Coupling a discrete state to a quasi-continuum: a model quantum-mechanical system that interpolates between Rabi oscillations and decay-revival dynamics
Enes Kutay Isgorur, Osman Cevheroğlu, Arkadaş Özakın
We formulate a quantum-mechanical system consisting of a single discrete state coupled to an infinite ladder of equally spaced states, with the coupling described by a Lorentzian profile. Various limiting cases of this system correspond to well-known models in quantum optics: the narrow-resonance limit yields the Rabi model, the wide-resonance limit yields the Bixon--Jortner model, the wide-resonance true continuum limit yields the Wigner--Weisskopf model, and the fixed-resonance true continuum limit yields a system typically studied using methods developed by Fano. We obtain a semianalytical solution to the eigenvalue problem by reducing it to a transcendental equation and demonstrate the limiting behaviors described above. We then investigate numerically the dynamics of the initial discrete state and show that, in various limiting regimes, the system exhibits a wide range of behaviors predicted by the asymptotic theory, including exponential decay, revivals, Rabi oscillations, and damped oscillations. The ability of the system to interpolate between such a rich variety of behaviors and established model systems, together with the accessibility of a semianalytical solution, makes it a useful model in quantum optics and related fields.
We formulate a quantum-mechanical system consisting of a single discrete state coupled to an infinite ladder of equally spaced states, with the coupling described by a Lorentzian profile. Various limiting cases of this system correspond to well-known models in quantum optics: the narrow-resonance limit yields the Rabi model, the wide-resonance limit yields the Bixon--Jortner model, the wide-resonance true continuum limit yields the Wigner--Weisskopf model, and the fixed-resonance true continuum limit yields a system typically studied using methods developed by Fano. We obtain a semianalytical solution to the eigenvalue problem by reducing it to a transcendental equation and demonstrate the limiting behaviors described above. We then investigate numerically the dynamics of the initial discrete state and show that, in various limiting regimes, the system exhibits a wide range of behaviors predicted by the asymptotic theory, including exponential decay, revivals, Rabi oscillations, and damped oscillations. The ability of the system to interpolate between such a rich variety of behaviors and established model systems, together with the accessibility of a semianalytical solution, makes it a useful model in quantum optics and related fields.
We formulate a quantum-mechanical system consisting of a single discrete state coupled to an infinite ladder of equally spaced states, with the coupling described by a Lorentzian profile. Various limiting cases of this system correspond to well-known models in quantum optics: the narrow-resonance limit yields the Rabi model, the wide-resonance limit yields the Bixon-Jortner model, the wide-resonance true continuum limit yields the Wigner-Weisskopf model, and the fixed-resonance true continuum limit yields a system typically studied using methods developed by Fano. We obtain a semianalytical solution to the eigenvalue problem by reducing it to a transcendental equation and demonstrate the limiting behaviors described above. We then investigate numerically the dynamics of the initial discrete state and show that, in various limiting regimes, the system exhibits a wide range of behaviors predicted by the asymptotic theory, including exponential decay, revivals, Rabi oscillations, and damped oscillations. The ability of the system to interpolate between such a rich variety of behaviors and established model systems, together with the accessibility of a semianalytical solution, makes it a useful model in quantum optics and related fields.
We formulate a quantum mechanical system consisting of a single discrete state coupled to an infinite ladder of equally-spaced states, the coupling between the two being given by a Lorentzian profile. Various limits of this system correspond to well-known models from quantum optics, namely, the narrow resonance limit gives the Rabi system, the wide resonance limit gives the Bixon-Jortner system, the wide resonance, true continuum limit gives the Wigner-Weisskopf system, and the fixed resonance, true continuum limit gives a system that is typically studied by methods developed by Fano. We give a semi-analytical solution of the eigenvalue problem by reducing it to a transcendental equation, and demonstrate the aforementioned limiting behaviors. We then study the dynamics of the initial discrete state numerically, and show that it gives a wide range of behaviors in various limiting cases as predicted by our asymptotic theory including exponential decay, revivals, Rabi oscillations, and damped oscillations. The ability of this system to interpolate between such a rich set of behaviors and existing model systems, and the accessibility of a semi-analytical solution, make it a useful model system in quantum optics and related fields.
When a discrete state is coupled to a continuum, the dynamics can be described either by the Weisskopf-Wigner exponential decay or by the Rabi oscillation, depending on the relative magnitudes of the continuum width and of the Rabi frequency. A continuous transition between these two regimes exists, as demonstrated in 1977 by C. Cohen-Tannoudji and P. Avan. Here, we describe a fully analogous transition in classical mechanics, by studying the dynamics of two coupled mechanical oscillators in the presence of damping. By varying the relative magnitudes of the damping and coupling terms, we observe a continuous transition between a regime analogous to the Rabi oscillation and a regime analogous to the Weisskopf-Wigner exponential decay.
The dynamics of a single quantum state embedded in one continuum or several (quasi)continua is one of the most studied phenomena in quantum mechanics. In this paper we investigate its discrete analog and consider short- and long-time dynamics based on numerical and analytical solutions of the Schr\"odinger equation. In addition to derivation of explicit conditions for initial exponential decay, it is shown that a recent model of this class [L. Guo, A. Grimsmo, A. F. Kockum, M. Pletyukhov, and G. Johansson, Phys. Rev. A 95, 053821 (2017)], describing a qubit coupled to a phonon reservoir with energy dependent coupling parameters, is identical to a qubit interacting with a finite number of parallel regularly spaced bands of states via constant couplings. As a consequence, the characteristic near periodic initial-state revivals can be viewed as a transition of probability between different continua via the reviving initial state. Furthermore, polynomial decay of the reviving peaks is present in any system with constant and sufficiently strong coupling.
Non-Markovian behavior in quantum systems is often studied in the context of bipartite systems consisting of a system of interest and an environment -- tracing over the environment results in non-Markovian behavior for the subsystem of interest. One may get a Markovian limit in certain regimes, which is studied using the Lindblad master equation, and corrections to this behavior can be obtained by techniques such as the Nakajima-Zwanzig formalism. In this paper, we obtain an exact non-Markovian equation for the dynamics of a simple model system that consists of a direct sum rather than a tensor product of two pieces, namely, a discrete state and an infinite ladder. This system, called the Bixon-Jortner model, was first developed in the quantum chemistry literature but has been utilized by the quantum optics community as a model system with interesting behavior, including a Wigner-Weisskopf limit of exponential decay. We attack the time evolution problem of this system directly in time-domain, and start with an integrodifferential equation describing the time evolution of the discrete state. Using tools from mathematical physics, we transform this equation to a delay differential equation, which makes the non-Markovianity completely transparent, and then we solve the delay equation using an intuitive ansatz. This allows us to obtain the analytic form of the dynamics directly in time domain, and demonstrate decay and revival behaviors coming from the aforementioned delay differential equation. We believe the explicit form of the time-domain non-Markovian equation we obtain and the accessibility the solution techniques we use make our results a useful case study of non-Markovianity in quantum systems.
The dynamics of a single quantum state embedded in one or several (quasi-)continua is one of the most studied phenomena in quantum mechanics. In this work we investigate its discrete analogue and consider short and long time dynamics based on numerical and analytical solutions of the Schrödinger equation. In addition to derivation of explicit conditions for initial exponential decay, it is shown that a recent model of this class [Phys. Rev. A 95, 053821, (2017)], describing a qubit coupled to a phonon reservoir with energy dependent coupling parameters is identical to a qubit interacting with a finite number of parallel regularly spaced band of states via constant couplings. As a consequence, the characteristic near periodic initial state revivals can be viewed as a transition of probability between different continua via the reviving initial state. Furthermore, the observation of polynomial decay of the reviving peaks is present in any system with constant and sufficiently strong coupling.