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arXiv:quant-ph/0211165 · Physical Review A

Comment on “Some implications of the quantum nature of laser fields for quantum computations”

Wayne M. Itano

Published Oct 14, 2003• 27 citations

Abstract

A recent discussion of quantum limitations to the fidelity with which superpositions of internal atomic energy levels can be generated by an applied, quantized, laser pulse, is shown to be based on unrealistic physical assumptions. This discussion assumed the validity of Jaynes-Cummings dynamics for an atom interacting with a laser field in free space, that is, when the atom is not surrounded by a resonant cavity. If the laser field is a multimode quantum coherent state, and the Rabi frequency is much greater than the spontaneous decay rate, then the total atomic decoherence rate is on the order of the spontaneous decay rate. With the use of a unitary transformation of the field states due to Mollow, it can be shown that the atomic decoherence rate is the same as if the laser field were treated classically, without any additional contribution due to the quantum nature of the laser field.

DOI: 10.1103/physreva.68.046301

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Reply I to “Comment on ‘Some implications of the quantum nature of laser fields for quantum computations’ ”

S. J. van Enk, Harry J. Kimble

We show that Itano's arguments confirm our conclusion, that entanglement of the atom with the laser field contributes a small but nonzero amount to the total decoherence of an atom in a laser field. This small effect scales as $1/\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ with $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ the average number of photons in the laser pulse.

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A spontaneously decaying two-level atom in an evacuated high-Q cavity undergoes Rabi oscillations. This phenomenon has been observed experimentally as enhanced or inhibited spontaneous emission. In the standard Jaynes-Cummings model the atom is assumed to interact with only a single cavity mode. By retaining the infinite set of cavity modes the propagation of the spontaneously emitted radiation may be calculated. Such an analysis yields a clear physical picture of Rabi oscillations. These vacuum-field Rabi oscillations are compared with Rabi oscillations of a one-electron atom induced by coherent radiation. We show that the same wave packet analysis explains both types of Rabi oscillation. The limit in which a multimode cavity may be approximated as a single-mode cavity is discussed.

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Reply II to “Comment on ‘Some implications of the quantum nature of laser fields for quantum computations’ ”

Julio R. Gea-Banacloche

I show that there is no contradiction between the results in my original paper [Phys. Rev. A 65, 022308 (2002)] and Itano's claims in the previous Comment. The error probabilities I calculated from quantum field fluctuations can equivalently be obtained by considering the probability of spontaneous emission by the atom into the field modes that make up the laser pulse.

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Some implications of the quantum nature of laser fields for quantum computations

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The quantum nature of the laser fields used in many proposed schemes for the manipulation of quantum information may have important consequences for some very large scale quantum computations. Some of these consequences are explored here, focusing especially on phase errors and their effects on error-correction schemes. Depending on the way the logical gates are performed, constraints are found on either the required number of photons per coherence time put out by the laser source, or the source's power

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A recent discussion of quantum limitations to the fidelity with which superpositions of internal atomic energy levels can be generated by an applied, quantized, laser pulse, is shown to be based on unrealistic physical assumptions. This discussion assumed the validity of Jaynes-Cummings dynamics for an atom interacting with a laser field in free space, that is, when the atom is not surrounded by a resonant cavity. If the laser field is a multimode quantum coherent state, and the Rabi frequency is much greater than the spontaneous decay rate, then the total atomic decoherence rate is on the order of the spontaneous decay rate. With the use of a unitary transformation of the field states due to Mollow, it can be shown that the atomic decoherence rate is the same as if the laser field were treated classically, without any additional contribution due to the quantum nature of the laser field.

Introduction So far, we have discussed closed systems involving a single quantized mode of the field interacting with atoms, as for example in the Jaynes–Cummings model in Chapter 4. As we saw in this model, the transition dynamics are coherent and reversible: the atom and field mode exchange excitation to and fro without loss of energy. As we add more modes for the atom to interact with, the coherent dynamics become more complicated as the relevant atom–field states come in and out of phase and beat together to determine the total state occupation probabilities. As time goes on, these beats get out of phase, leading to an apparent decay of the initial state occupation probability. But at later times, the beating eigenfrequencies get back in phase in a manner rather reminiscent of the Jaynes–Cummings revival discussed earlier in this book, and this leads to a partial recurrence or revival of the initial state probability. The time scale for this partial revival depends on the number of participating electromagnetic field modes and as these increase to the level appropriate for an open system in free space the recurrence disappears off to the remote future, and the exponential decay law appropriate for decay is recovered as an excellent approximation. We have already discussed the origin of spontaneous emission and the Einstein A coefficient using perturbation theory in Chapter 4.

A spontaneously decaying two-level atom in an evacuated high-Q cavity undergoes Rabi oscillations. This phenomenon has been observed experimentally as enhanced or inhibited spontaneous emission. In the standard Jaynes-Cummings model the atom is assumed to interact with only a single cavity mode. By retaining the infinite set of cavity modes the propagation of the spontaneously emitted radiation may be calculated. Such an analysis yields a clear physical picture of Rabi oscillations. These vacuum-field Rabi oscillations are compared with Rabi oscillations of a one-electron atom induced by coherent radiation. We show that the same wave packet analysis explains both types of Rabi oscillation. The limit in which a multimode cavity may be approximated as a single-mode cavity is discussed.

We present a general procedure to describe the dynamics of N degenerate quantized fields interacting resonantly with a two–level atom, all coupled with the same strength, within the rotating–wave approximation. Starting from the analysis of the two and three field cases, we generalize the method by identifying dynamical invariants that lead to a factorized form of the time–evolution operator. A unitary transformation reduces the problem to an effective Jaynes–Cummings Hamiltonian, where only one field interacts with the atom and the remaining modes contribute as free fields. Assuming initially coherent fields and an atomic superposition, we compute the atomic inversion and the mean photon number, revealing vacuum Rabi oscillations with a frequency determined by an effective coupling constant that exceeds the individual atom–field coupling, as well as the characteristic collapse-revival behavior.

The quantum nature of the laser fields used in many proposed schemes for the manipulation of quantum information may have important consequences for some very large scale quantum computations. Some of these consequences are explored here, focusing especially on phase errors and their effects on error-correction schemes. Depending on the way the logical gates are performed, constraints are found on either the required number of photons per coherence time put out by the laser source, or the source's power