On tomographic reconstruction of the quantum state of a two-mode light field

Abstract

We consider the state reconstruction of an optical two-mode light field from sum quadrature distributions measured with a single balanced homodyne detector. New explicit formulas for the pattern functions necessary to reconstruct the density matrix of the two-mode field in the photon-number basis are derived. Moreover, an expression of the measured quadature distribution in terms of the two-mode normally ordered moments is given and the determination of the moments from it is discussed.     

Quantum state reconstruction by multichannel unbalanced homodyning

Abstract

Multichannel unbalanced homodyning is proposed for measuring the density matrix of a single-mode optical field in the photon-number representation. Combining the signal beam and the local oscillator by a hightransmittance beam splitter, the interfering field is detected using multichannel photocounting. The density matrix is determined by direct statistical sampling of the coincident events recorded for various values of the local-oscillator phase. The usefulness of the method is demonstrated by computer simulations of measurements including error estimations.     

Preparation of an arbitrary density matrix of a harmonic oscillator

Abstract

We propose a deterministic method to generate an arbitrary (pure or mixed) density matrix of a harmonic oscillator. The general density matrices are achieved by manipulating quantum entanglement between the oscillator and an auxiliary oscillator. We discuss how our preparation scheme can be realized by cavity quantum electrodynamics interactions so that a general density matrix of a single-mode electromagnetic field can be created.     

Optical state measurement with a two-component probe

Abstract

A state of light which is a superposition of the vacuum and the one-photon number state is the simplest state containing phase information. Recently we have shown how a field in such a state might be generated and here we explore its usefulness as a probe for measuring unknown states of light. We find that this probe can be used reasonably simply both to determine completely some pure states of light and to measure the diagonal and nearest off-diagonal elements of the density matrix in the number state basis and hence to obtain the mean sine and cosine of the phase of an unknown mixed state. We suggest further how a field in a superposition of the vacuum and the two-photon number state might be generated and how this can be used as a probe, both to measure the off-diagonal matrix elements second nearest to the diagonal of a mixed state density matrix and to measure the variance of the cosine and the sine of the phase. We also examine the experimentally more likely case where the probe fields are in mixed states and show how the same information about the unknown state can still be retrieved.     

Radon transform and pattern functions in quantum tomography

Abstract

The two-dimensional Radon transform of the Wigner quasiprobability is introduced in canonical form and the functions playing a role in its inversion are discussed. The transformation properties of this Radon transform with respect to displacement and squeezing of states are studied and it is shown that the latter is equivalent to a symplectic transformation of the variables of the Radon transform with the contragredient matrix to the transformation of the variables in the Wigner quasiprobability. The reconstruction of the density operator from the Radon transform and the direct reconstruction of its Fock-state matrix elements and of its normally ordered moments are discussed. It is found that for finite-order moments the integration over the angle can be reduced to a finite sum over a discrete set of angles. The reconstruction of the Fock-state matrix elements from the normally ordered moments leads to a new representation of the pattern functions by convergent series over even or odd Hermite polynomials which is appropriate for practical calculations. The structure of the pattern functions as first derivatives of the products of normalizable and non-normalizable eigenfunctions to the number operator is considered from the point of view of this new representation.     

Determination of photon statistics and density matrix from double homodyne detection measurements

Abstract

We consider double balanced homodyne detection schemes with and without (local oscillator) phase-randomized detection. We discuss the reconstruction of the photon statistics from phase-randomized measurements. We show how the Wigner function of a photon-number state can be synthesized from phase-randomized double homodyne measurements of two properly prepared field states. Moreover we propose a new procedure to determine the whole density matrix from the Q function. Finally we express the Q function in terms of normally ordered moments and discuss their reconstruction.     

Direct sampling of density matrix in displaced Fock-state basis from quadrature distributions and reconstruction of quasiprobability distributions

Abstract

The sampling functions needed to reconstruct from quadrature distributions the density matrix elements in the displaced Fock-state basis are determined as scaled and shifted pattern functions f_mn used to reconstruct the density matrix elements Q _mn in the Fock-state basis. Having at hand the diagonal density matrix elements one can reconstruct any s-parametrized quasiprobability distribution via a simple weighted sum over these quantities. A smoothed Wigner function can be directly sampled from the measured quadrature distribution of the signal field. The corresponding sampling function is just a shifted and scaled version of f ₀₀.     

Correlation and joint density matrix of two spatial–temporal modes from balanced-homodyne sampling

Abstract

A theoretical treatment is given which establishes dual-mode balanced-homodyne detection as a practical and well characterized technique for measuring optical field correlations, photon-number correlations, or the full quantum state of a pair of optical modes. The definition of modes used includes temporal wave packets, Gaussian or other beam profiles, or two-frequency fields. The proposed method allows the measurement of two-time correlations on sub-picosecond scales, the disentangling of the statistics of signal light in two spatially overlapping modes, and the measurement of field correlations, such as squeezing, over 100 THz bandwidths. We show how to estimate from the data the statistical errors on the measured correlations and the density matrix arising from finite data sets, and the errors introduced by using finite numbers of phases and relative amplitudes of the two local oscillator fields.     

Long Time Behaviour of the Field in a Lossless Micromaser with Intensity-dependent Coupling Constant

Abstract

Using the asymptotic form of the density matrix the role of the non-diagonal elements of the density matrix is estimated in the lossless micromaser model based on the Jaynes-Cummings model with intensity dependent coupling constant. It is shown that a certain type of a (almost pure) multiphoton state can be generated in this system.     

State of Polarization in Open- and Closed-loop Helices of Single-mode Fibres

Abstract

The state of polarization in helically wound single-mode fibres is described in terms of coupled-mode equations and the Mueller matrix for an elliptically birefringent single-mode fibre in the quasi-monochromatic case. Possible depolarization has been accounted for by means of the mutual correlation function |γ| between eigenpolarization modes. The polarization state in closed-loop fibre-optic helices has been studied experimentally under single- and dual-mode operation. It has been shown that the closed-loop set-up can be used for the development of compact fibre-optic sensors.     

Characterization of Modal Noise,Splice and Bending Loss in Single-mode Depressed Cladding Fibres

Abstract

Using a Gaussian function to approximate the modal field of a single-mode depressed cladding (or W-) fibre, we derive a simple expression for the modal spot-size, which depends only on the normalized frequency and two parameters that characterize the refractive-index profile. This spot-size can then be used to determine the expected splice and bend loss. The formalism for determining the spot-size can be generalized to approximate higher-order modes, which allows for a simple characterization of modal noise due to misaligned joints in single-mode fibres operating above cut-off.     

Collapse-revival Phenomenon in the Jaynes-Cummings Model Interacting with the Multiphoton Holstein-Primakoff SU(2) Coherent State

Abstract

We have analysed the behaviour of the atomic population inversion of the two-level atom interacting with a single-mode field initially prepared in the multiphoton Holstein-Primakoff SU(2) coherent state. It is shown that the behaviour of the atomic inversion depends on the parameters characterizing the initial state of the field. In particular, the atomic inversion can exhibit periodical oscillations as well as the collapse-revival phenomenon.     

Modeling of two-dimensional random fields

This paper presents a method of conditional stochastic modeling of two-dimensional fields which can be used to predict values at certain field points at a given time, based on field values at other locations at the same time and on data about second order field moments at given points. For computer simulations, the Gaussian truncated distributions are used. The aim of this work is also to present a derivation of a formula for the probability density of an n-dimensional random variable with the Gaussian conditional truncated distribution. As a numerical example, a soil contamination field described by correlation functions corresponding to the white noise field, the Shinozuka field and the Markov field is analyzed. The acceptance-rejection method is applied to generate covariance matrices and vectors of field values. Then, conditional expected field values for adequate correlation functions are calculated.     

Reconstructing the Wavefunction in Quantum Optics

Abstract

The problem of reconstructing a wavefunction from probability distributions is re-examined in the context of whether a pure state vector of a single-mode optical field can be reconstructed from the photon number and phase probability distributions. An analytical solution is given for the case where the state of the mode is a superposition of a finite number of Fock states.     

Production of Macroscopic Schrödinger Cat States from Weak Quantized Cavity Fields Interacting with Atoms Driven by Classical Fields

Abstract

We show that macroscopic superposition (Schrödinger cat) states of a quantized single-mode cavity field can be produced via the interaction of this field with a two-level atom which is driven by a classical field even for small initial intensities of the quantized cavity mode. We show that with a properly chosen driving field an almost pure superposition state with arbitrary amplitudes and phases of component states can be produced.     

Ordered moments and relation to Radon transform of Wigner quasiprobability

Abstract

It is shown that the symmetrically ordered moments of boson operators for a single boson mode can be reconstructed from the corresponding moments of the Radon transform of the Wigner quasiprobability for discrete sets of equidistant inequivalent angles which solve the circle division problem. This reconstruction is sometimes simpler than the corresponding reconstruction of the normally ordered moments where one first has to multiply the Radon transform with Hermite polynomials in comparison to power functions for symmetrically ordered moments and then to integrate. The connection to the reconstruction for the general class of s-ordered moments is established. The transition from discrete sets of angles to integration over angles via averaging over the discrete angles is made. The results are applied to displaced squeezed thermal states. It is shown how the ordered moments for these states can be explicitly found from the calculated Radon transform of the Wigner quasiprobability. The obtained formulae for these moments possess independent interest since they contribute to the discussion of the properties of the most general class of states with quasiprobabilities of Gaussian form with many possible special cases as, for example, squeezed coherent states and squeezed thermal states.     

A Monte Carlo formulation of the Bogolubov theory

Abstract

We propose an efficient stochastic method to implement numerically the Bogolubov approach to study finite-temperature Bose-Einstein condensates. Our method is based on the Wigner representation of the density matrix describing the non-condensed modes and a Brownian motion simulation to sample the Wigner distribution at thermal equilibrium. Allowing it to sample any density operator Gaussian in the field variables, our method is very general and it applies both to the Bogolubov and to the Hartree-Fock Bogolubov approach, in the equilibrium case as well as in the time-dependent case. We think that our method can be useful to study trapped Bose-Einstein condensates in two or three spatial dimensions without rotational symmetry properties, as in the case of condensates with vortices, where the traditional Bogolubov approach is difficult to implement numerically due to the need to diagonalize very big matrices.     

Effect of losses on phase noise

Abstract

We study the effect of losses on the phase noise of single-mode field states. The losses are described by the standard loss master equation, and it is used to find an upper bound for the increase in the phase noise as a function of time. We compare the time dependence of the phase noise of an initial coherent state to that of a state that initially has very small phase noise. Both states have the same initial mean photon number. While the small-phase noise state is more susceptible to losses, the difference between its behaviour and that of the coherent state is not great.     

Pulse-mode study of the quantum correlation of photons in an N-photon Dicke model

Abstract

We find the N-photon state emitted by an N-step Dicke model and provide a method to construct the field coherence functions based on it. Our effort is concentrated on the second order coherence, or the one-photon density matrix. When expressed in its canonical representation, this matrix gives the photon number occupying each ‘pulse eigenmode’. This number serves as an indicator of the correlation between photons. By studying the evolution of the one-photon density matrix we can trace the creation of such correlation during the emission. From the asymptotic solution we are able to find approximate scaling law relations between the photon degeneracy in the eigenmodes and the total number of photons involved.