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.
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 2omega
(twice the resonance frequency) and for the translational mode at 3omega.
We have created a doubly tunable resonator, with the intention to simulate relativistic motion of the resonator boundaries in real space. Our device is a superconducting coplanar-waveguide microwave resonator, with fundamental resonant frequency ω 1 / (2 π ) ~ 5 GHz. Both of its ends are terminated to ground via dc-SQUIDs, which serve as magnetic-flux-controlled inductances. Applying a flux to either SQUID allows the tuning of ω 1 / (2 π ) by approximately 700 MHz. Using two separate on-chip magnetic-flux lines, we modulate the SQUIDs with two tones of equal frequency, close to 2 ω 1 . We observe photon generation, at ω 1 , above a certain pump amplitude threshold. By varying the relative phase of the two pumps we are able to control this threshold, in good agreement with a theoretical model. At the same time, some of our observations deviate from the theoretical predictions, which we attribute to parasitic couplings resulting in current driving of the SQUIDs.
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 thesis I present different types of manipulations of microwave fields using tunable superconducting resonators. A resonator is made tunable by adding one or more superconducting quantum interference devices, SQUIDs. The SQUID consists of a superconducting loop with two Josephson junctions and acts as a tunable nonlinear inductor. Modulation of the SQUID nonlinearity can be performed to induce different types of non-trivial oscillator dynamics. The first project I present is on subharmonic oscillations. Here, the SQUID in a tunable resonator is driven with an external signal at an integer multiple of the frequency w. When w is placed slightly below the first resonator mode, I show generation of radiation at w, which is known as frequency down-conversion. In my measurements, subharmonic oscillations have been detected from period doubling up to period quintupling. For the specific case of period tripling, theory is developed and I show good agreement between theory and experiments. The second project of the thesis is on a doubly tunable resonator. Here, I show creation of a superconducting resonator with two independently tunable boundary conditions. The idea with this system is to operate the resonator in a breathing or a translational mode and compare the two. For static dc magnetic flux the performance is very good. As a second step, I perform fast modulation of both SQUIDs to generate radiation. Even though some of the measurement results are promising, I also show some contradicting observations. These indicate that the actual modulation mechanism of the SQUID is not pumping of the magnetic flux in the SQUID loop as intended, but rather direct current driving of the SQUID. However, the system remains a promising idea for microwave manipulation and creation of interesting non-classical states. As a possible application of the doubly tunable resonator, I present a theory proposal on how to use it for measurements on relativistic effects. By using the tunability of the resonator boundary conditions, the resonator can simulate a space rocket and time dilation could be measured. However, due to the crosstalk problems in the doubly tunable resonator this experiment was never realized. Finally, I finish the thesis by presenting a design for a tunable microwave coupling. Papers E and F show a high on-off ratio for the coupler and demonstrate how it can be used for storage of microwave signals. Furthermore, the tunable coupler can be used to shape a microwave signal by tuning the strength of the coupling.
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.
When one of the parameters in the Euler-Lagrange equations of motion of a sys- tem is modulated, particles can be generated out of the quantum vacuum. This phenomenon is known as the dynamical Casimir efiect, and it was recently realized experimentally in systems of superconducting circuits, for example by using modulated resonators made of coplanar waveg- uides, or arrays of superconducting quantum intereference devices (SQUIDs) forming a Josephson metamaterial. In this paper, we consider a simple electrical circuit model for dynamical Casimir efiects, consisting of an LC resonator, with the inductor modulated externally at 10.8GHz and with the resonant frequency tunable over a range of §400MHz around 5.4GHz. The circuit is analyzed classically using a circuit simulator (APLAC). We demonstrate that if an additional source of classical noise couples to the resonator (on top of the quantum vacuum), for example via dissipative \internal modes, then the resulting spectrum of the photons in the cavity will present two strongly asymmetric branches. However, according to the theory of the dynamical Casimir efiect, these branches should be symmetric, a prediction which is conflrmed by our ex- perimental data. The simulation presented here therefore shows that the origin of the photons generated in our experiment with Josephson metamaterials is the quantum vacuum, and not a spurious classical noise source. The picture about vacuum that emerges from modern quantum fleld theory is very difierent from that ofiered by classical physics. In classical fleld theory, the vacuum is the zero-energy state of the fleld, deflned by the absence of any excitation. In quantum fleld theory, the vacuum state has a flnite zero-point energy associated with it, and the uncertainty principle indicates that ∞uctuations exist even in this state. Due to the existence of ∞uctuations, the quantum vacuum can become unstable under certain perturbations, and the energy of the perturbation is converted into creation of real particles (1). In the Schwinger efiect for example, a static intense electric fleld can create pairs of electrons and positrons. In very strong gravitational flelds, at the event horizon of black holes, the vacuum becomes unstable and energy is radiated away (Hawking efiect). By the equivalence principle, an accelerated observer in the Minkowski vacuum will detect a flnite-temperature fleld (Unruh efiect). In a previous paper, we reported the observation of the dynamical Casimir efiect using a ∞ux- biased Josephson metamaterial embedded in a microwave cavity at 5.4GHz (2). A non-adiabatic change in the index of refraction of the cavity (or in the electrical length) was realized by modulat- ing the ∞ux at values close to double the resonant frequency of the cavity. We measured also the frequency-correlated photons emitted from the cavity (which is kept at a temperature T = 50mK), and we obtained a power spectrum displaying a bimodal, \sparrow-tail distribution, with two sym- metric branches. Furthermore, we demonstrated that the generated photons are squeezed below the vacuum level and that the state of the two-photon fleld is nonseparable. The experimental results are in excellent agreement with the theoretical predictions. An alternative realization of the dynam- ical Casimir efiect consists in modulating an efiective boundary condition, realized experimentally as a SQUID structure placed at the end of a coplanar waveguide resonator (3). Remarkably, the experiments presented in Refs. (2,3) are the only existing measurements of the dynamical Casimir efiect, which was predicted theoretically in 1970 (4).
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.