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Relation Between the Entropy and the Purity Parameter in the Ion-Laser Interaction

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Relation Between the Entropy and the Purity Parameter in the Ion-Laser Interaction Definitions/entropy-logo-eps-converted-to.pdf Article Relation between the entropy and the purity parameter in the ion-laser interaction Raúl Juárez-Amaro, Leonardo Moya Rosales, Jorge A. Anaya-Contreras 2, Arturo Zúñiga-Segundo 1 and Héctor M. Moya-Cessa 3 1 Universidad Tecnológica de la Mixteca, Apdo. Postal 71, 69000 Huajuapan de León, Oax., Mexico 2 Instituto Politécnico Nacional, ESFM, Departamento de Física. Edificio 9, Unidad Profesional “Adolfo López Mateos,” CP 07738 CDMX, Mexico 3 Instituto Nacional de Astrofísica, Óptica y Electrónica, 72840 Sta. María Tonantzintla, Pue., Mexico * Correspondence: [email protected]; Tel.: +52-22-2266-3100 Received: 13 December 2019 Abstract: It is shown that, in the interaction of a field and a qubit, there exists a relation between the linear entropy and the von Neumann entropy. The Cayley-Hamilton theorem is used to obtain such relation. In the case we study, the qubit is given by the external degrees of freedom of an ion trapped in a Paul trap and the field given by its internal (vibrational) degrees of freedom. Keywords: entropy, entanglement, fluctuations 1. Introduction Nonclassical states of quantum systems [1–12] are of great importance not only because of fundamental aspects, such as the fact that they may present fluctuations that are below the quantum standard limit defined by coherent states, or a variety of linear combination of discrete variable systems and many-qubit system [13–22] but also because of practical reasons, for instance in metrology, quantum information and quantum computation [23]. Trapped ions interacting with laser fields and quantized fields interacting with two-level atoms have many common features as in both subjects it is possible to generate nonclassical states of the vibrational motion of the ion and of the quantized field, respectively, and to realize interactions of the Jaynes-Cummings [24–26] and anti-Jaynes-Cummings [27] type and multiphonon/multiphoton transitions in those systems. Trapped ions interacting with laser fields in the Lamb-Dicke regime may be described by the Jaynes-Cummings model, which in this case describes the interaction of an electronic transition and the quantized center-of-mass motion, assisted by a laser beam, in the resolved sideband regime [28,29]. The advantage in the ion-laser interaction, over the atom-field interaction, is the fact that decoherence processes do not affect the ion-laser interaction as much as it does to cavities [30]. Quantum systems such as the atom field and the ion-laser interactions and their non-linear generalizations may be modelled by using classical interactions [31] such as propagation of light through inhomogeneous media, namely waveguide arrays [32–35] Although some information about an initial state of the vibrational motion of the ion and/or the quantized field may extracted from atomic properties such as Rabi oscillations [36], for information about its degree of purity or mixedness we need some other quantities, like entropy [37] or linear entropy [38]. In next Section, we introduce the von Neumann entropy and the linear entropy. In Section 3 we present the ion-laser interaction Hamiltonian and give its solution. Still in Section 3 we use the Cayley-Hamilton theorem to write the powers of the vibrational density matrix in terms of powers of Entropy 2020, xx, 5; doi:10.3390/Entropyxx010005 www.mdpi.com/journal/entropy Entropy 2020, xx, 5 2 of 10 the spin density matrix. In Section 4 we give the relation between the von Neumann entropy and the linear entropy and Section 5 is left for the conclusions. 2. Entropy and linear entropy The Araki-Lieb inequality [37,38], jSA − SV j ≤ SAV ≤ SA + SV, may be of great help to obtain the entropy of the one subsystem (for instance the vibrational motion of the ion) from the entropy of another subsystem (ion’s electronic states) which is simpler to calculate. In the above expression, SAV denotes the total entropy, while SA is the entropy for the ion and SV is the vibrational entropy. From the above inequality one may note that, if the two subsystems are initially in pure states, the total entropy of the system is zero, implying that both subsystems entropies are equal after after both subsystems interact. The von Neumann entropy may be defined as the expectation value of the entropy operator, Sˆ = − ln r, S = TrfrSˆg, (1) where r is the density matrix of the quantum system. Linear entropy Linear entropy, originally called purity parameter, is a much simpler function of the density matrix, compared to entropy, and therefore much easier to calculate. It is given as x = 1 − Trfr2g. (2) The linear entropy is always lower than the entropy, being its limiting case. In this contribution we give a relationship between them. We next consider the ion-laser interaction and show that, for certain parameters, it may describe the atom-field interaction. We solve it and write the total density matrix in order to find the reduced density matrices for the vibrational and the internal degrees of freedom. We show that powers of the vibrational density matrix may be obtained from powers of the internal degrees of freedom density matrix, which, being a 2 × 2 matrix, its powers are easily obtained. This allows us to write a relation between the entropy and the linear entropy in the case we consider initial pure states for the wavefunctions associated to the vibrational motion and the internal degrees of freedom of the ion. 3. Ion-laser interaction We consider the Hamiltonian of a single ion trapped in a harmonic potential in interaction with laser light in the rotating wave approximation, which reads † (−) Hˆ = naˆ aˆ + weg Aˆ ee + [lE (xˆ, t)Aˆ ge + H.c.], (3) where aˆ and Aˆ ab are the annihilation operator of a quantum of the ionic vibrational motion and the electronic (two-level) flip operator for the jbi ! jai transition of frequency weg, respectively. The frequency of the trap is n, l is the electronic coupling matrix element, and E(−)(xˆ, t) the negative part of the classical electric field of the driving field. We assume the ion driven by a laser field tuned to the mth lower sideband, we may write E(−)(xˆ, t) as E(−)(xˆ, t) = Ee−i(kxˆ−weg+mn)t, (4) where k is the wave vector of the driving field. If m = 0 it would correspond to the driving field being on resonance with the electronic transition. The operator xˆ may be written as kxˆ = h(aˆ + aˆ†), (5) Entropy 2020, xx, 5 3 of 10 with aˆ and aˆ† are the annihilation and creation operators of the vibrational motion, respectively, and h is the so-called Lamb-Dicke parameter. In the resolved sideband limit, the vibrational frequency n is much larger than other characteristic frequencies and the interaction of the ion with the laser may be treated using a nonlinear Hamiltonian [1,8]. The Hamiltonian (3) in the interaction picture can then be written as Figure 1. We plot the atomic inversions as a function of time with W = 1 and a = 4 and for (a) h = 0.2, (b) h = 0.1 and (c) h = 0. 2 nˆ! (m) Hˆ = Aˆ We−h /2 L (h2)aˆm + H.c., (6) I eg (nˆ + m)! nˆ (m) 2 † where Lnˆ (h ) are the associated Laguerre polynomials that depend on the number operator, nˆ = aˆ aˆ, and W is the Rabi frequency. By solving the Schrödinger equation for the Hamiltonian we find the wavefunction [6], y(t), and from it the the total system’s density matrix, rˆ(t) = jy(t)ihy(t)j rˆ(t) = jeihj ⊗ jcihcj + jeihgj ⊗ jcihsj (7) + jgihj ⊗ jsihcj + jgihgj ⊗ jsihsj where the unnormalized wavefunctions of the vibrational motion of the ion are given by p p jci = cos lt nˆ + 1jai , jsi = −iVˆ † sin lt nˆ + 1jai , (8) and we have used m = 1 and have considered an initial vibrational state given by a coherent state, jai, and the ion in its excited state, jei. The operator 1 V = p aˆ (9) nˆ + 1 is the so-called London phase operator [39,40]. Entropy 2020, xx, 5 4 of 10 We find the vibrational reduced density matrix by tracing over the ion’s external degrees of freedom rˆV = jcihcj + jsihsj , (10) while the atomic density matrix is found by tracing over the internal degrees of freedom such that we obtain rˆA = jeihejhcjci + jeihgjhsjci (11) + jgihejhcjsi + jgihgjhsjsi = jeihejree + jeihgjreg + jgihejrge + jgihgjrgg. In Figure 1 we plot the atomic inversions, W(t) = ree − rgg, (12) for different values of the Lamb-Dicke parameter. The quantities ree and rgg are defined in equation (11). Figure 1 shows that, as the Lamb-Dicke parameter gets smaller, the ion-laser interaction becomes similar to the interaction between a two-level atom and a quantized field. In particular, in Figure 1 (c) the common revivals of oscillations [25] may be clearly observed. 3.1. Relation of the powers of reduced density matrices We use the Cayley-Hamilton theorem, this is, the fact that all square matrices obey their eigenvalue equation, to show that powers of the vibrational density matrix may be related to powers of the spin system. In order to be more specific, Cayley-Hamilton’s theorem states that, given an N × N matrix A, whose characteristic equation reads N N−1 N−2 x − qN−1x − qN−2x − · · · − q1x − q0 = 0, (13) the matrix A also obeys such equation, namely N N−1 N−2 A − qN−1A − qN−2A − · · · − q1A − q01ˆ = 0, (14) with 1ˆ the N × N unit matrix. It may be proved that for two subsystems initially in pure states, after interaction, it may be found a relation between the powers of the reduced density matrices [41] n+1 n rˆV = TrAfrˆrˆAg , (15) where the reduced (atomic) density matrix should be taken in tensor product with vibrational identity operator, that we have obviated.
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