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Physica Scripta

Violation of Cauchy-Schwarz inequality and creation of higher-order squeezing: quantum control of environment

Qing Yang Xu

Published Oct 16, 2026• 0 citations

Abstract

Abstract It is generally accepted that the ubiquitous interaction of the quantum system with its external world normally yields the randomization of phase relationships among the components of the superposition and causes the transition from quantum to classical behavior. We obtain both the steady-state Cauchy-Schwarz inequality violation and higher-order squeezing originating from the immediate consequence of the modified environment by the atom-driving-field interaction. Furthermore, the stronger dissipation enables the enhancement of the violation and amplification of the squeezing. We also show that the coherent control of environment results in the nontrivial locking of the phase relationships among the components of the superposition of quantized fields rather than their randomization in the stationary regime. This work underlines the transition of decohering systems by their coherently controlled environments to the quantum regime and constitutes a key step towards the steady-state perfect higher-order squeezing.

DOI: 10.1088/1402-4896/aeb592

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Abstract It is generally accepted that the ubiquitous interaction of the quantum system with its external world normally yields the randomization of phase relationships among the components of the superposition and causes the transition from quantum to classical behavior. We obtain both the steady-state Cauchy-Schwarz inequality violation and higher-order squeezing originating from the immediate consequence of the modified environment by the atom-driving-field interaction. Furthermore, the stronger dissipation enables the enhancement of the violation and amplification of the squeezing. We also show that the coherent control of environment results in the nontrivial locking of the phase relationships among the components of the superposition of quantized fields rather than their randomization in the stationary regime. This work underlines the transition of decohering systems by their coherently controlled environments to the quantum regime and constitutes a key step towards the steady-state perfect higher-order squeezing.

By studying the interaction of isolated two-level atom with two-mode entangled coherent state optical field, the time evolution of dipole squeezing quantum effect in the system is investigated in detail. The results show that the properties of interaction coherent mode determine the degree of atom dipole squeezing. Parameter ΔV of violation of Cauchy-Schwarz inequality of the interaction system is defined. Time evolution of parameter ΔVis studied, and it can be seen evidently from the results that the degree of violation of Cauchy-Schwarz inequality accords with the nonclassical property of interaction system. Through tuning the parameter of the coherent field, we can easily perform the remote control of the noncommercial properties.

We have observed a violation of the Cauchy-Schwarz inequality in the macroscopic regime by more than 8 standard deviations. The violation has been obtained while filtering out only the low-frequency noise of the quantum-correlated beams that results from the technical noise of the laser used to generate them. We use bright intensity-difference squeezed beams produced by four-wave mixing as the source of the correlated fields. We also demonstrate that squeezing does not necessarily imply a violation of the Cauchy-Schwarz inequality.

A review is given of correlation experiments in optics which explicitly distinguish between the predictions of classical and quantum theory. In particular, the Cauchy-Schwarz and Bell's inequalities and the phenomena of photon antibunching and squeezing are discussed. The violation of classical inequalities is shown to be associated with the nonexistence of a positive Glauber-Sudarshan P function. An example is given of a dissipative system where a violation of Bell's inequality occurs.

It is generally accepted that the role of quantum decoherence is to spoil the quantum superpositions and to yield the emergence of the classical behaviors from quantum mechanics. Contrary to the common knowledge of decoherence, we propose a scheme to generate a steady-state bright source of strongly sub-Poissonian squeezed light using the environmentally induced decoherence. The disappearance of the process of decoherence disables the generation of the squeezed state of the combination modes, while the presence of the decoherence results in the strongly nonclassical light source at the sub-shot-noise level and even near-perfect sub-Poissonian squeezed state of light. This excludes the use of the paradigm that the good insulation of a quantum system from its surrounding environment is highly desirable to explore its quantum property and fuel the growth of quantum information science. The interactions with the atom make no contribution to the preparation of squeezed light. These counterintuitive results are originated from the intrinsic quantum reservoir in the dressed-atom combination mode picture, which may open the door to exploring high levels of squeezing induced by the decoherence in the stationary regime.

Recently, violation of the Cauchy-Schwarz inequality by the second-order correlation function was reported in a collection of ultracold bosons [Kheruntsyan et al., Phys. Rev. Lett. 108, 260401 (2012)]. We show that the observation of this effect proves the presence of particle entanglement in a many-body system of bosons. Our derivation is based on the analogy between the coherent states of electromagnetic fields and separable states of massive particles. The presented argument applies to any quantum system of identical bosons with either a fixed or a fluctuating number of particles, provided that there is no coherence between different number states.