Coherent States of a Harmonic Oscillator in a Finite-dimensional Hilbert Space and Their Squeezing Properties

Abstract

New coherent states of a harmonic oscillator in a finite-dimensional Fock space are introduced. Some properties of these coherent states are discussed. The second-order squeezing of these coherent states with respect to the quadrature operators is studied in detail. In particular, for a two-state system the arbitrary higher-order squeezing of these states is investigated. It is shown that these coherent states exhibit much richer squeezing properties than the coherent states of a usual harmonic oscillator in an infinite-dimensional Fock space. It is found that these coherent states have not only second-order squeezing but also higher-order squeezing with respect to the quadrature operators of the field under consideration.     

Non-classical Properties of Two-mode SU(1, 1) Coherent States

Abstract

We examine the non-classical properties of two-mode coherent states based on different unitary irreducible representations of SU(1, 1). Such states are generated by the action of the two-mode squeezing operator on initial states of the field containing arbitrary numbers of photons in each of the two modes. If the initial state of the field is a two-mode vacuum state, the final state is of course the two-mode squeezed vacuum. An initial occupation generalizes the idea of a squeezed vacuum to the SU(1, 1) coherent states. We show that fields in such states have remarkable quantum properties such as sub-Poissonian statistics, violations of the Cauchy-Schwarz inequality, strong correlations in the photon number fluctuations and squeezing. Using information theory formalism, we show that these coherent states are less correlated than the usual two-mode squeezed vacuum. Moreover, we show that an initial coherent amplitude contribution, in a large amplitude limit, can result in the reduction of correlations between modes.     

Light Squeezing in the Jaynes-Cummings Model with Intensity-dependent Coupling

Abstract

We have found that in the intensity-dependent-coupling Jaynes-Cummings model with the coherent radiation field light squeezing exhibits periodical revivals. The influence of the initial states of the atom on the squeezing of light is investigated in detail.     

On the Squeezing of Coherent States by the Multiphoton Jaynes-Cummings Hamiltonian with an Intensity-dependent Coupling

Abstract

The squeezing properties of the multiphoton Hamiltonian with intensity-dependent coupling are evaluated for the [xcirc] and [pcirc] _x quadratures, for the initial state of a coherent electromagnetic field and an atom in the ground state. Two measures of squeezing: the percentage of total squeezing and the squeezing time-period percentage, are introduced. Interesting squeezing properties with respect to [xcirc] are observed for real coherent states when the time evolution of the above measures and of the time-averaged squeezing are analysed. The multiphoton intensity-dependent coupling Hamiltonian is found to be almost independent of the specific powers of the annihilation and creation operators, as long as the sum of the powers is kept constant.     

A simple model for a minimal environment: the two-atom Tavis–Cummings model revisited

Individual quantum systems may be interacting with surrounding environments having a small number of degrees of freedom. Here we discuss a simple toy model: a system constituted by a two-level atom (atom 1) interacting with a single mode cavity field which is (weakly) coupled to a small environment (atom 2). We investigate the influence of the minimal environment on the dynamics of the linear entropy and the atomic dipole squeezing of atom 1, as well as the entanglement between atom 1 and the field. We also obtain the full analytical solution of the two-atom Tavis–Cummings model for both arbitrary coupling strengths and frequency detunings, necessary to analyse the influence of the field-environment detuning on the evolution of the system’s quantum properties. For complementarity, we discuss the role of the degree of mixedness of the environment by analysing the time-averaged linear entropy of atom 1.     

Adding and subtracting energy quanta of the harmonic oscillator

Abstract

We propose a scheme to add/subtract excitations to/from an arbitrary quantum state or the harmonic oscillator. The method displaces the photon-number distribution and leaves its shape unchanged for a wide range of displacements. Mathematically this is realized by the action of phase operators of the Susskind-Glogower type onto the initial quantum state. Consequently, the shape of the phase distribution is preserved unless the number statistics are modified due to displacing it by subtraction onto the vacuum state. Starting with an initially coherent state one may realize pure quantum states displaying either amplitude or phase squeezing. The implementation of the method is based on interactions of the Jaynes-Cummings type, in the case of subtracting quanta one additionally needs to perform measurements on the electronic quantum state of the atoms. Our approach could be used for adding and subtracting both photons on a cavity field and motional quanta of a trapped ion.     

Squeezing in Multiphoton Absorption

Abstract

The conditions for obtaining squeezing by single-beam multiphoton absorption are examined. For an initial coherent light the amount of squeezing is calculated in the second order of interaction time using the master-equation formalism. A limit formula for the amount of squeezing is found in the case of an initial strong coherent beam. The results are compared with the exact numerical calculations and some remarks are made on the short-time approximation in multiphoton absorption.     

Population trapping in the Jaynes-Cummings model with a Kerr nonlinearity in the cavity

Abstract

An electromagnetic field state is found which maintains the population inversion of the atom stationary during the interaction with the field through a Jaynes-Cummings model (JCM) with a Kerr type nonlinearity in the cavity. The condition of stationarity of the population inversion includes the phase coupling of atomic dipole with the field. We have shown that the Kerr nonlinearity in the cavity field significantly modifies the photon statistics of the trapped field state through an intensity dependent detuning in the field compared to the normal JCM trapping state. We have also demonstrated the novel features of sub-Poissonian character and the squeezing of the trapped field state. The dynamics of the initial trapped field is studied in terms of squeezing and the Q-function.     

Magnus expansion method for two-level atom interacting with few-cycle pulse

Abstract

Using the Magnus expansion to the fourth order, we obtain analytic expressions for the atomic state of a two-level system driven by a laser pulse of arbitrary shape with small pulse area. We also determine the limitation of our obtained formulas due to limited range of convergence of the Magnus series. We compare our method to the recently developed method of Rostovtsev et al. (PRA 2009, 79, 063833) for several detunings. Our analysis shows that our technique based on the Magnus expansion can be used as a complementary method to the one in PRA 2009.     

SU(1, 1) and SU(2) Squeezing in Trilinear Optical Processes

Abstract

The squeezing properties in terms of SU(1, 1) and SU(2) operators for the case of trilinear processes are studied. The initial state of the system is supposed to be a coherent state in one of the modes and number states in the remaining modes. It is pointed out that in several cases a considerable amount of squeezing can be achieved. Due to the common mathematical structure the case of a two-mode coupler with intensity dependent coupling is also analysed.     

Two-photon Coherence and Steady-state Saturated and Inverted Populations in Three-level Systems

Abstract

We study the correlation between level populations, absorption and two-photon coherence in closed Λ- and V-shaped three-level systems interacting with two lasers of arbitrary intensity. Whereas it is well known that maximum two-photon coherence leads to population trapping in Λ-shaped three-level systems interacting with two equally detuned lasers, we demonstrate that minimum two-photon coherence, under saturation conditions, can lead to equal populations in all the levels of a Λ- or V-shaped system when the lasers are symmetrically detuned. Moreover we show that steady-state population inversion between one of the upper levels of a V-shaped system and the ground state can occur when either the detunings are asymmetrical or the upper states decay at different rates. It is shown that this population inversion does not lead to stimulated emission into the lasing modes. These effects are also correlated with two-photon coherence and are explained physically by a cross three-photon scattering process involving two laser photons of different frequencies.     

Entropy squeezing of a degenerate two-photon process with a nonlinear medium

Abstract

This paper presents new results on some aspects of the properties of the entropy squeezing in the time development of a degenerate two-photon of a single-mode interacting with a two-level system with a nonlinear medium. A general analytic expression for the density matrix is obtained on the basis of a Hamiltonian generalized in that it includes nonlinear effects and Stark shifts, by means of which we identify and numerically demonstrate the region of parameters where significantly large squeezing can be obtained. The influence of various parameters on the entropy squeezing are explored. It is shown that features of the entropy squeezing were influenced significantly by changing the nonlinear medium, Stark shift and the detuning. The results show that, certain amounts of the nonlinear effect yield the superstructure of atomic Rabi oscillation and changes the quasiperiod of the entropy evolution and atomic inversion.     

Quantum Theory of Hyper-Raman Scattering

Abstract

We develop the quantum theory of stimulated hyper-Raman scattering (HRS) and discuss the quantum statistics of pump and Stokes fields. We show that the initially coherent pump field acquires a strong squeezing that exceeds the similar squeezing in other nonlinear processes also involving two-photon absorption. Stokes field statistics are analysed in detail as a function of the initial statistics of both modes. Bunching effects are shown to be amplified in the short-time region, while in asymptotics the antibunching of Stokes photons becomes dominant, which in many cases results in sub-Poissonian statistics of Stokes mode. We also find a number of correlation effects between the photons of the two modes.     

Squeezing of Spectral Components in the Jaynes-Cummings Model

Abstract

Squeezing of spectral components of a fluorescence field in the framework of the Jaynes-Cummings model has been studied. It has been shown that squeezing is observed only in sidebands of the fluorescence field. The degree of squeezing increases as the intensity of the cavity mode is increased. Contrary to squeezing in the cavity mode (see Kuklinski and Madajczyk [14] a large degree of squeezing in sidebands of the fluorescence field is observed in the short-time limit.     

Two-level atom in a squeezed vacuum with finite bandwidth

Abstract

The resonance fluorescence of a two-level atom driven by a coherent laser field and damped by a finite bandwidth squeezed vacuum is analysed. We extend the Yeoman and Barnett technique to a non-zero detuning of the driving field from the atomic resonance and discuss the role of squeezing bandwidth and the detuning in the level shifts, widths and intensities of the spectral lines. The approach is valid for arbitrary values of the Rabi frequency and detuning but for the squeezing bandwidths larger than the natural line-width in order to satisfy the Markoff approximation. The narrowing of the spectral lines is interpreted in terms of the quadrature-noise spectrum. We find that, depending on the Rabi frequency, detuning and the squeezing phase, different factors contribute to the line narrowing. For a strong resonant driving field there is no squeezing in the emitted field and the fluorescence spectrum exactly reveals the noise spectrum. In this case the narrowing of the spectral lines arises from the noise reduction in the input squeezed vacuum. For a weak or detuned driving field the fluorescence exhibits a large squeezing and, as a consequence, the spectral lines have narrowed linewidths. Moreover, the fluorescence spectrum can be asymmetric about the central frequency despite the symmetrical distribution of the noise. The asymmetry arises from the absorption of photons by the squeezed vacuum which reduces the spontaneous emission. For an appropriate choice of the detuning some of the spectral lines can vanish despite that there is no population trapping. Again this process can be interpreted as arising from the absorption of photons by the squeezed vacuum. When the absorption is large it may compensate the spontaneous emission resulting in the vanishing of the fluorescence lines.     

Influence of squeezing bandwidths on resonance fluorescence

Abstract

We present an analytical technique for investigating the behaviour of a coherently driven atom damped by a squeezed vacuum with finite squeezing bandwidth. A master equation and analytic expressions for the fluorescent spectrum are derived for the simple case of a two-level atom exactly resonant with the frequencies of both the squeezed field and the driving field. Suitable choices of driving strength, phase and squeezing source lead to either sub-natural lincwidths of the Rabi sidebands or a sub-natural linewidth of the central spectral peak or indeed sub-natural linewidths of all three spectral peaks when squeezing bandwidths are taken into account. We also establish the remarkable result that in the strong driving regime, the width of the central peak of the fluorescent spectrum depends solely on the squeezing found at the Rabi sideband frequencies.     

Squeezing with Rydberg Atoms

Abstract

It is shown that some 17 Rydberg Na atoms initially placed into a coherent atomic state and super-radiating into a low-Q microwave cavity at temperatures T ? 0·4 K will show modest squeezing in the fluorescence field, the squeezing arising from terms oscillating at twice the cavity frequency. It is also shown that similar numbers of Rydberg atoms undergoing resonance fluorescence in a coherent single-mode microwave driving field and without any cavity will show more substantial squeezing in the fluorescence field.     

Effects of the Binomial Field Distribution on Collapse and Revival Phenomena in the Jaynes-Cummings Model

Abstract

The dynamical properties of a two-level atom interacting with a single non-decaying mode of an electromagnetic field in a binomial state are studied. The statistical aspects of the field, such as intensity-intensity correlation and squeezing, are also investigated. The binomial state reduces to a pure number state and a pure coherent state in different limits. Hence it enables us to study how the sinusoidal Rabi oscillations in a pure number state develop to give rise to the phenomenon of collapse and revival which has been studied extensively in the coherent-state field. In addition, the binomial state exhibits squeezing for certain values of parameters, but it is not a minimum-uncertainty-product state.     

Periodic Squeezing in the Tavis-Cummings Model

Abstract

By utilizing our previous operator solution [17], we have investigated the squeezing in the radiation field of the Tavis-Cummings model (collective N ? 1 two-level atoms interacting with a resonant single cavity quantized mode). With field and atoms initially in coherent field state strong or weak and atomic coherent state (of few excited atoms), periodic time-dependent squeezing in the field and the macroscopic polarization is expressed in terms of Jacobian elliptic functions of the first kind. The statistical investigations are carried out for the quasiprobability distribution functions (Wigner function and Q function). The distribution function of the field quadrature has a variance less (greater) than that for a coherent state if this quadrature is squeezed (unsqueezed).     

Squeezed States for Interferometric Gravitational-wave Detectors

Abstract

The possibility of using squeezed states for improving the shot-noise limit of the strain sensitivity of Michelson interferometers is discussed. We find that the spectrum of squeezing required depends on the method of stabilization used in the experiment. Details are given for the widely used phase-modulation technique (which also allows for recycling of the field), where we find an important application for broadband (‘two-mode’) squeezing.