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Modern Physics Letters B

SQUEEZING EFFECTS IN THE SUM AND DIFFERENCE OF THE FIELD AMPLITUDE IN THE RAMAN PROCESS

Dilip Kumar Giri, Partha Sarathi Gupta

Published Nov 10, 2005• 5 citations
Publisher

Abstract

Sum and difference squeezing of the field amplitude are higher-order squeezing effects. These effects are studied in the Raman process under the short-time approximation based on a fully quantum mechanical approach. It is shown that for uncorrelated modes, the normal squeezing in the sum and difference-frequency field depends on the sum and difference squeezing of input field modes respectively, which can generate normal squeezing in the sum and difference-frequency field mode. All the possibilities for obtaining sum and difference squeezing in two modes and its dependence on squeezing of individual field modes are investigated. We have also shown that if the high-frequency mode is in a coherent state and the low-frequency mode is squeezed, the field state will be difference squeezed if the amplitude of the high-frequency mode is large enough; otherwise the state may or may not be difference squeezed. If both modes are squeezed, then the state may or may not be difference squeezed. These higher-order squeezing effects are useful in the production of squeezing in the Raman process.

DOI
10.1142/s0217984905009146

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Sum and difference squeezing of the field amplitude are higher-order squeezing effects. These effects are studied in the Raman process under the short-time approximation based on a fully quantum mechanical approach. It is shown that for uncorrelated modes, the normal squeezing in the sum and difference-frequency field depends on the sum and difference squeezing of input field modes respectively, which can generate normal squeezing in the sum and difference-frequency field mode. All the possibilities for obtaining sum and difference squeezing in two modes and its dependence on squeezing of individual field modes are investigated. We have also shown that if the high-frequency mode is in a coherent state and the low-frequency mode is squeezed, the field state will be difference squeezed if the amplitude of the high-frequency mode is large enough; otherwise the state may or may not be difference squeezed. If both modes are squeezed, then the state may or may not be difference squeezed. These higher-order squeezing effects are useful in the production of squeezing in the Raman process.
Under the short-time approximation, Giri and Gupta1 investigated the sum and difference squeezing of the field amplitude in the Raman process. The relations between the normal squeezing and the sum and difference squeezing of the field amplitude are explicitly exhibited. They argued that the sum and difference squeezing of the field amplitude are essentially the squeezing in higher-order sense. Upon verification of the results of Giri and Gupta, we observe that the entire calculations and the conclusions thereof are misleading and baseless. To substantiate our claim, we give the correct expressions for the useful variances. Interestingly, there is no possibility of getting normal squeezing, sum squeezing and difference squeezing of the input state prepared in coherent state.
The multimode difference squeezing of the electromagnetic field in the four-wave mixing process is investigated under the short-time approximation based on a fully quantum mechanical approach. It is shown that for uncorrelated modes the normal squeezing in the difference-frequency field depends on the difference squeezing of input field modes, which can generate normal squeezing in the difference-frequency field mode. Various possibilities for obtaining difference squeezing in three modes and its dependence on squeezing of individual field modes are investigated. If one of the high-frequency modes is squeezed and the other two modes are in coherent states, difference squeezing can be generated when the amplitude of the low-frequency mode is large enough. Difference squeezing can be generated in the three modes if one of the high-frequency modes is in a coherent state and the other two modes are difference squeezed. It is also shown that when the low-frequency mode is in a coherent state and the high-frequency modes are sum squeezed, then difference squeezing can be generated for large amplitude of the low-frequency mode.
Sum squeezing of the field amplitude is studied in the nondegenerate and degenerate frequency upconversion process under the short interaction time. It is shown that sum squeezing can be converted into normal squeezing via sum-frequency generation in the nondegenerate frequency upconversion process, while the amplitude-squared squeezing of the fundamental mode directly changed into the squeezing of the harmonic in the degenerate frequency upconversion process. All reachable conditions of uncorrelated modes for obtaining a sum squeezing in two modes and its dependence on the squeezing of individual field modes are investigated. It is found that the squeezed states are associated with large number of pump photons. It is also confirmed that the higher-order squeezing (sum squeezing) is directly associated with coupling of the field and interaction time.
Sum and difference squeezing are both higher-order, two-mode squeezing effects. Sum squeezing is turned into normal squeezing by sum-frequency generation. The operators that are used to define it form a representation of the su(1,1) Lie algebra. Difference squeezing can generate normal squeezing through difference-frequency generation. The operators that characterize it form a representation of the su(2) Lie algebra. Both are nonclassical effects. For uncorrelated modes the presence of squeezing in at least one of the modes is a necessary condition for the existence of either sum or difference squeezing. If the modes are correlated this is no longer true. Both sum and difference squeezing can be generated by combining a squeezed mode with one in a coherent state. Sum squeezing is also produced by a parametric amplifier. In order to better illustrate some of these effects a pictorial representation of sum squeezing is presented.
Sum squeezing of the field amplitude is studied in the nondegenerate and degenerate frequency upconversion process under the short interaction time. A state is squeezed when the quantum fluctuation (amplitude noise or phase noise) in one variable is reduced below the symmetric limit at the expense of the increased quantum fluctuation in the conjugate variable such that the Heisenberg uncertainty relation is not violated. It is shown that sum squeezing can be converted into normal squeezing via sum-frequency generation in the nondegenerate frequency upconversion process, while the amplitude-squared squeezing of the fundamental mode directly changed into the squeezing of the harmonic in the degenerate frequency upconversion process. All reachable conditions of uncorrelated modes for obtaining a sum squeezing in two modes and its dependence on the squeezing of individual field modes are investigated. It is found that the squeezed states are associated with large number of pump photons. It is also confirmed that the higher-order squeezing (sum squeezing) is directly associated with coupling of the field and interaction time. The results obtained in this chapter are of interest for new experiments on the study of nonlinear optical processes in dielectric media using ultrashort intense laser pulses as exciting radiation, and the effects of damping and decoherence as well as higher-order time terms could be investigated.
The occurrence of fourth and fifth-order squeezing in the fundamental mode of the electromagnetic field in spontaneous and stimulated Raman processes under the short-time approximation is investigated on the basis of a fully quantum-mechanical approach. In an idealized model, the Raman process is looked upon as a two-photon on a one phonon interaction process. The coupled Heisenberg equations of motion involving real and imaginary parts of the quadrature operators are established. The dependence of fourth- and fifth-order squeezing in terms of the signal-to-noise ratio on the number of photons is investigated. We have also established the amplitude squeezing effects in the Stokes and vibrational modes, which is found to be dependent on higher-order difference squeezing showing a difference squeezing that can be converted to normal squeezing.