Strong and tunable couplings in flux-mediated optomechanics

We investigate a superconducting interference device (SQUID) with two asymmetric Josephson junctions coupled to a mechanical resonator embedded in the loop of the SQUID. We quantize this system in the case when the frequency of the mechanical resonator is much lower than the cavity frequency of the SQUID and in the case when they are comparable. In the first case, the radiation pressure and the cross-Kerr type interactions arise and are modified by the asymmetry. The cross-Kerr type coupling is the leading term at the extremum points where the radiation pressure is zero. In the second case, the main interaction is the single-photon beam splitter, which exists only at a finite asymmetry. Another interaction in this regime is of cross-Kerr type, which exists at all asymmetries, but is generally much weaker than the beam splitter interaction. Increasing magnetic field can substantially enhance the optomechanical couplings strength with a potential for the radiation pressure coupling to reach the single-photon strong coupling regime, even the ultrastrong coupling regime, in which the single-photon coupling rate exceeds the mechanical frequency.

Figure 1

(a) A schematic overview of the SQUID, which contains two Josephson junctions with different critical currents. The mechanical resonator is embedded into the SQUID loop. The magnetic field $B$ is applied under certain angle to the loop, such the displacement $X$ of the mechanical resonator results in a nonzero modulation of the magnetic flux through the loop. (b) The mechanical resonator couples inductively to the SQUID cavity via the total flux $\mathrm{\Phi }$. Here and below the color indicates the symmetry; 50% means ${\alpha }_I=0.5$. (c) The critical current and (d) cavity frequency are plotted as a function of renormalized bias flux of the symmetric and asymmetric SQUID. The cavity frequency of the symmetric SQUID is cut at realistic value of 2.5 GHz.

Figure 2

Light-matter couplings of symmetric and asymmetric SQUID for magnetic field $B=10$ mT: (a) radiation pressure, (b) cross-Kerr coupling. The maximum of radiation pressure for ${\alpha }_I=0.5$ is $g_{RP}^1=77$ kHz. The flux bias is shifted by $\pi BA/{\mathrm{\Phi }}_0=2\pi 72534$. We see that the radiation pressure coupling diverges at ${\alpha }_I=0$ and ${\varphi }_b=\pi /2$; this divergence is in reality smeared as discussed in the text.

Figure 3

The maximum value of the (a) radiation pressure and (b) cross-Kerr coupling as a function of the magnetic field at ${\alpha }_I=0.5$. The flux bias is fixed and corresponds to the sweet spot (maximum) of each coupling, respectively. The field dependence is linear in (a) and quadratic in (b), and thus both couplings can be significantly enhanced by magnetic field. For our chosen parameters, at 1 T the radiation pressure coupling becomes comparable with the mechanical frequency.

Figure 4

Interaction couplings for symmetric and asymmetric SQUID in the case of ${\omega }_c\left(0\right)\sim {\omega }_m^{\prime }$ and magnetic field $B=10$ mT: (a) cross-Kerr coupling, (b) single-photon beam splitter coupling. The flux bias is shifted by $12\pi BA/{\mathrm{\Phi }}_0=2\pi ·18$. Similarly to the dispersive case, the cross-Kerr coupling diverges close to ${\varphi }_b=\pi /2$ for a symmetric SQUID cavity. This divergence is cutoff in the same manner as we discussed for the dispersive regime. In contrast, the single-photon beam splitter coupling is finite and vanishes for a symmetric SQUID cavity.

Figure 5

The maximum of (a) the cross-Kerr coupling and (b) the single-photon beam splitter coupling at ${\alpha }_I=0.5$ as a function of the magnetic field corresponding to Fig. 4. The flux bias is fixed to the sweet spot of each coupling. Similarly to the dispersive case, the beam splitter coupling is linear in magnetic field, whereas the cross-Ker coupling is quadratic.

Figure 6

Potential energy of the SQUID cavity as a function of the phase and flux at ${\alpha }_I=0$ (left) and ${\alpha }_I=0.5$ (right). The minimum is shown in black and maximum in yellow. The corresponding cross sections of energy map for the different values of the flux are displayed on the bottom. See the text for the discussion.