We investigate the phase-space representation of quantum mechanics as a platform to perform Monte Carlo simulation of the non-interacting Jaynes-Cummings-Rabi (JCR) model involving a two-level hydrogen atom as a qubit and a quantum har-monic oscillator as a photonic cavity. Our algorithm starts by defining a trial wave function for solving the JCR model and then implement it on the “epistemically-restricted” phase-space representation, known as Budiyono-Rochrlich theory, with a Monte-Carlo sampling for the phase-space distribution, which is useful to obtain the average energy of our JCR model. We find that the numerical results are close to the analytical ones for every separate system in the JCR model, i.e., the hydrogen and the oscillator, calculated independently. Combining the two components in the non-interacting JCR model, the calculated energy values for the ground and the first excited states are then found to be —0.91 a.u. and 0.41 a.u. with corresponding standard errors of 9.12×10−8 a.u. and 1.95×10−8 a.u., respectively. We expect that the method presented in this paper can be an alternative approach to solving various qubit-photon systems.
We investigate the phase-space representation of quantum mechanics as a platform to perform Monte Carlo simulation of the non-interacting Jaynes-Cummings-Rabi (JCR) model involving a two-level hydrogen atom as a qubit and a quantum har-monic oscillator as a photonic cavity. Our algorithm starts by defining a trial wave function for solving the JCR model and then implement it on the “epistemically-restricted” phase-space representation, known as Budiyono-Rochrlich theory, with a Monte-Carlo sampling for the phase-space distribution, which is useful to obtain the average energy of our JCR model. We find that the numerical results are close to the analytical ones for every separate system in the JCR model, i.e., the hydrogen and the oscillator, calculated independently. Combining the two components in the non-interacting JCR model, the calculated energy values for the ground and the first excited states are then found to be —0.91 a.u. and 0.41 a.u. with corresponding standard errors of 9.12×10−8 a.u. and 1.95×10−8 a.u., respectively. We expect that the method presented in this paper can be an alternative approach to solving various qubit-photon systems.
The interaction of a two-level atom with a single-mode quantized field is one of the simplest models in quantum optics. Under the rotating wave approximation, it is known as the Jaynes-Cummings model and without it as the Rabi model. Real-world realizations of the Jaynes-Cummings model include cavity, ion trap and circuit quantum electrodynamics. The Rabi model can be realized in circuit quantum electrodynamics. As soon as nonlinear couplings are introduced, feasible experimental realizations in quantum systems are drastically reduced. We propose a set of two photonic lattices that classically simulates the interaction of a single two-level system with a quantized field under field nonlinearities and nonlinear couplings as long as the quantum optics model conserves parity. We describe how to reconstruct the mean value of quantum optics measurements, such as photon number and atomic energy excitation, from the intensity and from the field, such as von Neumann entropy and fidelity, at the output of the photonic lattices. We discuss how typical initial states involving coherent or displaced Fock fields can be engineered from recently discussed Glauber-Fock lattices. As an example, the Buck-Sukumar model, where the coupling depends on the intensity of the field, is classically simulated for separable and entangled initial states.
Almost all superconducting quantum technologies are built using a combination of qubits and microwave resonators. In this chapter, we develop the theory to study coherent qubit–photon interaction in such devices. We start with the equivalent of an atom in free space, studying a qubit in an open waveguide. We develop the spin-boson Hamiltonian, with specific methods to solve its dynamics in the limits of few excitations. Using these tools, we can study how an excited qubit can relax to the ground state, producing a photon, and how a propagating photon can interact with a qubit. We then move to closed environments where the photons are confined in cavities or resonators, developing the theory of cavity-QED. Using this theory, we study the Purcell enhancement of interactions, the Jaynes–Cummings model, Rabi oscillations, and vacuum Rabi splitting. We close the chapter illustrating some limits in which cavities can be used to control and measure qubits.
The scientific interest in the analytical solution of the quantum Rabi model is due to the widespread use of this simple model in quantum optics, quantum computing, cavity QED, and nanoelectromechanical systems. This interest is related to the need for the theoretical description of the interaction of a two-level system with a quantum oscillator in the case when the rotating wave approximation fails. In this Letter, we present an approach to the exact diagonalization of the quantum Rabi Hamiltonian. This approach is based on the properties of the Pauli operators and allows us to readily solve the stationary Schrodinger equation for a two-level system. First, we demonstrate the applicability of the approach to the Jaynes-Cummings Hamiltonian to get the well-known solution. Then, we obtain the eigenvalues and eigenstates for the quantum Rabi Hamiltonian using the proposed approach. It is shown that the obtained eigenstates can be represented in the basis of the eigenstates of the Jaynes-Cummings Hamiltonian.
The Jaynes-Cummings (JC) model is one of the most fundamental models for the interaction between light and matter. In this paper, we first derive the explicit expression for the time evolution operator of the JC model when the light field resonates with a two-level atom. Based on this, we calculate the analytical expressions for the system state and the population of each atomic energy level as a function of time when the two-level atom is initially in a superposition of two energy eigenstates and the light field is in a superposition of three photon number states and a coherent state, respectively. Finally, this paper introduces QuTiP, a useful extension library in Python for quantum mechanics simulation and numerical solution, and uses it to numerically solve the time evolution of this model, obtaining results consistent with the analytical solutions. These computational processes are of great significance in scientific research and university physics teaching.
The Jaynes-Cummings model (JCM), a soluble fully quantum mechanical model of an atom in a field, was first used (in 1963) to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in atomic excitation probabilities for fields with sharply defined energy (or photon number). For fields having a statistical distributions of photon numbers the oscillations collapse to an expected steady value. In 1980 it was discovered that with appropriate initial conditions (e.g. a near-classical field), the Rabi oscillations would eventually revive, only to collapse and revive repeatedly in a complicated pattern. The existence of these revivals, present in the analytic solutions of the JCM, provided direct evidence for discreteness of field excitation (photons) and hence for the truly quantum nature of radiation. Subsequent study revealed further non-classical properties of the JCM field, such as a tendency of the photons to antibunch. Within the last two years it has been found that during the quiescent intervals of collapsed Rabi oscillations the atom and field exist in a macroscopic superposition state (a Schrödinger cat). This discovery offers the opportunity to use the JCM to elucidate the basic properties of quantum correlation (entanglement) and to explore still further the relationship between classical and quantum physics. The relative simplicity of the JCM and the ease with which it can be extended through analytic expressions or numerical computation continues to motivate attention, as evidenced by the growing abundance of publications. We here present an overview of the theory of the JCM and some of the many extensions and generalizations that have appeared.
We theoretically investigate quantum Rabi oscillations in a system consisting of a two-level atom (qubit) strongly coupled to a one-dimensional open waveguide. In contrast to conventional cavity quantum electrodynamics, the qubit interacts with a continuum of propagating modes, which gives rise to fundamentally different dynamical behavior. Within the rotating-wave approximation, we express the multimode Jaynes-Cummings Hamiltonian in terms of collective bosonic operators and show that the system possesses two integrals of motion, enabling an exact diagonalization of the Hamiltonian in the single-excitation subspace. We show that for a strongly interacting qubit-photon system, the dynamics are captured by a reduced two-level model. In this framework, each level is defined as the product of the atomic excited state and a specific field mode, which takes the form of a multiphoton Fock-like state. In this picture, the Rabi oscillations represent a collective phenomenon corresponding to oscillations between multiphoton states differing by a single photon. We then extend our analysis to multiphoton processes in which the initial field is a coherent state with a continuous spectrum. In this case, the Rabi frequency is shown to be sensitive to the spectral profile of the function that generates the coherent state.