States of light via reducible quantization

Naudts

2006

Int J Theor Phys

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A covariant quantization scheme employing reducible representations of canonical commutation relations with positive-definite metric and Hermitian four-potentials (an alternative to the GuptaBleuler method) is tested on the example of quantum electromagnetic fields produced by a classical current. The Heisenberg dynamics can be consistently formulated since the fields are given by operators and not operator-valued distributions. The scheme involves a Hamiltonian whose free part is modified but the minimal-coupling interaction is the standard one. Solving Heisenberg equations of motion under the assumption that the fields are free for times t0 = ±∞ we arrive at retarded and advanced solutions. Once we have these solutions we can deduce the form of evolution of retarded and advanced fields between two arbitrary finite times. The appropriate unitary evolution operators are found and their generators are computed. Now the generators involve the same free part as before, but the interaction term turns out to be modified. For a pointlike charge localized on a worldline z a (t) we find the interaction term of the form −q A z(t) · v(t) − q d z · E where the integration is along those parts of the charge world-line where the charge velocity is nonzero. There is no selfenergy contribution. Next we compute photon statistics. Poisson statistics naturally results and infrared divergence can be avoided even for pointlike sources. Classical fields produced by classical sources can be obtained if one computes coherent-state averages of Heisenberg-picture operators. It is shown that the new form of representation automatically smears out pointlike currents. We discuss in detail Poincaré covariance of the theory and the role of Bogoliubov transformations for the issue of gauge invariance. The representation we employ is parametrized by a number that is related to Rényi's α. It is shown that the "Shannon limit" α → 1 plays here a role of a correspondence principle with the standard regularized formalism.

“…The part P I a is identical to the 4-momentum operator introduced in [6]. The 4-momentum for arbitrary N reads…”

Section: Poincaré Covariance Of Free Fieldsmentioning

confidence: 99%

Section: Construction Of the Reducible Representation Of Ccrmentioning

confidence: 99%

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Regularization as Quantization in Reducible Representations of CCR

Naudts

2006

Int J Theor Phys

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“…Recently, in [1], it was shown that the representation can be, in principle, directly tested in cavity QED. The N -representation of electromagnetic field operators was introduced in [10], and further analyzed and generalized in [11,12,13,14].…”

Section: N -Representation Calculationmentioning

confidence: 99%

Degree of entanglement as a physically ill-posed problem: The case of entanglement with vacuum

Pawłowski

2006

Phys. Rev. A

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We analyze an example of a photon in superposition of different modes, and ask what is the degree of their entanglement with vacuum. The problem turns out to be ill-posed since we do not know which representation of the algebra of canonical commutation relations (CCR) to choose for field quantization. Once we make a choice, we can solve the question of entanglement unambiguously. So the difficulty is not with mathematics, but with physics of the problem. In order to make the discussion explicit we analyze from this perspective a popular argument based on a photon leaving a beam splitter and interacting with two two-level atoms. We first solve the problem algebraically in Heisenberg picture, without any assumption about the form of representation of CCR. Then we take the ∞-representation and show in two ways that in two-mode states the modes are maximally entangled with vacuum, but single-mode states are not entangled. Next we repeat the analysis in terms of the representation of CCR taken from Berezin's book and show that two-mode states do not involve the mode-vacuum entanglement. Finally, we switch to a family of reducible representations of CCR recently investigated in the context of field quantization, and show that the entanglement with vacuum is present even for single-mode states. Still, the degree of entanglement is here difficult to estimate, mainly because there are N + 2 subsystems, with N unspecified and large.

“…Such a Hilbert space represents essentially a single harmonic oscillator of indefinite frequency (for physical motivation cf. [23,24] and the Appendix in [15]). An important property of the representation is that k I k = I is the identity operator in H. A vacuum of this representation is given by any state annihilated by all a k .…”

Section: N < ∞ Representationmentioning

confidence: 99%

Cavity-QED Tests of Representations of Canonical Commutation Relations Employed in Field Quantization

Wilczewski

2006

Int J Theor Phys

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Various aspects of dissipative and nondissipative decoherence of Rabi oscillations are discussed in the context of field quantization in alternative representations of CCR. Theory is confronted with experiment, and a possibility of more conclusive tests is analyzed.PACS numbers: 42.50. Pq, 42.50.Xa, 03.70.+k I. CAVITY QED IN DIFFERENT REPRESENTATIONS OF CCRAlthough the notion of entanglement between atomic and electromagnetic degrees of freedom plays a central role in quantum computing architecture based on cavity QED [1,2,3,4,5,6,7,8,9], the very concept of entanglement leads to conceptual difficulties if quantum vacuum comes into play (cf. uniqueness of the vacuum versus violation of the Bell inequality [10], problems with teleportation of quantum fields [11], ambiguous entanglement with vacuum [12,13]). One of the problems is that the electromagnetic field can be quantized in different representations of canonical commutation relations (CCR). As shown in [13] the degree of entanglement is a representation-dependent property, and it is not clear which representations are really physical. The problem is a part of a wider and ongoing discussion on different quantization paradigms [14]. Now, can the available experimental data distinguish between different representations of CCR? The answer is less obvious than one might expect. In this Letter we will try to clarify the status of some data from cavity QED, and then discuss possibilities of more definitive tests.We first analyze at a representation independent level the simple problem of Rabi oscillation of a two-level atom in an ideal cavity (for technicalities we refer to [15] The observed decoherence appears to be entirely of a nondissipative type, but it is not clear why the effect of dissipation is invisible. Perhaps the fact that a photon is with probability 1 absorbed by the atom at times separated by the Rabi period leads to a sort of Zeno effect. This point requires further experimental and theoretical studies, and is beyond the scope of the present paper.Assuming that Rabi oscillations indeed do not reveal observable damping due to energy dissipation we ask to what extent the experiment can distinguish between reducible and irreducible representations of CCR. In physical terms the question can be translated as follows: How many oscillators do we need to model quantum fields? The standard answer is that we need one oscillator per mode. We show that in reducible representations the data only set certain limitations on the number of oscillators, and this number is independent of the number of modes.Finally, we suggest that one should repeat the measurements reported in [18] with better cavities, finer time resolution, and monitor the Rabi oscillation for longer times. The point is that the decay due to experimental imprecisions may mask quantum beats of a completely new type and origin. In principle, the beats can be observed in a form of vacuum collapses and revivals, the effect occurring in reducible N -representations [19]. Observation of the revival...