Efficient single-photon frequency conversion in the microwave domain using superconducting quantum circuits

We present an approach to achieve efficient single-photon frequency conversion in the microwave domain based on coherent control in superconducting quantum circuits, which consist of a driven artificial atom coupled to a semi-infinite transmission line. Using the full quantum-mechanical method, we analyze the single-photon scattering process in this system and find that single-photon frequency up- or down-conversion with efficiency close to unity can be achieved by adjusting the parameters of the control field applied to the artificial atom. We further show that our approach is experimentally feasible in currently available superconducting flux qubit circuits.

Figure 1

(a) Schematics of a quantum emitter coupled to a one-dimensional waveguide, in which single photons propagate along the arrow direction. The upper part shows the directly coupled cases. The lower part shows the side-coupled cases. (b) Energy-level structure of the $\mathrm{\Delta }$-type emitter (artificial atom). The transitions $|1\rangle \leftrightarrow |2\rangle$ and $|1\rangle \leftrightarrow |3\rangle$ are coupled to the waveguide modes with strength $V_{21}$ and $V_{31}$, respectively. The transition $|2\rangle \leftrightarrow |3\rangle$ is driven by a classical control field with the Rabi frequency $\mathrm{\Omega }$.

Figure 2

Transmission spectra of the frequency down-conversion case. $|T_a{|}^2$ represents elastic transmission, and $|T_b{|}^2$ represents inelastic transmission. (a)–(c) Transmission spectra of the ideal case with no photon loss. (d)–(f) Transmission spectra with photon loss.

Figure 3

Frequency down-conversion efficiency for a Gaussian pulse given by Eq. (47) as function of pulse width $d$ for different values of $\eta ={\mathrm{\Gamma }}_{21}/{\mathrm{\Gamma }}_{31}$. (a) The ideal case with no photon loss. (b) The case with photon loss.

Figure 4

The circuit diagram of a three-junction flux qubit capacitively coupled to the end of a semi-infinite transmission line. The frequency-shifted photon can be measured in reflection. A circulator is used to separate the input and output fields.

Figure 5

Flux bias dependence of energy levels, transition matrix elements, and spontaneous emission rates, for the lowest three states. The circuit parameters are chosen as $\alpha =0.7,\beta =0.5,E_{\mathrm{J}}=80E_{\mathrm{C}}$, and $Z=50\mathrm{\Omega }$. (a) Energy levels, in units of $E_{\mathrm{J}}$, of the flux qubit vs reduced flux $f$. (b) Moduli $|n_{ij}|$ of the transition matrix elements between states $|i\rangle$ and $|j\rangle$ vs $f$. (c) Spontaneous emission rates, in units of $E_{\mathrm{J}}$, between states $|i\rangle$ and $|j\rangle$ vs $f$.

Figure 6

The conversion efficiencies for a monochromatic and on-resonance photon as functions of $f$. Here perfect coupling between flux qubit and transmission line is assumed. The solid line is the down-conversion efficiency, and the dashed line plots the up-conversion efficiency. The flux biases $f=0.4845$ and 0.5155, marked by the vertical thin lines in (a) and (b), are optimal working points for frequency conversion. (a) The conversion efficiency around $f=0.4845$. (b) The conversion efficiency around $f=0.5155$.

Figure 7

Input and output pulses for the case of frequency down-conversion. The input Gaussian pulses with different pulse widths, shown by the solid blue curves, are centered at frequency $20.318\mathrm{GHz}$. (a) The width of the input pulse is $0.05\mathrm{GHz}$. (b) The width of the input pulse is $0.005\mathrm{GHz}$. Dashed blue curves are the elastically scattered (i.e., frequency-unshifted) outputs, and dot-dashed red curves are the inelastically scattered (i.e., frequency-shifted) outputs. The flux bias is set as $f=0.4845$, the qubit-waveguide coupling rate is $99.9\%$, and other circuit parameters are the same as in Fig. 5.

Figure 8

Effects of nonlinearity on the transmission spectra (frequency down-conversion case). (a) Transmission spectra when ${\mathrm{\Omega }}_{\mathrm{p}}=0.01{\mathrm{\Gamma }}_{31}$. (b) Transmission spectra when ${\mathrm{\Omega }}_{\mathrm{p}}=0.5{\mathrm{\Gamma }}_{31}$. (c) Transmission coefficients as a function of ${\mathrm{\Omega }}_{\mathrm{p}}$. Other parameters are the same as in Fig. 7.