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Physical Review A

Quantum transitions in interacting fields

Evgueni Karpov, Gonzalo E. Ordonez, Tomio Yamakoshi Petrosky, Ilya Prigogine

Published Jul 30, 2002• 11 citations
Publisher

Abstract

In recent papers we have discussed the meaning of dressed excited quantum states. We have shown that these states can be formulated in terms of distribution functions outside the Hilbert space. Our approach applies to ``nonintegrable'' systems in the sense of Poincar\'e. It involves the analytic continuation of the unitary operator U describing the transformation from bare to dressed stable states. This leads to a ``star-unitary'' operator $\ensuremath{\Lambda}.$ Interacting fields are nonintegrable systems obtained as a result of resonances. It is therefore natural to expect that our previous results remain valid for interacting fields. We consider a simple example, which corresponds to an extension of the usual Friedrichs model. This involves a local field in interaction with a bilocal scalar field. This model has been previously studied by one of the co-authors (I.P.). It is a simplified version of the model describing $\stackrel{\ensuremath{\rightarrow}}{A}B+C$ transition with quadratic interaction. The usual Bogoliubov transformation eliminates the field corresponding to A, while our method leads to strictly exponential decay of unstable field. The dressed field A corresponds to a singular distribution function outside the Liouville-Hilbert space. As in the Friedrichs model there exists two time scales---one for the preparation of a dressed state and depending on initial conditions called ``Zeno period'' and the other universal for the decay.

DOI
10.1103/physreva.66.012109

Related papers

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arXiv:physics/0311048•Nov 10, 2003•0 citations
In recent papers we have discussed the meaning of dressed excited quantum states. We have shown that these states can be formulated in terms of distribution functions outside the Hilbert space. Our approach applies to nonintegrable'' systems in the sense of Poincar\'e. It involves the analytic continuation of the unitary operator U describing the transformation from bare to dressed stable states. This leads to a star-unitary'' operator \ensuremathΛ.\ensuremath{\Lambda}.\ensuremathΛ. Interacting fields are nonintegrable systems obtained as a result of resonances. It is therefore natural to expect that our previous results remain valid for interacting fields. We consider a simple example, which corresponds to an extension of the usual Friedrichs model. This involves a local field in interaction with a bilocal scalar field. This model has been previously studied by one of the co-authors (I.P.). It is a simplified version of the model describing A\ensuremath→B+C\stackrel{\ensuremath{\rightarrow}}{A}B+CA\ensuremath→​B+C transition with quadratic interaction. The usual Bogoliubov transformation eliminates the field corresponding to A, while our method leads to strictly exponential decay of unstable field. The dressed field A corresponds to a singular distribution function outside the Liouville-Hilbert space. As in the Friedrichs model there exists two time scales---one for the preparation of a dressed state and depending on initial conditions called ``Zeno period'' and the other universal for the decay.
We consider the problem of the meaning of quantum unstable states including their dressing. According to both Dirac and Heitler this problem has not been solved in the usual formulation of quantum mechanics. A precise definition of excited states is still needed to describe quantum transitions. We use our formulation given in terms of density matrices outside the Hilbert space. We obtain a dressed unstable state for the Friedrichs model, which is the simplest model that incorporates both bare and dressed quantum states. The excited unstable state is derived from the stable states through analytic continuation. It is given by an irreducible density matrix with broken time symmetry. It can be expressed by a superposition of Gamow density operators. The main difference from previous studies is that excited states are not factorizable into wave functions. The dressed unstable state satisfies all the criteria that we can expect: it has a real average energy and a nonvanishing trace. The average energy differs from Green's function energy by a small effect starting with fourth order in the coupling constant. Our state decays following a Markovian equation. There are no deviations from exponential decay neither for short nor for long times, as is the case for the bare state. The dressed state satisfies an uncertainty relation between energy and lifetime. We can also define dressed photon states and describe how the energy of the excited state is transmitted to the photons. There is another very important aspect: deviations from exponential decay would be in contradiction with indiscernibility as one could define, e.g., old mesons and young mesons according to their lifetime. This problem is solved by showing that quantum transitions are the result of two processes: a dressing process, discussed in a previous publication, and a decay process, which is much slower for electrodynamic systems. During the dressing process the unstable state is prepared. Then the dressed state decays in a purely exponential way. In the Hilbert space the two processes are not separated. Therefore it is not astonishing that we obtain for the unstable dressed state an irreducible density matrix outside the Liouville-Hilbert-space. This is a limit of Hilbert space states that are arbitrarily close to the decaying state. There are experiments that could verify our proposal. A typical one would be the study of the line shape, which is due to the superposition of the short-time process and the long-time process. The long-time process taken separately leads to a much sharper line shape, and avoids the divergence of the fluctuation predicted by the Lorentz line shape.
In a previous paper we have studied dressed excited states in the Friedrichs model, which describes a two-level atom interacting with radiation. In our approach, excited states are distributions (or generalized functions) in the Liouville space. These states decay in a strictly exponential way. In contrast, the states one may construct in the Hilbert space of wave functions always present deviations from exponential decay. We have considered the momentum representation, which is applicable to global quantities (trace, energy transfer). Here we study the space-time description of local quantities associated with dressed unstable states, such as, the intensity of the photon field. In this situation the excited states become factorized in Gamow states. To go from local quantities to global quantities, we have to proceed to an integration over space, which is far from trivial. There are various elements that appear in the space-time evolution of the system: the unstable cloud that surrounds the bare atom, the emitted real photons and the ``Zeno photons,'' which are associated with deviations from exponential decay. We consider a Hilbert space approximation to our dressed excited state. This approximation leads already to decay close to exponential in the field surrounding the atom, and to a line shape different from the Lorentzian line shape. Our results are compared with numerical simulations. We show that the time evolution of an unstable state satisfies a Boltzmann-like H\mathcal{H}H theorem. This is applied to emission and absorption as well as scattering. The existence of a microscopic H\mathcal{H}H theorem is not astonishing. The excited states are ``nonequilibrium'' states and their time evolution leads to the emission of photons, which distributes the energy of the unstable state among the field modes.
The problem of the meaning of quantum unstable states including their dressing is considered. The formulation given in terms of density matrices outside the Hilbert space is used. A dressed unstable state for the Friedrichs model, which is the simplest model that incorporates both bare and dressed quantum states is obtained. Due to resonance singularities that appear in the frequency denominators, quantum unstable systems are categorized as non-integrable systems in the sense of Poincaré. The excited unstable state is derived from the stable states through a suitable analytic continuation of the denominators. It is given by an irreducible density matrix with broken time-symmetry. Our state decays following a Markovian equation. There are no deviations from exponential decay neither for short nor for long times, as is the case for the bare state. The dressed state satisfies an uncertainty relation between energy and lifetime. There are experiments that could verify our proposal. A typical one would be the study of the line shape, which is due to the superposition of the short time process and the long time process. The long time process taken separately leads to a much sharper line shape, and avoids the divergence of the fluctuation predicted by the Lorentz line shape.
The interaction between matter and fields is a classical problem. Still there are difficulties. Time symmetry is broken. So the radiation damping belongs to a class of phenomena, which includes transport properties. We propose a radically different approach based on the extension of the dynamics of integrable systems. We consider a simple model, a harmonic oscillator interacting with a field. For integrable systems, it is well known that there exists a unitary transformation U. However, in the radiation damping we have resonances between the action of the particle and the actions of the field. This makes the system an example of Poincar\'e nonintegrable systems. We extend the unitary operator to a new star-unitary operator $\ensuremath{\Lambda}.$ This changes the dynamical description of radiation damping. Once we know the Hamiltonian, we can of course write the Hamilton equations. But we have the possibility to go to new descriptions. The invertible $\ensuremath{\Lambda}$ transformation gives many new aspects, which are hidden in the initial description. For example, we show that there are fluctuations, and that there exists an irreducible probability description. The transformation to $\ensuremath{\Lambda}$ representation corresponds to a transformation to Markovian probability equations. We can always come back to the initial representation by the inverse transformation. We have verified this remarkable prediction by detailed numerical calculations. We need the $\ensuremath{\Lambda}$ transformation to obtain the definition of a dressed unstable mode, which has a well-defined lifetime. In the initial representation there are various time scales and there is no strictly exponential lifetime. The situation is the same as the one we have studied in the quantum case in recent papers.
The desirability of considering "dressed" rather than "bare" states of atoms in strong electromagnetic fields has been noted by others. It is suggested here that such a treatment is conveniently carried out by performing a unitary transformation of the Hamiltonian so that the "dressed" states in the new basis are precisely the "bare" states of the usual basis. A perturbative method for constructing such a transformation, closely related to a procedure described by Heitler, is proposed. The method is applied through second order to a two-level atom model, and it is verified that the transformation makes the dressed ground atomic state stable, and the excited state unstable with the proper decay probability; furthermore, the energies of the two states are renormalized by the correct amounts. The method is then compared with the "nonperturbative" momentum-translation approximation, and shown to contain already in first order the entire operator content of that approximation, but to yield different transition probabilities. Finally, the transformation method is applied through second order to the standard scalar-field model and shown to produce the usual dressing transformation, mass renormalization, and induced nucleon-nucleon potential of that model, a result which tends to substantiate the hypothesis that the transformation produces the correct dressed states of a physical system.
We define a new unstable state in the Friedrichs model of a two-level atom. This unstable state is a complex eigenstate of the time evolution operator exp⁡(−iHt)\exp(-iHt)exp(−iHt) with a restricted test function space, which is obtained from causality conditions. The unstable state shows exact exponential decay for t≥0t\ge0t≥0. Its emitted field is confined inside the future light-cone. In this way the long-standing problem of exponential catastrophe is removed. This is an example of quantum mechanics outside Hilbert space, which consists of generalized eigenstates in a distribution space, and a dual (test function) space.