Optomechanics with cavity vacuum fluctuations: Self-alignment and collective rotation mediated by Casimir torque

We theoretically consider an ensemble of quantum dimers placed inside an optical cavity. We predict two effects. First, an exchange of angular momentum between the dimers mediated by the emission and reabsorption of the cavity photons leads to the alignment of dimers. Second, the optical angular momentum of the vacuum state of the chiral cavity is transferred to the ensemble of dimers, which leads to the synchronous rotation of the dimers at certain levels of light-matter coupling strength.

Figure 1

The geometry of the system. An ensemble of dimers is placed inside a chiral cavity. The orientation of each dimer is defined by the angle ${\phi }_i$. The dimers may exchange angular momentum via emission and reabsorption of cavity photons. Due to the breaking of time-reversal symmetry, cavity modes of opposite helicity have different energies, which leads to nonvanishing optical angular momentum in the ground state.

Figure 2

Dependence of the energy on the total angular momentum on magnetic field $B$ (left panel) and light-matter coupling strength $g$ (right panel). $g$ was set to 0.5 (left panel) and $B$ was set to 1.5 (right panel). The moment of inertia, $J$, was set to 1 for all cases

Figure 3

(a) Dependence of the optical angular momentum $L$ of a single dimer in the ground state on the magnetic field $B$ and coupling strength $g$. In the calculation, $J=1\times {10}^6$. (b) Dependence of the uncertainty of the angular momentum on the moment of inertia, $J$. The dashed horizontal line corresponds to the result obtained within the Born-Oppenheimer approximation (BO). In the calculation, $g=0.1,B=0.1$. The analytical result was computed with Eq. (5).

Figure 4

The dispersion of the difference of two angles, $\theta ={\phi }_1-{\phi }_2$, as a function of coupling strength $g$. Here it is assumed that $\mathrm{\Delta }=1,J=1\times {10}^7,B=0.3$. The subplots show mechanical wave functions for the two values of $g$ shown with red crosses on the main plot. On the subplots, solid blue and dashed orange lines show the confining potential obtained via numerical solution of ${\widehat{H}}_{\mathrm{Dicke}}$ within the RPA approximation, respectively, the red dashed line shows the position of the ground-state energy, and the solid black lines depict the profile of the wave function.

Figure 5

The angular momentum as a function of the field $B$. We supposed here that the incoming photon is resonant with the ground-state to excited-state transition of qubits, i.e., $\mathrm{\Delta }=1$, and fixed the coupling strength $g=0.5$.

Figure 6

Comparison of the analytically calculated angular momentum for the $N=1$ qubit as a function of coupling strength $g$ and numerically calculated for the cases (a) $B=0.01$ and (b) $B=0.1$, both for $\mathrm{\Delta }=1$. As can be seen, the angular momentum is proportional to $B$, even for big $g$. The biggest difference between the analytical and numerical results occurs in the vicinity of the maximal angular momentum.

Figure 7

The left part (II) region, i.e., the splitting levels due to the $\mathrm{\Delta }$ part are small compared to the excitations of the $l$ mode. The right part is the resonance regime, i.e., they are of the same order.

Figure 8

(a) The dependence between field $B$ and coupling strength $g$ assuming the incoming photons are resonant with the qubits, i.e., $\mathrm{\Delta }=1$. Red lines correspond to the analytical resonance relationships for $N=1$ and $N=2$, respectively, and blue lines correspond to the numerical solution. (b) The maximal possible angular momentum for the resonant photons, i.e., $\mathrm{\Delta }=1$, as a dependence on field $B$ for the $N=1$ system. The blue line corresponds to numerical solution, the black line is the simplest analytical model (C13), and the red lines are the solution of (C15) taking into account the interaction between the first $n-1$ excitations.

Figure 9

The numerical calculated angular momentum $L$ for (a) $N=1$ and (b) $N=2$ qubits. It is assumed that $\mathrm{\Delta }=1$.