Solitons in arbitrary dimensions stabilized by photon-mediated interactions

Abstract

We propose a scheme to generate solitons in arbitrary dimensions, in a matter-wave interferometer, without the need of quantum degeneracy. In our setting, solitons emerge by balancing the single-particle dispersion with engineered cavity-mediated exchange interactions between two wave packets, which, at the appropriate conditions, remain bound to each other and dispersion-free. For detection in thermal gases, we propose an interferometric probing scheme instead of traditional time-of-flight imaging.

Figures (6)

• Figure 1: (a) Schematic of the interferometry protocol in a cavity. (b) Cavity-mediated interactions between atoms in the two wave packets, each with average momentum $p_0-\hbar k$ and $p_0+ \hbar k$, momentum spread $\sigma_p$, and separated by an energy difference $\omega_Z$, create an energy gap, $\chi N$, which keeps the wave packets bound. (c) At the mean-field level, for $\chi <0$, the Bloch vector of an individual atom (initially pointing on the equator, along an initial angle $\phi$, $\hat{S}_{\phi}$ (black arrow), processes around the effective magnetic field (orange arrows) $\vec{B}_{p_r}= \vec{B}_0 + 2k p_r/M \hat{Z}$, with contributions from the self-generated field and kinetic energy (purple) and with $\vec{B}_0=\chi N \hat{S}_{\phi}$. (d) At the optimal interaction strength, in a frame moving with momentum $p_0$, the modified Doppler dispersion by the exchange interactions plus the free dispersion (dashed lines) generates a net flat dispersion (black solid line), enabling the formation of a soliton.
• Figure 2: Mean-field simulation of the soliton dynamics, for a given experimental realization sampled from the Wigner function. We show the separate evolution of the density distribution of each wave packet, ignoring interference terms. After a $\pi/2$ Bragg pulse, the $p_0 - \hbar k$ (red) and $p_0 + \hbar k$ (blue) wave packets spatially propagate according to three scenarios: (a) if $\chi=0$, the wave packets spatially separate while broadening over time as free particles. (b) if $\chi = \chi_{\rm{opt}}$, the wave packets merge and maintain their shape, demonstrating soliton behavior. (c) if $\chi = -\chi_{\rm{opt}}$, the wave packets merge but show enhanced broadening compared to the free evolution case. (d) Time evolution of the rms width of the spatial profile (using a Gaussian fit), normalized by the initial width. For $\chi=\chi_{\rm opt}$, the broadening is suppressed, consistent with the formation of a soliton.
• Figure 3: Wave-packet dynamics for arbitrary polar angle $\theta$ using mean-field simulations. (a) Schematic of the wave packets in position space for initial polar angle $\theta=\pi/4$ (green), $\pi/2$ (orange) and $3\pi/4$ (pink). (b) Normalized rms width $\sigma_z^*$ vs interaction strength $\chi$ at $t_d= 2\pi \times 30 /(|\chi_{\rm opt}| N)$. The dashed lines indicate $\chi=\chi_{\rm opt}(\pi/2)$. (c) Normalized rms width $\sigma_z^*$ vs $\theta$ and $\chi$ at $t_d$. The red curve is the optimal interaction strength as given by Eq. \ref{['eq:theta']}. The cuts for $\theta=\pi/4,\pi/2,3\pi/4$ are shown in panel (b).
• Figure 4: Proposed scheme to stabilize a soliton in 2D. (a) It uses a standing-wave cavity along the $z$-direction (orange) and a retro-reflected drive field along the $x$-direction (green). (b) The initial state is prepared via a Raman pulse as a superposition between $\ket{0,0;\downarrow}$ and $\ket{\pm \hbar k,\pm \hbar k;\uparrow}$ in the 2D momentum-spin coupled basis. We plot the initial momentum and spatial distributions for the internal states $\ket{\downarrow}$ and $\ket{\uparrow}$. (c) Normalized rms width $\sigma_{x,z}^*$ for $\ket{\downarrow}$ vs $\chi$ at $t_d = 2\pi \times 100/(|\chi_{\rm opt}| N)$, with the optimal interaction strength given by $\chi_{\rm opt}=-4E_R$. (d) Simulated spatial distribution vs time for $\ket{\downarrow}$. Top: $\chi=0$, the wave packets broaden over time. Bottom: $\chi=\chi_{\rm opt}$, the wave packets retain their initial shape without broadening.
• Figure 5: Mean-field coupling diagram in 2D for atom $r$. (a) The system consists of five states in the bare basis: $\ket{\Downarrow_{\vec{p}}} \equiv \ket{\vec{p},\downarrow}_r$ and four momentum states $\ket{\Uparrow_{\vec{p},\mu_x\mu_z}}_r \equiv \ket{p_x+\mu_x \hbar k, p_z+\mu_z \hbar k,\uparrow}_r$ with $\mu_x, \mu_z = \pm$. The single-particle energy difference between them is given by $\omega_Z + (p_x \mu_x k + p_z \mu_z k)/M$. The exchange interaction (orange arrows) couples $\ket{\Downarrow_{\vec{p}}}_r$ to all four momentum states $\ket{\Uparrow_{\vec{p},\mu_x\mu_z}}_r$. (b) Since the exchange interaction only couples $\ket{\Downarrow_{\vec{p}}}$ to the symmetric superposition of the four momentum states $\ket{\Uparrow_{\vec{p}}}_r$, it is convenient to use a different basis, defined in Eq. \ref{['eq:dress']}. In this basis, the single-particle inhomogeneity is no longer diagonal and has couplings between the states, $\ket{\Uparrow_{\vec{p},A_1}}_r$ and $\ket{\Uparrow_{\vec{p},A_2}}_r$, with strengths $k p_x / M$ and $k p_z / M$, respectively.