Stern, Aharonov, and Imry [Phys. Rev. A 41, 3436 (1990)] unified the two standard pictures of decoherence of a two-path interferometer — loss of overlap of the final environment states, and statistical uncertainty of the relative phase — by proving that the Feynman–Vernon influence functional equals ⟨eiϕ⟩\langle e^{i\phi}\rangle for a Hermitian phase operator ϕ\phi. Their construction, however, assumed effectively commuting path phases, two prescribed classical paths, and narrow or Gaussian phase distributions for quantitative estimates, and the paper explicitly deferred the general case. This problem formulates the surviving open core of that deferral: extend the operator-level phase-statistics picture to dynamical, back-reacting environments shared by both partial waves, in which the phase operators associated with different portions of the paths do not commute and the influence functional is an average of a product of noncommuting unitaries; determine how the presence of the environment modifies the selection of the interfering path pairs within the full sum over histories; and derive general quantitative conditions under which the dephasing (phase-breaking) rates of the h/eh/e and h/2eh/2e conductance oscillations of realistic rings differ. Later literature substantially resolved the non-Gaussian phase-distribution part (non-perturbative treatments of shot-noise dephasing) and supplied model-specific quantitative results for the electron–electron-interaction part (harmonic-dependent dephasing lengths in diffusive rings); the operator-level, path-selection, and general (model-independent) harmonic-dependent phase-breaking parts remain open.

Background

The Aharonov–Bohm (AB) effect — a phase shift acquired by a charged particle encircling a magnetic flux, observable as flux-periodic conductance oscillations in mesoscopic (intermediate between microscopic and macroscopic, where quantum coherence survives) rings — is a paradigmatic two-path interference experiment. In a ring, interference of partial waves that traverse opposite half-circumferences produces conductance oscillations of flux period h/eh/e (hh is Planck's constant and ee the electron charge), while interference of time-reversed partial waves that together encircle the full ring (the same pairs responsible for weak localization, the quantum correction to the conductivity of disordered conductors) produces oscillations of period h/2eh/2e.

When the interfering particle couples to environmental degrees of freedom (photons, electromagnetic fluctuations in a metal, magnetic impurities, or an adjacent electronic channel), the interference is suppressed — decoherence or dephasing. Two heuristic pictures were used before 1990: (i) the 'which-path' picture, in which the two partial waves leave the environment in different final states whose overlap multiplies the interference; and (ii) the phase-uncertainty picture, in which the environment randomizes the relative phase so that averaging cos⁡ϕ\cos\phi over a phase distribution P(ϕ)P(\phi) suppresses the interference. The formal object containing the environment's effect is the Feynman–Vernon influence functional, the environment trace of the product of forward and backward evolution operators.

Ady Stern, Yakir Aharonov, and Yoseph Imry [Phys. Rev. A 41, 3436 (1990)] proved the equivalence of the two pictures in a general setting: writing the interaction in the interaction picture (the standard representation in which the coupling evolves with the free dynamics) as VI(x(t),t)V_I(x(t),t), the factor multiplying the interference term equals ⟨χ0∣T^exp⁡[−i∫0τ0dt VI]∣χ0⟩\langle\chi_0|\hat{T}\exp[-i\int_0^{\tau_0}dt\,V_I]|\chi_0\rangle (with ∣χ0⟩|\chi_0\rangle the initial state of the environment, T^\hat{T} the time-ordering operator, and τ0\tau_0 the traversal time of the interfering partial waves), and since a time-ordered unitary can always be written as eiϕe^{i\phi} with ϕ\phi Hermitian, the influence functional is ⟨eiϕ⟩\langle e^{i\phi}\rangle, an average of eiϕe^{i\phi} over the initial environment state. For a dynamical environment this 'phase' is an operator, not a c-number; for a nondynamical environment it reduces to a classical phase distribution P(ϕ)P(\phi). When the phase distribution is narrow, the environment acts as a static average potential; when it is Gaussian (many weak uncorrelated contributions, or a linearly coupled oscillator bath), ⟨eiϕ⟩=exp⁡(i⟨ϕ⟩−⟨δϕ2⟩/2)\langle e^{i\phi}\rangle=\exp(i\langle\phi\rangle-\langle\delta\phi^2\rangle/2), i.e. phase diffusion with variance fixed by the environment potential–potential correlator. For the h/2eh/2e (full-circle, time-reversed) interference, both partial waves couple to the same environment and the interference factor is ⟨χ0∣e−iϕleiϕr∣χ0⟩\langle\chi_0|e^{-i\phi_l}e^{i\phi_r}|\chi_0\rangle with ϕr,ϕl\phi_r,\phi_l the phase operators of the two paths; if [ϕr,ϕl]=0[\phi_r,\phi_l]=0 this reduces to ⟨ei(ϕr−ϕl)⟩\langle e^{i(\phi_r-\phi_l)}\rangle, and interactions that make each partial wave individually uncertain but leave identical final environment states for the time-reversed pair dephase the h/eh/e but not the h/2eh/2e oscillations. The paper applied this machinery to dephasing by photon modes in coherent and thermal states and by the electromagnetic fluctuations of a metal (using the fluctuation–dissipation theorem, which relates a conductor's equilibrium current fluctuations to its dissipation, with a Drude conductivity σ\sigma — the classical model of a metal as an electron gas with a relaxation time — and skin depth δ(ω)\delta(\omega), the length over which an electromagnetic field of frequency ω\omega penetrates the metal).

The 1990 paper itself flagged what its construction does not cover. The equivalence proof and the h/2eh/2e reduction require the phase operators associated with different portions of the path to commute; for a genuinely dynamical environment that responds to and back-reacts on the interfering particle, this commutativity fails, and the paper deferred that case. It analyzed two prescribed classical paths, leaving open how, in the full sum over histories (path integral) with the environment included, the relevant interfering pairs of paths are selected and modified. The quantitative estimates assumed narrow or Gaussian phase distributions, while for broad, non-Gaussian P(ϕ)P(\phi) the value of ⟨eiϕ⟩\langle e^{i\phi}\rangle depends on the detailed shape of the distribution. The qualitative conditions it gave for differing phase-breaking times of h/eh/e and h/2eh/2e oscillations in realistic many-body environments were not quantitative. Finally, the metal correlator results were restricted to specific parameter regimes (ω≪σ\omega\ll\sigma, Dk2≪4πσDk^2\ll4\pi\sigma with DD the diffusion constant and kk a wave number, kδ(ω)≫1k\delta(\omega)\gg1, frequency cutoff at ω∼kBT\omega\sim k_BT). The last three of these gaps have since been substantially addressed by later literature, although for the harmonic-dependent phase breaking only model-specific results have been obtained; the noncommuting-phase and path-selection questions, together with the general harmonic-dependent phase-breaking conditions, remain open and constitute the surviving open core stated below.

Problem Statement

Consider a two-path interferometer (e.g., a mesoscopic ring or a Mach–Zehnder interferometer) in which both partial waves couple to the same dynamical, many-body environment that back-reacts on the interfering particle (e.g., the electromagnetic fluctuations of a metal with a Drude conductivity, dilute magnetic impurities, or an adjacent partitioned electronic channel). The factor multiplying the interference is ⟨χ0∣ T^exp⁡[i∫0τ0dt VI(xl(t),t)] T^exp⁡[−i∫0τ0dt VI(xr(t),t)] ∣χ0⟩\langle\chi_0|\,\hat{T}\exp[i\int_0^{\tau_0}dt\,V_I(x_l(t),t)]\,\hat{T}\exp[-i\int_0^{\tau_0}dt\,V_I(x_r(t),t)]\,|\chi_0\rangle (with ∣χ0⟩|\chi_0\rangle the initial state of the environment, T^\hat{T} the time-ordering operator, τ0\tau_0 the traversal time, and VI(xl,r(t),t)V_I(x_{l,r}(t),t) the interaction-picture coupling evaluated along the left and right classical paths), equivalently ⟨χ0∣e−iϕleiϕr∣χ0⟩\langle\chi_0|e^{-i\phi_l}e^{i\phi_r}|\chi_0\rangle with Hermitian path phase operators ϕr,ϕl\phi_r,\phi_l; for a dynamical shared environment these generally do not commute, so the factor cannot be reduced to an average ⟨eiΔϕ⟩\langle e^{i\Delta\phi}\rangle of a single relative phase over a phase probability distribution, and the statistical phase-uncertainty picture loses its direct meaning.

The problem, in the generality left open by Stern, Aharonov, and Imry (1990), is to complete their general picture by answering:

  1. (Noncommuting phases.) Construct an operator-level phase-statistics representation of the influence functional valid when the phase operators associated with different portions or paths of the interference do not commute — a quasi-probability or generalized-cumulant representation — or establish that no such reduction exists beyond the time-ordered product, identifying exactly what information about dephasing is and is not contained in the first two environment correlators in this regime. The construction must reproduce the known commuting limits: the classical phase distribution P(ϕ)P(\phi) for a nondynamical environment, the Gaussian (phase-diffusion) limit, and the exact ⟨eiϕ⟩\langle e^{i\phi}\rangle of non-Gaussian shot-noise environments.

  2. (Path selection in the sum over histories.) Within the full sum over histories of the interfering particle with the environment included — not restricted to two prescribed classical paths — determine how the coupling to the environment modifies which pairs of paths dominate the interference: derive general criteria for the dominant (e.g., stationary-phase or saddle) path pairs and their phase statistics in the presence of a dynamical environment, including the regime in which the particle's own path fluctuations couple to the environment.

  3. (Harmonic-dependent phase breaking.) Using the resulting framework, derive general quantitative conditions — not restricted to a single model — under which the dephasing rates (or lengths) governing the h/eh/e and h/2eh/2e conductance oscillations of realistic rings coupled to many-body environments differ, going beyond the two necessary qualitative conditions given in 1990 (the interaction must not distinguish the time-reversed paths, and the environment must not evolve appreciably between the two interactions) and beyond the model-specific quantitative results now available for electron–electron interactions and dilute magnetic impurities in diffusive rings.

Items 1–3 are one problem: each component supplies an ingredient (representation, dominant path pairs, quantitative criteria) of the single objective of extending the proven equivalence between the environment-overlap and phase-uncertainty pictures to realistic dynamical many-body environments.

References

  1. Ady Stern, Yakir Aharonov, and Yoseph Imry, Phase uncertainty and loss of interference: A general picture, Phys. Rev. A 41, 3436–3448 (1990). DOI: 10.1103/PhysRevA.41.3436, https://doi.org/10.1103/PhysRevA.41.3436 (source paper)
  2. Ady Stern, Yakir Aharonov, and Yoseph Imry, Dephasing of Interference by a Back Reacting Environment, in Quantum Coherence, World Scientific, pp. 201–219 (1991). DOI: 10.1142/9789814439251_0016, https://doi.org/10.1142/9789814439251_0016 (the authors' announced follow-up on the back-reacting environment)
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