Flux-modulated tunable interaction regimes in two strongly nonlinear oscillators

TL;DR

This work develops a flux-modulated coupler to selectively activate different interactions between two Kerr-nonlinear oscillators in a superconducting circuit. By DC+AC flux pumping and rotating-wave approximations, red-sideband modulation yields an effective beam-splitter-like hopping $J_1$, while blue-sideband modulation yields two-mode squeezing $J_2$, with cross-Kerr $V$ present throughout; the authors observe both level repulsion and level attraction in strongly nonlinear regimes and quantify the couplings via spectroscopy and master-equation simulations. The results expand the toolkit for analog quantum simulation of spin and many-body physics using nonlinear oscillators, enabling access to diverse interaction regimes and suggesting future directions with richer nonlinearities and driven-dissipative dynamics. The work points toward implementing XYZ spin-models and exploring non-Hermitian phenomena, nonreciprocal photonics, and quantum sensing in tunable nonlinear oscillator networks.

Abstract

The ability to efficiently simulate a variety of interacting quantum systems on a single device is an overarching goal for digital and analog quantum simulators. In circuit quantum electrodynamical systems, strongly nonlinear superconducting oscillators are typically realized using transmon qubits, featuring a wide range of tunable couplings that are mainly achieved via flux-dependent inductive elements. Such controllability is highly desirable both for digital quantum information processing and for analog quantum simulations of various physical phenomena, such as arbitrary spin-spin interactions. Furthermore, broad tunability facilitates the study of driven-dissipative oscillator dynamics in previously unexplored parameter regimes. In this work, we demonstrate the ability to selectively activate different dynamical regimes between two strongly nonlinear oscillators using parametric modulation. In particular, our scheme enables access to regimes that are dominated by photon-hopping, two-mode squeezing, or cross-Kerr interactions. Finally, we observe level repulsion and attraction between Kerr-nonlinear oscillators in regimes where the nonlinearities exceed the coupling strengths and decay rates of the system. Our results could be used for realizing purely analog quantum simulators to study arbitrary spin systems as well as for exploring strongly nonlinear oscillator dynamics in previously unexplored interaction regimes.

Figures (5)

• Figure 1: Device and measurement scheme. (a) Circuit diagram for the device. On the left and right sides are two flux-tunable transmon qubits consisting of SQUIDs with gate capacitances to ground. The tunable coupler in the center consists of a coupling capacitor and a symmetric SQUID. (b) Optical microscope image of the device, including transmission line, readout resonators ($R_i$), drive lines ($D_i$), flux lines ($\Phi_i$), two transmons ($Q_i$), and the tunable coupler. (c) Schematic of the experiment. The flux incident on the coupler SQUID loop is modulated at either the difference or sum frequency of the two transmons. (d) Optical microscope image of Qubit B.
• Figure 2: Single-photon hopping interaction induced by red sideband flux modulation of the coupler. (a) Change in transmission while driving transmon B and sweeping the modulation frequency of the DC signal incident on the tunable coupler through the red sideband of the two oscillators. The black dashed lines are guides for the eye. The horizontal dashed line is the first transition frequency of transmon B, and the diagonal dashed line is $(\omega_A-\omega_m)/2\pi$. (b) Eigenfrequencies obtained from fitting to the level repulsion model (markers) and the normalized expectation value of $\hat{b}$ obtained from a numerical quantum master equation simulation of the system.
• Figure 3: Two-mode squeezing interaction induced by blue sideband flux modulation of the coupler. (a) Change in transmission while driving transmon A and sweeping the modulation frequency of the DC bias incident on the tunable coupler through the blue sideband of the two oscillators. (b) The dashed lines are $(\omega_m-\omega_B)/2\pi$ and the same shifted vertically by $V/2\pi$. The dotted lines are the eigenfrequencies of the system determined from fits of the data to the analytical level attraction model. The underlying spectrum is the normalized expectation value of $\hat{a}$ obtained from a numerical simulation of a quantum master equation for the system.
• Figure 4: Interaction strengths and distinct response parameter regimes. (a) The calculated interaction strengths as the DC flux bias point of the coupler is changed. The gradient of curves in green shows the possible values of the cross-Kerr interaction for the range of $\Phi_{A,B}\in[0.0,0.5\Phi_0]$ at each value of $\Phi_{DC}$. The gradient of curves in gray shows the values of single-photon hopping or two-mode squeezing interactions for a range of modulation strengths between $\Phi_{AC} = \Phi_{DC}/100$ and $\Phi_{AC} = \Phi_{DC}/10$. The diamond ($V/2\pi$) and cross ($J_{AC}/2\pi$) markers indicate the extracted interaction strengths from Fig. \ref{['RSB Modulation']} (red) and Fig. \ref{['BSB Modulation']} (blue). (b), (c) The photon number expectation value for one mode of a system of two coupled Kerr-nonlinear oscillators subject to a drive-induced two-mode squeezing interaction where the strength of the interaction $J_2$, detuning of the drive from the sum frequency resonance condition $\delta$, and nonlinearity $\alpha$ are varied relative to the loss rates of the oscillators $\kappa$. (d), (e) The second-order correlation function of one of the oscillators as in (b), (c).
• Figure S1: Measurement setup for the experiment. (a) Wiring configuration from the top of the dilution refrigerator to the device. The input and flux lines are attenuated before reaching the device, while the outgoing signal passes through two isolators and is amplified before returning to the network analyzer. (b) Room temperature wiring configuration. The device probe and drive signals are sent from ports 1 and 3 of the vector network analyzer (VNA), while the returning signal is further amplified before arriving at port 2. The DC signals sent to the flux lines are produced by a current source, with the coupler current modulated by a signal from an additional microwave source. The microwave source and VNA share a common reference clock signal. (c) Legend for microwave components.