Flux-pumped Josephson circuits provide a versatile platform for parametric photon generation and multimode bosonic control. We identify an occupation-conditioned pair interaction that arises when two harmonic normal modes participate in the same driven SQUID coordinate. We consider a degenerate pair process in which two photons are created or annihilated in a signal mode while the occupation of a second harmonic mode remains unchanged. Although it does not exchange excitations, this controller mode modifies both the resonance frequency and the strength of the pair transition. The two effects have different physical origins: diagonal Josephson nonlinearities shift the pair resonance, whereas the nonlinear Josephson matrix element makes the avoided-crossing gap itself depend on the controller occupation. Starting from the full Josephson cosine, we develop complementary time-dependent Schrieffer--Wolff and Magnus descriptions that expose the direct and virtual processes responsible for this interaction, and benchmark the resulting effective theory against a numerically converged Floquet treatment of the unexpanded two-mode circuit Hamiltonian. The occupation dependence produces a two-dimensional pair spectrum that enables selective transitions within the bosonic Fock lattice. When the controller is prepared in a coherent superposition, the same interaction converts this conditional dynamics into signal--controller entanglement and controller-dependent non-Gaussian state generation. Observable-level comparisons further show that the variation of the pair-transition strength cannot be reproduced by occupation-dependent resonance shifts alone. These results establish a boson-controlled parametric interaction in which the state of one harmonic mode controls both the spectral position and the coupling strength of a nonlinear process in another.
Flux-pumped Josephson circuits provide a versatile platform for parametric photon generation and multimode bosonic control. We identify an occupation-conditioned pair interaction that arises when two harmonic normal modes participate in the same driven SQUID coordinate. We consider a degenerate pair process in which two photons are created or annihilated in a signal mode while the occupation of a second harmonic mode remains unchanged. Although it does not exchange excitations, this controller mode modifies both the resonance frequency and the strength of the pair transition. The two effects have different physical origins: diagonal Josephson nonlinearities shift the pair resonance, whereas the nonlinear Josephson matrix element makes the avoided-crossing gap itself depend on the controller occupation. Starting from the full Josephson cosine, we develop complementary time-dependent Schrieffer--Wolff and Magnus descriptions that expose the direct and virtual processes responsible for this interaction, and benchmark the resulting effective theory against a numerically converged Floquet treatment of the unexpanded two-mode circuit Hamiltonian. The occupation dependence produces a two-dimensional pair spectrum that enables selective transitions within the bosonic Fock lattice. When the controller is prepared in a coherent superposition, the same interaction converts this conditional dynamics into signal--controller entanglement and controller-dependent non-Gaussian state generation. Observable-level comparisons further show that the variation of the pair-transition strength cannot be reproduced by occupation-dependent resonance shifts alone. These results establish a boson-controlled parametric interaction in which the state of one harmonic mode controls both the spectral position and the coupling strength of a nonlinear process in another.
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.
Precise control of mechanical modes in the quantum regime is a key resource for quantum technologies, offering promising pathways for quantum sensing with macroscopic systems and scalable architectures for quantum simulation. In this work, we realize a multimode mechanical cavity coupled to a superconducting Kerr resonator, which induces nonlinearity in the mechanical modes. The Kerr mode is realized by a flux-tunable superconducting quantum interference device (SQUID) array resonator, while the mechanical modes are implemented by a surface acoustic wave cavity. Both mechanical and electromagnetic modes are individually addressable via dedicated measurement lines, enabling full spectroscopic characterization. We introduce a straightforward protocol to measure the SQUID array resonator’s participation ratio in the hybrid acoustic modes, quantifying the degree of hybridization. The participation ratio reveals that our device operates at the onset of the multimode coupling regime, where multiple acoustic modes simultaneously interact with the nonlinear superconducting element. Furthermore, this platform allows controllable Kerr-type nonlinearities in multiple acoustic modes, with the participation ratio serving as the key parameter determining both the dissipation rates and nonlinear strengths of these hybridized modes. Close to the resonant regime, we measure a cross-Kerr interaction between seven pairs of mechanical modes, which is controllable via the SQUID array resonator detuning. Finally, we apply a two-photon parametric drive to the SQUID array resonator and observe the resulting parametric down-conversion and metastable state switching in a mechanical mode. These results establish a platform for engineering nonlinear multimode mechanical interactions, offering potential for future integration with superconducting qubits and implementation of multiple mechanical qubits.
Spin-dependent forces enable independent control over the effective frequencies of bosonic motional modes and their coupling strengths to spins in trapped-ion spin-boson quantum simulators. We show that the relative signs of these effective frequencies provide an important programmable resource for engineering distinct multimode dynamics. Focusing on a two-mode quantum Rabi model, changing the frequencies of the two modes from the same sign to opposite signs reorganizes the resonance structure of the spin-mediated bosonic interaction, converting the dominant resonant process from excitation exchange to correlated pair creation. In the dispersive regime, the resulting effective interaction realizes a two-mode parametric amplifier with stable oscillatory, critical, and exponentially amplifying regimes separated by a spin-dependent instability threshold. Across these regimes, the dynamics generate collective-quadrature squeezing and bipartite motional entanglement. Preparing the spin in a coherent superposition further allows different spin components to experience distinct stability regimes, generating tripartite spin-boson entanglement, while spin postselection can coherently combine the conditional bosonic branches to generate non-Gaussian entangled bosonic states. Finally, we outline an implementation using parameters within reach of current trapped-ion platforms.
We consider the nonlinear effects in a Jahn–Teller (JT) system of two coupled resonators interacting simultaneously with a flux qubit using coupled SQUIDs. A two-frequency description of JT system that inherits the networked structure of both nonlinear Josephson junctions and harmonic oscillators is employed to describe the synchronous structures in a multifrequency scheme. Eigenvalue spectrum is used to show the switch between the effective single mode and two mode configuration in terms of frequency difference. The Rabi supersplitting is investigated by the spectral response of JT systems in different coupling regimes. Second-order coherence functions are employed to investigate antibunching effects in resonator mode. Synchronous structure between correlations of privileged mode and qubit is obtained in localization–delocalization and photon blockade regime controlled by the population imbalance.
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.
The realization of strong nonlinear coupling between single photons has been a long-standing goal in quantum optics and quantum information science, promising wide impact applications, such as all-optical deterministic quantum logic and single-photon frequency conversion. Here, we report an experimental observation of the strong coupling between a single-photon and a two-photon Fock state in an ultrastrongly-coupled circuit-QED system. This strong nonlinear interaction is realized by introducing a detuned flux qubit working as an effective coupler between two modes of a superconducting coplanar waveguide resonator. The ultrastrong light–matter interaction breaks the excitation number conservation, and an external flux bias breaks the parity conservation. The combined effect of the two enables the strong one–two-photon coupling. Quantum Rabi-like avoided crossing is resolved when tuning the two-photon resonance frequency of the first mode across the single-photon resonance frequency of the second mode. Within this new photonic regime, we observe the thresholdless second harmonic generation for a mean photon number below one. Our results represent a key step towards a new regime of quantum nonlinear optics, where individual photons can deterministically and coherently interact with each other in the absence of any stimulating fields. Systems with strong photon-photon interactions enable advanced quantum optical applications as well as the study of highly correlated light-matter states. Here the authors report strong coupling between single- and two-photon states in a superconducting circuit, enabling a new regime of nonlinear quantum optics.