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Physical Review A

Properties of displaced number states

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, V. Buek

Published Mar 1, 1990• 334 citations
Publisher

Abstract

Recent developments in quantum optics have led to new proposals to generate number states of the electromagnetic field using conditioned measurement techniques or the properties of atom-field interactions in microwave cavities in the micromaser. The number-state field prepared in such a way may be transformed by the action of a displacement operator; for the microwave micromaser state this could be implemented by the action of a classical current that drives the cavity field. We evaluate some properties of such displaced number states, especially their description in phase space. The photon number distribution is shown to display unusual oscillations, which are interpreted as interference in phase space, analogous to Franck-Condon oscillations in molecular spectra. The possibility of detecting these oscillations is discussed, through the photodetection counting statistics of the displaced number states. We show that the displaced-number-state quantum features are relatively robust when dissipation of the field energy is included.

DOI
10.1103/physreva.41.2645

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Recent developments in quantum optics have led to new proposals to generate number states of the electromagnetic field using conditioned measurement techniques or the properties of atom-field interactions in microwave cavities in the micromaser. The number-state field prepared in such a way may be transformed by the action of a displacement operator; for the microwave micromaser state this could be implemented by the action of a classical current that drives the cavity field. We evaluate some properties of such displaced number states, especially their description in phase space. The photon number distribution is shown to display unusual oscillations, which are interpreted as interference in phase space, analogous to Franck-Condon oscillations in molecular spectra. The possibility of detecting these oscillations is discussed, through the photodetection counting statistics of the displaced number states. We show that the displaced-number-state quantum features are relatively robust when dissipation of the field energy is included.
Experiments with the one-atom maser1 have demonstrated that it is possible to study the interaction of a single atom with a single-mode of a resonant electromagnetic field in a cavity. The atoms used in these experiments are Rydberg atoms with a very large principal quantum number. Another important ingredient is the superconducting cavity: The quality factor is high enough for a periodic energy exchange between atom and cavity field to be observed; i.e., the relaxation time of the cavity field is larger than the characteristic time of the atom-field interaction, which is given by the reciprocal of the Rabi frequency. It is therefore possible to perform experiments on the dynamics of the atom-field interaction. Some of the features are explicitly a consequence of the quantum nature of the electromagnetic field.2 With a new cavity (quality factor 3 × 1010) it could be demonstrated that the statistics of the photons in the cavity is sub-Poissonian; in suitable conditions it is even possible to generate number states. Using number states of the single-atom maser novel quantum interference experiments can be performed giving new insight in the wave particle duality.
Many sophisticated quantum states of light display non-classical, characteristic oscillations in their photon statistics altering with respect to the applied displacement. In order to detect these oscillations the mode matching between the studied state and the displacement field plays a crucial role. Only recently the developments in the photon counting techniques have allowed the direct measurements of the photon-number distributions of quantum states. We investigate the properties of displaced single photons by applying time-multiplexed detection of photon statistics and quantify the value of the mode overlap.
The time-evolution of a single two-level atom in a single-mode high-Q cavity is sensitive to the quantum fluctuations of the cavity radiation field and to its photon statistics: this sensitivity is realizable experimentally in the Rydberg atom micromaser. We study the effects of the interaction of a two-level atom with two new non-classical radiation fields: the squeezed number state and the displaced number state realizable by nonlinear and linear transformations of field number states which have an initially precise occupation number. The time-varying field fluctuations caused by the atomic interaction are described using the Q-function quasi-probability.
A numerical analysis of dispersive quantum nondemolition measurement of the photon number of a microwave cavity field is presented. Simulations show that a key property of the dispersive atom-field interaction used in Ramsey interferometry is the extremely high sensitivity of the dynamics of atomic and field states to basic parameters of the system. When a monokinetic atomic beam is sent through a microwave cavity, a qualitative change in the field state can be caused by an uncontrollably small deviation of parameters (such as atom path length through the cavity, atom velocity, cavity mode frequency detuning, or atom-field coupling constants). The resulting cavity field can be either in a Fock state or in a super-Poissonian state (characterized by a large photon-number variance). When the atoms have a random velocity spread, the field is squeezed to a Fock state for arbitrary values of the system’s parameters. However, this makes detection of Ramsey fringes impossible, because the probability of detecting an atom in the upper or lower electronic state becomes a random quantity almost uniformly distributed over the interval between zero and unity, irrespective of the cavity photon number.
Starting with early experiments about lifetimes of excited atoms near metallic boundaries, I review the evolution of Cavity Quantum Electrodynamics in the microwave and in the optical domains. The coherent coupling of single atoms with photons in high-Q cavities has led to the demonstration of fundamental quantum effects in light-matter interaction processes and to the generation of micromasers and microlasers emitting radiation with non-classical properties. It has made possible the development of single photon light sources and efficient detectors of single atoms as well as the implementation of quantum information and communication procedures in which atoms and photons play the roles of quantum bits. Mesoscopic superpositions of field states have been generated by entangling single atoms to fields made of many photons and the decoherence of these states has been studied, shedding new light on the quantum to classical boundary. Cavity Quantum Electrodynamics has also led recently to the quantum non-demolition counting of photons in a cavity, opening new perspectives for studying non-classical states of radiation. These atomic physics experiments are now extended - with promising applications in perspective - to the study of solid-state devices with microcavities built on substrates and coupled to artificial atoms such as quantum dots or superconducting junctions.
Using Pegg-Barnett formalism the author has studied the phase properties of displaced number states. It has been found that the phase probability distribution exhibits the same oscillations as the photon number distribution. Using the Hermitian phase operator the fluctuations of the phase of the displaced number states have been studied. The dependence of the fluctuations on the value of the reference phase have been analysed in detail. It has been found that phase of displaced number states exhibit squeezing in the high-intensity limit of the driving field. These states are approximately equivalent to phase states with precisely defined phase. It has been shown that displaced number states are not minimum uncertainty states with respect to the phase and the number operators.