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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Difference squeezing in four-wave difference-frequency generation

Ashok Kumar, Partha Sarathi Gupta

Published Jun 1, 1998• 13 citations
Publisher

Abstract

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.

DOI
10.1088/1355-5111/10/3/007

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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 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.
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.
Squeezing in the difference of the fields in degenerate four- and five-wave interaction processes is studied. It is shown that the difference squeezing can be led to normal squeezing of the field for uncorrelated modes. It is found that the amplitude-squared squeezing of the fundamental mode directly converted into the normal squeezing of the signal mode in the four-wave interaction process and amplitude-cubed of the fundamental directly converted into the normal squeezing of the signal mode in the five-wave interaction process. Detection of higher-order squeezing in these processes is also studied. It is observed that difference squeezing responds nonlinearly to the number of pump photons and found greater in the stimulated process than in spontaneous one. Difference squeezing exists only in certain domain value of pump photons. It is inferred that the multi-photon absorption process is more suitable for the generation of optimum squeezed light.
Squeezing of electro-magnetic field, which is a purely quantum phenomenon has attracted considerable attention owing to its low noise property with applications in high quality telecommunication. This quantum effect is expected to manifest itself in optical processes in which the nonlinear response of the system to the radiation field plays an important role. In this paper squeezing of electro-magnetic field in multi-wave mixing processes like Raman and hyper Raman processes and sum- frequency generation are investigated under short-time approximation. The coupled Heisenberg quantum mechanical equations of motion for field operators are set up and solved under short-time approximation. The occurrence of squeezing of field is investigated using the required conditions of squeezing in each of the cases. The squeezing is found to exist in the fundamental mode and the squeezing in the generated field depends on squeezing in the fundamental mode. Squeezing in the higher order amplitudes are also studied. This corresponds to the squeezing of the variables which describe the real and imaginary parts of square and cube of the complex amplitudes of the radiation field. The higher order multimode sum-squeezing of radiation field is dealt with for harmonic and sum frequency generation. It is shown that squeezing in the sum frequency field depends directly on the sum squeezing of fundamental modes. The results can be utilized in selecting a suitable material and suitable process which will generate a radiation field with optimum squeezing and can be useful in high quality tele-communication.
We studied squeezing in the difference of the field amplitude in degenerate six wave interaction process, which is a higher-order squeezing effect.It is shown that for uncorrelated modes the normal squeezing in the difference-frequency field directly depends on the difference squeezing of input field modes.It is established that the amplitude-squared squeezing of the fundamental feeds directly into the normal squeezing of the signal mode.Detection of difference and amplitude-squared squeezing in this process is also studied.It is found that squeezing is greater in stimulated process than in spontaneous interaction.
The concept of fourth-order squeezing of the electromagnetic field is investigated in the fundamental mode in spontaneous and stimulated four- and six-wave mixing processes under the short-time approximation based on a fully quantum mechanical approach. The coupled Heisenberg equations of motion involving real and imaginary parts of the quadrature operators are established. The possibility of obtaining fourth-order squeezing is studied. The dependence of fourth-order squeezing on the number of photons is also investigated. It is shown that fourth-order squeezing, which is a higher-order squeezing, allows a much larger fractional noise reduction than lower-order squeezing. It is shown that squeezing is greater in a stimulated process than the corresponding squeezing in spontaneous interaction. The conditions for obtaining maximum and minimum squeezing are obtained. We have also established the non-classical nature of squeezed radiation using the Glauber–Sudarshan representation.