Landau-Zener-Stückelberg-Majorana lasing in circuit quantum electrodynamics

We demonstrate amplification (and attenuation) of a probe signal by a driven two-level quantum system in the Landau-Zener-Stückelberg-Majorana regime by means of an experiment, in which a superconducting qubit was strongly coupled to a microwave cavity, in a conventional arrangement of circuit quantum electrodynamics. Two different types of flux qubit, specifically a conventional Josephson junctions qubit and a phase-slip qubit, show similar results, namely, lasing at the working points where amplification takes place. The experimental data are explained by the interaction of the probe signal with Rabi-like oscillations. The latter are created by constructive interference of Landau-Zener-Stückelberg-Majorana (LZSM) transitions during the driving period of the qubit. A detailed description of the occurrence of these oscillations and a comparison of obtained data with both analytic and numerical calculations are given.

Figure 1

(a) Normalized power transmission maximum in dB for the resonator coupled to the Josephson junction qubit as a function of the qubit energy bias and the driving amplitude at frequency $\omega /2\pi =7.444$ GHz. The transmission maxima were obtained by a Lorentzian fit of the measured transmission spectra. The transmission is quasiperiodically increased and suppressed, revealing characteristic LZSM interference patterns. Here, $\mathrm{\Delta }/h=12.2$ GHz, ${\omega }_{\mathrm{r}}/2\pi =2.481$ GHz, and the ratio $\mathrm{\Delta }/\hslash \omega \approx 1.6$, which is closer to the slow-passage limit (see main text). (b) Normalized power transmission spectra (data points) at driving amplitudes corresponding to the transmission maximums at zero bias and the corresponding Lorentzian fits (solid lines). Here, $\delta \omega$ is the detuning of the weak probe signal from the resonator's fundamental frequency $\delta \omega ={\omega }_{\mathrm{p}}-{\omega }_r$. (c) Spectral power density of the microwave radiation emitted by the resonator. The data points correspond to emission without driving (squares) and driving amplitude set to 2.40 V (circles) at zero bias. The black dashed lines corresponds Lorentzian fits. Similar to the transmission measurements, for driving turned ON, the resonator emission is increased and it's bandwidth is narrowed. To illustrate the amplification, a weak probe signal is applied in the bandwidth of the resonance in the absence of driving (gray dashed line) and with driving (red solid line). For driving set ON, the emission is locked to the probe frequency and energy is transferred, which is visible by the shrinkage of the Lorentzian shaped emission curve.

Figure 2

(a) Spectral power density in dB emitted by the resonator with the QPS qubit. LZSM lasing for small ratio $\mathrm{\Delta }/\hslash \omega =0.43$, which corresponds to the fast-passage limit (see main text). The position of the resonant amplification and attenuation points corresponds to the one- and two-photon Rabi oscillations, ${\mathrm{\Omega }}_{\mathrm{R}}^{\left(k\right)}={\omega }_{\mathrm{d}}$ with $k=1,2$. Here, $\mathrm{\Delta }/h=6.12$ GHz, $\omega /2\pi =16.3$ GHz and the emission is measured at ${\omega }_3/2\pi =6.967$ GHz. (b) Normalized power transmission through the resonator at ${\omega }_3$ without (black triangles) and with driving at 16.3 GHz (red open dots) and the measured emission under the same conditions with driving (gray squares). The lines correspond to Lorentzian fits.

Figure 3

(a) Energy levels $E$ of a two-level system as a function of the energy bias $\varepsilon$ of a superconducting qubit with energy level splitting $\mathrm{\Delta }$. The energy bias $\varepsilon$ is driven with a sinusoidal driving signal at frequency $\omega$. Under driving, the two-level system undergoes subsequent LZSM transitions. (b) Crossover between the subsequent LZ transitions and Rabi-like oscillation, resulting from constructive interference. The upper-level occupation probability is plotted as a function of time for many periods of the driving field. For ${\varepsilon }_0=0, A/\mathrm{\Delta }=15.71$, and $\hslash \omega /\mathrm{\Delta }\approx 0.05$, which corresponds to $P_{\mathrm{LZ}}=0.13\ll 1$ the time evolution shows destructive interference of subsequent LZ transitions (black curve). If the amplitude is slightly varied to $A/\mathrm{\Delta }=15.75$ the constructive interference leads to Rabi-like oscillations approximated by the dashed sinusoidal line. The frequency of the Rabi-like oscillations is given by $\mathrm{\Omega }\ll \omega$, see Eq. (1). Note that these Rabi-like oscillations appear far from resonance, at $\omega \ll \mathrm{\Delta }E/\hslash$. Position of the expected resonant interactions between the driven qubit and the weak probe signal, as defined by Eq. (2), are shown for the slow-passage limit in (c) and the fast-passage limit in (d). The following parameters were taken: ${\omega }_{\mathrm{p}}/2\pi =2.5$ GHz, $\omega =3{\omega }_{\mathrm{p}}, \mathrm{\Delta }/h=12.2$ GHz $>\omega /2\pi$ for (c) and $\mathrm{\Delta }/h=3\mathrm{GHz}\ll \omega /2\pi$ for (d). The inclined red lines in (c) and (d) mark the region of the validity of the theory: ${\varepsilon }_0<A$, which means that the system experiences avoided level crossings.

Figure 4

(a) Energy levels $E$ of a strongly coupled superconducting qubit-resonator system, as a function of the energy bias ${\varepsilon }_0$ of the qubit. In the AIM model, the system mimics an array of beam splitters placed at avoided-level crossings. (b) The average number of photons in the resonator in logarithmic scale, calculated by the AIM as a function of the qubit energy bias and the driving amplitude. The black dashed lines correspond to resonance condition given by Eq. (2). The obtained maximum number of photons in the resonator is approximately 8 for system parameters: ${\omega }_{\mathrm{r}}/2\pi =2.5$ GHz, $\omega =3{\omega }_{\mathrm{r}}, \mathrm{\Delta }/h=12.2$ GHz, and for simplicity ${\mathrm{\Omega }}_{m,m+5}/h=10$ MHz.

Figure 5

The upper-level occupation probability $P_{+}\left(t\right)$ as a function of time for nonzero bias. Parameters are the following: $A/\mathrm{\Delta }=15.75, \hslash \omega /\mathrm{\Delta }=0.05, {\varepsilon }_0/\mathrm{\Delta }=5.1$, and 5.17 for the black and red curves, respectively.

Figure 6

(a) The experimental setup scheme for transmission/emission measurements carried out by a vector network analyzer (VNA)/power spectrum analyzer (PSA). During an emission measurement, to demonstrate the amplification presented in Fig. 1, an additional probe signal was applied from a generator (GEN) at ${\omega }_s$. The qubit was biased by dc magnetic field of two superconducting coils mounted to a copper sample holder in Helmholtz geometry. The coils were fed by a dc current source (CS) filtered by carbon powder filters placed at 3K plate. The qubit was strongly driven by a microwave signal generator (GEN) at frequency $\omega /2\pi$ biasing an excitation loop through an additional coaxial line. (b) The simplified scheme of the measurement. The resonator's transmission was measured at ${\omega }_s$ by vector network analyzer (VNA). The qubit was inductively coupled to the resonator and to a separate excitation loop as well, to drive the qubit at ${\omega }_d$.