Semiclassical limits to the linewidth of an atom laser

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.

Figure 1

Two matter waves with wave vectors $k$ and $k+\delta k$ are incident on a step potential of height $V_0$.

Figure 2

An atom laser based on (a) rf outcoupling and (b) Raman outcoupling. In both cases trapped atoms in state ∣1⟩ are transferred to an untrapped state ∣2⟩ via electromagnetic fields. The ${\Omega }_{ij}$ represent the Rabi frequencies of the applied fields and the ${\Delta }_{ij}$ represent detuning from resonance.

Figure 3

(Color online) Naive description of the energy spread of outcoupled atoms due to a “balls on a hill” model. The atomic transition has a resonant width $\delta$ such that the coupling happens over a region $\Delta z$, corresponding to a change in potential of $\hslash \delta$. This leads to a spread in energies of $\hslash \delta$ for the outcoupled atoms. This model neglects the wavelike nature of the atoms and the fact that there is interference between the different energies.

Figure 4

(Color online) $F(\Delta \omega ,t)$ for $\gamma =1\mathrm{Hz}$. As $t\to \infty$, $F\left(\Delta \omega \right)$ asymptotically approaches a Lorentzian of FWHM $\gamma$.

Figure 5

${\mid A_n\left(k\right)\mid }^2$ for $n=0$ (top), $n=3$ (middle), and $n=8$ (bottom). $k_n=\sqrt{(\delta +{\omega }_n)2m∕\hslash }-k_0$, the place in the $k$-space wave function from where resonant outcoupling occurs, is indicated by a vertical black bar in each case. For $k_0={10}^5{\mathrm{m}}^{-1}$, $k_n$ follows the largest “lobe” of $A_n\left(k\right)$ for increasingly excited states. For $k_0={10}^8{\mathrm{m}}^{-1}$, $k_n$ remains approximately in the center of $A_n\left(k\right)$. $\delta =\hslash k_0^2∕2m-{\omega }_{t0}$ was chosen such that the outcoupling was perfectly on resonance for the zero momentum component of the ground state.

Figure 6

The relative intensities of the momentum components of the atom laser beam corresponding to the first 20 condensate eigenmodes for (a) $k_0={10}^5{\mathrm{m}}^{-1}$, (b) $k_0={10}^6{\mathrm{m}}^{-1}$, and (c) $k_0={10}^8{\mathrm{m}}^{-1}$. For small $k_0$ $(k_0={10}^5{\mathrm{m}}^{-1})$, the relative intensity of each mode decreases due to the spreading out of $A_n\left(k\right)$. For large $k_0$ $(k_0={10}^8{\mathrm{m}}^{-1})$, only even modes are present in the output. This is due to $k_n$ falling at nodes of $A_n\left(k\right)$ for the odd modes. In the intermediate case, there is complicated structure, with some odd modes and some even modes being attenuated.

Figure 7

(Color online) Momentum space density of an atom laser beam after (a) $20\mathrm{ms}$, (b) $100\mathrm{ms}$, (c) $400\mathrm{ms}$, and (d) $1000\mathrm{ms}$ of outcoupling. After $1000\mathrm{ms}$ 65% of the atoms in the condensate have been outcoupled. Parameters: $N_0={10}^6$, $\omega =150\mathrm{rad}{\mathrm{s}}^{-1}$, $a=4\times {10}^{-11}\mathrm{m}$, and $k_0=3.2\times {10}^6{\mathrm{m}}^{-1}$.

Figure 8

(Color online) Linewidth narrowing as a function of outcoupling time for a highly nonlinear atom laser with very weak outcoupling (log-log plot). Parameters: $N={10}^7$, ${\omega }_t=250\mathrm{rad}{\mathrm{s}}^{-1}$, $a=3\times {10}^{-9}\mathrm{m}$, and $k_0={10}^7{\mathrm{m}}^{-1}$.

Figure 9

(Color online) Comparison of linewidth over time for a condensate with significant nonlinearities that becomes depleted (log-linear plot). Solid line shows standard outcoupling; dashed line shows the effect of our chirp correction scheme. Parameters: $N_0={10}^6$, ${\omega }_t=150\mathrm{rad}{\mathrm{s}}^{-1}$, $a=4\times {10}^{-11}\mathrm{m}$, and $k_0=3.2\times {10}^6{\mathrm{m}}^{-1}$.