[Paper Review] Dynamically generated quadrature and photon-number variances for Gaussian states

Moorad Alexanian|arXiv (Cornell University)|Apr 6, 2017

Advanced Thermodynamics and Statistical Mechanics1 references3 citations

TL;DR

This paper derives the exact time-dependent Wigner quasiprobability distribution for a single-mode degenerate parametric amplifier in a displaced-squeezed thermal state, enabling precise calculation of quadrature and photon-number variances. The key finding is a phase transition in the Mandel parameter $Q_M(\tau)$ at a critical displacement $|\alpha_c|$, where the system transitions from strictly classical to mixed classical/nonclassical behavior, despite remaining nonclassical overall as per Glauber-Sudarshan $P(\beta)$ criteria.

ABSTRACT

We calculate exactly the quantum mechanical, temporal Wigner quasiprobability density for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state. The Wigner function allows us to calculate the fluctuations in photon number and the quadrature variance. We contrast the difference between the nonclassicality criteria, which is independent of the displacement parameter $α$, based on the Glauber-Sudarshan quasiprobability distribution $P(β)$ and the classical/nonclassical behavior of the Mandel $Q_{M}(τ)$ parameter, which depends strongly on $α$. We find a phase transition as a function of $α$ such that at the critical point $α_{c}$, $Q_{M}(τ)$, as a function of $τ$, goes from strictly classical, for $|α|< |α_{c}|$, to a mixed classical/nonclassical behavior, for $|α|> |α_{c}|$.

Motivation & Objective

• To derive the exact time-dependent Wigner quasiprobability density for a single-mode degenerate parametric amplifier in a displaced-squeezed thermal state.
• To calculate the temporal evolution of quadrature variance and photon-number variance from the Wigner function.
• To investigate the discrepancy between nonclassicality criteria based on the Glauber-Sudarshan $P(\beta)$ function and the time-dependent Mandel parameter $Q_M(\tau)$.
• To identify and characterize a phase transition in the Mandel parameter as a function of the displacement parameter $\alpha$.

Proposed method

• The time evolution of the system is governed by a degenerate parametric Hamiltonian, evolving an initial thermal state into a displaced-squeezed thermal state via unitary evolution.
• The Wigner quasiprobability density is derived exactly using the time-dependent quadrature means and variances, expressed in terms of $\alpha$, $\xi$, $\bar{n}$, and time $\tau$.
• Quadrature variance is computed by integrating the Wigner function over the conjugate quadrature, yielding a Gaussian probability distribution.
• Photon-number variance is calculated from the second moment of the photon number operator using the Wigner function and the time-dependent squeezing and displacement parameters.
• The Mandel parameter $Q_M(\tau)$ is evaluated as a function of time $\tau$ and displacement $\alpha$, revealing its dependence on $\alpha$.
• A phase transition is identified numerically by analyzing the behavior of $Q_M(\tau)$ as $|\alpha|$ crosses a critical threshold $|\alpha_c|$.

Experimental results

Research questions

• RQ1 How does the Mandel parameter $Q_M(\tau)$ behave over time for different values of the displacement parameter $\alpha$ in a dynamically evolving Gaussian state?
• RQ2 What is the critical displacement $|\alpha_c|$ at which the Mandel parameter transitions from strictly classical to mixed classical/nonclassical behavior?
• RQ3 Why does the Mandel parameter $Q_M(\tau)$ exhibit classical behavior for $|\alpha| > |\alpha_c|$ even though the system remains nonclassical according to the $P(\beta)$ criterion?
• RQ4 How do the time-dependent variances of quadrature and photon number evolve under degenerate parametric amplification without assuming statistical stationarity?
• RQ5 What is the role of the initial thermal state mean photon number $\bar{n}$ and squeeze parameter $r$ in determining the critical displacement $|\alpha_c|$?

Key findings

• The Mandel parameter $Q_M(\tau)$ transitions from strictly classical to mixed classical/nonclassical behavior at a critical displacement $|\alpha_c|$, marking a phase transition in the system's statistical dynamics.
• For $|\alpha| < |\alpha_c|$, $Q_M(\tau)$ remains positive for all $\tau$, indicating classical photon statistics.
• For $|\alpha| > |\alpha_c|$, $Q_M(\tau)$ becomes negative (nonclassical) at early times but eventually turns positive (classical) as $\tau$ increases, indicating a time-dependent classical-nonclassical crossover.
• The critical displacement $|\alpha_c|$ is determined by the condition where $Q_M(0)$ changes sign, and it depends on $\bar{n}$, $r$, and the phase $\theta$ of the squeeze parameter.
• Despite being nonclassical according to the Glauber-Sudarshan $P(\beta)$ criterion, the Mandel parameter $Q_M(\tau)$ can exhibit classical behavior for $|\alpha| > |\alpha_c|$, revealing a fundamental discrepancy between nonclassicality criteria.
• The exact time-dependent Wigner function is derived in closed form, explicitly showing the time evolution of the five real parameters: two quadrature means and three variances/covariances, enabling precise variance calculations.