Nonreciprocity Realized with Quantum Nonlinearity

Nonreciprocal devices are a key element for signal routing and noise isolation. Rapid development of quantum technologies has boosted the demand for a new generation of miniaturized and low-loss nonreciprocal components. Here, we use a pair of tunable superconducting artificial atoms in a 1D waveguide to experimentally realize a minimal passive nonreciprocal device. Taking advantage of the quantum nonlinear behavior of artificial atoms, we achieve nonreciprocal transmission through the waveguide in a wide range of powers. Our results are consistent with theoretical modeling showing that nonreciprocity is associated with the population of the two-qubit nonlocal entangled quasidark state, which responds asymmetrically to incident fields from opposing directions. Our experiment highlights the role of quantum correlations in enabling nonreciprocal behavior and opens a path to building passive quantum nonreciprocal devices without magnetic fields.

Figure 1

(a) Schematic of the quantum diode: two qubits embedded in a 1D waveguide, tuned to the optimal conditions for nonreciprocal behavior (${\omega }_1={\omega }_d-\delta \overline{{\gamma }_r}$, ${\omega }_2={\omega }_d$). An incoming field from the forward direction ($\alpha$ drive) at frequency ${\omega }_d$ is partially transmitted through the system, whereas a field incoming from the reverse direction ($\beta$ drive) is fully reflected. (b) Energy level diagram of the system. The quasidark state $|+⟩$ can be populated by the driving field either directly from the ground state $|gg⟩$ (purple path) or indirectly through the bright state $|-⟩$ (pink path). These two channels interfere either constructively or destructively depending on the driving direction. If interfering constructively, part of the population gets trapped in the quasidark state $|+⟩$, which in turn gives rise to the nonreciprocal behavior of the system. (c) Open 1D waveguide with embedded 3D transmons (dashed mint green boxes). Inset: Optical micrograph of one of the two identical 3D transmons. The scale bar corresponds to $500\mu \mathrm{m}$.

Figure 2

Nonreciprocity dependence on power. Experimental data (points) and theoretical fits (solid lines) for the forward driving transmission amplitude $|t_{\to }|$ (green), reverse driving transmission amplitude $|t_{\leftarrow }|$ (red), and diode efficiency $\mathcal{E}$ (blue). $\mathcal{E}\equiv |t_{\to }|(|t_{\to }|-|t_{\leftarrow }|)/(|t_{\to }|+|t_{\leftarrow }|)$ measures the nonreciprocal behavior of the system as well as its forward transmitting capabilities. The system is tuned to its optimal nonreciprocal configuration for two different detunings: (a) ${\delta }^2\simeq 0.001$, (b) ${\delta }^2\simeq 0.01$. In both cases the device behaves reciprocally and reflects most of the incoming radiation at the low power regime $p/\overline{{\gamma }_r}\ll {\delta }^2$. However, as the power increases past the onset of the diode regime, indicated by ${\delta }^2$, saturation of the quasidark state allows for an increase in $|t_{\to }|$, while $|t_{\leftarrow }|$ remains unchanged.

Figure 3

Power spectral densities of the forward driving transmitted field [$S_{\to \text{trans}}(\omega )$] and the reverse driving reflected field [$S_{\leftarrow \mathrm{ref}}(\omega )$]. Both spectra were taken after tuning the device to its optimum nonreciprocal configuration for ${\delta }^2\simeq 0.001$ [as in Fig. 2] and setting the driving power such that the diode efficiency $\mathcal{E}$ is maximum [corresponding to $p/\overline{{\gamma }_r}\simeq 0.05$ in Fig. 2]. Both scattered fields travel through the same amplification chain. The $\delta$-like peaks at the driving frequency ${\omega }_d$ correspond to the elastically scattered fields. Radiation inelastically scattered off the quasidark state $|+⟩$ generates an additional broader peak, which we fit to Lorentzians of width ${\gamma }_{\to \text{trans}}=2.92\mathrm{MHz}$ and ${\gamma }_{\leftarrow \mathrm{ref}}=1.76\mathrm{MHz}$ (solid lines). When driving the system in the forward direction, part of the population gets trapped in the quasidark state, consistent with a greater inelastically scattered radiation power: $\int S_{\to \text{trans}}d\omega \gg \int S_{\leftarrow \mathrm{ref}}d\omega$.

Figure 4

Power spectral densities of the inelastically scattered transmitted field. The system is driven in the forward direction for different detuning regimes (${\delta }^2\simeq 0.01$, 0.001, 0.0001). In every case, the spectra were taken at the optimum atomic detunings and driving powers that maximize the diode efficiency $\mathcal{E}$. The solid lines are fits to Lorentzians of width 1.43, 2.92, and 7.30 MHz, in increasing order of ${\delta }^2$. As predicted by theory, the linewidth ${\mathrm{\Gamma }}_{\mathrm{FWHM}}\equiv 2(3{\delta }^2\overline{{\gamma }_r}+2{\gamma }_{\varphi }+{\gamma }_{\mathrm{nr}})$ of the scattered field increases linearly with ${\delta }^2$ (see inset). However, in the experiment the width of the inelastically scattered radiation is broader than expected: ${\tilde{\mathrm{\Gamma }}}_{\mathrm{tot}}={\mathrm{\Gamma }}_{\mathrm{tot}}+{\mathrm{\Gamma }}_{\mathrm{exc}}$, with ${\mathrm{\Gamma }}_{\mathrm{exc}}/2\pi =1.22\mathrm{MHz}$. The elastic part of the scattered radiation at $\delta {\omega }_d=0\mathrm{MHz}$ has been omitted and the peaks are scaled for clarity.