Simulating moving cavities in superconducting circuits

We theoretically investigate the simulation of moving cavities in a superconducting circuit setup. In particular, we consider a recently proposed experimental scenario where the phase of the cavity field is used as a moving clock. By computing the error made when simulating the cavity trajectory with superconducting quantum interference devices (SQUIDs), we identify parameter regimes where the correspondence holds, and where time dilation and corrections due to clock size and to particle creation coefficients are observable. These findings may serve as a guideline when performing experiments on simulation of moving cavities in superconducting circuits.

Figure 1

Phase shift of a single trajectory of the type considered in Fig. 3 in [22] (see the Appendix), computed using the full Bogoliubov coefficients. The inset shows the absolute error $\epsilon$ made when computing the same phase shift using second-order Bogoliubov coefficients.

Figure 2

Extra effective length $\delta L_{\text{eff}}\left(\mathrm{\Phi }\right)$ at the end of a coplanar waveguide terminated by a dc SQUID, as a function of the external magnetic flux $\mathrm{\Phi }$ through the SQUID. The physical parameters of the SQUID and the waveguide are chosen as in [23], $L_0=0.44\mu \mathrm{F}/\mathrm{m}$ and $I_c=0.5\mu \mathrm{A}$.

Figure 3

The first two Robin mode functions (23) in a cavity of length $L_{\text{cav}}$ and Dirichlet mode functions (2) in a cavity of length $L=L_{\text{cav}}+d_l+d_r$. The solid blue (black) curve shows $u_1(0,x)$ in the Dirichlet (Robin) case and the dashed blue (black) curve shows $u_2(0,x)$ in the Dirichlet (Robin) case. In the bottom left figure, parameter values from [22] are used, corresponding to $d_l/L_{\text{cav}}=0.31$ and $d_r/L_{\text{cav}}=0.096$. Top left: $d_l/L_{\text{cav}}=2\times 0.31$ and $d_r/L_{\text{cav}}=2\times 0.096$. Top right: $d_l/L_{\text{cav}}=1.5\times 0.31$ and $d_r/L_{\text{cav}}=1.5\times 0.096$. Bottom right: $d_l/L_{\text{cav}}=0.5\times 0.31$ and $d_r/L_{\text{cav}}=0.5\times 0.096$.

Figure 4

Phase shift of a single cavity trajectory with $L=12.2\mathrm{cm}$, $t_a=1$ ns, and ${\delta }_{\text{min}}=0.0075\mathrm{mm}$. The blue (black) solid line shows the Dirichlet (Robin) phase shift $\mathrm{\Delta }{\theta }_D$ ($\mathrm{\Delta }{\theta }_R$) and the red dashed line shows the time dilation for a pointlike clock, scaled by the Dirichlet cavity mode frequency. The inset shows the relative error of the simulation, $\epsilon =(\mathrm{\Delta }{\theta }_D-\mathrm{\Delta }{\theta }_R)/\mathrm{\Delta }{\theta }_D$.

Figure 5

Phase shift of a single cavity trajectory with $L=9.5\mathrm{cm}$, $t_a=0.1$ ns, and ${\delta }_{\text{min}}=0.0075\mathrm{mm}$. The solid blue curve shows the full phase shift for a Dirichlet cavity, and the dashed blue curve shows the phase shift excluding the nonadiabatic effects. The red curve shows the result for a pointlike clock, scaled by the Dirichlet cavity mode frequency.

Figure 6

Phase shift of a single cavity trajectory with $L=2.38\mathrm{cm}$, $t_a=0.1$ ns, and ${\delta }_{\text{min}}=0.0075\mathrm{mm}$. The gray curves show the Robin phase shift when the trajectory has been approximated by using different numbers of Fourier harmonics.

Figure 7

Difference in the phase shift between the full trajectory transformation and a single-mode approximation where the nonadiabatic effects have been neglected. For the plot we considered 200 trajectories and the trajectory parameters are $t_a=0.1$ ns and ${\delta }_{\text{min}}=0.0075\mathrm{mm}$. The length at which the Bogoliubov coefficients are resonant is $L_{\text{res}}=2.38\mathrm{cm}$. For the plot, we keep the parameter $h$ fixed to the value $h=0.85\times {10}^{-2}$.

Figure 8

Difference in the phase shift between the cases with and without particle creation coefficients, after 500 trips. The trajectory parameters are $t_a=0.1$ ns and $a=2\times {10}^{16}\text{m}/{\text{s}}^2$ (corresponding to $h=0.34\times {10}^{-2}$ at resonance), and the resonance cavity length is $L_{\text{res}}=2.38\mathrm{cm}$.