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Quantum-classical correspondence in entanglement production: Entropy and classical tori

We analyze the connections between entanglement dynamics and classical trajectories in a semiclassi-cal regime for two systems: A pair of coupled oscillators and the Jaynes-Cummings model. We find that entanglement production depends on classical invariant tori and such phenomenon is closely related to the power spectra of classical trajectories. Classical power spectrum with a larger number of frequency com-ponents corresponds to larger entanglement. We introduce a frequency entropy to describe the classical frequency distribution, and find that there is good correspondence between the classical frequency entro-pies and the maximum von Neumann entropies.


I. INTRODUCTION
Problems of quantum-classical correspondence have been extensively investigated [1]. Recent studies have further shown the relations between entanglement production and the underlying classical dynamics for chaotic systems. In the early works on spin-boson system [2] and coupled kicked tops [3], it was suggested that entanglement production can be a good indicator of the regular to chaotic transition for classical chaotic systems. This claim has been affirmed in many subsequent studies [4,5,6]. Also it has been revised in some recent works [7][8][9]. An investigation on a pair of coupled kicked tops showed that increased chaos corresponds to a saturation of entanglement production rate in the weakly coupled regime [7]. Another study on Rydberg molecules [8] found that the quantum-classical correspondence in entanglement production is related to the inelastic scattering. Especially, a semiclassical approach shown by Jacquod [9] concluded that the entanglement production in bipartite system depends on both the degree of chaos and the coupling strength.
The aim of this paper is to further explore the connections between entanglement production and the underlying classical trajectories in a semiclassical regime for chaotic systems. In Section II, we investigate the dependence of entanglement production on the classical trajectories for the two coupled oscillators. We find that the production of entanglement is strongly related to the classical trajectories. The maximum values and rates of entanglement production correspond systematically to the classical invariant tori. When the initial state lies at the edge of regular islands for the regular case or in the chaotic sea for the mixed case, the production of entanglement is maximized. Furthermore, the above correspondences between the production of entanglement and the classical trajectories are closely related to the classical power spectrum. The more complicated classical power spectra correspond to the larger maximum values and rates of entanglement production. We thus define a frequency entropy to measure the frequency distribution of classical power spectrum. It is shown that there is a good correspondence between the frequency entropies and the maximum von Neumann entropies for different classical trajectories (especially for regular ones). Such correspondence is also found in the investigations on the Jaynes-Cumming model presented in Section III. This is because a classical power spectrum reveals the structure of the energy levels involved in the quantum evolution [10,11,12]. The latter is directly connected to the entanglement production [13]. Thus the classical power spectrum and the classical frequency entropy can indicate the behavior of entanglement production.

II. ENTANGLEMENT PRODUCTION IN A BI-PARTITE MODEL
We now study the entanglement of a bipartite system with a model of two coupled oscillators, which is often used to analyze decoherence [14] or construct qubit models [15,16]. The Hamiltonian reads 2 2 2 2 2 2 2 2 1 2 1 2 12 where m and ω are the mass and frequency of any of the two oscillators. λ is the coupling parameter. This system is proposed by Pullen and Edmonds [17] and used in many works to study the classical-quantum correspondence for chaotic systems [18]. In this section, using the model (1), we investigate the entanglement dynamics in both the regular and mixed cases to achieve a detailed picture of the relations between the entanglement production and the classical trajectories. In the calculations, we adopt the natural units for the harmonic oscillator, i.e., ω for energy, mω for length and mω for momentum. Accordingly, λ in the model (1) is measured in units of 2 3 m ω . The quantum dynamics is governed by the evolution operator . The connection between quantum and classical domains is done by the preparation of the initial coherent state , whose center is precisely on a classical phase space point 1 [20].

A. Entanglement production in the regular case
The classical dynamics of model (1) exhibits a regular phase space with 0.0075 λ = and total energy =58 E . In the classical phase space, we choose twenty one classical trajectories to explore the connections between entanglement production and classical dynamics. The Poincaré section of these trajectories is shown in Fig. 1. The three regular islands are marked with A, M and D. B (C) denotes the upper (lower) half part of the island M. In each regular island, there are seven tori.
From the tori in the island A, M and D, we pick up the centers of the initial coherent states (CICS) along an arrow line. As shown in Fig. 1, the arrow line starts from the origin in A, passes through the innermost torus in M and then reaches the minimum torus in D. Between the arrow line and the tori in A, B, C and D, there are twenty seven cross points. We pick up these cross points along the arrow line and display their coordinates in Table I  , the distance between any two adjacent cross points is approximately equal to 1 in the units used in this section. As mentioned above, the scales of the initial states in the same units are (1) O in both the length and momentum directions. Thus, the distance between any two adjacent CICS is approximately equal to the scales of the initial states. This makes sure that we are treating a semiclassical regime.
With the CICS chosen above, we calculate the von Neumann entropies and show the results in  Table I Table I To clarify the dependence of entanglement production on classical trajectories, we further calculate the maximum values of the entropy curves in Fig. 2 and show the results (i.e., the maximum von Neumann entropies) in Fig.  3. The maximum entropies M S in Fig. 3 are arranged in the same order as their corresponding CICS are picked up from the tori along the arrow line in Fig. 1 (i.e., the CICS of the k th M S in Fig. 3 correspond to the k th cross point picked up from the tori along the arrow line in Fig.  1).
In Fig. 3, the maximum von Neumann entropies M S vary significantly with the corresponding classical invariant tori. For instance, as the tori in the island A enlarge from the innermost to the outermost, the corresponding M S 's in Fig. 3 increase from the first square to the seventh square. Then, as the tori in B shrink from the outermost to the innermost, the values of M S in Fig. 3 decrease from the first diamond to the valley between 'B' and 'C'. For the CICS localized on the tori in C and D, the curve of M S also varies with the classical tori in a similar way. Roughly speaking, the value of M S increases (decreases) with the enlargement (shrink) of the classical tori and is maximized when the corresponding torus lies at the edge of the regular islands. Furthermore, since different entropy curves in Fig. 2 reach the saturation plateaus almost at the same time, a larger M S indicates a faster generation of entanglement. Thus, the rate of entanglement production also roughly increases (decreases) with the enlargement (shrink) of the classical tori.
Besides, in the investigations, we also note that there is good correspondence between the oscillations of the entropy ( ) V S t and those of the underlying classical coupling potential for some classical trajectories. The oscillations of classical coupling potential reveal the frequency distribution of classical trajectory. We thus perform frequency analyses of the classical trajectories and further find interesting connections between the en-tanglement production and the classical power spectrum.
Classical power spectrum is used to analyze the frequencies of classical motion [21]. For a classical trajectory, the classical motion is generally multiply periodic, admitting the Fourier expansion [22] [ ] With the EBK method shown in Ref. [21], we calculate the classical power spectra of ( ) q t α ( 1, 2 α = ) for the twenty one classical trajectories chosen above (the evolution time 17 2 t τ ω = = ). Figures 4 and 5 display the results for the trajectories whose tori lie in A and B (see Fig.  1), respectively. The results for the tori in D (C) are similar to those for the tori in A (B) and thus are not shown here. For comparison purpose, the strength axis in the power spectrum is plotted in logarithmic scale (i.e., Power spectra of (a) 1 q and (b) 2 q for the trajectories whose tori lie in the island A (see Fig.1). The seven pairs of spectra from top to bottom correspond to the seven tori in A from the innermost to the outermost. Unit of ω′ is ω . Fig. 4 but for the trajectories whose tori lie in the island M (see Fig.1). The seven pairs of spectra from top to bottom correspond to the seven tori in B or C from the innermost to the outermost. Unit of ω′ is ω .

FIG. 5. As in
In the spectra in Figs. 4 and 5, there are sharp lines at the frequencies ω′ . Each line corresponds to a transition from one energy level to another. Its strength is related to a physical process, which involves the motion of the two particles [22]. Thus, classical power spectrum indicates the energy levels involved in the corresponding quantum evolution. The latter can be revealed by quantum density spectrum.
Quantum density spectrum is expressed as ( ) is the quantum density at time t and n are the eigenvectors of the Hamiltonian with n E as the corresponding eigenvalues. According to Ghose et al. [13], a larger set of { } nn ρ correspond to a faster generation of entanglement in the regular to chaotic transition. Figures 6(a) and 6(b) display the quantum density spectra for the CICS localized on the tori in A and B, respectively. The seven spectra in Fig. 6(a) from top to bottom correspond to the seven pairs of classical power spectra in Fig. 4 from top to bottom. The same correspondence also holds between the quantum density spectra in Fig. 6(b) and the classical power spectra in Fig. 5. From the above comparisons, we can see that more complicated classical power spectrum corresponds to larger set of { } nn ρ . Since { } nn ρ is closely related to quantum entropy [13], the correspondence between the classical power spectrum and the quantum density spectrum indicates the connections between the entanglement production and the classical trajectories.  Table II. Similar to the regular case, with =150.75 E , the distance between any two CICS in Table II is also approximately equal to or larger than the length and momentum scales of the initial states. In addition, the initial states corresponding to the chaotic CICS are also fully contained in the chaotic sea.

B. Entanglement production in the mixed case
We calculate the von Neumann entropies for the CICS in Table II, and present the results in Fig. 8.The topmost entropy curves in Figs. 8(a) to 8(d) correspond to the four chaotic CICS in Table II from first to last. The lower six entropy curves in Fig. 8(a) [Fig. 8(c)] from bottom to top correspond to the six CICS in the row A (B) of Table II from first to last, while the lower six entropy curves in Fig. 8 Table II from first to last.
We further calculate the maximum values M S of the entropy curves in Fig. 8 and display the results in Fig. 9. The four M S 's marked with circles from first to last correspond to the four chaotic CICS in Table II from  gles correspond to the regular CICS localized on the tori in the island A, M and D, respectively. Similar to the regular case, the M S 's obtained with the regular CICS are arranged in Fig. 9 in the same order as their corresponding CICS are sampled from the tori along the arrow line in Fig. 7. For instance, the CICS of the M S 's numbered from one to six in Fig. 9 are sampled from the six tori in A from the innermost to the outermost. In Fig. 9, the curve of M S varies with the classical tori and is maximized when the CICS lies in the chaotic sea. So do the growth rates of the entropy curves as shown in Fig. 8.
We investigate the power spectra of all the above classical trajectories with the EBK methods (the evolution time 17 2 τ = ) [21]. In Fig. 10, we exhibits the results for the six tori in the island A and those for a chaotic trajectory with ( 1 1 10 , 10 q p )=(1 4 , 2.2427 ) (see Table II) as initial conditions. The upper six pairs of spectra in Fig.  10 from top to bottom correspond to the six tori in the island A from the innermost to the outermost. Also they correspond, from top to bottom, to the lower six entropy curves in Fig. 8(a) from bottom to top. The bottom pair is the power spectra of the chaotic trajectory and corresponds to the top entropy curve in Fig. 8(a). A comparison between Figs. 10 and 8(a) shows that more complicated power spectra correspond to larger and faster entanglement production. Such correspondence is also found in the investigations on the relation between the entanglement production and the classical power spectra for the other classical trajectories. Especially, for the chaotic trajectory, the power spectrum is continuous and the corresponding entanglement production is maximized.
In Fig. 8, there are relatively small differences in both the maximum values and the growth rates between the four chaotic entropy curves. This is because the instabilities of the chaotic trajectories are similar to each other, for the chaotic power spectra are all continuous and the Lyapunov exponents of the four chaotic trajectories are around 0.12 0.02 ± . This is in accord with the previous suggestion [3,9,23] that the enhancement of entanglement production induced by chaos depends on the Lyapunov exponent.  Table II Fig. 3 is similar to that in Fig. 9, though the classical dynamics transits from the regular case to the mixed case. Especially, the variation of M S in Fig. 9 through the transition to the chaotic trajectories is similar to that in Fig. 3 through the transition to the edges of the regular islands. This suggests that entanglement production is an indicator not only of the regular to chaotic transition but also of the transitions in different regular classical trajectories.
To further investigate the quantum-classical correspondence in entanglement production, we calculate the quantum density spectra corresponding to the CICS shown in Table II. In Fig. 11, we exhibit the results for the CICS shown in the row A of Table II and those for a chaotic CICS with ( 1 1 10 , 10 q p )=(1 4, 2.2427). The seven density spectra in Fig. 11 from top to bottom correspond to the seven entropy curves in Fig. 8(a) from bottom to top. Also they correspond, from top to bottom, to the classical power spectra in Fig. 10 from top to bottom. Similar to the regular case, a comparison of Figs. 10 and 11 reveals that a larger set of frequency components involved in the classical dynamics corresponds to a larger set of eigenstates involved in the quantum evolution.
C. Frequency entropy and quantum-classical correspondence In the above investigations on the coupled oscillators, we show that entanglement production depends systematically on classical invariant tori. When the initial state lies at the edge of regular islands for the regular case or in the chaotic sea for the mixed case, the entanglement production is maximized. As shown in the previous studies [3,9,23], the enhancement of entanglement production induced by chaos has been investigated and explained by the methods related to the Lyapunov exponents.
Here we further develop a practical method to describe the dependence of entanglement production on classical tori from the perspective of classical power spectra. We use ( ) k I α ω′ to stand for the amplitude of the k th component in the discrete power spectrum of q α ( 1, 2 α = ). To describe the discrete classical spectral distribution, we consider a new quantity as (4) We then define a frequency entropy Fr S as a measure of the classical frequencies, i.e., , ( )ln ( ) which is used to distinguish different regular trajectories and measure the stability of classical dynamics.
With the above method, we investigate the classical frequency entropies of all the regular trajectories chosen in the regular and mixed cases. The results are presented in Fig. 12 Fig. 12(a) [Fig. 12(b)] as they are shown in Fig. 3 (Fig. 9). In Figs. 12(a) and 12(b), there is good correspondence between the variations of M S and Fr S with the classical trajectories. This clearly indicates the quantum-classical correspondence in the production of entanglement.
For the continuous power spectrum of chaotic trajectories, the summation over k in Eqs. (4) and (5)   ρ nn E n placed by the integral over the frequency ω′ . Using this method, we can obtain the frequency entropies of classical chaotic trajectories. However, due to the continuity of chaotic power spectrum, the classical frequency entropies of chaotic trajectories are not consistent with those of regular trajectories. Further research will be needed to study the classical frequency entropies of chaotic trajectories. It is conceivable, however, that in Fig. 9 the M S 's corresponding to the chaotic trajectories are larger than those corresponding to the regular trajectories, for the chaotic power spectra are continuous and have more frequency components.
As mentioned above, the enhancement of entanglement induced by chaos has been discussed and explained by the methods related to the Lyapunov exponent. Here, to further understand the connections between the production of entanglement and the power spectrum of regular classical trajectories, we can get some clews from a general method of quantum-classical correspondence presented by P. Brumer et al. based on the Liouville mechanics [10,11,12]. Following this method, the time evolution of a classical property ( ) t x can be expressed as [10,12] , , where ( ) t x denotes some dynamical variable [e.g., position coordinate ( ) in the quantum counterpart, the frequency lines at ω′ = m ω i in the classical power spectrum reveal the structure of the energy levels involved in the quantum evolution. More complicated classical power spectrum indicates larger set of eigenstates extending over broader and denser quantum density spectrum (the latter can enhance the entanglement production [13]). Such connection is also revealed in the comparisons between the classical power spectra and the quantum density spectra shown above. Similar to the Lyapunov exponent, the classical power spectrum and the classical frequency entropy suggest a simple method to indicate the dependence of entanglement on classical dynamics.

III. ENTANLEMENT PRODUCTION OF THE JAYNES-CUMMINGS MODEL
To confirm the connection between the entanglement and the classical frequency entropy, we perform similar investigations on the Jaynes-Cummings model. The Hamiltonian reads [2] ( where the first term corresponds to the free single-mode field with frequency ω , the second term corresponds to the The connection between the classical and quantum domains is done by the choice of the initial states. Given a point ( 1 1 2 2 , ; , q p q p ) in the classical phase space, the corresponding quantum initial state is w v are the atomic (field)-coherent states given by (1 ) , where G G G ± ′ = ± .
Using the model (7), we investigate the connections between the von Neumann entropies and the frequency entropies for different classical trajectories in both regular and mixed cases. In the calculations, the evolution time for calculating M S is approximately equal to 350; and that for calculating Fr S is approximately equal to We calculate the maximum von Neumann entropies M S for the above sixteen CICS and show the results in Fig. 16. Here, the M S 's from first to last correspond to the above sixteen CICS from first to last. The curve of M S in Fig. 16 exhibits the variation of the maximum entanglement with the underlying classical trajectories. Among these classical trajectories, there are thirteen regular ones, whose tori lie in the island A, B and C (Fig.  15). Figure 17 shows the classical frequency entropies for these regular classical trajectories. In Fig. 17, the squares and circles stand for the classical frequency entropies and their corresponding maximum von Neumann entropies, respectively. From Fig. 17, one also can see that there is good correspondence between the maximum von Neumann entropies and the classical frequency entropies.
IV. CONCLUSION In this paper, using the model of coupled oscillators and the Jaynes-Cummings system, we show that the entanglement production of bipartite system changes systematically with classical invariant tori in both the regular and mixed cases. Furthermore, the dependence of entanglement on the underlying classical trajectories is closely related to the classical power spectrum. The larger the number of classical frequency components is, the larger the corresponding entanglement is. Such phenomena reveal the quantum-classical correspondence in the entanglement production from the perspective of classical trajectories. This sheds lights on the further understanding of the quantum-classical correspondence in the entanglement production of chaotic systems. ACKNOWLEDGEMENTS This work is supported by the National Science Foundation (Grant No. 1037504). . The three islands are marked along the arrow line with A, B and C. Units are as in Fig. 13.