[Paper Review] Single-mode nonclassicality measure from Simon-Peres-Horodecki criterion

Mehmet Emre Taşgın|arXiv (Cornell University)|Feb 3, 2015

Mechanical and Optical Resonators11 citations

TL;DR

This paper introduces a novel single-mode nonclassicality measure derived from the Simon-Peres-Horodecki criterion for two-mode inseparability, using only the expectation values ⟨â²⟩ and ⟨ↆâ⟩. The measure is both necessary and sufficient for Gaussian states, increases with squeezing, and exhibits a jump at the critical coupling for superradiant phase transitions, providing a direct link between nonclassicality and quantum phase transitions.

ABSTRACT

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 $\langle \hat{a}^2 angle$ and $\langle\hat{a}^\dagger\hat{a} angle$. 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, $|\langle \hat{a}^2 angle|>\langle\hat{a}^\dagger\hat{a} angle$, 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.

Motivation & Objective

• To establish a physically meaningful, operational measure of single-mode nonclassicality that connects to two-mode entanglement via beam-splitter transformation.
• To derive a necessary and sufficient condition for nonclassicality in Gaussian single-mode states using only second-order moments.
• To analyze the behavior of nonclassicality near critical points in quantum phase transitions, particularly in superradiant systems.
• To link the emergence of nonclassical output fields with the onset of non-causal linear optical responses in optomechanical cavities.

Proposed method

• Derive a single-mode nonclassicality measure by applying the Simon-Peres-Horodecki criterion to the two-mode output state generated by a beam-splitter acting on a single-mode input.
• Express the measure in terms of only ⟨â²⟩ and ⟨ↆâ⟩, enabling direct experimental access to the nonclassicality via second-order field moments.
• Use the Duan-Giedke-Cirac-Zoller criterion to derive a simple analytical condition |⟨â²⟩| > ⟨ↆâ⟩ as a sufficient condition for nonclassicality.
• Analyze the measure's behavior under varying squeezing strength and identify its divergence or discontinuity at the critical coupling for a superradiant phase transition.
• Apply the measure to predict non-causal optical responses in optomechanical cavities, linking nonclassicality to quantum response functions.

Experimental results

Research questions

• RQ1 How can single-mode nonclassicality be quantified using only second-order field moments?
• RQ2 What is the relationship between single-mode nonclassicality and the two-mode entanglement generated via a beam-splitter?
• RQ3 Does the derived nonclassicality measure correctly identify the critical coupling point in a superradiant phase transition?
• RQ4 Can the emergence of nonclassical output fields be linked to non-causal optical responses in optomechanical systems?

Key findings

• The proposed nonclassicality measure is both necessary and sufficient for Gaussian single-mode states, relying only on ⟨â²⟩ and ⟨ↆâ⟩.
• The measure increases monotonically with squeezing strength, reflecting growing nonclassical character.
• A discontinuous jump in the measure occurs precisely at the critical coupling for a superradiant phase transition, signaling a qualitative change in quantum behavior.
• The condition |⟨â²⟩| > ⟨ↆâ⟩ is derived as a simple, analytical criterion for nonclassicality using the Duan-Giedke-Cirac-Zoller criterion.
• The measure predicts that non-causal linear optical responses emerge in optomechanical cavities exactly at the same coupling strength where the output field becomes nonclassical.