Rabi oscillations of a qubit coupled to a two-level system

– The problem of Rabi oscillations in a qubit coupled to a fluctuator and in contact with a heath bath is considered. A scheme is developed for taking into account both phase and energy relaxation in a phenomenological way, while taking full account of the quantum dynamics of the four-level system subject to a driving AC field. Significant suppression of the Rabi oscillations is found when the qubit and fluctuator are close to resonance. The effect of the fluctuator state on the read-out signal is discussed. This effect is shown to modify the observed signal significantly. This may be relevant to recent experiments by Simmonds et al. (Phys. Rev. Lett., 93 (2004) 077003). Recent experiments have demonstrated Rabi oscillations in macroscopic quantum systems (qubits) [1–7]. These oscillations decay rather quickly due to interaction with the environment even in pure systems. An interesting behavior was observed in spectroscopic experiments with Josephson qubits [1]. Both the decay rate and the oscillation pattern were strongly dependent on the qubit eigenfrequency ramped by external parameters. Namely, the oscillations were significantly suppressed in the vicinity of certain eigenfrequencies. This was interpreted [1] as an influence of some two-level systems (fluctuators) located in the qubit environment. The suppression is strong when the fluctuator’s and qubit’s energy splittings are very close. Below we present a theory of a qubit interacting with an external AC field and a single fluctuator coupled to the qubit. In addition, we assume that the system interacts with some thermal bath providing both phase and energy relaxation. The account of the energy relaxation distinguishes our model from a similar approach used in ref. [8], whereas in the recent numerical work of ref. [9] no account is taken of the phase relaxation. By solving the kinetic equation for the density matrix we compute level populations vs. time. The results (∗) E-mail: [email protected] c © EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2005-10053-y 22 EUROPHYSICS LETTERS explain the strong influence of the resonant fluctuator on the Rabi oscillations of the qubit in agreement with experimental findings. An alternative interpretation of the same experiments was offered in ref. [10] and we do not believe that this can be ruled out on the basis of current experimental knowledge. We will characterize the qubit by a spin S interacting with a fluctuator represented by a spin s. The Hamiltonian in the form H̃(t) = Hq +Hf +Hq-f +Hman(t) (1) takes into account the qubit and the fluctuator (Hq and Hf) as well as the interaction between them (Hq-f). We also included into H̃(t) the “manipulation” part, Hman(t) = SxF cosωt, which in spin terms is an oscillating magnetic field in the x-direction applied to the qubit (F is the Rabi frequency). The other parts of the Hamiltonian can be written as Hq = E2 Sz , Hf = e 2 sz , Hq-f = u2 (Sxsx + Sysy). (2) Here E is the distance between the qubit levels, e is the distance between the fluctuator’s levels, Si and si are the Pauli matrices acting, respectively, in the spaces of the qubit and fluctuator, u is the off-diagonal coupling constant(). Below we will use the so-called rotating wave approximation replacing cosωt → (1/2) exp[iωt], see, e.g., Slichter [12]. This approximation, which neglects higher harmonics of the response, is valid close to the resonance, i.e., when |E − ω|, |e − ω|, |u| ω. We also omit the diagonal qubit-fluctuator interaction ∝ Szsz which is important only when the qubit and the fluctuator are far from the resonance. On the contrary, the off-diagonal part is important only when the qubit and the fluctuator have close energy splitting, |E − e| E, e. The importance of this interaction was stressed in ref. [1]. We will also assume that |u| ω T, (3) where ω and T are measured in energy units. Only in this case does the qubit act as a resonant system. We are not going to take the thermal bath explicitly into account. Instead we introduce damping phenomenologically into the equation for the density matrix evolution. The Hamiltonian (1) in the resonant approximation can be expressed as a 4× 4 matrix: H = 1 2   −E − e 0 Fe 0 0 −E + e u Fe Fe−iωt u∗ E − e 0 0 Fe−iωt 0 E + e   . (4) Shown in fig. 1 are the energy terms of the Hamiltonian vs. the qubit energy splitting E for a fixed fluctuator splitting e and u = 0.1e. Near the resonance, E = e, all 4 levels are involved in the AC-induced transitions. In this region one can expect strong influence of the fluctuator on the qubit response. The 4-level system qubit+fluctuator is characterized by a 4×4 density matrix ρ̃μν(t) whose diagonal matrix elements n↓↓, n↓↑, n↑↓, n↑↑ describe the occupations of each of the levels. In (1)We have taken into account only the off-diagonal coupling, (u/2)(S⊥·s⊥), between the qubit and fluctuator. One can also include Hz = (v/2)(Sz · sz). This interaction contributes to the decoherence of the qubit via random modulation of the Rabi frequency [11]. However, as we checked, it does not specifically affect the visibility of Rabi oscillations as long as the fluctuator is decoupled from the AC field, H12 = H34 = 0. Y. M. Galperin et al.: Rabi oscillations of a qubit 23