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Physical Review A

Laser theory with finite atom-field interacting time

Deshui Yu, Jingbiao Chen

Published Jul 30, 2008• 43 citations
Publisher

Abstract

We investigate the influence of atomic transit time $\ensuremath{\tau}$ on the laser linewidth by the quantum Langevin approach. With comparing the bandwidths of cavity mode $\ensuremath{\kappa}$, atomic polarization ${\ensuremath{\gamma}}_{ab}$, and atomic transit broadening ${\ensuremath{\tau}}^{\ensuremath{-}1}$, we study the laser linewidth in different limits. We also discuss the spectrum of fluctuations of output field and the influence of pumping statistics on the output field.The influence of atomic transit time $\ensuremath{\tau}$ on laser field has not been carefully discussed before, to our knowledge. In particular, a laser operating in the region of ${\ensuremath{\gamma}}_{ab}⪡{\ensuremath{\tau}}^{\ensuremath{-}1}⪡\ensuremath{\kappa}∕2$ appears not to have been analyzed in previous laser theories. Our work could be a useful complementarity to laser theory. It is also an important theoretical foundation for the recently proposed active optical atomic clock based on bad-cavity laser mechanism.

DOI
10.1103/physreva.78.013846

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We investigate the influence of atomic transit time \ensuremathτ\ensuremath{\tau}\ensuremathτ on the laser linewidth by the quantum Langevin approach. With comparing the bandwidths of cavity mode \ensuremathκ\ensuremath{\kappa}\ensuremathκ, atomic polarization \ensuremathγab{\ensuremath{\gamma}}_{ab}\ensuremathγab​, and atomic transit broadening \ensuremathτ\ensuremath−1{\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremathτ\ensuremath−1, we study the laser linewidth in different limits. We also discuss the spectrum of fluctuations of output field and the influence of pumping statistics on the output field.The influence of atomic transit time \ensuremathτ\ensuremath{\tau}\ensuremathτ on laser field has not been carefully discussed before, to our knowledge. In particular, a laser operating in the region of \ensuremathγab⪡\ensuremathτ\ensuremath−1⪡\ensuremathκ∕2{\ensuremath{\gamma}}_{ab}⪡{\ensuremath{\tau}}^{\ensuremath{-}1}⪡\ensuremath{\kappa}∕2\ensuremathγab​⪡\ensuremathτ\ensuremath−1⪡\ensuremathκ∕2 appears not to have been analyzed in previous laser theories. Our work could be a useful complementarity to laser theory. It is also an important theoretical foundation for the recently proposed active optical atomic clock based on bad-cavity laser mechanism.
In this chapter, we present a theory of the laser based on the Heisenberg–Langevin approach. This is a different, but completely equivalent approach to the density operator approach discussed in the previous chapter. In general, the density operator approach is better suited to study the photon statistics of the radiation field whereas the Heisenberg–Langevin approach has certain calculational advantages in the determination of phase diffusion coefficients, and consequently laser linewidth. In Section 12.1, a simple approach to determine laser linewidth based on a linear theory is presented. This analysis is especially interesting and useful in that it includes atomic memory effects, something that is difficult to do within a density matrix theory. In Sections 12.2–12.4, we consider the complete nonlinear theory of the laser and rederive all the important quantities related to the quantum statistical properties of the radiation field. A simple Langevin treatment of the laser linewidth including atomic memory effects† The full nonlinear quantum theory of the laser discussed in the previous chapter yields most of the interesting quantum statistical properties of the radiation field. In many problems of interest, however, we do not need such an elaborate treatment. For example, as we saw in the previous chapter, the natural linewidth of the laser can be determined from a linearized theory of the laser. That is, the full nonlinear theory serves to determine the amplitude of the field but the phase fluctuations about this operating point are described by a linear theory.
An ideal superradiant laser on an optical clock transition of noninteracting cold atoms is predicted to exhibit an extreme frequency stability and accuracy far below mHz-linewidth. In any concrete setup sufficiently many atoms have to be confined and pumped within a finite cavity mode volume. Using a magic wavelength lattice minimizes light shifts and allows for almost uniform coupling to the cavity mode. Nevertheless, the atoms are subject to dipole-dipole interaction and collective spontaneous decay which compromises the ultimate frequency stability. In the high density limit the Dicke superradiant linewidth enhancement will broaden the laser line and nearest neighbor couplings will induce shifts and fluctuations of the laser frequency. We estimate the magnitude and scaling of these effects by direct numerical simulations of few atom systems for different geometries and densities. For Strontium in a regularly filled magic wavelength configuration atomic interactions induce small laser frequency shifts only and collective spontaneous emission weakly broadens the laser. These interactions generally enhance the laser sensitivity to cavity length fluctuations but for optimally chosen operating conditions can lead to an improved synchronization of the atomic dipoles.
The theory of a laser system with a ladder-type configuration is studied in detail based on the quantum Langevin approach. By using an external field to link the lower lasing level with another atomic level, whose decay rate is much larger, laser intensity significantly increases and the quantum-limited linewidth can be quenched. We also discuss the spectrum of fluctuations of the output field, and the result shows that the fluctuations at low frequencies can be much suppressed too. On the other hand, this quenching approach can realize a laser output between two atomic levels, whose decay rates do not satisfy the usual lasing condition that the decay rate of the lower lasing level should be larger than that of the upper lasing level. It will be very useful to realize a laser output with the wavelength we want. This quenching approach has been widely used in the absorption spectrum of the ytterbium optical lattice clock and in the laser cooling approach for calcium atoms. Here we apply it in the stimulated emission of lasers.
We derive a general expression for the linewidth of inhomogeneously broadened traveling-wave gas lasers, without any assumption on the relative magnitude of the atomic and field decay constants, and for several types of atomic pumping statistics, ranging from Poissonian to regular. For small inhomogeneous broadening, an important linewidth narrowing occurs in the bad-cavity limit. This effect disappears, however, in the large inhomogeneous broadening limit (Doppler limit). \textcopyright{} 1996 The American Physical Society.
In this chapter we treat single-mode laser operation in which both the atoms and the field obey the laws of quantum mechanics. For simplicity, we suppose for most of the discussion that the field is tuned to atomic line center and that the active atoms are two-level systems which pass through the cavity in a time r, as in Sec. 16-1. To simulate a homogeneously broadened medium, we suppose that the transit times are distributed with the probability y exp(— yr). The more realistic (and complicated) calculation underlying the theory of Chap. 8 is given in Appendix I. We also suppose that excitation occurs only to the upper level; Probs. 17-13 and 17-14 consider excitation to the lower level as well. We describe cavity losses by an ensemble of nonresonant atoms injected into the lower level, as discussed in Sec. 16-1. The response of these “loss” atoms does not saturate (i.e., small coupling constant), as does that of the active atoms. The loss atoms here play the same role as the fictitious con­ ductivity in the semiclassical laser theory of Chap. 8. As shown in Sec. 16-5, an equivalent description is obtained by a reservoir of simple harmonic os­ cillators with a spread of frequencies. Our discussion depends on the reservoir theory of Sec. 16-1, in which the reduced density operator represents the ra­ diation field. In Chap. 20 we give a related development in which the Langevin (Heisenberg picture) approach is used.
The theory of a quantum-limited phase-matching effect in a $\ensuremath{\Lambda}$-type lasing system is studied in detail based on the quantum Langevin approach. Two quasimonochromatic fields are directly generated based on two lasing transitions. We find that the coherence between two lasers can well exceed the linewidth of either laser field. This result denotes that two field phases match each other although either laser field has a high phase fluctuation. Unlike the phase-matching effect based on atomic absorption, the final coherence between two laser fields here is not limited by saturation broadening, and the higher laser intensities lead to a higher coherence. Additionally, based on a linear stability analysis, we find that the instability of the field steady state can substantially restrict the occurrence of this phase-matching effect in the bad-cavity limit for a high pump rate. We also discuss the spectrum of amplitude fluctuations of output fields, and the result shows that the squeezing of amplitude fluctuations at low frequencies for a single field oscillating inside the cavity is damaged in the case of two fields oscillating.