Bashkirov E. K.

Bashkirov E. K., Evseev M. M., Gorchakova-Zakatova A. E., Othman A.

Evseev M. M.

Gorchakova-Zakatova A. E.

Othman A.

English

Samara University

Entanglement in two-atom Jaynes-Cummings modelwith Kerr nonlinearityE.K. Bashkirov1, M.M. Evseev1, A.E. Gorchakova-Zakatova1, A. Othman11Samara National Research University, Moskovskoe Shosse 34А, Samara, Russia, 443086Abstract. The dynamics of quantum system consisting of two superconductingcircuits interacting with one-mode quantum electromagnetic field of microwavecoplanar cavity is studied. The influence of the Kerr medium on qubit-qubitentanglement is examined. We showed that Kerr medium can greatly enhance theamount of qubit-qubit entanglement.1. IntroductionCoupling distant qubits is an important goal for quantum information and for its potentialapplications. This kind of coupling needs the study of interaction between qubits andphotons, which has been widely studied in a cavity quantum electrodynamics (CQED) [1].Resent years a lot of schemes are proposed for generating, controlling and protecting theatomic entanglement in CQED systems such as trapped and cooled ions or neutral atoms,superconducting circuits, spins in solids, quantum dots interacting with quantum fields of cavities[2]. The theoretical investigations of such schemes are based on a Jaynes-Cummings model(JCM) and its generalizations [3]. It is well known that JCM is the simplest physical modelthat describes the interaction of a natural or artificial two-level atom (qubit) with a single-modecavity field, and was used to understand a wide variety of phenomena in quantum optics andcondensed matter systems. During last decades the numerous generalizations of the JCM havebeen investigated (see references in [4]). Particularly, Buzek [5] studied the dynamics of atominteracting with field of one-mode cavity with a Kerr-like medium in a framework of nonlinearJCM [5]. A material whose refractive index depends on the intensity of the light field is called aKerr medium. A light beam travelling through such a material acquires a phase shift ϕ = χτI,where χ is the Kerr constant, τ is the interaction time of the light field with the material,and I is the intensity of the beam. The Kerr effect is a widely used phenomenon in nonlinearquantum optics, and has been successfully used to generate quadrature and amplitude squeezedstates, parametrically convert frequencies, and create ultra-fast pulses [3]. In practice, Kerrnonlinearities in atomic systems χ are, however, often small in comparison to photon loss rate κ,making the observation of these non-classical states of light difficult. As an alternative approach,strong photon-photon interaction can readily be realized in superconducting quantum circuits,with χ/κ 30 demonstrated experimentally [6]. Note, that in the field of quantum optics withmicrowave circuits, the direct analogue of the Kerr effect is naturally created by the nonlinearinductance of the Josephson junctions. Later the experimental and theoretical investigations ofsuperconducting circuits dynamics in coplanar cavities with Kerr media was investigated in alot of papers (see Ref. in [6]-[11]).Секция: Компьютерная оптика и нанофотоника VI Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2020)Resent years we studied the dynamics of atom-atom entanglement for different generalizationsof two-atom JCM. We explored the influence of dipole-dipole interaction, atomic coherence,detuning, Stark shift and non-degenerate and Raman two-photon transitions on atomicentanglement for superconducting qubits interacting with microwave fields of coplanar cavities[12]-[17]. It is of interest to expand our researches to more complicated CQED systems withstrong photon-photon interaction. In this paper, we study the influence of Kerr nonlinearity onatomic entanglement for system consisting of two identical superconducting qubits interactingwith one-mode microwave cavity field.2. Model and its exact solutionWe consider two identical superconducting circuits (qubits) resonantly interacting with cavitymode of 1D coplanar resonator. We suppose that qubit-field coupling constants are equal.The suppose also that there is an additional Kerr medium in the cavity. In a frame rotatingwith the twice field frequency, the interaction Hamiltonian for the system under rotating waveapproximation can be written asH =2∑i=1~γ(σ+i a+ a+σ−i ) + +χa+2a2, (1)where (1/2)σzi is the inversion operator for the ith qubit (i = 1, 2), σ+i = |+⟩ii⟨−|, andσ−i = |−⟩ii⟨+| are the transition operators between the excited |+⟩i and the ground |−⟩i statesin the ith qubit, a+ and a are the creation and the annihilation operators of photons of thecavity mode, γ is the coupling constant between qubits and the cavity mode, χ is the dispersivepart of the third- order nonlinearity of Kerr medium in the cavity. The initial qubits state isassumed to be separable such as|Ψ(0)⟩A = |+,−⟩ (2)or|Ψ(0)⟩A = |+,+⟩ (3).The initial cavity mode state is assumed to be Fock one-mode state |Ψ(0)⟩F = |n⟩, wheren = 1, 2, ...To obtain the exact dynamics of the model under consideration we use the so-called ”dressed”states or eigenfunctions of interaction Hamiltonian (1). Suppose that the excitation number ofthe qubits-field system is n (n ≥ 0). The evolution of the system is confined in the subspace|−,−, n+2⟩, |+,−, n+1⟩, |−,+, n+1⟩, |+,+, n⟩. Thus, the eigenfunctions of the Hamiltonian(1) can be written as|Ψin⟩ = win(Xi1n|−,−, n+ 2⟩+Xi2n|+,−, n+ 1⟩++Xi3n|−,+, n+ 1⟩+Xi4n|+,+, n⟩) (i = 1, 2, 3, 4), (4)wherewin = 1/√|Xi1n|2 + |Xi2n|2 + |Xi3n|2 + |Xi4n|2andX11,n = 0, X12,n = −1, X13,n = 1, X14,n = 0,Xi1,n = −2nχ̃− n2χ̃− n3χ̃+ 2εin + nεin√1 + n√2 + n(2χ̃+ 3nχ̃+ n2χ̃− εin),Xi2,n =√1 + n(2χ̃+ 3nχ̃+ n2χ̃− εin)2 + n+Секция: Компьютерная оптика и нанофотоника Entanglement in two-atom Jaynes-Cummings model with Kerr nonlinearityVI Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2020) 219+2 + n− (n(1 + n)χ̃− εin)(1 + n)(2 + n)χ̃− εin)(2χ̃+ 3nχ̃+ 5n2χ̃+ 2n3χ̃− 3εin − 2nεin)(2 + n)(√1 + n(2 + n)−√1 + n(2 + n− (n(1 + n)χ̃− εin)(1 + n)(2 + n)χ̃− εin)),Xi3,n = −2χ̃+ 3nχ̃+ 5n2χ̃+ 2n3χ̃− 3εin − 2nεin√1 + n(2 + n)−√1 + n(2 + n− (n(1 + n)χ̃− εin)((1 + n)(2 + n)χ̃− εin)),Xi4,n = 1,where χ̃ = χ/~γ. The corresponding scaled eigenvalues areε1n = E1n/~γ =(n+ n2)χ̃,ε2n = E2n/~γ =13Dn −21/3An3(Bn + Cn)1/3+(Bn + Cn)1/33 21/3,ε3n = E3n/~γ =13Dn +(1 + i√3)An3 22/3(Bn + Cn)1/3−(1− i√3)(Bn + Cn)1/36 21/3,ε4n = E4n/~γ =13Dn +(1− i√3)An3 22/3(Bn + Cn)1/3−(1 + i√3)(Bn + Cn)1/36 21/3,whereAn = −18− 12n− 4χ̃2 − 12nχ̃2 − 12n2χ̃, Bn = 72nχ̃+ 16χ̃3 + 72nχ̃3 + 72n2χ̃3,Cn =√4A3n +B2n, Dn = 2χ̃+ 3nχ̃+ 3n2χ̃.Assume that the system is initially prepared in the state |+,−, n+ 1⟩ (n ≥ 0). In this case,at time instant t, the whole system will evolve to|Ψ(t)⟩ = C12,n|−,−, n+ 2⟩+ C22,n|+,−, n+ 1⟩+ C32,n|−,+, n+ 1⟩+ C42,n|+,+, n⟩, (5)whereCi2,n = e−ıE1nt/~ w1n Y2in X1in + e−ıE2nt/~ w2n Y2inX2in+ (6)+e−ıE3nt/~ w3n Y2in X3in + e−ıE4nt/~ w4n Y2in X4in (i = 1, 2, 3, 4)andYijn = wjnX∗jin.Assume that the system is initially prepared in the state |−,+, n + 1⟩ (n ≥ 0). The time-dependent wave function can be presented as|Ψ(t)⟩ = C13,n|−,−, n+ 2⟩+ C23,n|+,−, n+ 1⟩+ C33,n|−,+, n+ 1⟩+ C43,n|+,+, n⟩, (7)where the coefficients Ci3,n can be obtained from (6) by replacing Y2in with Y3in (i = 1, 2, 3, 4).For initial states |+,+, n⟩ and |−,−, n+2⟩ (n ≥ 0) the time-dependent wave functions are|Ψ(t)⟩ = C11,n|−,−, n+ 2⟩+ C21,n|+,−, n+ 1⟩+ C31,n|−,+, n+ 1⟩+ C41,n|+,+, n⟩. (8)|Ψ(t)⟩ = C14,n|−,−, n+ 2⟩+ C24,n|+,−, n+ 1⟩+ C34,n|−,+, n+ 1⟩+ C44,n|+,+, n⟩. (9)The coefficients Ci1,n (Ci4,n) can be obtain from (6) by replacing Y2in with Y1in (Y4in)(i = 1, 2, 3, 4).Using expressions (4)–(9) one can obtain the density operator for the whole system ρ(t).Taking a trace over the field variables ρA(t) = TrFρ(t) one can also obtain the reduced atomicdensity operator.Секция: Компьютерная оптика и нанофотоника Entanglement in two-atom Jaynes-Cummings model with Kerr nonlinearityVI Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2020) 2203. Negativity calculationFor two-qubit system described by the density operator ρA(t), a measure of entanglement ornegativity can be defined in terms of the negative eigenvalues µ−i of partial transpose of a reducedatomic density matrix (ρT1A )ε = −2∑µ−i .For separable initial atomic states (2) or (3) the reduced atomic density matrix isρA(t) =ρ11(t) 0 0 00 ρ22(t) ρ23(t) 00 ρ23(t)∗ ρ33(t) 00 0 0 ρ44(t) . (10)The partial transpose of a reduced atomic density matrix (10) has the formρT1A (t) =ρ11(t) 0 0 ρ23(t)∗0 ρ22(t) 0 00 0 ρ33(t) 0ρ23(t) 0 0 ρ44(t) . (11)The elements of matrix (11) for initial atomic state |+,+⟩ areρ11 = |C41,n(, t)|2, ρ22 = |C21,n(t)|2,ρ33 = |C31,n(t)|2, ρ44 = |C11,n(, t)|2,ρ23 = C21,n(t)C31,n(t)∗.For initial atomic state |+,−⟩ these take the formρ11 = |C42,n−1(, t)|2, ρ22 = |C22,n−1(t)|2,ρ33 = |C32,n−1(t)|2, ρ44 = C12,n−1(t)|2,ρ23 = C22,n−1(t)C32,n−1(t)∗.Matrix (11) has only one eigenvalue, which may take a negative value. As a result we haveε(t) =√(ρ11(t)− ρ44(t))2 + 4|ρ23(t)|2 − ρ11(t)− ρ44(t). (12)The results of numerical calculations of negativity (12) are shown in Figs. 1-2.4. Results and discussionsThe negativity for a separable initial atomic state |+,−⟩ is plotted in Fig. 1 as a function of ascaled time γt for fixed values of photon number n = 1 and different values of Kerr nonlinearitycoefficient χ̃ = 0 (solid), χ̃ = 0.3 (dashed) and χ̃ = 0.5 (dotted). One can see from Fig. 1 thatwith the increase of Kerr non-linearity coefficient, the maximum value of negativity increases.Therefore, the Kerr medium can use to control the amount of entanglement between qubits.The negativity for a separable initial atomic state |+,+⟩ is plotted in Fig. 2 as a function of ascaled time gt for fixed values of photon number n = 1 and different values of Kerr non-linearitycoefficient χ̃ = 0.3 (solid) and χ̃ = 0.5 (dashed) . One can easily see from Fig. 2 that as the Kerrnon-linearity coefficient increases, larger entanglement is obtainable. Note that for zero Kerrnon-linearity coefficient the entanglement between atoms doesn’t arise during the evolution.This results is in accordance with previous investigations [14]-[17].Секция: Компьютерная оптика и нанофотоника Entanglement in two-atom Jaynes-Cummings model with Kerr nonlinearityVI Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2020) 221Figure 1. The negativity as a function of a scaled time γt for initial separable atomic state(2) and χ̃ = 0 (solid), χ̃ = 0.3 (dashed) and χ̃ = 0.5 (dotted) The cavity photon number n = 1.Figure 2. The negativity as a function of a scaled time γt for initial separable atomic state(3) and χ̃ = 0.3 (dashed), χ̃ = 0, 5 (dotted) The cavity photon number n = 2.5. ConclusionWe studied dynamics of two superconducting qubits interacting with cavity field in theframework of two-atom JCM with the Kerr medium, and examined the influence of the Kerrmedium on the atom-atom entanglement. The atomic entanglement behavior for separable initialatomic states and cavity Fock state was a subject of n our investigation. We derived that forinitial atomic state |+,−⟩ the Kerr nonlinearity enhances the amount of entanglement. We alsofound that for the initial state |+,+⟩ the Kerr nonlinearity causes entanglement of the atoms,while for a model without nonlinearity, this initial state does not cause atomic entanglementduring system evolution. These results may be useful for quantum information processing basedon the entanglement.Секция: Компьютерная оптика и нанофотоника Entanglement in two-atom Jaynes-Cummings model with Kerr nonlinearityVI Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2020) 2226. References[1] Buluta, I. Natural and artificial atoms for quantum computation / I. Buluta, S. Ashhab, F. Nori //Rep. Prog. Phys. – 2011. – Vol. 74. – P. 104401.[2] Georgescu, I.M. Quantum simulation / I.M. Georgescu, S. Ashhab, F. 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Influence of Stark shift on entanglement of two atoms with degenerate two-photontransitions for entangled and disentangled initial states / E.K. Bashkirov, M.S. Mastyugin // Optik. –2015. – Vol. 126(19). – P. 1787-1791.[15] Bashkirov, E.K. The dynamics of entanglement in two-atom Tavis-Cummings model with non-degenerate two-photon transitions for four-qubits initial atom-field entangled states / E.K. Bashkirov,M.S. Mastyugin // Opt. Commun. – 2014. – Vol. 313. – P. 170-174.[16] Bashkirov, E.K. Thermal Entanglement Between a Jaynes-Cummings Atom and an IsolatedAtom // Intern. J. Theor. Phys. – 2018. – Vol. 57(12). – P. 3761-3771.[17] Bashkirov, E.K. Entanglement Between Two Dipole-Coupled Qubits Interacting with TwoIndependent Slightly Detuned Cavity Modes // Intern. J. Theor. Phys. – 2019. – Vol. 58(7). – P.2346-2356.Секция: Компьютерная оптика и нанофотоника Entanglement in two-atom Jaynes-Cummings model with Kerr nonlinearityVI Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2020) 223

Entanglement in two-atom Jaynes-Cummings model with Kerr nonlinearity

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Entanglement in two-atom Jaynes-Cummings model with Kerr nonlinearity

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