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Bipartite Entanglement Induced by a Common Background (Zero-Point) Radiation Field

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This paper deals with an (otherwise classical) two-(non-interacting) particle system immersed in a common stochastic zero-point radiation field. The treatment is an extension of the one-particle case for which it has been shown that the quantum
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  Found Phys (2011) 41: 843–862DOI 10.1007/s10701-010-9527-y Bipartite Entanglement Induced by a CommonBackground (Zero-Point) Radiation Field A. Valdés-Hernández  · L. de la Peña  · A.M. Cetto Received: 27 May 2010 / Accepted: 13 December 2010 / Published online: 22 December 2010© Springer Science+Business Media, LLC 2010 Abstract  This paper deals with an (otherwise classical) two-(non-interacting) parti-cle system immersedin a commonstochastic zero-pointradiationfield. The treatmentis an extension of the one-particle case for which it has been shown that the quantumproperties of the particle emerge from its interaction with the background field un-der stationary and ergodic conditions. In the present case we show that non-classicalcorrelations—describable only in terms of entanglement—arise between the (nearby)particles whenever both of them resonate to a common frequency of the field. Foridentical particles the entanglement becomes maximum and must be described bytotally (anti)symmetric states. Keywords  Entanglement  ·  Zero-point radiation field  ·  Foundations of quantummechanics  ·  Symmetrization  ·  Stochastic processes 1 Introduction In previous work [1] it was shown that the quantum behaviour of one-particle sys-tems can be understood as an emergent property arising from the interaction of an A. Valdés-Hernández  ·  L. de la Peña (  )  ·  A.M. CettoInstituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364,Mexico City, Mexicoe-mail: luis@fisica.unam.mxA. Valdés-Hernándeze-mail: andreavh@fisica.unam.mxA.M. Cetto (  )e-mail: ana@fisica.unam.mx Present address: A.M. CettoIAEA, P.O. Box 200, Vienna Austria  844 Found Phys (2011) 41: 843–862 otherwise classical particle with the background zero-point radiation field, the lat-ter being a continuous (not quantized) field, whose energy per mode of frequency ω  is proportional to  ω . The statistical description of the mechanical system in thestationary, ergodic and radiationless regime corresponds to non-relativistic quantummechanics, a result that leads to important new insight into the srcin of quantization.However, for a more complete understanding of the mechanisms underlying quantummechanics it is necessary to inquire into a distinctive and unique trait of the quantumcomposite systems, namely entanglement.This phenomenon,which holds currently acentral place in physical research, should be regarded not only as an important sourceof possible revolutionary technological applications, but also as a key resource intothe search for a better understanding of quantum mechanics. Hence, in this paper weextend the one-particle theory developed in [1] to systems composed of two nearbynon-interacting particles embedded in a common background (zero-point) stochasticfield, to delve into the mechanism that generates and supports their entanglement. 1 We find that the mechanism responsible for entanglement is the coupling of theparticles through the common modes of the field to which they resonate, once thestationary, ergodic and radiationless regime has been attained. Specifically, wheneverthe particles share a resonance frequency, correlations arise between their motions,induced via the background field.When the description is reduced to one in termsof state vectors in an appropriate Hilbert space, the entangled states emerge natu-rally as the ones that reproduce such correlations. Moreover, for systems of identicalparticles the invariance properties of the field imply that entanglement is maximumand must be described by totally (anti)symmetric states. With these results the theoryreveals the mechanism and the srcin—both of them foreign to the usual quantumdescription—of the entanglement between non-interacting particles and the quantumsymmetrization postulate.For the sake of clarity the first part of the paper contains a schematic expositionof the one-particle theory; for a comprehensive discussion of it we refer the readerto [1]. In the second part the two-particle problem is discussed in detail. 2 The One-Particle System 2.1 The Background Field in Interaction with MatterWe consider a particle of (renormalized) mass  m  and electric charge  e  subject to anexternal force  f  ( x )  and immersed in an (otherwise classical, that is, continuous) sto-chastic radiation field, arbitrary except that it always includes the zero-point contribu-tion of average energy per normal mode E  (ω)  =    ω/ 2. 2 The equation of motion for 1 The very first version of the one-particle theory can be seen in [2], a short communication on the two-particle system in [3]; an extended version of the present theory is contained in [4]. 2 The mere introduction of this field already constitutes a clear departure from usual classical physics, itszero-point energy being contrary to classical equipartition of energy. It should be stressed that the zero-point spectrum requires no quantum assumptions (see, for example, [5]).  Found Phys (2011) 41: 843–862 845 the particle is given, in the non-relativistic approximation, by the Abraham-Lorentzequation m ·· x  =  f  ( x ) + mτ  ··· x  + e E ( x ,t), τ   =  2 e 2 3 mc 3 .  (1)The electric field is expanded as usual in terms of plane waves E ( x ,t) =  k ˜ E(ω k ) a k (x)e iω k t  + c.c., (2)where  a k (x)  is a stochastic vector (with random components) that results from an an-gular and polarization averaging over all field modes having frequency  ω k . Assumingthat the long-wavelength approximation is valid and restricting the study to the one-dimensional case (along an arbitrary direction  ˆ R ) (1) and (2) reduce to ( a k  =  a k  · ˆ R ) m ¨ x  =  f(x) + mτ  ... x  + eE(t),  (3a) E(t)  =  k ˜ E(ω k )a k e iω k t  + c.c. (3b)We will further assume that  f(x)  can be expanded as a power series of   x .2.2 The Family of Stationary Resonant SolutionsThe solutions of (3a) constitute a stochastic process  x (i) (t)  where the index  i —whichwill be omitted except when it is necessary—denotes the specific realization  i  of thefield. The ensemble  { i }  of all the realizations is a statistical set that can be reproducedby considering an ensemble of individual particles, each one subject to a differentrealization of the field. The average over the realizations ( ( · ) (i) ) can therefore bealternatively obtained by averaging over the ensemble of particles ( · ).A stationary state is reached when the mean power radiated by the particle bal-ances the mean power absorbed by it from the field. Thus, according to (3a), thestationary solutions satisfy the global condition  e ˙ xE  = mτ   ·· x 2  . We assume that  { i } can be decomposed as  { i } =  α { i } α  in such a way that the balance condition holdsseparately for each of the subensembles  { i } α , where  α  denotes the corresponding sta-tionary state, and hence each  { i } α  corresponds to those particles that have reached thestate  α.  Since we are dealing with a one-dimensional single-particle problem, suchstate is completely characterized by the energy attained by the particle, and so theindex  α  is directly associated with the mechanical energy.In what follows we focus on a given subensemble  { i } α . The quantities  x(t), E(t),f(t) , and in general all the time-dependent dynamical variables  A(t)  that can beexpressed as a power series of   x  and  ˙ x  in the form  h(x)  +  g( ˙ x) , must be thereforecharacterized with the index  α , so we write the generic expression (with  ˜ A αβ (t)  =  846 Found Phys (2011) 41: 843–862 ˜ A αβ e iω αβ t  ) : 3 A α (t) =  β ˜ A αβ a αβ e iω αβ t  =  β ˜ A αβ (t)a αβ .  (4)The index β  enumeratesthosefrequencies ω αβ  thatare relevantfor aspecificproblemgiven the (fixed) state  α . We refer to these frequencies, which are determined by thetheory (see (7)), as  relevant frequencies . The indices  αβ  appearing in the amplitudes ˜ A αβ  (which in principle depend on the field variables  { a (i)αβ } ) indicate that any suchquantity is associated with the frequency  ω αβ . Byintroducing x α (t),E α (t) and f  α (t) into(3a)(assumedtobesatisfiedseparatelyfor each relevant frequency) it can be shown that the response  x α  behaves resonantlyat certain frequencies, called  resonance frequencies , which are determined by theequation   αβ  ≡ ω 2 αβ  − iτω 3 αβ  +  ˜ f  αβ m ˜ x αβ   0 .  The resonances are extremely sharp due tothe small value of   τ   (10 − 23 s for electrons). Non-resonant contributions, correspond-ing to the remaining frequencies, constitute a small noise that will be disregarded.2.3 Ergodic Hypothesis and Matrix MechanicsBy demanding the stationary solutions to comply with the ergodic conditions (here ( · ) t  denotes a time average over long enough times)  | x (i)α  | 2  = | x (i)α  | 2 t  and  |˙ x (i)α  | 2  =|˙ x (i)α  | 2 t  ,  both  ˜ f  αβ  and  { ω αβ }  become independent of the field variables  { a (i)αβ } ,  andconsequently  f  α (t)  becomes a  linear   expansion in such variables (see (4)). The er-godic condition ultimately leads to the  chain rule , ω αβ 1  + ω β 1 β 2  + ω β 2 β 3  + ··· + ω β n β  = ω αβ ,  (5a) a (i)αβ 1 a (i)β 1 β 2 a (i)β 2 β 3 ··· a (i)β n β  = a (i)αβ  =  exp (iϕ (i)αβ ),  (5b)where the  ϕ (i)αβ  also satisfy the rule (5a). The solutions for  ω αβ  and  ϕ (i)αβ  are therefore(here  φ (i)α  are random variables and   α  non-stochastic numbers) ω αβ  =  α  −  β , ϕ (i)αβ  = φ (i)α  − φ (i)β  .  (6)A major implication of (5) is that they lead to the matrix multiplication rule for theamplitudes  ˜ x αβ , whence the coefficients in the expansion for  x α (t)  define a hermiteanmatrix  ˆ x . This allows us to associate a hermitean matrix  ˆ A(t)  to the dynamical vari-able  A,  whose elements  αβ  are precisely the amplitudes  ˜ A αβ (t)  appearing in (4). Asa result the srcinal stochastic equation (3a) transforms into a non-stochastic matrixequation. 3 Note that the c.c. term has been omitted. Thus, the quantities  A α  are not exactly the physical (real)variables, but complex variables associated to them.  Found Phys (2011) 41: 843–862 847 2.4 Radiationless Approximation: Establishing Contact with the Formalismof Quantum MechanicsOnce the background field and the radiation reaction have played their fundamen-tal role in leading the system to the stationary regime, the corresponding terms inthe matrix version of (3a) represent just radiative corrections that can be neglectedin a first approximation. Though this eliminates from the description any explicitreference to the zero-point field, its footprint reappears—via the constant     thatdetermines the zero-point energy—in the canonical commutator which is found tobe [1]  [ˆ x,  ˆ p ] =  i I .  For the matrix  ˆ A(t)  one obtains the Heisenberg evolution law i   d   ˆ Adt   = [ ˆ A,  ˆ H  ] ,  with  ˆ H   the matrix representing the mechanical Hamiltonian. 4 Fur-ther, the energy E  α  attained by the particle in state  α , is found to satisfy the relation   ω αβ  = E  α  − E  β ,  (7)which allows us to identify the resonance frequencies with transition frequencies. 5 On the other hand, the matrix  ˆ A(t)  can be expanded in terms of a canonical basis {| e α  e β |}  as usual,  ˜ A αβ (t)  being the corresponding coefficients of the expansion. Byperforming the transformation | e α  → | α(t)  = e − i( E  α /  )t  | e α  ,  (8)the evolution of the system is transferred to the state vectors  {| α(t) } , directly re-lated to  E  α . From (8) and the orthonormal property of the basis it can be shown that  ˜ A αβ (t)  =  e α | ˆ A(t) | e β  =  α(t) | ˆ A( 0 ) | β(t)  ,  or in a more condensed notation, ˜ A αβ  =  α | ˆ A | β  .  These results allow us to recover the whole of the basic Hilbertspace formalism of quantum mechanics. 3 The Bipartite System 3.1 The Field in the Vicinity of the ParticlesWe consider a pair of particles located at  x 1  and  x 2  and subject to the externalforces  f  1 ( x 1 )  and  f  2 ( x 2 )  respectively, with no external interaction potential betweenthem. The equations of motion, again in the non-relativistic approximation, are (here i,j   =  1 , 2 with  i    = j  ) m i ·· x i  =  f  i ( x i ) + m i τ  i ··· x i  +  e i e j  m j  τ  j  ··· x j   + e i E ( x i ,t).  (9) 4 It can be shown [4] that  ˆ A  follows the more general evolution law  i d   ˆ Adt   = [ ˆ A,  ˆ D  ˆ H  ] , where the elements of  ˆ D ,  D αβ  = d  α δ αβ ,  relate the mechanical energy  E  α  to the parameters   α  defining the relevant frequencies(see (6)) according to   α  = d  α E  α . It is in the radiationless approximation that  ˆ D  reduces to  ˆ D  =   − 1 I . 5 Those relevant frequencies that are not resonance frequencies, though important for the description of dynamical variables, do not induce transitions. This observation serves to understand why the selectionrules (or Einstein’s  A  coefficients) involve the elements of   ˆ x  and not other matrix elements.
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