<i>su</i>(1,1) intelligent states

Joanis, Patrick; Mahler, Dylan H.; Guise, Hubert de

doi:10.1088/1751-8113/43/38/385304

Cited by 9 publications

(6 citation statements)

References 28 publications

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“…The concept of fluctuation state vectors, which arises from the evaluation of expectation values of time evolving operators within the Heisenberg picture, constitutes an exact quantum state engineering process [17] and the corresponding density matrices can be used to study the entanglement properties of the fully quantized parametric amplification process. The positive and negative helicity intensity operators, together with the polarization operators, are useful in the construction of intelligent state vectors on an SU(1, 1) manifold [18]. In particular, polarization operators are currently employed in characterization and tomography of photon polarization states, mostly under parametric interactions [19]- [22].…”

Section: Discussionmentioning

confidence: 99%

Some New Features of Photon Statistics in a Fully Quantized Parametric Amplification Process

Omolo¹

2014

JMP

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An exact quantum treatment reveals that signal and idler photon number operators are not wellbehaved dynamical operators for studying photon statistics in parametric amplification/downconversion processes. Contrary to expectations, the mean signal-idler photon number difference () () 1 2 n t n t − varies with time, while the corresponding signal-idler photon number cross-correlation function () () 1 2 n t n t is complex and experiences an interference phenomenon driven by the interaction parameters. The intensity operators and related polarization operators of the polarized signal-idler photon pair in positive and negative helicity states are identified as the appropriate operators specifying a conservation law and the dynamical symmetry group (SU(1, 1)) of the parametric amplification process. The conservation of the mean positive and negative helicity photon intensity difference and the purely real positive-negative helicity intensity cross-correlation function correctly account for the simultaneous production of polarized signal and idler photons in positive and negative helicity states.

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Section: Discussionmentioning

confidence: 99%

Some New Features of Photon Statistics in a Fully Quantized Parametric Amplification Process

Omolo¹

2014

JMP

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“…Conversely, any two-mode Fock state |𝑛 𝑎 , 𝑛 𝑏 can be mapped as states in an irrep of SU(1,1), although that is more involved as the state can appear in several irreps. 77 We always can consider 𝑛 𝑎 > 𝑛 𝑏 , since the opposite can be obtained by just a relabelling of modes, with no physical consequences. The total Hilbert space of the two oscillators decomposes then as…”

Section: The Su(11) Wigner Functionmentioning

confidence: 99%

Local sampling of the SU(1,1) Wigner function

Fabre¹,

Klimov²,

Leuchs³

et al. 2023

Preprint

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Despite the indisputable merits of the Wigner phase-space formulation, it has not been widely explored for systems with SU(1,1) symmetry, as a simple operational definition of the Wigner function has proved elusive in this case. We capitalize on the unique properties of the parity operator, to derive in a consistent way a bona fide SU(1,1) Wigner function that faithfully parallels the structure of its continuous-variable counterpart. We propose an optical scheme, involving a squeezer and photon-number-resolving detectors, that allows for direct point-by-point sampling of that Wigner function. This provides an adequate framework to represent SU(1,1) states satisfactorily. I. INTRODUCTIONThe phase-space formulation of quantum theory 1-11 stands as a self-contained alternative to the conventional Hilbert-space formalism. In this approach, observables become 𝑐-number functions instead of operators and quantum mechanics appears as a statistical theory. Moreover, it is the most convenient construct for visualizing quantum states and processes.The foundations of the method were laid by Weyl 12 and Wigner. 13 Subsequently, Groenewold 14 and Moyal 15 developed all the indispensable tools that have evolved into an accomplished discipline with applications in many diverse fields.The phase-space picture of systems described by continuous variables, such as Cartesian position and momentum of a harmonic oscillator, gained popularity among the quantum optics community, mostly influenced by the authoritative work of Glauber. 16 In particular, the celebrated quasiprobability distributions, such as the 𝑃 (Glauber-Sudarshan), 17,18 𝑊 (Wigner), 13 and 𝑄 (Husimi) 19 representations, are nothing but the functions connected with the density operator.The formalism has been extended in a natural way to other dynamical symmetries (recall that a Lie group 𝐺, with Lie algebra 𝔤, is a dynamical symmetry if the Hamiltonian of the system under consideration can be expressed in terms of the generators of 𝐺; that is, the elements of 𝔤). Perhaps, the most significant example is that of SU(2), with the Bloch sphere as the underlying phase space, [20][21][22] which is of significance in dealing with two-level systems. [23][24][25][26][27][28] Entrhalling results have also been reported for the Euclidean group E(2), now with the cylinder as phase space: [29][30][31][32] this is of relevance in treating, e.g., the orbital angular momentum of twisted photons. 33,34 Additional developments for more general symmetries have appeared in the literature. [35][36][37][38] Moreover, the basic ideas have been adapted to discrete qudits, where the phase space becomes a finite grid. [39][40][41][42][43][44][45] It is apparent from the previous discussion that a satisfactory description of a physical phenomenon requires the use of a suitable geometric arena. Surprisingly, the phase-space description of systems with SU(1,1) dynamical symmetry has received comparatively little attention, 46,47 even if SU(1,1) plays a crucial role in connection with what has been br...

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“…Although they have been investigated from various perspectives [51][52][53], we follow here the comprehensive approach of Ref. [54], which starts by noting that the coherent state exp(iτ ky )|k, k is intelligent provided λ = cosh τ [with eigenvalue Λ = −(k + M) sinh τ, and M = 0, 1, . .…”

Section: Radial Intelligent and Squeezed Statesmentioning

confidence: 99%

“…[54], which starts by noting that the state exp(iτk y )|k,k is intelligent provided λ = cosh τ [with eigenvalue = −(k + M) sinh τ , and M = 0,1, . .…”

mentioning

confidence: 99%