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Physical Review A

Time evolution of squeezing and antibunching in an optically bistable two-photon medium

Christopher C. Gerry, Stanley Rodrigues

Published Dec 1, 1987• 37 citations
Publisher

Abstract

We use a numerical technique to study the time development of squeezing and antibunching resulting from the interaction of coherent light with a two-photon medium. While we find that enhanced squeezing occurs for short time, the squeezing is eventually revoked. We also find that antibunching develops but also is eventually revoked.
DOI
10.1103/physreva.36.5444

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The time evolution of squeezed vacuum states, taken to be SU(1,1) coherent states, interacting with a two-photon medium is examined using numerical methods. The non-photon-conserving medium is modeled by a Hamiltonian which consists of the free-field term, a non-photon-conserving term in the form of a degenerate parametric amplifier, and an anharmonic term which gives rise to optical bistability. This system has previously been shown to produce enhanced squeezing when interacting with ordinary coherent light. In the present work we find that the squeezing property is generally revoked by the anharmonic term and increased by the nonconserving term. The combined effects are not additive, however, and a quantum interference effect is observed in the calculations.
With the help of second-order nonlinear interactions and hence Hamiltonian, we construct the equations of motion corresponding to pump and signal (second harmonic) fields in a second harmonic generation. The corresponding coupled differential equations involving the non-commuting field operators are not solvable in closed analytical forms. With the help of a perturbation method, we obtain analytical solutions of these field operators up to cubic orders in the interaction constant. In an appropriate limit (truncating the solution up to the second order in the interaction constant), these solutions lead to the existing solutions under the short-time approximation. The present analytical solutions are exploited to investigate the squeezing and the antibunching of photons of the input coherent light coupled to the said second-order nonlinear medium. We report the squeezing and antibunching effects of the pump field even for solutions up to the second order and hence the present results are consistent with earlier results. However, for the second harmonic mode, we report the squeezing involving the leading cubic term in the interaction constant.
In Chapter 1 we consider the question of squeezing in a two-photon laser. A two-photon laser is modelled as N two-level atoms interacting via a two-photon transition with a single resonant cavity field mode. A Fokker-Planck equation containing all leading noise terms is derived. A linearised fluctuation analysis reveals that there is no squeezing in a two-photon laser with injected signal. A small amount of squeezing is present in two-photon optical bistability. In Chapter 2 a microscopic model of degenerate four wave mixing including the quantisation of the medium is presented. Time-dependent solutions show ideal squeezing to be obtained in the limits of large detuning, low saturation and zero loss. The effect of nonideal atomic parameters is analysed. Equations describing the spatial propagation of the field through the medium are also derived. In Chapters 3 and 4 we are concerned with quantum effects, in particular violation of Bell's Inequalities, present in optical systems. An optical field produced by an intracavity four wave mixing is shown to exhibit the nonclassical features of photon antibunching, squeezing and violation of Cauchy-Schwarz and Bell's Inequalities. These intrinsic quantum mechanical effects are shown to be associated with the nonexistence of a positive normalizable Glauber-Sudarshan P-function.
We study the time evolution of SU(1,1) coherent states in an exactly solvable model of a single-mode quantized electromagnetic field in a four-photon medium that has previously been shown to give squeezed states from ordinary coherent states. If the SU(1,1) coherent state initially exhibits squeezing in the X1{X}_{1}X1​ quadrature, the squeezing is revoked. We also show that for certain choices of phase in the initial state, enhanced squeezing may appear in the X2{X}_{2}X2​ quadrature. Numerical results are also presented for the case when the model is not exactly solvable.
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.