The most widely applied model in the study of quantum optics is the Jaynes–Cummings model [*V* configurations, and 1- or 2-mode field was examined [

Journal of the European Optical Society-Rapid Publications, Volume. 19, Issue 2, 2023039(2023)

Atom-field system: Effects of squeezing and intensity dependent coupling on the quantum coherence and nonclassical properties

Recently, Kumar Gerry et al. [Phys. Rev. A 90, 033427 (2014)

Keywords

1 Introduction

The most widely applied model in the study of quantum optics is the Jaynes–Cummings model [*V* configurations, and 1- or 2-mode field was examined [

Many quantum phenomena have been produced as resources for carrying out numerous tasks in the field of quantum optics and information, both theoretically and practically [

Quantum entanglement is the aspect of a composite quantum system that most captivates our attention as a fascinating occurrence. If the combined state of two particles cannot be represented as a product of the states of their constituent subsystems, then the particles are said to be entangled [

Motivated by the recent development of atom-field system in quantum optics, we analyze the interaction between a SLAS and 1-mode field initially in squeezed coherent states. The field has one mode and the interaction is affected by five photons. We extend the Jeans–Cummings model to describe the interaction between the atom and the SF and the system dynamics. We analyze the time evolution of the atomic coherence, non-local correlation, statistical properties within the bipartite system in the presence and absence of intensity-dependent coupling for different squeezing regimes of the field. A full understanding of the dynamics of atomic systems that interact with a quantized field is becoming crucial, especially with respect to applications for quantum optics and information science. To implement the quantum computer the most important ingredient is the quantum dynamics, in which one subsystem undergoes a coherent evolution that depends on the state of another subsystem.

The present manuscript is outlined as follows: In Section 2, we present the model which describes SLAS in a cascade type interacting with a quantized field initially defined in a SF. Section 3 describes the quantumness measures and the main results. Finally, a summary and conclusions are illustrated in the last section.

2 Physical model and wave function

Here, we introduce the quantum scheme of a SLAS interacting with a nonlinear field. The SLAS is considered with transition energy between the six levels *ω*_{j} (*j* = 1,…, 6) where ω_{1} > ω_{2} > .... > ω_{6}. The SLAS states |j〉 are ordered from the upper state |1〉 to the lower state |6〉. We assume that the SLAS begins in its upper state and that the field in the squeezed coherent states (SCS) denoted by |*α*, *r*〉. The state of the system at = 0 is |*U*(0)〉 = |*α*, *r*, 1〉 and the state |*α*, *r*〉 is defined by [*r* is the squeezed parameter and the amplitude *b*_{n,r} is given by

The interaction Hamiltonian can be expressed as [*λ*_{j} designs the atom-field coupling constant, and we focus here on the case of identical coupling with *ξ*_{j} = *λ*_{j+1} = *λ*, *j* = 1…4.

The SLAS-field wave function at *t* > 0 and *ξt* being the scaled time can be formulated as:

In the case of absence of the squeezing (i.e. *r* = 0), the field is in the coherent state and

3 Quantum quantifiers and numerical results

3.1 Atomic quantum entropy

Here, we provide a brief description of the von Neumann entropy, quantum coherence, and nonclassical features of the radiation field based on the variation of the Mandel parameter.

In this manuscript, the SLAS-SF entanglement can be detected through the von Neumann according to the atomic basis or field basis. For a quantum system, the von Neumann entropy is determined by [*x* is the density operator describing the given quantum state that gives zero for all pure states with *x*^{2} = *x*. In our case, it can be written as a function of the eigenvalues SLAS density matrix

Let us now analyze the effect of the system parameters on the dynamics of the quantum entanglement of the SLAS-SF state. In *S*_{SLAS} against the normalized time *λt* in the absence and presence of squeezing and I-DC effect. Generally, we can see that the entanglement measure exhibits an oscillator behavior with amplitudes that depend on the values of *f* and *r*. This indicates that the SF can help to achieve and stabilize the amount of entanglement of the SLAS-SF state at a high level. The increase in the squeezed parameter *r* organizes and stabilizes the dynamical behavior of *S*_{SLAS} and diminish the amplitude of the oscillations. On the other hand, the presence of the I-DC effect can decrease the amount of entanglement and augment the oscillations amplitude.

Figure 1.Temporal evolution of

3.2 Quantum coherence

The measure of coherence for a quantum state *ρ* is related to its off-diagonal elements by [*M*, respectively, describes the norm and the set of incoherent states. For different quantum states, the function *Q*_{C} satisfies the monotonicity property.

Now, we consider the dynamics of atomic coherence in the presence of SF. *Q*_{C} with and without squeezing and I-DC effect. We can observe that the atomic coherence measure exhibits oscillations during the evolution accompanied by amplitudes that depend on *f* and *r*. The augmentation in the value of the parameter *r* reduces the amplitudes of the oscillations and stabilize the dynamical behaviour of the coherence measure. On the other side, the the I-DC effect can increase the amount of atomic coherence and enhance the amplitudes of the oscillations. By comparing the quantum entropy and coherence, associated with the SLAS-SF model investigated here, we find that the parameter *r* and the function *f* act on similar way on the measures of entanglement and coherence. This indicates that the coherence can be viewed as a correlation function, which can capture the correlation between subsystems.

Figure 2.Temporal evolution of ^{2} = 25. The subfigures (a, b, c) are for

3.3 Statistical properties

The Mandel *Q*_{M} parameter can be used to measure the deviation of the occupation number distribution from the Poissonian statistics. This parameter was introduced by L. Mandel in quantum optics [

Negative values of *Q*_{M} corresponds to state which a photon number variance that is less than the mean. The minimal value *Q*_{M} = −1 is obtained for the case of Fock states. *Q*_{M} > 0 corresponds to the case of super-Poissonian statistics (classical fields) and *Q*_{M} = 0 is for the Poissonian distribution and corresponds to case of the standard coherent state.

Let us now consider the time variation of the statistical properties of the initial SF. For this purpose, the temporal evolution of the Mandel’s parameter is plotted in *Q*_{M} variation shows that the SF statistical properties exhibit a different order in terms of the squeezed parameter *r*. Depending on the value of *r*,*r* gets close to zero, the parameter

Figure 3.Temporal evolution of Q_{M} for the field initially prepared in a SCS with |α|^{2} = 25. The subfigures (a, b, c) are for

4 Conclusion

In this manuscript, we have investigated the interaction between six-level atomic systems and 1-mode field initially in a squeezed coherent state. We have extended the Jeans-Cummings model to explore the interaction between the atom and the squeezed field as well as the system dynamics. We have analyzed the time evolution of the atomic coherence, non-local correlation, statistical properties within the bipartite system in the presence and absence of the intensity-dependent coupling for different squeezing regimes of the field. We have introduced the von Neumann entropy to detect the time evolution of SLAS-SF entanglement and the

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Mariam Algarni, Kamal Berrada, Sayed Abdel-Khalek. Atom-field system: Effects of squeezing and intensity dependent coupling on the quantum coherence and nonclassical properties[J]. Journal of the European Optical Society-Rapid Publications, 2023, 19(2): 2023039

Paper Information

Category: Research Articles

Received: Aug. 17, 2023

Accepted: Sep. 20, 2023

Published Online: Dec. 23, 2023

The Author Email: Berrada Kamal (berradakamal@ymail.com)