## Abstract

We find the conditions for one quantum system to function as a classical controller of another quantum system: the controller must be an open system and rapidly diagonalized in the diagonal basis of the controller variable that is coupled to the controlled system. This causes decoherence in the controlled system that can be made small if the rate of diagonalization is fast. We give a detailed example based on the quantum optomechanical control of a mechanical resonator. The resulting equations are structurally similar to recently proposed models for consistently combining quantum and classical stochastic dynamics.

## 1. Introduction

The results of measurements made on quantum systems are classical random variables extracted from classical instruments. A great deal of effort has been spent in trying to reconcile this fact with the apparent universality of quantum theory. In one approach, referred to as the decoherence programme [1], the open, dissipative nature of a measurement apparatus becomes central. An equivalent, although often ignored, problem of consistency arises when we ask for the conditions under which the control parameters in a quantum system can be regarded as entirely classical: why not simply include the control apparatus as part of the quantum system as well? In this study, it will be shown that the classical controllability of a quantum system also requires us to make explicit reference to the open and dissipative nature of control devices (e.g. lasers).

The question of specifying the conditions under which we can treat the control of quantum systems using classical signals has arisen in a variety of fields. Most recently, it has engendered some debate in the field of quantum information [2,3]. Quantum computing requires one to be able to implement rather complex time-dependent Hamiltonians for many interacting quantum systems (qubits). If any of these subsystems became entangled with the control apparatus, it would not be possible to perform unitary control, and errors would necessarily enter the quantum computation. For example, in ion trap quantum computing, we do not concern ourselves with the fundamental quantum nature of the controlling laser fields, or the radio frequency trapping fields, which are treated entirely classically. A very elegant discussion of this problem has been given by Carmichael and co-workers [4,5], who draw attention to the fact that lasers are open quantum systems of a very special kind and the conditions for classical controllability turn out to be very similar to the conditions for the classical stochastic nature of measurement results. In the emerging field of hybrid-engineered quantum systems, we will need to consider a greater variety of quantum controllers than simply a laser.

There is another control problem in which the control is intrinsically classical and cannot be treated as an appropriate limit of a quantum interaction; this is the case of control implemented by a gravitational field. This is related to the long-debated inconsistencies that arise when we combine classical and quantum dynamics [6–8]. One resolution has been proposed by Diosi [9]. It requires that we add noise to both the classical and the quantum dynamical systems so that the noise spectral densities in both cases are linked. Adding noise to a classical dynamical system can be quite harmless, but adding noise to a quantum system necessarily implies decoherence.

Wiseman & Warszawski [10,11] discuss a related measurement problem that requires one to combine classical and quantum stochastic dynamics. They consider realistic measurement scenarios in quantum optics in which the actual observed stochastic process is conditioned not only on the quantum nature of the source but also by additional classical noise processes occurring in the detection circuits. The resulting equations are structurally similar to those obtained in this study, as well as those postulated to provide a consistent hybrid classical and quantum dynamics. A similar motivation was used by Reginatto & Hall [12] in order to consistently combine classical and quantum stochastic processes. An interesting application of their model was recently given by Chua *et al.* [13]. A different kind of classical–quantum hybrid dynamics is discussed by Tsang [14], wherein the objective is to optimally estimate the classical, and possibly stochastic, controls on a quantum system by making continuous measurements on the quantum system. The starting point of this study is different from these models. Here, we seek the conditions under which two interacting quantum systems can be regarded not as a measurement scenario but rather as a classical control scenario.

A simple example will suffice to illustrate the relationship between classical controllability and the role of decoherence in measurement. Consider the case of applying a classical impulsive force to a free particle, that is to say, a force applied on a time scale much shorter than the time scale of the intrinsic dynamics of the particle. The change in the state of the system owing to the impulsive force is well described by the unitary transformation |*ψ*_{out}〉=*U*(*X*)|*ψ*_{in}〉, where
1.1where *X* is a classical control parameter equal to the change in momentum of the particle owing to the impulse and where are the quantum canonical position and momentum operators of the quantum system.

Now suppose the impulse was, in fact, due to a collision with another free particle—call it the *control particle* for ease of reference—with canonical position and momentum operators . In this case, we know that there will be a mutual exchange of momentum, and the change in the state through the collision may be described by the unitary operator
1.2Under what conditions are we justified in describing this interaction by the unitary transformation in equation (1.1)?

The answer of course depends on how we prepare the state of the control particle, |*Φ*〉_{c}, for it is easy to see that the change in the state of the target particle is given by
1.3where
1.4with *Φ*_{c}(*Q*)=〈*Q*|*Φ*〉_{c} being the position probability amplitude of the control particle immediately before the collision. In this form, we see that we may regard the transformation as a mixture of classical controls with a random control parameter distributed according to *P*(*X*).

As an example, consider the case for which |*Φ*_{c}(*Q*)|^{2} is a Gaussian sharply peaked at a particular value with variance *σ*. We first change the variable of integration to , where . We can then factor out an overall unitary transformation,
If we expand the unitary operators , we can integrate term by term. If , we can truncate this expansion to second order. The output state in equation (1.3) may be written as
1.5and . We see that provided *Γ*≪1, we can treat the control as entirely classical. The correction term adds noise to the momentum of the target particle, as one easily sees that
1.6

The mixed state in equation (1.3) begs an interpretation in terms of a measurement performed on the control. Suppose that, immediately after the impulsive interaction between the control and the target, the position of the control is measured with extreme accuracy. The resulting conditional state of the target system, given a result *Q* from the measurement, is given by projecting the output state onto |*Q*〉_{c} and tracing out the control. Thus,
1.7which indicates a classical control of the target subject to intrinsic quantum fluctuations of the control system position coordinate. In order to make this small, we need to ensure that *σ* is small, i.e. the position coordinate of the controller is well defined.

It is worth contrasting this model with how one would use this kind of impulsive interaction to make a measurement of the system position variable, . We now regard the control particle not as a controller but as a measurement apparatus, a meter. Clearly, there is no point in measuring the position of the controller immediately after the impulse as this does not change. Instead one must measure the canonically conjugate momentum coordinate . For this to be a good measurement, we need to prepare the state of the controller not with a well-defined position but rather with a well-defined momentum; exactly the opposite to what is required to use the system as a classical control. A quantum controller and a quantum meter are in this sense complementary devices. In the models we discuss later, this kind of complementarity is determined by what controller variable is coupled to its environment.

## 2. Quantum stochastic model

The impulsive model of §1 is a pump to our intuition; however, it is rather limited. In practice, we are more likely to be interested in a system evolving continuously in time subject to a classical controller. In this section, we show that we can still regard the classical controller as a quantum system subject to continuous measurement of the controller variable coupled to the system. This of course means we need to treat the quantum controller as an open quantum system. It is the nature of this interaction that enables us to describe the target system dynamics in terms of a purely classical stochastic control.

Consider two interacting quantum systems: a *controller* and a *target* (figure 1). The control system is also coupled to an environment in such a way that one particular degree of freedom in the environment may be monitored, providing a continuous weak measurement of the controller variable that is coupled to the target system. The Hamiltonian describing the dynamics is
2.1where is a (dimensionless) Hermitian operator for the control and is a (dimensionless) Hermitian operator on the system. We will assume that has a continuous spectrum. The Hamiltonians *H*_{e} and *H*_{ce} represent, respectively, the Hamiltonian for the environment coupled to the control and the interaction Hamiltonian for the control and its environment. We will assume that *H*_{ce} is simply linear in so that, in principle, the environment can be regarded as a measurement apparatus for the control variable . What we are interested in is the dynamics of the target system alone, given the interaction with the control system and given the entire measurement record of the control variable .

The theory of weak continuous measurement is now well established [15]. In the case of measurement of the control system observable , we use the theory described in Milburn [16]. There are two ways to view the measurement, which we might loosely refer to as the inside view and the outside view. The first of these simply gives the unconditional dynamics of the measured system when no account is taken of the measured results. The dynamics is given in terms of a master equation for the measured system density operator by averaging over the state of the environment. The inside view gives the stochastic conditional dynamics of the measured system conditioned on a particular measurement record. The conditional dynamics is given by a nonlinear stochastic master equation and is sometimes called a quantum trajectory. The reason the dynamics is nonlinear is to ensure that the future measurement records are conditioned on what was seen in the past.

We will take the unconditional dynamics to be given by
2.2where *Γ* is the *decoherence rate* and determines the rate of decay of the off diagonal matrix elements of *ρ* in the basis of the measured quantity. It simultaneously adds noise to the observable canonically conjugate to . The corresponding conditional dynamics is given by the Ito stochastic master equation,
2.3where
2.4and *dW*(*t*) is the Wiener stochastic process [17] and the superscript on the state serves to remind us that this state is conditional on the entire measurement record up to time *t*. The classical stochastic process corresponding to the observed measurement record is
2.5where is the conditional mean. The decoherence rate *Γ* has a dual role. It determines both the decoherence rate and the signal-to-noise ratio. Indeed, *Γ*≫1 implies rapid decoherence and a large signal-to-noise ratio, which we simply refer to as the good measurement limit. In what follows, we set *H*_{c}=0 so that the control variable has no dynamics other than what is implied by the continuous measurement made upon it. We will remove this assumption in the next section, when we consider simultaneous measurement of canonically conjugate observables on the control system.

The conditional stochastic master equation tends to drive the system to eigenstates of the measured quantity. To see this let us suppose for simplicity that the coupling to the target is turned off and that we ignore the free dynamics of the control (that is to say, we assume that it is very slow compared with the measurement dynamics). The unconditional dynamics of the controller is 2.6and the conditional dynamics is 2.7

In the good measurement limit, the unconditional state will be rapidly diagonalized in the eigenstates of . We can thus assume that the unconditional state for the controller, on time scales of interest, can be written as
2.8Substituting this into the unconditional master equation (2.6), we see that it implies d*ρ*_{c}(*t*)=0 and thus *P*(*X*,*t*)=*P*(*X*,0). On the other hand, substituting this into the conditional master equation (2.7), we find that the solution has the same form but that the *conditional* distribution *P*^{c}(*X*,*t*) obeys the classical conditional dynamics
2.9In this sense, the control system (in the good measurement limit) may be described as a classical stochastic process subject to continuous observation. Note that averaging over the noise, we have that *dP*(*X*,*t*)=0, which is consistent with the unconditional dynamics d*ρ*_{c}(*t*)=0 and *P*(*x*,*t*)=*P*(*x*,0). This simply says that the diagonal matrix elements of the controller, in the basis that diagonalizes the measured quantity, do not change. If we further assume that the initial state has a Gaussian Wigner function, the conditional dynamics will remain Gaussian and we can derive the following equations for the conditional mean and conditional variance of . Taking care with the Ito calculus, we find that the conditional mean and conditional variance satisfy
2.10and
2.11The conditional mean is stochastic, but the conditional variance is deterministic and decays exponentially to zero at the decoherence rate. In other words, for a good measurement, the conditional state will localize on a random eigenstate of the measured quantity.

Turning now to the unconditional dynamics including the target system, we make the ansatz that the solution is well approximated, on time scales of interest to the target dynamics, by
2.12We further assume that the controller dynamics is little changed from what it would be in the absence of the coupling to the target. This assumes that *Γ* is much larger than all the system frequencies. It also accords with our intuition for a classical controller, that is to say, a classical controller should not be affected by the system it is used to control. Taking the partial trace of this state over the target system gives the state of the control. Consistency then requires that we set
2.13We can then write the unconditional state as
2.14where
2.15The unconditional state of the target system alone is obtained by taking the partial trace of *ρ*(*t*) in equation (2.14)
2.16Substituting this into the unconditional master equation, we find that
2.17where
2.18To simplify this equation, we use *P*(*X*,*t*)=*P*(*X*,0) and approximate
2.19and
2.20where *X*_{0}, *σ*_{0} are the mean and variance of *P*(*X*,0), respectively. We now proceed iteratively in powers of *σ*. We first use the zeroth-order solution *ρ*_{s}(*t*)=*ρ*_{s}(*t*|*X*_{0}) to get
2.21which may be solved as
2.22which in turn gives the next order approximation,
2.23We can pause here to note an important feature already present in the simple model of the introduction. The double commutator in will cause decoherence in the position basis and drive fluctuations into the momentum of the controlled system. We get classical control, provided that we can make this term small enough over the time scales of interest. Provided that *σκ*^{2}, which has units of frequency, is much smaller than the inverse time scale of the system, we can disregard the decoherence and take the dynamics to be very closely approximated by the Hamiltonian with *X*_{0} a classical control parameter.

On the other hand, given that we have assumed a measurement of the controller, we do have access to the actual measurement record and can use this to describe the conditional dynamics of the target system. We now postulate that the conditional state of the total system is given by
2.24where *P*^{c}(*X*,*t*) satisfies the classical conditional equation (2.9). It is easier to interpret this equation if we write *P*^{c}(*X*,*t*) in functional form:
2.25where 𝒫^{c}(*X*_{t}) is a distribution over stochastic paths. We can write the conditional state of the combined control and target system as
2.26where, in order to ease notation, we have set *ρ*_{s}(*t*|*X*_{t})≡*ρ*(*X*_{t}). It is then a simple matter to see that
2.27where *X*_{t} is the observed classical control process in equation (2.9). In the example of a Gaussian process, this is given by equation (2.11). In the case of a very good measurement, *X*_{t} will settle down to a constant with vanishing fluctuations.

The key feature of this model is the rapid decoherence of the controller in the basis of the control variable acting on the system (). This occurs because the classical controller was assumed to be continuously monitored in this basis. In the language of decoherence, the eigenstates of are the *pointer basis* for the measurement. In general, the pointer basis is determined by how the controller is coupled to its environment. In order to make a quantum system behave as a classical controller, we must engineer it so that the required control variable that is to act on the system is the pointer variable for controller. In the simple example of this section, the system–controller interaction Hamiltonian commutes with the pointer basis variable of the controller.

It is interesting to contrast this result with the role of the pointer basis in a continuous measurement model [15]. In the quantum theory of continuous measurement, a quantum system can play the role of the first stage of a measuring instrument (let us call it the meter), provided that the meter variable that is coupled to the system does *not commute* with the pointer basis variable. In the example of this section, we could use the controller as a meter by choosing the interaction Hamiltonian to be of the form , where is canonically conjugate to the pointer basis variable . In this case, a continuous measurement of will give information on the system variable . A quantum system engineered to function as a classical controller can be regarded as *complementary* to a quantum system engineered to be a measurement apparatus. In both cases, a key role is played by the pointer basis of the controller/meter. In other words, it is how a quantum system is coupled to its environment that fixes its use as either a classical controller or as a measurement apparatus.

## 3. An optomechanical example

In this section, we give a less-idealized example for how to engineer a classical controller as an open quantum system. The example is taken from the field of quantum optomechanics in which the mechanical action of the electromagnetic field can be used to control or measure the motion of a mechanical resonator. In particular, we will consider the case of a single-mode optical cavity field interacting with a mechanical resonator comprising one mirror of the cavity [18].

The interaction between the field and the mechanics is known as the radiation pressure interaction, as it corresponds to a force applied to the mechanical element that is simply proportional to the photon number inside the cavity. The Hamiltonian is given by [18]
3.1where *a*,*a*^{†} are the annihilation and creation operators for the single-mode cavity field, are the canonical displacement and momentum operators for the mechanical resonator, *ω*_{c} is the frequency of the optical cavity mode, *ω*_{m} is the mechanical frequency of the moving mirror with effective mass *m* and *L* is the length of the optical cavity. It will be more convenient to write the displacement and momentum of the mechanical element in terms of the lowering (*b*) and raising operators (*b*^{†}):
3.2and
3.3Then,
3.4where
3.5is known as the optomechanical vacuum coupling rate. For most systems currently in operation, *G*_{0} varies from a few hertz [19] to fractions of a megahertz [20]. We will regard the cavity field as the controller, and the mechanical resonator as the system to be controlled.

At first sight, it would appear from the Hamiltonian in equation (3.4) that the classical control variable of interest is the photon number inside the cavity and the required pointer variable is the photon number. However, this is misleading, as we need first to consider how the cavity is coupled to its environment. The (approximate) pointer basis variable for a damped cavity mode is not the intracavity photon number but rather the amplitude of the field in the cavity [21]. We thus need to drive the cavity in such a way that a steady state field can build up. The small position-dependent detuning that results from the radiation pressure interaction is then turned into a small displacement of the cavity field amplitude that does define the pointer basis for the damped cavity.

The quantum theory of a single-damped cavity mode is well known [15,17] and is described by the master equation
3.6where
3.7and *κ* is the decay rate of the photon number inside the cavity. The pointer basis states for this master equation are the coherent states |*α*〉 defined by *a*|*α*〉=*α*|*α*〉 [15,21]. In the absence of any coherent driving of the cavity, the steady state is simply the vacuum state |0〉, which is a coherent state. More interesting is the case of a cavity driven by a coherent laser source with carrier frequency *ω*_{L}. The steady state of the cavity in this case is a coherent state with coherent amplitude *α*_{0} given by
3.8where Δ=*ω*_{c}−*ω*_{L} is the detuning of the driving laser from the cavity and units have been chosen so that |ℰ_{L}|^{2} are in units of photon flux (i.e. number per second). This result can be obtained by including in the master equation a Hamiltonian term of the form
3.9

At this point, we note that we are describing the laser by a classical control parameter and this might be regarded as question begging. It looks like we need to go back one stage to describe the laser itself as a classical control field. This reasoning will lead to an infinite regress very much like the infinite regress that results if we insist on regarding every stage of a measurement as a quantum system. We need to put a quantum–classical cut somewhere, and in the case of measurement, this is determined by how certain key degrees of freedom of the measurement apparatus are coupled to the environment. However, we do not face this problem here. As discussed in some detail by Noh & Carmichael [5], the description of laser driving used here is a good approximation, given the particular kind of open irreversible system that is a laser. Their explanation parallels the view taken in this study. We will not consider this further here as our objective is to determine under what conditions we can regard an optomechanical interaction as a classical controller for the mechanical resonator.

If we now include the driving Hamiltonian and the damping of the cavity mode, the dynamics of the optomechanical system, in an interaction picture with respect to the cavity field at frequency *ω*_{L}, is given by
3.10where
3.11and which now has units of frequency.

We now follow the approach of the previous section; we first consider the steady state of the controller itself and then use this to get approximate equations of motion, including fluctuations, for the target system under the assumption that the irreversible dynamics of the controller is much faster than the target system dynamics. We then consider the conditional dynamics, given that the state of the controller is continuously monitored. In the case of the optomechanical model, this will be performed via heterodyne detection, which is a means of monitoring the cavity field amplitude.

The steady state of the master equation for the controller, in the absence of the coupling to the target system, is a coherent state given by
3.12where *α*_{0} is given in equation (3.8). In the presence of the coupling to the mechanical system, we assume that the total unconditional state may be written as
3.13Taking the partial trace over the controller (i.e. the cavity) gives the unconditional state of the mechanical resonator as
3.14Taking the partial trace over the mechanical resonator gives the unconditional state of the controller as
3.15where
3.16The form of equation (3.15) implies that we are defining a Glauber–Sudarshan representation for the cavity state. We may need to admit some unusual distributions for this function (e.g. derivatives of delta-functions) in order to account for non-classical effects. Note that *ρ*_{b}(*t*,*α*) is not a normalized state of the mechanical resonator. We may define a normalized unconditional state for the mechanical resonator as
3.17and we have written the normalized state *as if* it were a conditional state conditioned on a random variable *α*, with distribution *P*(*t*,*α*). However, this assumes that *P*(*t*,*α*) is a regular probability density, which may not always be the case.

Substituting the assumption, equation (3.13), into the master equation, and using the correspondence rules for the P-representation [22], we find that
3.18If we take the trace of this equation over the mechanical resonator states, we find that
3.19where we have defined the *conditional* mean value
3.20The form of equation (3.19) is almost the classical Liouville for the controller density *P*(*t*,*α*), but the conditional mean in the last term gives this a more complicated dynamics, as this term needs to be found by also solving equation (3.18). The classical and quantum dynamical systems are tied together in a particular way. We shall return to this interpretation in §4 when we discuss the connection to work on hybrid quantum–classical dynamics.

We can now find the master equation for the unconditional state of the mechanical resonator, the controlled system, by integrating both sides of equation (3.18) over the complex variable *α*. This takes a rather simple form after integrating by parts,
3.21This again begs an obvious interpretation: it looks like the system is evolving with a random Hamiltonian. However, the random variable itself is *not* independent as it is tied to the system through the last term in equation (3.19).

In order to proceed, we need to make some assumptions about the rate of dissipation in the controller, i.e. the cavity field damping rate, *κ*. We now assume that the cavity is strongly damped so that *κ*≫Δ,*G*_{0}. Under this assumption, we can expand the total system state near the cavity vacuum state as
3.22where the operators *ρ*_{jk}(*t*,*α*) act only on the Hilbert space of the mechanical system. This can be written in the form given in equation (3.13) using known correspondences [22] for the P-representation of the raising and lowering operators on the cavity field. We find that
3.23The state of the mechanical resonator is then obtained by tracing out the control in equation (3.22) to get
3.24If we substitute the expression in the integrand on the right-hand side of equation (3.23) into equation (3.21) and integrate by parts, we find that
3.25

On the other hand, by substituting the approximation in equation (3.22) into the master equation, equation (3.10), and by equating coefficients, we obtain equations for d*ρ*_{jk}/d*t*. These show that the off-diagonal terms rapidly approach a steady state (i.e. ), and we can solve for the off-diagonal terms in terms of the diagonal terms,
3.26and
3.27This procedure is equivalent to the adiabatic elimination of the cavity field as described in Wiseman & Milburn [23]. By substituting these into equation (3.25), we find that to lowest order in 1/*κ*,
3.28where
3.29and . The target system is then found to behave like a classically driven harmonic oscillator with driving determined by replacing and a decoherence term in . This decoherence term tends to diagonalize the target system in the eigenbasis of and simultaneously adds noise to the conjugate variable . This is the same result that we obtained for the simple model of the previous section. Provided that *κ*≫*ω*_{m},(*G*_{0}*α*_{0})^{2}, we can neglect this decoherence and treat the control as entirely classical. We again see that because the control system, the cavity field, is an open system with a particular pointer basis (the coherent states), we can find a limit in which the control may be treated classically.

To complete the discussion, we turn now to the conditional dynamics that results when we continuously monitor the cavity field. A continuous measurement of the coherent field in a damped-driven cavity requires a knowledge of two real numbers and can be carried out by heterodyne detection [15]. In essence, this is a continuous simultaneous measurement of the two canonically conjugate quadrature phase operators of the cavity field. The resulting measurement record is described by two real stochastic processes, which may be combined to form a single complex stochastic process. The conditional master equation for a single damped cavity mode subject to heterodyne detection is [24]
3.30where the conditioning superoperator, ℋ, is given in equation (2.4), and the complex valued Wiener process and
3.31and
3.32The classical measurement record is given by the in-phase and quadrature-phase heterodyne currents, *J*_{x},*J*_{y}, which obey the classical stochastic (Ito) differential equations
3.33and
3.34which may be more conveniently written in terms of the complex stochastic current, *J*=(*J*_{x}+*iJ*_{y})/2, as
3.35Note that the superoperator ℋ gives zero when acting on coherent states—another indication that the coherent states are the pointer states for the damped cavity field.

Given that we do have access to the measurement results, we can use them to classically control the mechanical resonator. We now show that this is consistent with the irreversible, unconditional dynamics of the controlled system we have already described. With this in mind, we consider the stochastic Hamiltonian
3.36Over a small time step d*t*, the system then evolves as
3.37where
3.38

If we expand *dH*_{I}(*t*) out to second order in the stochastic increment, taking care to keep the Ito increment *dZ* *dZ**, and linearize around the steady-state value 〈*a*〉^{c}=*α*_{0}, we find the effective stochastic increment of the unitary evolution operator is
3.39Expanding this to second order in the stochastic increment and averaging over the noise, we find that the unconditional dynamics is given by the master equation in equation (3.28). This indicates that it is entirely consistent to describe the dynamics conditionally using a classical stochastic control Hamiltonian based on the appropriate measurement record and the unconditional dynamics that results when we treat the open-systems dynamics of the total system by adiabatically eliminating the controller. This is schematically illustrated in figure 1.

## 4. Discussion and conclusion

We have given two examples of interacting quantum systems in which one of the systems can be regarded as a classical controller for the other. The conditions necessary for this are, firstly, that the controller is open in such a way as to define a pointer basis for the controller operator that is coupled to the controlled system, and secondly that the open-system dynamics of the controller is such that the approach to the pointer basis is very rapid compared with the time scales of interest for the controlled system. In the optomechanical example, we saw that the key features are captured by two linked equations (3.18) and (3.19). One equation gives the unconditional dynamics of the controlled quantum system alone in terms of an equation for a state operator *ρ*(*t*,*α*) that depends on a classical phase space variable, *α*. The other equation gives the evolution for the classical distribution function *P*(*α*,*t*) that depends on a conditional mean of the controlled system variable occurring in the interaction 〈*b*+*b*^{†}〉^{(α)}. This structure parallels closely an approach to consistently combining quantum and classical dynamics developed by Diosi and others (see Diosi [9] and references therein).

In Diosi [9], a classical system with canonical variables (*q*,*p*) is interacting with a quantum system. The description begins by defining an operator *ρ*(*t*,*q*,*p*) acting on the quantum system Hilbert space such that the actual state of the quantum system is given by
4.1The operator *ρ*(*q*,*p*) is called the *hybrid state*. The corresponding classical Liouville distribution for the classical system is defined by
4.2A hybrid observable is then defined as an operator on the quantum state space as *A*(*q*,*p*) with average
4.3

This is all well and good but do the dynamics correspond to anything that would make physical sense? Diosi [9] shows that it does provided the quantum system undergoes decoherence in just the right way and the classical system suffers a corresponding noise contribution. To see how this works, we define a hybrid Hamiltonian , 4.4where the last term is a hybrid observable intended to describe an interaction between the quantum and classical subsystems. The dynamics is then given by the quantum Liouville equation for the quantum system with an additional decoherence term. The classical evolution is given by a classical Fokker–Planck equation that includes a classical drift term via a noise-averaged Aleksandrov bracket and an additional classical diffusion term. The quantum decoherence rate and the classical diffusion rate are related so as to ensure the overall dynamics is a positive stochastic process. In the optomechanical example, the positivity is ensured by the construction as both the classical noise and the quantum decoherence ultimately originate from a positive quantum evolution equation for the controller and the target system.

The hybrid dynamics defined by Diosi and others was motivated by the problem of reconciling classical gravitation with quantum dynamics. We have seen in this study that similar models arise when a quantum system is used as a classical controller for another target quantum system. The possibility of classical control emerges as a result of the interaction of the controller with its environment. This is very similar to the way in which a measurement device works: a classical stochastic output is possible only if the measurement apparatus is coupled to the environment in this right way (i.e. define a pointer basis). It would be interesting to see whether a similar possibility were open for gravity. Could the apparent classical nature of gravity be telling us something important about the nature of the underlying quantum theory? Perhaps the classical gravitational degrees of freedom are coupled to hidden environmental degrees of freedom in just the right way to make general relativity possible. If this were true, there would be an unavoidable decoherence in the dynamics of every quantum system moving in a gravitational field. The real question however is, how big is the effect?

## Acknowledgements

I gratefully acknowledge the support of the Australian Research Council Centre of Excellence for Engineered Quantum Systems.

## Footnotes

One contribution of 11 to a Theme Issue ‘Decoherence’.

- This journal is © 2012 The Royal Society