Quantum theory of the cold-atom micromaser including gravity

The quantum theory of the cold-atom micromaser including the effects of gravity is considered. We show that gravity does not break the special properties of the induced emission probability for the micromaser in the cold atom regime and rather new effects are predicted. In particular, we show that the cavity acts in the gravity field as an additional repulsive and attractive potential, resulting in quasibound states of the atomic motion. This feature gives rise to fine resonances in the induced emission probability that are not restricted to any particular cavity mode function, in contrast to the usual cold-atom micromaser. It is also shown that the atom is able to emit a photon inside the cavity, though classically it does not reach the interaction region. Predictions about the photon number statistics when the cavity is pumped by a flux of excited atoms are finally given. Unusual highly nonclassical “dragon” distributions are still predicted in the vertical geometry.

Figure 1

(Color online) Scheme of the horizontal micromaser.

Figure 2

(Color online) Scheme of the vertical micromaser.

Figure 3

Schematic representation of the energy $E$ (solid line) of the atoms launched upwards upon a micromaser cavity of length $L$. The cavity containing $n$ photons acts in the gravity linear field $m\mathcal{G}z$ as an additional repulsive potential (dashed) and an attractive one (dotted). The picture illustrates the case where the atom-field coupling is contant inside the cavity. Tilted symbols are dimensionless variables [see Eq. (9)].

Figure 4

Induced emission probability ${\mathcal{P}}_{\mathrm{em}}$ as a function of the cavity length with (a) and without (b) quantization of the atomic motion (hot atom regime: ${\tilde{h}}_E=1000$ and ${\tilde{h}}_{\mathrm{int}}=1$). Tilted variables are dimensionless [see Eq. (9)].

Figure 5

Induced emission probability ${\mathcal{P}}_{\mathrm{em}}$ with respect to the particle energy for $\tilde{L}=10$ and ${\tilde{h}}_{\mathrm{int}}=10$.

Figure 6

Induced emission probability ${\mathcal{P}}_{\mathrm{em}}$ with respect to the cavity length for ${\tilde{h}}_E=-1$ and ${\tilde{h}}_{\mathrm{int}}=10$.

Figure 7

$V_n^{-}\left(z\right)$ quasibound state energy levels for $\tilde{L}=10$ and ${\tilde{h}}_{\mathrm{int}}=10$.

Figure 8

Stationary probability density ${\mid {\varphi }_{E,n}^{-}\left(z\right)\mid }^2$ (in a.u.) for $E$ matching a quasibound state energy of the potential $V_n^{-}\left(z\right)$ ($\tilde{L}=10$ and ${\tilde{h}}_{\mathrm{int}}=10$).

Figure 9

Stationary probability density ${\mid {\varphi }_{E,n}^{-}\left(z\right)\mid }^2$ (in a.u.) for $E$ different than the quasibound state energies of the potential $V_n^{-}\left(z\right)$ ($\tilde{L}=10$ and ${\tilde{h}}_{\mathrm{int}}=10$).

Figure 10

Induced emission probability ${\mathcal{P}}_{\mathrm{em}}$ with respect to the particle energy for $\tilde{L}=10$ and ${\tilde{h}}_{\mathrm{int}}=10$.

Figure 11

$V_n^{+}\left(z\right)$ quasibound state energy levels for $\tilde{L}=10$ and ${\tilde{h}}_{\mathrm{int}}=10$.

Figure 12

Induced emission probability ${\mathcal{P}}_{\mathrm{em}}$ with respect to the cavity length for ${\tilde{h}}_E=8.5$ and ${\tilde{h}}_{\mathrm{int}}=1000$.

Figure 13

Induced emission probability ${\mathcal{P}}_{\mathrm{em}}$ with respect to the particle energy for $\tilde{L}=10$ and ${\tilde{h}}_{\mathrm{int}}=10$, sine mode.

Figure 14

Induced emission probability ${\mathcal{P}}_{\mathrm{em}}$ with respect to the particle energy for $\tilde{L}=10$ and ${\tilde{h}}_{\mathrm{int}}=10$, Gaussian mode.

Figure 15

$V_n^{-}\left(z\right)$ quasibound state energy levels for $\tilde{L}=10$ and ${\tilde{h}}_{\mathrm{int}}=10$, sine mode.

Figure 16

Induced emission probability ${\mathcal{P}}_{\mathrm{em}}$ with respect to the particle energy for $\tilde{L}=10$ and ${\tilde{h}}_{\mathrm{int}}=10$, mesa mode [the dashed line represents the classical limit given by Eq. (58)].

Figure 17

Steady-state photon number distribution $\mathcal{P}\left(n\right)$ ($r∕C={10}^4$, $n_b=20$, ${\tilde{h}}_g=\tilde{L}=100$, and ${\tilde{h}}_E=-2$, mesa mode).