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 Schrödinger equation. Decoherence appears as a dynamical effect, in agreement with a recent proposal (Frasca 2001 Phys. Lett. A 283 271). To prove that this effect is quite general, we show that this same behaviour 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 wavefunction.
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 Schrödinger equation. Decoherence appears as a dynamical effect, in agreement with a recent proposal (Frasca 2001 Phys. Lett. A 283 271). To prove that this effect is quite general, we show that this same behaviour 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 wavefunction.
We consider a large number of two-level atoms interacting with the mode of a cavity in the rotating-wave approximation (Tavis–Cummings model). We apply the Holstein–Primakoff transformation to study the model in the limit of the number of two-level atoms, all in their ground state, becoming very large. The unitary evolution that we obtain in this approximation is applied to a macroscopic superposition state showing that, when the coherent states forming the superposition are distant enough, then the state collapses on a single coherent state describing a classical radiation mode. This appears as a true dynamical effect that could be observed in experiments with cavities.
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 model 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 283, 271 (2001)]. The thermodynamic limit can be very effective in generating 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.
We prove a very general theorem on decoherence in the thermodynamic limit, widely extending the scope of our preceding results [Phys. Lett. A 308, 135 (2003)]. The concept of a singular limit in time is used in the proof. Recent experimental findings [1, 2, 3, 4, 5] in NMR with organic molecular crystals have shown that an intrinsic decoherence effect is present in a many-body system in the thermodynamic limit. These experiments, that are known just in the NMR community, give a strong support to our understanding of decoherence as an effect arising from the thermodynamic limit applied to unitary evolution [6, 7, 8, 9, 10, 11, 12]. These theoretical results are based on a concept of singular limit in time. This concept seems to have been pioneered by Bohm [13, 14]. Indeed, the existence of such an effect would give a relevant answer both to the measurement problem and the question of the irreversibility: both are essential to the understanding of a classical world emerging from quantum mechanics. Besides, recent experiments with cavities realized by Haroche’s group [15, 16] have produced asymptotic states with a large number of photons as foreseen by Gea-Banacloche [17, 18] and further analyzed by Knight and Phoenix [19, 20]. As firstly pointed out by Gea-Banacloche, these states support a view of quantum measurement described by decoherence in the thermodynamic limit, in agreement with our view. In this case one has that
A reason is discussed (may be not the only one) for why we do not see any superposition of macroscopic states in the real world. Under the general assumption that quantum macrostates are statistical ensembles of microstates, it is shown that any superposition of macrostates is reduced in a very short time by the unitary dynamics of the ordinary Schrödinger equation, deducing the Born rule without having to postulate it. In more detail, the macroscopic and microscopic degrees of freedom are decoupled in the Schrödinger equation, yielding an effective stochastic equation for the macroscopic variables, with the ensemble average of the microscopic amplitudes that acts as a self-generated internal white noise. The stochastic equation is shown to be a reducing Itô equation if some general causality conditions are met, predicting a very quick collapse of any macroscopic superposition upon formation, with probabilities which satisfy the Born rule. In the context of the von Neumann measurement scheme, the relevance of the result is discussed as a simple dynamical solution of the measurement problem.
It is shown how classical states, meaning states representing a classical object, can be produced in the thermodynamic limit, retaining the unitary evolution of quantum mechanics. Also, 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 that is formally infinity that, from a physical standpoint, can be proved equivalent to a time average.
Environment induced decoherence [1] has taught us that unitary evolution can be very fragile. Recent experiments with cavities [2], ion traps [3] and interference [4] confirm this expectation. All the experiments carried out so far do not help to answer a fundamental question: Can we maintain unitary evolution down to the Planck time? Different formulations of intrinsic decoherence [5-9] foresee that effects on smaller scales permit us to understand the breaking of the unitary evolution and in some way point out possible answers to our question. But, one can ask if some experiments in quantum optics can be devised to find an answer to the question with the present technology. Decoherence in the thermodynamic limit is a proposal to respond to these questions assuming that the Planck time should be a boundary between different physical theory, that is, we have a singular limit [10]. This singular limit is due to blurring in time and an experimental device will always introduce an average with its proper timescale. We show, with some examples recently presented in literature [11-15], how quantum optics can help to give a definite answer to the main question about unitary evolution, providing in this way direct hints for new physics. This approach to decoherence appears as a new understanding of the measurement problem in quantum mechanics.