Resonant excitations of single and two-qubit systems coupled to a tank circuit

The interaction of flux qubits with a low-frequency tank circuit is studied. It is shown that changes in the state of the interacting qubits influence the effective inductance and resistance of the circuit, which is the essence of the so-called impedance measurement technique. The multiphoton resonant excitations in both single flux qubits and pairs of coupled flux qubits are investigated. In particular, we compare our theoretical results with recent spectroscopy measurements, Landau-Zener interferometry, and the multiphoton fringes.

Figure 1

Two-qubit system coupled to a tank circuit. The flux qubits are pierced by magnetic fluxes ${\Phi }_x^{\left(a,b\right)}$ induced by the currents in the controlling coils (not shown in the scheme) and by the current in the tank inductor. The qubits are coupled to each other and to the tank circuit. The resonant tank circuit consists of the inductor $L_T$, capacitor $C_T$, and resistor $R_T$; the circuit is biased with a rf current $I_{\text{bias}}$. The tank voltage $V$ is the measurable value.

Figure 2

(Color online) Influence of temperature on the ground-state measurement: the dependence of the tank phase shift on the flux detuning $f_{\text{dc}}={\Phi }_{\text{dc}}/{\Phi }_0-1/2$ when the qubit is thermally excited. The numbers next to the curves stand for the temperature in gigahertz. Inset: temperature dependence of the width $\Delta f_{\text{dc}}$ of the dip at half depth in the phase shift shown in the main panel.

Figure 3

(Color online) Curves for a resonantly excited flux qubit: (a) the phase shift and (b) the amplitude of the tank voltage versus flux detuning $f_{\text{dc}}$. In panel (a) the numbers next to the curves stand for the driving amplitude $f_{\text{ac}}$ multiplied by $1\times {10}^3$; in panel (b) the curves are for the amplitudes $f_{\text{ac}}\left(1\times {10}^3\right)=1$, 1.5, and 3 from bottom to top.

Figure 4

(Color online) Multiphoton excitations of a single flux qubit. (a) Dependence of the phase shift $\alpha$ and (b) the amplitude $v$ on the bias current frequency ${\omega }_{\text{rf}}$ and the flux detuning $f_{\text{dc}}$; the parameters are $f_{\text{ac}}=8\times {10}^{-3}$, $\omega /2\pi =4.15\text{GHz}$, and $S=0.8$.

Figure 5

(Color online) Landau-Zener interferometry: dependence of the tank phase shift on the flux detuning $f_{\text{dc}}={\Phi }_{\text{dc}}/{\Phi }_0-1/2$ and on the driving flux amplitude $f_{\text{ac}}={\Phi }_{\text{ac}}/{\Phi }_0$.

Figure 6

(Color online) Impact of the finite bias current: dependence of the phase shift $\alpha$ on the bias flux $f_{\text{dc}}$ for different values of the bias current amplitude $I_A$, for $\omega /2\pi =4\text{GHz}$ and $f_{\text{ac}}=8\times {10}^{-3}$.

Figure 7

(Color online) Multiphoton excitations of coupled flux qubits: (a) energy levels, (b) total probability of the currents in qubits to flow clockwise $Z$, (c) the tank phase shift $\alpha$, and (d) the entanglement measure $\mathcal{E}$ versus the bias $f_{\text{dc}}=f_a=f_b$.

Figure 8

(Color online) Resonant excitation of two coupled flux qubits: the tank phase shift versus partial bias fluxes in two qubits, $f_a$ and $f_b$.

Figure 9

(Color online) Multiphoton resonances in coupled qubits: dependence of the tank phase shift on the flux detuning in qubits $a$ and $b$, $f_a$ and $f_b$.

Figure 10

(Color online) Experimental demonstration of multiphoton excitations in a single flux qubit. (a) Dependence of the phase shift $\alpha$ and (b) the amplitude $v$ on the bias current frequency ${\omega }_{\text{rf}}$ and the flux detuning $f_{\text{dc}}$ recorded for a microwave excitation with $\omega /2\pi =4.15\text{GHz}$ and amplitude $f_{\text{ac}}=4.5\times {10}^{-3}$.

Figure 11

(Color online) Experimental dependence of the phase shift $\alpha$ on the bias flux $f_{\text{dc}}$: (a) multiphoton resonances for driving amplitudes $f_{\text{ac}}/{10}^{-3}=0.5$, 2.5, 4.5, and 6.5 from top to bottom; (b) influence of the bias current (red line $I_{A}M/{\Phi }_0=2\times {10}^{-6}$ and blue line $I_{A}M/{\Phi }_0=8\times {10}^{-6}$) for $\omega /2\pi =4.15\text{GHz}$.