Microwave Down-Conversion with an Impedance-Matched $\mathrm{\Lambda }$ System in Driven Circuit QED

By driving a dispersively coupled qubit-resonator system, we realize an “impedance-matched” $\mathrm{\Lambda }$ system that has two identical radiative decay rates from the top level and interacts with a semi-infinite waveguide. It has been predicted that a photon input from the waveguide deterministically induces a Raman transition in the system and switches its electronic state. We confirm this through microwave response to a continuous probe field, observing near-perfect (99.7%) extinction of the reflection and highly efficient (74%) frequency down-conversion. These proof-of-principle results lead to deterministic quantum gates between material qubits and microwave photons and open the possibility for scalable quantum networks interconnected with waveguide photons.

Figure 1

Experimental setup at the 10 mK stage of a dilution refrigerator. The resonator-qubit chip and the JPA chip are mounted in separate sample packages with independent coils for dc flux bias.

Figure 2

Energy-level diagram of the coupled system. (a) Lowest four energy levels of the qubit-resonator coupled system. $|k,l⟩$ denotes the eigenstates of the Jaynes-Cummings Hamiltonian, which are close to the product states $|k{⟩}_q|l{⟩}_r$ of the qubit and the resonator, where $k=g,e$ and $l=0,1,\cdots$. (b) Dressed-state energy levels in the frame rotating at ${\omega }_d$. $|\tilde{i}⟩$ represents the dressed state of the qubit-resonator coupled system, and ${\tilde{\kappa }}_{ij}$ is the radiative decay rate for the $|\tilde{i}⟩\to |\tilde{j}⟩$ transition. The nesting regime is realized when ${\omega }_{\text{ge}}-2\chi <{\omega }_d<{\omega }_{\text{ge}}$. The impedance matching occurs when ${\tilde{\kappa }}_{41}={\tilde{\kappa }}_{42}$ or ${\tilde{\kappa }}_{31}={\tilde{\kappa }}_{32}$. (c) Raman processes in the impedance-matched $\mathrm{\Lambda }$ system (solid arrows). The $|\tilde{2}⟩\to |\tilde{1}⟩$ decay (dotted line) is mainly caused by qubit relaxation.

Figure 3

Perfect absorption of the probe field. (a) Reflection coefficient $|r|$ as a function of the probe frequency ${\omega }_p$ and the qubit drive power $P_d$. The probe power is fixed at $-146.2\mathrm{dBm}$, and the detuning $\delta {\omega }_d$ ($\equiv {\omega }_d-{\omega }_{\text{ge}}$) is fixed at $2\pi \times (-64)\mathrm{MHz}$. The dashed lines show the drive powers ($P_{d3}$ and $P_{d4}$) where the depth of the dips is maximized. (b) $|r|$ as a function of ${\omega }_p$ for $P_d=P_{d4}$. The red solid and blue dashed lines depict the data with the qubit drive on and off, respectively. (c) Numerical simulations corresponding to (a). The dashed lines indicate transition frequencies between the dressed states. (d) Calculated radiative decay rates ${\tilde{\kappa }}_{ij}$ as a function of $P_d$. The dashed line shows the drive power $P_{d0}$, where ${\tilde{\kappa }}_{41}={\tilde{\kappa }}_{42}$ and ${\tilde{\kappa }}_{31}={\tilde{\kappa }}_{32}$.

Figure 4

Tunability of the $\mathrm{\Lambda }$ system. (a) Reflection coefficient $|r_2|$ of probe 2 as a function of ${\omega }_{p2}$. Here, ${\omega }_p/2\pi =10.681\mathrm{GHz}$, $P_p=-141.2\mathrm{dBm}$, $P_{p2}=-135\mathrm{dBm}$, and $\delta {\omega }_d/2\pi =-64\mathrm{MHz}$. The dashed curves are fits with Lorentzian functions and the arrows represent center frequencies of the peak and dips. They indicate the transitions depicted in the energy-level diagram in the inset by corresponding arrows. Vertical black arrow in the inset represents the transition induced by probe 1. (b) Center frequencies of the observed peaks and dips as a function of $\delta {\omega }_d$. The solid symbols and dashed curves represent the measured and calculated transition frequencies, respectively.

Figure 5

Down-conversion of the microwave field. (a) Output power of the JPA. Blue dots are the measurement data and the solid line is a fit with Eq. (3). A JPA with $G_{\mathrm{s}}=21\mathrm{dB}$ and a bandwidth of $2\pi \times 3.3\mathrm{MHz}$ was used. The black arrow indicates the band center of the JPA. (b) Extracted power spectral density of the down-converted signal. The solid line shows calculated $S_{\text{in}}$ with parameters obtained by the fitting in (a).