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Marco Merkli

Publications and source records attributed to Marco Merkli.

At least 37 records · Page 2Linked to original sources
arXiv2021

Dynamics of Open Quantum Systems I, Oscillation and Decay

We develop a framework to analyze the dynamics of a finite-dimensional quantum system $\rm S$ in contact with a reservoir $\rm R$. The full, interacting $\rm SR$ dynamics is unitary. The reservoir has a stationary state but otherwise dissipative dynamics. We identify a main part of the full dynamics, which approximates it for small values of the $\rm SR$ coupling constant, uniformly for all times $t\ge 0$. The main part consists of explicit oscillating and decaying parts. We show that the reduced system evolution is Markovian for all times. The technical novelty is a detailed analysis of the link between the dynamics and the spectral properties of the generator of the $\rm SR$ dynamics, based on Mourre theory. We allow for $\rm SR$ interactions with little regularity, meaning that the decay of the reservoir correlation function only needs to be polynomial in time, improving on the previously required exponential decay. In this work we distill the structural and technical ingredients causing the characteristic features of oscillation and decay of the $\rm SR$ dynamics. In the companion paper [27] we apply the formalism to the concrete case of an $N$-level system linearly coupled to a spatially infinitely extended thermal bath of non-interacting Bosons.

quant-ph↗
arXiv2020

Qubit dynamics with classical noise

We study the evolution of a qubit evolving according to the Schrödinger equation with a Hamiltonian containing noise terms, modeled by random diagonal and off-diagonal matrix elements. We show that the noise-averaged qubit density matrix converges to a final state, in the limit of large times $t.$ The convergence speed is polynomial in $1/t$, with a power depending on the regularity of the noise probability density and its low frequency behaviour. We evaluate the final state explicitly. We show that in the regimes of weak and strong off-diagonal noise, the process implements the dephasing channel in the energy- (localized) and the delocalized basis, respectively.

quant-ph↗
arXiv2020

Dark Matter Spin-Spin Interaction through the Pseudo-Scalar Vacuum Field

We suggest that the pseudo-scalar vacuum field (PSV) in the dark matter (DM) sector of the Universe may be as important as the electromagnetic vacuum field in the baryonic sector. In particular, the spin-spin interaction between the DM fermions, mediated by PSV, may represent the strongest interaction between the DM fermions due to the absence of the electric charge and the magnetic dipole moment. Based on this assumption, we consider the influence of the spin-spin interaction, mediated by PSV, on the spin precession of the DM fermions (e. g. neutralino). In the secular approximation, we obtain the exact expression describing the frequency of the precession and estimate the decoherence rate.

hep-ph↗
arXiv2020

Two-Component Axionic Dark Matter Halos

We consider a two-component dark matter halo (DMH) of a galaxy containing ultra-light axions (ULA) of different mass. The DMH is described as a Bose-Einstein condensate (BEC) in its ground state. In the mean-field (MF) limit we have derived the integro-differential equations for the spherically symmetrical wave functions of the two DMH components. We studied numerically the radial distribution of the mass density of ULA and constructed the parameters which could be used to distinguish between the two- and one-component DMH. We also discuss an interesting connection between the BEC MF ground state of a one-component DMH and Hawking temperature and entropy of a corresponding Black Hole, and Unruh temperature.

hep-ph↗
arXiv2020

Adiabatic transitions in a two-level system coupled to a free Boson reservoir

We consider a time-dependent two-level quantum system interacting with a free Boson reservoir. The coupling is energy conserving and depends slowly on time, as does the system Hamiltonian, with a common adiabatic parameter $\varepsilon$. Assuming that the system and reservoir are initially decoupled, with the reservoir in equilibrium at temperature $T\ge 0$, we compute the transition probability from one eigenstate of the two-level system to the other eigenstate as a function of time, in the regime of small $\varepsilon$ and small coupling constant $λ$. We analyse the deviation from the adiabatic transition probability obtained in absence of the reservoir.

math-ph↗
arXiv2019

Quantum Electron Transport in Degenerate Donor-Acceptor Systems

We develop a mathematically rigorous theory for the quantum transfer processes in degenerate donor-acceptor dimers in contact with a thermal environment. We calculate explicitly the transfer rates and the acceptor population efficiency. The latter depends critically on the initial donor state. We show that quantum coherence in the initial state enhances the transfer process. If the electron is initially shared coherently by the donor levels then the efficiency can reach values close to 100%, while an incoherent initial donor state will significantly suppress the efficiency. The results are useful for a better understanding of the quantum electron transport in many chemical, solid state, and biological systems with complex degenerate and quasi-degenerate energy landscapes.

quant-ph↗
arXiv2018

Noise-Assisted Quantum Exciton and Electron Transfer in Bio-Complexes with Finite Donor and Acceptor Bandwidths

We present an analytic and numerical study of noise-assisted quantum exciton (electron) transfer (ET) in a bio-complex, consisting of an electron donor and acceptor (a dimer), modeled by interacting continuous electron bands of finite widths. The interaction with the protein-solvent environment is modeled by a stationary stochastic process (noise) acting on all the donor and acceptor energy levels. We start with discrete energy levels for both bands. Then, by using a continuous {limit} for the electron spectra, we derive integro-differential equations for ET dynamics between two bands. Finally, we derive from these equations rate-type differential equations for ET dynamics. We formulate the conditions of validity of the rate-type equations. We consider different regions of parameters characterizing the widths of the donor and acceptor bands and the strength of the dimer-noise interaction. For a simplified model with a single energy level donor and a continuous acceptor band, we derive a generalized analytic expression and provide numerical simulations for the ET rate. They are consistent with Wigner-Weisskopf, Förster-type, and Marcus-type expressions, in their corresponding regime of parameters.Our analytic results are confirmed by numerical simulations. We demonstrate how our theoretical results are modified {when both the donor and the acceptor are described by finite bands}. We also show that, for a relatively wide acceptor band, the efficiency of the ET from donor to acceptor can be close to 100% for a broad range of noise amplitudes, for both "downhill" and "uphill" ET, for sharp and flat redox potentials, and for reasonably short times. We discuss possible experimental implementations of our approach with application to bio-complexes.

physics.bio-ph↗
arXiv2018

Mean field dynamics of some open quantum systems

We consider a large number $N$ of quantum particles coupled via a mean field interaction to another quantum system (reservoir). Our main result is an expansion for the averages of observables, both of the particles and of the reservoir, in inverse powers of $\sqrt{N}$. The analysis is based directly on the Dyson series expansion of the propagator. We analyze the dynamics, in the limit $N\rightarrow\infty$, of observables of a fixed number $n$ of particles, of extensive particle observables and their fluctuations, as well as of reservoir observables. We illustrate our results on the infinite mode Dicke model and on various energy conserving models.

math-ph↗
arXiv2016

Completely positive dynamical semigroups and quantum resonance theory

Starting form a microscopic system-environment model, we construct a quantum dynamical semigroup for the reduced evolution of the open system. The difference between the true system dynamics and its approximation by the semigroup has the following two properties: It is (linearly) small in the system-environment coupling constant for all times, and it vanishes exponentially quickly in the large time limit. Our approach is based on the quantum dynamical resonance theory.

math-ph↗
arXiv2015

On the irreversible dynamics emerging from quantum resonances

We consider the dynamics of quantum systems which possess stationary states as well as slowly decaying, metastable states arising from the perturbation of bound states. We give a decomposition of the propagator into a sum of a stationary part, one exponentially decaying in time and a polynomially decaying remainder. The exponential decay rates and the directions of decay in Hilbert space are determined, respectively, by complex resonance energies and by projections onto resonance states. Our approach is based on an elementary application of the Feshbach map. It is applicable to open quantum systems and to situations where spectral deformation theory fails. We derive a detailed description of the dynamics of the spin-boson model at arbitrary coupling strength.

math-ph↗
arXiv2015

Representations of canonical commutation relations describing infinite coherent states

We investigate the infinite volume limit of quantized photon fields in multimode coherent states. We show that for states containing a continuum of coherent modes, it is mathematically and physically natural to consider their phases to be random and identically distributed. The infinite volume states give rise to Hilbert space representations of the canonical commutation relations which we construct concretely. In the case of random phases, the representations are random as well and can be expressed with the help of Itô stochastic integrals. We analyze the dynamics of the infinite state alone and the open system dynamics of small systems coupled to it. We show that under the free field dynamics, initial phase distributions are driven to the uniform distribution. We demonstrate that coherences in small quantum systems, interacting with the infinite coherent state, exhibit Gaussian time decay. The decoherence is qualitatively faster than the one caused by infinite thermal states, which is known to be exponentially rapid only. This emphasizes the classical character of coherent states.

math-ph↗
arXiv2015

Multiscale dynamics of open three-level quantum systems with two quasi-degenerate levels

We consider a three-level quantum system interacting with a bosonic thermal reservoir. Two energy levels of the system are nearly degenerate but well separated from the third one. The system-reservoir interaction constant is larger than the energy difference of the degenerate levels, but it is smaller than the separation between the latter and the remaining level. We show that the quasi-degeneracy of energy levels leads to the existence of a manifold of quasi-stationary states, and the dynamics exhibits two characteristic time scales. On the first, shorter one, initial states approach the quasi-stationary manifold. Then, on the much longer second time scale, the final unique equilibrium is reached.

quant-ph↗
arXiv2015

Quantum Measurements of Scattered Particles

We investigate the process of quantum measurements on scattered probes. Before scattering, the probes are independent, but they become entangled afterwards, due to the interaction with the scatterer. The collection of measurement results (the history) is a stochastic process of dependent random variables. We link the asymptotic properties of this process to spectral characteristics of the dynamics. We show that the process has decaying time correlations and that a zero-one law holds. We deduce that if the incoming probes are not sharply localized with respect to the spectrum of the measurement operator, then the process does not converge. Nevertheless, the scattering modifies the measurement outcome frequencies, which are shown to be the average of the measurement projection operator, evolved for one interaction period, in an asymptotic state. We illustrate the results on a truncated Jaynes-Cummings model.

math-ph↗
arXiv2014

Overlapping Resonances in Open Quantum Systems

An $N$-level quantum system is coupled to a bosonic heat reservoir at positive temperature. We analyze the system-reservoir dynamics in the following regime: The strength $λ$ of the system-reservoir coupling is fixed and small, but larger than the spacing $σ$ of system energy levels. For vanishing $σ$ there is a manifold of invariant system-reservoir states and for $σ>0$ the only invariant state is the joint equilibrium. The manifold is invariant for $σ=0$ but becomes quasi-invariant for $σ>0$. Namely, on a first time-scale of the order $1/λ^2$, initial states approach the manifold. Then they converge to the joint equilibrium state on a much larger time-scale of the order $λ^2/σ^2$. We give a detailed expansion of the system-reservoir evolution showing the above scenario.

math-ph↗
arXiv2014

Suppression of Decoherence by Periodic Forcing

We consider a finite-dimensional quantum system coupled to a thermal reservoir and subject to a time-periodic, energy conserving forcing. We show that, if a certain dynamical decoupling condition is fulfilled, then the periodic forcing counteracts the decoherence induced by the reservoir: for small system-reservoir coupling $λ$ and small forcing period $T$, the system dynamics is approximated by an energy conserving and non-dissipative dynamics, which preserves coherences. For times up to order $(λT)^{-1}$, the difference between the true and approximated dynamics is of size $λ+T$. Our approach is rigorous and combines Floquet and spectral deformation theory. We illustrate our results on the spin-fermion model and recover previously known, heuristically obtained results.

quant-ph↗
arXiv2013

Repeated interactions in open quantum systems

Analyzing the dynamics of open quantum systems has a long history in mathematics and physics. Depending on the system at hand, basic physical phenomena that one would like to explain are, for example, convergence to equilibrium, the dynamics of quantum coherences (decoherence) and quantum correlations (entanglement), or the emergence of heat and particle fluxes in non-equilibrium situations. From the mathematical physics perspective, one of the main challenges is to derive the irreversible dynamics of the open system, starting from a unitary dynamics of the system and its environment. The repeated interactions systems considered in these notes are models of non-equilibrium quantum statistical mechanics. They are relevant in quantum optics, and more generally, serve as a relatively well treatable approximation of a more difficult quantum dynamics. In particular, the repeated interaction models allow to determine the large time (stationary) asymptotics of quantum systems out of equilibrium.

math-ph↗
arXiv2012

A rigorous derivation of the transition rate for the dissipative two-state system

The transition rate for a two-state system interacting with a bosonic heat bath, from the initial state `up' at time t=0 to `down' at time t>0, was derived formally in the seminal paper [12] by Leggett, Chakravarty, Dorsey, Fisher, Garg and Zwerger. The expression was obtained for small tunneling parameter of the two-level system using time-dependent perturbation theory to second order. It contains all orders in the parameter of coupling between the system and the bath. We give a rigorous derivation of the transition rate, using a quantum resonance approach.

math-ph↗