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arXiv:cond-mat/0602391 · Physical Review B

Interference in the presence of dissipation

B. Horovitz, Pierre Le Doussal

Published Aug 31, 2006• 16 citations
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Abstract

We study a particle on a ring in the presence of various dissipative environments. We develop and solve a variational scheme assuming low frequency dominance. We analyze our solution within a renormalization group (RG) scheme to all orders which reproduces a two loop RG for the Caldeira-Legget environment. In the latter case the Aharonov-Bohm (AB) oscillation amplitude is exponential in $\ensuremath{-}{R}^{2}$, where $R$ is the ring's radius. For either a charge or an electric dipole coupled to a dirty metal we find that the metal induces dissipation, however the AB amplitude is $\ensuremath{\sim}{R}^{\ensuremath{-}2}$ for large $R$, as for free particles. Cold atoms with a large electric dipole may show a crossover between these two behaviors.

DOI
10.1103/physrevb.74.073104

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We study a particle on a ring in the presence of various dissipative environments. We develop and solve a variational scheme assuming low frequency dominance. We analyze our solution within a renormalization group (RG) scheme to all orders which reproduces a two loop RG for the Caldeira-Legget environment. In the latter case the Aharonov-Bohm (AB) oscillation amplitude is exponential in \ensuremath−R2\ensuremath{-}{R}^{2}\ensuremath−R2, where RRR is the ring's radius. For either a charge or an electric dipole coupled to a dirty metal we find that the metal induces dissipation, however the AB amplitude is \ensuremath∼R\ensuremath−2\ensuremath{\sim}{R}^{\ensuremath{-}2}\ensuremath∼R\ensuremath−2 for large RRR, as for free particles. Cold atoms with a large electric dipole may show a crossover between these two behaviors.
We study a particle on a ring in presence of various dissipative environments. We develop and solve a variational scheme assuming low-frequency dominance. Our solution produces a renormalization-group (RG) transformation to all orders in the inverse dissipation strength and, in particular, reproduces known two-loop results. Our RG leads to a weak dissipation parameter, for which a weak-coupling expansion for the position correlation function shows a $1/{\ensuremath{\tau}}^{2}$ decay in imaginary time.
The amplitude of the Bohm-Aharonov oscillations of a particle moving around a ring threaded by a magnetic flux and coupled to different dissipative environments is studied. The decay of the oscillations on increasing the radius of the ring is shown to depend on the spatial features of the coupling. When the environment is modeled by the Caldeira-Leggett bath of oscillators, interference effects are suppressed beyond a finite length, even at zero temperature. A nontrivial renormalization of the Aharonov-Bohm oscillations is also found when the particle is coupled by the Coulomb potential to a dirty electron gas. A finite renormalization of the Aharonov-Bohm oscillations is obtained for other models of the environment.
The motion of a particle in a ring of length L is influenced by a dirty metal environment whose fluctuations are characterized by a short correlation distance ℓ<<L\ell << Lℓ<<L. We analyze the induced decoherence process, and compare the results with those obtained in the opposing Caldeira-Leggett limit (ℓ>>L\ell >> Lℓ>>L). A proper definition of the dephasing factor that does not depend on a vague semiclassical picture is employed. Some recent Monte-Carlo results about the effect of finite temperatures on "mass renormalization" in this system are illuminated.
Abstract.- We consider a particle coupled to a dissipative environment and derive a perturbative formula for the dephasing rate based on the purity of the reduced probability matrix. We apply this formula to the problem of a particle on a ring, that interacts with a dirty metal environment. At low but finite temperatures we find a dephasing rate ∝ T 3/2, and identify dephasing lengths for large and for small rings. These findings shed light on recent Monte Carlo data regarding the effective mass of the particle. At zero temperature we find that spatial fluctuations suppress the possibility of having a power law decay of coherence. Introduction. – The problem of dephasing of a particle coupled to a dissipative environment at temperature T, and in particular in the limit T → 0 has fascinated the mesoscopic community during the last two decades [1–7]. It has been shown [8,9] that the Caldeira-
We report measurements of Aharonov-Bohm oscillations in normal metal rings in the presence of high-frequency electromagnetic fields. The power dependence of the decoherence time scale ${\ensuremath{\tau}}_{\ensuremath{\phi}}(P)$ agrees well with the anticipated power law ${\ensuremath{\tau}}_{\ensuremath{\phi}}\ensuremath{\propto}{P}^{\ensuremath{-}1∕5}$ when the field-induced decoherence rate ${\ensuremath{\tau}}_{\mathit{ac}}^{\ensuremath{-}1}$ is large compared to the intrinsic decoherence rate ${\ensuremath{\tau}}_{o}^{\ensuremath{-}1}$, measured in the absence of external fields. As theoretically expected, we observe a decline in field-induced decoherence when ${\ensuremath{\tau}}_{\mathit{ac}}^{\ensuremath{-}1}\ensuremath{\leqslant}{\ensuremath{\tau}}_{o}^{\ensuremath{-}1}$. The frequency dependence of ${\ensuremath{\tau}}_{\ensuremath{\phi}}$ shows a minimum in the oscillation amplitude at a characteristic frequency ${\ensuremath{\omega}}_{\mathit{ac}}\ensuremath{\simeq}1∕{\ensuremath{\tau}}_{o}$, where ${\ensuremath{\tau}}_{o}$ is evaluated from the oscillation amplitude using the standard mesoscopic theory. Both the suppression in the oscillation amplitude and the concomitant change in conductivity allow a direct measurement of the intrinsic decoherence time scale.
We study complementarity in a two-terminal closed-loop Aharonov-Bohm interferometer with a quantum dot embedded in an arm of the ring and a nearby charge detector capacitively coupled to the quantum dot. We show that perfect charge detection does riot give full information of the electronic path, so it does not completely suppress the interference. We propose that this property can be verified even in the limit of weak charge detection by investigating the dephasing of the first and the second harmonics of the Aharonov-Bohm oscillation. The first harmonic of the AB oscillation is reduced monotonically as a function of the charge detection strength. In contrast, the second harmonic interference remains unaffected due to the fact that charge detection is not able to detect the electronic path responsible for the second harmonic. Our theory confirms that the particle-like behavior of an electron emerges from the path information itself rather than from the disturbance.