† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 61873186 and 11902220), the Natural Science Foundation of Tianjin City of China (Grant No. 17JCZDJC38300), the Provincial Foundation for Excellent Young Talents of Colleges and Universities of Anhui Province of China (Grant No. GXYQ2017014), and the Anhui University Humanities and Social Sciences Research Project of China (Grant No. SK2019A0116).
Quantum-classical correspondence is affirmed via performing Wigner function and a classical-quantum chaotic system containing random variables. The classical-quantum system is transformed into a Kolmogorov model for force and energy analysis. Combining different forces, the system is divided into two categories: conservative and non-conservative, revealing the mechanical characteristic of the classical-quantum system. The Casimir power, an analysis tool, is employed to find the key factors governing the orbital trajectory and the energy cycle of the system. Detailed analyses using the Casimir power and an energy transformation uncover the causes of the different dynamic behaviors, especially chaos. For the corresponding classical Hamiltonian system when Planck’s constant ħ → 0, the supremum bound of the system is derived analytically. Difference between the classical-quantum system and the classical Hamiltonian system is displayed through trajectories and energies. Quantum-classical correspondences are further demonstrated by comparing phase portrait, kinetic, potential and Casimir energies of the two systems.
Research on quantum applications has attracted a great deal of attention recently, especially in quantum computation,[1] quantum chip,[2] quantum communication, quantum application in image binarization[3] using quantum entanglement[4] or quantum protocols,[5] and so on. Quantum mechanics does not describe the phase space of the system motion as classical mechanics because of the uncertainty and stochasticity. However, quantum systems become classical Hamiltonian systems in the limit when Planck’s constant ħ → 0.[6] Alternatively, researchers have demonstrated that under decoherence a quantum system corresponds to a conservative chaotic Hamiltonian system. In the last two decades, researchers have investigated the correspondence between quantum and classical Hamiltonian systems. For example, Brack et al.[7] demonstrated the existence of quantum beats in a classical system, i.e., the Hénon–Heiles system, which is a conservative chaotic Hamiltonian system. Hou and Hu[8] as well as Song et al.[9,10] derived the relationship between quantum entanglement and classical chaos through the average linear entropy using the Dicke Hamiltonian model. Eckhardt et al.[11] presented a conservative chaotic Hamiltonian function in which there appears quantum mechanical behavior. Gong and Brumer[12] pointed out that the decoherence, i.e., the loss of coherence, of a quantum system corresponds to this conservative chaotic Hamiltonian system. Taking into account the quantum-classical correspondence, the classical chaotic system is called a classical-quantum chaotic system. Furthermore, researchers examined the classical-quantum correspondence in the chaotic system using a uniform structure measure for the distribution functions of both the classical and quantum phase spaces.[13] When the ratio of two parameters in the chaotic system is large, the maximum Lyapunov exponent is larger than that of many other existing conservative chaotic systems, such as the classical Hénon–Heiles system.[7] Therefore, the objective of this paper is the study of the classical-quantum chaotic system, particular in relation to the quantum mechanics of decoherence and study of corresponding classical Hamiltonian conservative system in the limit when Planck’s constant ħ → 0.
Chaotic systems mainly divide into two major categories: dissipative systems for which the divergence of the system is less than zero and conservative systems for which it is equal to zero.[14,15] Conservative systems preserve the volume in phase space, its dimension being integer, whereas dissipative systems do not conserve the volume, its dimension being factional. From the Liouville theorem, a Hamiltonian system must conserve this volume.[16] Hence, the conservative chaotic Hamiltonian system is restricted more than the conservative chaotic non-Hamiltonian system, because it conserves both Hamiltonian energy and volume. To our knowledge, research on the conservative chaos dynamics is scant, especially concerning the conservative chaotic Hamiltonian systems.
More and more researchers pay attention to nonlinear dynamic analyses,[17–20] especially dynamic analyses of chaotic systems. The methodologies employed entail numerical calculations, analyses of boundaries, aperiodic solutions, sensitivity to initial, bifurcation,[21–24] circuit implementation,[25] Lyapunov exponent calculations,[26,27] fractional order,[28] Melnikov analysis,[29–33] Poincaré map,[34,35] system control, and synchronization.[36] Both the Melnikov method and the mechanical analysis method divide a Hamiltonian system with disturbance into a conservative part and a non-conservative part, similarities between the two methods are corresponding to the conservative part, the rate of change of the Melnikov function in the Melnikov method and the Casimir function of the mechanical analysis method are both zeros. The Melnikov function is analyzed from a mathematical perspective, and the Casimir function has a physical background. The Melnikov method uses the homoclinic orbit to find the existence conditions of chaos in the sense of Smale horseshoe. The mechanical analysis method first obtains the extreme surface and then analyze trajectory characteristics of the system via extreme surface. Other methods also do not cover the mechanics of the chaotic systems such as the conservation of energy, the physical underpinnings and background, the conversion among the internal energy, the dissipation, and the external force. Recently, Pelino et al.,[37] Qi et al.,[38,39] Yang and Qi[40] have studied the mechanics including energy transformation, force for some chaotic systems. For instance, using the Kolmogorov model, Pelino et al.[37] presented the energy cycle of the Lorenz system. For the brushless dc motor chaotic system, Qi[38] decomposed the forces and derived the energy cycling by employing the Kolmogorov model and the Casimir function. Using the mechanics analysis method, Yang and Qi[40] decomposed the vector field of the plasma chaotic system into four types of torque: inertial, internal, dissipative, and external, and comparison of mechanics analysis and generalized competitive mode analysis. For the chaotic system of a permanent-magnet synchronous motor, the Casimir energy as stored energy and its rate of change as the power difference between the dissipative energy and the energy supplied to the motor are employed to shed some insight into the mechanisms of the system.[39] Moreover, details of mechanical analysis or energy cycling of many chaotic systems are revealed by converting these systems into the Kolmogorov form.[41–45] The mechanical analysis is able to uncover the causes of sinks, periodic orbits, and chaos produced by systems. However, studies concerning the mechanism have only focused on dissipative chaotic systems. To date, there has been no analysis of causes of different dynamical behaviors for a quantum, classical-quantum or conservative Hamiltonian system.
A classical-quantum chaotic Hamiltonian system can display both classical chaotic and quantum features. Therefore, the mechanical analysis is a good technique to uncover the mechanism behind the production of different dynamical behaviors.
This paper identifies a classical-quantum chaotic system and quantum-classical correspondence as the object of study. The classical-quantum chaotic system is transformed into a Kolmogorov-type system characterized by a vector field of three forces: inertial, internal, and external. Correspondingly, three energies, i.e., kinetic, potential and supplied, are found for the classical-quantum Hamiltonian system. The mechanism behind the dynamical behaviors, such as periodic orbits, pseudo-periodic orbits, sources and chaos, are revealed using a combination of forces and energy. The Casimir power is employed to find the cause for the different dynamical behaviors. For the corresponding classical Hamiltonian system, the supremum bound of the system is derived analytically and a variety of dynamics is shown in the Lyapunov spectrum. By comparing trajectories and energies of the system, difference between the classical-quantum system and the classical Hamiltonian system is demonstrated. Quantum-classical correspondences are demonstrated by comparing trajectories, 3D view of the phase space trajectories, phase portrait, energies of the two systems.
The structure of this paper is as follows. In Section
In this study, the classical Hamiltonian function chosen to investigate the corresponding quantum mechanical system is[11–13]
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![](cpb_29_2_020502/cpb_29_2_020502_ieqn1.gif)
For convenience, let x1 = x, x2 = y, x3 = px, x4 = py, then H(
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In the quantum world, the notion of a certain point in the phase space does not make sense because the positions x1 and x2, the momenta x3 and x4 cannot be measured instantly (Heisenberg’s uncertainty principle). To describe the behavior of quantum, we have to introduce a function of quasi-probability density, called the Wigner function, to measure the random property of quantum state. The Wigner function has a good performance for analyzing quantum unstable variables, and is written as
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![](cpb_29_2_020502/cpb_29_2_020502_ieqn2.gif)
The Wigner function of a given state can be calculated by Eqs. (
The Wigner distribution is presented in Figs.
![]() | Fig. 1. The Wigner distribution in 3D space: (a) (x1, x3, W) and (b) (x2,x4,W), 2D plane: (c) (x1,x3) and (d) (x2,x4), (e) W–x3, (f) frequency spectrum of the rectangular pulse signal. |
The Wigner function is employed to calculate the mean of quantum behavior or trajectory 〈xi⟩ based on the Hamiltonian equation. Equation (
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Eckhardt et al.[11] studied a quantum mechanics of the corresponding Hamiltonian function (
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Furthermore, the quantum-classical corresponding to the Hamiltonian function (
By Eq. (
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Because the dynamics of quantum system is quite hard to investigate, we can firstly confirm whether the stochastic dynamical behavior of system (
Adopting the values of all parameters ħ = 0.1, a = 1, b = 0.01, D = 6 × 10−4. Here the stochastic variables η1 and η2 satisfying mean M(η1) = M(η2) = 0, variance E(η1) = E(η2) = 0.001. Comparisons of the results between classical-quantum system (
![]() | Fig. 2. Time series for (a) 〈x1⟩, (b) 〈x2⟩, (c) 〈x3⟩, (d) 〈x4⟩ and 3D view of the phase space trajectory 〈x1 − x2 − x3⟩, (e) classical result, (f) quantum result. |
In addition to the simulation demonstration of similarity between the two systems, indeed, the quantum system becomes a classical Hamiltonian system in the limit ħ → 0 from the Schrödinger equation.
The classical Hamiltonian system corresponding to the Hamiltonian function is
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![](cpb_29_2_020502/cpb_29_2_020502_ieqn7.gif)
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![](cpb_29_2_020502/cpb_29_2_020502_ieqn8.gif)
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The Casimir function, C(
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![](cpb_29_2_020502/cpb_29_2_020502_ieqn10.gif)
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Considering the Casimir energy and Casimir power,from Refs. [38,40,41] we have:
If If If If If If
Therefore, the Casimir power provides a criterion to determine whether a system can generate chaotic motion.
Next, we discuss the effects of the different types of forces on the system that alter its behavior from simplicity to complexity, and we reveal the key causes producing chaos. In the following, we set a = 1, b = 0.01, and impose the initial condition
In this section, we mainly discuss the situation when the system contains inertial force K1(
When the system contains only the inertial force term (describing the kinetic energy released K1(
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![]() | Fig. 3. System containing only the inertial force term, (a) 3D view of 〈x1 − x2 − x3⟩. Time series of (b) the Hamiltonian energy 〈H1⟩, (c) the Casimir energy, and (d) the Casimir power. |
The reason why the Casimir energy of system (
When the system contains both inertial force (kinetic energy released K1(
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![](cpb_29_2_020502/cpb_29_2_020502_ieqn20.gif)
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The Casimir power is
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Now, the system is governed by
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The derivative of the Hamiltonian is
From system (
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![](cpb_29_2_020502/cpb_29_2_020502_ieqn24.gif)
![](cpb_29_2_020502/cpb_29_2_020502_ieqn25.gif)
![](cpb_29_2_020502/cpb_29_2_020502_ieqn26.gif)
![](cpb_29_2_020502/cpb_29_2_020502_ieqn27.gif)
We need to demonstrate how the two external forces,
The Casimir power
In general, the bounds of a chaotic system are difficult to find, especially in an analytic form. Nevertheless, we find an analytic bound for the chaotic trajectory of this conservative Hamiltonian.
In Fig.
Let Lyapunov exponents of the system are L1, L2, L3, L4, respectively, and L1 ⩾ L2 ⩾ L3 ⩾ L4.
(a) Setting b = 0.01 and
![]() | Fig. 6. Lyapunov exponent of system ( |
The Lyapunov exponents are symmetric about the zero point [Fig.
(b) Setting a = 1 and initial value
When the system contains the terms of inertial, internal, external force and random variables
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In this section, quantum-classical correspondence is further revealed via comparing energies. In Fig.
Quantum-classical correspondence has been conducted via the Wigner function and a classical-quantum chaotic system containing random variables. A dynamic mechanical analysis has been used effectively in analyzing a classical-quantum system exhibiting strongly chaotic flow. The system is transformed into an equivalent Kolmogorov-type system. The vector field of the chaotic Hamiltonian system is decomposed into three different forces. We find that combinations of different forces (or different energies) determine the dynamical behaviors of the system, such as periodic orbits. The Casimir energy and power are effective in finding insights into causes of different dynamical behaviors produced. For the corresponding classical Hamiltonian system, the supremum bound of the system is analytically derived and verified, and rich dynamics of the system are explored through the Lyapunov spectrum. Differences between the classical quantum system and the classical Hamiltonian system are displayed by trajectories and energies. Quantum-classical correspondences are further demonstrated by comparing phase portrait, energies of the two systems.
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