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arXiv:quant-ph/0201040 · Journal of Optics B Quantum and Semiclassical Optics

Classical states and decoherence produced by unitary evolution in the thermodynamic limit

Marco Frasca

Published Aug 1, 2002• 8 citations
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Abstract

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.

DOI
10.1088/1464-4266/4/4/336

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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.
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
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.
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.
Time evolution operator in quantum mechanics can be changed into a statistical operator by a Wick rotation. This strict relation between statistical mechanics and quantum evolution can reveal deep results when the thermodynamic limit is considered. These results translate in a set of theorems proving that these effects can be effectively at work producing an emerging classical world without recurring to any external entity that in some cases cannot be properly defined. In a many-body system has been recently shown that Gaussian decay of the coherence is the rule with a duration of recurrence more and more small as the number of particles increases. This effect has been observed experimentally. More generally, a theorem about coherence of bulk matter can be proved. All this takes us to the conclusion that a well definite boundary for the quantum to classical world does exist and that can be drawn by the thermodynamic limit, extending in this way the deep link between statistical mechanics and quantum evolution to a high degree.
Time evolution operator in quantum mechanics can be changed into a statistical operator by a Wick rotation. This strict relation between statistical mechanics and quantum evolution can reveal deep results when the thermodynamic limit is considered. These results translate in a set of theorems proving that these effects can be effectively at work producing an emerging classical world without recurring to any external entity that in some cases cannot be properly defined. For a many-body system it has been recently shown that Gaussian decay of the coherence is the rule with a duration of recurrence more and more small as the number of particles increases. This effect has been observed experimentally. More generally, a theorem about coherence of bulk matter can be proved. All this takes us to the conclusion that a well defined boundary for the quantum to classical world does exist and that can be drawn by the thermodynamic limit, extending in this way the deep link between statistical mechanics and quantum evolution to a high degree.
In this review, we discuss the decoherence and thermalization of a quantum spin system interacting with a spin bath environment, by numerically solving the time-dependent Schrödinger equation of the whole system. The effects of the topologic structure and the initial state of the environment on the decoherence of the two-spin and many-spin system are discussed. The role of different spin-spin coupling is considered. We show under which conditions the environment drives the reduced density matrix of the system to a fully decoherent state, and how the diagonal elements of the reduced density matrix approach those expected for the system in the microcanonical or canonical ensemble, depending on the character of the additional integrals of motion. Our demonstration does not rely on time-averaging of observables nor does it assume that the coupling between system and bath is weak. Our findings show that the microcanonical distribution (in each eigenenergy subspace) and canonical ensemble are states that may result from pure quantum dynamics, suggesting that quantum mechanics may be regarded as the foundation of quantum statistical mechanics. Furthermore, our numerical results show that a fully decoherent quantum system prefers to stay in an equilibrium state with a maximum entropy, indicating the validity of the second law of thermodynamics in the decoherence process.