Electromagnetically Induced Transparency in Circuit Quantum Electrodynamics with Nested Polariton States

Quantum networks will enable extraordinary capabilities for communicating and processing quantum information. These networks require a reliable means of storage, retrieval, and manipulation of quantum states at the network nodes. A node receives one or more coherent inputs and sends a conditional output to the next cascaded node in the network through a quantum channel. Here, we demonstrate this basic functionality by using the quantum interference mechanism of electromagnetically induced transparency in a transmon qubit coupled to a superconducting resonator. First, we apply a microwave bias, i.e., drive, to the qubit-cavity system to prepare a $\mathrm{\Lambda }$-type three-level system of polariton states. Second, we input two interchangeable microwave signals, i.e., a probe tone and a control tone, and observe that transmission of the probe tone is conditional upon the presence of the control tone that switches the state of the device with up to 99.73% transmission extinction. Importantly, our electromagnetically induced transparency scheme uses all dipole allowed transitions. We infer high dark state preparation fidelities of $>99.39\%$ and negative group velocities of up to $-0.52\pm 0.09\mathrm{km}/\mathrm{s}$ based on our data.

Figure 1

(a) Optical micrograph (with false color) of the device including a capacitively coupled $\lambda /2$ microstrip resonator and a concentric transmon qubit. ${\omega }_d$, ${\omega }_p$, and ${\omega }_c$ are the frequencies of the polariton drive field, the transmission spectrum (or EIT) probe field, and the EIT control field, respectively. (b) Generation of polariton states (red solid lines) in the nesting regime from the Jaynes-Cummings ladder (green solid lines). Black dashed double-headed arrows indicate the photon-number-dependent qubit transitions, black solid arrows show the polariton drive, and the shaded regions denote the nesting regime. ${\omega }_q$ and $\chi$ represent the bare qubit frequency and the effective dispersive shift, respectively. Microwave fields are applied through the input coupler in (a).

Figure 2

Transmission spectrum of polariton states in the nesting regime. (a) The four transitions (dashed double-headed arrows) between the polariton states. (b) Transmission spectrum of polariton states in arbitrary units shows the four different transitions in (a). ${\mathrm{\Omega }}_d$ and ${\omega }_p$ are the Rabi strength of the polariton drive and the probe frequency, respectively. Dashed curves denote predicted transmission peaks using the ac Stark shift model. (c) A line cut on (b) at ${\mathrm{\Omega }}_d/2\pi =1.46\mathrm{MHz}$. (d) The lowest three levels $|1⟩$, $|2⟩$, and $|3⟩$ form the $\mathrm{\Lambda }$-type transition for implementing EIT. ${\mathrm{\Omega }}_p$ (${\mathrm{\Omega }}_c$) and ${\omega }_p$ (${\omega }_c$) are the Rabi strength and frequency of the probe (control) field in an EIT experiment, respectively.

Figure 3

Transmission magnitude (a) and phase (b) of EIT with varying control field strength. (c) and (d) are line cuts on (a) and (b) with ${\mathrm{\Omega }}_c/2\pi =0.04$, 0.2, 0.4, and 0.82 MHz, respectively. The black dash-dotted lines in (a) and (b) denote the boundary of EIT set by ${\mathrm{\Omega }}_c/2\pi ={\gamma }_c/2\pi =0.82\mathrm{MHz}$. Note that traces are offset vertically from the ${\mathrm{\Omega }}_c/2\pi =0.04\mathrm{MHz}$ case for (c) and (d), and $S_{21}$ data include all round-trip amplification and attenuation.

Figure 4

Fitting the logarithm of transmission magnitude (a) and phase (b) data (blue dots) with the EIT model (red solid curves) and the ATS model (green dash-dotted curves) at ${\mathrm{\Omega }}_c/2\pi =0.82\mathrm{MHz}$. (c) Calculated weights of EIT and ATS by AIC-based testing. Vertical black dash-dotted line is the EIT boundary given by ${\mathrm{\Omega }}_c/2\pi ={\gamma }_c/2\pi =0.82\mathrm{MHz}$. Shaded regions are where both EIT and ATS model fits do not yield meaningful results.

Figure 5

(a) The time for microwaves to traverse the chip with control field strength ${\mathrm{\Omega }}_c/2\pi =0.2$, 0.4, and 0.82 MHz. Inset shows an enlarged image of (a) around the center frequency. (b) Calculated group velocity in the center of the transmission suppression window as a function of the control field strength.