Dynamics of probing a quantum-dot spin qubit with superconducting resonator photons

The hybrid system of electron spins and resonator photons is an attractive architecture for quantum computing owing to the long coherence times of spins and the promise of long-distance coupling between arbitrary pairs of qubits via photons. For the device to serve as a building block for a quantum processer, it is also necessary to readout the spin qubit state. Here we analyze in detail the measurement process of an electron spin singlet-triplet qubit in quantum dots using a coupled superconducting resonator. We show that the states of the spin singlet-triplet qubit lead to readily observable features in the spectrum of a microwave field through the resonator. These features provide useful information on the hybrid system. Moreover, we discuss the working points which can be implemented with high performance in the current state-of-the-art devices. These results can be used to construct the high fidelity measurement toolbox in the spin-circuit QED system.

Here σ z and σ x are the Pauli matrix defined in the subsapce of |(1, 1)S ⟩ and |(0, 2)S ⟩, ϵ and T c are the detuning and tunneling betwen the two dots, respectively.
The superconducting cavity is usually given as a resonator with the characterized frequency ω r : Here a + and a are the creation and annihilation operators. The electric voltage induced by the resonator field is quantized as with the length L and capacitance C R of the resonator. When the double quantum dot is fabricated inside the superconducting cavity, the interaction between the double dot and the resonator can be given as [1]: Here C c is the capacitive coupling between the dot and the resonator, and C QD is the capacitance of the dot system. Introducing the coupling coefficient we can rewire the interaction Hamiltonian as Taking into account all the terms of the combined system, we obatin For the quantum dot system, the Hamiltonian in the basis of |(1, 1)S ⟩ and |(0, 2)S ⟩ is Diagonalizating the above matrix, there are two eigen states and . (S10) with eigen energies and Here we simplify the algerba by letting the mixing angle θ as and (S14) Working in the eigen basis of |e⟩ and |g⟩, we can transform the Hamiltonian (S7) into Here the energy splitting ω s = √ ϵ 2 + 4T 2 c , σ z and σ x are the Pauli matrix in the basis of |e⟩ and |g⟩, the transverse and longitudinal coupling strength are defined as g x = 1 2 g sin 2θ, g z = 1 2 g cos 2θ, respectively. These results are the Eq. (11) in the main text.

SIGNAL TO NOISE IN THE MEASUREMENT PROCESS
For quantum information processing, a fast and high fidelity readout of qubit states plays an essential role. The readout strategy prsented in this paper depends on the setup in which the electron spin qubit in quantum dots is coupled to microwave resonstor. With the interaction between the qubit and resonator, the resonator field is displaced in a qubit state-dependent fashion.
For the qubit is in state |0⟩ = |(1, 1)T 0 ⟩ and |1⟩ = |(1, 1)S ⟩, the phase shift of the resonator field corresponds to ϕ |0⟩ and ϕ |1⟩ . Thus the signal of the resonator field contains the complete information of the qubit states.
The measurement can be characterized by its signal-to-noise (SNR). The magnitude of the phase shift is proportional to the number of photons in the resonator Here n is the mean number of photons in the resonator, κ is the rate of photons leaving the resonator, t m is the integration time during which the signal accumulates. A conservaticve estimate of the noise of the resonator field is given as δn = k B T n / ω r . (S17) Here T n is the noise temperature of the readout circuit, k B and are the Boltzmann constant and Planck constant. Since the intergation time is set by the decay time of the qubit T 1 = 1/Γ 1 , we obtain the formula SNR = n δn κ 2Γ 1 . (S18)