Theory of Quantum Dynamics During a Qubit Readout Process with a Josephson Bifurcation Amplifier

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Hayato Nakano, Kosuke Kakuyanagi, Shiro Saito, Kouichi Semba, and Hideaki Takayanagi*
Physical Science Laboratory, *Tokyo University of Science

@The readout of a qubit state is a typical indirect quantum measurement that is performed with a probe. We report here our successful theoretical analysis of a superconducting qubit readout with a Josephson bifurcation amplifier (JBA) as the probe. We also use this readout method in our experiments.
JBA is a non-linear oscillator, which has two resonance modes (high-amplitude mode E state, and low-amplitude mode G state). Small changes in operational parameters (driving frequency, driving amplitude, etc.) determine which mode is realized. When we make a JBA interact with a qubit, the resulting JBA state reflects a small change in the effective operational conditions depending on the qubit state. So, we can readout the qubit state (microscopic information) as the realized JBA state (macroscopic information) [1]. This constitutes a form of quantum signal amplification. The JBA readout method is suitable for quantum processing because it has almost no detrimental effect on the post-measurement qubit state. The readout process is certainly the quantum time-evolution (quantum dynamics) of a qubit-JBA coupled system. We have clarified the process theoretically. In particular, we have shown how a superposed qubit is projected into one of the measurement basis states probabilistically during this readout process [2].
@As the driving force is increased, the interaction between the qubit and the JBA gradually increases, and the unitary evolution makes the coupled system become an entangled state consisting of two qubit-JBA correlated states (0<t< 1: see figures below). In one of them the qubit is in the excited (e) state, and the JBA is in the G state. In the other, the qubit is in the ground (g) state, and the JBA is in the Gf. As the driving force is further increased, the g-Gf pair starts to transit to g-E. At this point, the decoherence in the JBA destroys the entanglement (t~ 2). In due cource, the coupled system becomes a mixture consisting of an e-G and g-E pair. This means that the coupled system is now (t~ 3) one of the two possible states although we do not know which is realized until we perform a measurement. By judging whether the JBA state is G or E with a usual classical measurement, we can know into which state (g or e) the qubit is projected.

[1] I. Siddiqi et al., Phys. Rev. Lett. 93 (2004) 207002.
[2] H. Nakano, S. Saito, K. Semba, and H. Takayanagi, Phys. Rev. Lett. 102 (2009) 257003.
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Fig. 1. Time-evolution of JBA (a) (b) (c) (d)
(Q-representation).
Fig. 2. Time variation of entanglement strength
between qubit and JBA.

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