the coming of a classical world

10 RESONANCE September 2004

GENERAL ARTICLE

The Coming of a Classical World

Anu Venugopalan

Anu Venugopalan is alecturer with the School

of Basic and AppliedSciences, GGS

Indraprastha University,Delhi. Her primary

research interests are inthe field of foundations of

quantum mechanics,quantum optics and

quantum information.She is also actively

involved in teaching.

KeywordsQuantum superpositions, quan-tum measurement, entangledstates, quantum coherence,decoherence.

Quant um t heory 's unusual predict ions st em fromit s basic formalism which involves concept s l iket he wavefunct ion or probabil i t y amplitudes inst eadof probabi l it ies. M any ser ious doubt s have beenraised about quant um t heory 's connect ion wit hperceived classical dynamics. H ow does quan-t um mechanics, wit h all i t s st range ideas, un-fold t o give us t he `reali t y ' of t he familiar phys-ical wor ld? W hat is t he connect ion between t heclassical and t he quantum? I f quant um mechan-ics is, indeed, t he fundament al t heory of nat ure,as is widely accept ed, t hen how does it explainclassicality? I n t he following, some of t he fasci-nat ing concept ual problems of quant um mechan-ics are highlight ed. T he `environment -induced-decoherence' approach is t hen discussed as onepract ical at t empt at explaining t he emergenceof a classical wor ld from an under lying quant umsubst rat e.

1. I nt roduct ion

The emergence of quantum mechanics as a fundamentaltheory of nature symbolizes one of the greatest revolu-t ions in physics. Since its incept ion a century ago, thetheory has introduced stunning new ideasand madepre-cise predictions about a wide range of physical phenom-ena. Throughout its development, quantum mechanicshas been subjected to the most intense scrut iny by thescient i¯c establishment. Yet, experiment after experi-ment has con¯rmed the predict ions of quantum theory,no matter how preposterous they might have sounded.Quantum mechanics has the unique dist inct ion of be-ing the most successful working theory of nature and

11RESONANCE September 2004

GENERAL ARTICLE

there is no known example of any con° ict between itspredict ions and experimentally observed results. How-ever, in spite of its impressive successes and the factthat it is current ly accepted as one of the best tools forunderstanding the physics of the micro world, quantumphysics remains an enigma. The predict ions of quantummechanics are often at odds with what people regard as`common sense'. The concepts of quantum mechanicsseem absurd when related to the world of our `experi-ence' { the familiar physical world, and thephilosophicalimplicat ions of the theory remain highly controversial.

Quantum mechanics, it seems, fails to provide a nat-ural framework to accommodate our `classical' percep-t ions of the physical world. Our normal percept ionsof the world around us are classical in the sense thatthey are based on classical ideas like Newton's laws,and well-understood concepts like precise posit ions, mo-menta and trajectories for moving objects. A centralconcept of classical dynamics is predictability. In theclassical way of looking at things, the world is consid-ered to be made up of observable objects, like part icles,° uids, ¯elds, etc., and they obey de nite laws of force.Classical systems evolve determinist ically and are ex-pected to possess de nite propert ies. Our classical per-cept ions help us form simple, direct , mental pictures inspace and t ime of the way the world works. Quantummechanics, however, calls for a radically di®erent visionof the world. Triggered by Max Planck's revolut ionaryconcept of `quanta' in 1900, the theory was born andgrew in the years that followed. In its current form,quantum mechanics is stunningly powerful. However ithas compelled us to completely reshape and revise ourideas of reality and not ions like posit ion, momentum,cause and e®ect. The quantum view appears abstractand counterintuit ive to our classically coloured minds.Though the pract it ioners of quantum theory have nodoubt that it is one of the most elegant and sat isfying

Quantum mechanicshas the uniquedistinction of beingthe most successfulworking theory ofnature and there isno known exampleof any conflictbetween itspredictions andexperimentallyobserved results.

The predictions ofquantum mechanicsare often at oddswith what peopleregard as ‘commonsense’.


GENERAL ARTICLE

According to theBorn interpretation,

the wave functionis understood in

terms ofprobabilities.

descript ions of phenomena at the atomic scale, its clashwith the more comfortable classical scheme cont inues tobother physicists.

At the heart of quantum mechanics is the wave func-t ion, Ã, a mathemat ical ent ity that contains all pos-sible informat ion about the system to which it is at-t ributed. The wave funct ion evolves in t ime accordingto the SchrÄodinger equation, which is linear and deter-minist ic:

i ¹hddt

jÃi = H jÃi : (1)

Here H is called the Hamiltonian of the system and ¹his h=2¼, h being Planck's constant .The wave funct ion,however, does not have a physical counterpart , since ititself is not an observable. According to the Born inter-pretat ion, the wave funct ion is understood in terms ofprobabilit ies. For example, if one measures the posit ionof an electron, the probability of ¯nding it in a givenregion depends on the intensity' of the wave funct ionthere. This means that in spite of the apparent deter-minism manifested in the SchrÄodinger equat ion, knowl-edge of Ã, the wave funct ion, does not ensure a preciseknowledge of the observable propert ies of the system {the kind we are familiar with in the classical world, e.g.posit ion, momentum, etc. There seems to be a funda-mental randomness built into the laws of quantum me-chanics. This challenge to our cherished common-sensepercept ions of, and preference for, a determinist ic uni-verseperhapsmadeEinstein ut ter hisoft quoted remark,\ I can't believe that God plays dice" .

An intriguing consequence of the linearity of SchrÄodin-ger'sequat ion is that wavefunct ionscould describecom-binat ions of di®erent states or `superposit ions'. For ex-ample, an electron could be described by a wave func-t ion, which describes a superposit ion of several di®er-ent locat ions. When this linear superposit ion principleof quantum mechanics is extrapolated to macroscopic


GENERAL ARTICLE

An apparentresolution of thequantummeasurementparadox is oftenforced by the notionof a sudden collapseof the state vectorinto one of theeigenstates of thedynamical operator.

systems, which are convent ionally described by classicalmechanics, we are faced with the famous counterintu-it ive example of `SchrÄodinger's cat ' . SchrÄodinger's catis the unfortunate vict im of a nasty contrapt ion wherethe decay of a radioact ive atom triggers a device whichkills the cat. The quantum mechanical descript ion ofthis scenario demands that a superposit ion state of `de-cayed' and `not decayed' for the atoms leads to the catbeing both dead and alive in superposit ion! This bizarrestate of suspended animat ion of SchrÄodinger's cat is aclassic illustrat ion of theclash between thepredict ionsofquantum theory and familiar classical insights. A closelyrelated problem is that of quantum measurement. Ifquantum theory is accepted to be the fundamental the-ory of nature, then the process of measurement and allits ingredientsmust bedescribed quantum mechanically.In a quantum measurement, the coupling between a mi-croscopic system and a macroscopic measuring appara-tus results in an entangled state where quantum me-chanics seems to allow the apparatus (meter) to existin a coherent superposit ion of macroscopically dist inctstates{ a SchrÄodinger'scat likesituat ion, which isclassi-cally unimaginable. Such concepts raise many problemsabout quantum theory's connection with the emergenceof classicality and the elusive boundary between quan-tum and classical worlds.

Over the years, there have been several at tempts to un-derstand and explain the strange and novel concepts ofquantum mechanics, the quantum measurement prob-lem, and the nature of classicality as an emergent prop-erty of an underlying quantum system. von Neumannpostulated that an irreversible reduction process takesthequantum superposit ion to a statistical mixturewhichis classically interpretableand meaningful. Thus, an ap-parent resolut ion of the quantum measurement paradoxis often forced by the not ion of a sudden collapse of thestate vector into one of the eigenstates of the dynami-


GENERAL ARTICLE

B ox 1. Quant um M easurement

The quantum mechanical descript ion of a system is contained in its wave funct ion jÃi ,which lives in an abstract Hilbert space. The state jÃi , in general, can be representedas a linear combinat ion of an orthonormal set of basis vectors in Hilbert space. Thedynamics of the wave funct ion are governed by the SchrÄodinger equat ion:

i ¹hddt

jÃi = H jÃi :

Here H is called the Hamiltonian. The mathemat ical st ructure of the SchrÄodinger equa-t ion is such that it is linear and deterministic. A consequence of it being deterministicis that if jÃ(0)i is known, then the equat ion can specify jÃ(t)i for all t imes. Anotherproperty is that quantum evolut ion as governed by the SchrÄodinger equat ion is unitary.

Box 1. continued...

The quantum andthe classical

undoubtedly sharean intimate bond.

cal operator. However, the non-unitary nature of thisreduct ion is at odds with the inherent unitary natureof the SchrÄodinger equat ion (see Box 1). This seemsto imply that the mechanism lies outside the realm ofquantum mechanics, thus quest ioning the validity of thetheory itself. What, then, is the connect ion betwen the`classical' and the `quantum' worlds? Is there a de niterelat ionship? Are classical mechanics and quantum me-chanics two mutually exclusive incompat ible theories orare they two aspects of thesame underlying philosophy?Classical objects are eventually composed of elementsof the microworld which can be described quantum me-chanically. So, how and where can there be a boundarybetween the two worlds? The quantum and the classicalundoubtedly share an int imate bond. Among the vari-ousexplanat ions that seek a resolut ion to theconceptualproblems of quantum mechanics are the Copenhageninterpretat ion, the Hidden Variable theory, the Many-Worlds interpretat ion, the de-Broglie Bohm theory, theenvironment-induced decoherence theory, etc. Thoughall thesedi®erent interpretat ionsseem mutually con° ict-ing at the philosophical level, they all accurately explainthe outcomes of known experiments and correct ly pre-dict the outcomes of new experiments! Each physicist 'spreference, therefore, for a part icular interpretat ion of


GENERAL ARTICLE

This means that if we have some arbit rary quantum system, U, that takes as input astate jÁi which evolves to a di®erent state UjÁi , then mathemat ically we can describeU as a unitary linear tranformation. This means that U+ U = 1, where U+ is the com-plex transpose of U. Examples of unitary t ransformat ions are rotat ions and re° ect ions.Unitary transformat ions preserve inner products, and, therefore, norms:

(Ã; Á) = (UÃ; UÁ):

Inspite of the apparent determinism manifested in the SchrÄodinger equat ion, a preciseknowledge of jÃi does not ensure a precise knowledge of the observable propert ies of thesystem- the kind that we are familiar with in the classical world, e.g. posit ion, momen-tum etc. Dynamical variables or observables are represented in quantum mechanics bylinear Hermitian operators which act on the state vector. An operator A correspond-ing to a dynamical quant ity A is associated with eigenvalues ai s and correspondingeigenvectors f j®i i g, which form a complete orthonormal set. A consequence is that anyarbit rary statevector jÃi can be expanded as a linear superposit ion of theseeigenvectors:jÃi =

Pci j®i i . How does standard quantum mechanics describe measurement? A basic

postulate of quantum mechanics is that a measurement of A can only yield one of theeigenvalues ai s, but the result is not de nite in the sense that di®erent measurements forthe quantum state jÃi can yield di®erent eigenvalues. Quantum mechanics predicts theprobabili ty of obtaining the eigenvalue ai to be jci j2. In quantum mechanics one writesthe expectation value of A in the state jÃi as:

hAi = hÃjA jÃi =X

ai jci j2:

An addit ional postulate of quantum mechanics is that the measurement of an observableA, which yields one of the eigenvalues ai (with probability jci j2) culminates with thereduction or collapse of the state vector jÃi to the eigenstate j®i i . This means that everyother term in the linear superposit ion vanishes except one. Such a transit ion is obviouslyimpossible to achieve via the norm preserving unitary evolut ion discussed above. This iswhere the measurement postulates clash with unitary SchrÄodinger evolut ion.

If a system has a state vector jÃi , then it is possible to de ne the density operator forthe system by the outer product

½= jÃi hÃj:

Then the equivalent formula for the expectat ion value of an operaor A is

hA i = Tr acef A½g:

Thedensity operator, thus, is essent ially an object which is equivalent to thestate vector.It is a convenient formal tool to compare and cont rast quantum and classical systems interms of probabilit ies. The conceptual problems of quantum measurement become moret ransparent when analyzed in the language of density mat rices, and hence this is oftenthe preferred approach.


GENERAL ARTICLE

Classicality is anemergent property

triggered in opensystems by their

environments.

The achievements ofthe decoherence

theory seemparticularly relevant in

the light of severalrecent proposals to

exploit purely quantummechanical features

such as the linearsuperposition principle,

and quantumentanglement to buildhigh speed quantum

computers.

quantum mechanics remains a matter of taste unt il theday some irrefutable experimental evidence chooses oneinterpretat ion over all others.

In the following we will highlight the `environment in-duced decoherence theory'{ one attempt to explain theemergence of classicality from quantum dynamics. Thistheory employs the methods developed by several au-thors to analyse the quantum mechanics of a systemin interact ion with its environment. The central idea ofthisapproach is that classicality is an emergent propertytriggered in open systems by their environments. Thetheory explains why we do not rout inely see quantumsuperposit ion in the familiar physical world around us.Though this theory might seem to some as less dramat iccompared to its rival counterparts, it is being widely ac-cepted as a successful and pract ical solut ion for explain-ing the emergence of a classical world. The strength ofthis approach is that it provides explanat ions within therealm of quantum mechanics and does not call for anymodi¯cat ion of the exist ing framework of theory.

Recent ly, interest in the understanding of decoherencehas been heightened by stunning advances on the exper-imental front , where some of the predict ions of the deco-herence approach have been successfully demonstrated.The achievements of the decoherence theory seem par-t icularly relevant in the light of several recent propos-als to exploit purely quantum mechanical features suchas the linear superposit ion principle, and quantum en-tanglement to build high speed quantum computers.There have also been several at tempts to implementother interest ing ideas from this novel ¯eld of quan-tum informat ion like quantum cryptography and quan-tum teleportat ion in the laboratory. Since environmen-tal in° uence is often unavoidable in a real life labora-tory scenario, the ubiquitous decoherence mechanismcan ruin the funct ioning of such futurist ic systems whichrely heavily on the maintenance of quantum coherence.


GENERAL ARTICLE

Thus, for the morepract ical physicist , an understandingof decoherence and its quant itat ive and qualitat ive pre-dict ions holds more signi¯cance when looking for realis-t ic explanat ions for the emergence of classical featuresin a quantum world.

2. T he Problem: Quant um M easurement , En-t angled St at es and t he Point er Basis

Consider a measurement-like scenario, which is to bedescribed quantum mechanically. A simple and illustra-t ive example is when both the system whose propert iesare being measured and the apparatus, which measuresthese propert ies, are quantum-mechanical spin-1/ 2 par-t icles. The measurement interact ion between the twoshould be such that the property of the system gets cor-related with some property of the apparatus. This way,by looking at the state of the apparatus (meter) one caninfer what state the system is in. Let ¾and L representthe Pauli spin operators for the system and the appara-tus, respect ively. Let j " z i and j #z i be theeigenstates of¾z and j * z i and j +z i be the eigenstates of L z. It is rea-sonable to assume that the system and the apparatusare init ially uncoupled. Their init ial combined wave-funct ion (state) can then be writ ten as a direct productstate of the form:

jÃ(t = 0)i = (aj " zi + bj #z i ) (cj * z i + dj +z i ); (2)

where the system and apparatus are, in general, in thesuperposit ion states (aj " z i + bj #z i ), and (c * z i + d +z i ),respect ively. a; b; c, and d are complex numbers. Themechanism of a measurement process demands that thesystem and the apparatus interact. Let us assume thatthis happens via a Hamiltonian of the form

HSA = g¾zL y ; (3)

where g is the strength of the system-apparatus cou-pling. It can be easily shown that the Hamiltonian evo-


GENERAL ARTICLE

lut ion takes an init ial state of the kind (2) into an en-tangled state (see Box 2) of the form:

jÃS+ A (t)i = k1j " z i j +z i + k2j #z i j * z i ; (4)

where k1 and k2, in general, depend on the couplingstrength, g, the complex coe± cients a; b; c; d and t ime,t, in a complicated way. For a certain choice of c; d andt, we can have the simpli¯ed form of (4), which we will

B ox 2. Superposit ions and Ent angled St at es

One of the consequences of the linearity of the SchrÄodinger equat ion is that its solut ionsare subject to the linear superposition principle. This means that the equat ion allowsfor solut ions which are linear superposit ions of other solut ions. For example, if Ã1 andÃ2 are solut ions of the equat ion for a certain system, then

Ã = c1Ã1 + c2Ã2

is also a perfect ly legit imate solut ion for the same system, where c1 and c2 are someconstant complex numbers. Such states are specially intriguing and counterintuit ivewhen one considers situat ions where Ã1 and Ã2 each represent macroscopical ly distinctstates of the system. When the superposit ion principle is applied to more than onequantum part icle, say, to a system of two interact ing part icles, then one has what areknown as entangled states. This phenomenon is even more peculiar than a superposit ionstate of a single part icle as described above. SchÄodinger's famous cat-paradox deals withsuch an entangled state between the states of the atom (decayed and undecayed) and thecat (dead and alive):

Ã = c1 jdecayedi jdeadi + c2 jundecayedi jal ivei :

Like the cat and the atom in the state above, the members of an entangled state donot have their own individual pure quantum states. Only the combinat ion as a wholehas a well-de ned state. In a measurement scenario, the entangled state exists betweenthe system and the measuring apparatus. This is completely at odds with anything weknow in classical physics. Entangled objects behave as if they were connected with oneanother no matter how far apart they are and always contain quantum correlat ions. Ein-stein Podolsky and Rosen were the ¯rst to draw at tent ion to the possibility of what theycalled \ spooky act ion-at-a distance" or nonlocal e®ects involving entangled part icles. Inthe nineteen sixt ies, John Bell predicted that entangled quantum states allow crucial ex-perimental tests that dist inguish between classical and quantum physics. Subsequent ly,experimenters have con¯rmed the existence of these \ nonlocal" correlat ions. More re-cent ly, e®orts have gone into exploit ing the peculiar nature of entangled part icles in theexcit ing ¯eld of quantum informat ion (see [2]).


GENERAL ARTICLE

The system andapparatus developquantum correlationswhich makes themone inseparablequantum unit.

The off-diagonalterms are signaturesof the quantumcorrelationscontained in theentangled state.

use to illustrate the measurement problem. Equat ion(4) is no longer a direct product state of the systemand apparatus but an entangled state where the sys-tem and apparatus develop quantum correlations whichmakes them one inseparable quantum unit. Such a sit -uat ion, like the state of Einstein, Podolsky and Rosen(EPR) (see Box 2) is a uniquely quantum mechanicalstate of a®airs, which is di± cult to comprehend classi-cally. Consider the density matrix corresponding to theentangled state (4). The density matrix correspondingto such a state is, in general, `non-diagonal' and con-structed as the project ion operator of the form:

½pur e = jÃi hÃj = jk1j2j " z i h" z jj +z i h+z j + jk2j2j #z ih#z jj * z i h* z j + k¤

1k2j #z i h" z jj * z ih+z j + k1k¤

2j " z i h#z jj +z i h* z j: (5)

While (4) and (5) represent mathemat ically legit imatesolut ions of the SchrÄodinger equat ion for the Hamil-tonian evolut ion, there is a serious di± culty in theirphysical interpretat ion, when one appeals to their usualmeanings in the language of probabilit ies. How do theyexplain theoutcomesof a real-lifemeasurement process?If one appeals to the usual de nit ion of density matri-ces as probabilit ies, then the o® diagonal elements (thethird and fourth terms in (5)) are puzzling. They do notmake sense to our classically tuned minds like the diag-onal elements do. The diagonal terms conform with ourunderstanding of classical probabilit ies. In the measure-ment scenario described here, they can be interpretedas the probabilit ies corresponding to the cases wherethe spin system is in the `up' or `down' states. Theo®-diagonal elements make no such sense and we wouldbe happy with an expression where they did not ex-ist . In fact , the o®-diagonal terms are signatures of thequantum correlations contained in the entangled state.The only way, it seems, to reconcile this situat ion witha classical measurement-scenario is to accept that theundesirable `o®-diagonal' elements somehow disappear.


GENERAL ARTICLE

ρpure = |ψ⟩ ⟨ψ| containsapparent system-

apparatus correlationswhich ‘suffer’ from

what is often referredto as the ‘ambiguity of

basis’.

Subsequent ly, thedensity matrix is reduced to a diagonalstatistical mixture of the form:

½r ed = jk1j2j " z i h" z jj +z i h+z j + jk2j2j #z i h#z jj * z i h* z j:(6)

This is equivalent to von Neumann's non-unitary reduc-t ion process or the collapse postulate. ½r ed makes clas-sical sense. Its coe± cients may be interpreted as classi-cal probabilit ies corresponding to the two potent ial out-comes, ùp' and `down' of the spin-systems. Thus ½r ed

represents classical correlat ions between the system andthe apparatus in a familiar measurement situat ion.

It might be argued that both ½pur e and ½r ed contain thepotent ial outcomes of ùp' and `down' spins. Why can'twe, then, look at just the diagonal elements of ½pur e andsimply ignore its o®-diagonal elements? Why is it nec-essary to have the non-unitary reduct ion process thatcompletely gets rid of the o®-diagonal elements?. Inother words, doesn't ½pur e through its diagonal elementsalready contain the descript ion of a completed measure-ment process? The answer is, no! The fact is, ½pur e =jÃi hÃj contains apparent system-apparatus correlat ionswhich `su®er' from what is often referred to as the àm-biguity of basis' . To illustrate this point , let us choosethecoe± cientsk1 and k2 such that k1 = ¡ k2 = 1p

2 . Thismakes the density operator ½pur e a project ion operatorconstructed from the following pure state:

jÃi =1p2

hj " z i j +z i ¡ j #z i j * z i

i: (7)

The state (7) is rotationally invariant. This means thatif the state is re-writ ten in a di®erent basis, e.g. in theeigenstates of ¾x and ÃLx for the system and apparatus,then

j " zi =1p2

hj " x i + j #x i

i; j #z i =

1p2

hj " x i ¡ j #x i

i;

j * z i =1p2

hj * x i + j +x i

i; j +zi =

1p2

hj * x i ¡ j +x i

i


GENERAL ARTICLE

gives

jÃi =¡ 1p

2

hj " x i j +x i ¡ j #x i j * x i

i; (8)

and similarly, in the eigenstates of ¾y and ÃLy ,

jÃi =¡ ip

2

hj " y i j +y i ¡ j #y i j * y i

i: (9)

½pur e = jÃi hÃj in each case would have the same formas (5). However, everyt ime we change the basis, ½pur e =jÃi hÃj would end up with o® diagonal terms which con-tain correlat ions of di®erent states of the system andapparatus. The democrat ic world of quantum mechan-ics gives us the freedom to choose any basis since allbases are alike and equivalent . The price to pay for thisfreedom is that ½pur e ends up being an ambiguous andconfusing object when one appeals to it to describe acompleted measurement process. By simply looking atthe diagonal elements of ½pur e, one cannot assert thatthe apparatus as well as the system is each separatelyin a de nite state with one-to-one correlat ions. As wehave seen, a mere change in the basis will t ransformthe picture. This makes the interpretat ion of its diago-nal termsas probabilit ies for the two potent ial outcomesquitemeaningless. For a classically meaningful situat ionone must be able to talk of the system/ apparatus exist-ing in speci¯c states with classical probabilit ies. Thisnecessarily requires a descript ion in terms of a densitymatrix that is diagonal in a uniquely speci¯ed pointerbasis which is preferred over all others. The crux of themeasurement problem and associated di± cult ies is toexplain the transit ion from ½pur e to ½r ed in a preferredpointer basis, which, in essence is a sat isfactory explana-t ion for the transit ion from `quantum' to `classical'. Isit possible to explain the mechanism of reduct ion withinthe framework of the theory of quantum mechanics?

3. T he Decoherence A pproach

Asstated earlier, theo®-diagonal elementsof thedensity

The democraticworld of quantummechanics givesus the freedom tochoose any basissince all bases arealike andequivalent.

The crux of themeasurementproblem andassociated difficultiesis to explain thetransition from ρpure toρred in a preferredpointer basis, which,in essence is asatisfactory explana-tion for the transitionfrom ‘quantum’ to‘classical’.


GENERAL ARTICLE

The Schrödingerequation driving

unitary evolutions isactually applicable to

completely isolatedsystems.

The environment‘washes away’

quantum coherence,leaving behind a

system which looksand behaves like aclassical object of

our cherishedcommon-sense

world.

matrix are the signatures of quantum correlations. (4)and (5), in fact represent the bizarre state of SchrÄodin-ger's cat when extrapolated to the macroscopic domain.In quantum mechanics, wave funct ions evolve via theSchrÄodinger equat ion which is linear and determinist icand this evolut ion is unitary in nature. Unitary evolu-t ions ensure that eigenvalues are preserved. There is noway that some terms of the density matrix can vanishin the course of a unitary evolut ion. How, then, can`classical behaviour' (as discussed above) ever emergefrom this substrate of the quantum world where entan-glements and coherences are ubiquitous and inevitable?

Theanswer lies in realizing the fact that theSchrÄodingerequat ion driving unitary evolut ions is actually applica-ble to completely isolated systems. The real world, how-ever, is far from this prist ine pure situat ion. Macro-scopic systems are almost never isolated from their sur-roundings. They are constantly interact ing with a com-plex environment. In a measurement like situat ion, theapparatus is almost always a macroscopic object fromwhich one reads out the measured property of the sys-tem. In fact , theapparatusisnot only considered macro-scopic, it is also regarded as `classical' in its dynamics.

The `decoherence' explanat ion is that it is the in° uenceof the environment that makes a quantum system ap-pear `classical'. The environment `washes away' quan-tum coherence, leaving behind a system which looksandbehaves like a classical object of our cherished common-sense world. The `open system' is coupled to a largenumber degrees of freedom which const itute the envi-ronment. However, one is always monitoring only a fewdegrees of freedom, which are of relevance. In mathe-mat ical language, this amounts to tracing over all otherdegress of freedom. This tracing over has the e®ect ofcausing the much desired transit ion:

½pur e ¡ ! ½r ed : (10)


GENERAL ARTICLE

The randommotion of thesuspended particlecan be statisticallyexplained by takinginto account itsinteraction with alarge number ofparticles whichconstitute thereservoir of watermolecules.

An illuminat ing and popular paradigm for understand-ing this is the phenomenon of Brownian mot ion, namedafter the Brit ish botanist , Thomas Brown, who ¯rst dis-covered it in pollen grains suspended in water. We allknow now that such a suspended part icle, when exam-ined, is seen to bounce around in a random, irregular,`zig-zag' fashion. Einstein showed that this pat tern ofbehaviour is exact ly what should be expected if thepollen grain is being repeatedly `kicked' by other un-seen submicroscopic part icles (water molecules). Therandom motion of the suspended part icle can be stat is-t ically explained by taking into account its interact ionwith a large number of part icles which const itute thereservoir of water molecules. This is the environmentwhose in° uenceon thesuspended part iclecauses its ran-dom zig zag mot ion. When we see Brownian mot ion, weare only focussing on the dynamics of the suspendedpart icle and do not monitor each and every particle ofthe environment (the water bath). Mathemat ically, wetrace over all the degrees of freedom of the environmentand look only at the reduced system, the suspended par-t icle. As a consequence, the tagged part icle is foundto show a dynamics that contains dissipation (a steadyloss of energy or relaxat ion) and di®usion (the randomzig-zag motion).

Quantum Brownian motion describes a similar situat ionat the quantum mechanical level. The quantum mea-surement situat ion can be described in the followingmanner. The microscopic system couples to a macro-scopic apparatus, which in turn is interact ing with alargenumber of degreesof freedom which const itutes theenvironment. SchrÄodinger's equat ion is applied to theent ire closed universe of system-apparatus-environment.Hamiltonian evolut ion drives this closed system from aninit ial uncoupled state into a gigant ic entangled statecontaining all the degrees of freedom. A tracing over allthe environmental degrees of freedom salvages the re-

The phenomenon ofBrownian motion,named after theBritish botanist,Thomas Brown, wasfirst discovered inpollen grainssuspended in water.


GENERAL ARTICLE

After a characteristictime, the apparatus,

impacted by theenvironment, appears

classical in itsdynamics.

duced system-apparatus combine from this mess. Aftera characterist ic t ime, the apparatus, impacted by theenvironment, appears classical in its dynamics. Thus,the environment causes a general quantum state to de-cay into a stat ist ical mixture of pointer states which canbe understood and interpreted as classical probabilitydistribut ions. This in essence is the approach of the de-coherencetheory to explain theemergenceof classicality.A commonly used model to describe the environment isto consider it as a reservoir of quantum oscillators, eachof which interacts with the quantum system in quest ion.The Hamiltonian for this would look like the following:

H = Hs + xX

kgkQk +

X

k

12M k

³P2

k + M 2k ! 2

kQ2k

´; (11)

where Hs is the system Hamiltonian. The third termrepresents the Hamiltonian of the collect ion of oscil-lators which const itutes the environment. Pk and Qk

are the momentum and posit ion coordinates of the kth

oscillator. The second term represents the coordinate-coordinate coupling between the system and the envi-ronment, x being the posit ion coordinate of the quan-tum system. The dynamics for the closed universe ofthe system and environment is governed by the Hamil-tonian evolut ion via (11) and the SchrÄodinger equat ion(1). Subsequent ly, the process of t racing over all the de-grees of freedom of the environment results in an equa-t ion describing the dynamics of the reduced density ma-trix: ½(x; x0; t) = Tracek

h½

³x; (Qk ); x0; (Q0

k ); t´ i

.

The reduced density matrix evolves according to a mas-ter equation which isobtained by solving theSchrÄodingerequat ion for the ent ire universe of the system and theenvironment and then tracing over the environment de-grees of freedom. Several authors like Deiter Zeh andWoejeich Zurek, among others, have worked extensivelyon the decoherence approach using the master equation.Without going into the details of the master equation,


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it su± ces to point out that the equat ion naturally sep-arates into three dist icnt terms, namely (i) a term de-scribing the von Neumann equat ion which can be de-rived from the SchrÄodinger equat ion and thus representsthe pure quantum evolut ion, (ii) a term that causes dis-sipation, which can be understood as a steady loss inenergy or a relaxation process, and (iii) a term causingdi®usion, which can beunderstood asthe° uctuat ionsorzig-zag movement seen in Brownian mot ion. The init ialwork of Zeh, Zurek and subsequent work where thedeco-herence approach has been applied to typical quantum-measurement-like situat ions like the Stern Gerlach ex-periment have shown that coherent quantum superpo-sit ions persist for a very short t ime. They are rapidlydestroyed by the act ion of the environment. In fact ,results show that the larger the quantum superposit ion,the faster thedecoherence. For truly macroscopic super-posit ions such as SchrÄodinger's cat , decoherence occurson such a short t ime scale that it is impossible to ob-serve these quantum coherences. A simple example byZurek illustrates this point (see Figure 1). Here an ini-t ial state constructed as a coherent superposit ion of twospat ially separated gaussian wavepackets decoheres intoa stat ist ical mixture (diagonal density matrix) when itsdynamics is governed by the master equation discussedabove. Several authors have studied many more exam-ples and decoherence calculat ions for these studies areseen to contain two main featues which can be seen assignatures of decoherence: (a) the decoherence t ime, ¿D ,

Figure 1. Decoherence ofthe density matrix for aninitial coherent superposi-tion state of two spatiallyseparated gaussian wave-packets: (a) The initial den-sity matrix in the positionrepresentation showingthe diagonal and off-diago-nal terms of the quantumsuperposition. (b) Thedecohered density matrixwhere the off-diagonalpeaks have decayed toleave the statistical mix-ture of diagonal termswhich is classically inter-pretable.

(a) (b)

Coherent quantumsuperpositionspersist for a veryshort time.The largerthe quantumsuperposition, thefaster thedecoherence.


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These experimentshave done no less

than createSchrödinger-cat-likeentangled states inthe laboratory and

seen them transforminto classically

recognizable objectsunder the influence of

environmentalcoupling.

¿D , over which the superposit ions decay is much muchshorter than any characterist ic t ime scale of the system,e.g., the relaxation time, ° ¡ 1. In the example chosen byZurek, ¿D =° ¡ 1 » 10¡ 40 , and (b) ¿D varies inversely asthe square of a quant ity that indicates the `size' of thequantum superposit ion. In the example of Zurek, thisis ¢ x, the spat ial separat ion between the two gaussianwavepackets (see Figure 1).

4. Decoherence and Exper iment s

In recent years, the predict ions of the decoherence the-ory have been tested in several spectacular experimentswhich put the theory on a ¯rm foot ing. Of these, two ex-perimentsarepart icularly noteworthy and wewill brie° yment ion them. Both these experiments have succeededin monitoring the decoherence mechanism, i.e., the ac-tual transition from a pure entangled state to a sta-t ist ical mixture. Moreover, the experiments also givea quant itat ive est imate of the decoherence t ime, ¿D .These experiments have done no less than create SchrÄo-dinger-cat-like entangled states in the laboratory andseen them transform into classically recognizableobjectsunder the in° uence of environmental coupling.

Among the¯rst successful at tempts is a beautiful exper-iment by Brune et al. at the ¶Ecole Normale Sup¶erieurein Paris. Using Rubidium atomsand high technology su-perconduct ing microwave cavit ies, Brune et al. createda superposit ion of quantum states involving radiat ion¯elds. Thesuperposit ion was theequivalent of a `system+ measuring apparatus' situat ion in which the `meter'was point ing simultaneously towards two di®erent di-rect ions. This is a SchrÄodinger-cat-like entangled state.Through a series of ingenious `atom-interferometry' ex-periments, Brune et al. managed to not only read'this pure state but also monitored the decoherence phe-nomenon as it unfolded, t ransforming this superposit ionstate to a stat ist ical mixture. Besides providing a direct


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The decoherenceexplanation hascertainly providedvaluable insights intothe actualmechanism of theloss of quantumcoherences. Itsqualitative andquantitativepredictions are highlyrelevant to allexperimentalimplementations ofthe novel ideas ofquantum information.

insight into the role of the environment in a quantummeasurement process, their experiment also con¯rmedthe basic tenets of the decoherence theory. The twomain signatures of the decoherence theory were clearlyobserved in this classic experiment. The environment inthis experiment are the `modes' of the electromagnet ic¯eld in the cavity.

At the Nat ional Inst itute of Standards and Technology,Boulder, Colorado, thegroup headed by Wineland chosea di®erent experimental candidate to con¯rm the signa-tures of the decoherence mechanism. They created aSchrÄodinger-cat-like state using a series of laser pulsesto entangle the internal (electronic) and the external(motional) states of a Beryllium ion in a `Paul t rap'.The mot ion of this trapped ion couples to an electric¯eld which changes randomly, thus simulat ing an en-vironment. Wineland et al. call this environment anengineered reservoir whose state and coupling can becontrolled. Through their measurements, Wineland etal. have successfully demonstrated the two importantsignatures of the decoherence mechanism.

The above two experiments, along with several othershave provided important insights into the role of theenvironment in bringing about classicality. The deco-herence theory is strengthened by these spectacular ob-servat ions.

5. A re t he Problems Resolved?

At the end of the day can we say that the decoherenceexplanat ion makesquantum mechanics appear less enig-mat ic and mysterious? Have we explained away some ofits conceptual problems? Physicists part ial to the deco-herence approach think so. The experiments discussedin the previous sect ion clearly show how decoherencewashesaway quantum coherences. This is fairly convinc-ing evidence for explaining the absence of SchÄodinger's


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Suggested Reading

[1] Max Tegmark and John Archibald Wheeler, 100 years of QuantumMysteries, Scientific American, p.68, February 2001.

[2] R Simon, From Shannon to Quantum Information Science, ResonanceVol.70, No.2, p.66, February 2002; Vol.70, No.5, p.66, May 2002.

Address for CorrespondenceAnu Venugopalan

School of Basic and AppliedSciences

GGS Indraprastha University

Kashmere Gate, Delhi 110 006,India.

Email:[email protected]

There are many whofind the decoherence

theory too prosaic aresolution to the

complex and uncannyworld of quantum

mechanics.

cats in the real world. The decoherence explanat ionhas certainly provided valuable insights into the actualmechanism of the loss of quantum coherences. Its qual-itat ive and quant itat ive predict ions are highly relevantto all experimental implementat ions of thenovel ideas ofquantum information. Asfor therealm of quantum mea-surements, to quoteTegmark and Wheeler,\ decoherenceproduces an e®ect which looks and smells like a col-lapse..." . This is an opinion shared by many physicistswho see in the decoherence explanat ion a sat isfactoryset t lement of the quantum measurement problem.

However, there are many who ¯nd the decoherence the-ory too prosaic a resolut ion to the complex and un-canny world of quantum mechanics. To them, manyof the conceptual problems of quantum mechanics arest ill unresolved and deeply disturbing. This spectacu-larly powerful theory is st ill believed to conceal manysecrets. Whatever one chooses to think, quantum me-chanics remains a theory of great beauty and mystery,and a wonderful experience for anyone who has encoun-tered it .

The aim of science is not to open the doorto infinite wisdom, but to set a limit toinfinite error.

Berolt BrechtFrom: The Life of Galileo

the coming of a classical world

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