the stochastic description, which leads to a number of qualitatively different unravelings. As relevant examples, we show that a fixed basis of post-jump states can be selected under some definite conditions, or that the deterministic evolution can be set by a chosen time-independent non-Hermitian Hamiltonian, even in the presence of external driving. Our approach relies on the definition of rate operators, whose positivity equips each unraveling with a continuous-measurement scheme and is related to a long known but so far not widely used property to classify quantum dynamics, known as dissipativity. Starting from formal mathematical concepts, our results allow us to get fundamental insights into open quantum system dynamics and to enrich their numerical simulations.