Decoherence and Bohmian Mechanics · Decoherence and Bohmian Mechanics ... D.Dürr,S.Teufel(2009): Bohmian Mechanics. The Physics and Mathematics of Quantum Theory,Springer,Berlin

Bohmian Mechanics Decoherence Scattering

Decoherence and Bohmian Mechanics(Towards a Bohmian classical limit in bounded regions)

Davide Romano

University of Lausanne

18th June 2014, Rome


Summary of the talk

(1) Measurement theory in Bohmian Mechanics

(2) Classical limit in bounded regions

(3) Decoherence induced by scattering

(4) Conclusions


What is Bohmian Mechanics

Bohmian Mechanics is a quantum theory in which the completestate of a system is described by the couple (Q,Ψ), where:

Q is the actual configuration of particles of the systemΨ is the wave function of the system

Dynamical evolution for a N-particle Bohmian system:

Schrödinger equation for the time evolution of Ψ

(1) i~∂Ψ

∂t= HΨ, with H = −

N∑k=1

~2

2m∇2 + V

Guidance equation for the time evolution of Q=(q1,...,qN)

(2)dQdt

=~m

ImΨ∗∇Ψ

Ψ∗Ψ(Q)


What is Bohmian Mechanics

There is a precise mathematical relation between Ψ and Q:

given a system with wave function Ψ, the statisticaldistribution of the actual configuration is ρ (q) = |Ψ (q) |2(Quantum Equilibrium Hypothesis)

Moreover, from equations (1) + (2), it follows that:

if ρ (q, 0) = |ψ (q, 0) |2, then ρ (q, t) = |Ψ (q, t) |2 for all times(Equivariance of ρ)

QEH + Equivariance =⇒ same empirical predictions of StandardQuantum Mechanics.


Measurement Theory in BM

Measurement-type interaction between a system Ψ (x) and anapparatus Φ (y):

X= actual configuration of the systemY= actual configuration of the pointer-stateΨ (x) = αψ1 (x) + βψ2 (x), with |α|2 + |β|2 = 1

We introduce the notion of support of a wave function:A support of a function is the domain on which it is not equalto zero (support of WF ⇐⇒ configuration space)

In particular, if m = degrees of freedom of the system and n =degrees of freedom of the apparatus:

supp(Ψ (x)) lives in the m-dimensional configuration space ofthe systemsupp(Φ (y)) lives in the n-dimensional configuration space ofthe apparatus



At time t = t0, Ψ and Φ do not interact (the total WF isfactorized)

Ψ (x , t0) Φ (y , t0) = (αψ1 (x , t0) + βψ2 (x , t0)) Φ (y , t0)

The interaction follows the linear Schrödinger evolution for a lapseof time T (4tint = (0,T ))

The apparatus becomes entangled with the the system: pointerstates and eigenfunctions are now correlated

Ψ (x , t0) Φ (y , t0)→ αψ1 (x ,T )φ1 (y ,T ) + βψ2 (x ,T )φ2 (y ,T )

In the context of Standard Quantum Mechanics, this physicalprocess originates the measurement problem.

How does Bohmian Mechanics solve it?



For BM, the latter equation does not describe all the story: beyondthe wave function, a Bohmian system is always composed of anactual configuration of particles.

The final result of the pointer will depend exactly on the state ofthe actual configuration of the pointer (Y):

if Y ∈ supp(φ1), then the pointer points to the result “1”if Y ∈ supp(φ2), then the pointer points to the result “2”

Since the two supports are macroscopically disjoint, i.e. supp(φ1) ∩supp(φ2) ∼= 0 , the final result is unique and the superpositiondisappears.



We suppose that, after the interaction, Y ∈ supp(φ1), then X ∈supp(ψ1) and we will say that:

ψ1 is the effective wave function of the system (effectivecollapse), namely, the wave function which contains and guidesthe particleψ2 is the empty wave function, and can FAPP be ignored afterthe interactionFrom QEH (and Equivariance) + disjointness of supports =⇒Correct probabilities for the measurement outcomes (effectiveBorn Rule)

NB: The support of a wave function is typically unbounded, so that“disjointness of supports” means, more precisely, that the overlapbetween the wave functions has to be negligible over anymacroscopic region



A pictorial representation of the pointer-states in physical spaceand their supports in configuration space (Dürr, Teufel 2009)



A pictorial representation of the measurement process in BohmianMechanics (Dürr, Teufel 2009)


Classical limit in bounded regions

Classical limit in Bohmian Mechanics: under which conditions dothe Bohmian trajectories become Newtonian?

Serious issue: formation of caustics in bounded regions (Allori,Dürr, Goldstein, Zanghi’ 2002, section 8):

We consider an infinite potential well of size L in onedimension and a well-localised ψ in the center of the wellWe suppose that ψ is a superposition of two wave packetswith definite momenta p and -pIf λ (ψ)� L, then the wave packets move classically in bothdirections (Allori, Dürr, Goldstein, Zanghi’ 2002, sections 3-6)At the time tr , they reach the walls and are reflected fromthemAt the time tc = 2tr (first “caustic time”), the two packetsinterfere in the middle of the well and classical motiondisappears


Classical limit in bounded regions

Solution proposed by ADGZ: when we take the environmentinto account, the problem is solvedThe system particle will interact with an external particle (aneutrino, a photon, an air molecule,...) before the time tcThis interaction is a measurement-like interaction on theposition of the system: it will select only one of the twobranches, which will continue to move classically


Classical limit and Decoherence

“These interactions [...] should produce entanglement between thecenter of mass x of the system and the other degrees of freedom y ,so that their effective role is that of “measuring” the position Xand suppressing superpositions of spatially separated wave functions(taking these interactions into account is what people nowadayscall decoherence)”. (Allori, Dürr, Goldstein, Zanghi’ 2002, section8, p. 12)


Classical limit and Decoherence

My aim is to discuss whether or not this position is completelytenable:

first, we shall analyze a specific model of decoherence inducedby scattering (Joos and Zeh 1985, Joos et alii 2002,Schlosshauer 2007)then, we may consider if this model could be successfullyapplied to solve the issue of the classical limit in boundedregions


Decoherence

The theory of decoherence is the study of spontaneous interactionsbetween a system and its environment.

Main results of decoherence:

(1) Interference terms between different “branches” of a quantumsuperposition are locally destroyed

(2) The interaction selects “classical” states, those which are stableunder the influence of the environment (einselection)


Density matrix

The density matrix of a quantum system is a mathematical toolwhich permits to calculate the correct probabilities for themeasurement outcomes:

〈A〉 = Tr (ρA), where A is an hermitian operator and ρ is thedensity matrix, or statistical operator, of the system.

Given a quantum system

|ψ〉 = α |ψ1〉+ β |ψ2〉

We easily obtain the density matrix for the system

ρψ = |ψ〉〈ψ| =|α|2 |ψ1〉〈ψ1|+ αβ∗ |ψ1〉〈ψ2|+ α∗β |ψ2〉〈ψ1|+ |β|2 |ψ2〉〈ψ2|

More precisely, this formula represents the density operator. Thedensity matrix refers to the matrix representation of the densityoperator in a particular basis.


Density matrix

Now, we consider a system |ψ〉 + its environment |E 〉: when theyare physically independent, the global wave function is factorized

|Ψ〉 = |ψ〉 |E 〉=(α |ψ1〉+ β |ψ2〉) |E0〉

And the density matrix of the global system is

ρΨ = |Ψ〉〈Ψ| = |ψ〉 |E 〉〈E | 〈ψ| = ρψ ⊗ ρE


Reduced density matrix

When the system interacts with the environment, the two systemsbecome entangled.

New quantum state: entangled state between system andenvironment

|Ψ〉 = α |ψ1〉 |E1〉+ β |ψ2〉 |E2〉

Tracing out the degrees of freedom of the environment, we obtainthe reduced density matrix for the system

ρψred = TrE |Ψ〉〈Ψ| ∼= |α|2 |ψ1〉〈ψ1|+ |β|2 |ψ2〉〈ψ2| , if 〈Ei |Ej〉 ∼= δij

The reduced density matrix formally appears as a mixture of states.


Reduced density matrix

Nevertheless, ρψred does not represent a proper mixture of states,but an improper one:

it is impossible to assign an individual quantum state to asubsystem of a larger entangled system (Schlosshauer 2009)

ρψred only describes the statistical distribution of measurementsoutcomes for an observer who locally performs a measure onthe systemDecoherence does not select one particular branch of thesuperposition (all the different branches of the superpositionremain real!) =⇒ it does not solve the measurement problem


Einselection

During the interaction between the system and the environment,the relevant dynamics is encoded by the interaction Hamiltonianoperator:

Hint = |n〉〈n|+ An , where |n〉 is an eigenstate of the “observable”measured by the interaction, and An is an hermitian operator whichacts only in the Hilbert space of |E 〉.

States “einselected” by the environment:Eigenstates of Hint

Eigenstates of B , where[B, Hint

]= 0 (the observable B is a

constant of motion under the dynamical interaction with theenvironment)


Scattering

We consider a specific model of decoherence, namely, decoherenceinduced by scattering :

The system S scatters a collection of environmental particles(environment E )At initial time t = 0, S and E are uncorrelated:ρSE (0) = ρS (0)⊗ ρE (0)

(1) We analyze the case of a single scattering event, in order to seeits influence on ρS

(2) We analyze the time evolution of ρS under the influence ofmany scattering events


Scattering

Initial state of the center of mass of the system: |x〉Initial state of the incoming environmental particle: |χi 〉The effect of the scattering event is encoded by the action ofthe scattering operator S :

|x〉 |χi 〉 → S |x〉 |χi 〉 = |x〉 Sx |χi 〉 ≡ |x〉 |χ (x)〉

where |χ (x)〉is the final state of the particle scattered at x


Scattering

A pictorial image of a single scattering event (Schlosshauer 2009)

Scattering process = measurement-like interaction whichestablishes correlations between |x〉 and |χ (x)〉

|χ (x)〉 = pointer-states which encode information about theposition x of the scattering center


Decoherence induced by scattering

The scattering process transforms the initial density operator of thecomposite system:

ρSE (0) = ρS (0)⊗ρE (0) =´

dx´

dx ′ρS (x , x ′, 0) |x〉〈x ′| ⊗|χi 〉〈χi |

into the new density operator:

ρSE =´

dx´

dx ′ρS (x , x ′, 0) |x〉〈x ′| ⊗ |χ (x)〉〈χ (x ′)|


Decoherence induced by scattering

The final reduced density operator of the system after thescattering event will be:

ρS = TrE ρSE =´

dx´

dx ′ρ (x , x ′, 0) |x〉〈x | 〈χ (x ′) |χ (x)〉

The evolution of the reduced density matrix of the system inducedby scattering will be:

ρS (x , x ′, 0)→ ρS (x , x ′, 0) 〈χ (x ′) |χ (x)〉

The local suppression of spatial coherence is quantified by theoverlap of the relative final states of the scattered particle


Orthogonality of states

We note that the overlap 〈χ (x ′) |χ (x)〉 is nothing but a rephrase inthe specific scattering-model of the more general term 〈Ei |Ej〉.

Some remarks about ortoghonality of states...:in the general framework, the condition 〈Ei |Ej〉 ∼= δij is usuallyassumedin the scattering framework, the condition 〈χ (x ′) |χ (x)〉 ∼= δijis not assumed in a strong formThe ortoghonality of the environmental pointer-states isrelated to the width of the system 4x =|x − x ′|, since〈χ (x ′) |χ (x)〉 =

⟨χi |S†x ′ Sx |χi

⟩∼= 0 if |x − x ′| is large (Joos,

Zeh 1985)


Disjointness of supports

...and the disjointness of supports:In Bohmian Mechanics, in order to have a realmeasurement-like interaction, the supports of different pointerstates have to be disjointThis is a stronger condition respect to the simple ortoghonalityof statesMoreover, the latter is not strictly necessary in order to havedecoherence induced by scattering in the standard framework


Time evolution

Now, we consider a system described by a coherent superposition oftwo well-localized wave packets a distance 4x = |x − x ′| apart. (7Steps’ model)

How fast will the superposition become decohered by theenvironmental scattering?

Since the two wave packets are macroscopically separated, weassume that the short-wavelength limit holds.

Short-Wavelength limit = the typical wavelength λ of the scatteredparticle is much shorter than the distance 4x


Time evolution

Following Schlosshauer (2009, section 3.3) and Joos (2002, section3.2), we arrive to the final relevant result:

Time evolution of the reduced density matrix of the system

∂ρS (x , x ′, t)

∂t= −ΓρS (x , x ′, t)

with Γ= total rate of scattering events

By integration of the equation above, we finally obtain

ρ (x , x ′, t) = ρ (x , x ′, 0) e−Γ t


Time evolution

Time evolution of ρS (x , x ′, t) during the scattering process forshort-λ limit:

ρ (x , x ′, t) = ρ (x , x ′, 0) e−Γ t

Spatial coherence between the two wave packets becomesexponentially suppressed

The “localization rate” depends, in this model, simply from thetotal scattering rate Γ and not from the initial coherencelength 4x


Conclusive remarks

Ortoghonality of states VS Disjointness of supports

A typical incoming particle in the scattering model can berepresented by a plane wave: is it possible to maintain thecondition of disjoint supports?

A single scattering event cannot locally destroy thesuperposition between two well-localised wave packets (7Steps’ model too optimistic)


Conclusive remarks

Decoherence induced by scattering is an asymptotic limitingprocess: classical world emerges only as an approximation

Solution proposed by ADGZ in 7 Steps’ ←→ disjointness ofsupports of the final states of the scattered particle. Is it a toostrong condition?

I think decoherence could be a great weapon for the classical limitin Bohmian Mechanics...but very much work has to be done!


References

V. Allori, D. Dürr, S. Goldstein, N. Zanghi’ (2002): SevenSteps Towards the Classical World, Journal of Optics B 4.G. Bacciagaluppi (2011): Measurement and Classical Regimein Quantum Mechanics , in R. Batterman (ed.): The OxfordHandbook of Philosophy of Physics, Oxford University Press,Oxford, 2013.D. Dürr, S. Teufel (2009): Bohmian Mechanics. The Physicsand Mathematics of Quantum Theory, Springer, Berlin.E. Joos, H. D. Zeh (1985): The Emergence of ClassicalProperties Through Interaction with the Environment,Zeitschrift für Physik B - Condensed Matter 59.E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, I. O.Stamatescu (2003): Decoherence and the Appearance of aClassical World in Quantum Theory, Springer, Berlin.M. Schlosshauer (2007): Decoherence and the Quantum toClassical Transition, Springer, Berlin.W. H. Zurek (2002): Decoherence and the Transition fromQuantum to Classical - Revisited, Los Alamos Science 27.

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Decoherence and Bohmian Mechanics · Decoherence and Bohmian Mechanics ... D.Dürr,S.Teufel(2009): Bohmian Mechanics. The Physics and Mathematics of Quantum Theory,Springer,Berlin