Quantum approach to coupling classical and quantum dynamics

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Quantum approach to coupling classical and quantum dynamics

DIÓSI, Lajos, GISIN, Nicolas, STRUNZ, Walter

Abstract

We present a consistent framework of coupled classical and quantum dynamics. Our result allows us to overcome severe limitations of previous phenomenological approaches, such as evolutions that do not preserve the positivity of quantum states or that allow one to activate quantum nonlocality for superluminal signaling. A “hybrid” quantum-classical density is introduced, and its evolution equation derived. The implications and applications of our result are numerous: it incorporates the back-reaction of quantum on classical variables, and it resolves fundamental problems encountered in standard mean-field theories, and remarkably, also in the quantum measurement process; i.e., the most controversial example of quantum-classical interaction is consistently described within our approach, leading to a theory of dynamical collapse.

DIÓSI, Lajos, GISIN, Nicolas, STRUNZ, Walter. Quantum approach to coupling classical and quantum dynamics. Physical Review. A , 2000, vol. 61, no. 2

DOI : 10.1103/PhysRevA.61.022108

Available at:

http://archive-ouverte.unige.ch/unige:37040

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Quantum approach to coupling classical and quantum dynamics

Lajos Dio´si,^1,2Nicolas Gisin,³ and Walter T. Strunz⁴

1Institute for Advanced Study, Wallotstraße 19, D-14193 Berlin, Germany

2Research Institute for Particle and Nuclear Physics, P.O.B. 49, 1525 Budapest 114, Hungary

3Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland

4Fachbereich Physik, Universita¨t GH Essen, 45117 Essen, Germany 共Received 23 February 1999; published 10 January 2000兲

We present a consistent framework of coupled classical and quantum dynamics. Our result allows us to overcome severe limitations of previous phenomenological approaches, such as evolutions that do not preserve the positivity of quantum states or that allow one to activate quantum nonlocality for superluminal signaling.

A ‘‘hybrid’’ quantum-classical density is introduced, and its evolution equation derived. The implications and applications of our result are numerous: it incorporates the back-reaction of quantum on classical variables, and it resolves fundamental problems encountered in standard mean-field theories, and remarkably, also in the quantum measurement process; i.e., the most controversial example of quantum-classical interaction is consis- tently described within our approach, leading to a theory of dynamical collapse.

PACS number共s兲: 03.65.Bz, 03.65.Sq, 42.50.Lc

Opinions vary about the coexistence of and interaction between classical and quantum systems. In orthodox quan- tum theory, classical macrosystems and quantized microsys- tems coexist; their interaction is described asymmetrically.

The influence of macrovariables upon microsystems is pre- cisely taken into account as external forces. The back- reaction of quantized microsystems upon classical macrosys- tems is largely ignored, except for detector variables which are typically sensitive to certain microvariables. The theory of this specific back-reaction, called measurement theory关1兴, predicts the statistics of the final states after the interaction.

However, the interpretation of quantized dynamics is exclu- sively based on this nondynamical model of back-reaction 共cf. collapse of the quantum state兲; without it we could not test the validity of quantized dynamics at all. Possibly, quan- tization extends to macrosystems, indeed the criteria of being macroscopic or microscopic are loosely if ever defined. Con- trary to quantized microvariables, quantized macrovariables may have significant back-reactions on generic classical macrovariables as well. This becomes apparent in the widely used mean-field approximation关2兴which, however, has sev- eral fundamental drawbacks 关3兴, as we shall recall in this paper. The measurement theory also describes back-reaction, but only for idealized detectors. In some attempts to define quantum gravity, matter and some fields are quantized, while other fields 共gravity in particular兲 are treated as classical, thus requiring a definite hybrid—i.e., a coupled classical and quantum—dynamics. Thus a general model of the back- reaction is desirable. Such a theory would describe the ‘‘col- lapse’’ of the wave function dynamically关4兴, would replace mean-field approximations in a systematic way 关5–7兴, and could have deep implications for quantum cosmology. The first conceptual attempts 关4–6兴 were followed by ups and downs关8兴, until severe limitations were clarified关9兴. In this paper we use a straightforward transformation of the problem which automatically leads to a consistent model of hybrid dynamics.

The difficulties to overcome in our paper are unrelated to the high complexity of the emblematic mean-field equation

of quantum cosmology. Actually, they lie in foundational principles. To illustrate these difficulties, we assume a quan- tum Pauli spin␴ˆ interacting with a classical harmonic oscil- lator of Hamiltonian H_C(x, p)⫽¹2( p²⫹x²). The spin inter- acts with the ‘‘magnetic’’ field of the oscillator via the Hamiltonian

Hˆ_I⫽␬␴^ˆ3p. 共1兲

Regarding Hˆ⫽H_C⫹Hˆ

I as the total Hamiltonian, it is natural to prescribe the Heisenberg equation of motion ⳵t␴^ˆ

⫽i关Hˆ ,␴^ˆ兴⫽i␬^p关␴^ˆ3,␴^ˆ兴 to the spin. The classical oscillator momentum satisfies the Hamilton equation ⳵tp⫽⫺⳵xHˆ

⫽⫺x, but the coordinate cannot satisfy ⳵tx⫽⳵pHˆ⫽p

⫹␬␴^ˆ3since x, being a real number, should not evolve into a matrix. The obvious way out is to replace the operator␴^ˆ3 by its quantum expectation value, i.e., to apply some mean-field approximation. Yet, if taken literally, this implies that quan- tum expectations can be deduced with arbitrary precision from a measurement of the classical variables x and p. Hence the message is that the classical oscillator should, in some way, inherit quantum fluctuations from the spin. It comes to one’s mind that x and p should be random variables, but not arbitrary ones. As we shall demonstrate, mathematical con- sistency imposes that the classical variables x or p must never take sharp values: The consistent theory assumes an unremovable coarse-graining 关10,11兴.

The mathematical issue is the following. The phase-space density␳C(x, p) of a classical canonical systemCsatisfies the Liouville equation of motion

⳵t␳_C共x, p兲⫽兵^HC共x, p兲,␳_C共x, p兲其P, 共2兲 where H_C(x, p) is the Hamilton function and 兵^{f ,g}其^P

⫽⳵xf⳵pg⫺⳵pf⳵xg stand for the Poisson bracket. On the other hand, the density operator␳^ˆ_Q^{of a}共canonical, or maybe discrete兲 quantum system Q evolves according to the von Neumann equation

⳵t␳^ˆ_Q⫽⫺i关Hˆ

Q,␳^ˆ_Q兴, 共3兲 where Hˆ

Qis the Hamilton operator. To introduce interaction between Q and C, we assume a hybrid Hamiltonian in the form

Hˆ共x, p兲⫽Hˆ

Q⫹H_C共x, p兲⫹Hˆ

I共x, p兲, 共4兲

where the interaction term 共and thus the total Hamiltonian too兲is a Hermitian operator forQ, depending on the phase- space coordinates ofC.

The mean-field approach assumes sharp classical coordi- nates x_t, and p_t at each time, and the current quantum ex- pectation value H_{M F}(x, p;t)⫽tr关Hˆ (x,p)␳^ˆ_Q^(t)兴 of the Hamiltonian is regarded as the effective Hamilton function for the classical subsystem C. The coupled evolution equa- tions then take this forms:

⳵t␳^ˆQ共t兲⫽⫺i关Hˆ共x_t, p_t兲,␳^ˆQ共t兲兴, 共5兲

⳵tx_t⫽⳵pH_{M F}共x_t, p_t兲, ⳵tp_t⫽⫺⳵xH_{M F}共x_t, p_t兲. 共6兲 This approach has well-known deficiencies. In particular, it gives no account of the indeterminacy of the classical states x and p inherited from the quantum uncertainties of ␳^ˆQ. There is thus an essential nonlinearity in the mean-field von Neumann equation共5兲which leads to fundamental conflicts with principles of locality 关12兴. Furthermore, the mean-field effective Hamilton function H_{M F}(x, p) will never be the proper representative of the interaction when quantum uncer- tainties in␳^ˆ_Q^{are large.}

A promising conceptual approach关5兴uses the hybrid den- sity␳ˆ (x,p) to represent the state of the composite system C

⫻Q. If the subsystems are uncorrelated, then the hybrid den- sity simply factorizes as␳_C^{(x, p)}␳^ˆ_Q. In the general case, the hybrid density ␳ˆ (x,p) should be an (x,p)-dependent non- negative operator, satisfying an overall normalization condi- tion tr兰␳ˆ (x,p)dxdp⫽1. One interprets the marginal distri- bution ␳_C^{(x, p)}⬅tr␳ˆ (x,p) as the phase-space density of the classical subsystem C, while the density operator ␳^ˆQ

⬅兰␳ˆ (x,p)dxdp represents the unconditional state of the quantum subsystemQ. Conditional quantum states are natu- ral to introduce at fixed canonical coordinates (x, p) of the classical subsystem关10兴:

␳^ˆx p⬅␳^ˆ共x, p兲/␳C共x, p兲. 共7兲 Aleksandrov 关5兴 proposed the following evolution equation for the hybrid density:

⳵t␳^ˆ共x, p兲⫽⫺i关Hˆ共x, p兲,␳^ˆ共x, p兲兴⫹1

2兵^Hˆ共x, p兲,␳^ˆ共x, p兲其^P

⫺1

2兵␳^ˆ共x, p兲,Hˆ共x, p兲其P. 共8兲 If Hˆ

I(x, p)⫽0, then this equation splits into the standard equations 共2兲and共3兲.

Let us test Aleksandrov’s equation on the spin-oscillator system共1兲. The generic form of the hybrid state is

␳^ˆ共x, p兲⫽¹2关1⫹sជ_共x, p兲␴ជ^ˆ_兴␳_C共x, p兲, 兩sជ_兩⭐1, 共9兲 where sជ(x, p) is the spin vector correlated with the oscilla- tor’s state. We read off the conditional quantum state 共7兲of the spin:␳^ˆx p⫽¹2关1⫹sជ(x, p)␴ជ^ˆ_兴. For the hybrid state共9兲and interaction Hamiltonian 共1兲, Aleksandrov’s equation 共8兲 reads

⳵t␳^ˆ⫽共x⳵p⫺p⳵x兲␳^ˆ⫺i␬^p关␴^ˆ3,␳^ˆ兴⫺¹2␬关␴^ˆ3,⳵x␳^ˆ兴_⫹. 共10兲 This equation easily violates the positivity condition 兩sជ_兩⭐1 on␳ˆ . That is, the initial normalized polarization vector

sជ_⫽ ¹

x²⫹p²⫹1共2x,2p,x²⫹p²⫺1兲, 共11兲 leads to 兩sជ_兩⬎1 for all x⬎0 under the evolution关Eq. 共10兲兴. So, the naive Eq. 共8兲is inconsistent since it does not guar- antee the positivity of the hybrid density ␳^{ˆ (x,p)} 关6兴. One sees that the mathematical textures of the classical (C) and quantum (Q) systems, though well understood separately, are not at all trivial to couple.

A royal road offers itself nonetheless. Let us quantize canonicallyCas well. We do so temporarily and, at the end of the day, we regard it classical again. The hybrid Hamil- tonian 共4兲transforms into the total Hamilton operator of the fully quantized systemC丢Q

:Hˆ共xˆ, pˆ兲ªHˆ_Q⫹:H_C共xˆ, pˆ兲:⫹:Hˆ_I共xˆ, pˆ兲:, 共12兲 where : . . . : stand for normal ordering in terms of the usual annihilation and creation operators (xˆ⫾i pˆ)/

冑

tively. Let the equation of motion for the total system’s den- sity operator ␳ˆ be the standard von Neumann one:

⳵t␳^ˆ⫽⫺i关:Hˆ共xˆ, pˆ兲:,␳^ˆ兴. 共13兲 Our royal road is based on coherent states 关13兴. Coherent states 兩x, p典 are eigenstates of the annihilation operator:

共xˆ⫹i pˆ兲兩x, p典⫽共x⫹i p兲兩x, p典^. 共14兲 Using Bargmann’s convention 关13兴, the coherent states sat- isfy the following differential relation:

共⳵x⫺i⳵p兲兩x, p典^{⫽共xˆ⫺i pˆ}兲兩x, p典^{, 共15兲} 共⳵x⫹i⳵p兲兩x, p典^{⫽0. 共16兲} The latter relation expresses the fact that the bras兩x, p典 ^are entire analytic functions of the complex canonical variable x⫹i p, a crucial fact as we shall see. The coherent states form an overcomplete basis, with normalization

LAJOS DIO´ SI, NICOLAS GISIN, AND WALTER T. STRUNZ PHYSICAL REVIEW A 61 022108

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I⫽

冕

: f共xˆ, pˆ兲:兩x, p典^{⫽: f}

冉

冊

共18兲 for an arbitrary normal ordered function of the quantized variables on the left-hand side. On the right-hand side, the same symbols : . . . : mean that all derivations must be done first.

We apply a projection to the density operator ␳^{ˆ of the} fully quantized systemC丢Q, and thus reintroduce the hybrid density

␳^ˆ共x, p兲⬅tr关共兩x, p典具^{x, p}兩丢I兲␳^ˆ兴exp关⫺¹2共x²⫹p²兲兴 2␲ ^{. 共19兲} Indeed, this can formally be considered the hybrid density of the composite system C⫻Q as ifC were unquantized 共i.e., classical兲 again. This is what we are going to do. We can thus derive the closed equation of motion of the hybrid den- sity共19兲from the exact von Neumann equation共13兲. Using the basic relation 共18兲 and the identity 共16兲, we obtain the desired evolution equation

⳵t␳^ˆ共x, p兲⫽⫺i:Hˆ

冉

冊

共20兲 This hybrid dynamic equation is our proposal to couple clas- sical systems to quantum ones canonically. Note that the hybrid density␳ˆ (x,p) incorporates our statistical knowledge of Q’s quantum state, of C’s classical state, and of their correlations. If Hˆ

I(x, p)⫽0, then by integrating both sides of Eq. 共20兲over x and p we obtain the standard von Neumann equation共3兲, as it should be. The trace of both sides, how- ever, does not lead to the standard classical dynamics 关Eq.

共2兲兴. Instead, we obtain共for obvious reasons兲 the evolution equation of a Husimi function 关14兴. To lowest order in the derivatives, however, this is the classical Liouville equation 共2兲. Hence classical dynamics is recovered if both the Hamil- ton function and the state distribution change slowly with x and p共see, e.g., Ref.关15兴and references therein兲.

Consistency of the hybrid equation of motion共20兲is, con- trary to the case of the naive Eq.共8兲, assured by construction.

It preserves the positivity of the hybrid density␳ˆ (x,p) along with a certain analyticity property. In fact, projection 共19兲 always leads to hybrid densities of the form

␳^ˆ共x, p兲⫽exp关⫺¹2共x²⫹p²兲兴

2␲

兺

共21兲 where␸n(x⫺i p) are unnormalized nonorthogonal state vec- tors for Q, being complex entire functions of the combina-

tions x⫺i p of C’s classical variables. This positive form is then preserved by our hybrid equation 共20兲. The analyticity condition共21兲restricts the possible hybrid states: sharp val- ues of x and p and, generally, characteristic phase-space de- pendences inside single Planck cells, are excluded.

In particular, the ‘‘pure state’’ form of Eq.共21兲, i.e. with a single ␸␸^† dyadic term, is also preserved. The hybrid Schro¨dinger equation of the hybrid state vector ␸^(x⫺i p) follows from Eq. 共20兲 关16兴. For completeness, we mention that the polarization 共11兲 can be reproduced by ␸^(x⫺i p)

⫽关(x⫺i p)兩⫹典^⫹兩⫺典^]/

冑

A post-mean-field approximation, incorporating some of the fluctuations of system Q into its back-reaction on the systemC, is worth deriving. Consider the first-order expan- sion of Eq.共20兲in the derivatives of Hˆ (x,p):

⳵t␳^ˆ共x, p兲⫽⫺i关Hˆ共x, p兲,␳^ˆ共x, p兲兴⫹1

2兵^Hˆ共x, p兲,␳^ˆ共x, p兲其P

⫺1

2兵␳^{ˆ共x, p兲,Hˆ共x, p兲}其^P

⫺i

2关⳵xHˆ共x, p兲,⳵x␳^ˆ共x, p兲兴

⫺i

2关⳵pHˆ共x, p兲,⳵p␳^ˆ共x, p兲兴. 共22兲 This equation has two additional terms with respect to共math- ematically inconsistent兲Eq.共8兲. These two additional terms reduce the domain of inconsistency. More importantly, it can be shown that the above equation is equivalent to the exact Eq.共20兲ifCis harmonic and its coupling toQis linear in x and p. In particular, Eq.共22兲gives a mathematically consis- tent theory of fully quantized atomic systems (Q) interacting with the fully developed classical radiation field (C). More- over, its physics is equivalent to the true fully quantized radiation theory 关11兴.

By taking the trace of Eq. 共22兲 one can show that the evolution of the classical states is a flow:

⳵tx⫽具⳵pHˆ共x, p兲典x p, ⳵tp⫽⫺具⳵xHˆ共x, p兲典x p, 共23兲

where 具•••典^{x p} stands for the expectation values in the cur- rent conditional quantum state␳^ˆx p. Obviously this flow re- sembles locally the naive mean-field equation 共6兲. Here, however, the classical state is randomly distributed: it inher- its the quantum fluctuations of Q. On the other hand, the deterministic evolution of the state’s distribution is a remark- able fact with significant consequences being discussed else- where.

Dynamical collapse of the quantum state is encoded in our hybrid evolution equation 共20兲. It is a most spectacular feature. The ‘‘presence’’ of standard collapse will be dem- onstrated on the spin-oscillator model. For the hybrid state 关Eq. 共9兲兴, our evolution equation共20兲reads

⳵t␳^ˆ⫽共x⳵p⫺p⳵x兲␳^ˆ⫺i␬^p关␴^ˆ3,␳^ˆ兴

⫺1

2␬关␴^ˆ3,⳵x␳^ˆ兴⫹⫺ i

2␬关␴^ˆ3,⳵p␳^ˆ兴, 共24兲 which differs from the naive Eq.共10兲by the presence of the fourth term on the right-hand side. This term guarantees the positivity of the hybrid state␳ˆ (x,p) for all times. The oscil- lator plays the role of the Stern-Gerlach apparatus detecting the quantized spin-component ␴^ˆ3. The pointer variable of the apparatus is x, it is set initially to zero with precision⌬

⫽1. Accordingly, we choose ␳_C^{0(x, p)}⫽(2␲^)⫺1exp„⫺¹2(x²

⫹p²)…for the oscillator’s initial state. We shall switch on the interaction Hamiltonian at t⫽0 for a short time␶ ^compared to the oscillator period, keeping the effective coupling as strong as g⫽␬␶Ⰷ1. Actually, we replace ␬ ^{by g}␦^(t) 关1兴. Let us assume that the spin’s initial state is the superposition 兩␺典^⫽c⫹兩⫹典^⫹c⫺兩⫺典, and the initial hybrid state 共9兲 is the uncorrelated 兩␺典具␺兩␳_C⁰(x, p). Immediately after the interac- tion the hybrid state␳ˆ (x,p) becomes

兩c_⫹兩²兩⫹典具⫹兩␳_C⁰共x⫹g, p兲⫹兩c_⫺兩²兩⫺典具⫺兩␳_C⁰共x⫺g, p兲

⫹c_⫹c_⫺^쐓兩⫹典具^⫺兩exp

冉

冊

The trace yields the pointer’s state distribution

␳C共x, p兲⫽兩c_⫹兩²␳_C⁰共x⫺g, p兲⫹兩c_⫺兩²␳_C⁰共x⫹g, p兲. 共26兲 Since gⰇ1, we shall ignore the overlap between the two terms on the right-hand side, so we can say that the pointer x has swung out to g⫾⌬or to⫺(g⫾⌬) with the probabilities predicted by the standard measurement theory. Invoking Eq.

共7兲, we can easily read out the conditional quantum state of the spin from Eq. 共25兲 关since the off-diagonal terms are damped by exp(⫺¹2g²), we ignore them兴:

␳^ˆx p⫽兩⫾典具^{⫾兩, x⬇⫾g, p⬇0. 共27兲}

This shows the standard collapse of the spin’s quantum state:

the quantum state is correlated with the classical pointer’s position.

Discussing this paper’s results, we repeat that we are aware of the ambiguous contemporary views concerning the concept of genuine hybrid systems. Nevertheless, the old Copenhagen interpretation as well as recent quantum gravity and quantum cosmological models assume such hybrid sys- tems. We made the necessary compromises to neutralize strict no-go theorems. Our Eq.共20兲is an example of hybrid dynamics which is both mathematically consistent and physi- cally relevant. The applications of our equation are numer- ous, for example as phenomenological models whenever the mean-field approximation is poor. Moreover, we derive post- mean-field equations which describe the back-reaction of quantum fluctuations to first order. On the foundational level, we point out that our hybrid dynamics关Eq.共20兲兴reproduces the ideal quantum measurement, including the collapse of the wave function and the motion of the classical pointer. Let us also stress the close connection to current phenomenological theories of dynamic collapse which follow from our hybrid dynamics 关16兴. Our hybrid theory is likely to be an integrat- ing concept for treating quantum measurement dynamically and to overcome the inconsistent mean-field method in quan- tum cosmology.

We are grateful to Jonathan Halliwell, Ian C. Percival, and Ting Yu for useful conversations. L.D. was partly sup- ported by EPSRC and by OTKA T016047, N.G. by the Swiss National Science Foundation, and W.T.S. by the Deut- sche Forschungsgemeinschaft through the SFB 237 ‘‘Unor- dnung und große Fluktuationen.’’

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