Spontaneous Crystallization of Light and Ultracold Atoms

Coherent scattering of light from ultracold atoms involves an exchange of energy and momentum introducing a wealth of nonlinear dynamical phenomena. As a prominent example, particles can spontaneously form stationary periodic configurations that simultaneously maximize the light scattering and minimize the atomic potential energy in the emerging optical lattice. Such self-ordering effects resulting in periodic lattices via bimodal symmetry breaking have been experimentally observed with cold gases and Bose-Einstein condensates (BECs) inside an optical resonator. Here, we study a new regime of periodic pattern formation for an atomic BEC in free space, driven by far off-resonant counterpropagating and noninterfering lasers of orthogonal polarization. In contrast to previous works, no spatial light modes are preselected by any boundary conditions and the transition from homogeneous to periodic order amounts to a crystallization of both light and ultracold atoms breaking a continuous translational symmetry. In the crystallized state the BEC acquires a phase similar to a supersolid with an emergent intrinsic length scale whereas the light field forms an optical lattice allowing phononic excitations via collective backscattering, which are gapped due to the infinte-range interactions. The system we study constitutes a novel configuration allowing the simulation of synthetic solid-state systems with ultracold atoms including long-range phonon dynamics.

Popular Summary

It has been known since the time of Albert Einstein that photons can carry momentum and induce mechanical forces on atoms. These forces enable optical trapping and cooling, which is an essential tool required for quantum simulations and computing with (ultra)cold atoms in lattices. Here, we introduce a new regime of optical lattice physics in one dimension. At sufficiently high densities and low temperatures, mechanical forces induce a phase transition from homogeneous to periodic spatial order, whereby the lattice appears as a self-adjusted crystalline structure of both photons and atoms.

We consider ultracold atoms forming a Bose-Einstein condensate illuminated by two counterpropagating lasers that do not interfere. We study the conditions for the emergence of such a self-consistent crystalline phase and its dynamical properties. Because particles can shape the light into a standing-wave pattern in which they trap themselves, the dynamics are much more complex and richer than those of standard optical lattices. We find that intrinsic interactions at all length scales generate acoustic phonons. Our setup constitutes a well-controlled quantum system for simulating crystallization phenomena, as well as solid-state Hamiltonians with long-range interactions, with built-in precise monitoring provided by the scattered light.

We anticipate that our findings will motivate new schemes for ultraprecise accelerometers and rotation sensors.

Figure 1

Schematic representation of the considered setup. An elongated BEC interacting with two counterpropagating, noninterfering laser beams of orthogonal polarization. The two beams are far detuned from any atomic resonance in order to avoid mixing between the two polarizations. Both polarizations are assumed to be equivalent with respect to the considered atomic transition, the latter thus involving a spherically (or at least cylindrically) symmetric ground state. Alternatively to the use of two different polarizations, sufficiently different frequencies of the two counterpropagating lasers can be chosen.

Figure 2

Excitation spectrum Eqs. (8)] in the homogeneous phase for different field intensities: $I^{L,R}=2.0$ (red), $I^{L,R}=20.0$ (green), and $I^{L,R}=60.0$ (blue) ($\zeta =0.1$, $L=100{\lambda }_0$, $g_{c}N/A{\lambda }_0=E_{\mathrm{rec}}$).

Figure 3

$\zeta$ dependence of the critical intensity. The solid blue line depicts the analytical result defined by Eq. (9), whereas the red dots depict numerical threshold estimations for large system sizes ($L=120{\lambda }_0$) ($\zeta =0.1$, $g_{c}N/A{\lambda }_0=E_{\mathrm{rec}}$).

Figure 4

Dependence of the reflection coefficient of the BEC on the incoming field amplitudes for different atom-field couplings $\zeta =0.1$ (dashed red line) and $\zeta =0.2$ (solid blue line). The remaining parameters are the same as in Fig. 2.

Figure 5

(a) Crystal ground state for $\zeta =0.1$, $I_l=I_r=200$ and (b) corresponding intensity distribution for the field from left (green line) and right (red line). The solid blue line depicts the sum of both intensities. A zoom into the yellow shaded region can be found in Fig. 8. The remaining parameters are $g_{c}N/A{\lambda }_0=E_{\mathrm{rec}}$ and $L=10{\lambda }_0$.

Figure 6

Excitation spectrum of the atom-light crystal. The blue points are the eigenvalues of the GP and Helmholtz equations linearized about the crystalline stationary state (see Appendix pp1). The numerical diagonalization is performed with a momentum-space discretization $dq=2\pi /L$. The parameters are the same as in Fig. 2 except for $L=50$ and a fixed drive intensity $I^{L,R}=50$ (slightly above threshold). $q_{\max }$ is the momentum corresponding to the largest component of the eigenvector of each eigenvalue. The insets show examples of eigenvectors (unormalized probability in momentum space) for three different eigenvalues representative of each region of the spectrum, from left to right: ${\lambda }_0{q}_{\max }=1.38<{\lambda }_0{k}_{\mathrm{eff}}$, ${\lambda }_0{k}_{\mathrm{eff}}<{\lambda }_0{q}_{\max }=11.9<2{\lambda }_0{k}_{\mathrm{eff}}$, and ${\lambda }_0{q}_{\max }=12.7>2{\lambda }_0{k}_{\mathrm{eff}}$. The latter region corresponds to lattice phonons, characterized by two symmetric pairs of peaks about a finite momentum. This phononic branch $q_{\max }>2k_{\mathrm{eff}}$ has a gap ${\mathrm{\Delta }}_{\mathrm{ph}}$. Its analytical estimate in Eq. (11) yields ${\mathrm{\Delta }}_{\mathrm{ph}}\simeq 2\sqrt{2}{E}_{\mathrm{rec}}$, in reasonable agreement with the numerical data.

Figure 7

(a) Real-time evolution of the kinetic energy for $\zeta =0.1$, $I_l=I_r=100$, $g_{c}N=1$. (b) Real-time evolution of the reflection coefficient for the same parameters as in (a). The solid black line shows the mean value of the corresponding functions.

Figure 8

Zoom into the yellow shaded region of Fig. 5. The blue dots mark the maxima of the field from left (green line) and right (red line), whereas the black dot marks the maximum of the total field intensity (blue line). The red dot shows the actual position of the particles.

Figure 9

Real-time dynamics of the (a) BEC density distribution and (b) the total light intensity for the same parameters as in Fig. 5. The solid black lines in (b) show the time evolution of the intensity maxima.

Figure 10

Real-time evolution of the maxima of the intensity distribution as it is shown in Fig. 9. To simplify the comparison between the single curves, the maxima positions are shifted so that they all start at $x=0$. Panel (a) shows the total time evolution where one can clearly recognize collective phononlike excitations of the lattice after ${\omega }_{\mathrm{rec}}t\gtrsim 5$. Panel (b) shows the zoom into the yellow marked area in (a) in order to demonstrate the slight dephasing between the oscillations of the maxima. All parameters are chosen as in Fig. 5.

Figure 11

(a) Crystal ground state and (b) corresponding intensity distribution for the field from left (green line) and right (red line) for the same parameters as in Fig. 5 with an additional external potential $V_{\mathrm{ext}}(x)=(E_{\mathrm{trap}}/2)x^2/{\lambda }_0^2$ with $E_{\mathrm{trap}}=1.0E_{\mathrm{rec}}$ and for different pump intensities from left and right $I_l=200$ and $I_r=150$. The solid blue line depicts the sum of both intensities.