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Journal of the Optical Society of America B

Two-Mode Formalism for Two-Photon Optics

Bonny L. Schumaker, Carlton M. Caves

Published Jan 1, 1984• 0 citations
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Abstract

In this paper we discuss a new, two-mode formalism1 for analyzing two-photon optical devices (e.g., parametric amplifiers and four-wave mixers)–devices whose output light is generated by the simultaneous emission of two photons into two of the output modes. Such devices can produce in principle squeezed-state light, in which the two modes of each pair of output modes have correlated noise. Two-photon devices are to be contrasted with one-photon devices, such as the laser, whose output light is generated by the emission of independent single photons.

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In this paper we discuss a new, two-mode formalism1 for analyzing two-photon optical devices (e.g., parametric amplifiers and four-wave mixers)–devices whose output light is generated by the simultaneous emission of two photons into two of the output modes. Such devices can produce in principle squeezed-state light, in which the two modes of each pair of output modes have correlated noise. Two-photon devices are to be contrasted with one-photon devices, such as the laser, whose output light is generated by the emission of independent single photons.
This paper introduces a new formalism for analyzing two-photon devices (e.g., parametric amplifiers and phase-conjugate mirrors), in which photons in the output modes are created or destroyed two at a time. The key property of a two-photon device is that it excites pairs of output modes independently. Thus our new formalism deals with two modes at a time; a continuum multimode description can be built by integrating over independently excited pairs of modes. For a pair of modes at frequencies \ensuremath{\Omega}\ifmmode\pm\else\textpm\fi{}\ensuremath{\epsilon}, we define (i) quadrature-phase amplitudes, which are complex-amplitude operators for modulation at frequency \ensuremath{\epsilon} of waves ``cos[\ensuremath{\Omega}(t-x/c)]'' and ``sin[\ensuremath{\Omega}(t-x/c)]'' and (ii) two-mode squeezed states, which are the output states of an ideal two-photon device. The quadrature-phase amplitudes and the two-mode squeezed states serve as the building blocks for our formalism; their properties and their physical interpretation are extensively investigated.
A new formalism for analyzing two-photon devices, such as parametric amplifiers and phase-conjugate mirrors, is proposed in part I, focusing on the properties and the significance of the quadrature-phase amplitudes and two-mode squeezed states. Time-stationary quasi-probability noise is also detailed for the case of Gaussian noise, and uncertainty principles for the quadrature-phase amplitudes are outlined, as well as some important properties of the two-mode states. Part II establishes a mathematical foundation for the formalism, with introduction of a vector notation for compact representation of two-mode properties. Fundamental unitary operators and special quantum states are also examined with an emphasis on the two-mode squeezed states. The results are applied to a previously studied degenerate limit (epsilon = 0).
It has been known for a while that coherent high-intensity photon number eigenstate fields can be generated through nonlinear optical parametric processes,1 the same ones that give rise to quadrature squeezing.2,3 In contrast to quadrature squeezing which is essentially a single-mode phenomenon, although it is widely misunderstood that it is a two-mode phenomenon, the generation of single-mode number states through three- or four-wave mixing depends on two separate signal and idler modes that are perfectly correlated in photon numbers, in other words a twin beam. The nature of such twin beams is elucidated in both the Schrodinger and the Heisenberg pictures. In particular, it is shown that strong photon number correlation in a twin beam can still be obtained in an amplifier rather than an oscillator configuration, another widely missed point.
The level of quantum noise in measurements is bounded from below by the Heisenberg uncertainty principle, but it can be unequally distributed between two non-commuting observables — it can be “squeezed”. Since 2019, all gravitational-wave observatories have been using squeezed light for increasing the astronomical reach. Squeezed laser light is efficiently produced by degenerate parametric down-conversion in a nonlinear crystal located inside an optical resonator. A spontaneously generated initial pair of indistinguishable photons is amplified to a squeezed vacuum state. Overlapped with bright coherent light, the photo electric measurement shows a sub-Poissonian photon statistics. Squeezed states have ample applications in nonlocal quantum sensing, device-independent quantum key distribution, and quantum computing. Here, we present our continuous-wave 1550 nm ‘squeeze laser’ with a footprint of 80 cm × 80 cm. The well-defined output beam has an interference contrast of≳99%\gtrsim 99\%≳99%with an overlapped 10 mW beam being in an almost perfect TEM00 mode. The interference result shows 13 dB squeezing of the photon shot noise in balanced detection.
This paper provides the mathematical foundation for the two-mode formalism introduced in the preceding paper. A vector notation is introduced; it allows two-mode properties to be written as compactly as the comparable properties for a single mode. The fundamental unitary operators of the formalism are described and their properties are examined; particular attention is paid to the two-mode squeeze operator. Special quantum states associated with the formalism are considered, with emphasis on the two-mode squeezed states.