Twin paradox with macroscopic clocks in superconducting circuits

We propose an implementation of a twin-paradox scenario in superconducting circuits, with velocities as large as a few percent of the speed of light. Ultrafast modulation of the boundary conditions for the electromagnetic field in a microwave cavity simulates a clock moving at relativistic speeds. Since our cavity has a finite length, the setup allows us to investigate the role of clock size as well as interesting quantum effects on time dilation. In particular, our theoretical results show that the time dilation increases for larger cavity lengths and is shifted due to quantum particle creation.

Figure 1

Cavity trajectories. Minkowski diagram showing the cavity trajectories in the laboratory frame. Alice's cavity (green [light gray]) stays static. The mirrors of Rob's cavity move along trajectories composed of segments of constant proper acceleration (red [gray]) and inertial motion (blue [dark gray]). The dashed trajectories are those of Alice and Rob themselves and the black line is a line of constant Rindler time. Both cavities have the same proper length $L$.

Figure 2

Experimental setup. Top: The flux-tuned SQUIDs generate time-dependent boundary conditions for the cavity field, equivalent to Dirichlet boundary conditions at different effective positions. The external fluxes ${\Phi }_{\pm }\left(t\right)$ correspond to effectively moving the boundary conditions the distances $d_{\pm }\left(t\right)$. Bottom: Sketch of the circuit setup. The signal from a coherent microwave source is used to represent Alice's clock and to fill Rob's cavity with photons. When Rob's cavity has been filled a set of travels is performed by flux tuning the two SQUIDs, using the external magnetic fluxes ${\Phi }_{-}\left(t\right)$ and ${\Phi }_{+}\left(t\right)$. After the trips, the field in the cavity leaks out and is down-converted by a mixer using Alice's clock. A phase difference between the two clocks is then detected as a dc change at the output of the mixer.

Figure 3

Time dilation. Relative phase shift between Rob's and Alice's cavities in an experimentally feasible regime. The parameter values used are $t_a=1$ ns, $t_i=0$, and $L=1.1$ cm, leading to an effective cavity displacement of $1.7$ mm and a maximal velocity of $0.014c$. With $t_t=4$ ns and the trajectory being repeated 500 times, the total travel time is $2\mu s$. Left inset: Difference between the time dilation shown by the cavity clock and a pointlike clock as a function of $L$, normalized to the total time dilation between Alice and Rob. The blue (red) [dark gray (light gray)] curve is excluding (including) the effects of mode mixing and particle creation. Right inset: Difference in time dilation between the cases with and without particle creation, again normalized to the total time dilation. The parameter values used in the inset plots are $t_a=1$ ns, $t_i=0$, and $a=1.7\times {10}^{15}\text{m}/{\text{s}}^2.$