Bifurcation and chaos in nonlinear Lindblad equations

The Lindblad equation describes the dissipative time evolution of a density matrix that characterizes an open quantum system in contact with its environment. The widespread ensemble interpretation of a density matrix requires its time evolution to be linear. However, when the density matrix is obtained from a mean field theory of interacting quantum systems or from a top-down control by a changing classical environment, the ensemble interpretation is inappropriate and nonlinear dynamics arise naturally. We therefore study the dynamical behavior of nonlinear Lindblad equations using the example of a two-level system. By using techniques developed for classical dynamical systems we show that various types of bifurcations and even chaotic dynamics can occur. We also discuss experimental situations for which our results could be relevant.


Introduction
The description of a quantum system by a Schrödinger equation is based on the assumption that the system is sufficiently isolated from the rest of the world that no uncontrollable influences affect its time evolution [1]. In such a situation, the external environment needs to be taken into account only via boundary conditions, electromagnetic potentials or other potential energy terms. However, when environmental influences that cannot be controlled in detail affect the quantum system, the Schrödinger equation is not appropriate any more [2,3]. As is familiar from statistical mechanics, an environment can induce random transitions between the states of a quantum system so that it does not undergo a unitary time evolution any more. A frequently used equation that captures this effect of the environment is the Lindblad equation [4]. It is a Master equation for the density matrix ρ of the quantum system that contains transitions between states in addition to the von Neumann term that describes unitary time evolution. There are two general ways to derive the Lindblad equation: the first one starts from a quantum description of the system and the bath and takes the trace over the states of the bath in order to obtain a time evolution equation for the reduced density matrix that describes only the system [2]. This derivation requires a couple of ad-hoc assumptions that cannot really be justified, in particular that the combined state of system and bath can be written as a product state. The second way to derive the Lindblad equation consists in writing down the most general equation that satisfies the requirement that the density matrix remains a density matrix, i.e., that it remains hermitian, positive semidefinite, and with trace 1. This means that the time evolution must be completely positive and trace preserving [5]. Furthermore, in order to stick to the ensemble interpretation of the density matrix, the Lindblad equation is linear in ρ. This second derivation makes no assumption about the nature of the environment, except that its influence on the quantum system depends only on the present state of the quantum system and not on its past, as no memory terms are included.
While the Schrödinger equation is linear in the wave function, the Lindblad equation is nonlinear in the wave function, but is linear in the density matrix. But these two types of equations are not the only ones that are used to describe quantum systems. In fact, various nonlinear equations for the wave function or the density matrix can be found in the literature. The best known examples are the Hartree and Hartree-Fock theory for many-particle systems, where the Schrödinger equation becomes nonlinear in the wave function, as the influence of the other particles on a given particle is taken into account via the potential generated by the other particles, which in turn is proportional to their charge density [6]. Applying this method to bosons gives the Gross-Pitaevskii equation used for Bose condensates. These equations are mean-field equations that replace the explicit interaction terms by an effective, average influence. Nonlinear, mean-field-like equations are also used for the density matrix: Breuer and Petruccione (see Section 3.7 in [7]) list three classes of such equations: (i) nonlinear Boltzmann equations, which describe the time evolution of a one-particle density matrix due to collisions with other particles, (ii) mean-field Master equations for n identical interacting quantum systems, where the average influence of the other systems on the time evolution of the density matrix of one system is included in a similar way as in the Hartree method, and (iii) nonlinear Schrödinger equations, which we have mentioned before. When the nonlinear Schrödinger equation is translated into a von Neumann equation, one obtains a term that is nonlinear in the density matrix. While all those equations have been developed in order to describe a particular quantum mechanical system, a more abstract and general approach that investigates the possible dynamical behaviors of nonlinear equations for the density matrix has not yet been taken. However, such an approach can reveal a wealth of phenomena, as nonlinear dynamical systems are known to undergo various bifurcations and to show various types of attractors, including limit cycles and strange attractors.
In this paper, we will therefore investigate a nonlinear version of the Lindblad equation, where the transition rates are made dependent on the density matrix. There are essentially two ways to provide a physical justification for such nonlinear Lindblad equations, both of which will be discussed in more detail in the concluding part of this paper. The first way is again a mean-field assumption. If we assume that there are several identical quantum systems (for instance the spins associated with the atoms of a solid) that are coupled to each other and to a shared environment (for instance the phonons of the solid), the time evolution of a spin is affected by the the temperature of the solid and by average state of the other spins, and a mean-field consideration leads to nonlinear equation for the density matrix of one quantum system. The second way is to interpret the density matrix as that of a single quantum system, with the environment determining the density matrix in a top-down manner, and with the finite temperature of the environment being responsible for the description of the system by a density matrix. Changes in time of this density matrix are then correlated with corresponding changes in the environment.
In the following, we will consider a general class of nonlinear Lindblad equations for the smallest possible Hilbert space, which is a two-level system. For this nonlinear model, we will show that it can undergo a variety of bifurcations known from classical nonlinear dynamical systems, and we will discuss the relevance of these findings.

Model
The Lindblad equation describes the time evolution of the density matrix ρ = ρ(t) of an open quantum system in the limit where no memory terms are required, [7] Here, i denotes the imaginary unit, = h 2π is the Planck constant, [·, ·] and {·, ·} denote a commutator and anticommutator respectively, the γ k ∈ R ≥0 are non-negative transition rates, and (A k ) form an orthonormal basis of the complex N × N matrices, that is trace(A † k A j ) = δ kj . W.l.o.g. we can assume that A N 2 is proportional to the identity, A N 2 = 1 N , so that the terms with A N 2 can be absorbed in the von-Neumann term by introducing a modified Hamilton operator [4]. The first term on the right hand side (which is the von Neumann term) describes a unitary time evolution of the quantum mechanical states, whereas the dissipator term describes a probability flow between states. In the following, we consider a two-state model, so we have N = 2. The model can represent any two-level system, but the examples discussed below are best suited for a spin-1/2 system. Let B = {|0 , |1 } be an orthonormal basis of C 2 . We define the three operators L i expressed in the basis B as The operators L 1 and L 2 applied to the basis states induce transitions between the states |0 and |1 , and L 3 yields the eigenvalues ± 1 for the eigenstates |0 and |1 . The transition operators A k can then be written as With this replacement, the Lindblad equation becomeṡ with h ij = h * ji and (h ij ) being positive semi-definite. We introduce the nonlinearities by making the coefficient matrix (h ij ) dependent on the density matrix. This means that the transition rates between quantum states depend on the densities of the quantum states. This is a natural assumption when we consider the model as a mean-field description of many identical interacting quantum systems coupled to the same environment: the state of these systems affects via the direct and indirect interaction the transition rates within each system.
In order to make the dependence of the transition rates on the density matrix explicit, we write ρ in terms of three parameters, where x, y and z are real-valued. The only constraint on their values is that (x, y, z) must stay inside the admissible region U , which assures that ρ stays positive semi-definite, By defining Γ := 1 2 (h 11 + h 22 + 4 h 33 ), we can write the Lindblad equation in the following form, The last terms are due to the von Neumann term, with H ij being the matrix elements of the Hamiltonian. Later, we will consider only special cases where these terms vanish or have a simpler form. In equations (5), we assume that the transition rates depend on the density matrix, i.e., h ij = h ij (z, x, y).

Results
The three-dimensional system of equations (5) has the form of a nonlinear dynamical system with three variables. Such nonlinear dynamical systems are known to show various types of bifurcations and attractors [8]. In this section, we will explore the different types of bifurcations that can occur in equations (5). To this purpose, we consider the model first for the case that it is effectively one-dimensional, then two-dimensional, and finally three-dimensional. With increasing dimension, the dynamical behavior becomes more complex and allows for more types of bifurcations and attractors.

One-dimensional case
The system of equations (5) becomes one-dimensional when the transition matrices A k make transitions between and measurements of the two eigenstates of the Hamilton operator. If we choose as basis states the two eigenstates of the Hamilton operator, the matrices A k become identical to the operators L 1 to L 3 , and the coefficient matrix h becomes diagonal. Furthermore, a diagonal density matrix ρ remains diagonal under time evolution.
With a diagonal h, the coupling between the three differential equations in (5) is lifted. With s = x + i y, we geṫ In the special case that the h ii are independent of ρ, we obtain the solution where the initial conditions are z(0) = z 0 and s(0) = s 0 . This means that In the case where h 11 = 0 = h 22 , the diagonal elements of the density matrix remain constant, whereas the off-diagonal elements still vanish.
When the h ii depend on ρ, we still have Γ > 0 at all times, and the off-diagonal elements of ρ still vanish for t → ∞ and can be neglected when considering the long-term dynamics. This means that the interesting time evolution is contained in the one-dimensional differential equation for ż Here, 1+z 2 is the probability for being in state |0 , while 1−z 2 is the probability for being in state |1 . In the following, we will explore the bifurcations that can occur in a system described by the one-dimensional nonlinear Lindblad equation (8). With a polynomial ansatz for h ii (z), the right hand side of (8) becomes nonlinear in the variable z. The only boundary condition isż (z = −1) ≥ 0 ≥ż (z = 1), since 1+z 2 is a probability and must remain in the interval [0, 1] for all times. Figure 1 illustrates the qualitative change in the phase portrait that results from making the time evolution nonlinear in the density matrix.  In such a system, three types of bifurcations are possible, namely a pitchfork bifurcation (when the system has an intrinsic symmetry), a saddle-node bifurcation, and a transcritical bifurcation. Such bifurcations can occur as a control parameter that affects the functions h 11 and h 22 is changed. In the following, we give examples for all three types of bifurcations in our model.

Pitchfork bifurcation
When the system has an intrinsic symmetry around z = 0, equation (8) contains only odd powers when written as a polynomial of z. Such a symmetry occurs for instance for a two-level spin system when none of the two spin orientations is preferred by the environment. The minimum model showing this pitchfork bifurkation is given by a polynomial of the order 3. Through an appropriate choice of the time scale, this polyomial can be brought to the forṁ which is the normal form of the pitchfork bifurcation and depends only on the parameter t. The bifurcation occurs when t changes its sign, see figure 2.
The bifurcation equation (9) can indeed be written in the form of equation (8) with non-negative functions h ii (q) on the interval [0, 1], for instance by setting This ansatz respects the symmetry between the states |0 and |1 and their transition rates, namely h 11 (z) = h 22 (−z).
A sufficient criterion for h ii (z) being non-negative is when the parameters t and α satisfy α ∈ (0, 2), |t − 2 α| ≤ √ 8 α. A plot of the parameter space can be seen in figure 3. The admissible region is marked by gray stripes. The actual bifurcation happens, when t crosses the threshold t = 0. With the additional constraints that the coefficient matrix is at most quadratic in z, equation (10) is the most general expression for h ii (z) which respect the symmetry between the states and leads to the normal form of a pitchfork bifurcation, equation (9).

Saddle-node bifurcation
When the system contains no symmetry around z = 0, we can add a constant term to the right-hand side of equation (9), resulting inż When the parameter b crosses a critical value b c , a saddle-node bifurcation happens as the unstable fixed point collides with one of the stable fixed points, with the outcome that both of them vanish, see figure 4.
(a) The bifurcation equation (11) can be expressed in the form of equation (8) with non-negative functions h ii (q), for instance by setting The parameters α, t and b must satisfy α ∈ n(0, 2), In order to demonstrate that the saddle-node bifurcation can occur with these restrictions, a stability diagram is shown in figure 5. A saddle node bifurcation occurs when the parameters cross the dashed line. For our choice of parameters, this is indicated by the arrow. The thick lines mark the admissible region for the parameters, guaranteeing that the remains non-negative. The critical value b c can be calculated as |b c | = 2 |t|  Figure 5: Stability diagram for model (12). The dashed line indicates the critical value b c , which depends on the second bifurcation parameter t. We have three fixed point in the gray area and one fixed points in the white areas. The saddle node bifurcation occurs when crossing the dashed line.

Transcritical bifurcation
A transcritical bifurcation requires a fixed point at that always remains a fixed point. Typically, such fixed points lie at the boundaries of the state space. We choose in the following the boundary z = −1 and introduce the variable q = (z +1)/2, which lies in the interval [0, 1]. With this variable, the fixed point lies at q = 0. The transcritical bifurcation occurs when this stable fixed point becomes unstable and a new attracting fixed point q * ∈ (0, 1) (representing a mixed state) appears, see figure 6. By using a minimal polynomial of degree 2 and by scaling time properly, one can write the model as the normal form of a transcritical bifurcation,q = − q (q − c) . (14) A system that is described by this model remains in the ground state if all its two-level subsystems are in the ground state. When c becomes positive, this fixed point becomes unstable and a stable fixed point at a value q = 0 is generated. This means that the system undergoes a time evolution to the second fixed point if at least one if its subsystems is initially in the excited state |1 .
The bifurcation equation (14) can be expressed in the form of equation (8) with non-negative functions h ii (q), for instance by setting The parameters c and α must satisfy α > 0 and c ∈ (−α, 1), which again ensures that h ii (q) are non-negative on the interval [0, 1]. Equation (15) is the most general ansatz when assuming that the h ii (q) are polynomials with degree 1.

Two-dimensional case
When in equation (5) y = 0 is an attracting value for the y dynamics irrespective of the values of z and x, we obtain an effectively two-dimensional system. A sufficient criterion for this to happen is that h ij ∈ R for all i, j ∈ {1, 2, 3} and Re[H mn ] = 0 for all m, n ∈ {1, 2}. Then the system of equations (5) becomes (if we set y = 0) From a physical point of view, a decaying value for y means that there is no equilibrium polarization along the y axis. The probabilities for measuring spin-up in the y direction becomes identical to that for measuring spin-down, namely 1/2. In the geometric view of the Bloch sphere, this means that the dynamics becomes restricted to a circle of radius one, namely the intersection of the Bloch sphere with the y = 0 plane.
First, we consider the situation that H 10 = 0. In this case, the fixed point of the system is a stable node, since the corresponding Jacobian matrix of equation (16) has real eigenvalues with a negative trace and a positive determinant [8]. When we have two different eigenvalues λ 1 < λ 2 < 0, we call the eigenspaces corresponding to λ 1 and λ 2 the fast and slowly decaying directions. When we change the basis such that the slow direction becomes the new z direction, we obtain an effectively one-dimensional system along the slow direction, with the fast direction relaxing during a short time interval. This is similar to the situation given in equations (6) above. When a bifurcation occurs, the larger of the two eigenvalues goes through zero, which means that the bifurcation occurs within the one-dimensional subspace given by the slow direction, and all results obtained in the previous section can be applied also to this two-dimensional situation. In the degenerate case where the two eigenvalues coincide, the fixed point is a "star", and the dynamics follows a straight trajectory with the distance to the fixed point fully characterizing the dynamics, i.e., the dynamics is again effectively one-dimensional [8].
In addition to the one-dimensional bifurcations, a two-dimensional dynamical system can exhibit a Hopf bifurcation, of which we will construct in the following an explicit example. A Hopf bifurcation occurs when a stable spiral becomes unstable and a limit cycle (a closed, isolated trajectory) is generated. Such a limit cycle can be physically realized by precessing spins. We therefore introduce a magnetic field along the y axis (and hence not changing the y component), resulting in the Hamiltonian where all constant factors are included in b. This means that now the last terms on the right-hand side of equations (16) become nonzero, and we have the dynamical systeṁ where Γ is again defined as Γ = 1 2 (h 11 + h 22 + 4 h 33 ). We now make again the coefficient matrix (h ij ) dependent on the variables z and x. We choose the dependency h ij = h ij (z, x) such that the right-hand side becomes the normal form of a Hopf bifurcation.
Let δ ∈ (0, 1), < min{ δ 2 , δ (2−δ) 2 }, r := √ z 2 + x 2 and let the coefficient matrix depend on ρ in such a way, that The special choice for the functions h ij (x, y, z) ensures that the coefficient matrix h remains positive semi-definite for all times. A sufficient criterion is that all principal minors of h are non-negative [9]. Hence, the following inequalities must be satisfied for all times: It can easily be verified, that these conditions are indeed fulfilled. Then equations (17) becomė In radial coordinates, equations (20) have the following form, which is the desired normal form of the Hopf bifurcation: A stable fixed point at the origin r * = 0 (which is a stable spiral in the two-dimensional case) becomes unstable and a stable fixed point away from the origin r * = √ (which corresponds to the attracting limit cycle) appears.
We know that the density matrix ρ(t) can be diagonalized for all times t > 0. This means that there exists a (time Hence, changing into the time dependent basis B t makes the system one-dimensional. In the above example for a limit cycle, the rotating frame has an angular velocity of b. Within this rotating frame dynamics is one-dimensional with the relevant variable being r, which approaches the fixed point r * = √ . The dynamics on the limit cycle is given by the density matrix ρ limit cycle (t) := 1 2 The projection of the spin on the two basis states oscillates according to the equation

Three-dimensional case
We now look at the full system of equations (5) with no constraint forcing the dynamics onto a lower-dimensional manifold. A nonlinear three-dimensional dynamical system can exhibit the same types of bifurcations as the lowerdimensional systems, but in addition it can make the transition to chaos and show a strange attractor [8]. One of the simplest examples of a strange attractor is the Roessler attractor [8], which is an attractor of the following model: We show that the dynamics of equations (5) can lead to a strange attractor by explicitly writing down the ρ-dependent coefficient matrix h ij = h ij (ρ) that yields the dynamics (22). In addition to the three real-valued parameters a, b, and c of the Roessler system, we introduce a scaling factor M and a parameter > 0 that allows us to shift the variable x so that the attractor lies in the admissible region U . We assume that the Hamiltonian vanishes (H = 0) and specify the entries of the coefficient matrix h as By inserting this into equations (5), we obtain the following time evolution of the density matrix: Other strange attractors, such as the Lorenz attractor, can be obtained with similar methods. There is no particular reason why we chose the Roessler attractor apart from its simplicity.

Discussion and Conclusion
In this section we want to summarize our work and integrate it into a broader context. We started with the Lindblad equation, which describes the time evolution of a density matrix. When assuming that the time evolution is a one-parameter cntinuous semigroup (which is equivalent of saying that the time evolution for the density matrix is markovian), then the Lindbladian time evolution is the most general time evolution that is completely positive, trace preserving and linear in the density matrix. Whereas the first two requirements are needed for the mathematical structure of a density matrix, the last one has to do with the interpretation of density matrices. Every density matrix can be written as a convex combination (a linear superposition, where the coefficients are non negative numbers which add up to 1) of pure states. When interpreting the density matrix only as a device to reflect our limited knowledge of the system, one assumes implicitly that "in reality" the system is in a pure state (which represents the only possible reality). In this case, the time evolution must be linear since then the different pure states cannot interact with one another but represent together an ensemble similar to those used in statistical mechanics: However, while this interpretation of the density matrix applies to a variety of situations, it is not the only possible view one can take about density matrices. In most experimental situations, the preparation of a quantum mechanical state cannot be controlled in all detail. One reason for this is the nonzero temperature of the preparation device. In this situation, it is impossible even in principle to prepare a pure state that can be verified experimentally, for instance by quantum state tomography [10]. When considering this imprecision of a quantum state as fundamental, the density matrix does not only reflect our limited knowledge but is the best description of the system we can possibly give. This means that the density matrix now describes one system and not an ensemble of systems. Furthermore, when not just the preparation, but also the subsequent time evolution of the quantum system is subject to uncontrollable stochastic influences, this density matrix does not evolve according to the von Neumann equation but requires a Lindblad equation (or a non-Markovian equation, but we do not consider this situation here). Since now the density matrix represents only one system and not an ensemble, there are no more reasons why the Lindblad equation should be linear in the density matrix.
In the introduction, we mentioned two possible interpretations of nonlinear Lindblad equations, both of which shall be considered in the following in the light of the examples discussed in the previous section. The first interpretation is that of a mean-field equation for identical quantum systems that are embedded in a shared environment, which provides an interaction between the quantum systems and includes a heat bath that exerts a stochastic influence. Our model (5) is particularly suited to spin-1/2 systems as its Hilbert space is 2-dimensional. Indeed, the mean-field theory of ferromagnetism gives the same dynamical equation as our model (9): According to the Landau theory of phase transitions, the free energy of a uniaxial ferromagnet is with m being the order parameter (magnetisation) and with the parameters satisfying u > 0 and r ∝ (T − T C ). Relaxation dynamics toward equilibrium takes the form Our choice of the transition rates h 11 and h 22 between the two spin orientations is the simplest one that satisfies the symmetry between the two spin orientations and includes a cooperative effect. Of course, it is well known that the pitchfork bifurcation and the Landau theory of phase transitions describe the same type of phenomenon, namely the spontaneous breaking of a reflection symmetry.
Similarly, our example for a saddle-node bifurcation (11) is equivalent to the relaxation dynamics of the Landau theory in the presence of a magnetic field, which is given by F = rm 2 + um 4 − hm and The saddle-node bifurcation corresponds to the vanishing of the metastable state that has the higher free energy when the magnetic field becomes too large.
The transcritical bifurcation is not a plausible bifurcation for coupled spin-1/2 systems. It occurs in classical models for population dynamics with a linear growth rate and a nonlinear death rate. A population size of 0 is an absorbing state since no birth processes can occur when no individual is present. A quantum system that shall show such a behavior must have similar properties to such population models. One textbook example for a transcritical bifurcation in a two-level system is a simple laser model, where the medium is represented as a set of two-level systems. Usually, this model is written in terms of the photon population in the cavity (see for instance [8]), and in this notation the equivalence to a classical population model is obvious. In this model q(t) is the occupation probability of the excited level. The change of q with respect to time isq where P > 0 is the pumping rate, n(t) ≥ 0 the number of photons and G > 0 the "gain" factor. Since n(t) is proportional to the number of non-exited atoms, n(t) = n 0 (1 − q(t)), we obtaiṅ The transcritical bifurcation occurs when P = G n 0 . So for a small pumping rate (P ≤ G n 0 ), q * = 0 is an attractive fixed point, but for a large enough pumping rate (P > G n 0 ) the system goes to a stable state with a finite fraction of excited atoms.
Similar to the transcritical bifurcation, realizing a Hopf bifurcation requires an ongoing energy input into the system. If a system of coupled spins in a magnetic field is left to itself and its interaction with a heat bath, an equilibrium state with detailed balance must be reached, and this is not compatible with a sustained rotation of the magnetization of the coupled spin system. The thermal equilibrium state of model (17) would have a preferred spin orientation in the y direction, as imposed by the static field, and this means that it is inconsistent to presume that the y coordinate of the average spin orientation vanishes. There are essentially two ways for obtaining an ongoing rotation in the x-z plane: the most straightforward one is an external driving which imposes on the system a rotating preferred direction. In this case, not a magnetic field in y direction, but a rotating field in the x-z plane would be required, and this would make the model explicitly time dependent. The other way is an active feedback process that couples to the y component and drives it back to zero. Such a stabilization of an unstable fixed point can be done in a minimally invasive way [11]. Since equations (17) do not account for the y component, the dissipative processes that leads to a nonzero y component and the active driving that reduces this component are not made explicit in the model.
The discussion so far has led us to the conclusion that nondriven, mean-field like models of coupled spins can only give rise to simple one-dimensional bifurcations, namely the pitchfork and saddle-node bifurcation. On the other hand, driving the system in an appropriate way can lead more complex dynamics of the density matrix. In fact, it is in principle possible to obtain any imaginable dynamics of the density matrix that preserves its hermiticity, positivity, and unit trace. But of course not all of these time evolutions are simple or natural in a suitable sense. In order to discuss this further, we now proceed to a different interpretation of the density matrix. So far, we interpreted it as a mean-field description of a set of identical, coupled quantum systems in a shared environment. From now on, we interpret it as representing a single quantum two-level system in an environment that has a magnetic field and a temperature that can change with time. Interpreting the density matrix as describing a single system is also advocated by other authors [12]. If we write the two energy values of a spin-1/2 in a magnetic field of strength B = |B| as ± µ B and introduce the parameter β = 1/(k B T ), we obtain the following expression for the vector r in (3): By adjusting the direction and strength of B, any vector r can be realized. Consequently, by building a suitable environment that leads to a changing B (and possibly also T ), any time evolution of the density matrix ρ can be realized. The only requirement is that there is a time scale separation between a fast thermal flipping of the spin and a slow change of B and T so that the spin is always in thermal equilibrium with the present values of B and T . In such a situation, the density matrix is well defined at every moment in time, as is also argued by authors working in the field of NMR [13]. A system that leads to a chaotic equation of motion for the density matrix of the spin can be built in the following way: First, one constructs a electromechanical device that leads to a chaotic equation of motion in spherical coordinates r, θ, φ of the end point of an arm of variable length and orientation. Then one mounts a magnet on the end of this arm such that the field points inwards in radial direction, and one places the quantum spin system in the center at r = 0. The direction and strength of the magnetic field at the location of the spin are then fixed by the angle (θ, φ) and radius r respectively. This means that the equation of motion of the electromechanical device translates into an equation of motion of the magnetic field at the center, and this in turn translates into an equation of motion of the type (5) for the density matrix of the spin.
The last example has demonstrated most clearly that the density matrix is determined by the environment of the quantum system. In this example, the environment is given by a classical device that determines the magnetic field and the temperature to which the quantum system is exposed. There is no other way in which a quantum system can be controlled or influenced apart from changing classical variables, which then can cascades down to the quantum system. When the environment is simple, for instance a heat bath that does not change in time, this top-down determination of the density matrix does not impose itself so clearly. However, when the environment is not in equilibrium but undergoes a nontrivial time evolution, the density matrix that characterizes the quantum system also undergoes a nontrivial time evolution, which is in general nonlinear.
For the mean-field model of interacting quantum systems, the situation is somewhat different. Here, the other quantum systems act as part of the environment and mediate a feedback mechanism from the changes of one quantum system back to itself. However, this type of nonlinear dynamics is simpler as it is only transient, until the fixed point is reached.
In order to trigger a new burst of dynamical activity, again a change in the classical environment is required that then cascades down to the coupled quantum systems. According to various authors [14,15,16], the top-down influence from the classical world on the quantum world is an irreducible feature of nature that must be taken into account when one wants to find a solution to the puzzles surrounding the interpretation of quantum mechanics.