Rotating and spiraling spatial dissipative solitons of light and cold atoms

Clouds of cold neutral atoms driven by a coherent light beam in a ring cavity exhibit self-structured states transversely with respect to the beam axis due to optomechanical forces and the backaction of the atomic structures on the beam. Below the instability threshold for extended hexagonal structures, localized solitonlike excitations can be stable. These constitute peaks or holes of atom density, depending on the linear susceptibility of the cloud. Complex rotating and spiraling motion of coupled atom-light solitons, and, hence, atomic transport, can be achieved via phase gradients in the input field profile. We also discuss the stability of rotating soliton chains in view of soliton-soliton interactions. The investigations are performed in a cavity scheme but expected to apply to other longitudinally pumped schemes with diffractive coupling.

Figure 1

Optomechanical single-mode ring cavity in a bow tie configuration. The intracavity field (dotted arrow) $\mathcal{E}(\mathbf{r},t^{\prime })$ is driven by a phase structured beam of amplitude ${\mathcal{A}}_{\mathrm{in}}\left(\mathbf{r}\right)$, where $\mathbf{r}=(x,y)$ is the transverse coordinate. We assume one imperfect mirror with transmittivity $\tau$ and unidirectional propagation (e.g. by adding a Faraday rotator). An ensemble of overdamped laser-cooled two-level atoms of temperature $T$ and optical density at resonance $b_0$ is placed within the cavity, coupled to the optical field via a Smoluchowski equation for the atom density $n(\mathbf{r},t^{\prime })$. CSs arise from the bistability of patterned and homogeneous states below threshold [2, 38].

Figure 2

Rotation of bright density optomechanical CSs at different radii for $\mathrm{\Delta }<0, \kappa t_{\max }=300$, and OAM index $l=1$. (The rotation is counterclockwise for $l>0$.) The white dots track the peak position at different times. (a) A localized peak of both the cavity field and the atom density, initially defined at a position ${\mathbf{r}}_0=(0,y_0)$ with $y_0>y_d/4$, covers approximately a quarter of its orbit. (b) For an inner radial initial position $y_0<y_d/4$, the CS achieves a faster rotation speed [see Eq. (9)], completing a cycle within $\kappa t_{\max }$. Model parameters chosen as follows: $I/I_0\approx 0.668, \theta =5.1, \mathcal{C}\mathrm{\Delta }=-2.25$, and $\sigma =25$.

Figure 3

Spiraling trajectories of optomechanical CSs for $\alpha \ne 0$ and $\kappa t_{\max }=750$. (a) Atom-density peak undergoing inward spiraling motion for $\alpha =1.2$. (b) Total input phase $\alpha \psi \left(r\right)+\varphi$ (OAM index $l=1$), including gradient field, representing the soliton drift velocity. (c) Outward spiraling trajectory for $\alpha =-1.2$. (d) Corresponding phase and velocity field. Model parameters: $I/I_0\approx 0.672, \theta =5.1, \mathcal{C}\mathrm{\Delta }=-2.25$, and $\sigma =25$.

Figure 4

Average displacement $\langle \eta \rangle$ across the inversion symmetry point $\langle \eta \rangle =0$ of the atom density $n_{\mathrm{eq}}\left(\mathbf{r}\right)$ from the homogeneous value $n_{\mathrm{eq}}=1$ for $\theta =-1$ (triangle down), $\theta =-0.5$ (triangle up), $\theta =0$ (square), and $\theta =0.5$ (circle) at fixed $\sigma =50$. Data are measured the steady-state atom density according to Eq. (11) by spanning values of $\mathcal{C}\mathrm{\Delta }$ for a chosen value of $\theta$. The error bars reflect local variations of the value $\langle \eta \rangle$ across the domain $\mathrm{\Omega }$.

Figure 5

Dark light optomechanical CSs for blue light-atom detuning with plane-wave pumping. (a) and (b) Soliton spatial profiles with model parameters $I/I_0=0.968, \theta =-0.85, \mathcal{C}\mathrm{\Delta }=0.2$, and $\sigma =100$. (c) Oscillating soliton behavior obtained by tracking maximum density peak over time for pump values: $I/I_0=0.964$ blue (lower) line, $I/I_0=0.966$ yellow (middle) line, and $I/I_0=0.964$ green (upper) line. Each curve is vertically shifted for ease of understanding. The profiles in (a) and (b) are plotted in correspondence of an atom density peak maximum at $\kappa t\approx 40$ for the corresponding green curve.

Figure 6

Counterclockwise rotation of a dark-light optomechanical CS and its trajectory measured within a period of $\kappa t_{\max }={10}^3$ and higher OAM index $l=5$. Parameters chosen as follows: $I/I_0\approx 0.495, \theta =-0.43, \mathcal{C}\mathrm{\Delta }=0.16$, and $\sigma =100$. An additional radial trap prevents atoms from accumulating in the dark regions of optical intensity.

Figure 7

A 3-CS chain rotational dynamics with OAM index $l=1$. Rotating CSs initially excited in a bound state evolve into a triangle. Snapshots of atom density $n_{\mathrm{eq}}\left(\mathbf{r}\right)$ at (a) $\kappa t={10}^3$, (b) $\kappa t=3\times {10}^3$, (c) $\kappa t=3.75\times {10}^3$, (d) $\kappa t=4.25\times {10}^3$ for $\mathcal{C}\mathrm{\Delta }=-2.5, \sigma =25,I/I_0\approx 0.66,\mathrm{and}\theta =5.1$. (e) Stability of the rotating 3-soliton chain together with corresponding single CSs profiles for different values of the susceptibility $\mathcal{C}\mathrm{\Delta }$. Model parameters chosen as above except for the pump intensity $I$ which is slightly adjusted when varying $\mathcal{C}\mathrm{\Delta }$. This result suggests that the stability of CS chains increases with decreasing susceptibility $\left|\mathcal{C}\mathrm{\Delta }\right|$. The error bars measure the observed duration of the chain collapse in units of $\kappa t$.