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Physical Review A

Measure of the nonclassicality of nonclassical states

Ching Tsung Lee

Published Sep 1, 1991• 361 citations
Publisher

Abstract

A continuous parameter introduced into the convolution transformation between P and Q functions leads to a measure of how nonclassical quantum states are with values ranging from 0 to 1: For photon-number states, the value is 1, the maximum possible. For squeezed vacuum states, it is a monotonically increasing function of the squeeze parameter with values varying from 0 to 1/2. This measure is identical to the minimum number of thermal photon necessary to destroy whatever nonclassical effects existing in the quantum states.

DOI
10.1103/physreva.44.r2775

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A continuous parameter introduced into the convolution transformation between P and Q functions leads to a measure of how nonclassical quantum states are with values ranging from 0 to 1: For photon-number states, the value is 1, the maximum possible. For squeezed vacuum states, it is a monotonically increasing function of the squeeze parameter with values varying from 0 to 1/2. This measure is identical to the minimum number of thermal photon necessary to destroy whatever nonclassical effects existing in the quantum states.
The concept of photon antibunching as a manifestation of nonclassical character in single-mode radiation is extended to two-mode radiation. A simple criterion for the existence of nonclassical effects in two-mode radiation is established. It implies that the intermode photon bunching, rather than antibunching, can play a key role in rendering two-mode radiation nonclassical. The two-mode squeezed vacuum states are used as simple examples to illustrate the essential point of the criterion. The criterion is then used to study the nonclassical properties of two types of two-mode squeezed states. A state generated by applying the squeeze operator to the two-mode vacuum first, followed by the displacement operator, is called a two-mode coherent squeezed state. If the order of the two operators is reversed, a two-mode squeezed coherent state is obtained. It is found that the latter has a much stronger tendency to become a nonclassical state than the former, assuming that the parameters involved are all the same. A measure of nonclassical depth of radiation is adopted. It is found that, according to this measure, the effect of squeezing in making two-mode coherent squeezed states nonclassical is not monotonic. Such phenomena do not occur in squeezed coherent states.
We investigate nonclassical properties of the field states generated by subtracting any number of photons from the squeezed thermal state (STS). It is found that the normalization factor of photon-subtracted STS (PSSTS) is a Legendre polynomial of squeezing parameter rrr and average photon number $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ of the thermal state. Expressions of several quasiprobability distributions of PSSTS are derived analytically. Furthermore, the nonclassicality is discussed in terms of the negativity of the Wigner function (WF). It is shown that the WF of single PSSTS always has negative values if $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}<\mathrm{sinh}{}^{2}r$ at the phase space center. The decoherence effect on PSSTS is then included by analytically deriving the time evolution of WF. The results show that the WF of single PSSTS has negative value if $2\ensuremath{\kappa}t<\mathrm{ln}{1\ensuremath{-}(2\mathrm{n\ifmmode \bar{}\else \={}\fi{}}+1)(\mathrm{n\ifmmode \bar{}\else \={}\fi{}}\ensuremath{-}\mathrm{sinh}{}^{2}r)/[(2\mathfrak{N}+1)(\mathrm{n\ifmmode \bar{}\else \={}\fi{}}\mathrm{cosh}2r+\mathrm{sinh}{}^{2}r)]}$, which is dependent not only on average number N\mathfrak{N}N of thermal photons in the environment, but also on $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ and rrr.
We discuss nonclassical properties of single-photon subtracted squeezed vacuum states in terms of the sub-Poissonian statistics and the negativity of the Wigner function. We derive a compact expression for the Wigner function from which we find the region of phase space where Wigner function is negative. We find an upper bound on the squeezing parameter for the state to exhibit sub-Poissonian statistics. We then study the effect of decoherence on the single-photon subtracted squeezed states. We present results for two different models of decoherence, viz. amplitude decay model and the phase diffusion model. In each case we give analytical results for the time evolution of the state. We discuss the loss of nonclassicality as a result of decoherence. We show through the study of their phase-space properties how these states decay to vacuum due to the decay of photons. We show that phase damping leads to very slow decoherence, slower than the photon-number decay.
Nonclassicality is a useful quantum resource in quantum information and has important applications in various quantum tasks such as quantum computing, quantum communication. The robustness of nonclassicality is used to quantify the nonclassicality. The nonclassicality of number states, squeezed states and cat states has been known. We consider the nonclassicality of the squeezed number states. Several different upper and lower bounds for the nonclassicality of squeezed single photon state are obtained through analytical calculation, modeling analysis and numerical calculation. By comparing all these bounds, we find that the nonclassicality of squeezed single photon state can be well determined.
We propose a model for computing of a certain set of analytical functions based on estimating the output distribution of multiphoton outcomes in an optical scheme with an initial single-mode squeezed vacuum (SMSV) state and photonic states measuring the number of photons in one of the output modes of the beam splitter (BS) by photon number resolving (PNR) detector. The set of considered analytical functions is polynomial expressions including arbitrary derivatives of certain functions which can take on very large values even on small interval in their argument and small values of the parameter indicating the number of the subtracted photons. The large values that the analytic functions can take are offset by a very small term including the factorial of the number of subtracted photons, which guarantees an output normalized distribution of multiphoton measurement outcomes. The quantum computing algorithm makes it possible to find the values of the analytical functions for each number of extracted photons after a sufficiently large number of trials that would allow replacing the measurement repetition rate of multiphoton events by their probabilities. Changing the initial parameters (squeezing amplitude of the SMSV state and BS parameter) makes it possible to implement calculations of the functions over the entire (or, at least, significant) continuous interval of alteration in their argument. The potential of optical quantum computing based on nonclassical states of a certain parity can be expanded both by adding new optical elements such as BSs, and by using other continuous variable (CV) states of definite parity.