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arXiv:1007.1058 · Physical Review A

Dynamical Casimir effect in superconducting microwave circuits

J. R. Johansson, Göran Johansson, Christopher M. Wilson, Franco Nori

Published Nov 11, 2010• 194 citations
arXivHTMLPDF

Abstract

We theoretically investigate the dynamical Casimir effect (DCE) in electrical circuits based on superconducting microfabricated waveguides with tunable boundary conditions. We propose implementing a rapid modulation of the boundary conditions by tuning the applied magnetic flux through superconducting quantum-interference devices that are embedded in the waveguide circuits. We consider two circuits: (i) An open waveguide circuit that corresponds to a single mirror in free space, and (ii) a resonator coupled to a microfabricated waveguide, which corresponds to a single-sided cavity in free space. We analyze the properties of the DCE in these two setups by calculating the generated photon-flux densities, output-field correlation functions, and the quadrature squeezing spectra. We show that these properties of the output field exhibit signatures unique to the radiation due to the DCE, and could, therefore, be used for distinguishing the DCE from other types of radiation in these circuits. We also discuss the similarities and differences between the DCE, in the resonator setup, and the down-conversion of pump photons in parametric oscillators.

DOI
10.1103/physreva.82.052509

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•Jan 1, 2012•3 citations
We theoretically investigate the dynamical Casimir effect (DCE) in electrical circuits based on superconducting microfabricated waveguides with tunable boundary conditions. We propose implementing a rapid modulation of the boundary conditions by tuning the applied magnetic flux through superconducting quantum-interference devices that are embedded in the waveguide circuits. We consider two circuits: (i) An open waveguide circuit that corresponds to a single mirror in free space, and (ii) a resonator coupled to a microfabricated waveguide, which corresponds to a single-sided cavity in free space. We analyze the properties of the DCE in these two setups by calculating the generated photon-flux densities, output-field correlation functions, and the quadrature squeezing spectra. We show that these properties of the output field exhibit signatures unique to the radiation due to the DCE, and could, therefore, be used for distinguishing the DCE from other types of radiation in these circuits. We also discuss the similarities and differences between the DCE, in the resonator setup, and the down-conversion of pump photons in parametric oscillators.
We investigate the dynamical Casimir effect in a coplanar waveguide (CPW) terminated by a superconducting quantum interference device (SQUID). Changing the magnetic flux through the SQUID parametrically modulates the boundary condition of the CPW, and thereby, its effective length. Effective boundary velocities comparable to the speed of light in the CPW result in broadband photon generation which is identical to the one calculated in the dynamical Casimir effect for a single oscillating mirror. We estimate the power of the radiation for realistic parameters and show that it is experimentally feasible to directly detect this nonclassical broadband radiation.
In this thesis I present different experiments on superconducting circuits exploring parametric interactions with external signals and the vacuum in the microwave regime. These parametric processes are the result of the periodic modulation of a property of a system which results in different interactions. The systems used in this thesis are circuits where the nonlinear inductance of a superconducting quantum interference device (SQUID) is parametrically driven.I present the first experimental observation of the dynamical Casimir effect (DCE), since it was predicted in 1970. The DCE is an interaction between the vacuum and a periodically modulated boundary condition of the electromagnetic field, here implemented by a flux-tuned SQUID. In essence the modulated boundary will parametrically amplify the vacuum fluctuations which results in the pairwise generation of photons over a broad frequency range. I have characterized the system and measured the statistical properties of the emitted radiation to show that the radiation exhibits two-mode squeezing.Next, I present measurements on a superconducting multimode resonator containing a SQUID. I show that it is possible to get parametric amplification by driving the SQUID at either twice the frequency of one of the modes or by driving it at the sum of two mode frequencies. In both cases I show that it is possible to reach quantum-limited noise performance.In the same system I also demonstrate frequency conversion which occurs when the system is pumped at the difference frequency between two modes. Microwave photons are coherently transferred from one mode to the other. I show that the coupling strength depends linearly on the applied pump amplitude.The thesis also contains a linearized theoretical model to describe and analyze the flux-pumped SQUID. The model describes an equivalent circuit element called the pumpistor, with an impedance which depends on the pump phase. I show that under specific conditions the impedance becomes real and negative allowing pump power to be injected into the circuit, providing gain.Finally I also present a demonstration of an on-chip Mach-Zehnder interferometer. This experiment uses the tunability of the SQUID to provide a controllable phase shift in one of the interferometer arms. The transmission through the device can be modulated with a maximum change of 45 dB.
In this work we study simulations of photon generation due to the dynamical Casimir effect (DCE) in a one-dimensional ( 1 + 1 ) 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 L , the difference in length between the two cavities Δ L , and the electric susceptibility χ and conductivity v 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 k c = v / χ . 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 χ is not being tuned. However, when the susceptibility χ 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.
A fundamental property of the quantum vacuum is that a time-varying boundary condition for the electro-magnetic field can generate photons from the vacuum, the so-called the dynamical Casimir effect (DCE). The photons are created pairwise and should possess the particular quantum correlations referred to as entanglement. In particular, broadband entanglement sources are useful for two reasons: i) they can be very bright, generating a large number of entangled photons, and ii) the wide frequency content allows shaping the emitted radiation in time. Here we present the observation of entanglement generated in a broadband system using the DCE. The measured sample consists of a superconducting quantum interference device (SQUID) terminating a microwave transmission line. A magnetic flux modulates the SQUID's Josephson inductance at approximately 9~GHz, resulting in a time-varying boundary condition for the quantum vacuum in the transmission line. We detect both quadratures of the microwave radiation emitted at two different frequencies separated by 0.7~GHz. Analysing the measured data and using a careful calibration of the power level emitted from the sample, we determine the purity and type of entanglement using two different methods. We estimate an entanglement rate of 5~Mebit/s (mega entangled bits per second).
We study the Bell's inequality violation by dynamical Casimir radiation with pseudospin measurement. We consider a circuit quantum electrodynamical setup where a relativistically moving mirror is simulated by variable external magnetic flux in a superconducting quantum interference device terminating a superconducting-microwave waveguide. We analytically obtain expectation values of the Bell operator optimized with respect to channel orientations in terms of the system parameters. We consider the effects of local noise in the microwave field modes, asymmetry between the field modes resulting from nonzero detuning, and signal loss. Our analysis provides ranges of the above experimental parameters for which the Bell violation can be observed. We show that the Bell violation can be observed in this setup up to 40mK40\phantom{\rule{0.16em}{0ex}}\text{mK}40mK temperature as well as up to 65% signal loss.
In November 2011 the first experimental observation of the dynamical Casimir effect was published [1]. Superconducting circuits were used to show that a single mirror moving in vacuum could generate photons, by parametric amplification of vacuum uctuations. This project is a side track of this research where a second mirror is introduced to investigate photon generation in-between the mirrors. The work includes both theory studies and experimental practice. Resonators have been fabricated on chip in a cleanroom and measured in a dilution refrigerator. The focus has been on how the different boundary conditions of the resonator field affect the photon generation. A superconducting circuit has been used including a = 2/2 resonator with two SQUIDs inducing tunability of the resonance frequency via on-chip ux tuning lines. The two SQUIDs are placed in each end of the cavity in order to simulate moving mirrors. The mirrors are used to modulate the boundary conditions of the cavity. This is done by changing the SQUID inductance which is achieved by applying a magnetic ux to the SQUID loop. Depending on if the mirrors move in or out of phase, a breathing or translational mode can be created in the cavity. Theory predicts photon generation in different modes depending on if the pump frequency is two or three times the resonance frequency [2]. Measurements have been done at a temperature of around 50mK, which is low enough to have a negligible amount of thermal photons and quasi-particles in the cavity. The static tuning of the resonator have been probed by sweeping the uxes applied to the two SQUIDs and at each point the resonance frequency has been extracted. Good results have been achieved and two dimensional tuning is shown. A high frequency, ux modulation have been applied to the SQUIDs both one at a time and simultaneously to generate photons. In the two mirror modulation case qualitative agreement with theory has been observed for driving in the breathing and translational mode respectively. Generation occurs first for the breathing mode at 2*omega (twice the resonance frequency) and for the translational mode at 3*omega.