Glauber-Sudarshan P-representation Research Articles

Glauber P-representations for fermions

The Glauber-Sudarshan P-representation for bosons is well-known within quantum optics, and is widely applied to problems involving photon statistics. Less familiar, perhaps, is its fermionic counterpart, introduced by Cahill and Glauber. We present a derivation of both the bosonic and fermionic distributions and, in doing so, demonstrate the reason for the existence of two distinct fermionic forms and the relationship between these. We consider both single mode systems and also multiparticle systems with many modes. Expressions for the moments involving products of mode annihilation and creation operators are obtained. For simplicity only one type of boson or fermion will be considered, but generalising to more types is straightforward.

Density operator formulation for magnetic systems: Physical and mathematical aspects

In this work, the density operator diagonal representation in the coherent states basis, known as the Glauber–Sudarshan-P representation, is used to study harmonic oscillator quantum systems and models of spinless electrons moving in a two-dimensional noncommutative space, subject to a magnetic field background coupled with a harmonic oscillator. Relevant statistical properties such as the Q-Husimi distribution and the Wehrl entropy are investigated.

Differences and similarities between lasing and multiple-photon subtracted states

We examine the effect that the subtraction of multiple photons has on the statistical characteristics of a light field. In particular, we are interested in the question whether an initial state transforms into a lasing state, i.e.,~a (phase diffused) coherent state, after infinitely many photon subtractions. This question is discussed in terms of the Glauber P-representation $P(\alpha)$, the photon number distribution $P[n]$, and the experimentally relevant autocorrelation functions $g^{(m)}$. We show that a thermal state does not converge to a lasing state, although all of its autocorrelation functions at zero delay time converge to one. This contradiction is resolved by the analysis of the involved limits, and a general criterion for an initial state to reach at least such a pseudo-lasing state ($g^{(m)}\to 1$) is derived, revealing that they can be generated from a large class of initial states.

Operational Resource Theory of Continuous-Variable Nonclassicality

Genuinely quantum states of a harmonic oscillator may be distinguished from their classical counterparts by the Glauber-Sudarshan P-representation -- a state lacking a positive P-function is said to be nonclassical. In this paper, we propose a general operational framework for studying nonclassicality as a resource in networks of passive linear elements and measurements with feed-forward. Within this setting, we define new measures of nonclassicality based on the quantum fluctuations of quadratures, as well as the quantum Fisher information of quadrature displacements. These lead to fundamental constraints on the manipulation of nonclassicality, especially its concentration into subsystems, that apply to generic multi-mode non-Gaussian states. Special cases of our framework include no-go results in the concentration of squeezing and a complete hierarchy of nonclassicality for single mode Gaussian states.

Gaussian and Non-Gaussian operations on non-Gaussian state: engineering non-Gaussianity

AbstractMultiple photon subtraction applied to a displaced phase-averaged coherent state, which is a non-Gaussian classical state, produces conditional states with a non trivial (positive) Glauber-Sudarshan Prepresentation. We theoretically and experimentally demonstrate that, despite its simplicity, this class of conditional states cannot be fully characterized by direct detection of photon numbers. In particular, the non-Gaussianity of the state is a characteristics that must be assessed by phase-sensitive measurements. We also show that the non-Gaussianity of conditional states can be manipulated by choosing suitable conditioning values and composition of phase-averaged states.

On nonlinear coherent states properties for electron-phonon dynamics

This work addresses a construction of a dual pair of nonlinear coherent states (NCS) in the context of changes of bases in the underlying Hilbert space for a model pertaining to an electron-phonon model in the condensed matter physics, obeying a f-deformed Heisenberg algebra. The existence and properties of reproducing kernel in the NCS Hilbert space are studied and discussed; the probability density and its dynamics in the basis of constructed coherent states are provided. A Glauber-Sudarshan P-representation of the density matrix and relevant issues related to the reproducing kernel properties are presented. Moreover, a NCS quantization of classical phase space observables is performed and illustrated in a concrete example of q-deformed coherent states. Finally, an exposition of quantum optical properties is given.

An alternative approach for deriving a positive P-representation

Using the technique of integration within an ordered product of operators and adopting similar procedures as for the theory for quantizing a gauge field, we derive the Drummond-Gardiner positive P-representation directly from the Glauber-Sudarshan P-representation. The analogy between the extra phase-space dimensions in the positive P-representation and the gauge transformation freedoms for gauge field quantization is pointed out.

Positivization and regularization of quantum phase space distributions

It is shown that, to any quasiprobability distribution corresponding to a given density operator, one can associate a class of positive distributions in an extended space. In situations where the quasiprobability distributions are singular, these distributions are shown to provide regularized versions thereof. The positive distribution introduced by Drummond and Gardiner (1981) in the context of the Glauber-Sudarshan P-representation is shown to arise as a special case. One can thus associate a positive distribution to the Wigner function such that its moments give the averages of Weyl-ordered operators with respect to the given density operator. It is also found that one can associate to the Glauber-Sudarshan P-function, a distribution in the extended space which, though not positive, interestingly, involves the Wigner function (1932). A measurement scheme which directly yields these positive distributions is presented.

Strong linear amplification of quantum fields

Going ahead with investigations started by ideas of H. Paul it is examined how different quantum properties in the initial field state show up in the strongly amplified output field. As examples of extreme quantum fields the number state and the recently introduced phase optimized state are used. For the linear laser amplifier the inevitable noise is fully taken into account. These results are compared with those for a coherent input field. The characteristic differences in the macroscopic output fields allow unique conclusions onto the initial field. For the calculations the positive definite Glauber's P-representation after strong amplification is applied.

Generalised P-representations in quantum optics

A class of normal ordering representations of quantum operators is introduced, that generalises the Glauber-Sudarshan P-representation by using nondiagonal coherent state projection operators. These are shown to have practical application to the solution of quantum mechanical master equations. Different representations have different domains of integration, on a complex extension of the usual canonical phase-space. The 'complex P-representation' is the case in which analytic P-functions are defined and normalised on contours in the complex plane. In this case, exact steady-state solutions can often be obtained, even when this is not possible using the Glauber-Sudarshan P-representation. The 'positive P-representation' is the case in which the domain is the whole complex phase-space. In this case the P-function may always be chosen positive, and any Fokker-Planck equation arising can be chosen to have a positive-semidefinite diffusion array. Thus the 'positive P-representation' is a genuine probability distribution. The new representations are especially useful in cases of nonclassical statistics.

Quantum Statistics of Multiple-Mode, Superposed Coherent and Chaotic Radiation

This paper is concerned with the determination of the quantum statistics of the photoelectric detection, and the E-field detection of multiple-mode superposed chaotic and coherent radiation. The Glauber P-representation is utilized extensively, and results are obtained by a fully quantum-mechanical treatment. The generating function, the mean, and the variance of the photocount statistics are derived for illumination times of arbitrary duration and for modes of arbitrary statistical interdependence. For short illumination times, the complete photocount distribution is obtained. The distribution has the same form as that for a single mode of superposed chaotic and coherent radiation. The multiple-mode distribution, however, is nonstationary. The difference between experimental determinations of the mean and variance which utilize periodic sampling and those which utilize random sampling is derived. A method for determining the P-representation for modulated fields is described, and the detection statistics are derived. The statistics of the E field are found to be Gaussian, and it is found that for multiple-mode classical statistics and quantum statistics differ.