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AIP conference proceedings

Dephasing caused by the Thin Spectrum in a BCS Superconductor

Jasper van Wezel, Jeroen van den Brink

Published Jan 1, 2007• 2 citations
Publisher

Abstract

Recently we have shown that the thin spectrum associated with the process of spontaneous symmetry breaking will lead to a maximum coherence time for many‐particle solid state qubits. In this paper we will discuss how the general mechanism of dephasing caused by a thin spectrum can be applied to the case of BCS superconductors. We will find that the BCS superconducting state is a state of broken global U(1) symmetry, and that the thin spectrum associated with the breaking of this phase symmetry induces dephasing and decoherence within the fundamental timescale τ = 2πNℏ/kBT.

DOI
10.1063/1.2752005

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Recently we have shown that the thin spectrum associated with the process of spontaneous symmetry breaking will lead to a maximum coherence time for many‐particle solid state qubits. In this paper we will discuss how the general mechanism of dephasing caused by a thin spectrum can be applied to the case of BCS superconductors. We will find that the BCS superconducting state is a state of broken global U(1) symmetry, and that the thin spectrum associated with the breaking of this phase symmetry induces dephasing and decoherence within the fundamental timescale τ = 2πNℏ/kBT.
We show that superconductors have a thin spectrum associated with spontaneous symmetry breaking similar to that of antiferromagnets, while still being in full agreement with Elitzur's theorem, which forbids the spontaneous breaking of local (gauge) symmetries. This thin spectrum in the superconductors consists of in-gap states that are associated with the spontaneous breaking of a global phase symmetry. In qubits based on mesoscopic superconducting devices, the presence of the thin spectrum implies a maximum coherence time which is proportional to the number of Cooper pairs in the device. Here we present the detailed calculations leading up to these results and discuss the relation between spontaneous symmetry breaking in superconductors and the Meissner effect, the Anderson-Higgs mechanism, and the Josephson effect. Whereas for the Meissner effect a symmetry breaking of the phase of the superconductor is not required, it is essential for the Josephson effect.
We have recently shown that there is a limit to quantum coherence in many-particle spin qubits due to spontaneous symmetry breaking. These results were derived for the Lieb-Mattis spin model. Here we will show that the underlying mechanism of decoherence in systems with spontaneous symmetry breaking is in fact more general. We present here a generic route to finding the decoherence time associated with spontaneous symmetry breaking in many-particle qubits, and subsequently we apply this approach to two model systems, indicating how the continuous symmetries in these models are spontaneously broken and discussing the relation of this symmetry breaking to the thin spectrum. We then present in detail the calculations that lead to the limit to quantum coherence, which is due to energy shifts in the thin spectrum.
We present a mathematically simple procedure to explain spontaneous symmetry breaking in quantum systems. The procedure is applicable to a wide range of models and can be easily used to explain the existence of a symmetry broken state in crystals, antiferromagnets, and even superconductors. It has the advantage that it automatically brings to the fore the main players in spontaneous symmetry breaking: the symmetry-breaking field, the thermodynamic limit, and the global excitations of a “thin” spectrum.
We investigate the influence of spontaneous symmetry breaking on the decoherence of a many-particle quantum system. This decoherence process is analyzed in an exactly solvable model system that is known to be representative of symmetry broken macroscopic systems in equilibrium. It is shown that spontaneous symmetry breaking imposes a fundamental limit to the time that a system can stay quantum coherent. This universal time scale is t(spon) approximately = 2piNH/(kBT), given in terms of the number of microscopic degrees of freedom N, temperature T, and the constants of Planck (h) and Boltzmann (kB).
By performing experiments with thin-film resonators of NbSi, we elucidate a decoherence mechanism at work in disordered superconductors. This decoherence is caused by charged Two Level Systems (TLS) which couple to the conduction electrons in the BCS ground state; it does not involve any out-of-equilibrium quasiparticles, vortices, etc. Standard theories of mesoscopic disordered conductors enable making predictions regarding this mechanism, notably that decoherence should increase as the superconductor cross section decreases. Given the omnipresence of charged TLS in solid-state systems, this decoherence mechanism affects, to some degree, all experiments involving disordered superconductors. In particular, we show it easily explains the poor coherence observed in quantum phase slip experiments and may contribute to lowering the quality factors in some disordered superconductor resonators.
In superconductors gauge U(1) symmetry is spontaneously broken. According to Goldstone theorem this breaking of a continuous symmetry establishes the existence of the Bogoliubov phase mode while the gauge-invariant response also includes the amplitude fluctuations of the order parameter. The latter, which are also termed ’Higgs’ modes in analogy with the standard model, appear at the energy of the spectral gap 2Δ when the superconducting ground state is evaluated within the weak coupling BCS theory and are therefore damped. Previously we have shown that in the time-dependent Gutzwiller approximation (TDGA) Higgs modes appear inside the gap with a finite binding energy relative to the quasiparticle continuum. Here we show that the binding energy of the Higgs mode becomes exponentially small in the weak coupling limit converging to the BCS solution. On the other hand, well-defined undamped amplitude modes exist in strongly coupled superconductors when the interaction energy becomes of the order of the bandwidth.