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arXiv:quant-ph/0207034

Quantum fluctations of coherent light in nonlinear media

Anirban Pathak

Published Jul 5, 2002• 0 citations
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Abstract

The interaction of coherent light with a nonlinear medium is modeled here by a general quantum anharmonic oscillator. The model is not exactly solvable in a closed analytical form. But we need operator solutions of the equations of motion corresponding to these models in order to study the quantum fluctuations of coherent light in nonlinear media. In the present work we derive approximate operator solutions. From these solutions we observe that there exists an apparent discrepancy between the solutions obtained by different techniques. We compare different solutions and conclude that all correct solutions are equivalent and the apparent discrepancy is due to the use of different ordering of the operators. We use these solutions to investigate the possibilities of observing different optical phenomena in a nonlinear dielectric medium. To be precise, we have studied quantum phase fluctuations of coherent light in a third order inversion symmetric nonlinear medium. Fluctuations in phase space quadrature for the same system are studied and the possibility of generating squeezed state is reported. Fluctuations in photon number are studied and the nonclassical phenomenon of antibunching is predicted. We have generalized the results obtained for third order nonlinear medium and have studied the interaction of an intense laser beam with a general (m-1)-th order nonlinear medium. Aharanov Anandan nonadiabatic geometric phase is also discussed in the context of (m-1)-th order nonlinear medium.

DOI
10.48550/arxiv.quant-ph/0207034

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The interaction of coherent light with a nonlinear medium is modeled here by a general quantum anharmonic oscillator. The model is not exactly solvable in a closed analytical form. But we need operator solutions of the equations of motion corresponding to these models in order to study the quantum fluctuations of coherent light in nonlinear media. In the present work we derive approximate operator solutions. From these solutions we observe that there exists an apparent discrepancy between the solutions obtained by different techniques. We compare different solutions and conclude that all correct solutions are equivalent and the apparent discrepancy is due to the use of different ordering of the operators. We use these solutions to investigate the possibilities of observing different optical phenomena in a nonlinear dielectric medium. To be precise, we have studied quantum phase fluctuations of coherent light in a third order inversion symmetric nonlinear medium. Fluctuations in phase space quadrature for the same system are studied and the possibility of generating squeezed state is reported. Fluctuations in photon number are studied and the nonclassical phenomenon of antibunching is predicted. We have generalized the results obtained for third order nonlinear medium and have studied the interaction of an intense laser beam with a general (m-1)-th order nonlinear medium. Aharanov Anandan nonadiabatic geometric phase is also discussed in the context of (m-1)-th order nonlinear medium.
An analytical approach is adopted for the calculation of squeezed property of coherent light interacting with a nonlinear medium of inversion symmetry. A medium is called inversion symmetric if the potential energy remains invariant under parity transformation. Thus the even order susceptibilities would vanish for the medium and the leading contribution to the nolinear polarization due to the medium is the third order susceptibility. Therefore interaction Hamitonian of the single mode light field with the medium is proportional to the fourth power of the annihilation operator plus creation operator. This means is to take care all the non conserving energy terms for the calculation of the squeezing property of coherent light passing through the nonlinear medium of inversion symmetry. For a single mode light, two dimensionless quadrature operators are defined which are proportional to the usual position and momentum operators respectively. Using the Heisenberg operator equation of motion, the differential equation of the quadrature operators and the approximate solutions to the equations are got. Then the quantum fluctuations of the two quadratures are calculated. The result shows that the squeezing property of the two quadrature operators is controllable through changing the following parameters: coupling constant, mean photon number, light field frequency, and interaction time.
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.
Nonlinear optical media of Kerr type are described by a particular version of an anharmonic quantum harmonic oscillator. The dynamics of this system can be described using the Moyal equations of motion, which correspond to a quantum phase-space representation of the Heisenberg equations of motion. For the Kerr system we derive exact solutions of the Moyal equations for an irreducible set of observables. These Moyal solutions incorporate the asymptotics of the classical limit in a simple, explicit form. An unusual feature of these solutions is that they exhibit periodic singularities in the time variable. These singularities are removed by the phase-space averaging required to construct the expectation value for an arbitrary initial state. Nevertheless, for strongly number-squeezed initial states the effects of the singularity remain observable.
The statistical properties of radiation passing through a nonlinear medium modelled as a third-order nonlinear dissipative oscillator interacting with squeezed light have been investigated. Whereas the photon statistics, being insensitive to the nonlinearity, are determined exactly, an approximative approach has been adopted in the description of squeezing in order to involve fluctuations. As a consequence of the general treatment, new results for a damped linear oscillator with initial squeezed light are provided.
The systematic theory of the formation of the short light pulses in the squeezed state during the propagation in a medium with inertial Kerr nonlinearity is developed. The algebra of time-dependent Bose-operators is elaborated and the normal-ordering theorem for them is formulated. It is established that the spectral region where the quadrature fluctuations are weaker than the shot-noise, depends on both the relaxation time of the nonlinearity and the magnitude of the nonlinear phase shift. It is also shown that the frequency at which suppression of the fluctuation is greatest can be controlled by adjusting the phase of the initial coherent light pulse. The spectral correlation function of photons is introduced and photon antibunching is found.
A quantum treatment of a coherently driven dispersive cavity is given based on a cubic nonlinearity in the polarisability of the internal medium. This system displays bistability and hysteresis in the semiclassical solutions. Quantum fluctuations are included via a Fokker-Planck equation in a generalised P representation. The transmitted light shows a transition from a single-peaked spectrum to a double-peaked spectrum above the threshold of the lower branch. Fluctuations in the field are reduced on the upper branch and both photon bunching and photon antibunching are predicted, for different operating points. An exact solution obtained for the steady-state generalised P function shows decidedly non-equilibrium behaviour, e.g. the lack of a Maxwell construction.