Breaking Lorentz Reciprocity with Frequency Conversion and Delay

We introduce a method for breaking Lorentz reciprocity based upon the noncommutation of frequency conversion and delay. The method requires no magnetic materials or resonant physics, allowing for the design of scalable and broadband nonreciprocal circuits. With this approach, two types of gyrators—universal building blocks for linear, nonreciprocal circuits—are constructed. Using one of these gyrators, we create a circulator with $>15\mathrm{dB}$ of isolation across the 5–9 GHz band. Our designs may be readily extended to any platform with suitable frequency conversion elements, including semiconducting devices for telecommunication or an on-chip superconducting implementation for quantum information processing.

Figure 1

(a) Signal flow graph of an up-down converter used in nonreciprocal circuits [13, 20]. Signals incident on the left port are down-converted while those incident on the right port are up-converted. (b) Signal flow graph of an ambidirectional modulator, configured as a down-converter. Signals incident on either port are down-converted. (c) Schematic of a “ladder gyrator,” constructed from two frequency converters concatenated around a delay. (d) Frequency-time diagram of the ladder gyrator. Oscillating arrows indicate directed signal flow, and circled arrows indicate the signal’s phase in a frame rotating at the input frequency. Signals acquire a nonreciprocal dynamical phase. (e) Schematic representation of the “balanced gyrator,” which is constructed using a “symmetric” frequency converter. Operations $\cos (\mathrm{\Omega }t)$ and $\pm \sin (\mathrm{\Omega }t)$ convert an input signal into both its upper and lower sideband. (f) Measured forward-reverse phase difference, $\angle s_{21}-\angle s_{12}$, as a function of frequency for both gyrators. For visual clarity, the balanced gyrator trace has been translated down by $2\pi$. Performance is broadband, in the sense that the bandwidth is comparable to the operation frequency.

Figure 2

Circuit schematics of a ladder gyrator (a) and a balanced gyrator (b) realized with $IQ$ mixers (crossed circles) as the frequency modulation elements and coaxial transmission-line cables (coil symbols) for delay $\tau$. Signals in the balanced gyrator are split into two paths and recombined using reciprocal summation elements (labeled $\mathrm{\Sigma }$). In an $IQ$ mixer, bias tones $I(t)$ and $Q(t)$ multiply the incident signal by $I(t)-iQ(t)$ leading to frequency conversion of fields moving between ports $L$ and $R$. In the balanced gyrator, $Q(t)=0$. (c),(d) For both circuits, the measured device transmission, normalized to the maximum forward transmission of each device $|s_{21}{|}_{\max }$ (c) and the measured phase difference (d) are plotted, as a function of the relative phase $\mathrm{\Omega }\tau$ acquired in the delay. Orange and black boxes indicate operation points where the ladder circuit acts as a gyrator or as a reciprocal element, which we call a “reciprocator.”

Figure 3

(a) A circulator is constructed from a ladder gyrator and a reciprocator placed between two microwave beam splitters: linear, reciprocal elements that transform signals incident on ports 1 and 2 into their sum and difference, exiting at ports $\mathrm{\Sigma }$ and $\mathrm{\Delta }$, respectively. (b) Forward (solid lines) and reverse transmission (solid lines with circles) between two adjacent ports of the circuit, when the device is configured as a clockwise circulator (gray traces) and a counter-clockwise circulator (green traces). Relative isolation greater than 15 dB is achieved between 5 and 9 GHz. A moving average filter (span 15 MHz) has been applied to the isolated traces to smooth a several-dB ripple resulting from reflections within the delay lines.