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arXiv:1502.00992 · Journal of Physics B Atomic Molecular and Optical Physics

Measuring nonclassicality of single-mode systems

Mehmet Emre Tasgin

Published Jun 15, 2020• 3 citations
arXivHTMLPDF

Abstract

Abstract Nonclassicality of a single-mode state can be quantified by the entanglement (potential) the state creates at the output of a beam splitter. Thus, beam splitter transformation can be utilized for obtaining single-mode nonclassicality conditions (quantifications) from two-mode entanglement criteria (measures). Here, (i) we employ this relation to quantify the single-mode nonclassicality (SMNc) of a nonlinear crystal field generating second harmonic signals. (ii) We also manage to witness the SMNc of superradiant emission, a non-Gaussian state, in a Dicke phase transition. Entanglement potential, at the beam splitter output, is obtained in terms of measurable single-mode quantities. Thus, presence of a beam splitter in an experiment is not necessary. Employing the beam splitter method, one can also utilize hybrid entanglement criteria as SMNc conditions working well in different classes of single-mode states. (iii) We also work on essentials of the entanglement-SMNc relation: one obtains the generalized squeezing condition ⟨ â † â ⟩ < |⟨ â 2 ⟩| as the SMNc condition when an optimized form of Duan–Giedke–Cirac–Zoller criterion [PRL 84, 2722] is used at the output of the beam splitter.

DOI
10.1088/1361-6455/ab9d02

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Abstract Nonclassicality of a single-mode state can be quantified by the entanglement (potential) the state creates at the output of a beam splitter. Thus, beam splitter transformation can be utilized for obtaining single-mode nonclassicality conditions (quantifications) from two-mode entanglement criteria (measures). Here, (i) we employ this relation to quantify the single-mode nonclassicality (SMNc) of a nonlinear crystal field generating second harmonic signals. (ii) We also manage to witness the SMNc of superradiant emission, a non-Gaussian state, in a Dicke phase transition. Entanglement potential, at the beam splitter output, is obtained in terms of measurable single-mode quantities. Thus, presence of a beam splitter in an experiment is not necessary. Employing the beam splitter method, one can also utilize hybrid entanglement criteria as SMNc conditions working well in different classes of single-mode states. (iii) We also work on essentials of the entanglement-SMNc relation: one obtains the generalized squeezing condition ⟨ â † â ⟩ < |⟨ â 2 ⟩| as the SMNc condition when an optimized form of Duan–Giedke–Cirac–Zoller criterion [PRL 84, 2722] is used at the output of the beam splitter.
Nonclassicality of a single-mode field is equivalent to the rank of two-mode entanglement this field generates at the output of a beam-splitter [Phys. Rev. A 89, 052302 (2014)]. We derive a measure for the nonclassicality of a single-mode field using the two-mode inseparability measure of Simon-Peres-Horodecki. Degree of the single-mode nonclassicality requires the knowledge of only ⟨a^2⟩\langle \hat{a}^2\rangle⟨a^2⟩ and ⟨a^†a^⟩\langle\hat{a}^\dagger\hat{a}\rangle⟨a^†a^⟩. This condition/measure is both a necessary and sufficient condition for Gaussian single-mode states. We show that this measure increases with squeezing strength and display a jump at the critical coupling for a superradiant phase transition. Additionally, we derive a simple analytical condition, ∣⟨a^2⟩∣>⟨a^†a^⟩|\langle \hat{a}^2\rangle|>\langle\hat{a}^\dagger\hat{a}\rangle∣⟨a^2⟩∣>⟨a^†a^⟩, for nonclassicality from Duan-Giedke-Cirac-Zoller criterion. In a forthcoming study, we use the derived measure to show the following. Non-causal linear optical response of an optomechanical cavity emerges at the same critical cavity-mechanical coupling where output field becomes nonclassical.
We study the relation between single-mode nonclassicality and two-mode entanglement in a beam splitter. We show that single-mode nonclassicality (the entanglement potential) of incident light cannot be transformed into two-mode entanglement completely after a single beam splitter. Some of the entanglement potential remains as single-mode nonclassicality in the two entangled output modes. Two-mode entanglement generated in the process can be equivalently quantified as an increase in the minimum uncertainty widths (or decrease in the squeezing) of the output states compared to the input states. We use the nonclassical depth and logarithmic negativity as single-mode nonclassicality and entanglement measures, respectively. We realize that a conservation relation between the two quantities can be adopted for Gaussian states, if one works in terms of uncertainty width. This conservation relation is extended to many sets of beam splitters.
If a single-mode nonclassical light is combined with the vacuum on a beam splitter, then the output state is entangled. As proposed in [Phys. Rev. Lett. 94, 173602 (2005)], by measuring the output-state entanglement for a balanced lossless beam splitter, one can quantify the input-state nonclassicality. These measures of nonclassicality (referred to as entanglement potentials) can be based, in principle, on various entanglement measures, leading to the negativity (NP) and concurrence (CP) potentials, and the potential for the relative entropy of entanglement (REEP). We search for the maximal nonclassicality, which can be achieved by comparing two entanglement measures for arbitrary two-qubit states and those which can be generated from a photon-number qubit via a balanced lossless beam-splitter, where the qubit basis states are the vacuum and single-photon states. Surprisingly, we find that the maximal nonclassicality, measured by the REEP for a given value of the NP, can be increased (if NP<0.527) by using either a tunable beam splitter or by amplitude damping of the output state of the balanced beam splitter. We also show that the maximal nonclassicality, measured by the NP for a given value of the REEP, can be increased by dephasing. The entanglement itself is not increased by these local losses, but the possible ratios of different measures are affected. Moreover, we show that partially-mixed states can be more nonclassical than both pure states and completely-mixed states, by comparing the NP for a given value of the REEP. This implies that not all entanglement measures can be used as entanglement potentials. Alternatively, a single balanced lossless beam splitter is not always transferring the whole nonclassicality of its input state into the entanglement of its output modes. Applying a lossy beam splitter can solve this problem at least for the cases analyzed in this paper.
The behavior of general nonclassical two-mode Gaussian states at a beam splitter is investigated. Single-mode nonclassicality as well as two-mode entanglement of both input and output states are analyzed suggesting their suitable quantifiers. These quantifiers are derived from local and global invariants of linear unitary two-mode transformations such that the sum of input (or output) local nonclassicality measures and entanglement measure gives a global invariant. This invariant quantifies the global nonclassicality resource. Mutual transformations of local nonclassicalities and entanglement induced by the beam splitter are analyzed considering incident noisy twin beams, single-mode noisy squeezed vacuum states, and states encompassing both squeezed states and twin beams. A rich tapestry of interesting nonclassical output states is predicted.
If a single-mode nonclassical light is combined with the vacuum on a beam splitter, then the output state is entangled. As proposed in [Phys. Rev. Lett. 94, 173602 (2005)], by measuring this output-state entanglement for a balanced lossless beam splitter, one can quantify the input-state nonclassicality. These measures of nonclassicality (referred to as entanglement potentials) can be based, in principle, on various entanglement measures, leading to the negativity (NP) and concurrence (CP) potentials, and the potential for the relative entropy of entanglement (REEP). We search for the maximal relative nonclassicality, which can be achieved by comparing two entanglement measures for (i) arbitrary two-qubit states and (ii) those which can be generated from a photon-number qubit via a balanced lossless beam splitter, where the qubit basis states are the vacuum and single-photon states. Surprisingly, we find that the maximal relative nonclassicality, measured by the REEP for a given value of the NP, can be increased (if NP<0.527\mathrm{NP}<0.527NP<0.527) by using either a tunable beam splitter or by amplitude damping of the output state of the balanced beam splitter. We also show that the maximal relative nonclassicality, measured by the NP for a given value of the REEP, can be increased by phase damping (dephasing). Note that the entanglement itself is not increased by these losses (since they act locally), but the possible ratios of different measures are affected. Moreover, we show that partially dephased states can be more nonclassical than both pure states and completely dephased states, by comparing the NP for a given value of the REEP. Thus, one can conclude that not all standard entanglement measures can be used as entanglement potentials. Alternatively, one can infer that a single balanced lossless beam splitter is not always transferring the whole nonclassicality of its input state into the entanglement of its output modes. The application of a lossy beam splitter can solve this problem, at least for the cases analyzed in this paper.
Abstract We study different representations of a many-particle system. We reveal the connections among the nonclassicalities of many-particle, single-mode and two-mode systems in the infinite particle number limit. First, we demonstrate that (separable) atomic coherent states, of a many-particle system, mimic the coherent states of single-mode light as the number of particles goes to infinity. This enables the utilization of a many-particle entanglement (MPE) criterion as a single-mode nonclassicality (SMNc) condition via Holstein–Primakoff transformation in this limit. As an example, we show that spin-squeezing criterion, for MPE, can be transformed into quadrature-squeezing, a nonclassicality condition for a single-mode system. Second, we demonstrate a similar connection between the two-mode separable states and separable atomic coherent states of a many-particle system for the infinite particle number limit. The second connection makes us be able to obtain an SMNc condition from a two-mode entanglement (TME) criterion, using the many-particle system as a bridge. As an example, we obtain Mandel’s Q -parameter from the TME criterion of Hillery and Zubairy.