Superconducting quantum circuit to simulate the dynamical Casimir effect in a double cavity

In this work we study simulations of photon generation due to the dynamical Casimir effect (DCE) in a one-dimensional (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mn>1</a:mn><a:mo>+</a:mo><a:mn>1</a:mn></a:mrow></a:math>) double superconducting cavity. To achieve this, we propose the use of a superconducting quantum circuit so as to be able to experimentally perform quantum simulations of the DCE. The simulated cavity consists of two perfectly conducting mirrors and a dielectric membrane of infinitesimal depth that effectively couples two cavities. The total length of the double cavity <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>L</b:mi></b:math>, the difference in length between the two cavities <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi mathvariant="normal">Δ</c:mi><c:mi>L</c:mi></c:mrow></c:math>, and the electric susceptibility <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>χ</e:mi></e:math> and conductivity <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>v</f:mi></f:math> of the dielectric membrane are tunable parameters. All four parameters are treated as independent and are allowed to be tuned at the same time, even with different frequencies. We analyze the cavity's energy spectra under different conditions, finding a transition between two distinct regimes that is accurately described by <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:msub><g:mi>k</g:mi><g:mi>c</g:mi></g:msub><g:mo>=</g:mo><g:msqrt><g:mrow><g:mi>v</g:mi><g:mo>/</g:mo><g:mi>χ</g:mi></g:mrow></g:msqrt></g:mrow></g:math>. In particular, a lowest-energy mode is forbidden in one of the regimes while it is allowed in the other. We compare analytical approximations obtained through the multiple-scale analysis method with exact numerical solutions, obtaining the typical results when <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>χ</h:mi></h:math> is not being tuned. However, when the susceptibility <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mi>χ</i:mi></i:math> is tuned, different behaviors (such as oscillations in the number of photons of a cavity prepared in a vacuum state) can arise if the frequencies and amplitudes of all parameters are adequate. These oscillations might be considered as adiabatic shortcuts where all generated photons are eventually destroyed.