Resonant excitation of single and coupled qubits for coherent quantum control and microwave detection

Abstract

Resonant driving enables coherent control of quantum systems, including single and coupled qubits. From a complementary perspective, transitions of a quantum system can be exploited for the detection of microwave photons. In this work, we theoretically investigate resonant multiphoton excitations in a system of qubits. When the energy of K photons matches the energy splitting of the qubit system, the absorption of these photons leads to collective excitation of the qubits. We focus on the case of two coupled qubits and analyze the quantum dynamics of both excitation and relacation processes. In the particular case where only a single qubit is relevant and the remaining qubits can be neglected, the dynamics admits an analytical treatment. We examine multiphoton resonances, the Bloch-Siegert shift, and population inversion, phenomena that are central to both coherent quantum control and microwave photon detection.

Figures (5)

• Figure 1: Sketch of the model system. Two flux qubits are coupled with the interaction energy $J$. The constant biases are different for the two qubits ($\varepsilon_{01}$ and $\varepsilon_{02}$), while the microwave driving $A\cos(\omega t)$ is the same. The qubits are also coupled to a dissipative environment.
• Figure 2: Levels and occupations of the microwave driven two-qubit system. Upper panel shows the eigenenergies of the Hamiltonian (\ref{['4-level-Hamiltonian']}) (the solid curves) as functions of $\varepsilon_{02}$ at a constant bias $\varepsilon_{01}=2\,\Delta_2$. Other parameters are $\Delta_1=1.5\,\Delta_2$, $J=0.82\,\Delta_2$. The dashed straight lines stand for the energies in the diabatic basis (at $\Delta_1=\Delta_2=0$). Middle and lower panels present the Rabi oscillations calculated in the basis of the eigenfunctions of the stationary Hamiltonian (\ref{['4-level-H0']}) with dissipation taken into account. Middle panel corresponds to the frequency of the external signal resonant with the levels 2 and 3, i.e., $\omega=(E_3-E_2)/\hbar$. For lower panel we took $\omega=(E_2-E_1)/\hbar$. Other parameters are $\varepsilon_{02}=4.8\,\Delta_2$, $A=0.2\,\Delta_2$, the dissipation factor $\alpha=0.01$, $k_\text{B}T=\Delta_2$.
• Figure 3: The averaged over time occupations (in the diabatic basis) as functions of the driving frequency $\omega$. The circles correspond to the numerical solution of Eq. (\ref{['2-level-system-of-equations']}), while the black and blue curves stand for ${\overline\rho}_{\downarrow\downarrow}$ and ${\overline\rho}_{\uparrow\uparrow}$, respectively, calculated analytically by using Eq. (\ref{['averaged-probability']}). For panel (a) the parameters are chosen so that the RWA works rather well, namely, the offset $\varepsilon_0=62\,\text{GHz}\cdot h$, the driving amplitude $A=70\,\text{GHz}\cdot h$, the tunneling amplitude $\Delta=0.2\,\text{GHz}\cdot h$, the photon number $K=8$, and the relaxation rate $\Gamma=5\cdot 10^{-3}\,\text{GHz}\cdot h$. In panel (b) the green and red curves are obtained from the analytical solution (\ref{['averaged-probability']}) by shifting the abscissa by $s(\tilde{n})$ [the Bloch-Siegert shift, see the discussion below Eq. (\ref{['least-squares-method']})], and this shifted analytical solution coincides with the numerical solution (circles). The parameters used are the following: $\varepsilon_0=62\,\text{GHz}\cdot h$, $A=45\,\text{GHz}\cdot h$, $\Delta=1\,\text{GHz}\cdot h$, $K=8$, and $\Gamma=5\cdot 10^{-3}\,\text{GHz}\cdot h$.
• Figure 4: The averaged over time qubit occupation probabilities as functions of the driving frequency for different values of the driving amplitude. The green and red colors correspond to ${\overline\rho}_{\downarrow\downarrow}$ and ${\overline\rho}_{\uparrow\uparrow}$. Top to bottom: $A=100\,\text{GHz}\cdot h$, $A=120\,\text{GHz}\cdot h$, $A=150\,\text{GHz}\cdot h$, $A=200\,\text{GHz}\cdot h$. Other parameters are the following: the offset $\varepsilon_0=100\,\text{GHz}\cdot h$ (which corresponds to the photon number $K\approx 100$), the tunneling amplitude $\Delta=10\,\text{GHz}\cdot h$, and the relaxation rate $\Gamma=0.01\,\text{GHz}\cdot h$.
• Figure 5: The time-averaged qubit occupation probabilities as functions of the driving frequency for the case of the frequency-dependent amplitude [as in Eq. (\ref{['A-on-omega']})] and for different values of the overall factor $A$. The green and red colors correspond to ${\overline\rho}_{\downarrow\downarrow}$ and ${\overline\rho}_{\uparrow\uparrow}$. The amplitude $A$ is as follows: (a) $30\,\text{GHz}\cdot h$, (b) $48\,\text{GHz}\cdot h$, (c) $60\,\text{GHz}\cdot h$, and (d) $80\,\text{GHz}\cdot h$. Other parameters are the following: the offset $\varepsilon_0=40\,\text{GHz}\cdot h$, the tunneling amplitude $\Delta=7\,\text{GHz}\cdot h$, the relaxation rate $\Gamma=0.004\,\text{GHz}\cdot h$, the resonator frequency $\omega_0/2\pi=6.884\,\text{GHz}$, and the FWHM $\kappa=2\pi\times 0.006\,\text{GHz}$ (the quality factor $Q\approx 1990$).