Generation, manipulation, and detection of snake-state trajectories of a neutral atom in a ring cavity

We propose a setup to create and detect the atomic counterpart of snake-state trajectories which occur at the interface where the magnetic field reverses direction. Such a synthetic magnetic field is generated by coupling two counterpropagating modes of a ring cavity to a two-level atom. The spatial distribution and the strength of the induced magnetic field are controlled by the transverse- mode profile of the cavity modes and the number of photons in the two modes, respectively. By analyzing the atomic motion in such a magnetic field while including the cavity back-action, we find that the atom follows snake-state trajectories which can be nondestructively detected and reconstructed from the phase and the intensity of the light field leaking from the cavity. We finally show that the system parameters can be tuned to modify the transport properties of the snake states and even amplify the effect of cavity feedback, which can completely alter their topology.

Figure 1

(a) Schematic of a single two-level atom trapped inside a ring cavity with two counterpropagating running-wave modes which are orthogonally polarized and are described by the photon annihilation operators ${\widehat{a}}_1$ and ${\widehat{a}}_2$. The two modes are respectively pumped with strengths ${\eta }_1$ and ${\eta }_2$. $\kappa$ is the cavity decay rate. For (b)–(d) ${\eta }_1\approx 2\pi \times 52$ MHz, and ${\eta }_2=0$, which gives time-averaged photon numbers in the two cavity modes of $\langle n_1\rangle \approx 342$ and $\langle n_2\rangle \approx 8$. (b) Vector potential $A$ (blue solid curve) and magnetic field $B$ (orange dash-dotted curve) as a function of $y$. $B_0$ and $l_B$ are, respectively, the natural magnetic field and magnetic length scales of the system, and $w_0$ is the waist size of the cavity modes (see text). (c) The snake-state trajectory of the atom in the $x\text{−}y$ plane, with ${\lambda }_p$ being the pump wavelength. The black arrows indicate the direction of increasing time. (d) The $x$ position $x_a$ (blue dashed curve) and $y$ position $y_a$ (red solid curve) of the atom as a function of time $t$. The dotted curves in (c) and (d) show the atomic trajectory, $x$ position, and $y$ position for fixed photon numbers $\langle n_1\rangle$ and $\langle n_2\rangle$ in the two cavity modes and thus exclude cavity back-action.

Figure 2

(a) Fourier transform for $y_a/w_0$, which shows a peak at ${\nu }_0\approx 2$ Hz. (b) Photon number $n_2$ (red dotted curve) and the corresponding phase ${\varphi }_{2,\mathrm{ss}}$ (blue solid curve) in cavity mode 2 as a function of time $t$. (c) Fourier transform of $n_2$, which shows peaks at $2{\nu }_0, 4{\nu }_0, 6{\nu }_0$, and so on. (d) Fourier transform of $e^{i{\varphi }_{2,\mathrm{ss}}}$, which overlaps with the Fourier transform of $e^{2ik{x}_a}$.

Figure 3

Average drift velocity $\langle v_x\rangle$ (blue solid curve) and peak-to-peak $y_a$ variation $y_a^{\mathrm{pp}}$ (red dotted curve) as a function of (a) the initial velocity $v_{y0}$ (with ${\eta }_1=80\kappa$ and ${\eta }_2=0$) and (b) the pump strength ${\eta }_1$ (with $v_{y0}=0.06v_0$ and ${\eta }_2=0$). Here, $v_0$ is a natural velocity scale of the system (see text), $w_0$ is the cavity-mode waist, and the initial speed along the $x$ direction is fixed to zero. The normalized maximum photon number in cavity mode 2 $n_2^{\max }/\langle n_2\rangle$ (red dotted curve) and the average time derivative of the corresponding phase $\langle d{\varphi }_{2,\mathrm{ss}}/dt\rangle$ (blue solid curve) are shown in (c) as a function of $v_{y0}$ and in (d) as a function of ${\eta }_1$. The vertical dash-dotted green lines mark the boundary where the snake-state trajectory ceases to exist, and the particle is not trapped along the $y$ direction.

Figure 4

Signal-to-noise ratio (SNR) as a function of (a) the initial velocity $v_{y0}$ (with ${\eta }_1=80\kappa$ and ${\eta }_2=0$) and (b) the pump strength ${\eta }_1$ (with $v_{y0}=0.06v_0$ and ${\eta }_2=0$). The blue solid curve shows the numerically obtained SNR. The yellow dash-dotted curve represents the analytical approximation of the SNR for $y_a^{\mathrm{pp}}\gg w_0$, and the magenta dotted curve represents the analytical approximation of the SNR for $y_a^{\mathrm{pp}}\ll w_0$, as discussed in Sec. 5. The green dash-dotted vertical lines represent the boundaries where the snake-state-trajectory picture breaks down.

Figure 5

(a) Average photon number variation in cavity modes 1 (magenta curve with dashes) and 2 (red curve with pluses) as a function of $g_0$. The shaded region denotes the corresponding peak-to-peak amplitude around the average photon number. The vertical gray solid lines correspond to $g_0=(0.55,1.1,1.6)g_{0c}$ and represent the parameters used for plotting (c) and (f), (d) and (g), and (e) and (h), respectively. (b) Average drift velocity $\langle v_x\rangle$ (blue solid curve) is plotted as a function of the coupling strength $g_0$, which is normalized by $g_{0c}=2\pi \times 90$ MHz (see text). The dotted blue curve shows the average drift speed in the absence of cavity feedback. The green vertical shaded region in (a) and (b) marks the region where the snake-state trajectory is destroyed due to dynamical cavity feedback. (c)–(e) The evolution of the photon numbers in cavity mode 1, $n_1$ (magenta solid curves), and mode 2, $n_2$ (red dotted curves), with time $t$ and (f)–(h) the corresponding particle trajectories (solid curves) in the $x\text{−}y$ plane for $g_0=(0.55,1.1,1.6)g_{0c}$, respectively. The dotted trajectories in (f)–(h) depict the particle trajectories without cavity back-action, and the black arrows indicate the direction of increasing time in the time evolution. For all the calculations here, ${\eta }_1=80\kappa , {\eta }_2=0, v_{x0}=0$, and $v_{y0}=0.06v_0$ are considered, which give $g_0=0.29g_{0c}$ as the minimum coupling parameter required to get a trapped atomic trajectory.

Figure 6

Comparison of scalar potential $W$ (black dash-dotted curve) and vector potential $A$ (green solid curve).

Figure 7

The solid blue line shows the snake-state trajectory, and the maroon dash-dotted line shows the reconstructed snake-state trajectory using parameters of the output cavity field ${\alpha }_2$. Refer to the text for details.

Figure 8

(a) Photon numbers in cavity mode 1 $n_1$ as a function of time $t$. (b) Fourier transform of $n_1$, which shows a peak at $2{\nu }_0$. (c) Phase ${\varphi }_1$ of photons in mode 1 as a function of time $t$. (d) Fourier transform of $e^{i{\varphi }_1}$, which also shows a peak at $2{\nu }_0$.

Figure 9

The reconstructed snake-state trajectory including the effect of the Doppler shift.

Figure 10

Full phase ${\varphi }_2$ (purple solid curve) of cavity mode 2 as a function of time $t$, showing the contributions of ${\varphi }_{2,\mathrm{ss}}$ (blue dash-dotted curve) and ${\varphi }_{2,ds}$ (magenta dotted curve), including the effect of the Doppler shift.

Figure 11

(a) Oscillation frequency ${\nu }_0$ (blue solid curve) of atomic position $y_a$ along the $y$ direction plotted as a function of initial velocity $v_{y0}$ along the $y$ direction for fixed $v_{x0}=0$. The green dash-dotted curve shows the analytical frequency obtained from the $y_{\mathrm{period}}$ formula in Eq. (9). We consider four points, $\left(v_1\right), \left(v_2\right), \left(v_3\right)$, and $\left(v_4\right)$, of the initial velocity and show the corresponding trajectories and photon numbers. The $v_{y0}$ velocity at point $\left(v_2\right)$ was considered in the main text, and therefore, its trajectories are not shown here. Snake-state trajectories for $v_{y0}=v_1, v_3$, and $v_4$ are shown in (b), (d), and (f), respectively. Correspondingly, the photon number (purple solid curve) in mode 2 (left $y$ axis) and $y_a/w_0$ (red dotted curve) variation (right $y$ axis) as a function of time are shown in (c), (e), and (g). For parameters and explanation, refer to the text.

Figure 12

Phase diagram for the atomic trajectories (blue shaded regions); the particle follows cyclotron orbits (orange shaded region), the particle follows snake-state trajectories, and the white region indicates the regime where the atom is not trapped by the synthetic magnetic field. $y_{a0}$ and $v_{y0}$ are the initial position and speed of the atom along the $y$ direction, respectively. $w_0$ is the waist of the cavity mode, and $v_0$ is a natural scale of the particle speed (see text). The red arrows indicate the direction of particle evolution with increasing time.

Figure 13

For $g_0=1.1g_{0c}$, we show the trajectories of the atom with and without feedback ($\langle n_1\rangle =946$ and $\langle n_2\rangle =894$). The initial velocity $v_{x0}=0$, and $v_{y0}=0.06v_0$.