Figure 1

A two level qubit is coupled to a many level harmonic oscillator, investigated for two different oscillator energies. First, the oscillator resonant frequency is set to $1.36\mathrm{GHz}$, this more resembles the conventional configuration such that the fundamental component of the oscillator does not drive the qubit. However, we also investigate the use of a high frequency oscillator of $3.06\mathrm{GHz}$ which can excite this qubit. In addition, qubit is constantly driven by a microwave field at $3.49\mathrm{GHz}$ to generate Rabi oscillations and in this paper we examine the relation between these three fields.Reuse & Permissions

Figure 2

(Color online) An abstract layout of a simple single junction Cooper-pair box, with the biasing voltage connection to the first filter stage modeled by a resistor and inductor. The Cooper pair box is modeled by an equivalent circuit network, consisting of three capacitors: $C_J$—the capacitance of the Josephson junction, $C_g$—the capacitance of the island gate, and $C_p$—the parasitic capacitance formed between the biasing electrodes. These capacitors, together with the resistor and inductor, form an RLC oscillator circuit, the inductance value is changed to investigate the two oscillator frequencies (1.36 and $3.06\mathrm{GHz}$).Reuse & Permissions

Figure 3

(Color online) Time averaged Floquet qubit energy levels, with one and two photon transitions (3.49 and $6.98\mathrm{GHz}$). The following results focus on the region surrounding the single photon transition $\left(3.49\mathrm{GHz}\right)$ at $\text{bias}=0.5187$. The solid upper and lower lines are the time averaged energies of the two states during microwave drive, the dashed upper and lower lines are the undisturbed energies. The energy diagram clearly indicates the bias values near which the microwave transitions exist. Time averaged Floquet energies are necessary to conveniently show the transitions as the microwave drive is a time varying field.Reuse & Permissions

Figure 4

(Color online) Oscillator and qubit power spectra slices for $\text{bias}=0.5187$, using the low frequency oscillator circuit $f_{\mathit{osc}}=1.36\mathrm{GHz}$. The solid lines overlay the energy level separations found in Fig. 3 $(\kappa =5\times {10}^{-5})$. As one would expect, the bias oscillator peak at $1.36\mathrm{GHz}$ is clearly observed in the oscillator PSD, but only weakly in the qubit PSD. Likewise the qubit Rabi frequency is found to be stronger in the qubit PSD. However, it is important to note that the qubit dynamics such as the Rabi oscillations are indeed coupled to the bias oscillator circuit and so can be extracted. In addition, it is recommended to compare the layout of the most prominent features with Fig. 6.Reuse & Permissions

Figure 5

(Color online) (A) Oscillator power spectra when the coupled qubit is driven at $f_{\mathit{mw}}=5.00\mathrm{GHz}$. An increase in bias noise power $(f_{\mathit{osc}}=1.36\mathrm{GHz})$ can be observed when Rabi oscillations occur, the more frequent quantum jump noise couples back to the oscillator. (B) Bias noise power peak position changes as a function of $f_{\mathit{mw}}$, the microwave drive frequency. Therefore it is possible to probe the qubit energy level structure by using the power increase in the oscillator which is already in place, eliminating the need for additional measurement devices. However, it should be noted that the surrounding oscillator harmonics may mask the microwave driven peak $(\kappa =1\times {10}^{-3})$.Reuse & Permissions

Figure 6

(Color online) Oscillator PSD as a function of bias, for microwave amplitudes $A_{\mathit{mw}}=0.0025$ (A) and $A_{\mathit{mw}}=0.0050$ (B). The red lines track the positions (in frequency) of significant power spectrum peaks ($+10\text{to}+15\mathrm{dB}$ above background), the overlaid black and blue lines are the qubit energy and microwave transition (Fig. 3). Unlike Fig. 4, in these figures the $3.06\mathrm{GHz}$ oscillator circuit can now drive the qubit (Fig. 3) and so creates excitations which mix with the microwave driven excitations creating a secondary splitting centered on $f_{\mathit{mw}}-f_{\mathit{osc}}$ $\left(430\mathrm{MHz}\right)$. This feature contains the Rabi frequency information in the sidebands of the splitting, but now in a different and controllable frequency regime. In addition, the intersection of the two differently driven excitations (illustrated in the magnified sections) opens the possibility of calibrating the biased qubit against a fixed engineered oscillator circuit, using a single point feature $(\kappa =5\times {10}^{-5})$.Reuse & Permissions

Figure 7

(Color online) Oscillator PSD as a function of the applied microwave drive frequency $f_{\mathit{mw}}$, for microwave amplitudes $A_{\mathit{mw}}=0.0050$ (A) and $A_{\mathit{mw}}=0.0100$ (B). It is important to notice that there are now two frequency axes per plot, a drive (H) and a response (V). Of particular interest is the magnified section which shows clearly the distinct secondary splitting in the sub-GHz regime. This occurs due to a high frequency interaction seen in the upper plots, where the lower Rabi sideband of the microwave drive passes through the high frequency oscillator signal. The maximum splitting occurs when the Rabi amplitude is a maximum, hence this is observed for a very particular combination of bias and drive, which is beneficial for characterizing the qubit. Most importantly, this would not be observed with a conventional low frequency oscillator configuration as the $f_{\mathit{mw}}-f_{\mathit{osc}}$ separation would be too large for the Rabi frequency $(\kappa =5\times {10}^{-5})$.Reuse & Permissions