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arXiv:quant-ph/0612191 · Physical Review A

Multimode quantum limits to the linewidth of an atom laser

Mattias Johnsson, J. J. Hope

Published Apr 20, 2007• 10 citations
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Abstract

The linewidth of an atom laser can be limited by excitation of higher energy modes in the source Bose-Einstein condensate, energy shifts in that condensate due to the atomic interactions, or phase diffusion of the lasing mode due to those interactions. The first two are effects that can be described with a semiclassical model, and have been studied in detail for both pumped and unpumped atom lasers. The third is a purely quantum statistical effect and has been studied only in zero-dimensional models. We examine an unpumped atom laser in one dimension using a quantum field theory using stochastic methods based on the truncated Wigner approach. This allows spatial and statistical effects to be examined simultaneously, and the linewidth limit for unpumped atom lasers is quantified in various limits.

DOI
10.1103/physreva.75.043619

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The linewidth of an atom laser can be limited by excitation of higher energy modes in the source Bose-Einstein condensate, energy shifts in that condensate due to the atomic interactions, or phase diffusion of the lasing mode due to those interactions. The first two are effects that can be described with a semiclassical model, and have been studied in detail for both pumped and unpumped atom lasers. The third is a purely quantum statistical effect and has been studied only in zero-dimensional models. We examine an unpumped atom laser in one dimension using a quantum field theory using stochastic methods based on the truncated Wigner approach. This allows spatial and statistical effects to be examined simultaneously, and the linewidth limit for unpumped atom lasers is quantified in various limits.
We investigate the linewidth of a quasicontinuous atom laser within a semiclassical framework. In the high flux regime, the lasing mode can exhibit a number of undesirable features such as density fluctuations. We show that the output therefore has a complicated structure that can be somewhat simplified using Raman outcoupling methods and energy-momentum selection rules. In the weak outcoupling limit, we find that the linewidth of an atom laser is instantaneously Fourier limited, but, due to the energy ``chirp'' associated with depletion of the condensate, the long-term linewidth of an atom laser is equivalent to the chemical potential of the condensate source. We show that correctly sweeping the outcoupling frequency can recover the Fourier-limited linewidth.
We present a steady-state analysis of a quantum-mechanical model of an atom laser. A single-mode atomic trap coupled to a continuum of external modes is driven by a saturable pumping mechanism. In the dilute flux regime, where atom-atom interactions are negligible in the output, we find an analytic form for the linewidth and frequency shift of the laser. This result does not make the Born-Markov approximation, but is based on the far less restrictive ``self-consistent Markov approximation.'' The more exact treatment has a different effective damping rate and occupation of the lasing mode, as well as a shifted frequency and linewidth of the output. We examine gravitational damping numerically, finding linewidths and frequency shifts for a range of pumping rates. We treat mean-field damping analytically, finding a memory function for the Thomas-Fermi regime. The occupation and linewidth are found to have a nonlinear scaling behavior which has implications for the stability of atom lasers.
An atom laser is a device which would ideally produce a coherent excitation of a single mode of the Bose-atom field. That is to say, an atom laser field would have a well defined number of atoms, and a slow rate of phase diffusion. Here we consider an atom laser model recently proposed by Holland et al, based on evaporative cooling of atoms. We extend their model, and give approximate analytical results for the time-dependent intensity and phase statistics of the atom laser in different regimes. Although the intensity fluctuations may be relatively small, the laser linewidth is always in excess of the bare linewidth of the laser mode. Nevertheless, in one regime (weak collisions), the laser linewidth may be less than the output flux of atoms, which is a minimum requirement for considering the output to be coherent.
Essentially closed-form formulae are given for the linewidth of inhomogeneously broadened lasers, with arbitrary cavity detuning and pumping statistics (Poissonian or quiet pumps). The so-called `bad-cavity' mode of operation is treated. The general formula is obtained from a semiclassical theory first derived from a linearized symmetrically ordered quantum theory in the limit of a large number of atoms. This semiclassical theory is most easily explained in terms of independent quantum jumps performed by emitting and absorbing atoms. The effect of inhomogeneous broadening on linewidth is found to be negligible for some pumping schemes, but when the atoms are independently pumped the linewidth is proportional (at some constant power level) to the gain-medium spectral width. We consider, in particular, an intermediate situation in which the rate at which atoms are promoted from the lower to the upper level (pump rate) is proportional to the lower-state population for each atom class. For that particular model, pump fluctuations do not affect the linewidth, even when the cavity is detuned from the centre of the atomic transition frequency distribution.