On-Chip Superconducting Microwave Circulator from Synthetic Rotation

We analyze a design for a microwave circulator which could replace many of the commercial ferrite circulators that are ubiquitous in contemporary quantum superconducting microwave experiments. The lossless, lumped-element design is capable of being integrated on chip with other superconducting microwave devices, thus circumventing the many performance-limiting aspects of ferrite circulators. The design is based on the dynamic modulation of dc superconducting microwave quantum interference devices that function as nearly linear, tunable inductors. The connection to familiar ferrite-based circulators is a simple frame boost in the internal dynamics’ equation of motion. In addition to the general, schematic analysis, we also give an overview of many considerations necessary to achieve a practical design with a tunable center frequency in the 4–8-GHz frequency band, a bandwidth of 240 MHz, reflections at the $-20\text{−}\mathrm{dB}$ level, and a maximum signal power of approximately 100 microwave photons per inverse bandwidth.

Figure 1

(a) Schematic of a four-port ferrite turnstile circulator [3, 4]. Arrows represent the electric field polarization of the four waveguide and two resonator modes. The electric fields of ports 1 and 3 couple directly to resonator mode $x$ (blue arrows), and the electric fields of ports 2 and 4 couple directly to resonator mode $y$ (red arrows). (b) Depiction of the circulator’s steady-state operation: A traveling-wave signal incident through waveguide port 1 induces a steady-state response in the resonator modes that is rotated by angle $\psi$ relative to the $x$ mode. When $\psi =\pi /4$, all of the incident power is emitted out port 2. (c) Concept of an alternate realization, based on Eq. (5). The ferrite Faraday rotator is now removed, and reciprocity is now broken by the resonant cylinder mechanically rotating at rate $\mathrm{\Omega }$ relative to the waveguide junction.

Figure 2

(a) Electrical schematic of a Wheatstone bridge-based $LC$ resonator that serves as the basic module of our four-port circulator design. Analogous to the turnstile-type structure depicted in (d), the resonant mode couples to the two transmission lines 1 and 3 (characteristic impedance $r$) with a magnitude and sign that depend on $\theta$. The resonant mode center frequency is $\theta$ independent, which is also true of (d). (b) Two such modules placed in parallel, with “orthogonal” bridge unbalancing. (c) Four such modules, configured to emulate the mechanically spinning turnstile depicted in Fig. 1. As mentioned in the text, the inductances in (c) are modulated linearly in Sec. 2, but the inverse inductances are modulated linearly in Sec. 3. (d) Turnstile analogue of the circuit in (a).

Figure 3

Schematic of a dc SQUID-based realization of the tunable bridge network. In the right-hand figure, the wire configuration (black lines) is critical, as it determines the magnetic flux through the five depicted loops (four SQUID loops, one loop of the bridge itself; the four small circles represent unimportant network terminals). The total magnetic flux $\mathrm{\Phi }$ through each SQUID loop is the sum of the flux from a uniform background field (${\mathrm{\Phi }}_{\mathrm{\Sigma }}$) and from a magnetic-flux control current that flows vertically through the “twisted” bridge (${\mathrm{\Phi }}_{\mathrm{\Delta }}$). Ideally, the net magnetic flux through the bridge loop is always zero.

Figure 4

(a) The inductance bridge as a network with two ports, 1 and $q$. (b) Four inductance bridges configured as a six-port network. Attaching a capacitor across ports $q$ and $p$ and transmission lines across ports 1–4 realizes the circulator network depicted in Fig. 2.

Figure 5

Simulation of the frequency-domain response of various models of the circulator, as described in the text. The dependence on detuning $\mathrm{\Delta }$ for the other scattering parameters is identical to the traces shown above, with a cyclic permutation of the port indices, confirming proper circulation. (a) IO network. (b) Lumped-element network with $l{\omega }_0/r=0.4$. (c) Lumped-element network with $l{\omega }_0/r=0.8$.

Figure 6

(a) Schematic of a dc SQUID with $I_0$ critical current Josephson junctions, $l_g$ geometric inductance in each branch, and a magnetic flux $\mathrm{\Phi }$. (b) A series array of $N$ such SQUIDs.

Figure 7

Numerical network simulations. (a) Network output response in time and frequency, driving port 1, given parameters from Fig. 5, and the ideal inductance modulation given in Fig. 2 and Eq. (37). (b) The same network simulation, but where the linear inductances are modulated in a “SQUID-like” manner, i.e., using Eq. (38), as described in the text.