Circuit analog of quadratic optomechanics

We propose a superconducting electrical circuit that simulates a quadratic optomechanical system. A capacitor placed between two transmission-line (TL) resonators acts like a semitransparent membrane, and a superconducting quantum interference device (SQUID) that terminates a TL resonator behaves like a movable mirror. Combining these circuit elements, it is possible to simulate a quadratic optomechanical coupling whose coupling strength is determined by the coupling capacitance and the tunable bias flux through the SQUIDs. Estimates using realistic parameters suggest that an improvement in the coupling strength could be realized, to five orders of magnitude from what has been observed in membrane-in-the-middle cavity optomechanical systems. This leads to the possibility of achieving the strong-coupling regime of quadratic optomechanics.

Figure 1

Schematic design of an analog circuit for quadratic optomechanics. Resonator A, described by the annihilation operator ${\widehat{a}}_n$ and the mode frequency ${\omega }_n$, consists of two capacitively coupled SQUID-terminated TL resonators. Resonator B, described by the annihilation operator ${\widehat{b}}_m$ and the mode frequency ${\Omega }_m$, is a TL resonator. Resonator A and resonator B provide optical and pseudomechanical degrees of freedom, respectively. The current distribution of resonator B is chosen to be antisymmetric to ensure opposite flux variations $\pm \delta {\widehat{\Phi }}_{\mathrm{ext}}$ through the SQUIDs forming resonator A.

Figure 2

Two capacitively coupled TL resonators expressed as a lumped-element circuit. The TL resonators on the left and the right are labeled with $\alpha =\mathrm{L}$ and $\mathrm{R}$, respectively. Each TL resonator can be modeled as an infinite number of LC circuits, each with node capacitance $c_k^{\alpha }\Delta x$ and node inductance ${\ell }_k^{\alpha }\Delta x$ $\left(1\le k\le N_{\alpha }\right)$. In the continuum limit, the discrete node flux ${\Phi }_k^{\alpha }\left(t\right)$, node capacitance per unit length $c_k^{\alpha }$, and node inductance per unit length ${\ell }_k^{\alpha }$ converge to continuous functions inside each TL resonator ${\Phi }^{\alpha }(x,t),c^{\alpha }\left(x\right)$, and ${\ell }^{\alpha }\left(x\right)$, respectively. In the middle, there is a capacitor $C_{\mathrm{c}}$ which couples the two TL resonators. If both TL resonators are uniform and homogeneous, this capacitor can be thought of as a partially transparent membrane (shown in blue) giving rise to a linear transformation between the wave amplitudes of different regions.

Figure 3

Normal-mode frequencies of the first six modes $(n=0$ to 5, bottom to top), calculated from Eq. (15a). The four panels show the mode frequencies, as a function of the location of the capacitor, for decreasing coupling strengths: (a) ${\omega }_{\mathrm{c}}={10}^{-1}{v}_0/d$, (b) ${\omega }_{\mathrm{c}}=v_0/d$, (c) ${\omega }_{\mathrm{c}}=10v_0/d$, (d) ${\omega }_{\mathrm{c}}={10}^2{v}_0/d$. The dotted lines are the normal-mode frequencies for the completely decoupled case $(C_{\mathrm{c}}=0$ or ${\omega }_{\mathrm{c}}\to \infty )$. The regions inside the dashed boxes are zoomed in Fig. 4.

Figure 4

Enlargement (with the same scale) of the dashed boxes in Figs. 3 and 3. Here, (a), (c), (e) and (b), (d), (f) correspond to the dashed boxes in Figs. 3 and 3, from top to bottom, respectively. The dashed lines are plotted with Eq. (16) up to third order in $\xi$, and the vertical markers ❘ on each dashed line show the 99% validity range of this approximation, obtained from Eq. (17).

Figure 5

Normal-mode functions of two capacitively coupled TL resonators as a function of the position $y$ inside the resonator for characteristic frequencies of the capacitive coupling: (a) ${\omega }_{\mathrm{c}}={10}^{-1}{v}_0/d$, (b) ${\omega }_{\mathrm{c}}=10v_0/d$, (c) ${\omega }_{\mathrm{c}}={10}^3{v}_0/d$, and displacements of the capacitor: (i) $\xi =0$, (ii) $\xi =0.1d$, (iii) $\xi =-0.3d$. The capacitive coupling is decreased from (a) to (c), and the asymmetry is increased from (i) to (iii). In each panel, the four curves represent the first four normal modes $\left(n=0,1,2,3\right)$ of the system from bottom to top. For clarity, the vertical axes are displaced for different modes. The coordinate describing the position in the resonator is shifted with $y=x+d_{\mathrm{L}}$ in such a way that $y=0$ and $d$ correspond to both ends of the resonator, i.e., $x=-d_{\mathrm{L}}$ and $d_{\mathrm{R}}$, respectively. The position of the capacitor $y=d/2+\xi$ is marked with vertical dashed lines.

Figure 6

SQUID-terminated TL resonator expressed as a lumped-element circuit. The SQUID consists of two Josephson junctions on a loop through which an external flux ${\Phi }_{\mathrm{ext}}$ is applied. Each junction in the SQUID has the capacitance and the Josephson energy $C_{\mathrm{J}\beta }$ and $E_{\mathrm{J}\beta }$ $\left(\beta =1,2\right)$, respectively. The flux across each junction is denoted as ${\Phi }_{\mathrm{J}1}$ and ${\Phi }_{\mathrm{J}2}$. One side of the SQUID is grounded, and the opposite side is connected to a TL resonator with characteristic capacitance per unit length $c\left(x\right)$ and inductance per unit length $\ell \left(x\right)$ in the continuum limit. $\Phi (x,t)$ is the flux of the TL resonator at position $x$ and time $t$.

Figure 7

Normal-mode frequencies of the SQUID-terminated resonator as a function of the external flux ${\Phi }_{\mathrm{ext}}$, for (a) $L_{\mathrm{J}0}={\ell }_{0}d$, (b) $L_{\mathrm{J}0}={10}^{-1}{\ell }_{0}d$, (c) $L_{\mathrm{J}0}={10}^{-2}{\ell }_{0}d$ and (i) $C_{\mathrm{J}}=c_{0}d$, (ii) $C_{\mathrm{J}}={10}^{-1}{c}_{0}d$, (iii) $C_{\mathrm{J}}={10}^{-2}{c}_{0}d$. The three lines of each panel correspond to the first three modes $\left(n=1,2,3\right)$, from bottom to top. The numerical values of the normal-mode frequencies obtained from Eq. (26) (solid line), and the analytical result from Eq. (28) (dashed line) are compared. The three markers ★ in the [(c), (ii)] panel, which are the fundamental-mode frequencies for ${\Phi }_{\mathrm{ext}}/{\Phi }_0=\{0.32,0.38,0.44\}$, correspond to the mode functions depicted in Fig. 8.

Figure 8

The $n=1$ mode functions of the SQUID-terminated resonator system, under the condition of $L_{\mathrm{J}0}={10}^{-2}{\ell }_{0}d,C_{\mathrm{J}}={10}^{-1}{c}_{0}d$, and (a) ${\Phi }_{\mathrm{ext}}=0.32{\Phi }_0$, (b) ${\Phi }_{\mathrm{ext}}=0.38{\Phi }_0$, (c) ${\Phi }_{\mathrm{ext}}=0.44{\Phi }_0$ {marked with ★ in Fig. 7[(c), (ii)]}. The real mode functions $(x>0$, blue line) are distinguished from the virtual mode functions $(x<0$, red line) by the real end $(x=0$, dashed line) of the SQUID-terminated resonator. The effective length $\Delta d$ obtained from Eq. (27a) is marked with vertical dotted lines. The panels on the left correspond to the shaded areas of the panels on the right. It is seen that the $x$ intercept of each panel is in good agreement with $\Delta d$.

Figure 9

Schematic diagram of the analogy of (a) a semitransparent membrane and (b) a movable mirror in electrical circuits.

Figure 10

Combining the principles shown in Fig. 9, an analog circuit [(a), up] of the system of [(a), down] can be designed to give the quadratic coupling of optomechanics. The motion of the virtual ends of the tunable resonators is synchronized so as to maintain the total effective length of the resonator unchanged. Therefore, in the comoving frame with the effective cavity, (b) is equivalent to a cavity consisting of fixed mirrors with a semitransparent membrane moving inside [(a), down].

Figure 11

Detailed layout of Fig. 1, composed of resonator A (two capacitively coupled tunable resonators) and resonator B (a TL resonator). The SQUID loops that belong to tunable resonators are denoted as loop L and loop R. The total length, the characteristic capacitance, and inductance per unit length of the resonator $\alpha =\mathrm{A},\mathrm{B}$ are given by $d_{\alpha },c_{\alpha }$, and ${\ell }_{\alpha }$, respectively. All Josephson junctions are equal with junction capacitance $C_{\mathrm{J}}/2$ and Josephson energy $E_{\text{J}0}$. The three coordinate axes $(x,z$, and $s)$ specify positions in the system.

Figure 12

The normalized coupling strength $|g_{nm}|/{\Omega }_m$ as a function of the bias flux, for $m=2$ and different $n$ modes $(n=0$ to 9, from bottom to top). The parameters $d_{\mathrm{A}}=d_{\mathrm{B}}/20=20\mathrm{mm},A/d_{\mathrm{B}}{s}_1={10}^{-3},{\ell }_{\mathrm{A}}={\ell }_{\mathrm{B}}=4.57\times {10}^{-7}\mathrm{H}{\mathrm{m}}^{-1},c_{\mathrm{A}}=c_{\mathrm{B}}=1.46\times {10}^{-10}\mathrm{F}{\mathrm{m}}^{-1},C_c=1\mathrm{fF},C_{\mathrm{J}}=30\mathrm{fF}$, and $E_{\mathrm{J}0}=6.17\times {10}^{-22}\mathrm{J}$ were used to evaluate Eq. (63). The two modes labeling a single line $(n=8,9$ and the uppermost line, for instance) are in fact different but too close to be distinguishable in the plot. The three markers + at ${\Phi }_{\mathrm{ext}}^0/{\Phi }_0=0.4$ correspond to those in Fig. 13.

Figure 13

The normalized coupling strength $|g_{nm}|/{\Omega }_m$ as a function of the coupling capacitance $C_{\mathrm{c}}$ of resonator A, for $m=2$ and different $n$ modes $(n=0$ to 5, from bottom to top). All parameters used are the same as in Fig. 12, except for the fixed bias flux ${\Phi }_{\mathrm{ext}}^0/{\Phi }_0=0.4$ and varying capacitance $C_{\mathrm{c}}$. The three markers + at $C_{\mathrm{c}}=1\mathrm{fF}$ correspond to those in Fig. 12. The upper horizontal axis is the reflectivity of $n=1$ mode obtained from $C_{\mathrm{c}}$.

Figure 14

Maximal amplitudes $X_{nm*}$ of the position quadrature defined in Eq. (64), as a function of bias flux ${\Phi }_{\mathrm{ext}}^0$, for $m=2$ and different values of $n$. All the parameters used are the same as in Fig. 14. Each curve corresponds to a discrete $n$ mode. Note that the two modes labeling a single curve $(n=0,1$ and the uppermost curve, for instance) are in fact different but so close to each other as to look degenerate when the reflectivity is high. The upper-right region (yellow) and the lower-left region (green) correspond to the higher-order-coupling regime and the quadratic-coupling regime, respectively.