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Annual Meeting Optical Society of America

Photocount statistics of broadband squeezed states

Carlton M. Caves

Published Jan 1, 1985• 0 citations
Publisher

Abstract

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.

DOI
10.1364/oam.1985.fm1

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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.
A squeezed state of light is usually taken to mean a state in which the fluctuations of one of the quadratures are below the level associated with a coherent or vacuum state. These squeezed fluctuations are phase sensitive in the sense that they vary periodically as the field evolves, but the relationship between squeezing and phase fluctuations is not trivial. While squeezed states can exhibit an arbitrarily low level of phase fluctuations, the phase states, which are eigenstates of the Hermitian optical phase operator are not themselves quadrature squeezed. A phasesqueezed state is one with a lower level of phase fluctuations than a coherent state with the same mean photon number. Phase squeezing is fundamentally different from quadrature squeezing in much the same way that sub-Poissonian photon statistics is different. For this reason optimally squeezed states and states with minimum phase noise will be distinct. We derive the form of the states which minimum phase noise for a given intensity. These have an Airy function behavior for the photon number probability distribution. These phase squeezed states have a far better phase resolution than coherent states of similar intensity, but do not seem to offer a significant reduction of phase fluctuations compared with conventional squeezed states.
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.
A photodetector's electronic bandwidth is the frequency range over which beats between different frequencies in the input can be observed in the output. The output of a balanced homodyne detector, the prototypical device for detecting squeezed light, consists of beats between a signal to be measured and a local oscillator. Beats between the signal and quantum local-oscillator fluctuations lead to noise that degrades the observed squeezing. Through wide-band calculations, we show that this noise contains contributions from the entire signal bandwidth, even when this bandwidth is orders of magnitude larger than the detector's electronic bandwidth. This allows us to place an upper limit on the total power that can enter the detector's signal port (both signal and extraneous light) within the detector's optical bandwidth in order for a given level of squeezing to be observable.
Nonclassical effects such as squeezing, antibunching, and sub-Poissonian statistics of photons have been attracting attention in quantum optics over the last decade. Up to now most theoretical and experimental investigations have been carried out exclusively in the time domain while neglecting the spatial aspects by considering only one spatial mode of the electromagnetic field. In many situations such an approximation is well justified. There are, however, problems that do not allow in principle a single-mode consideration. This is the case when one wants to investigate the quantum fluctuations of light at different spatial points in the plane perpendicular to the direction of propagation of the light beam. Such an investigation requires a complete description of quantum fluctuations of light in both time and space and cannot be done within a single-mode theory. This space-time description brings about a natural generalization into the spatial domain of such notions as the standard quantum limit, squeezing, antibunching, etc. It predicts, for example, the possibility of generating a light beam with sub-Poissonian statistics of photons not only in time but also in the beam's transverse plane. Of particular relevance to the applications is a situation in which the cross section of the light beam contains several nonoverlapping areas with sub-Poissonian statistics of photons in each. Photodetection of such a beam produces several sub-shot-noise photocurrents depending on the number of independent areas with sub-Poissonian statistics. This is in marked contrast to the case of a single-mode sub-Poissonian light beam in which any attempt to collect light from only a part of the beam deteriorates the degree of shot-noise reduction. This property of multimode squeezed light opens a range of interesting new applications in optical imaging, optical parallel processing of information, parallel computing, and many other areas in which it is desirable to have a light beam with regular photon statistics across its transverse area. The aim of this review is to describe the recent development in this branch of quantum optics.