Does decoherence in the thermodynamic limit exist?
Paper withdrawn due to an error into the equations of the probability amplitudes.
quant-ph↗
arXiv subjects
Marco Frasca
Publications and source records attributed to Marco Frasca.
Strong Coupling Perturbation Theory in Quantum Mechanics
We present a full introduction to the recent devised perturbation theory for strong coupling in quantum mechanics. In order to put the theory in a proper historical perspective, the approach devised in quantum field theory is rapidly presented, showing how it implies a kind of duality in perturbation theory, from the start. The approach of renormalization group in perturbation theory is then presented. This method permits to resum secularities in perturbation theory and makes fully algorithmical the resummation, transforming the perturbation calculations in a step by step computational procedure. The general theorem on which is founded a proper application of the strong coupling expansion, based on a result in the quantum adiabatic theory, is then exposed. This theorem gives the leading order of a strong coupling expansion. Then, after the introduction of the principle of duality in perturbation theory that puts in a proper context the quantum field theory method, the resulting theory of the strong coupling expansion and the free picture are presented. An algorithm for the computation of the perturbation series is finally given. This approach has a lot of applications in fields as quantum optics, condensed matter and so on, extending the original expectations of the quantum field theory method. So, we give some examples of application for a class of two-level systems that, in recent years, proved to be extremely important. One of the most interesting concepts that can be obtained in this way is that of a Quantum Amplifier (QAMP) that permits to obtain an amplification to the classical level of the quantum fluctuations of the ground state of a single radiation mode.
quant-ph↗
Dynamical disappearance of superposition states in the thermodynamic limit
It is shown that a macroscopic superposition state of radiation, strongly interacting with an ensemble of two-level atoms, is removed generating a coherent state describing a classical radiation field, when the thermodynamic limit is taken on the unitary evolution obtained by the Schroedinger equation. Decoherence appears as a dynamical effect in agreement with a recent proposal [M. Frasca, Phys. Lett. A 283, 271 (2001)]. To prove that this effect is quite general, we show that this same behavior appears when a superposition of two Fock number states is also considered. Higher order corrections are computed showing that this result tends to become exact in the thermodynamic limit. It appears as a genuine example of intrinsic collapse of the wave function.
quant-ph↗
A modern review of the two-level approximation
The paradigm of the two-level atom is revisited and its perturbative analysis is discussed in view of the principle of duality in perturbation theory. The models we consider are a two-level atom and an ensemble of two-level atoms both interacting with a single radiation mode. The aim is to see how the latter can be actually used as an amplifier of quantum fluctuations to the classical level through the thermodynamic limit of a very large ensemble of two-level atoms [M. Frasca, Phys. Lett. A {\bf 283}, 271 (2001)] and how can remove Schrödinger cat states. The thermodynamic limit can be very effective for producing both classical states and decoherence on a quantum system that evolves without dissipation. Decoherence without dissipation is indeed an effect of a single two-level atom interacting with an ensemble of two-level atoms, a situation that proves to be useful to understand recent experiments on nanoscale devices showing unexpected disappearance of quantum coherence at very low temperatures.
quant-ph↗
General Theorems on Decoherence in the Thermodynamic Limit
We extend the results on decoherence in the thermodynamic limit [M. Frasca, Phys. Lett. A {\bf 283}, 271 (2001)] to general Hamiltonians. It is shown that N independent particles, initially properly prepared, have a set of observables behaving classically in the thermodynamic limit. This particular set of observables is then coupled to a quantum system that in this way decoheres so to have the density matrix in a mixed form. This gives a proof of the generality of this effect.
quant-ph↗
Dephasing by two-level systems at zero temperature by unitary evolution
We analyze the unitary time evolution of a conduction electron, described by a two-level system, interacting with two-level systems (spins) through a spin-spin interaction and prove that coherent spin states of the conduction electron are obtained in the strong coupling regime, when the number of the spins is taken to be, formally, infinitely large (thermodynamic limit). This model describes a spin interacting with a spin-bath in a strong coupling regime and gives a dephasing time at zero temperature that agrees with the recent experimental results for quantum dots. Dephasing is proved to occur as the conduction electron oscillates between the two states with frequency going to infinity in the thermodynamic limit, that is, increasing the number of spins. Then, it is shown that the only meaning that can be attached to such oscillations with infinite frequency is by an average in time, eliminating the off-diagonal terms of the density matrix. This model is in agreement with a recent proposal of appearance of classical states in quantum mechanics due to the large number of components of a quantum system, if properly prepared for the states of each component part. The strong coupling study of the model is accomplished through the principle of duality as recently introduced in perturbation theory [M.Frasca, Phys. Rev. A {\bf 58}, 3439 (1998)].
cond-mat.mes-hall↗
Ultraviolet analysis of one dimensional quantum systems
Starting from the study of one-dimensional potentials in quantum mechanics having a small distance behavior described by a harmonic oscillator, we extend this way of analysis to models where such a behavior is not generally expected. In order to obtain significant results we approach the problem by a renormalization group method that can give a fixed point Hamiltonian that has the shape of a harmonic oscillator. In this way, good approximations are obtained for the ground state both for the eigenfunction and the eigenvalue for problems like the quartic oscillator, the one-dimensional Coulomb potential having a not normalizable ground state solution and for the one-dimensional Kramers-Henneberger potential. We keep a coupling constant in the potential and take it running with a generic cut-off that goes to infinity. The solution of the Callan-Symanzik equation for the coupling constant generates the harmonic oscillator Hamiltonian describing the behavior of the model at very small distances (ultraviolet behavior). This approach, although algorithmic in its very nature, does not appear to have a simple extension to obtain excited state behavior. Rather, it appears as a straightforward non-perturbative method.
quant-ph↗
Rabi oscillations and macroscopic quantum superposition states
A two-level atom interacting with a single radiation mode is considered, without the rotating-wave approximation, in the strong coupling regime. It is shown that, in agreement with the recent results on Rabi oscillations in a Josephson junction (Y. Nakamura, Yu. A. Pashkin and J. S. Tsai, Phys. Rev. Lett. {\bf 87}, 246601 (2001)), the Rabi frequency is indeed proportional to first kind integer order Bessel functions in the limit of a large number of photons and the dressed states are macroscopic quantum superposition states. To approach this problem analytically use is made of the dual Dyson series and the rotating-wave approximation.
quant-ph↗
Classical states and decoherence by unitary evolution in the thermodynamic limit
It is shown how classical states, meant as states representing a classical object, can be produced in the thermodynamic limit, retaining the unitary evolution of quantum mechanics. Besides, using a simple model of a single spin interacting with a spin-bath, it is seen how decoherence, with the off-diagonal terms in the density matrix going to zero, can be obtained when the number of the spins in the bath is taken to go formally to infinity. In this case, indeed, the system appears to flop at a frequency being formally infinity that, from a physical standpoint, can be proved equivalent to a time average.
quant-ph↗
Non-relativistic quantum electrodynamics for strong laser-atom interaction
A formulation of quantum electrodynamics is given that applies to atoms in a strong laser field by perturbation theory in a non-relativistic regime. Dipole approximation is assumed. The dual Dyson series, here discussed by referring it to the Birkhoff theorem for singularly perturbed linear differential equations, can be applied and a perturbation series obtained transforming the Hamiltonian by a Pauli-Fierz transformation. But, if just few photons are present high-order harmonics cannot be generated. So, it is proven that odd high-order harmonics only appear when the laser field is intense and one can substitute the creation and annihilation operators by the square root of the mean number of photons taken to be huge, the field retaining its coherency property as observed experimentally for harmonics. In this case, the Hamiltonian for perturbation theory comes to the Kramers-Henneberger form. The theory has a dipolar contribution when the free-electron quiver motion amplitude is larger than the atomic radius. For a Coulomb potential one has that the outer electron is periodically kicked, and so a prove is given that the same should happen to Rydberg atoms in intense microwave fields. The distribution representing the kicking has a Fourier series with just odd terms. Using a modified Rayleigh-Schrödinger perturbation theory, it is shown that under the same condition of validity of the quiver motion amplitude to atomic radius ratio, the atomic wave function is only slightly modified by the laser field due to the way the energy levels rearrange themselves. This gives a prove of stabilization in the limit of laser frequency going to infinity. Then, perturbation theory can be applied when the Keldysh parameter becomes smaller with respect to the shifted distance between the energy levels of the atom.
quant-ph↗
U(1) Field Theory in the Limit of a Large Coupling Constant
We consider a massive fermion interacting with a U(1) gauge field in the limit of a large coupling constant. It is found that the current has a generalized London term that can originate massive excitations for two of the three components of the gauge field, which disappear for a free particle at rest. The origin of the superconductive term is due to a partial breaking of the gauge symmetry in the limit of a large coupling constant. Beside, the scalar potential generated by the particle, the only component of the gauge field that keeps gauge invariance, increases with the square of the distance. These results should give a path towards the derivation of quark confinement from QCD.
hep-th↗
Perturbative study of multiphoton processes in the tunneling regime
A perturbative study of the Schrödinger equation in a strong electromagnetic field with dipole approximation is accomplished in the Kramers-Henneberger frame. A prove that just odd harmonics appear in the spectrum for a linear polarized laser field is given, assuming that the atomic radius is much lesser than the free-electron quiver motion amplitude. Within this approximation a perturbation series is obtained in the Keldysh parameter giving a description of multiphoton processes in the tunneling regime. The theory is applied to the case of hydrogen-like atoms: The spectrum of higher order harmonics and the above-threshold ionization rate are derived. The ionization rate computed in this way determines the amplitudes of the harmonics. The wave function of the atom proves to be rigid with respect to the perturbation so that the effect of the laser field on the Coulomb potential in the computation of the probability amplitudes can be neglected as a first approximation: This approximation improves as the ratio between the amplitude of the quiver motion of the electron and the atom radius becomes larger. The semiclassical description currently adopted for harmonic generation is so rederived by solving perturbatively the Schrödinger equation.
quant-ph↗
Theory of dressed states in quantum optics
The dual Dyson series [M.Frasca, Phys. Rev. A {\bf 58}, 3439 (1998)], is used to develop a general perturbative method for the study of atom-field interaction in quantum optics. In fact, both Dyson series and its dual, through renormalization group methods to remove secular terms from the perturbation series, give the opportunity of a full study of the solution of the Schrödinger equation in different ranges of the parameters of the given hamiltonian. In view of recent experiments with strong laser fields, this approach seems well-suited to give a clarification and an improvement of the applications of the dressed states as currently done through the eigenstates of the atom-field interaction, showing that these are just the leading order of the dual Dyson series when the Hamiltonian is expressed in the interaction picture. In order to exploit the method at the best, a study is accomplished of the well-known Jaynes-Cummings model in the rotating wave approximation, whose exact solution is known, comparing the perturbative solutions obtained by the Dyson series and its dual with the same approximations obtained by Taylor expanding the exact solution. Finally, a full perturbative study of high-order harmonic generation is given obtaining, through analytical expressions, a clear account of the power spectrum using a two-level model, even if the method can be successfully applied to a more general model that can account for ionization too. The analysis shows that to account for the power spectrum it is needed to go to first order in the perturbative analysis. The spectrum obtained gives a way to measure experimentally the shift of the energy levels of the atom interacting with the laser field by looking at the shifting of hyper-Raman lines.
quant-ph↗
Duality in Perturbation Theory and the Quantum Adiabatic Approximation
Duality is considered for the perturbation theory by deriving, given a series solution in a small parameter, its dual series with the development parameter being the inverse of the other. A dual symmetry in perturbation theory is identified. It is then shown that the dual to the Dyson series in quantum mechanics is given by a recent devised series having the adiabatic approximation as leading order. A simple application of this result is given by rederiving a theorem for strongly perturbed quantum systems.
hep-th↗
Quantum Kicked Dynamics and Classical Diffusion
We consider the quantum counterpart of the kicked harmonic oscillator showing that it undergoes the effect of delocalization in momentum when the classical diffusional threshold is obeyed. For this case the ratio between the oscillator frequency and the frequency of the kick is a rational number, strictly in analogy with the classical case that does not obey the Kolmogorov-Arnold-Moser theorem as the unperturbed motion is degenerate. A tight-binding formulation is derived showing that there is not delocalization in momentum for irrational ratio of the above frequencies. In this way, it is straightforwardly seen that the behavior of the quantum kicked rotator is strictly similar to the one of the quantum kicked harmonic oscillator, although the former, in the classical limit, obeys the Kolmogorov-Arnold-Moser theorem.
chao-dyn↗
The Higher Orders of the Theory of Strong Perturbations in Quantum Mechanics and the Secularity Problem
We solve the higher order equations of the theory of the strong perturbations in quantum mechanics given in M. Frasca, Phys. Rev. A 45, 43 (1992), by assuming that, at the leading order, the wave function goes adiabatically. This is accomplished by deriving the unitary operator of adiabatic evolution for the leading order. In this way it is possible to show that at least one of the causes of the problem of phase-mixing, whose effect is the polynomial increase in time of the perturbation terms normally called secularities, arises from the shifts of the perturbation energy levels due to the unperturbed part of the hamiltonian. An example is given for a two-level system that, anyway, shows a secularity at second order also in the standard theory of small perturbations. The theory is applied to the quantum analog of a classical problem that can become chaotic, a particle under the effect of two waves of different amplitudes, frequencies and wave numbers.
quant-ph↗
The Leading Order of the Theory of Strong Perturbations in Quantum Mechanics
We prove that, for a quantum system that undergoes a strong perturbation, the solution of the leading order equation of the strong field approximation (M.Frasca, Phys. Rev. A, {\bf 45}, 43 (1992)) can be derived by the adiabatic approximation. In fact, it is shown that greatest is the perturbation and more similar the quantum system is to an adiabatic one, the solution being written as a superposition of eigenstates of the time-dependent perturbation.A direct consequence of this result is that the solution of the Schrödinger equation in the interaction picture, in the same approximation for the perturbation, coincides with the one of the leading order of the strong field approximation. The limitation due to the requirement that the perturbation has to commute at different times is so overcome. Computational difficulties could arise to go to higher orders. Beside, the method is not useful for perturbations that are constant in time. In such a case a small time series is obtained, indicating that this approximation is just an application to quantum mechanics of the Kirkwood-Wigner expansion of statistical mechanics. The theory obtained in this way is applied to a time-dependent two-level spin model, already considered for the study of the Berry's phase, showing that a geometrical phase could arise if a part of the hamiltonian is considered as a strong perturbation. No adiabatic approximation is taken on the parameters of the hamiltonian, while their cyclicity is retained.
quant-ph↗
The Strong Perturbation Approach for the Dirac Equation in a Gauge Field
We discuss a possible approach to the problem of a gauge theory with a strong coupling constant. It is seen that, instead of plane waves, we have to consider the adiabatic eigenstates of the perturbation in order to get a meaningful perturbation approach.
hep-th↗