Representation of Operators in Quantum Optics

Miller, Marvin M.; Mishkin, E.

doi:10.1103/physrev.164.1610

Cited by 28 publications

(9 citation statements)

References 15 publications

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“…It governs vanishing of the off-diagonal part in (12) in the trace-norm [5]. A closed formula for |Γ X,X ′ (t)| for general initial states ̺ 0k is possible, using the fact [21] that one can always write ̺ 0k = (1/π) d 2 αP k (α)|α α|, where |α are the usual coherent states [19] and P k (α) is in general a distributional Glauber-Sudarshan P -representation:…”

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confidence: 99%

Dynamical objectivity in quantum Brownian motion

Tuziemski¹,

Korbicz²

2015

EPL

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-Classical objectivity as a property of quantum states-a view proposed to explain the observer-independent character of our world from quantum theory, is an important step in bridging the quantum-classical gap. It was recently derived in terms of spectrum broadcast structures for small objects embedded in noisy photon-like environments. However, two fundamental problems have arisen: a description of objective motion and applicability to other types of environments. Here we derive an example of objective states of motion in quantum mechanics by showing a formation of dynamical spectrum broadcast structures in the celebrated, realistic model of decoherence-Quantum Brownian Motion. We do it for realistic, thermal environments and show their noise-robustness. This opens a potentially new method of studying quantum-to-classical transition.Introduction. -Reconciliation of quantum theory with the classical world of everyday experience has been one of the central problems in our understanding of Nature [1,2], touching such deep questions as is there any 'reality' out there [3]. One of its aspects has been how to explain the objective character of our world with fragile quantum systems, inevitably disturbed by measurements. As quantum state is to date our most fundamental description of Nature, it is natural to look for an explanation at this level. Indeed, recently specific quantum state structuresspectrum broadcast structures (SBS) [4,5], have been identified as responsible for the perceived objectivity, suggesting that the latter is, in fact, a property of quantum states. Building on the quantum Darwinism idea [2, 6]-a realistic form of decoherence theory [2] where the system of interest S interacts with multiple environments E 1 , . . . , E N and observers acquire information about S through them, it has been shown in [4] (see also [7]) in a model-and dynamics-independent way that the only, in a certain sense, states that encode objective states of the system are precisely the SBS:

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confidence: 99%

Dynamical objectivity in quantum Brownian motion

Tuziemski¹,

Korbicz²

2015

EPL

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“…For instance, the Q function of the following operator, A = k|0 2| + |1 1| + k * |2 0|, which is the unit trace and Hermitian but not positive, has the The convex geometry of separable and classical states leads the entanglement parameter that takes into account the nonclassicality, the NcDE in Eq. (7).…”

Section: Geometry Of Non-classical Statesmentioning

confidence: 99%

“…Positive operators(P ) are only a subset from τ = 0 to τ = 1, including the set of separable states(S) which consists of all classical states. (B):The convex geometry of separable and classical states leads the entanglement parameter that takes into account the nonclassicality, the NcDE in Eq (7)…”

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confidence: 99%

Entanglement, detection, and geometry of nonclassical states

Kim

Bae

2010

Phys. Rev. A

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Non-classical states that are characterized by their non-positive quasi-probabilities in phase space are known to be the basis for various quantum effects. In this work, we investigate the interrelation between the non-classicality and entanglement, and then characterize the non-classicality that precisely corresponds to entanglement. The results naturally follow from two findings: one is the general structure among non-classical, entangled, separable, and classical states over Hermitian operators, and the other a general scheme to detect non-classical states.

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“…It was later rediscovered [193]. Of special interest has been another highly singular and non-positive "distribution function" associated with the density operator, the so-called "Sudarshan-Glauber diagonal Prepresentation" [194,196] in terms of the coherent states |α : In general P (α, ᾱ) can be negative in certain regions of the phase space and highly singular [194,196,197,199,200], namely a generalized function (linear functional) belonging to the space Z ′ , which contains, e.g. infinite series where the n-th term is proportional to the n-th derivative of a delta-function [201,202]!…”

Section: A Lot Of Efforts Has Been Put Into Attempts To Find Ways For...mentioning

confidence: 99%

Quantization of the optical phase space 𝒮² = {ϕ mod 2π, I > 0} in terms of the group SO^↑(1, 2)

Kastrup¹

2004

Fortschritte der Physik

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The problem of quantizing properly the canonical pair "angle and action variables ", ϕ and I, is almost as old as quantum mechanics itself and since decades an intensively debated but still unresolved issue in quantum optics. The present paper proposes a new approach to the problem, namely quantization in terms of the group SO(1, 2): The crucial point is that the phase space S 2 = {ϕ mod 2π, I > 0} has the global structure S 1 × R + (a simple cone) and cannot be quantized in the conventional manner. As the group SO(1, 2) acts transitively, effectively and Hamilton-like on that space its irreducible unitary representations of the positive discrete series provide the appropriate quantum theoretical framework. The phase space S 2 has the conic structure of an orbifold R 2 /Z 2 . That structure is closely related to a Z 2 gauge symmetry which corresponds to the center of a 2-fold covering of SO(1, 2), the symplectic group Sp(2, R). The basic variables on the phase space are the functions h 0 = I , h 1 = I cos ϕ and h 2 = −I sin ϕ the Poisson brackets of which obey the Lie algebra so(1, 2). In the quantum theory they are represented by the self-adjoint Lie algebra generators K 0 , K 1 and K 2 of a unitary representation, where K 0 has the spectrum {k + n, n = 0, 1, . . . ; k > 0}. A crucial prediction is that the classical Pythagorean relation h 2 1 + h 2 2 = h 2 0 can be violated in the quantum theory. For each representation one can define three different types of coherent states the complex phases of which may be "measured" by means of the operators K 1 and K 2 alone without introducing any new phase operators! The SO(1, 2) structure of optical squeezing and interference properties as well as that of the harmonic oscillator are analyzed in detail. The additional coherent states can be used for the introduction of (Husimi type) "Q" distributions and (Sudarshan-Glauber type) "P" representations of the density operator. The three operators K 0 , K 1 and K 2 are fundamental in the sense that one can construct composite position and momentum operators out of them! The new framework poses quite a number of fascinating experimental and theoretical challenges.

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Representation of Operators in Quantum Optics

Abstract: 164As a result of the last equation, the expansion of an arbitrary state vector |/) is invariant under the 3 R.

Cited by 28 publications

References 15 publications

Dynamical objectivity in quantum Brownian motion

Dynamical objectivity in quantum Brownian motion

Entanglement, detection, and geometry of nonclassical states

Quantization of the optical phase space 𝒮² = {ϕ mod 2π, I > 0} in terms of the group SO^↑(1, 2)

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Representation of Operators in Quantum Optics

Abstract: 164As a result of the last equation, the expansion of an arbitrary state vector |/) is invariant under the 3 R.

Cited by 28 publications

References 15 publications

Dynamical objectivity in quantum Brownian motion

Dynamical objectivity in quantum Brownian motion

Entanglement, detection, and geometry of nonclassical states

Quantization of the optical phase space 𝒮2 = {ϕ mod 2π, I > 0} in terms of the group SO↑(1, 2)

Contact Info

Product

Resources

About

Quantization of the optical phase space 𝒮² = {ϕ mod 2π, I > 0} in terms of the group SO^↑(1, 2)