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Journal of Modern Optics

Squeezing and photon antibunching in second harmonic generation: an analytical approach

Biswajit Sen, Swapan Mandal, Jan Peřina

Published Dec 8, 2011• 8 citations
Publisher

Abstract

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.

DOI
10.1080/09500340.2011.640952

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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.
The squeezing and sub-Poissonian eects of light in third harmonic generation are investigated based on the fully quantum mechanical approach up to the rst order Hamiltonian interaction in gt, where g is the coupling constant between the modes per second and t is the interaction time between the waves during the process in a nonlinear medium.The coupled Heisenberg equations of motion involving real and imaginary parts of the quadrature operators are established.The occurrence of amplitude squeezing eects in both the quadratures of the radiation eld in the fundamental mode is investigated and found to be dependent on the selective phase values of the eld amplitude.The photon statistics of the pump mode in this process have also been investigated and found to be sub-Poissonian in nature.It is shown that for particular phase values the amplitude squeezing and sub-Poissonian photon statistics of light occur simultaneously.It is observed that there is no possibility to produce squeezed light in the harmonic mode up to rst-order interaction in gt.Further, it is found that the normal squeezing in the harmonic mode directly depends upon the amplitude-cubed squeezing of the initial pump eld to the case of second-order interaction in gt.This gives a method of converting higher-order (amplitude-cubed) squeezing of the fundamental mode into normal squeezing of the harmonic mode and vice versa.
We studied photon antibunching in the pump and the harmonic modes of the sixth harmonic generation process. The generalized interaction Hamiltonian is solved for several particular cases in the Heisenberg picture, and the possibility of observing photon antibunching is investigated using the short-time approximation technique. It is shown that the photon antibunching in the pump field depends on the number of pump photons, the interaction time, and the coupling of the field between the modes. With the same amount of pump photons, we deduced that third-order photon antibunching is more nonclassical than second- and first-order photon antibunching. In this process, the effect of photon antibunching is not seen in the harmonic mode over pump mode. It is shown that photon antibunching is more noticeable with shorter interaction times as the depth of nonclassicality increases and the second-order correlation function decreases. The first-order Hamiltonian interaction, which stimulates both pump and harmonic fields, is demonstrated to be more nonclassical than the second-order Hamiltonian interaction. It is clear that the coherent state, or vacuum state, of a pump field with a nonzero harmonic field creates photon clusters because the pump field causes interaction, which leads to bunching effects. It is interpreted that the degree of photon antibunching is maximum in the regions where the second correlation function is minimum. Photon antibunching has been shown to be one of the quantum properties of light.
An initially prepared coherent state coupled to a second-order nonlinear medium is responsible for stimulated and spontaneous hyper Raman processes. By using an intuitive approach based on perturbation theory, the Hamiltonian corresponding to the hyper Raman processes is analytically solved to obtain the temporal development of the field operators. It is true that these analytical solutions are valid for small coupling constants. However, the interesting part is that these solutions are valid for reasonably large time. Hence, the present analytical solutions are quite general and are fresh compared to those solutions under short-time approximations. By exploiting the analytical solutions of field operators for various modes, we investigate the squeezing, photon antibunching and nonclassical photon statistics for pure modes of the input coherent light responsible for hyper Raman processes. At least in one instance (stimulated hyper Raman processes for vibration phonon mode), we report the simultaneous appearance of classical (photon bunching) and nonclassical (squeezing) effects of the radiation field responsible for hyper Raman processes.
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Higher-order nonclassical properties of fields propagating through a codirectional asymmetric nonlinear optical coupler which is prepared by combining a linear waveguide and a nonlinear (quadratic) waveguide operated by second harmonic generation are studied. A completely quantum mechanical description is used here to describe the system. Closed form analytic solutions of Heisenberg's equations of motion for various modes are used to show the existence of higher-order antibunching, higher-order squeezing, and higher-order two-mode and multimode entanglement in the asymmetric nonlinear optical coupler. It is also shown that nonclassical properties of light can transfer from a nonlinear waveguide to a linear waveguide.
The study investigates antibunching, sub-Poissonian statistics, and photon correlation functions in the pump and harmonic modes of the seventh harmonic generation process.We investigated various approaches to photon antibunching by solving the generalised interaction Hamiltonian in the Heisenberg picture using short-time approximation.It is inferred that the number of pump photons, coupling of the field strength, and interaction times significantly influence photon antibunching.It is observed that third-order antibunching in the pump mode has greater nonclassicality than second-and first-order antibunching.In the harmonic mode, the nonclassical effect is found to be less pronounced and becomes more evident as the interaction time decreases.The first-order Hamiltonian interaction induces a stronger nonclassical effect than the second-order interaction.The interaction of a coherent pump field with a nonzero harmonic field leads to photon clustering and bunching effects.Finally, photon antibunching appears most pronounced, indicating nonclassicality, as the second-order correlation function decreases.This confirms that antibunching is a fundamental quantum property of light.