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Physical Review A

Effects of a squeezed vacuum on a laser exhibiting phase locking: An application to a laser with injected atomic coherence

Ning Lü, János A. Bergou

Published Jul 1, 1989• 8 citations
Publisher

Abstract

We study the steady-state operation and noise properties of a laser (or maser) exhibiting phase locking when an ordinary vacuum is replaced by a squeezed vacuum. Such a phase-sensitive laser can be realized by preparing the active medium initially in a coherent superposition of levels involved in the laser transition. We develop a general formalism to treat number-phase fluctuations and quadrature fluctuations. When applied to a one-photon laser with injected atomic coherence, we find that not only is a complete quenching of the (spontaneous-emission) quantum noise possible, but the field can actually be in a squeezed state. The variance in either amplitude or phase quadrature can be below the vacuum noise level by a factor of 2 (50%) at the expense of enhanced fluctuations in the other quadrature.

DOI
10.1103/physreva.40.250

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Digest on Nonlinear Optics: Materials, Phenomena and Devices•Jan 1, 1990•0 citations
We study the steady-state operation and noise properties of a laser (or maser) exhibiting phase locking when an ordinary vacuum is replaced by a squeezed vacuum. Such a phase-sensitive laser can be realized by preparing the active medium initially in a coherent superposition of levels involved in the laser transition. We develop a general formalism to treat number-phase fluctuations and quadrature fluctuations. When applied to a one-photon laser with injected atomic coherence, we find that not only is a complete quenching of the (spontaneous-emission) quantum noise possible, but the field can actually be in a squeezed state. The variance in either amplitude or phase quadrature can be below the vacuum noise level by a factor of 2 (50%) at the expense of enhanced fluctuations in the other quadrature.
A quantum theory of a two-level single-mode laser with injected atomic coherence is developed by generalizing the Scully-Lamb laser theory to a form appropriate for the analysis of a coherently pumped laser. We assume that the active atoms are prepared initially in a coherent superposition of the upper and lower levels, and we derive the master equation for the field density operator by treating the interaction of the laser field with many active atoms simultaneously. It is shown that the photon-number distribution can be exactly Poissonian. The laser operation is analyzed in terms of the Fokker-Planck equation for the laser field. Both the intensity and phase diffusion coefficients are phase sensitive and, for stable laser operation, become much smaller than those of an ordinary laser. Consequently, the injected atomic coherence reduces both the photon-number noise and phase noise simultaneously. The intensity diffusion coefficient can vanish exactly, and at the same time the phase diffusion coefficient can become very small. This leads to spontaneous-emission noise quenching in the photon-number distribution, and the laser field can become very close to a coherent state. A scheme to generate the proper form of the initial atomic coherence necessary for the quantum noise quenching is proposed and analyzed.
The influence of injected atomic coherence on a laser's operation is discussed by analyzing the semiclassical equations of motion of this system. Both stationary-steady-state and time-dependent regimes of operation are investigated. We present the dependence of the stationary laser intensity and phase on the external parameters (population inversion, amplitude of atomic coherence, and detuning). For small cavity-field detuning and large atomic coherence, both frequency and phase locking occurs. The laser frequency is locked to the atomic frequency, and the phase is locked to a value determined primarily by the phase of the atomic coherence. Under certain conditions, in the inverted regime, the output intensity is an ssS-shaped function of the atomic coherence, leading to the possibility of a bistable behavior. However, only a portion of the bistable curve gives stable stationary operation, and in critical points (the location of which depends crucially on the external parameters) time-dependent instabilities branch away. When the detuning is larger than a critical value there is neither phase nor frequency locking. We find that the time-dependent behavior of the laser intensity in this case is oscillatory (quasiperiodic), and show that there is a stable limit cycle in the phase plane of the quadratures. There is a small parameter region where a stationary steady state and an oscillatory state may coexist. We also consider the nonlinear quantum theory of a laser with injected atomic coherence, and include the effect of pumping statistics. We derive the Fokker-Planck equation for the P representation, and express the noise in terms of moments. We find that in the steady state the intensity noise can be suppressed below the shot-noise limit but that the phase fluctuations are not affected by pump regularity. For nonzero detuning, we find that transient squeezing of the phase fluctuations is possible.
We investigate the effect of injected squeezed vacuum on the phase and intensity fluctuations of a laser. By using a stochastic simulation of the Langevin equations, we find that the phase diffusion noise of a laser with injected squeezed vacuum can be transiently reduced by a factor of 2, in agreement, with earlier predictions. Further, we find that the laser linewidth is broadened. The key observation is that the phase-diffusion rate is time dependent. We find that the steady-state intensity fluctuations are increased by the injected squeezing, in qualitative (but not quantitative) agreement with an earlier prediction. The problem of locking is examined. We find that the injected squeezed vacuum does not cause the laser to lock to any particular phase. We present an approximate solution of the Fokker-Planck equation in steady state, and a simple geometrical, vector kick model from which most of our results can be obtained.
The fundamental source of noise in laser and optical amplifier is spontaneous emission. However, it is possible to reduce this noise by preparing the gain medium (atoms) in coherent superposition of states. There are essentially two schemes for generating atomic coherence in optical amplifiers. \n \nFirst scheme involves the preparation of atoms in a coherent superposition of states before injecting them inside the cavity. In the other scheme atoms (initially pumped incoherently) are continuously driven by an external pump field inside the cavity. Almost all the studies concerning the quenching of spontaneous emission noise in optical amplifiers, based on the notion of atomic coherent superposition, ignore the finite bandwidth of the field that is responsible for generating coherence. In this dissertation, we have examined the effect of the finite bandwidth associated with the phase fluctuations of the driving field on a number of noise quenching schemes. \n \nIt is known that coherent preparation of atoms can, under certain conditions, lead to noise-free amplification and almost 50 percent phase squeezing in a two-photon correlated emission laser. We study the effect of phase fluctuations in the injected atoms on the phase noise of the field in these systems. We discuss the conditions under which squeezing is preserved in the two-photon linear amplifiers. We also show that, even for very large phase fluctuations in the injected atoms, almost 50 percent phase squeezing can be achieved. The corresponding gain is however significantly reduced. \n \nWe also study a system of phase-sensitive amplifier where atoms are prepared in a coherent superposition of states by a strong classical field. We study the role of pump phase fluctuations associated with the finite linewidth on the quantum statistical properties of the radiation field. A study of output field characteristics reveals that the presence of a weak pump field gives rise to equal variances in both the field quadratures which are independent of linewidth. On the other hand, the presence of strong pump field alters the output field characteristics with the generation of squeezing which is later on destroyed due to the presence of phase fluctuations
An intense single-mode correlated-spontaneous-emission laser or maser can be realized by driving the active atoms coherently with an injected external field. The scheme involves single-photon transitions unlike the two-photon or quantum-beat correlated-emission lasers that utilize correlations between successive photon-cascade emissions or between simultaneous emissions into modes of the field. Quenching of both the amplitude and phase noise and, in certain cases, squeezing of the amplitude fluctuations are found.
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.