Dissipative effects in Josephson qubits☆
Introduction
Among various realizations of quantum bits for quantum information processing [1] solid-state devices appear particularly promising because of their scalability, established fabrication techniques, and flexibility in design. Superconducting devices [2] have a great potential since they combine intrinsic coherence with control possibilities of single-charge systems and the SQUID technology. Recent research in the field produced several breakthrough experiments, such as observation of quantum coherent oscillations in charge Josephson-junction qubits [3], [4], phase qubits (current-biased junctions) [5], [6], and magnetic-flux qubits [7]. Very recently first successful coherent manipulation of two-qubit systems have been demonstrated [8], [9].
In solid-state devices dissipative effects play a prominent role. In these systems the discrete quantum degrees of freedom are coupled to many environmental modes, typically with a dense low-frequency spectrum. The fluctuations of these modes affect the qubits’ dynamics and reduce their quantum coherence. Recent studies, both experimental and theoretical, suggest that the most serious noise sources in Josephson devices are linear fluctuations in the electromagnetic environment – typically with Johnson–Nyquist, or ohmic, spectrum – and various sources of low-frequency noise due to fluctuations in the charge channel (the “background charge fluctuations”) and/or magnetic flux and critical current channels – typically with 1/f spectrum. The latter are particularly dangerous since reducing their level turns out to be difficult in experiments.
In view of the difficulties in reducing the noise level, the alternative approach, namely reducing the effect of the noise, is important. This may be achieved by tuning the system to special “symmetry” points where the lowest-order effects of the noise are suppressed and only higher-order contributions survive. As demonstrated in recent experiments [4], [7] this procedure can increase the coherence time by up to three orders of magnitude as compared to the first demonstration [3] of quantum coherent oscillations in solid-state devices.
Further contributions to the noise and dephasing are produced by quantum detectors, which are necessarily coupled to the qubits in order to monitor the quantum state. Even in the off-state, during coherent manipulations of the qubits’ states, the detectors cannot be decoupled completely from the quantum circuit. Thus they induce additional noise, also typically with ohmic or 1/f spectrum. One of the important parameters, which characterize the efficiency of quantum detectors, is its dephasing effect on the qubit in the off-state.
The coupling to the environment (incl. detectors) leads to dephasing and relaxation processes. In this paper we extend earlier work in this field by analyzing the effect of nonlinear coupling and higher-order contributions of various noise sources. These extensions are especially important at the “optimal” points of the qubit operation and for singular low-frequency noise spectra. After a brief description of a Josephson-junction quantum bit we discuss a diagrammatic technique, similar in spirit to the Keldysh technique, which allows for a systematic description of decoherence effects. Then, after illustrating its application by a few simple examples (incl. the Bloch–Redfield description) in Section 4.3, we proceed to the analysis of nonlinear coupling and higher-order effects.
Section snippets
Josephson qubits, the Hamiltonian and the dissipation
An example of a “Josephson charge qubit” (a superconducting charge box) is shown in Fig. 1. Josephson qubits are nanoscale electronic elements embedded into and manipulated by electrical circuits. The unavoidable electromagnetic environment leads also to dissipation. This environment plays a crucial role in many contexts, e.g., the physics of the Coulomb blockade in tunnel junctions, for which the appropriate theory is reviewed in [10]. In the spirit of this analysis one may also study the
Lowest order: Bloch equations
In the eigenbasis of the spin (qubit) the Hamiltonian (2) readswhere and tanη=Bx/Bz. We denote the ground and excited states of the free qubit by and , respectively. The coupling to the bath is a sum of a transverse (∝sinη) and a longitudinal (∝cosη) term. Only the transverse part can cause spin flips. In the weak-noise limit we consider X as a perturbation and apply Fermi’s golden rule to obtain the relaxation rate, , and excitation rate,
1/f noise
Recent experiments with Josephson circuits have revealed the presence of 1/f noise at low frequencies. While the origin of this noise may be different in different circuits, it appears that in the charge devices it derives from “background charge fluctuations”. It is convenient to present this noise as an effective noise of the gate charge (see Eq. (1)), i.e., SQg(ω)=α1/fe2/|ω|. Recent experiments (cf. [22] and references therein) yield α1/f∼10−7–10−6. We translate this noise into fluctuations
Summary
In this article, we discussed several issues of dissipation in solid-state qubits due to the coupling to the environment. Using the Keldysh diagrammatic technique we analyzed several situations motivated by recent experiments with Josephson-junction quantum circuits. These include the case of a singular low-frequency noise (the 1/f noise) and higher-order effects. The latter are especially important at the so called optimal points of qubit operation when the coupling to the environment is
Acknowledgements
The work is part of the EU IST Project SQUBIT and of the CFN (Center for Functional Nanostructures) which is supported by the DFG (German Science Foundation). Y.M. was supported by the Humboldt foundation, the BMBF, and the ZIP Programme of the German government.
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Dedicated to Ulrich Weiss on the occasion of his 60th birthday.