Multimode quantum limits to the linewidth of an atom laser

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.

Figure 1

Atomic level scheme. Using a Raman process, trapped atoms in the condensate (∣1⟩) are transferred to the untrapped state (∣2⟩) via an intermediate state (∣3⟩).

Figure 2

(Color online) Momentum space power spectrum of the output beam after $5\mathrm{ms}$ (a) and $10\mathrm{ms}$ (b). Over time, the atoms build up outside the coupling region, and gain a small extra momentum kick due to the BEC-beam repulsion. Parameters: $N=5\times {10}^6$, $\omega =250\mathrm{rad}{\mathrm{s}}^{-1}$, $a=1\times {10}^{-9}\mathrm{m}$, and $k_0=2\times {10}^7{\mathrm{m}}^{-1}$.

Figure 3

Power spectrum of the output beam at $t=0.35\mathrm{s}$ according to the stochastic simulation (solid line) as well as a Gaussian fitting profile (dashed line). Parameters: $N={10}^7$, $\omega =250\mathrm{rad}{\mathrm{s}}^{-1}$, $a=3\times {10}^{-9}\mathrm{m}$, and $k_0={10}^7{\mathrm{m}}^{-1}$.

Figure 4

Linewidth narrowing as a function of outcoupling time (log-log plot). Dashed line shows the semiclassical result; solid line shows the result of the stochastic simulation. Horizontal line indicates the fundamental linewidth limit according to twice Eq. (8). Parameters: $N={10}^7$, $\omega =250\mathrm{rad}{\mathrm{s}}^{-1}$, $a=1\times {10}^{-9}\mathrm{m}$, and $k_0={10}^7{\mathrm{m}}^{-1}$.

Figure 5

Linewidth narrowing as a function of outcoupling time (log-log plot). Dashed line shows the semiclassical result; solid line shows the result of the stochastic simulation. Horizontal line indicates the fundamental linewidth limit according to twice Eq. (8). Parameters: $N={10}^7$, $\omega =250\mathrm{rad}{\mathrm{s}}^{-1}$, $a=3\times {10}^{-9}\mathrm{m}$, and $k_0={10}^7{\mathrm{m}}^{-1}$.

Figure 6

(Color online) Momentum space density of a two-dimensional atom laser, with $p_x=\hslash k_x$ and $p_z=\hslash k_z$. The parabolic shape is due to energy conservation as per Eq. (30), with the thickness of the line tracing out the parabola giving the linewidth. Parameters: $N=5\times {10}^6$, $\omega =1500\mathrm{rad}{\mathrm{s}}^{-1}$, $a=1\times {10}^{-8}\mathrm{m}$, and $k_0=4\times {10}^6{\mathrm{m}}^{-1}$.

Figure 7

Results for a two-dimensional simulation. Linewidth narrowing as a function of outcoupling time, with energy spread measured in the longitudinal direction (log-log plot). Dashed line shows the semiclassical result; solid line shows the result of the stochastic simulation. Horizontal line indicates the fundamental linewidth limit according to Eq. (9). Parameters: $N=5\times {10}^6$, $\omega =2000\mathrm{rad}{\mathrm{s}}^{-1}$, $a=1\times {10}^{-8}\mathrm{m}$, and $k_0=2\times {10}^7{\mathrm{m}}^{-1}$.

Figure 8

(Color online) Lower bound on linewidth as a function of condensate number for squeezed and nonsqueezed condensates. Solid lines represent stochastic simulations; dashed lines represent the prediction according to Eqs. (8, 36). The upper pair of lines assume a nonsqueezed condensate; the lower pair represent a condensate with quadrature squeezing of magnitude $r=0.69$.