Master equation methods provide useful tools for analyzing the statistics of quantum optical systems. They focus attention on a simple system of interest (a single cavity mode, for example), accounting for its interaction with its environment in parameters describing decay rates and environmental fluctuations. Intracavity systems for generating squeezed light are prime candidates for analysis by these methods. It has been shown, however, that it is important in such analysis to distinguish between the statistics of the intracavity field and the statistics of the field generated outside the cavity by transmission through the cavity mirrors.1,2 It is desirable to perform calculations with the single mode inside the cavity. Then the statistics of the multimode field outside the cavity must be related to those of the internal mode. The situation is analogous to that in resonance fluorescence, where the statistics of the fluorescent field can be calculated in terms of the source dipole operator. In this paper the field outside an optical cavity is expressed in terms of the intracavity oscillator as a source. The measurement of squeezing in the external field is then related to the correlation properties of the cavity source operator. Measurement by photon counting of homodyned light is related to the spectrum of squeezing calculated by Walls and co-workers.
Master equation methods provide useful tools for analyzing the statistics of quantum optical systems. They focus attention on a simple system of interest (a single cavity mode, for example), accounting for its interaction with its environment in parameters describing decay rates and environmental fluctuations. Intracavity systems for generating squeezed light are prime candidates for analysis by these methods. It has been shown, however, that it is important in such analysis to distinguish between the statistics of the intracavity field and the statistics of the field generated outside the cavity by transmission through the cavity mirrors.1,2 It is desirable to perform calculations with the single mode inside the cavity. Then the statistics of the multimode field outside the cavity must be related to those of the internal mode. The situation is analogous to that in resonance fluorescence, where the statistics of the fluorescent field can be calculated in terms of the source dipole operator. In this paper the field outside an optical cavity is expressed in terms of the intracavity oscillator as a source. The measurement of squeezing in the external field is then related to the correlation properties of the cavity source operator. Measurement by photon counting of homodyned light is related to the spectrum of squeezing calculated by Walls and co-workers.
The detection of squeezed optical fields generated by intracavity nonlinear-optical interactions is described. The relationship between quantum-statistical properties of the cavity mode and those of the field at a detector placed outside the cavity discussed. The detected field is composed of a source field from the cavity plus a free external field. The free field couples weakly to the cavity so that it is correlated with the source field. The photocurrent spectrum in optical homodyne detection is calculated, and the spectrum of squeezing is defined. This spectrum can be calculated entirely in terms of intracavity fields without requiring knowledge of the correlations between source and free-field contributions in the cavity output. This follows from an explicitly normally ordered, time-ordered treatment of the photodetection problem. Contrasting earlier treatments of photodetection for squeezed fields, in a normally ordered approach, shot noise arises naturally from the self-correlation of photocurrent for pulses. The derived spectrum is converted into nonnormally ordered, non-time-ordered form to recover the results of these earlier treatments and their interpretation of shot noise in terms of local-oscillator quantum noise, signal and quantum noise, and detector-efficiency quantum noise.
The interaction between matter and squeezed light has mostly been treated within the approximation that the field-correlation time is small. Methods for treating squeezed light with more general correlations currently involve explicitly modeling the systems producing the light. We develop a general purpose input-output theory for a particular form of narrowband squeezed light, a squeezed wave-packet mode, that only concerns the statistics of the squeezed field and the shape of the wave packet. This formalism allows us to derive the input-output relations and the master equation. We also consider detecting the scattered field using photon counting and homodyne measurements, which necessitates the derivation of the stochastic master equation. The non-Markovian nature of the field manifests itself in the master equation as a coupled hierarchy of equations. We illustrate these with consequences for the decay and resonance fluorescence of two-level atoms in the presence of such fields.
A quantum well with a single exciton mode in a microcavity driven by squeezed vacuum is studied in the low exciton density regime. By solving the quantum Langevin equations, we study the intensity, spectrum, and intensity correlation function for the fluorescent light. An expression for the Q function of the field inside the cavity is derived from the solutions of the quantum Langevin equations. Using the Q function, the intracavity photon number distribution and the quadrature fluctuations for both the cavity and fluorescent fields are studied. Several interesting and new effects due to squeezed vacuum are found.
In this thesis, we investigate the quantum optical properties of a single cavity mode interacting with an external classical coherent driving field inside a closed cavity, which is coupled to a thermal reservoir via a single-port mirror. We establish a mathematically comprehensive framework utilizing both the Master Equation under the density operator formalism and the Quantum Langevin Equation under the Heisenberg picture. By analyzing the system with an arbitrary ordering of reservoir noise operators, we derive the steady-state P-representation of the quasi-probability distribution. Employing this quasi-probability distribution, we analytically compute the mean photon number, photon number variance, photon number distribution, and the second-order intensity correlation function. Our findings establish that the interaction of the cavity mode with the coherent drive in the presence of thermal fluctuations exhibits strictly super-Poissonian statistics. The mean photon number and photon number variance scale monotonically with both the strength of the external coherent drive and the average thermal photon occupation of the reservoir. Furthermore, we prove that the produced cavity field does not exhibit quadrature squeezing, confirming that the presence of thermal noise effectively destroys quantum squeezing in this configuration. Finally, the input-output is evaluated to verify the preservation of the commutation relations of the formalism output field. Keywords: Quantum Optics, Master Equation, Fokker-Planck, Quantum Langevin Equation, P- function, Photon Statistics, Super-Poissonian, Squeezing.
In this thesis, we investigate the quantum optical properties of a single cavity mode interacting with an external classical coherent driving field inside a closed cavity, which is coupled to a thermal reservoir via a single-port mirror. We establish a mathematically comprehensive framework utilizing both the Master Equation under the density operator formalism and the Quantum Langevin Equation under the Heisenberg picture. By analyzing the system with an arbitrary ordering of reservoir noise operators, we derive the steady-state P-representation of the quasi-probability distribution. Employing this quasi-probability distribution, we analytically compute the mean photon number, photon number variance, photon number distribution, and the second-order intensity correlation function. Our findings establish that the interaction of the cavity mode with the coherent drive in the presence of thermal fluctuations exhibits strictly super-Poissonian statistics. The mean photon number and photon number variance scale monotonically with both the strength of the external coherent drive and the average thermal photon occupation of the reservoir. Furthermore, we prove that the produced cavity field does not exhibit quadrature squeezing, confirming that the presence of thermal noise effectively destroys quantum squeezing in this configuration. Finally, the input-output is evaluated to verify the preservation of the commutation relations of the formalism output field. Keywords: Quantum Optics, Master Equation, Fokker-Planck, Quantum Langevin Equation, P- function, Photon Statistics, Super-Poissonian, Squeezing.
In the preceding paper we developed a formalism based on the true modes of the universe containing a leaky cavity to analyze the relationship of quantum noise inside that cavity to that outside. We illustrate this formalism here by studying several active systems. We analyze both an ordinary laser and a laser operating in the phase-locked regime. We calculate the extracavity quadrature variances for the phase-locked laser and find that the spectrum of noise reduction in the squeezed quadrature is Lorentzian. Our formalism is also applied to a two-photon correlated-spontaneous-emission laser. We find the familiar result that with 50% squeezing in the phase quadrature of the field inside the cavity, one has nearly 100% phase squeezing outside the cavity. However, under different conditions the phase quadrature of the intracavity field is nearly perfectly squeezed, but the extracavity field shows almost no squeezing in its phase quadrature. We also analyze the effect of finiteness of measurement time on the extracavity quadrature variances.