Linear entropy and squeezing of the interaction between two quantum system described by su (1, 1) and su(2) Lie group in presence of two external terms M Sebawe Abdalla, E M Khalil, A S.-F Obada, J Peřina, and J Křepelka Citation: AIP Advances 7, 015013 (2017); doi: 10.1063/1.4973916 View online: http://dx.doi.org/10.1063/1.4973916 View Table of Contents: http://aip.scitation.org/toc/adv/7/1 Published by the American Institute of Physics AIP ADVANCES 7, 015013 (2017) Linear entropy and squeezing of the interaction between two quantum system described by su (1, 1) and su(2) Lie group in presence of two external terms M Sebawe Abdalla,1,a E M Khalil,2,3 A S.-F Obada,2 J Peˇrina,4 and J Kˇrepelka5 Mathematics Department, College of Science, King Saud University, P.O Box 2455, Riyadh 11451, Saudi Arabia Mathematics Department, Faculty of Science, Al-Azher University, Nassr City, Cairo 11884, Egypt Mathematics Department, Faculty of Science, Taif University, P.O Box 888 Taif 21974, Saudi Arabia Department of Optics and Joint Laboratory of Optics of Palack´ y University and Institute of Physics of AS CR, Faculty of Science, Palack´y University, 17 listopadu 12, 771 46 Olomouc, Czech Republic Joint Laboratory of Optics of Palack´ y University and Institute of Physics of Physics of Academy of Sciences of the Czech Republic, 17 listopadu 50a, 771 46 Olomouc, Czech Republic (Received October 2016; accepted 28 December 2016; published online January 2017) A Hamiltonian, that describes the interaction between a two-level atom (su(2) algebra) and a system governed by su(1, 1) Lie algebra besides two external interaction, is considered Two canonical transformations are used, which results into removing the external terms and changing the frequencies of the interacting systems The solution of the equations of motion of the operators is obtained and used to discuss the atomic inversion, entanglement, squeezing and correlation functions of the present system Initially the atom is considered to be in the excited state while the other systems is in the Perelomov coherent state Effects of the variations in the coupling parameters to the external systems are considered They are found to be sensitive to changing entanglement, variance and entropy squeezing © 2017 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4973916] I INTRODUCTION It is well known that the interaction between atom and electromagnetic field plays crucial role in the quantum optics On the other hand the interaction between three electromagnetic fields represents an important nonlinear parametric interaction, which has played a significant role in several physical phenomena of interest, such as stimulated and spontaneous emissions of radiation, coherent Raman and Brillouin scattering In Brillouin scattering one finds that an intense monochromatic laser source induces parametric coupling between the two scattered electromagnetic fields and the acoustical phonons in the scattering medium In Raman scattering a similar coupling occurs between the scattered Stokes and anti-Stokes waves and the optical phonons of a Raman active medium.1–3 Note that the interaction between three electromagnetic fields (which is of nonlinear type) can be transformed into either parametric amplifier or parametric frequency converter; the first type leads to amplification of the system energy while the second type leads to the energy exchanges between modes.1–8 This depends on the nature of the used approximation The most familiar Hamiltonian representing such a system is given by Hˆ ˆ c + aˆ bˆ † cˆ † , = ω1 aˆ † aˆ + ω2 bˆ † bˆ + ω3 cˆ † cˆ + g aˆ † bˆ (1) a E-mail: m.sebaweh@physics.org 2158-3226/2017/7(1)/015013/12 7, 015013-1 © Author(s) 2017 015013-2 Abdalla et al AIP Advances 7, 015013 (2017) ˆ cˆ where ωi , i = 1,2,3 are the field frequencies and g is the coupling parameter, while aˆ † , bˆ † , cˆ † aˆ , b, are creation (annihilation) operators fulfilling the relations ˆ bˆ † = cˆ , cˆ † = aˆ , aˆ † = b, (2) ˆ Jˆ− = aˆ bˆ † and Jˆz = (ˆnb − nˆ a ), nˆ a Now if we consider the Schwinger transformations Jˆ+ = aˆ † b, † ˆ ˆ nˆ b = b b then we have = aˆ † aˆ , Jˆ+ , Jˆ− = 2Jˆz , Jˆz , Jˆ± = ±Jˆ± (3) Jˆ± and Jˆz are the angular momentum operators which belongs to su(2) Lie algebra In this case the Hamiltonian1 is converted to the Tavis-Cummings model ω0 ˆ J z + g cˆ † Jˆ− + cˆ Jˆ+ (4) This Hamiltonian is a generalization of the two-level atom interaction with a single mode field, known as the Jaynes-Cummings model, described by the following form Hˆ = ω2 cˆ † cˆ + ω0 σ ˆ z + g aˆ † σ ˆ − + aˆ σ ˆ+ , (5) where σ ˆ ± and σ ˆ z are the standard Pauli operators which satisfy [σ ˆ +, σ ˆ −] = σ ˆ z , [σ ˆ z, σ ˆ ± ] = ±2σ ˆ ± Thus the interaction between the fields is transformed into atoms-field interaction On the other hand if we define9 ˆ c, Kˆ z = bˆ † bˆ + cˆ † cˆ + , Kˆ + = bˆ † cˆ † , Kˆ − = bˆ (6) where Kˆ + , Kˆ − = −2Kˆ z , Kˆ z , Kˆ ± = ±Kˆ ± (7) Hˆ = ω3 aˆ † aˆ + and Kˆ ± and Kˆ z are the generators of the su(1, 1) Lie algebra, the Casimir operator of which is given by ˆ ˆ Kˆ = Kˆ z2 − K + K − + Kˆ − Kˆ + , Kˆ , Kˆ z = 0, Kˆ , Kˆ ± = (8) Then the Hamiltonian (1) is transformed into the form describing the interaction between a field and an su(1,1) group system, given by Hˆ = ω4 aˆ † aˆ + g aˆ † Kˆ − + aˆ Kˆ + (9) In fact the su(1, 1) Lie algebra plays an important role in the description of linear dissipative processes in the Liouville space.10,11 Now if we use the Holstein-Primakoff representation12 Kˆ + = 2s − + nˆ c cˆ † , Kˆ − = cˆ 2s − + nˆ c , Kˆ z = s + nˆ c , (10) where nˆ c = cˆ † cˆ and s is a c-number, the Hamiltonian (5) takes the form Hˆ ω0 = ωKˆ z + σ ˆ z + λ Kˆ + σ ˆ − + Kˆ − σ ˆ+ , (11) where λ = g(2s − + nˆ c )− It has to be mentioned that (11), without the rotating wave approximation, is converted to Hˆ ω0 = σ ˆ z + ωKˆ z + λ Kˆ + − Kˆ − (σ ˆ−−σ ˆ +) (12) The main purpose of this communication is to modify the above model to include two classical terms, one of them is described by su(1, 1) Lie algebra operators, while the other obeys the su(2) Lie algebra These two extra classical terms can be interpreted as the exhibition of the effect of the parametric amplification represented by su(1, 1) Lie algebra, while the other term plays the role of external driving force in sense of su(2) Lie algebra This will be achieved by studying the degree of entanglement and the entropy as well as the different types of squeezing.13 In fact quantum 015013-3 Abdalla et al AIP Advances 7, 015013 (2017) entanglement represents the correlations between physical systems that cannot be accounted for using classical physics For example, quantum information is qualitatively different from its classical counterpart, which is the origin of new ideas of quantum cryptography and quantum computing Information is defined in terms of the probabilities for certain events to occur, and the difference between classical and quantum information arises from the different ways in which the probabilities are calculated It is widely recognized that the control of quantum entanglement leads to new classes of measurements, communication and computational systems, and in some cases it dramatically changes non-quantum analogies The research on quantifying entangled states has been considered by several authors, for example.14–17 Also in the present paper we shall consider the Glauber second order correlation function in order to examine the classical and non-classical effects of the system However, we start our study of the system by considering the atomic inversion where we can discuss the collapses and revival phenomenon and show how it differs from the standard case The suggested model can be realized in a cavity containing a two-level atom and nonlinear optical parametric medium, both of which are pumped by external strong coherent beams The organization of the paper is as follows: in the sec II we modify the Hamiltonian (12) to include two external terms Sec III is devoted to the introduction of the Heisenberg equations of motion and finding their solutions In sec IV we consider the atomic inversion behavior examining the collapses and revivals phenomenon The discussion of the degree of entanglement is given in sec V, which is followed by a study of squeezing, introducing three kinds of the squeezing in sec VI The correlation function is studied in sec VII and our conclusions are given in sec VIII II THE HAMILTONIAN MODEL We devote this section to introduce the Hamiltonian model which represents the interaction between an su(1, 1) and an su(2) under the action of two classical terms To so we consider the Hamiltonian which represents the interaction between two fields of the parametric amplifier type This Hamiltonian is given by Hˆ = ωKˆ z + iλ Kˆ + − Kˆ − (13) Now suppose that a two-level atom is injected into the cavity Then the system will be affected by the atom and Hamiltonian takes the form of eq.(12) On the other hand when we consider the effect of two external quantum systems, one of su(1, 1) and the other of su(2), the Hamiltonian is given by the following form Hˆ = ω0 σ ˆ z + ωKˆ z + λ Kˆ + − Kˆ − (σ ˆ−−σ ˆ + ) + λ1 Kˆ + + Kˆ − + λ2 (σ ˆ−+σ ˆ +) (14) The first and the last terms describe the two-level atom and its interaction with a classical field, while the second and the fourth term display the su(1, 1) system, and finally the third term describes the interaction between the two-level atom and the su(1, 1) amplifier system As one can see it is difficult to tackle such problem, therefore to simplify matters we eliminate the terms multiplied by λi (i = 1,2) For this reason we introduce two kinds of the transformations, the first is Kˆ + cosh2 ϑ sinh2 ϑ − sinh 2ϑ Rˆ + (15) Kˆ − = sinh2 ϑ cosh2 ϑ − sinh 2ϑ Rˆ − ˆ Kz Rˆ z − sinh 2ϑ − 21 sinh 2ϑ cosh 2ϑ and the second transformation is given by σ ˆ+ cos2 η − sin2 η 21 sin 2η σ ˆ − = − sin2 η cos2 η 21 sin 2η σ ˆz − sin 2η − sin 2η cos 2η Sˆ + Sˆ − , Sˆ z (16) here we take 2λ1 ϑ = tanh−1 , ω η= 2λ2 tan−1 ω0 (17) 015013-4 Abdalla et al AIP Advances 7, 015013 (2017) From eqs (15) and (16) taking together, the Hamiltonian (14) is simplified to Hˆ = Ω0 ˆ S z + Ωz Rˆ z + λ Rˆ + Sˆ − + Rˆ − Sˆ + (18) with Ωz = ω2 − 4λ21 and Ω0 = ω02 + 4λ22 (19) From eq (19) we note that the value of the frequency Ωz is less than ω due to the coupling parameter λ1 , while the frequency Ω0 increases compared to ω0 by the coupling parameter λ2 Consequently we can control the behavior of the present system through the couplings to the external fields λ1 and λ2 In Sec III we employ the Heisenberg equations of motion with the Hamiltonian (18) to derive the dynamical operators III THE EQUATIONS OF MOTION AND THEIR SOLUTION ˆ is given by The Heisenberg equation of motion for any dynamical operator Q ˆ ∂Q d ˆ ˆ ˆ Q= Q, H + dt i ∂t (20) Therefore the Heisenberg equations of motion for the operators appearing in eq (18) lead to the following equations: dRˆ z dSˆ z = −iλ Rˆ + Sˆ − − Rˆ − Sˆ + , = 2iλ Rˆ + Sˆ − − Rˆ − Sˆ + , dt dt dSˆ + dRˆ + = iΩz Rˆ + + 2iλRˆ z Sˆ + , = iΩ0 Sˆ + − iλRˆ + Sˆ z , dt dt dRˆ − dSˆ − = −iΩz Rˆ − − 2iλRˆ z Sˆ − , = −iΩ0 Sˆ − + iλRˆ − Sˆ z dt dt (21) These equations can be converted to the second order differential equations d2 Rˆ z ˆ + 4Cˆ Rˆ z = 4Cˆ Nˆ − ∆C, dt d2 Sˆ z ˆ + 4Cˆ Sˆ z = 2∆C dt (22) Also we have d2 Sˆ + dSˆ + Ωz − 2i Ωz + Cˆ − 2λ2 Nˆ − + 2Ωz Cˆ + dt 2 dt Sˆ + = 0, (23) d2 Sˆ − dSˆ − Ωz + 2i Ωz − Cˆ + 2λ2 Nˆ + + 2Ωz Cˆ − dt 2 dt Sˆ − = (24) ˆ d2 Rˆ + ˆ dR+ − 2λ2 Nˆ − + 2Ωz Cˆ + Ωz − 2i Ω + C z dt 2 dt Rˆ + = 0, (25) d2 Rˆ − dRˆ − Ωz + 2i Ωz − Cˆ + 2λ2 Nˆ + + 2Ωz Cˆ − dt 2 dt Rˆ − = (26) and Furthermore finally Here we defined ∆ Cˆ = Sˆ z + λ Rˆ + Sˆ − + Rˆ − Sˆ + , Nˆ = Rˆ z + Sˆ z , ∆ = Ω0 − Ωz (27) 2 ˆ hence they are constants of It is to be noted that Cˆ and Nˆ commute with each other and with H, ˆ motion In the following we look for the solution C of these equations 015013-5 Abdalla et al AIP Advances 7, 015013 (2017) A The general solution In this subsection we introduce the general solution of the above equations in terms of the new operators Rˆ z (t) and Rˆ ± (t) as well as Sˆ z (t) and Sˆ ± (t) From the above equations we have the following solutions ˆ sin 2Ct Rˆ + (0)Sˆ − (0) − Rˆ − (0)Sˆ + (0) 2Cˆ ˆ ∆ sin Ct − Rˆ − (0)Sˆ + (0) + Nˆ − Cˆ Cˆ ˆ − iλ Rˆ z (t) = Rˆ z (0) cos 2Ct (28) and ˆ sin 2Ct Rˆ + (0)Sˆ − (0) − Rˆ − (0)Sˆ + (0) ˆ 2C ˆ sin Ct − Rˆ − (0)Sˆ + (0) + ∆Cˆ Cˆ ˆ + iλ Sˆ z (t) = Sˆ z (0) cos 2Ct (29) For the other operators we get ˆ Rˆ − (t) = e−i(Ωz −C)t Cˆ λ sin µˆ t Rˆ − (0) − 2i sin µˆ t Rˆ z (0)Sˆ − (0) µˆ µˆ (30) ˆ sin µˆ t ˆ (∆ + C) sin µˆ t Sˆ − (0) + iλ R− (0)Sˆ z (0) µˆ µˆ (31) cos µˆ t − i and ˆ Sˆ − (t) = e−i(Ωz −C)t cos µˆ t − i The Hermitan conjugate quantities for the last two equations are given by ˆ Rˆ + (t) = ei(Ωz +C)t λ Cˆ sin µˆ t Rˆ + (0) + 2i sin µˆ t Rˆ z (0)Sˆ + (0) µˆ µˆ (32) ˆ (∆ − C) sin µˆ t ˆ sin µˆ t Sˆ + (0) − iλ R+ (0)Sˆ z (0) , µˆ µˆ (33) cos µˆ t − i and ˆ Sˆ + (t) = ei(Ωz +C)t cos µˆ t + i where µˆ 21 = Cˆ + 2λ2 Nˆ + , µˆ 22 = Cˆ − 2λ2 Nˆ − (34) The discrete representation of the su(1, 1) Lie algebra satisfies Rˆ |m; k = k (k − 1) |m; k , Kˆ z |m; k = (m + k) |m; k , √ Rˆ + |m; k = (m + 1) (m + 2k)|m + 1; k , √ Rˆ − |m; k = m (m + 2k − 1)|m − 1; k , (35) where |m; k are the state vectors, for more details see ref ? On the other hand the inverse transformation of eq (15) takes the form Rˆ + Rˆ − = Rˆ z cosh2 ϑ sinh2 ϑ sinh 2ϑ sinh2 ϑ sinh 2ϑ cosh2 ϑ sinh 2ϑ sinh 2ϑ cosh 2ϑ Kˆ + Kˆ − Kˆ z (36) In Sec IV we consider the atomic inversion so that we can discuss the phenomenon of collapse and revival in the present system 015013-6 Abdalla et al AIP Advances 7, 015013 (2017) IV THE ATOMIC INVERSION The main task of this section is to discuss the phenomenon of collapse and revival This is achieved by calculating the expectation value of the operator σ ˆ z To so we use the modified Perelomov coherent state as the initial state for su(1, 1) given by |R, m = exp β+ Rˆ + + β0 Rˆ z + β− Rˆ − |0, k , where β0 = z∗ − z sinh 2ϑ, β+ = zcosh2 ϑ − z∗ sinh2 ϑ, and β− = −z∗ cosh2 ϑ + z sinh2 ϑ (37) The exponential term in the state is factorized to take the form exp β+ Rˆ + + β0 Rˆ z + β− Rˆ − = exp A+ Rˆ + exp ln(A0 )Rˆ exp A− Rˆ − , (38) where A± = β± sinh φ φ cosh φ − with φ = β0 β0 sinh φ −1 , A0 = cosh φ − β0 sinh φ 2φ −2 − β+ β− (39) In this case we have |R, m = exp A+ Rˆ + exp ln(A0 )Rˆ |0, k , β0 |R, m = cosh φ − sinh φ 2φ −2k ∞ m=0 Γ (2k + m) m!Γ (2k) Am + |m, k (40) To reach our goal we consider the atom to be initially in its excited state while the quantum system is in the generalized coherent state given by the eq (40), this would enable us to find the atomic inversion using the equation σ ˆ z (τ) = Sˆ z (τ) cos 2η − Sˆ x (τ) sin 2η, (41) from which we can discuss the phenomenon of collapses and revivals For this reason we plot fig for different values of the coupling parameters λ1 and λ2 , and for fixed value of the other FIG Atomic inversion for a two-level atom system initially in the excited state in interaction with su(1, 1) quantum system which is initially in Peremelov coherent state for fixed values of r = 7, θ = π/4, k = and ω = ω0 = 40, where a) λ1 = λ2 = 0, b) λ2 /λ = 20 and λ1 = 0, c) λ2 = 0, λ1 /λ = 19.99 and d) λ2 /λ = 20 and λ1 /λ = 19.99 015013-7 Abdalla et al AIP Advances 7, 015013 (2017) parameters For example, we take ω = ω0 = 40λ, k = and z = r exp(iθ) with r = and θ = π/4 In fig 1a we consider the case of absence of both external fields in which λ1 = λ2 = where the function shows periods of collapses as well as sharp revivals As one can see the function shows collapse period after onset of the interaction, this is followed by a sharp revival where the function fluctuates around zero between ☞1 and 0.5 In the second period of the revival the function shifts its amplitude and fluctuates between ☞0.6 and 0.9, as the time increases the function repeats this behavior We also note that the revivals are very short On the other hand when we consider absence of the su(1,1), λ2 /λ = 20, while λ1 = 0, different observation is seen, in this case the function decreases its amplitude and shows regular slowly oscillating envelope, however during the revivals period there are small spread of the fluctuations which increases as the time increases This means that the energy is almost concentrated in the quantum system, see fig 1b For the case of absence of the su(2) external term in which λ1 /λ = 19.99 while λ2 = 0, the function shows dense fluctuations with small amplitudes (pseudocollapses) and slight shift up and displays a dense spread of fluctuations around the revival periods This is due to the atomic effect of the external su(1,1) term, during these periods, see fig 1c Finally we discuss the effect of both external terms λ2 /λ = 20 and λ1 /λ = 19.99, in this case it also shows slowly oscillatory envelope as well as revival periods For instance the function decreases its amplitude and starts with a short period of revival but the revival gets pronounced at the maximum points of the envelope curve of‘ partial collapse However, the function has larger amplitude at the second period of revival which occurs at the minimum point of the envelope curve This behavior is repeated as the time increases, see fig 1d It may be concluded that the external su(2) term adds an oscillating while the su(1,1) external term makes very dense small fluctuations making pseudo-collapses V THE ENTANGLEMENT We devote this section to investigate the entanglement in the system through the linear entropy More precisely, we demonstrate the effect of the existence of the coupling parameters λ1 and λ2 on the entanglement To quantify the entanglement we write down the definition of the linear entropy as18,19 χ(τ) = − ξ (τ), (42) where ξ(τ) is the well-known Bloch sphere radius, which has the form ξ (τ) = σ ˆ x (τ) + σ ˆ y (τ) = Sˆ x (τ) + Sˆ y (τ) 2 + σ ˆ z (τ) 2 + Sˆ z (τ) (43) The Bloch sphere is a tool in quantum information, where the simple qubit state is successfully represented, up to an overall phase, by a point on the unit sphere, whose coordinates are the expectation values of the atomic set operators of the system This means that the entanglement is strictly related to the behavior of observables of clear physical meaning The function χ(τ) ranges between for disentangled bipartite and for maximally entangled ones In order to discuss the degree of entanglement we plot fig against the scaled time τ = λt For this reason we apply the same parameters used to examine the atomic inversion When we consider the case λ1 = λ2 = the function displays maximum entanglement located near the revivals of the atomic inversion as well as disentanglement through located around middle of the collapses and at the revivals periods of the atomic inversion (see fig 1a) for all the considered time, however we can see small periods of partial entanglement, see fig 2a For λ2 /λ = 20 while λ1 = the case of absence of the su(1,1) external term, the function decreases its amplitude and shows partial entanglement and does not reach its maximum value (full entanglement) or minimum value (disentanglement), see fig 2b This means that the interaction between atom and quantum system gets stronger than the previous case which is result of the existence of λ1 Also we examine the case in which λ2 /λ = 19.99 and λ1 = 0, as one can see the function reduced its minimum and shows disentanglement as well partial entanglement However, as the time increases the minimum value increases, we also noted that there are rapid fluctuations when the function approaches the minimum value and shows disentanglement, see fig 2c Finally we discuss the case when λ1 /λ = 20 and λ2 /λ = 19.99 In this case the function decreases its amplitude and shows partial entanglement, as the time increases the minimum value gets 015013-8 Abdalla et al AIP Advances 7, 015013 (2017) FIG The linear entropy against the scaled time τ to quantify the entanglement for a two-level atom system initially in the excited state in interaction with su(1, 1) quantum system which is initially in Peremelov coherent state for fixed values of r = 7, θ = π/4, k = 2, ω = ω0 = 40 and where a) λ1 = λ2 = 0, b) λ2 /λ = 20 and λ1 = 0, c) λ2 = 0, λ1 /λ = 19.99 and d) λ2 /λ = 20 and λ1 /λ = 19.99 far from the disentanglement points where the energy concentrated at both minimum and maximum values of the function, see fig 2d VI THE SQUEEZING In this section we discuss one of the important nonclassical phenomenon in the field of quantum optics, that is the squeezing phenomenon Here we consider three different types of squeezing, namely variance squeezing, entropy squeezing and normal squeezing A Variance squeezing The variance squeezing is built up on the concept of the uncertainty relations that discuss the quantum fluctuations In this case we have to use the entropic uncertainty relations for two-level system instead of the Heisenberg uncertainty relations The discussion of this argument is given in the refs 20, 21 As is well known for the quantum mechanical system with two physical observables ˆ B] ˆ = iC, ˆ the represented by the Hermitian operators Aˆ and Bˆ satisfying the commutation relation [A, Heisenberg uncertainty takes the form ˆ (∆A) ˆ ≥ | Cˆ | , (∆B) (44) ˆ = ( Aˆ − Aˆ ) Consequently, the uncertainty relation for a two-level atom characwhere (∆A) terized by the Pauli operators σ ˆ x, σ ˆ y and σ ˆ z , satisfying the commutation relation [σ ˆ x, σ ˆ y ] = iσ ˆ z , can also be written as ∆σ ˆ x ∆σ ˆ y ≥ 21 | σ ˆ z | Fluctuations in the component ∆σ ˆ α of the atomic dipole is said to be squeezed if σ ˆ α satisfies the condition σ ˆz V (σ ˆ α ) = ∆σ ˆα− < 0, α = x or y (45) In this case we use different data to those in the previous sections, for example we consider a fixed value of k = and take ω = ω0 = 1, consequently we have to change the value of the coupling parameter, this is to avoid the discontinuity which would appear in the augmented frequency To 015013-9 Abdalla et al AIP Advances 7, 015013 (2017) discuss the variance squeezing we plot fig against this scaled time τ = λt for different values of λ1 and λ2 For λ1 = λ2 = there is no squeezing where the quadrature Vx (τ) (red solid line) reaches only the value zero for a short period of time, while the other quadrature Vy (τ) (blue dashed line) is far away even from the zero value, see fig 3a When we consider λ2 /λ = 0.3 and λ1 = 0, the squeezing is observed in the second quadrature three times for a short period and no squeezing occurs in the first quadrature, see fig 3b When we examine the case in which λ1 /λ = 0.3, although the first quadrature approaches zero, however, the squeezing occurred in the second quadrature once at the middle of the considered time, see fig 3c Different observation is in the case λ1 /λ = 0.3 and λ2 /λ = 0.3 where the squeezing can be seen in both quadratures, but it starts in Vy (τ) and as the time increases it is observed in Vx (τ) until it gets to be pronounced and then disappeared before the end of the considered time, see fig 3d The examinations show that increasing the coupling of the external system results in increasing the amounts of squeezing in both quadratures B Entropy squeezing We now turn our attention to consider another kind of squeezing that is the entropy squeezing which is one of the important nonclassical phenomena To examine the entropic squeezing we use the Shannon information entropies of the two-level atom operators given by22,23 1 1 (σ (1 − σ ˆ α + 1) ln ˆ α + 1) − (1 − σ ˆ α ) ln ˆ α) , H(σ ˆ α ) = − (σ 2 2 α = x, y, z (46) The fluctuations in component σ ˆ α (α = x, y) of the atomic dipole are said to be squeezed in entropy if the information entropy H(σ ˆ α ) of σ ˆ α satisfies the condition24,25 E(σ ˆ α ) = δH(σ ˆ α) − δH(σ ˆ z) < 0, with δH(σ ˆ α ) = exp H(σ ˆ α ), where α = x or y (47) We plot fig against the scaled time τ = λt using the same data in Subsection VI A For the case in the absence of λ1 and λ2 , the squeezing occurs in both quadrature Ex (τ) (red solid line) and Ey (τ) (blue dashed line) where we can see exchange between the squeezing in the quadratures, see fig 4a Similar behavior is occurred when we consider λ1 /λ = 0.3 and λ2 = 0, however the squeezing FIG The variance squeezing against the scaled time τ for a two-level atom system initially in the excited state in interaction with su(1, 1) quantum system which is initially in Peremelov coherent state for fixed value of r = 7, k = 2, ω = ω0 = and θ = π/4 where a) λ1 = λ2 = 0, b) λ1 /λ = 0.3 and λ2 = 0, c) λ1 = and λ2 /λ = 0.3 and d) λ1 /λ = 0.3, λ2 /λ = 0.3 015013-10 Abdalla et al AIP Advances 7, 015013 (2017) FIG 4: The entropy squeezing against the scaled time τ for a two-level atom system initially in the excited state in interaction with su(1, 1) quantum system which is initially in Peremelov coherent state for fixed value of r = 7, θ = π/4, k = 2, ω = ω0 = 1, where a) λ1 = λ2 = 0, b) λ2 /λ = 0.3 and λ1 = 0, c) λ2 = 0, λ1 /λ = 0.3 and d) λ2 /λ = 0.3 and λ1 /λ = 0.3 gets pronounced in the second quadrature where the maximum value of squeezing approaches ☞0.3 and then it decreases as the time increases, see fig 4b For the case in which λ1 /λ = 0.3 and λ2 = 0, the squeezing starts in the second quadrature Ey (τ) and later starts to appear in the first quadrature Ex (τ) Note that the maximum value of the squeezing in this case takes on the value 0.4 in Ex (τ), however as the time increases the value of the squeezing decreases, see fig 4c Finally when we consider the case in which λ1 /λ = λ2 /λ = 0.3, the squeezing is seen in Ey (τ) for a short period of the time Increasing the time it appears for periods shorter than the previous case, and then it starts to appear and to get pronounced in Ex (τ) It is also realized in Ey (τ) for several short periods of times, see fig 4d Therefore, coupling of the effects of both the external systems improves the squeezing in entropy C Normal squeezing Finally we discuss the normal squeezing for two cases The first case when the system is at exact resonances, while the second when it is at off resonances To measure squeezing we define the functions Nj = ∆Kˆ j 2| − 21 | Kˆ z | Kˆ z | , j = x, y (48) In what follows we plot the quadratures Nx and Ny against the scaled time τ = λt, see fig In fig 5a the function shows squeezing for the exact resonances case (blue solid curve) and for the off resonance case (red dashed curve) However, the squeezing occurs in both cases, and it is more pronounced for the off resonance case A small amount of the squeezing at short periods of time appears in the second quadrature Ny , this can be seen in fig 5b Thus we may conclude that the coupling parameter λ2 is responsible of the squeezing phenomenon and the first quadrature acquires more squeezing compared with the second quadrature Furthermore, the squeezing at off resonance case is more pronounced compared to the amount at exact resonance case 015013-11 Abdalla et al AIP Advances 7, 015013 (2017) FIG The normal squeezing against the scaled time τ for a two-level atom system initially in the excited state in interaction with su(1, 1) quantum system which is initially in Peremelov coherent state for fixed value of r = 7, θ = π/4, k = 2, ω = ω0 = 1, λ2 /λ = 0.3 (exact resonance case, red-dashed curve) and ω0 = 1, ω = 20 and λ2 /λ = (off-resonance case, blue solid curve); a) the first quadrature, b) the second quadrature FIG The correlation function against the scaled time τ for a two-level atom system initially in the excited state in interaction with su(1, 1) quantum system which is initially in Peremelov coherent state for fixed value of r = 0.2, θ = π/4, k = and ω = ω0 = 1, a) λ1 = λ2 = 0, b) λ2 = and λ2 /λ = 0.042 VII CORRELATION FUNCTION We devote this section to consider the correlation function from which we are able to discuss non-classical behavior of the present system The normalized second order correlation function g(2) (t) is used to measure the classical and non-classical behavior as well as the coherence behavior It is defined by10 K+2 K−2 g(2) (t) = , (49) Kˆ + Kˆ − where the function shows non-classical behavior when g(2) (t) < and classical effect when g(2) (t) > 1, while it shows coherence for g(2) (t) = For the present case we take r = 0.2, k = and ω0 = ω = 0.1 while we consider no effect of the coupling parameters such that λ1 = λ2 = In this case the function shows non-classical behavior most of the considered periods of time Furthermore the classical behavior is observed for shorter periods, see fig 6a On the other hand the non-classical behavior is decreased when the coupling parameter λ2 is taken into account even there is a decrease observed in the classical behavior, see fig 6b Therefore we can conclude that the classical and the non-classical phenomena are sensitive to the parameters of the external system The parametric processes can really reduce as well as support quantum noise, including super-chaotic behavior VIII CONCLUSION In the present paper we have introduced the problem of the interaction between a two levelatom and su(1, 1) quantum system in the presence of two external classical terms The Hamiltonian model of such a quantum system contains three coupling parameters Two transformations are used to simplify the Hamiltonian and reduce it to an operator that describes the interaction between 015013-12 Abdalla et al AIP Advances 7, 015013 (2017) a two-level atom and a system governed by an su(1,1) algebra with augmented frequencies The solutions of the Heisenberg equations of motion are given Then we discussed different phenomena We considered the atomic inversion to examine the collapse and revival phenomenon where we found these non-classical behavior for large values of frequencies, and they were sensitive to the variation of the coupling parameters, the collapses periods are no longer straight lines, but slowly oscillating envelopes carrying the revivals Furthermore, we studied the degree of entanglement using linear entropy where the maximum as well as disentanglement are obtained when λ1 and λ2 are zero For λ1 0, while λ2 = the function shows partial entanglement and when λ1 = and λ2 the disentanglement appears again at middle of collapses periods of the atomic inversion For the case in which λ1 λ2 the function shows only partial entanglement We also considered three types of squeezing For variance squeezing there is no squeezing in the first quadrature for λ1 = but the squeezing occurs in the second quadrature just for λ2 0, however, the squeezing is observed in both quadratures when λ1 λ2 We discussed the entropy squeezing when the phenomenon occurs in the E x (first quadrature) and E y (second quadrature) whatever the values of λ1 and λ2 , but it is pronounced in E y when λ1 0, and in E x when λ2 For λ1 λ2 there is an increase in the squeezing periods but it decreases as the time increases The normal squeezing is found sensitive to the variations in both coupling parameter and the frequencies of the system The second-order correlation function for su(1,1) system shows classical and non-classical as well as coherence behavior at exact resonance case, but it shows a small amount of it for the off resonance case ACKNOWLEDGMENTS 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Abdalla ,1, a E... given by σ ˆ+ cos2 η − sin2 η 21 sin 2? ? σ ˆ − = − sin2 η cos2 η 21 sin 2? ? σ ˆz − sin 2? ? − sin 2? ? cos 2? ? Sˆ + Sˆ − , Sˆ z (16 ) here we take 2? ?1 ϑ = tanh? ?1 , ω η= 2? ?2 tan? ?1 ω0 (17 ) 015 013 -4 Abdalla... introduce two kinds of the transformations, the first is Kˆ + cosh2 ϑ sinh2 ϑ − sinh 2? ? Rˆ + (15 ) Kˆ − = sinh2 ϑ cosh2 ϑ − sinh 2? ? Rˆ − ˆ Kz Rˆ z − sinh 2? ? − 21 sinh 2? ? cosh 2? ? and the second