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Physical Review A

Kinetics of photon correlation functions under the time-dependent quadratic Hamiltonian

Shan-Wen Tsai, A. F. R. de Toledo Piza

Published May 1, 1996• 5 citations
Publisher

Abstract

The dynamic decoupling of the correlation function hierarchy under the general quadratic Hamiltonian is exploited to extend the results obtained for coherent pumping and squeezing in Ref. . Closed c-number dynamic equations are given explicitly for three- and four-operator averages, for an arbitrary (pure or mixed) initial state of the field. At this level an independent c-number constant of motion appears, related to two-photon correlation properties of the initial state. Extension to higher correlation functions is straightforward. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.3683

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The dynamic decoupling of the correlation function hierarchy under the general quadratic Hamiltonian is exploited to extend the results obtained for coherent pumping and squeezing in Ref. . Closed c-number dynamic equations are given explicitly for three- and four-operator averages, for an arbitrary (pure or mixed) initial state of the field. At this level an independent c-number constant of motion appears, related to two-photon correlation properties of the initial state. Extension to higher correlation functions is straightforward. \textcopyright{} 1996 The American Physical Society.
We derive an exact analytical expression for the characteristic function of work in closed many-body bosonic systems governed by general time-dependent quadratic Hamiltonians. This is made possible by adopting a unified prescription in which all operators appearing in the characteristic function are treated as elements of the inhomogeneous Metaplectic group. The esulting expression involves a single symplectic matrix, a single phase-space displacement, and a single accumulated phase. It holds for any number of degrees of freedom, arbitrary time dependence of the quadratic Hamiltonian, and any initial thermal equilibrium state of the system. The formalism trades the explicit time-ordering problem associated with the unitary evolution for the determination of a symplectic path in phase space, from which the corresponding driving Hamiltonian can be reconstructed. We also discuss the generalization to arbitrary nonequilibrium initial states within the two-projective-measurement scheme for quantum work. The same construction yields Loschmidt echoes and mixed-state total phases. Using the derived characteristic function, we obtain formulas for the Helmholtz free energy, the Jarzynski equality, the mean work, the work variance, and the nonequilibrium lag. Undisplaced Hamiltonians, sudden quenches, constant-Hessian protocols, and the adiabatic limit follow as special cases of the general result. As an illustration, we analyze a one-dimensional system constructed from an unconventional symplectic path and exhibiting physically relevant limiting cases. We then show how affine and purely symplectic contributions affect work statistics and irreversibility.
The quantum dynamics of a parametric interaction of the electromagnetic field with a nonlinear medium is considered. The three nonlinear coupled Heisenberg equations of motion are solved under the short-time approximation. The characteristic function for the normal ordering rule of association is evaluated. This function is then used to obtain the time dependence of the density operator. Assuming that the initial state of the system is a coherent state, explicit expressions for the diagonal coherent state representation of the reduced density operators for the pump as well as for the signal mode are obtained. For the pump mode, the initially coherent state remains coherent, whereas for the signal mode the diagonal coherent state representation is found to be a gaussian distribution whose variance is proportional to the average number of photons initially present in the pump mode.
Derivation of the procedures that can be applied in evaluating two-time correlation function in terms of coherent-state propagator and corresponding Q-function is presented. On the basis that the involved functions are generally exponential in nature, obtaining the two-time second-order correlation function is essentially claimed to be reduced to carrying out relatively simple integrations. Fundamentally, the time dependence of the operators is transferred to the density operator. Moreover, manipulation in reordering the operators is performed by applying the usual trace operation. With all details, it is basically observed that the two-time correlation can be readily determined once the pertinent coherent-state propagator or Q-function is known. Since working with c-number equation is far more handy than the associated operator equation, it is expected that the results derived in this contribution can aid in easing the otherwise involving mathematical rigor.
We summarise the main results that follow from our theory of frequency- and time-resolved photon correlations, which provides exact computations in nontrivial systems. We illustrate the theory with original results as interesting cases can be picked up from the endless configurations that abound in any quantum optical system. In particular, we compare our exact results with previous approximations, upgrade correlations to the case of photon bundles as otherwise no clear physical picture emerge, highlight the interest of measuring away from the peaks in rich landscapes of correlations and emphasize the joint time and frequency aspect of our theory.
Under the assumption of strong (classical) pump condition, the coherent light interacting with a medium of second-order nonlinearity gives rise to a model represented by a quadratic Hamiltonian. By exploiting this model, we obtain the squeezing and the higher-order squeezing of the input coherent light. These phenomena are found to depend on the classical intensity but do not depend on the number of photons present in the input coherent radiation field. For the same physical model, it is found that the photons are bunched in the input vacuum field provided that two-photon processes dominate over one-photon processes. Depending on the suitable choice of phases and interaction time, we show both photon bunching and photon antibunching phenomena for the vacuum field and for the field with a large number of photons as well. The results obtained in this paper are completely analytical in nature and provide a feeling for the physical situation in an explicit manner.