Design of an On-Chip Superconducting Microwave Circulator with Octave Bandwidth

We present a design for a superconducting, on-chip circulator composed of dynamically modulated transfer switches and delays. Design goals are set for the multiplexed readout of superconducting qubits. Simulations of the device show that it allows for low-loss circulation (insertion loss $<0.35$ dB and isolation $>20$ dB) over an instantaneous bandwidth of 2.3 GHz. This design improves on the bandwidth of previous superconducting circulators by more than an order of magnitude, making it ideal for integration with broadband quantum-limited amplifiers.

Figure 1

Circulation with actively modulated transfer switches and delays. (a) A four-port circulator is constructed from two dynamically modulated transfer switches and a pair of delay lines. The left (blue) switch is shown in the crossed state, and the right (red) switch is shown in the through state. The delay is set to one quarter of the modulation period $\tau =T/4$. (b) The left and right switches are modulated with period $T$ and a relative phase of $\pi /2$. For visual clarity, the modulation period is divided into quarter-period intervals indicated by the pastel background colors. (c) The schematics depict the way the switches will route a right-propagating signal, depending on its arrival time. Signals incident on port 1 are always routed to port 2, and signals incident on port 3 are routed to port 4. Note that these are not “snapshots” of the switches’ configurations, as the signal is delayed by a quarter period of the modulation as it propagates through the network. (d) The same depiction as (c), but for left-traveling waves incident on ports 2 or 4. To keep the device’s orientation the same as in (b), time flows from right to left in this diagram.

Figure 2

Transfer switch with tunable inductors. A symmetric lattice of tunable inductors forms a transfer switch. (a) When the inductors are tuned such that $l_t\ll l_c$, the switch is in its through state. Tuning $l_c\ll l_t$ realizes the switch’s crossed state. Parallel capacitances $c$ are used to match the switch to a characteristic impedance $Z_0$. (b) Calculated scattering parameters of the network in (a) parametrized by Eq. (3), with $l_0=0.94$ nH, $c=270$ fF, $ϵ=2.5\times {10}^{-2}$. Solid lines are exact solutions. Dashed lines are the approximations to first order in $ϵ$ given in Eq. ((B5)). Insertion loss is $<0.03$ dB between 4 and 8 GHz. The gray rectangle shows the Bode-Fano criterion [Eq. (2)] between 2 and 10 GHz. Its proximity to $S_{11}$ attests to the efficacy of the single-pole matching network.

Figure 3

Optimal modulation rate. (a) A possible layout for the proposed circulator. Delay lines occupy the majority of the chip’s area. (b) Estimated chip area (left axis, blue trace) and total insertion loss (right axis, thick red trace) as a function of the modulation frequency. Contributions to the insertion loss from dispersive, dielectric, and finite modulation bandwidth effects are also shown. These assume a fractional group-velocity dispersion of $3\times {10}^{-4}$ per $80$ MHz, an internal quality factor of $Q_i={10}^5$, and a modulation bandwidth of ${\mathrm{\Omega }}_b/(2\pi )=2$ GHz.

Figure 4

Simulated performance. Time-domain numerical simulations of the circulator, with transfer switches formed from symmetric lattices of tunable inductors modulated at $\mathrm{\Omega }=2\pi \times 80$ MHz. (a) First column of the simulated scattering matrix, when the device is configured for counter-clockwise circulation. The horizontal line at $-20$ dB is a guide to the eye. (b) Power spectrum of the signal emerging from the circulator’s second port, when the circulator is driven by a 6-GHz tone incident on its first port. Power is normalized to the incident signal power $P_{\mathrm{inc}}$. The thin black trace shows the same quantity in a simulation with infinite modulation bandwidth.

Figure 5

Performance in the presence of nonidealities. The effects of group-delay dispersion (a),(b) and finite modulation bandwidth (c),(d) on the circulator’s insertion loss (a),(c) and isolation (b),(d), calculated with Eq. ((A14))

Figure 6

Tunable inductor design. (a) Top view of a ladder of alternating rf SQUIDs, as studied in Refs. [20, 29]. The linear inductors may be realized either with the geometric inductance of galvanic connections, or with high-critical-current Josephson junctions. When biased close to frustration the impedance of the array can exceed a resistance quantum [29]. (b) Top view of a series array of dc SQUIDs, with SQUID loops in the plane of the chip. The tunability of such arrays may be limited by the geometric inductance of the SQUID loops. (c) Side view of a series array of dc SQUIDs, with SQUID loops orthogonal to the plane of the chip. In this “vertical” geometry, the vertical dimension of the loop is set by the interlayer spacing, which can substantially reduce the loop’s geometric inductance [41].

Figure 7

Noise from active flux control. (a) Circuit model for noise from a dynamically modulated flux ${\mathrm{\Phi }}_e$. (b) Alternative representation for the circuit in (a). The pink path traces an Amperian loop, which encloses a time-varying flux. The resulting electromotive force drives current through the resistors. (c) Steady-state photon number vs photon frequency for the circuit in (b), calculated with Eq. ((D1)). The three traces show the resulting occupations when the external flux ${\mathrm{\Phi }}_e(t)$ is modulated with a Fourier-series approximation of an $\mathrm{\Omega }=2\pi \times 80$ MHz square wave capped at $2$ GHz (olive), a sigma approximation [44] of an $\mathrm{\Omega }=2\pi \times 80$ MHz square wave capped at $2$ GHz (purple), and a sigma approximation of an $\mathrm{\Omega }=2\pi \times 15$ MHz square wave capped at $750$ MHz (teal).