Why macroscopic quantum tunnelling in Josephson junctions differs from tunnelling of a quantum particle

We show that the macroscopic quantum tunnelling of a fluxon in a Josephson junction cannot be described, even qualitatively, as the tunnelling of a quantum particle in a potential U(φ), where the phase difference φ plays the role of particle position, if the length of the junction d exceeds a fluxon length. We calculate the probability per unit time of tunnelling (or escape rate), Γ, which has a form Γ=A exp(−B). In contrast to particles, where the B is proportional to d, our field-theory predicts a different behavior of B for either usual, 0–π, or stacks of Josephson junctions, giving rise to a renormalization of Γ by many orders of magnitude. Copyright c © EPLA, 2007 Macroscopic quantum tunnelling (MQT) is one of the few manifestations of quantum effects in macroscopic systems [1]. MQT was experimentally observed [1] in Josephson junctions (JJs) in 1980s and has been studied for different Josephson systems. General interest on MQT is now fuelled not only by its fundamental interest but as a readout mechanism for phase qubits [2]. A new surge of interest on MQT occurred after the recent discovery of MQT in high-temperature layered superconductors [3,4]. The observed giant enhancement of MQT could be attributed to the spatial structure of the tunnelling fluxon [5]. This is in contrast to the standard approach, where MQT is associated with a quantum particle tunnelling through an effective potential barrier. The latter approach is correct only for very short junctions, when one can ignore the spatial dependence of the phase difference φ across the junction. A naive guess would be that the spatial dependence of φ should suppress the MQT because of an increase of the potential energy U(φ) from (∇φ)2. This is in analogy to the positive energy of a domain wall in a ferromagnet. However, further analysis shows that the total potential energy can decrease similarly to the decrease of energy for a ferromagnet having several domains. Our work was motivated by the MQT recently observed in high-temperature layered superconductors. However, we emphasize that the problem we are considering is far more general and is also applicable to MQT in long Josephson junctions (see, e.g., [6]). Moreover, MQT in long Josephson junctions is of relevance for the so-called phase qubits which might provide basic elements for future quantum information processing devices (see ref. [2] for a review). These phase qubits are current-biased JJs driven by microwaves, and exhibiting MQT. Controlling the motion of Josephson vortices is also of interest for the design of novel types of THz and ratchet devices [7]. Using a field-theory approach, here we show that the fluxon tunnels through the formation of single nucleus followed by a classical motion through the contact. For a long junction this results in a huge renormalization of the tunnelling escape rate Γ, with respect to the “particle” approximation. The tunnelling probability differs for the usual and 0–π junctions: for usual junctions, log Γ(d) is proportional to d for d λJ and shows a maximum at d∼ λJ ; while log Γ(d) for 0–π junctions increases slower with d for d λJ and does not exhibit a maximum within the studied range of parameters. We also demonstrate that the strong renormalization of Γ for stacks of intrinsic JJs occurs even for very short junctions of about 1μm, in agreement with very recent experiments [4]. This offers potentially useful flexibility when designing readouts for phase qubits [2].