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Physical Review A

Photon-counting statistics of multimode squeezed light

Jianming Huang, Prem Kumar

Published Aug 1, 1989• 22 citations
Publisher

Abstract

We present photon-counting distributions of multimode squeezed light of bandwidth B. Such light, for example, is generated by the intracavity parametric down-converter of Wu et al. [Phys. Rev. Lett. 57, 2520 (1986)], which has now become a standard source for generating highly squeezed light. By expanding the electric-field operator into relevant modes over the counting interval T and applying the Bogoliubov transformation that generates squeezing, we calculate the photon-counting distributions under both BT\ensuremath{\gg}1 and BT\ensuremath{\ll}1 conditions. We find that the probability for counting an odd number of photons is identically zero under the former condition, whereas the latter is equivalent to having a photodetector with quantum efficiency that is much less than unity.

DOI
10.1103/physreva.40.1670

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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.

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We present photon-counting distributions of multimode squeezed light of bandwidth B. Such light, for example, is generated by the intracavity parametric down-converter of Wu et al. [Phys. Rev. Lett. 57, 2520 (1986)], which has now become a standard source for generating highly squeezed light. By expanding the electric-field operator into relevant modes over the counting interval T and applying the Bogoliubov transformation that generates squeezing, we calculate the photon-counting distributions under both BT\ensuremath{\gg}1 and BT\ensuremath{\ll}1 conditions. We find that the probability for counting an odd number of photons is identically zero under the former condition, whereas the latter is equivalent to having a photodetector with quantum efficiency that is much less than unity.
For continuous-wave (cw) squeezed light that has a monochromatic mean field and a rectangular squeezing spectrum covering bandwidth B, we calculate the photocount distribution for a counting time T. The nonclassical oscillations in the photon-number distribution of single-mode squeezed light have reduced visibility in the cw photostatics and often disappear entirely. The even-odd oscillations associated a with single-mode squeezed vacuum are absent for BT\ensuremath{\ll}1, but for BT\ensuremath{\gtrsim}1 they appear in the cw photocount distribution with reduced visibility under appropriate conditions. The oscillations associated with a single-mode squeezed state that has a mean complex amplitude disappear from the cw photostatistics for all counting times.
Consider a photodetector characterized by two bandwidths: (i) an optical bandwidth over which it has nearly unity quantum efficiency and (ii) a much smaller electronic bandwidth, the bandwidth over which it produces beats between different frequencies. Direct onto the photodetector a light beam which is the superposition of two parts: (i) an intense, nearly monochromatic carrier wave at some carrier frequency and (ii) a squeezed state with zero mean field, which is squeezed in the broadband (multimode) sense (correlation between frequencies located symmetrically above and below the carrier frequency) within some bandwidth about the carrier. The standard statements are that, if the squeezed quadrature is in phase with the carrier (amplitude-modulation squeezing), one will see sub-Poissonian photocount statistics, and if the squeezed quadrature is out of phase with the carrier (phase-modulation squeezing), one will see super-Poissonian statistics. Bandwidth considerations impact these statements in the following way: the only part of the squeezing that is relevant for decreasing or increasing the counting noise is that part that lies within an electronic bandwidth of the carrier frequency; to see the maximum decrease or increase, one must squeeze over the entire electronic bandwidth. These conclusions require modification if the carrier wave is not sufficiently intense.
Photon number probabilities for a single squeezed mode can exhibit sub-Poissonian or super-Poissonian statistics and can display sharp oscillations in appropriate conditions. We consider photocount probabilities for cw squeezed light occupying a single transverse mode. The squeezing spectrum is assumed to be flat over a bandwidth B, centered at optical frequency Ω/2π. We compute single-fold photoelectron probabilities for a counting interval of duration T. We proceed by decomposing the single transverse mode into a complete set of longitudinal wave-packet modes, defined in terms of prolate spheroidal wave functions. Each of these wave-packet modes is in a separate squeezed state. The Kelley-Kleiner photocount probability can then be written in terms of single-mode photon-number probabilities for these wave packet modes, which are detected with varying efficiencies. We need consider only a finite number of modes, because only about BT of the modes have high enough efficiency to contribute significantly to the photocount probability.
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.