A Condition for Entropy Exchange Between Atom and Field

Transcription

1 Commun. Theor. Phys. 57 (2012) Vol. 57, No. 2, February 15, 2012 A Condition for Entropy Exchange Between Atom and Field YAN Xue-Qun ( ) and LÜ Yu-Guang (ù ½) Institute of Physics and Department of Physics, Tianjin Polytechnic University, Tianjin , China (Received August 8, 2011; revised manuscript received October 24, 2011) Abstract We study the entropy correlations between subsystems in a bipartite system. Our results show that there is an entropy exchange effect (i.e., anticorrelation phenomenon) between atom and field. Especially we prove that there exists a condition for complete entropy exchange between atom and field. PACS numbers: a, Ct, Ud Key words: quantum correlation, entropy exchange, atom and field 1 Introduction One of the most challenging features of quantum theory has been to understand the natures and implication of quantum correlations. Several measures of these quantum correlations have been investigated in the literature, [1 6] and among them the quantum discord has recently received a great deal of attention. [7 22] Quantum discord is defined as the difference between quantum mutual information and classical correlation in a bipartite system. In general, this correlation is different from entanglement, which is known as a key resource in quantum information processing. [23] Quantum discord was shown theoretically [24 26] and experimentally, [27] that some separable states may also speed up certain tasks over their classical counterparts. Therefore, this has motivated the study of other measures of capturing quantum correlation. Recently, by extending the analysis by Phoenix and Knight, Boukobza and Tannor [28] presented examples of qualitatively different entropy correlations. Specifically, they explored the regime of entropy exchange between light and matter; that is, where the rate of change of the two are anticorrelated. This behavior contrasts with the case of pure light-matter states in which the rate of change of the two entropies is positively correlated. They also explored the relationship between the entropy correlations and entanglement. In addition, similar works on this issue have been considered by other authors. [29 31] However, all of entropy exchanges stated above are not complete. In the present paper we will mainly prove that, under certain conditions, the atomic and field partial entropy changes are completely anticorrelated so that the net change in entropy of the system is zero. It is formally similar to that of the thermodynamics reversible processes, which can be understood from thermodynamics textbook. To view this from another angle, there is in fact a conservation law of entropy between light and matter. 2 Formulation of Problem The entropy of a quantum system can be defined in terms of the S = Tr(ˆρ ln ˆρ), (1) where ˆρ is the density operator for a given quantum system and we have set Boltzman s constant equal to 1. It is most easily calculated from the nonzero eigenvalues λ i of ˆρ as S = λ i ln λ i. (2) i The main object in this paper is to investigate the entropy correlations between subsystems in a bipartite system, which is an interesting work in a sense that it attempts to link thermodynamics and information entropy. We shall label these sub-systems by the suffixes a and b. The total state of the two subsystems is described by a density operator ˆρ and the states of the sub-systems are described by the partial density operators ˆρ a and ˆρ b. The partial density operators for the a(b) subsystems are constructed from the full two subsystems density operator ˆρ by tracing over one of the subsystems, that is, ˆρ a(b) = Tr b(a)ˆρ. (3) The entropies for a and b can then be expressed as S(ˆρ a,b ) = Tr a,b (ˆρ a,b ln ˆρ a,b ). (4) Using the quantum entropy, Boukobza and Tannor explored entropy correlations between a quantized cavity mode and a single atom in the framework of the JCM by considering both pure and mixed atomic and field states. [28] They found that there is an entropy transfer process going on, consistent with classical thermodynamic concepts. And they showed that if the quantum state of Supported by the Natural Science Foundation of Tianjin under Grant No. 09JCYBJC xqyan867@yahoo.cn c 2011 Chinese Physical Society and IOP Publishing Ltd

2 210 Communications in Theoretical Physics Vol. 57 total system is initially mixed, then the rate of change of both the atomic and field partial entropies can have opposite signs; that is, there is entropy exchange between the atom and the field. This behavior contrasts with the case of pure atom-field states in which the rate of change of the two entropies are positively correlated and in fact identical. Motivated by their analysis we would like to prove that there exists a condition for the substantial entropy exchange between light and matter. To make our results more general, we consider a generalized JC model with additional forms of nonlinearities of the field and the intensity-dependent atom-field coupling. The Hamiltonian of this system in the rotating wave approximation can be written as [32] Ĥ =ωâ â ω 0ˆσ z + R(â â) + λ[â f(â â)ˆσ + f(â â)âˆσ + ], (5) where ω and ω 0 are the field and atomic transition frequencies, respectively, λ is the real atom-field coupling constant, λf(â â) represents an arbitrary intensity-dependent atom-field coupling, and we have set equal to unity for convenience. The detuning = ω 0 ω should satisfy that ω 0, ω in order to maintain the reliability of the rotating-wave approximation. For simplicity we just discuss the effect of the nonlinearity of the single-mode field with a Kerr-type medium (that is, R(â â) = χâ 2 â 2 ) on the partial entropy changes. The Kerr cavity is used to modify the photon statistics of the micro-maser field towards the state with a low number of photons, and also applied to reduce the amplitude noise. We assume that initially the field is prepared in a single-mode thermal state ˆρ f (0) = P n n n, P n = [m n /(m + 1) n+1 ], (6) where m is the average number of photons in the cavity, and the atom is initially in the mixed state ˆρ a (0) = P e e e + P g g g, (7) Here P e, 0 P e 1, and P g = 1 P e, is the mixed state parameter of the atom. P e = 0 (or 1) means the atom is initially in the ground (or excited) state and P e = 0.5 in the maximally mixed state. The density operator for the total system at initial moment is taken to be a factored mixed state ˆρ af (0) = ˆρ f (0) ˆρ a (0) = P e Pn n, e n, e + P g Pn n, g n, g. (8) The time evolution of the joint density operator ˆρ af (t) at any time t > 0 can be obtained by dˆρ af (t) dt = i[ĥ(t), ˆρ af(t)]. (9) From Eqs. (8) and (9), we can obtain the expressions for the density matrix considered here. For discussing the entropy correlations between the two subsystems, the atom and the field, we need to have the knowledge of the eigenvalues spectrum of the partial density operator. The partial density operators of the atom and the field are given respectively by ˆρ f (t) = Tr aˆρ af (t) = λ n f (t) n n, (10) ˆρ a (t) = Tr f ˆρ af (t) = λ 1 (t) e e + λ 2 (t) g g, (11) where e and g stand for the excited and ground states of the atom, λ 1 (t), λ 2 (t) and λ n f (t) are respectively eigenvalues of the atomic and the field partial density operator. By substituting λ 1, λ 2, and λ n f into Eq. (4) we obtain the partial entropies of the atom and the field in the following forms: S a (t) = (λ 1 lnλ 1 + λ 2 ln λ 2 ), (12) S f (t) = λ n f lnλn f. (13) With these formulae we are in a position to calculate the numerical results of partial entropy changes for the field S f (t) = S f (t) S f (0) and the atom S a (t) = S a (t) S a (0) for the system under consideration. 3 Numerical Investigations Based on the above theoretical calculations, we can next present some numerical results to describe the partial entropy changes of the field and the atom in the present model. Firstly, we are going to prove that, in some cases, the atomic and the field partial entropy changes are completely anticorrelated; that is, there exists a conservation law of entropy exchange between light and matter. On the other hand, it can also be shown that there is no entanglement when the completely entropy transfer occurs. For the sake of simplicity and the focuses to the essence problems discussed, here we will restrict our attention to the case when f(â â) = 1. In Fig. 1(a), we exhibit the dynamical behavior of the field and atomic partial entropy changes S(t) against the scaled time λt for the field initially in a weakly excited thermal state (m = 0.2) and the atom close to excited state (Pe = 0.9) with /λ = 5, χ/λ = 1. The solid line represents field partial entropy changes and the dotted line represents atomic partial entropy changes. From the figure we observe that both partial entropy changes rise and lower together. This case is quite similar to the situation discussed by Boukobza and Tannor in Ref. [28] (see Fig. 1(a) of the reference), where the atom is initially in the excited state and the field is in a weakly excited thermal state. In Fig. 1(b), the parameters are same as Fig. 1(a) but the atom is initially in a mixed state, that is, Pe = 0.3. Comparing Figs. 1(b) with 1(a) one can see that both partial entropies of the atom and the field no longer fluctuate together in time, whereas are opposite in

3 No. 2 Communications in Theoretical Physics 211 sign to each other. It clearly shows that there is entropy exchange between the atom and the field, but here the entropy is not conserved, that is, S a + S f 0 (> 0), which is shown by the dash-dot line in Fig. 1(b). In addition to these stated above, the results can also confirm, as expected, that the sum of partial entropy change of the atom and field is always nonnegative, which is independent of the values of Pe and m. We see that although the entropy of one part of the system decreased, the entropy of the other part increased by a greater amount so that the net change in entropy of the total system is positive. of generality, let us choose a point which is located at Pe = 0.42 along with suitable value of m on the curve of Fig. 2. For the chosen point, the m on the curve is situated in the interval from 2.62 and We note that when m = 2.62, the field partial entropy change is negative ( S f < 0) and the one of the atomic is positive ( S a > 0) (see Fig. 3(a)). The situation is completely changed when we take m = 2.63, where we note that the field partial entropy change is positive, that is, S f > 0 ( S f has changed from a negative to the positive value) and the atomic is negative, that is, S a < 0 ( S a has changed from a positive to the negative value) (see Fig. 3(b)). Also, we can see that in the two cases the sum of partial entropy change of the atom and field is completely zero (see the dash-dot line in the figures). This fact therefore implies that there exists a point for Pe = 0.42 and m taking a suitable value, in which the sum of partial entropy change of the atom and field is completely zero, that is, S a + S f = 0. Similarly, other points (the circle points in Fig. 2) can be given the same proof. Thus, we have proved that there exists a substantial entropy exchange process between the atom and field when Pe and m are taken suitable values (the points on the curve of Fig. 2). In other words, the points of the curve imply that the sum of the atomic and field entropy changes is conversed. It should be pointed out that the situation is formally similar to classical thermodynamic reversible process. And our numerical results reveal that the condition for the substantial entropy exchange between light and matter is shown to hold as 0 < Pe < 0.5 and 0 < m < 6. Fig. 1 Plots of the time variation of the partial entropy changes S(t) for / λ = 5, χ/λ = 1 and f(â â) = 1. (a) The field is initially in a weakly excited thermal field, i.e., m = 0.2 and atom close to excited state, i.e., Pe = 0.9 (field solid line, atomic dotted line); (b) The field is initially in a weakly excited thermal field, i.e., m = 0.2 and atom in the mixed state, i.e., Pe = 0.3 (atomic solid line, field dotted line, atomic plus field dash-dot line). However, it is interesting to find that there exists a substantial entropy exchange process between the atom and field for Pe and m taking suitable values (the values are the points on the curve of Fig. 2), that is, S a = S f (in region A, S f < 0 and S a > 0; in region B, S f > 0 and S a < 0). In other words, the points of the curve represent that the sum of the atomic and field entropy is conversed, that is, S a (t) + S f (t) = S a (0) + S f (0). This is what we will prove next. Now we are ready to prove the law of conservation to take the situation of Fig. 3 as an example. Without loss Fig. 2 The solid line represents a conservation law of entropy between light and matter (S a(t) + S f (t) = S a(0) + S f (0)) and line connecting the numerical experimental points (circle points) is a guide to the eye. In region A, S f < 0 and S a > 0; in region B, S f > 0 and S a < 0. To investigate more general results we examine another example (see Fig. 4) where the changes in atomic and field partial entropies are shown for the atom initially in a mixed state Pe = 0.42 and the field in an excited thermal state m = 2.62 as well as detuning parameter /λ = 6

4 212 Communications in Theoretical Physics Vol. 57 with χ/λ = 0, f(â â) = 1. One can observe from the figure that the detuning leads to collapse-revival of the field and atomic partial entropy changes occurring. It is worth mentioning that even for the complicated oscillations, a complete entropy exchange can take place and, it occurs when m = 2.62, Pe = 0.42 for this example. Indeed, in this case the sum of the two partial entropies is completely conserved. The point to note is that the result disagrees with that of Ref. [28] where substantial entropy exchange occurs when the field is in a weakly excited thermal state (m = 0.1) and the atom is initially close to the ground state. In fact, we find that when Pe and m deviate from the values of the curve of Fig. 2 the conserved is no longer complete, that is, the substantial entropy exchange vanishes. Additionally, using concurrence to measure the atomfield entanglement we can also show that the concurrence tends to zero when the reversible entropy transfer occurs. This will be reported in detail in a forthcoming paper. Fig. 4 Partial entropy change (atomic solid line, field dotted line, atomic plus field dash-dot line) for the atom in a mixed state (Pe = 0.42) and the field in the state m = 2.62, with /λ = 6, χ/λ = 0, and f(â â) = 1. Fig. 3 Partial entropy change (field solid line, atomic dotted line, atomic plus field dash-dot line) for /λ = 0, χ/λ = 0, and f(â â) = 1. (a) Pe = 0.42; m = 2.62; (b) Pe = 0.42; m = Finally, we also investigate the effect of the parameters of the nonlinear medium, the intensity of nonlinear coupling and detuning on the entropy exchange. We find that the substantial entropy exchange is independent of the above parameters. Our results show that the substantial entropy exchange is dependent strongly on the choices of the initial state of both the field and the atom. We note that the farther away the two initial states are from the substantial entropy exchange, the less entropy exchange becomes. 4 Summary and Concluding Remarks We have investigated the partial entropy changes of the field and the atom in a system of an atom interacting with a single quantized cavity field with additional forms of nonlinearities of the field and the intensity-dependent atom-field coupling. We demonstrate that there exist two qualitatively different entropy correlations, that is, positive correlation and anticorrelation, between the field and the atom. More importantly, we have verified that there exists a condition for the substantial entropy exchange between light and matter. It is well known that if the total system is prepared in a pure state then the component systems have equal entropies throughout subsequent unitary evolution. Thus, in this case partial entropies are not an additive (extensive) quantity. However, as we have found in the previous section that there exists the situation where the partial entropy changes of light and matter are additive; that is, the sum of the field and atomic entropy changes is conserved when light and matter are initial in mixed states, and this situation occurs when the mixed state parameter of the atom and the average number of photons in the cavity are respectively bounded in the regions: 0 < Pe < 0.5 and 0 < m < 6. In addition, it should be pointed out, that there is no entanglement when the substantial entropy transfer occurs. We also find that the detuning leads to collapse-revival of the field and atomic partial entropy changes occurring. Our results show that the substantial entropy exchange is dependent strongly on the choices of the initial state of both the field and the atom, and independent of the parameters of the non-linear medium, the intensity of non-linear coupling and detuning.

5 No. 2 Communications in Theoretical Physics 213 References [1] L. Henderson and V. Vedral, J. Phys. A 34 (2001) [2] H. Olliver and W.H. Zurek, Phys. Rev. Lett. 88 (2001) [3] J. Oppenheim, et al., Phys. Rev. Lett. 89 (2002) [4] B. Groisman, S. Popescu, and A. Winter, Phys. Rev. A 72 (2005) [5] S. Luo, Phys. Rev. A 77 (2008) [6] K. Modi, et al., Phys. Rev. Lett. 104 (2010) [7] W.H. Zurek, Phys. Rev. A 67 (2003) [8] M. Horodecki, et al., Phys. Rev. A 71 (2005) [9] C.A. Rodriguez-Rosario, et al., J. Phys. A 41 (2008) [10] A. Datta, A. Shaji, and C. Caves, Phys. Rev. Lett. 100 (2008) [11] M. Piani, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 100 (2008) [12] J. Maziero, et al., Phys. Rev. A 80 (2009) [13] A. Shabani and D.A. Lidar, Phys. Rev. Lett. 102 (2009) [14] A. Datta and S. Gharibian, Phys. Rev. A 79 (2009) [15] M. Piani, et al., Phys. Rev. Lett. 102 (2009) [16] T. Werlang, et al., Phys. Rev. A 80 (2009) [17] J. Maziero, et al., Phys. Rev. A 81 (2010) [18] D. Cavalcanti, et al., Phys. Rev. A 83 (2011) [19] A. Brodutch and D.R. Terno, Phys. Rev. A 83 (2011) (R). [20] A. Streltsov, H. Kampermann, and D. Bruß, Phys. Rev. Lett. 106 (2011) [21] A. Datta, Phys. Rev. A 80 (2009) [22] J.L. Guo, Y.J. Mi, J. Zhang, and H.S. Song, J. Phys. B: At. Mol. Opt. Phys. 44 (2011) [23] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2000). [24] S.L. Braunstein, C.M. Caves, R. Jozsa, N. Linden, S. Popescu, and R. Schack, Phys. Rev. Lett. 83 (1999) [25] D.A. Meyer, Phys. Rev. Lett. 85 (2000) [26] A. Datta, A. Shaji, and C.M. Caves, Phys. Rev. Lett. 100 (2008) [27] B.P. Lanyon, M. Barbieri, M.P. Almeida, and A.G. White, Phys. Rev. Lett. 101 (2008) [28] E. Boukobza and D.J. Tannor, Phys. Rev. A 71 (2005) [29] X.Q. Yan, B. Shao, and J. Zhou, Chaos, Solitons & Fractals 37 (2008) 835. [30] H. Kayhan, Commun. Theor. Phys. 56 (2011) 139. [31] J.L. Guo, Y.B. Sun, and Z.D. Li, Opt. Commun. 284 (2011) 896. [32] M. Abdel-Aty, S. Furuichi, and A.S.F. Obada, J. Opt. B 4 (2002) 37.