Triangular and honeycomb lattices of cold atoms in optical cavities

We consider a two-dimensional homogeneous ensemble of cold bosonic atoms loaded inside two optical cavities and pumped by a far-detuned external laser field. We examine the conditions for these atoms to self-organize into triangular and honeycomb lattices as a result of superradiance. By collectively scattering the pump photons, the atoms feed the initially empty cavity modes. As a result, the superposition of the pump and cavity fields creates a space-periodic light-shift external potential and atoms self-organize into the potential wells of this optical lattice. Depending on the phase of the cavity fields with respect to the pump laser, these minima can either form a triangular or a hexagonal lattice. By numerically solving the dynamical equations of the coupled atom-cavity system, we have shown that the two stable atomic structures at long times are the triangular lattice and the honeycomb lattice with equally populated sites. We have also studied how to drive atoms from one lattice structure to another by dynamically changing the phase of the cavity fields with respect to the pump laser.

Figure 1

Schematic drawing of the setup that can be used to create 2D triangular and honeycomb lattices of bosonic cold atoms. A cloud of atoms (blue disk) is loaded inside two initially empty optical cavities symmetrically oriented by the angle $\theta =\pi /3$ about axis $Oy$. The atoms are then pumped by a retro-reflected classical laser field ($Oy$ direction) and release photons into the modes of the cavities. In return, the coherent superposition of the pump and cavity fields creates an effective periodic potential and atoms self-organize themselves into the corresponding potential wells. A triangular or honeycomb lattice structure of atoms thus emerges as a result of superradiance processes.

Figure 2

(Left panel) Triangular Bravais lattice associated with the effective potential $V_{\text{eff}}(\vec{\rho },\tau )$, Eq. (9), and its Bravais vectors ${\vec{a}}_1$ and ${\vec{a}}_2$, Eq. (12). (Right panel) The corresponding triangular reciprocal lattice spanned by vectors ${\widehat{\mathbf{b}}}_1$ and ${\widehat{\mathbf{b}}}_2$, Eq. (4). The initial zero-momentum state of the atomic cloud (black disk at the center) is directly coupled to 18 points of the reciprocal lattice (shown as numbered colorful circles) by Schrödinger's equation, Eq. (5). Circles with the same color and number refer to a given coupling term in Eq. (5): purple circles 1 and blue circles 2 correspond to pump-cavity coupling terms ${\eta }_1$ and ${\eta }_2$ respectively; red circles 3 correspond to intercavity coupling term $U_{12}$; green circles 4 and brown circles 5 correspond to intracavity coupling terms $U_1$ and $U_2$ respectively; orange circles 6 correspond to the pump coupling term $U_p$.

Figure 3

Effective potential $V_{\text{eff}}(\vec{\rho },\tau )$ given by Eq. (9) and its lattice structure of minima (in units of the common wavelength $\lambda$ associated to the cavity and pump fields) for three values of the cavity phase $\varphi$ appearing in the mode functions $F_j\left(\vec{r}\right)$ (see text). (Right panels) For $\varphi =\pi /2$ a honeycomb structure is obtained for the potential minima which have all same depths. (Middle panels) For lower values, e.g., $\varphi =\pi /4$, a honeycomb structure is obtained but with energy-imbalanced minima. (Left panels) For $\varphi =0$, a triangular lattice is obtained for the potential minima. The dimensionless coupling parameters are $U_p=-4,U_j=U_{12}=-0.02,{\eta }_j=-\sqrt{2}/5$ [see Eq. (6)] and ${\alpha }_j\left(\tau \right)=\langle a_i\left(\tau \right)\rangle =15 \left(j=1,2\right)$. With these values, the pump and cavity fields are almost balanced since $U_p/U_j=200$ and $n_j={\left|{\alpha }_j\right|}^2=225$; see text after Eq. (11).

Figure 4

Imaginary (top panels) and real parts (bottom panels) of the eigenvalues $\omega$ of stability matrix $\mathcal{M}$ as functions of the (negative) dimensionless pump coupling parameter $U_p$ (red-detuned pump). We have used 37 reciprocal lattice vectors to compute and diagonalize $\mathcal{M}$; see discussion end of Sec. 3. Positive imaginary parts signal instability of our initial state (empty cavities, homogeneous atomic cloud). The dimensionless cavity parameters are $U_j=U_{12}=-1.74\times {10}^{-2},{\mathrm{\Delta }}_j=-4.7\times {10}^3$, and ${\kappa }_j=480 \left(j=1,2\right)$. The total number of atoms is $N=1.5\times {10}^5$.

Figure 5

As a result of superradiance, an initial homogeneous atomic cloud loaded inside empty identical cavities self-organizes into a triangular lattice when the cavity phase is set to $\varphi =0$. The pump dimensionless coupling strength is $U_p=-3$ and all other parameters are the same as in Fig. 4. Atoms scatter pump photons into the cavity modes and, after a short while, superradiance and self-organization take place. (Top left panel) Cavity photon numbers $n_j={\left|{\alpha }_j\right|}^2 \left(j=1,2\right)$ as functions of time $\tau ={\omega }_{R}t$. The insets show the logarithms of $n_1$ and $n_2$ as a function of time and the horizontal grid lines mark the values 1, 10, and 100. From a mathematical point of view, the symmetry of the dynamical equations and of the initial state enforces $n_1=n_2$ at all times as confirmed here by the numerical simulation. (Top right panel) Atomic density distribution at time $\tau =2$, before superradiance takes place. The atomic cloud self-organizes according to the minima of the pump standing-wave potential alone since the cavity fields are almost zero. [Bottom left $\left(\tau =25\right)$ and bottom right $\left(\tau =130\right)$ panels] Atomic density distributions, and their corresponding color codes, after superradiance and self-organization have taken place. The atomic lattice, just like the lattice of minima of the effective potential, is clearly triangular.

Figure 6

Same as Fig. 5 but with a cavity phase set to $\varphi =\pi /4$. (Left panels) Cavity photon numbers $n_j={\left|{\alpha }_j\right|}^2 \left(j=1,2\right)$ as functions of time $\tau ={\omega }_{R}t$. [Middle $\left(\tau =25\right)$ and right $\left(\tau =130\right)$ panels] After superradiance and self-organization have taken place, an atomic honeycomb lattice with density-imbalanced consecutive sites is formed.

Figure 7

Same as Fig. 5 but with a cavity phase set to $\varphi =\pi /2$. (Left panels) Cavity photon numbers $n_j={\left|{\alpha }_j\right|}^2 \left(j=1,2\right)$ as functions of time $\tau ={\omega }_{R}t$. [Middle $\left(\tau =25\right)$ and right $\left(\tau =125\right)$ panels] After superradiance and self-organization have taken place, an atomic honeycomb lattice with density-balanced consecutive sites is formed.

Figure 8

Long-time dynamics of the system shown in Fig. 5 $\left(\varphi =0\right)$. (Left panel) Cavity photon numbers $n_j={\left|{\alpha }_j\right|}^2 \left(j=1,2\right)$ as functions of time $\tau ={\omega }_{R}t$. The cavity steady states are reached around a time $\tau =2000$. (Middle panel) Contour plot of the (stable) atomic density distribution obtained at time $\tau =3000$. (Right panel) Effective lattice potential $V_{\text{eff}}$ at $\tau =3000$.

Figure 9

Long-time dynamics of the system shown in Fig. 6 $\left(\varphi =\pi /4\right)$. (Left panel) Cavity photon numbers $n_j={\left|{\alpha }_j\right|}^2 \left(j=1,2\right)$ as functions of time $\tau ={\omega }_{R}t$. The cavity steady states are reached around a time $\tau =4000$. (Middle panel) Contour plot of the atomic density distribution obtained at time $\tau =1500$. (Right panel) $V_{\text{eff}}$ at $\tau =1500$. As one can see, the density-imbalanced atomic honeycomb lattice obtained at short times in Fig. 6 is not a stable structure at long times. The atoms reorganize in the deepest potential wells of the effective potential shown in the right panel and form a stable triangular lattice.

Figure 10

Dimensionless latency time $T$ against the cavity phase $\varphi$. After $T$, atoms have reorganized into the deepest wells of the energy-imbalanced effective potential obtained for $\varphi \in ]0,\pi /2[$ and exhibit a triangular lattice structure. This time $T$ increases when the energy mismatch between the principal and secondary potential minima decreases, i.e., when $\varphi$ increases. It diverges when $\varphi \to \pi /2$. The atomic honeycomb lattice obtained at $\varphi =\pi /2$ is a stable structure.

Figure 11

Long-time dynamics of the system described in Fig. 7 $\left(\varphi =\pi /2\right)$. (Left panel) Cavity photon numbers $n_j={\left|{\alpha }_j\right|}^2 \left(j=1,2\right)$ as functions of time $\tau ={\omega }_{R}t$. As one can see, the cavity photon numbers fluctuate a lot and, in turn, so does the atomic honeycomb lattice. At the longest time $\tau ={10}^4$ of our numerical simulation, the system has not yet fully reached its steady state. (Middle panel) Corresponding contour plot of the atomic density. (Right panel) Effective lattice potential $V_{\text{eff}}$ obtained at this longest time. The atomic honeycomb lattice is a stable structure but it takes a much longer time to reach the steady state.

Figure 12

Transition from the density-imbalanced honeycomb atomic lattice to the triangular atomic lattice. (Top left panel) Pump and cavity phase temporal sequences and time evolution of the cavity photon numbers. The cavity phase changes linearly from ${\varphi }_i=\pi /4$ to ${\varphi }_f=0$ over the time interval $\mathrm{\Delta }\tau =200$. Meanwhile, the pump strength $U_p$ decreases from 0 to $U_f=-20$ over the time window $3\mathrm{\Delta }\tau =600$. The other parameters are the same as in Fig. 4. Contour plots show the atomic density obtained at time $\tau ={\omega }_{R}t=200$, before the cavity phase starts to decrease (top right panel), at time $\tau =260$, just before the triangular lattice become the dominant structure (bottom left panel), and at time $\tau =400$, when the cavity phase has reached its target value ${\varphi }_f=0$ (bottom right panel). The triangular lattice becomes the prominent structure around $\tau \approx 280$, well before the cavity phase reaches the target value 0. At this time, the cavity photon numbers rise considerably and then increase linearly in time like the pump strength for the rest of the time sequence.

Figure 13

Transition from the triangular atomic lattice to the density-balanced atomic honeycomb lattice. (Top left panel) Pump and cavity phase temporal sequences and time evolution of the cavity photon numbers. The cavity phase changes linearly from ${\varphi }_i=0$ (triangular lattice) to ${\varphi }_f=\pi /2$ (energy-balanced honeycomb lattice) over the time interval $\mathrm{\Delta }\tau =200$. Meanwhile, the pump strength $U_p$ decreases from 0 to $U_f=-20$ over the time window $3\mathrm{\Delta }\tau =600$. The other parameters are the same as in Fig. 4. When the cavity phase departs from 0, secondary minima start to grow and the effective potential becomes an energy-imbalanced honeycomb potential. However, the atomic density keeps its triangular lattice structure where atoms are sitting at the deeper minima of the effective potential. When the cavity phase reaches $\pi /2$ then the atomic density starts oscillating in time between the two triangular sublattices of the honeycomb lattice which now have the same depth. (Top right panel) Shows the atomic density at time $\tau =400$ when phase reaches $\pi /2$. (Bottom left panel) Atomic density distribution at time $\tau =402$. The initial triangular lattice is still dominant. (Bottom right panel) Atomic density distribution at time $\tau =404$. The atomic distribution now lives mainly on the other triangular sublattice of the honeycomb lattice.

Figure 14

Transition from the density-balanced atomic honeycomb lattice to the triangular atomic lattice. (Top left panel) Pump and cavity phase temporal sequences and time evolution of the cavity photon numbers. The cavity phase changes linearly from ${\varphi }_i=\pi /2$ to ${\varphi }_f=0$ over the time interval $\mathrm{\Delta }\tau =200$. Meanwhile, the pump strength $U_p$ decreases from 0 to $U_f=-20$ over the time window $3\mathrm{\Delta }\tau =600$. The other parameters are the same as in Fig. 4. (Top right panel) Atomic density distribution at time $\tau =100$, forming the expected density-balanced honeycomb lattice. As the cavity phase departs from $\pi /2$, consecutive potential wells start to have different depths. As long as the potential mismatch is weak enough, the density-balanced honeycomb lattice resists the phase change but, with increasing potential mismatch, finally gives in and becomes a fading density-imbalanced honeycomb lattice around $\tau =250$ (bottom left panel). When the cavity phase becomes even smaller, before it reaches $\varphi =0$ at $\tau =300$, the atoms reorganize into the stable triangular structure. (Bottom right panel) Atomic density distribution at time $\tau =600$.

Figure 15

Transition from the density-imbalanced to the density-balanced atomic honeycomb lattice. (Top left panel) Pump and cavity phase temporal sequences and time evolution of the cavity photon numbers. The cavity phase increases linearly from ${\varphi }_i=\pi /4$ to ${\varphi }_f=\pi /2$ over the time interval $\mathrm{\Delta }\tau =200$. Meanwhile, the pump strength $U_p$ decreases from 0 to $U_f=-20$ over the time window $3\mathrm{\Delta }\tau =600$. The other parameters are the same as in Fig. 4. (Top right panel) Atomic density at time $\tau =200$ where the cavity phase starts to increase. Later on, as the cavity phase increases, the shallower wells become deeper, however, the atoms in deeper wells do not move to the shallower wells until phase becomes exactly $\pi /2$ and all sites have same depth. (Bottom left panel) Atomic density at time $\tau =400$ when the cavity phase reaches $\pi /2$ one gets an almost density-balanced atomic honeycomb lattice. From this time on, and for the rest of the time sequence, the atomic density starts oscillating fast in time between the two stable triangular sublattices. (Bottom right panel) Hinting at this oscillating behavior, the atomic density at time $\tau =401$.

Figure 16

Transition from the density-balanced to the density-imbalanced atomic honeycomb lattice. (Top left panel) Pump and cavity phase temporal sequences and time evolution of the cavity photon numbers. The cavity phase decreases linearly from ${\varphi }_i=\pi /2$ to ${\varphi }_f=\pi /4$ over the time interval $\mathrm{\Delta }\tau =200$. Meanwhile, the pump strength $U_p$ decreases from 0 to $U_f=-30$ over the time window $4.5\mathrm{\Delta }\tau =900$ (same rate as previous cases). The other parameters are the same as in Fig. 4. (Top right panel) Atomic density at time $\tau =100$ when the cavity phase is $\pi /2$ and one gets a density-balanced atomic honeycomb lattice. When the phase departs from $\pi /2$, similar to what happens in Fig. 14, the initial density-balanced honeycomb lattice resists destabilization, even though the lattice potential is imbalanced, but finally gives in around $\tau \approx 600$, well after the cavity phase has reached its final value $\varphi =\pi /4$. (Bottom left panel) Atomic distribution at $\tau =650$. (Bottom right panel) Atomic distribution at $\tau =900$. The atomic lattice transits to the triangular lattice at long times through a density-imbalanced honeycomb lattice.