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arXiv:cond-mat/9710079 · Physical Review Letters

Quantum Decoherence in Disordered Mesoscopic Systems

Dmitry S. Golubev, Andrei D. Zaikin

Published Aug 1, 1998• 154 citations
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Abstract

We point out that the low temperature saturation of the electron phase decoherence time in a disordered conductor can be explained within the existing theory of weak localization provided the effect of quantum (high frequency) fluctuations is taken into account. Making use of the fluctuation-dissipation theorem we evaluate the quantum decoherence time, the crossover temperature below which thermal effects become unimportant, and the weak localization correction $\ensuremath{\delta}\ensuremath{\sigma}$ at $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$. For 1D systems the latter is found to be $\ensuremath{\delta}\ensuremath{\sigma}/\ensuremath{\sigma}\ensuremath{\propto}1/\sqrt{N}$, where $N$ is the number of conducting channels.

DOI
10.1103/physrevlett.81.1074

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We point out that the low temperature saturation of the electron phase decoherence time in a disordered conductor can be explained within the existing theory of weak localization provided the effect of quantum (high frequency) fluctuations is taken into account. Making use of the fluctuation-dissipation theorem we evaluate the quantum decoherence time, the crossover temperature below which thermal effects become unimportant, and the weak localization correction \ensuremathδ\ensuremathσ\ensuremath{\delta}\ensuremath{\sigma}\ensuremathδ\ensuremathσ at T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0. For 1D systems the latter is found to be \ensuremathδ\ensuremathσ/\ensuremathσ\ensuremath∝1/N\ensuremath{\delta}\ensuremath{\sigma}/\ensuremath{\sigma}\ensuremath{\propto}1/\sqrt{N}\ensuremathδ\ensuremathσ/\ensuremathσ\ensuremath∝1/N​, where NNN is the number of conducting channels.
We point out that even at the absolute zero of temperature environmental decoherence limits the destructive interference between time-reversed paths for an electron in a disordered metal, and thus causes the leading (`weak localization') quantum correction to the conductivity to saturate at T=0. Our calculation, which is intended to be illustrative rather than complete, uses a model in which an electron interacts with the fluctuations of the mean voltage in the sample. The average of the fluctuations produces the steady damping well known in Brownian motion, introduces a direction of time, ensures that arbitrarily long time-reversed electron paths lose phase coherence, and is consistent with the experimental observation of a finite low temperature phase-breaking lifetime.
Recently von Delft (JvD) attempted to revise our theory of low temperature decoherence of electrons in disordered conductors with electron-electron interactions [D.S. Golubev and A.D. Zaikin, Phys. Rev. B 59, 9195 (1999)] and argued [Intl. J. Mod. Phys. B 22, 727 (2008)] that by 'more accurate treatment of recoil efiects' he was able to obtain the electron decoherence rate that vanishes at T = 0. In this paper we demonstrate that this JvD's claim is in error, as it is based on ambiguous and uncontrolled manipulations violating basic principles of quantum theory, such as energy-time uncertainty relation, causality, fluctuation-dissipation theorem, detailed balance and the like. While our theory has a simple and transparent relation to the well known P ( E ) theory (which exactly accounts for all recoil efiects) JvD's approach is inconsistent with this theory thus being unable to correctly treat even recoil efiects it was explicitly aimed at.
We develope a theory of a fundamental effect of the interaction-induced decoherence of the electron wave function in a disordered metal. With the aid of the Keldysh technique and the path integral formalism we derive a formally exact equation of motion for the electron density matrix in the presence of interaction. We demonstrate that the effect of interaction of the electron with other electrons and lattice ions in a disordered metal is equivalent to that of an effective dissipative environment. Quantum noise of this environment causes quantum decoherence even at T=0. Our analysis explicitely accounts for the Pauli principle which plays an important role for inelastic scattering processes but turns out not to affect quantum decoherence. Our results seriously challenge the existence of strong localization in low dimensional disordered metals.
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.
The low-temperature phase diagram of one-dimensional weakly disordered quantum systems such as charge or spin density waves and Luttinger liquids is studied by a full finite temperature renormalization-group (RG) calculation. For vanishing quantum fluctuations this approach is amended by an exact solution in the case of strong disorder and by a mapping onto the Burgers equation with noise in the case of weak disorder, respectively. At zero temperature we reproduce the quantum phase transition between a pinned (localized) and an unpinned (delocalized) phase for weak and strong quantum fluctuations, respectively, as found previously by Fukuyama or Giamarchi and Schulz. At finite temperatures the localization transition is suppressed: the random potential is wiped out by thermal fluctuations on length scales larger than the thermal de Broglie wavelength of the phason excitations. The existence of a zero-temperature transition is reflected in a rich crossover phase diagram of the correlation functions. In particular we find four different scaling regions: a classical disordered, a quantum disordered, a quantum critical, and a thermal region. The results can be transferred directly to the discussion of the influence of disorder in superfluids. Finally we extend the RG calculation to the treatment of a commensurate lattice potential. Applications to related systems are discussed as well.