Interference in the presence of dissipation

Abstract

We study a particle on a ring in 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 2 loop RG for the CaldeiraLegget environment. In the latter case the Aharonov-Bohm (AB) oscillation amplitude is exponential in −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 ∼ R −2 for large R, as for free particles. Cold atoms with a large electric dipole may show a crossover between these two behaviors.PACS numbers: 73.23. Ra, The problem of interference and dephasing in presence of dissipative environments is of significance for a variety of experimental systems and a fundamental theoretical issue. The experimental systems include mesoscopic rings embedded on various surfaces where Aharonov-Bohm (AB) oscillations can be measured 1,2 , and the related problem of decoherence at low temperatures 3 . A different type of experimental systems are cold atom traps created by atom chips 4,5,6 , or trapped excited atoms with huge electric dipoles 7 . The atom chip that produces a magnetic or electric trap for the cold atoms necessarily also produces noise. Our problem is then relevant for evaluating the interference amplitude of cold atoms or molecules in presence of such noise.As an efficient tool for monitoring the effect of the environment we follow a suggestion by Guinea 8 to find the AB oscillation amplitude as function of the radius R of the ring, as measured by the curvature 9,10 1/B c of the ground state energy E 0 at external flux φ x = 0, i.e. 1/B c = ∂ 2 E 0 /∂φ 2 x | 0 . For free particles of mass M this amplitude is the mean level spacing ∼ 1/M R 2 . Two types of environments were suggested to lead to an anomalous suppression, i.e. a stronger decrease of the oscillation amplitude than 1/R 2 : system (i) is that of a Caledeira Legget bath and system (ii) of a charge in a dirty metal environment.System (i) is relevant to the Coulomb blockade problem 9,10,11,12 as well as to quantum dots at a distance from metallic gates 13 . This problem has been extensively investigated by instanton methods 14,15,16,17 , by RG methods 8,9,18 , and by Monte Carlo (MC) methods 9,10,19 . All methods show that B c increases exponentially with the dissipation strength α, i.e. B c ∼ α −µ e π 2 α , with differences in the exponent µ. In 2nd order renormalization group 9 (RG) µ = 2 , real time RG gives 18 µ = 6.5, instanton methods give either 14 µ = 2 or 15,17 µ = 3, or 16 µ = 4, while MC suggests 19 µ = 5. This system was also studied by a variational approach 21 which was solved numerically showing a nonperturbative regime at strong α. Since α = γR 2 where γ is a friction coefficient, a length scale π/ √ γ is identified 8 , beyond which the AB oscillations decay.The system (ii) was investigated by RG methods 8 finding B c ∼ R 2+µ ′ with µ ′ 1 nonunivers...

Dephasing of a particle in a dissipative environment

“…We further discuss the significance of the long time decay later in this section after Eq. (18) and in the Summary be demonstrated using simple examples: For an environment that consists of static scatterers we have P survival < 1 but P ϕ = 1 thanks to the second term. For a particle in a ring that interacts with a q=0 environmental mode P survival < 1 but P ϕ = 1 thanks to the third term.…”

“…2 and further discussed below. Using a field theoretical approach [18] it is argued that the renormalized value of the high frequency cutoff is ω c effective = max{ γ, ∆ } (41) ¶ As discussed in section 9 the coherence measure is Γϕ/∆eff. In the regime of main interest Γϕ ≪ T .…”

“…The reasoning is as follows: all the higher frequencies contribute to mass renormalization only, and do not affect the dephasing process. In more details: The significant renormalization starts only below ω c where the linear |ω| dispersion of the dissipation term dominates and leads to ln ω terms in perturbation theory and to the need of either renormalization group or an equivalent variational method [18]. We would like to remark that if we do not apply Eq.…”

“…However in recent works [16,17,18] the mass renormalization concept appears in a new context. The free energy F(T, Φ) of a particle in a ring is calculated, where T is the equilibrium temperature and Φ is the Aharonov Bohm flux the through the ring.…”

Dissipation-Driven Quantum Phase Transitions in a Tomonaga-Luttinger Liquid Electrostatically Coupled to a Metallic Gate

“…We discuss here transitions induced by dissipation, as in [10,11,12]. Some aspects of this work are also related to previous research on 1D systems coupled to environments [13, 14, 0000000000000000000000000000000000000 0000000000000000000000000000000000000 0000000000000000000000000000000000000 0000000000000000000000000000000000000 0000000000000000000000000000000000000 0000000000000000000000000000000000000 0000000000000000000000000000000000000 0000000000000000000000000000000000000 0000000000000000000000000000000000000 0000000000000000000000000000000000000 0000000000000000000000000000000000000 0000000000000000000000000000000000000 0000000000000000000000000000000000000 0000000000000000000000000000000000000 0000000000000000000000000000000000000 1111111111111111111111111111111111111 1111111111111111111111111111111111111 1111111111111111111111111111111111111 1111111111111111111111111111111111111 1111111111111111111111111111111111111 1111111111111111111111111111111111111 1111111111111111111111111111111111111 1111111111111111111111111111111111111 1111111111111111111111111111111111111 1111111111111111111111111111111111111 1111111111111111111111111111111111111 1111111111111111111111111111111111111 1111111111111111111111111111111111111 1111111111111111111111111111111111111 15, 16,17,18]. In particular, the system studied here can be considered a microscopic realization of the model studied in [10].…”