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Physical Review A

Amplitude squeezing through photon fractioning

Giacomo Mauro D’Ariano

Published Mar 1, 1990• 8 citations
Publisher

Abstract

A prototype mechanism for producing amplitude-squeezed states, based on the so-called ``photon fractioning'' procedure, is investigated. The coherent state is connected with a quasinumber eigenstate through an intermediate density matrix involving multiphoton processes. The enhancement in the phase fluctuations and the simultaneous reduction of the noise in the photon number result from two competing limits; the ``bright limit,'' which is semiclassical and requires highly excited states, and the photon fractioning limit, which is pure quantum. Asymptotical evaluations of the number probability distribution and of the Q function show that the mixed states are very close to the exact number-phase minimum-uncertainty states.

DOI
10.1103/physreva.41.2636

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Physical Review A•Feb 1, 1989•15 citations
A prototype mechanism for producing amplitude-squeezed states, based on the so-called photon fractioning'' procedure, is investigated. The coherent state is connected with a quasinumber eigenstate through an intermediate density matrix involving multiphoton processes. The enhancement in the phase fluctuations and the simultaneous reduction of the noise in the photon number result from two competing limits; the bright limit,'' which is semiclassical and requires highly excited states, and the photon fractioning limit, which is pure quantum. Asymptotical evaluations of the number probability distribution and of the Q function show that the mixed states are very close to the exact number-phase minimum-uncertainty states.
The number-phase squeezing mechanisms related to the multiphoton and photonfractioning procedures are analyzed. The main ingredients in these approaches are nonlinear multimode realizations of group-theoretical states and nonunitary subdynamics of the field modes. The states which are squeezed in the number of photons are suited to optical communications in local area networks; the states squeezed in the phase of the field optimize high-sensitive interferometry. Corresponding to the different kinds of squeezing and to the pertaining grouptheoretical states one has different types of amplifying devices with the same dynamical group of the states. In particular, two recently proposed devices – the photon-number amplifier (PNA) and the photon-number duplicator (PND) – attain both the multiphoton and photon-fractioning transformations. The PNA amplifies the field without degrading the direct-detection signal-to-noise ratio and thus can be used to produce number-squeezed states; the PND provides two copies of the same input number-state and, therefore, can connect two-mode states with one-mode states. Both the PNA and PND allow the realization of virtually lossless optical taps, which are essential in designing highly transparent local area networks. The PND can also be used in series with a conventional phase-insensitive amplifier to produce phase-squeezed states.
A number-phase interaction is suggested that is analogous to the usual quadrature squeezing Hamiltonian for a mode of the electromagnetic field. Applied to an initial coherent state, it produces number-phase squeezed states. For moderate squeezing in either direction the result is close to a minimum uncertainty state. Two different ${\mathrm{\ensuremath{\chi}}}^{(3)}$ approximations to the ideal interaction are found, appropriate for the different directions of squeezing. Even with the approximate interaction, the fluctuations in photon number can be reduced to less than half a photon for an arbitrarily intense field.
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.
The evolution of a quadrature-squeezed state through a Kerr medium is studied from the point of view of amplitude squeezing. We propose a Mach-Zehnder interferometer configuration, which incorporates a squeeze- state generator and a nonlinear Kerr medium, to generate and subsequently displace these states. The displaced states show amplitude squeezing for a proper choice of the initial quadrature squeezed state, and we study two such cases here in detail. In the limit where the evolution through a Kerr medium results in a macroscopic superposition, an initial phase-squeezed coherent state shows significant amplitude squeezing, and we have shown that the photon-number uncertainty 〈(\ensuremath{\Delta}n^)2{)}^{2})2〉 can be minimized to a value smaller than 〈n^〉2/3{\mathrm{〉}}^{2/3}〉2/3. On the other hand, an initial quadrature-squeezed state, whose direction of squeezing is along the direction of excitation, upon evolving through a Kerr medium for a suitable small duration of time shows substantial amplitude squeezing for an appropriate displacement, and we show that the photon-number uncertainty in this case can be minimized to 〈n^〉1/5{\mathrm{〉}}^{1/5}〉1/5, which is far below the previously achieved limits. \textcopyright{} 1996 The American Physical Society.
The origin of quantum noise is addressed and it can be eliminated when the quantum state is optimally prepared for the single measured observable. Minimum uncertainty states are reviewed using Q (α) -representation, which include squeezed states and number-phase minimum uncertainty states. Squeezing is a concept of trading quantum uncertainties between two conjugate observables to optimize the state for the measurement. Need for optical nonlinearity to achieve squeezing is induced. Principles and experimental results for squeezing quadrature amplitude and photon number are delineated. Applications to optical communications and optical precision measurement are briefly stated.
We construct a large class of squeezed states realized in terms of density matrices. The moments of the canonical variable q^ and the number operator n^ distributions are analytically evaluated. The new states can be simultaneously squeezed in q^ and n^ to any desired amount and exhibit almost Gaussian shape in the former and strongly sub-Poisson distribution in the latter. The probability distributions can be synthetically characterized in terms of the fractional-photon index introduced by Katriel, Rassetti, and Solomon [Phys. Rev. D 35, 1248 (1987)].