Generation of Nonclassical Microwave States Using an Artificial Atom in 1D Open Space

We have embedded an artificial atom, a superconducting transmon qubit, in a 1D open space and investigated the scattering properties of an incident microwave coherent state. By studying the statistics of the reflected and transmitted fields, we demonstrate that the scattered states can be nonclassical. In particular, by measuring the second-order correlation function, $g^{(2)}$, we show photon antibunching in the reflected field and superbunching in the transmitted field. We also compare the elastically and inelastically scattered fields using both phase-sensitive and phase-insensitive measurements.

Figure 1

(A) A micrograph of our artificial atom, a superconducting transmon qubit embedded in a 1D open transmission line. (Zoom In) Scanning-electron micrograph of the SQUID loop of the transmon. (B) Schematic setup for measurement of the second-order correlation function. This setup enables us to do Hanbury–Brown–Twiss measurements between output ports 1 and 2. Depending on the choice of input port, we can measure $g^{(2)}$ of the reflected or transmitted field. (C) Transmittance on resonance as a function of incident power. (Inset) A weak, resonant coherent state is reflected by the atom.

Figure 2

Second-order correlation function of a thermal state, a coherent state, and the states generated by the artificial atom. (A) $g^{(2)}$ of a thermal state and a coherent state as a function of delay time $\tau$. (B) $g^{(2)}$ of the resonant transmitted microwaves as a function of delay time for four different incident powers. Inset: $g^{(2)}(0)$ as a function of incident power. For comparison, the result for a thermal state and a coherent state are also plotted. We see that the transmitted field statistics (red curve) approach that of a coherent field at high incident power, as expected. For a coherent state, $g^{(2)}(0)=1$ is independent of incident power (blue). The peculiar feature of $g^{(2)}$ around zero in the solid theory curves is due to the trigger jitter model (see text). (C) Comparison of $g^{(2)}$ for the transmitted field with and without trigger jitter. (D) $g^{(2)}$ of a resonant reflected field as a function of delay time for two different incident powers. The antibunching behavior reveals the quantum nature of the field. The curves shown here had a digital filter with a 55 MHz bandwidth applied to each detector. Inset: Power dependence of $g^{(2)}(0)$, resulting from a finite BW and temperature. At 0 mK and with infinite BW, $g^{(2)}(0)=0$, independent of incident power (for the power levels considered here). (E) $g^{(2)}$ of a resonant reflected field as a function of delay time at $-131\mathrm{dB}\mathrm{m}$ for different filter bandwidths. As the bandwidth decreases, the antibunching dip vanishes. The solid curves in (D) and (E) are the theory curves, including the trigger jitter model and stray fields. The stray fields arise from background reflections in the line (5%) and leakage through circulator 1 [Fig. 1b] (the same as in previous work [2]), assuming the phase between the leakage field and the field reflected by the atom is $\pi /2$. We extract a temperature of 50 mK from these fits in (B), (D) and (E), with no additional free-fitting parameters. The error bar indicated for each data set is the same for all the points. (F) The progression of $g^{(2)}(\tau )$ degradation, due to temperature, BW, trigger jitter, and stray fields. Locally around $g^{(2)}(0)$, the red, green, and dark blue curves exhibit a tiny bunched feature arising from the 50 mK thermal field.

Figure 3

Elastic vs inelastic scattering from the artificial atom. (A) A microwave pump is applied at ${\omega }_{01}$. As the power of the ${\omega }_{01}$ pump increases, the Mollow triplet appears in the spectrum with peak separation equal to the Rabi frequency ${\Omega }_p$. (inset) Dressed state picture of the energy levels. (B) The coherently or elastically reflected power (phase-sensitive average, red curve) or total reflected power (phase-insensitive average, green, purple, and blue curves) as a function of resonant incident powers for different bandwidths (BW). The total power reflected is the sum of both the elastic and inelastic fields, after we subtract off the amplifier noise power. Solid curves are the theory fits to experimental data. The black dash curve shows the input power for comparison. At low powers, $N\ll 1$, we observed that the reflected field is mostly first-order coherent: the elastically reflected power is the same as the total reflected power. At high powers, $N>1$, more and more photons are inelastically scattered as the Mollow triplet begins to emerge. The wider the measurement bandwidth, the more of the Mollow triplet we capture. Note that the output power includes the 79 dB gain of the amplifiers.