DOE OSTI2011
We analyze the dynamics of a qubit-resonator system coupled with a thermal bath and external electromagnetic fields. Using the evolution equations for the set of Heisenberg operators that describe the whole system, we derive an expression for the resonator field, that includes the resonator-drive, the resonator-bath, and resonator-qubit interactions. The renormalization of the resonator frequency, caused by the qubit-resonator interaction, is accounted for. Using the solutions for the resonator field, we derive the equation that describes the qubit dynamics. The dependence of the qubit evolution during the measurement time on the fidelity of a single-shot measurement is studied. The relation between the fidelity and measurement time is shown explicitly. We proposed a novel adiabatic method for the phase qubit measurement. The method utilizes a low-frequency, quasi-classical resonator inductively coupled to the qubit. The resonator modulates the qubit energy, and the back reaction of the qubit causes a shift in the phase of the resonator. The resonator phase shift can be used to determine the qubit state. We have simulated this measurement taking into the account the energy levels outside the phase qubit manifold. We have shown that, for qubit frequencies in the range of 8-12GHZ, a resonator frequency of 500 MHz and a measurement time of 100 ns, the phase difference between the two qubit states is greater than 0.2 rad. This phase difference exceeds the measurement uncertainty, and can be detected using a classical phase-meter. A fidelity of 0.9999 can be achieved for a relaxation time of 0.5 ms. We also model and simulate a microstrip-SQUID amplifier of frequency about 500 MHz, which could be used to amplify the resonator oscillations in the phase qubit adiabatic measurement. The voltage gain and the amplifier noise temperature are calculated. We simulate the preparation of a generalized Bell state and compute the relaxation times required for achieving high fidelities. We consider two capacitively coupled phase qubits similar to that used in the experimental work of R. Bialczak et al. (Nature Physics, 6, 409, 2010). The Bell state was created using a NOT gate and a universal entangling gate. We solved the equation of motion for the density matrix with the Bloch relaxation times, T{sub 1} and T{sub 2}. For every qubit we took into consideration all the states in the shallow well including the states outside the qubit manifold. Also we took into consideration unwanted interaction between the qubits during the application of the NOT gate. The NOT gate was implemented using a Gaussian pulse assuming the frequency difference between the two qubits is 200 MHz. An entangling gate was implemented reducing this frequency difference to zero. We have found the optimal parameters for the Gaussian pulse and the optimal duration for the entangling gate. We have shown that the maximum fidelity of 0.99 can be achieved for the minimum relaxation time, T{sub 1} = 1.2 {micro}s, if T{sub 2} = 2T{sub 1}. By slightly increasing the required value of T{sub 1}, one can sharply reduce the required value for T{sub 2}.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗