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arXiv:quant-ph/9812084 · Physical Review A

Optimal squeezing, pure states, and amplification of squeezing in resonance fluorescence

Peng Zhou, S. Swain

Published May 1, 1999• 11 citations
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Abstract

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.

DOI
10.1103/physreva.59.3745

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It is shown that 100100%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=1/8N=1/8N=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 \ensuremathΦ\ensuremath{\Phi}\ensuremathΦ, the Rabi frequency \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ, and the atomic detuning \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ. Pure states are described for arbitrary \ensuremathΦ\ensuremath{\Phi}\ensuremathΦ, not just \ensuremathΦ=0\ensuremath{\Phi}=0\ensuremathΦ=0 or \ensuremathπ\ensuremath{\pi}\ensuremathπ 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.
The squeezing properties of the fluorescence field emitted by a two-level atom driven by a coherent laser field in a squeezed vacuum are calculated. We show that in the region of the anomalous resonance fluorescence the emitted field exhibits squeezing that is much larger than that in the input squeezed vacuum. The squeezing spectrum attains a minimum value that corresponds to 75% squeezing. We also find that, in the total fluorescence field, squeezing attains an optimum achievable value in the fluorescence field emitted by a two-level atom. The optimum squeezing is associated with the collapse of the system into a pure state.
We discuss the conditions which give rise to anomalous features in the resonance fluorescence of a two-level atom interacting with a resonant classical field and a squeezed vacuum. A simple expression is obtained, which is shown to coincide closely with the conditions for amplification of a probe beam and with the collapse of the atomic system into a pure state. We thereby illustrate striking observable consequences of the evolution of a driven system into a pure state. This situation also provides a method for preparing an atom in a pure state whose nature depends on the squeezing phase and Rabi frequency. The conditions for hole burning are obtained analytically, and the sensitivity of these effects to the two-photon correlations produced by the squeezed vacuum is demonstrated.
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.
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.
The authors present analytical and numerical results for resonance fluorescence produced by a two-level atom damped by a broad-band squeezed vacuum. Previous theoretical analyses of this model have assumed that the laser frequency and the carrier frequency of the squeezed vacuum are equal. The authors introduce a non-zero detuning between the laser frequency and the carrier frequency while maintaining resonance between the laser frequency and atomic resonance. In the absence of the driving field there is a squeezing threshold: above threshold the decay rate of the atomic polarization can be reduced, and below threshold a squeezing-induced two peak spectrum and frequency shift, which is analogous to the Bloch-Siegert shift, are predicted. In the presence of the driving field, the spectrum is asymmetric for Rabi frequencies smaller or comparable to the detuning of the squeezed vacuum. A symmetric spectrum is obtained for large Rabi frequencies. An interpretation is predicted by means of a dressed-state analysis. The authors find that for not too large Rabi frequencies a remarkably large population inversion can be induced in the dressed states, even at exact resonance of the driving field, provided that the laser frequency and carrier frequency are detuned. Moreover, for detection times larger than the inverse detuning, the spectrum is insensitive to the phase between the driving field and the squeezed vacuum, in marked contrast to the case of no detuning.