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arXiv:1802.09259 · Applied Physics Letters

Period multiplication in a parametrically driven superconducting resonator

Ida-Maria Svensson, Andreas Bengtsson, Jonas Bylander, Vitaly Shumeiko, Per Delsing

Published Jul 9, 2018• 38 citations
arXivHTMLPDF

Abstract

We report on the experimental observation of period multiplication in parametrically driven tunable superconducting resonators. We modulate the magnetic flux through a superconducting quantum interference device, attached to a quarter-wavelength resonator, with frequencies nω close to multiples, n = 2, 3, 4, and 5, of the resonator fundamental mode and observe intense output radiation at ω. The output field manifests n-fold degeneracy with respect to the phase, and the n states are phase shifted by 2π/n with respect to each other. Our demonstration verifies the theoretical prediction by Guo et al. [Phys. Rev. Lett. 111, 205303 (2013)] and paves the way for engineering complex macroscopic quantum cat states with microwave photons.

DOI
10.1063/1.5026974

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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.

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In this licentiate thesis, I present the design, fabrication, and characterization of superconducting parametric resonators, for use in quantum information processing. These devices are quarter-wavelength coplanar waveguide resonators (~5 GHz), terminated at one end by a non-linear inductance provided by a SQUID (superconducting quantum interference device). The SQUID acts as a flux-controlled boundary condition, which effectively changes the electrical length of the resonator. This enables the modulation of the resonant frequency by coupling microwave magnetic flux into the SQUID, using on-chip transmission lines. The modulation occurs on a timescale much faster than the photon loss out of the resonator. The non-linearity provides several different regimes for operating this device. I focused on the parametric regime, in which the non-linear element was being modulated (pumped) at or near twice the resonant frequency. This pumping can lead to amplification and frequency conversion of an incoming signal. It can also give rise to instabilities and self-oscillations "parametric oscillations" - that is, the generation of an intense electric field in the resonator (the creation of photons). Parametric oscillations set in when the pumping strength exceeds a threshold value, and also depends on the pump-frequency detuning from twice the resonance and on the static flux through the SQUID. I characterized the devices over a wide parameter range by doing homodyne detection of the reflected signal or of the emitted field oscillations. In particular, I investigated the damping and two leading nonlinearities, dominating in different operating regimes, which influence the dynamics of the parametric resonator. First, I extracted the Duffing nonlinearity by studying the line width of the resonator as a function of the input signal (without pumping the SQUID). Second, with high pump amplitude, I extracted the pump-induced nonlinearity by determining the threshold parameters (pump amplitude and detuning) for the onset of parametric oscillations; this nonlinearity leads to the generation of higher-order pump terms that complicate the system dynamics, but can be mitigated now that they have been understood. My results validate a recent theoretical model for the classical, non-linear dynamics of parametric resonance, and helps determine workable design parameters. Finally, I helped develop a linearized model $-$ the "pumpistor" - describing this device in terms of a flux-controlled impedance. This will be useful when designing more complex circuits. One goal of this work is to demonstrate single-shot measurements on superconducting qubits (quantum bits of information). We will use the parametric resonator as a threshold discriminator, associating the qubit's two energy eigenstates with the parametric oscillations either turning on or not. This would be a very useful device in the quantum engineer's toolbox when designing systems for quantum information processing and communication.

•Jan 1, 2013•1 citation
We report on the experimental observation of period multiplication in parametrically driven tunable superconducting resonators. We modulate the magnetic flux through a superconducting quantum interference device, attached to a quarter-wavelength resonator, with frequencies nω close to multiples, n = 2, 3, 4, and 5, of the resonator fundamental mode and observe intense output radiation at ω. The output field manifests n-fold degeneracy with respect to the phase, and the n states are phase shifted by 2π/n with respect to each other. Our demonstration verifies the theoretical prediction by Guo et al. [Phys. Rev. Lett. 111, 205303 (2013)] and paves the way for engineering complex macroscopic quantum cat states with microwave photons.
We investigate nondegenerate parametric oscillations in a superconducting microwave multimode resonator that is terminated by a superconducting quantum interference device (SQUID). The parametric effect is achieved by modulating magnetic flux through the SQUID at a frequency close to the sum of two resonator-mode frequencies. For modulation amplitudes exceeding an instability threshold, self-sustained oscillations are observed in both modes. The amplitudes of these oscillations show good quantitative agreement with a theoretical model. The oscillation phases are found to be correlated and exhibit strong fluctuations which broaden the oscillation spectral linewidths. These linewidths are significantly reduced by applying a weak on-resonant tone, which also suppresses the phase fluctuations. When the weak tone is detuned, we observe synchronization of the oscillation frequency with the frequency of the input. For the detuned input, we also observe an emergence of three idlers in the output. This observation is in agreement with theory indicating four-mode amplification and squeezing of a coherent input.
We have observed period-tripling subharmonic oscillations in a driven superconducting coplanar waveguide resonator operated in the quantum regime, ${k}_{B}T\ensuremath{\ll}\ensuremath{\hbar}\ensuremath{\omega}$. The resonator is terminated by a tunable inductance that provides a Kerr-type nonlinearity. We detected the output field quadratures at frequencies near the fundamental mode, $\ensuremath{\omega}/2\ensuremath{\pi}\ensuremath{\sim}5\phantom{\rule{0.16em}{0ex}}\mathrm{GHz}$, when driving the resonator with a current at $3\ensuremath{\omega}$, with amplitude exceeding an instability threshold. We observed three stable radiative states with equal amplitudes, phase shifted by $2\ensuremath{\pi}/3$ rad, red detuned from the fundamental mode. The down-conversion from $3\ensuremath{\omega}$ to $\ensuremath{\omega}$ is strongly enhanced by near-resonant excitation of the second mode of the resonator and the cross-Kerr effect. Our experimental results are in quantitative agreement with a model for the driven dynamics of two coupled modes.
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.
In this Master thesis, I present a theoretical description and experimental measurements of a tunable microwave resonator operated as a parametric amplifier. Superconducting parametric amplifiers have attracted great interest in recent years as it has been demonstrated that they can operate with a noise performance at the standard quantum limit. \nParametric amplification is achieved in the devices by modulating the electrical length of the resonator at twice the resonance frequency. The device measured in this thesis consists of a quarter-wavelength microwave resonator fabricated in a thin-film aluminum coplanar waveguide geometry. With a termination to ground through a Superconducting Quantum Interference Device (SQUID), the boundary condition of the resonator becomes sensitive to an applied magnetic field. This enables the tuning of the resonance frequencies, using a DC magnetic field, over a wide frequency range of 670MHz. Similarly, an AC magnetic field then allows for the modulation of the resonator electrical length at microwave frequencies. \nWe characterized the device by first measuring the DC tuning curves of the 2nd and 3rd harmonics of the resonator, around 4 GHz and 6 GHz respectively. We then studied parametric amplification in the device when the electrical length was modulated at approximately twice the resonance frequency of one of the modes. We scanned the pump frequency and pump power over a wide range to study the region in parameters which gives parametric gain. We performed an in-depth parameter sweep over this region, identifying pump parameters that provided maximum gain and maximum system noise improvements compared the following HEMT amplifier. \nWe find that the use of the parametric amplifier can improve the system noise temperature by more than 10 dB compared to a start-of-the-art commercial HEMT amplifier. This implies a performance near the standard quantum limit.
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 licentiate thesis, I present the design, fabrication, and characterization of superconducting parametric resonators, for use in quantum information processing. These devices are quarter-wavelength coplanar waveguide resonators (~5 GHz), terminated at one end by a non-linear inductance provided by a SQUID (superconducting quantum interference device). The SQUID acts as a flux-controlled boundary condition, which effectively changes the electrical length of the resonator. This enables the modulation of the resonant frequency by coupling microwave magnetic flux into the SQUID, using on-chip transmission lines. The modulation occurs on a timescale much faster than the photon loss out of the resonator. The non-linearity provides several different regimes for operating this device. I focused on the parametric regime, in which the non-linear element was being modulated (pumped) at or near twice the resonant frequency. This pumping can lead to amplification and frequency conversion of an incoming signal. It can also give rise to instabilities and self-oscillations "parametric oscillations" - that is, the generation of an intense electric field in the resonator (the creation of photons). Parametric oscillations set in when the pumping strength exceeds a threshold value, and also depends on the pump-frequency detuning from twice the resonance and on the static flux through the SQUID. I characterized the devices over a wide parameter range by doing homodyne detection of the reflected signal or of the emitted field oscillations. In particular, I investigated the damping and two leading nonlinearities, dominating in different operating regimes, which influence the dynamics of the parametric resonator. First, I extracted the Duffing nonlinearity by studying the line width of the resonator as a function of the input signal (without pumping the SQUID). Second, with high pump amplitude, I extracted the pump-induced nonlinearity by determining the threshold parameters (pump amplitude and detuning) for the onset of parametric oscillations; this nonlinearity leads to the generation of higher-order pump terms that complicate the system dynamics, but can be mitigated now that they have been understood. My results validate a recent theoretical model for the classical, non-linear dynamics of parametric resonance, and helps determine workable design parameters. Finally, I helped develop a linearized model −-− the "pumpistor" - describing this device in terms of a flux-controlled impedance. This will be useful when designing more complex circuits. One goal of this work is to demonstrate single-shot measurements on superconducting qubits (quantum bits of information). We will use the parametric resonator as a threshold discriminator, associating the qubit's two energy eigenstates with the parametric oscillations either turning on or not. This would be a very useful device in the quantum engineer's toolbox when designing systems for quantum information processing and communication.