The definitions of homogeneous squeezing, inhomogeneous squeezing, spectral component squeezing, and degree of squeezing are introduced for an optical field that is not necessarily squeezed in the full sense. They are then applied to three different examples of squeezed states. It is shown that in resonance fluorescence from a two-level atom the squeezing is homogeneous when the relative Rabi frequency Ω/β < 1, inhomogeneous when 1 < Ω / β < 2 , and spectral component-type squeezing when Ω / β > 2 , and the maximum degree of squeezing is 28%. For the process of downconversion in a resonant cavity the squeezing is homogeneous with a 100% maximum degree of squeezing. In the process of four-wave mixing the squeezing is always inhomogeneous, and there is a domain of spectral component squeezing, but the degree of squeezing can be 100% in certain circumstances.
The definitions of homogeneous squeezing, inhomogeneous squeezing, spectral component squeezing, and degree of squeezing are introduced for an optical field that is not necessarily squeezed in the full sense. They are then applied to three different examples of squeezed states. It is shown that in resonance fluorescence from a two-level atom the squeezing is homogeneous when the relative Rabi frequency Ω/β < 1, inhomogeneous when 1 < Ω / β < 2 , and spectral component-type squeezing when Ω / β > 2 , and the maximum degree of squeezing is 28%. For the process of downconversion in a resonant cavity the squeezing is homogeneous with a 100% maximum degree of squeezing. In the process of four-wave mixing the squeezing is always inhomogeneous, and there is a domain of spectral component squeezing, but the degree of squeezing can be 100% in certain circumstances.
Certain two-time correlation functions of squeezed optical fields and their Fourier conjugates are investigated. These functions are related to the autocorrelation function and the spectral density of the photoelectric current that appears in measurements made of the squeezed light by a homodyne technique. Because the field is often assumed to contain a monochromatic carrier, it is only quasi-stationary, and modified quantum correlation functions, specially constructed to be time independent, are used to describe its properties. The symmetries of the corresponding conjugate spectral functions, which are not always spectral densities, are investigated. The concepts of homogeneous squeezing, inhomogeneous squeezing, and spectral component squeezing are introduced, and it is shown that the condition for cross-spectral purity, when satisfied, leads to homogeneous squeezing. A frequency-dependent parameter called the degree of squeezing, which lies between 0 and −1 for a squeezed state and is simply related to photoelectric measurements, is introduced. These ideas are then applied to the squeezing produced in resonance fluorescence, in parametric downconversion, and in four-wave mixing, and the degrees of squeezing under different conditions are compared.
The nonclassical nature of the so-called squeezed states of electromagnetic fields has recently been discussed in connection with the phenomenon of resonance fluorescence from a coherently driven two-level atom.1 It has been shown that either the in-phase component E1 or the out-of-phase component E2 of the fluorescent field can become squeezed if the Rabi frequency Ω of the ex- citing field is sufficiently low. Squeezing is defined as a negative value of the normally ordered variance 〈: (ΔE1)2:〉 (or 〈:(ΔE2)2:〈). To satisfy this condition in the steady state, if the exciting field is tuned perfectly to the atomic transition, the Rabi frequency Ω has to be less than √2y, with 2y = A—the Einstein coefficient for spontaneous emission. Off-resonance excitation further lowers this critical value of 0. Walls and Zoller1 have indicated, without giving the details, that, in a transient regime, the squeezing effect can become twice as large as in the steady state.
Light emitted from laser-driven atoms often has squeezed quadrature fluctuations reflecting the phase dependence of the atomic source excited into a coherent superposition by the driving field. In this paper, we study the maximum squeezing in the emitted light fields from three-level atoms, paying particular attention to the role of atomic coherences in governing the optimum squeezing which is possible. Some phase dependent nonlinear optical processes have the potential for substantial quantum noise reduction at useful intensities: for example, four-wave mixing involving atomic systems has already been used to generate quadrature-squeezed light. The threoretical description of such a source is generally complicated with damping processes, coherent and incoherent pumping, and so on, all needing to be taken into account. Fully quantum treatments are available, but the basic physics of the process is often difficult to grasp in the necessarily complicated analysis. Although less important from the point of view of efficient practical sources of squeezed light, studying the generation of squeezed light in simpler systems such as two- or three-level atoms interacting with one or two modes (as in Jaynes-Cummings models) or in spontaneous emission or resonance fluorescence gives greater insight into the fundamentals of the squeezing process. In particular, the relationship between the atomic coherences and the degree of squeezing in the generated fields can be explored. We study the maximum (optimized) squeezing that can be obtained in the emitted fields from the irreversible decay from three-level atoms for any choice of initial conditions, excitations and decay process, for all quadrature components and for all possible configurations (V, \ensuremath{\Lambda}, and ``ladder'' systems).We show that for V and \ensuremath{\Lambda} systems, optimum squeezing is actually associated with a two-level state (in a suitable basis) involving a single one-photon coherence, but for ladder systems the state for maximum noise reduction is not equivalent to a two-level system and involves a single two-photon coherence and no intermediate state population. We pay particular attention to whether the total field is in a minimum uncertainty state, and the nature of the atomic state associated with maximum squeezing, especially whether it is a mixed or pure state. In the case where the initial free field has a zero amplitude at the detector and the source atoms are confined to a region small compared to the wavelength (Dicke source) we show that the squeezing in the total field is given in terms of the squeezing in the source field, and hence related to atomic populations and atomic one-photon and two-photon coherences for the case of three level atom. Previously unconsidered source-free field interferences terms are shown to be zero. The choice of quadrature phase and frequency is optimized to minimize the source field normally ordered variance. We then minimize further with respect to the choice of atomic density matrix elements subject to the constraints that the density matrix is Hermitean, positive, and has a trace equal to unity. In all cases we find that the optimum squeezing is produced when the source field is in a minimum uncertainty state.
It is shown that 100 squeezed output can be produced in the resonance fluorescence from a coherently driven two-level atom interacting with a squeezed vacuum. This is only possible for N=1/8 squeezed input, and is associated with a pure atomic state, i.e., a completely polarized state. The quadrature for which optimal squeezing occurs depends on the squeezing phase $\ensuremath{\Phi}$, the Rabi frequency $\ensuremath{\Omega}$, and the atomic detuning $\ensuremath{\Delta}$. Pure states are described for arbitrary $\ensuremath{\Phi}$, not just $\ensuremath{\Phi}=0$ or $\ensuremath{\pi}$ as in previous work. For small values of N, there may be a greater degree of squeezing in the output field than the input, i.e., we have squeezing amplification.
We study the effects of equal and unequal Rabi frequencies as well as cavity bandwidth on the optimal noise of the output fields from a two-mode optical cavity, which is fed by two coherent fields and encloses a collection of three-level Λ atoms. Each of the output modes is shown to be highly squeezed in the case of smaller symmetrical atomic detunings compared to the spontaneous-emission rates, where the system is in the vicinity of perfect coherent population trapping (CPT). We find that the squeezing on each output mode is larger than that reported by Dantan et al (2006 Phys. Rev. Lett. 97 023605) when adjusting the equal Rabi frequencies. The cavity bandwidth affects, not the degree of squeezing, but the linewidth of the spectral components. In an alternative way, the squeezing of each outgoing mode can be understood as a consequence of the enhanced cross-phase modulation, different from the nonlinear Faraday effect in Dantan's paper (2006 Phys. Rev. Lett. 97 023605). The application of squeezed light is to obtain high-precision measurement, optical communication, and interferometric detection of gravitational radiation.
The quadrature squeezing spectra produced in the resonance fluorescence of a two-level atom are investigated, including the effects of a finite laser bandwidth due to phase diffusion. For resonant excitation at low intensities, the fluorescence field exhibits narrow bandwidth squeezing in the out-of-phase quadrature, centered at the laser frequency. Otherwise, squeezing occurs only in the in-phase quadrature. Specifically, for slightly off-resonance and weak excitation, the squeezing still centers at the laser frequency, with a finite bandwidth of the order of $\ensuremath{\gamma}.$ More importantly, for far-off-resonance and strong excitation, the resonance fluorescence exhibits two-mode squeezing around the Rabi sideband frequencies. Thus we have a source of two-mode squeezing that is very easily frequency tuned simply by adjusting the generalized Rabi frequency. The presence of a laser linewidth substantially reduces the degree of squeezing obtainable. The fluorescence shows no squeezing at all when the laser linewidth is greater than the atomic spontaneous emission linewidth.