We have now reached a point where it is possible to consider more advanced and currently intensively developing topics in quantum optics. These particular topics have been chosen because of their importance in the development of modern laser physics, quantum atom optics, quantum information and computation. We start by considering the dynamics of an atom coupled in a free space to a multi-mode electromagnetic (EM) vacuum field. In free space an atom is coupled to many modes of the EM field and then an initial excitation in the system is irreversibly emitted from the atom into the field modes, never again to be reabsorbed by the atom, in contrast to a close system such as the Jaynes-Cummings model where an excitation was periodically exchanged between the atom and the field. The evolution of an excited atom coupled to a multi- mode EM field is a particular example of an open quantum system.
We have now reached a point where it is possible to consider more advanced and currently intensively developing topics in quantum optics. These particular topics have been chosen because of their importance in the development of modern laser physics, quantum atom optics, quantum information and computation. We start by considering the dynamics of an atom coupled in a free space to a multi-mode electromagnetic (EM) vacuum field. In free space an atom is coupled to many modes of the EM field and then an initial excitation in the system is irreversibly emitted from the atom into the field modes, never again to be reabsorbed by the atom, in contrast to a close system such as the Jaynes-Cummings model where an excitation was periodically exchanged between the atom and the field. The evolution of an excited atom coupled to a multi- mode EM field is a particular example of an open quantum system.
Background. The study of entangled states in Jaynes–Cummings models and its multiqubit generalizations and extensions remains an important and relevant topic in modern quantum optics, quantum information science and solid state physics. This is due to the fact that such models naturally describe the dynamics of interaction of logical elements of quantum devices, such as quantum computers or quantum networks (qubits) with microwave fields of resonators used to control the states of qubits. In this regard, it is very important to search for the most efficient schemes of generation, control and monitoring of entangled qubit states within the framework of Jaynes–Cummings type models. Aim. To investigate the entanglement dynamics of a pair of qubits, each of which is locked in a single-mode resonator and interacts non-resonantly with the vacuum field mode, in the presence of detuning between frequencies of transitions in the qubits and frequencies of resonator modes and dipole-dipole interaction of the qubits. Methods. To analyze the dynamics of the considered system, the solution of the time-dependent quantum Schrödinger equation is investigated. The exact solution of the above equation in the case of entangled Bell-type initial qubits states is found. This is used to calculate the criterion of entanglement of a pair of qubits – negativity. Numerical modeling of the time dependence of negativity for different values of the parameters of the considered model has been carried out. Results. It is shown that the presence of detuning, the difference in the qubit-photon couplings and the intensity of the dipole-dipole interaction significantly affect the maximum degree of entanglement of the qubit subsystem during its evolution. It is found that for certain parameters of the model under consideration, the initial Bell entangled states of qubits can be considered as long-lived stable states. Conclusion. The possibility of realization of long-lived and robust two-qubit entangled states in the system under consideration has been established. The obtained results can be used for effective control and manipulation of the degree of entanglement of qubits interacting with microwave fields of resonators.
A note to the reader Quantum optics has grown from a sub-discipline in atomic, molecular, and optical physics to a broad research area that bridges several branches of physics and that captures the imagination of the public. Quantum information science has put quantum optics into the spotlight of modern physics, as has the physics of ultracold quantum gases with its many spectacular connections to condensed-matter physics. Yet through all these exciting developments quantum optics has maintained a characteristic core of ideas that I try to explain in this slim volume. Quantum optics focuses on the simplest quantum objects, usually light and few-level atoms, where quantum mechanics appears in its purest form without the complications of more complex systems, often demonstrating in the laboratory the thought experiments that the founders of quantum mechanics dreamed of. Quantum optics has been, and will be for the foreseeable future, quintessential quantum mechanics, the quantum mechanics of simple systems, based on a core of simple yet subtle ideas and experiments. One of the strengths of quantum optics is the close connection between theory and experiment. Although this book necessarily is theoretical, many of the theoretical ideas I describe are guided by experiments or, in turn, have inspired experiments themselves. Another strength of quantum optics is that it is done by individuals or small teams. One single person or a small group can build and perform an entire experiment.
The multiphoton Jaynes-Cummings model is investigated and applications in quantum information science are explored. Considering the strong atom-field coupling regime and an N-photon interaction, a nonlinear driving field can perform an arbitrary rotation in the Fock space of a bosonic mode involving the vacuum and an M-Fock state, with M<N. In addition, driving a bosonic mode with a linear coherent field (superposition of many Fock states), only the cavity states within the Fock subspace {|0⟩,|1⟩,…,|N-1⟩} can be populated; i.e., we show how to implement a Fock state filter, or quantum scissor, that restricts the dynamics of a given bosonic mode to a limited Hilbert space. Such a device can be employed as a generator of finite-dimensional quantum-optical states and also as a quantum-optical intensity limiter, allowing as a special case the generation of single-photon pulses. On the other hand, our system also provides a very rich physics in the weak atom-field coupling regime, in particular, multiphoton electromagnetically induced transparencylike phenomena, inducing a narrow (controllable) reflectivity window for nonlinear probe fields. These results are useful for applications in quantum information processing and also motivate further investigations, e.g., the use of an N-photon Jaynes-Cummings system as a qudit with harmonic spectrum and the exploration of multiphoton quantum interference.
Introduction So far, we have discussed closed systems involving a single quantized mode of the field interacting with atoms, as for example in the Jaynes–Cummings model in Chapter 4. As we saw in this model, the transition dynamics are coherent and reversible: the atom and field mode exchange excitation to and fro without loss of energy. As we add more modes for the atom to interact with, the coherent dynamics become more complicated as the relevant atom–field states come in and out of phase and beat together to determine the total state occupation probabilities. As time goes on, these beats get out of phase, leading to an apparent decay of the initial state occupation probability. But at later times, the beating eigenfrequencies get back in phase in a manner rather reminiscent of the Jaynes–Cummings revival discussed earlier in this book, and this leads to a partial recurrence or revival of the initial state probability. The time scale for this partial revival depends on the number of participating electromagnetic field modes and as these increase to the level appropriate for an open system in free space the recurrence disappears off to the remote future, and the exponential decay law appropriate for decay is recovered as an excellent approximation. We have already discussed the origin of spontaneous emission and the Einstein A coefficient using perturbation theory in Chapter 4.
In this work we shall study the interaction of matter and light. In so doing we shall rely heavily on the description of such processes provided by quantum mechanics. This appears on a number of levels: firstly, a quantum description of matter is indispensable if one wants to understand on the microscopic scale the different kinds of interaction processes that can occur. Secondly, a quantum description of light often turns out to be useful, sometimes necessary, to better understand these processes. We shall study phenomena such as spontaneous emission, which can only be properly treated by a theory taking into account the quantum nature of both light and matter. In the following chapters we shall address, amongst others, the following question: ‘given an atom prepared at a given time in a particular state and subjected from this time onwards to electromagnetic radiation, what is the state of the atom and radiation at any later moment in time?’ In order to be able to answer this question it will be necessary for us to know how to calculate the evolution of a quantum system in a small number of typical situations. These methods we shall demonstrate in the first chapter. The evolution of the coupled atom–light system depends on the temporal dependence of the applied light field, which could, for example, be applied from a given moment and thereafter remain unchanged in intensity, or, perhaps, be appreciable only for a finite period of time (pulsed excitation).