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Physical Review Letters

Collapse and revival of the state vector in the Jaynes-Cummings model: An example of state preparation by a quantum apparatus

Julio R. Gea-Banacloche

Published Dec 31, 1990• 373 citations
Publisher

Abstract

The evolution of the atomic state in the resonant Jaynes-Cummings model (a two-level atom interacting with a single mode of the quantized radiation field) with the field initially in a coherent state is considered. It is shown that the atom is to a good approximation in a pure state in the middle of what has been traditionally called the ``collapse region.'' This pure state exhibits no Rabi oscillations and is reached independently of the initial state of the atom. For most initial states a total or partial ``collapse of the wave function'' takes place early during the interaction, at the conventional collapse time, following which the state vector is recreated, over a longer time scale.

DOI
10.1103/physrevlett.65.3385

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Atom- and field-state evolution in the Jaynes-Cummings model for large initial fields

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Schrödinger-cat states in the resonant Jaynes-Cummings model: Collapse and revival of oscillations of the photon-number distribution

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Physical Review A•Jun 1, 1992•275 citations
The evolution of the atomic state in the resonant Jaynes-Cummings model (a two-level atom interacting with a single mode of the quantized radiation field) with the field initially in a coherent state is considered. It is shown that the atom is to a good approximation in a pure state in the middle of what has been traditionally called the collapse region.'' This pure state exhibits no Rabi oscillations and is reached independently of the initial state of the atom. For most initial states a total or partial collapse of the wave function'' takes place early during the interaction, at the conventional collapse time, following which the state vector is recreated, over a longer time scale.
An asymptotic result is derived for the Jaynes-Cummings model of a two-level atom interacting with a quantized single-mode field, which is valid when the field is initially in a coherent state with a large average photon number. It is shown that for certain initial atomic states the joint atom-field wave function factors into an atomic and a field part throughout the interaction, so that each system remains separately in a pure state. The atomic part of the wave function displays a crossing of trajectories in the atom Hilbert space that leads to a unique state for the atom, independent of its initial state, at a specific time t0{\mathit{t}}_{0}t0​ (equal to half the revival time). The field part of the wave function resembles a crescent squeezed state. The well-known collapses and revivals are investigated from this perspective. The collapse appears to be associated with a ``measurement'' of the initial state of the atom with the field as the measuring apparatus. The measurement is not complete for finite average photon number: the system is instead left in a coherent superposition of macroscopically distinct states. At the half-revival time t0{\mathit{t}}_{0}t0​ this superposition of states is entirely in the field part of the state vector, so that the (pure) state of the field at that time is of the form sometimes referred to as a ``Schr\"odinger cat.'' The revivals of the population inversion are seen to be entirely due to the fact that the linear superposition of the two macroscopically distinct field states is coherent (i.e., a pure state), as opposed to an incoherent mixture.
We consider the time evolution of the radiation field (R) and a two-level atom (A) in a resonant microcavity in terms of the Jaynes-Cummings model with an initial general pure quantum state for the radiation field. It is then shown, using the Cauchy-Schwarz inequality and also a Poisson resummation technique, that perfect coherence of the atom can in general never be achieved. The atom and the radiation field are, however, to a good approximation in a pure state ∥ψ〉A⊗R=∥ψ〉A×∥ψ〉R in the middle of what has been traditionally called the ‘collapse region’, independent of the initial state of the atoms, provided that the initial pure state of the radiation field has a photon number probability distribution which is sufficiently peaked and phase differences that do not vary significantly around this peak. An approximate analytic expression for the quantity Tr[ρ2 A(t)], where ρA (t) is the reduced density matrix for the atom, is derived. We also show that under quite general circumstances an initial entangled pure state will be disentangled to the pure state ∥ψ〉PSA⊗R.
We consider the time evolution of the radiation field (R) and a two-level atom (A) in a resonant microcavity in terms of the Jaynes-Cummings model with an initial general pure quantum state for the radiation field. It is then shown, using the Cauchy-Schwarz inequality and also a Poisson resummation technique, that perfect coherence of the atom can in general never be achieved. The atom and the radiation field are, however, to a good approximation in a pure state ∥ψ〉A⊗R=∥ψ〉A×∥ψ〉R in the middle of what has been traditionally called the ‘collapse region’, independent of the initial state of the atoms, provided that the initial pure state of the radiation field has a photon number probability distribution which is sufficiently peaked and phase differences that do not vary significantly around this peak. An approximate analytic expression for the quantity Tr[ρ2 A(t)], where ρA (t) is the reduced density matrix for the atom, is derived. We also show that under quite general circumstances an initial entangled pure state will be disentangled to the pure state ∥ψ〉PSA⊗R.
The Jaynes–Cummings model (JCM) is used to give a detailed analysis of the long-time behavior of a two-level atom interacting with a (Glauber) coherent-state field. This work is based on the earlier work of J. H. Eberly and collaborators, who drew attention to the collapse and revival of the Rabi oscillations of the atomic inversion. We use the three components S1, S2, and S3 of the pseudospin vector S to analyze the dynamics of the atom within the rotating-wave approximation and exact resonance. We show that, at the times when the revivals peak, the variables σ = |S| and η = S3/σ become independent of (1) the transition dipole moment of the atom, (2) the intensity of the coherent-state field, and (3) the (common) frequency of the atom and the field. Thus these special values of σ and η are a universal feature of the JCM that should be observed in experiments with any two-level atom in an arbitrarily intense (but resonant) coherent-state field. The derivation of the above results was facilitated by transforming into a rotating coordinate frame in which S1 = 0 so that S2 and S3 are the only dynamical variables. The two-dimensional nature of S in this frame makes it easy to visualize the dynamical changes in the atom. The dynamics of the JCM is clarified by comparison with the closely related Buck–Sukumar model. The relationship of the phenomena discussed here to photon echoes and to quench echoes (in condensed media) is pointed out.
In the Jaynes–Cummings model a two-level atom interacts with a single-mode electromagnetic field. Quantum mechanics predicts collapses and revivals in the probability that a measurement will show the atom to be excited at various times after the initial preparation of the atom and field. In retrodictive quantum mechanics we seek the probability that the atom was prepared in a particular state given the initial state of the field and the outcome of a later measurement on the atom. Although this is not simply the time reverse of the usual predictive problem, we demonstrate in this paper that retrodictive collapses and revivals also exist. We highlight the differences between predictive and retrodictive evolutions and describe an interesting situation where the prepared state is essentially unretrodictable.
The Jaynes-Cummings model of optical resonance describes the simplest fully quantized interaction between two quantum systems of different nature: a two-level atom (fermionic system) and a quantized field mode (bosonic system). This interaction leads to extreme quantum entanglement of the atom and field. However, the model also predicts that, at precisely half of the revival time, the atom and field become asymptotically disentangled. This disentanglement becomes more exact as the coherent-state amplitude increases. In this paper we investigate the nature of the pure-field-state superposition generated at such times. We show that this superposition is of distinguishable states of the field with the same amplitude but opposite phase. Interference between these components leads to nonclassical oscillations in photon-number distributions and squeezing in quadratures of the field. The Schr\"odinger-cat states of the field are highly transient, and depend very sensitively on the interaction time, the initial intensity of the field, and the atom-field detuning.