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Proposal for an experimental test of the many-worlds interpretation of quantum mechanics

arXiv:quant-ph/9510007v3 10 Nov 1995

R.Plaga Viktoriastr. 8 80803 Munchen, Germany ¨

Abstract The many-worlds interpretation of quantum mechanics predicts the formation of distinct parallel worlds as a result of a quantum mechanical measurement. Communication among these parallel worlds would experimentally rule out alternatives to this interpretation. A procedure for "interworld" exchange of information and energy, using only state of the art quantum optical equipment, is described. A single ion is isolated from its environment in an ion trap. Then a quantum mechanical measurement with two discrete outcomes is performed on another system, resulting in the formation of two parallel worlds. Depending on the outcome of this measurement the ion is excited from only one of the parallel worlds before the ion decoheres through its interaction with the environment. A detection of this excitation in the other parallel world is direct evidence for the many-worlds interpretation. This method could have important practical applications in physics and beyond.



plaga@hegra1.mppmu.mpg.de

1


1

Intro duction

There has b een a renewed intense interest in the quantum mechanical measurement problem recently[1]. The reason for this is a growing dissatisfaction with the orthodox[2] and statistical [3] interpretations which do not allow to derive the prop erties of the classical reality from the Schrodinger equation even in principle. A further problem is that b oth ¨ interpretations use concepts ("reduction of the state vector" in the former and "conceptual ensemble of similarly prepared systems" in the latter) that are describ ed only by words and not mathematically, so their meaning remains vague. Moreover in the orthodox interpretation the human consciousness has to play a sp ecial role in physics (in the words of Bohr the purp ose of physics is "... not to disclose the real essence of phenomena but only to track down ... relations b etween the manifold asp ects of exp erience" [4]), a notion that does not go easy with the ma jority of physicists. For simplicity I will consider in this pap er only the simple case of measurements with two discrete results. A generalization to the case of more than two outcomes is straightforward. According to the classical b ook on quantum measurements in the orthodox interpretation by von Neumann[5], a quantum mechanical measurement consists of a "process 1" or "collapse of the wavefunction": a coherent wave function (which contains a complete description of the quantum mechanical system and of the measurement apparatus), is suddenly converted to a statistical mixture of 1 which describ es one p ossible outcome of the measurement, and 2 which describ es the other outcome. This state reduction is not derived from the Schrodinger equation (called "process 2" by von Neumann) but introduced ad hoc to ex¨ plain the observed facts. An imp ortant progress during the last decade was the realization that "decoherence" plays a decisive role in a quantum mechanical measurement[6]. Decoherence explains "process 1" as a loss of phase relations in the wavefunction of the measuring apparatus while it interacts with the quantum system. This loss is a continuous process and can b e quantitatively calculated in a variety of situations[6] without going b eyond the Schrodinger ¨ equation. Process 1 needs a finite amount of time in this view b ecause of its continuous nature, the so called "decoherence time" tdec , which is very short in most "usual" measurement situations (i.e. the measurement apparatus is macroscopic and interacts strongly with the quantum system). The sudden reduction envisioned by von Neumann is a very good approximation which suffices for a description of practical situations up to now. A complete statistical mixture is never reached, but if one takes into account that macroscopical measurement apparati always interact with a large environment, the assumption of a statistical mixture b ecomes extremely good and can explain all observational facts. There remains one question (quoted here directly from Omn`s[7]): after decoherence has e taken place..."why or how does it happ en that an apparatus shows up unique and precise data (in our case: either 1 or 2 is actually observed) whereas the theory seems only to envision all p ossibilities on the same footing?". This necessity of some mechanism in 2


addition to "process 1" (sometimes called "ob jectification"or "actualization") was already recognized by von Neumann; he calls the measurement apparatus "I I" and the apparatus "with the actual observer" "I I I". He only states that the interaction b etween "I I" and "I I I" "remains outside the calculation"[5](chapt.VI.1). Prop osals to answer Omn`s question can e b e group ed in three categories: · there are so called "hidden variables", arising from some extension to the Schrodinger ¨ equation which causes actualization (not necessarily in a deterministic way)[8]. A violation of the Bell inequalities in EPR typ e exp eriments has b een shown with great precision in a variety of set ups recently[9]. If one does not want to take recourse to contrived conspiracies (see Ref.[10] how to exclude even these), any hidden variable theory has to introduce nonlocal interactions as a consequence; this would require a revision of many physical concepts. · the question is declared "meaningless"; "actualization" occurs without any mechanism. e.g. Hartle states [11] "We do not see it (i.e. actualization) as a "problem" for quantum mechanics." This standp oint is logically consistent and leads to the so called "logical" [1] and "many histories" [12] interpretation of quantum mechanics. These (quite similar) interpretations include decoherence in their description of nature and thus go far b eyond the orthodox interpretation. Actualization is obviously crucial for our p erception of nature, but it is not considered to b e a part of physics in this view. Therefore these interpretations (like the orthodox interpretation) have to renounce the existence of an "indep endent reality", a physical universe which exists indep endent of our consciousness, Omn`s states: "physics e is not a complete explanation of reality...theory ceases to b e identical with reality at their ultimate encounter..."[1]. · a very radical and elegant answer was given by Everett[13]: after decoherence has taken place, the orthogonal states 1 and 2 (each also describing an indep endent "split" observer) continue to evolve according to the Schrodinger equation and have "equal rights". ¨ In this view "actualization" is explained as an illusion in the brain of a human observer: after a few decoherence times, his weak senses and crude measuring devices are unable to detect the increasingly weak influences of the other "outcome". He therefore calls the one outcome he can see "the world". The same happ ens with the other outcome. For this reason DeWitt termed the name "many-worlds interpretation"(MWI) for this view of nature [14]. I will use the word "universe" to indicate space time together with all "worlds" existing in it. I call the two outcomes of a measurement "parallel worlds" b elow, b ecause they exist in the same Minkowskian space time. The worlds which form as a result of a measurement with a finite numb er of discrete outcomes are usually called "branches". In Hilb ert space the parallel worlds are orthogonal of course. Together with decoherence (a concept still unknown when Everett wrote his thesis) this idea leads to a deterministic view of the universe in which the human mind plays no sp ecial role outside physics[15]. Section 2 contains a general discussion of the method for an exp eriment to test Everett's interpretation. Sections 3 provides a detailed analysis of a decoherence process which is of critical imp ortance for the exp eriment. A reader mainly interested in the practical re3


alization of the exp eriment can skip this somewhat technical part and proceed directly to section 4. Here a concrete example for a p ossible technical setup is given. In the conclusion (section 5) the predictions of the various interpretations of quantum mechanics for the outcome of the prop osed exp eriment are compared, and the p otential practical imp ortance of a result confirming the MWI is stressed.

2

Prop osal of an exp eriment to test the many-worlds interpretation

The MWI together with decoherence corresp onds to the conceptually very simple view that nonrelativistic quantum mechanics can b e understood by assuming only the existence of ob jectively real wavefunctions whose evolution is governed by the usual Schrodinger equa¨ tion, together with the second quantization conditions of the underlying wave field, without any sub jective or non-local elements. It is therefore imp ortant to find exp erimental tests for this interpretation. Indep endent of what one thinks ab out the MWI a priori, this is also a very systematic way to make exp erimental progress in the question of the interpretation of quantum mechanics, b ecause in the MWI the predictions for any conceivable exp eriment are free from philosophical subtelties (which can b e a problem in the orthodox interpretation) or free parameters (which often occur in one of the many prop osed hidden variable models). I already mentioned that decoherence only leads to approximate mixtures (though the approximation is extremely good in most situations)[14]. The separation of worlds in the MWI is never quite complete therefore, and there should b e small influences from a parallel world even after decoherence, which must be measurable in principle. This has b een most clearly p ointed out by Zeh[16, 17]. In Ref.[16] he discusses the p ossibility to observe "probability resonances" (later further discussed by Albrecht[18]), which occur at a singular p oint when the amplitudes of 1 and 2 have exactly the same magnitude. An exp eriment to test the MWI against the orthodox interpretation along similar lines was prop osed by Deutsch [19]. Unfortunately it is still far from practical realization, as it requires a computer which remains in a quantum mechanically coherent state during its op erations and in addition p ossesses artificial intelligence comparable to that of humans. I will describ e an exp eriment for testing the MWI with state of the art technology. Imagine a human called Silvia which is programmed to p erform different actions in dep endence on the outcome of a quantum mechanical measurement. For our purp oses Silvia might just as well b e imagined e.g. as a suitably programmed commercially available computer connected to the exp erimental equipment via a CAMAC crate instead of as a human. As an example Silvia sends a linearly p olarized photon through a linear p olarization filter. Let the photon b e in a state |P , such that the filter axis of complete transmission is at 45o to the linear p olarization plane of the photon. She is programmed (decides) to switch on a microwave

4


emitter if she will measure that the photon passed through the linear-p olarization filter into photomultiplier tub e and to refrain from doing so if she will find that the photon was absorb ed by the filter. If one assumes detectors with 100 % efficiency for simplicity, the probablity for either outcome is 50 %. In the MWI there are two indep endent humans after the measurement was p erformed and decoherence took place: one which detected a photon and switched on the emitter (called "Silvia1" b elow) and the other that didn't ("Silvia2"). Could these humans (Silvia1 and 2) communicate with each other? The standard answer in the MWI is no, b ecause decoherence is so complete after very short time scales that no one of them can influence the world of the other, which is of course necessary for communication. One could isolate a small part of the original apparatus (b efore the measurement takes place) so p erfectly that it does not immediately participate in the decoherence. It is now p ossible in principle to change the state of this isolated part before it is completely decohered by means of an influence from only one of the two worlds. In this way it could act as a "gateway state" b etween the parallel worlds. Because it is only partially decohered, it can still b e influenced by b oth worlds (and in turn can influence b oth worlds), thus making p ossible communication. For humans an isolation on a time scale of at least seconds would b e necessary for real communication. For the current electronic computers a time scale of µsecs and for simple macroscopic logic electronic (e.g. in the commercial NIM standard) nsecs would b e enough to verify the existence of the parallel world. This prop osition is not realistic if the "gateway state" is macroscopic, b ecause the required isolation would b e difficult to achieve technically (see however recent exp eriments with macroscopic quantum systems e.g. Ref.[20]). Since the late 1970s it has b ecome p ossible to p erform precision exp eriments on single ions stored for long times in electromagnetic traps[21]. I will show in section 4 that these single ions are isolated from the environment to such a degree that the decoherence timescale is on the order of seconds or longer with existing technical ion-trap equipment. Moreover it is p ossible to excite these atoms b efore they are correlated with the environment to such a degree that complete decoherence took place. In our example ab ove Silvia1 switches on the microwave emitter long enough to excite an ion in a trap with a large probability. After that, Silvia2 measures the state of the ion and finds that it is excited with some finite probability, though Silvia verified it was in the ground state b efore the branching took place. From that Silvia2 infers the existence of Silvia1. In an obvious way Silvia1 and 2 can exchange informations (bit strings of arbitrary length), e.g. by preparing more than one isolated ion. Single ions in traps can act as "gateway states" and communication b etween parallel worlds is p ossible. Let us write down the evolution of the wave function describing the prop osed exp eriment explicitly in several time steps. We write the initial wave function |t0 of our system (the lab oratory with all its contents shortly b efore the exp eriment b egins at time t0 ) as a direct product of several "subsystems" (in the sense of Zurek [6]). The chosen factoring is somewhat arbitrary, the final results are indep endent of the choice to a good approximation, 5


however. (1) |
t0

= |P |f

ilter

| |A

Here |P stands for the initial state of the photon which can b e represented by the coherent 1 sup erp osition 2 (|P1 +|P2 ) of the two p olarization states of the photon (the subindex 1 indicates a p olarization plane parallel to the transmission direction of the filter, and 2 at a 90 o angle to this direction). |f ilter describ es the p olarization filter, | describ es the lab oratory including all further exp erimental equipment, p ossibly produced microwave fields and Silvia. The isolated ion in its trap is symb olized by |A . A commerical linear p olarization filter is macroscopic and its Poincar´ recurrence time is much larger than any e other time scale in the exp eriment. Therefore it qualifies as "environment" [6] and some time after the photon |P has interacted with the filter (at time t1 ) the two comp onents of |P have decohered and we obtain to very good precision two distinct decohered subsystems ("worlds"). Let us call this time, when |f ilter has already decohered but the other subsystems | and |A did not yet interact with |P "t1 " (such a time can surely b e found, even if it would b e only b ecause of the finite c). At this time the state of the subsystem "photon and filter" no longer corresp onds to any one ray in Hilb ert space (it is describ ed by a mixture). Rather the decoherence process has selected two sp ecial states. While the exact nature of these states is not yet entirely clear, current research suggest that they are characterized by maximal thermodynamical stability, i.e. they are states with minimal increase in entropy[22]. Let us symb olized these two orthogonal vectors in Hilb ert space in the following way: (2) (3) |W |W
1 2

= |P1 f = |P2 f

ilter 1 ilter 2

I left out the direct product symb ol b etween the symb ols to indicate that they are in an entangled state.These functions are very nearly orthogonal to each other and will stay like that forever. However one should not conclude that the process of decoherence is already finished. It is finished only later when all subsystems are decohered. The rest of the lab oratory and the ion can still b e describ ed by pure states as can the state of the total system at time t1 : (4) |
t1

1 = (|W 2

1

+ |W2 ) | |A the deld') can

Just like the p olarizer "measured" the two states of the photon |P via decoherence, subsystem | (including Silvia) "measures" the state of the p olarizer. The resulting coherence leads to two distinct subsystems: |W1 =|P1 f ilter1 1 (`photon detected wor and |W2 = |P2 f ilter2 2 ("no photon detected world"). The final state at a time t2 b e written as: 1 (5) |t2 = (|P1 f ilter1 1 + |P2 f ilter2 2 ) |A 2 6


The "branches" |W1 and |W2 are orthogonal to a very high precision, this also guarantees the stability of the records whether the p olarized photon was detected in the further evolution of the system. To reach a final state at time t3 in which also |A is decohered into two comp onents (see b elow and section 3 for a more detailed discussion of this decoherence process), the ion has to interact with the rest of our system. It is p ossible to excite the ion during the decoherence process, i.e. the interaction during the time interval tdec =t3 -t2 can excite A. When I fine tune the technical set up I can make sure that the time interval texc necessary to excite |A to |A is much smaller than tdec . These two time scales have no direct relation to each other. In this case we have for the final state: 1 (6) |t3 = (|P1 f ilter1 1 A + |P2 f ilter2 2 A ) 1 2 2 It is of course also p ossible not to excite |A in the course of decoherence. The p ossibility of this choice allows for communication. The excitation of an internal degree of freedom of a subsystem does not necessarily lead to decoherence as the reader might think at first. A counter example are Welcher Weg detectors[23], in which atoms can b e excited in micromasers without momentum transfer and necessary loss of quantum coherence. Let us discuss in more detail what happ ens when |A is excited from only one world. Immediately after the excitation, at time t2 +texc (texc tdec ), only a part of the phase space in which the ion resides is excited. It is the part corresp onding to the one macroscopic world |W1 exciting the ion (macroscopic states are very well localized in phase space[25]). After unitary evolution of |A for a short time interval of the order of tmix = dcoh m/p dcoh dm/h, the excited part of phase space b egins to overlap with the unexcited one and it is no longer p ossible to treat their temp oral evolution indep endently. Here dcoh is the coherence length of the system in the branch exciting the ion, which is extremely small for macroscopic ob jects[25], m is the mass of the ion and p is the momentum uncertainty of a region in phase space with extension dcoh . The momentum uncertainty p is approximately given as h/d where d is the spatial extension of the trap. A time scale analogous to tmix ("duration of reduction") in a somewhat different situation was introduced by Dicke[24 ]. tmix can b e shown to b e negligibly small for all exp erimental purp oses (very roughly O(10-15 sec) for typical trap sizes (µm) and decoherence lengths as quoted by Tegmark[25]). Because of the mentioned overlap a measurement of the excitation of |A from the other world |W2 , which measures another part of phase space than a measurement from |W1 , also finds the ion in an excited state. Only after complete decoherence of |A the parts of phase space seen by |W1 and |W2 have an indep endent temp oral evolution.

3

Determination of the decoherence timescale of the single ion

I now quantitatively calculate the time scale tdec if the decoherence of the ion wavefunction |A into |A1,2 as defined in the previous section. For this I will analyze the transition from 7


eq.(5) to eq.(6) in greater detail than b efore. This analysis is indep endent of whether the ion is excited b etween t2 and t3 or not. I will use the "dilute gas" approximation develop ed by Harris and Stodolsky [26, 27]. The interaction of systems is treated in terms of a series of distinct collisions b etween the ion in the trap and particles from the rest of the system. The correctness of this simplification in the case of weak coupling has b een verified with a full second quantized calculation by Raffelt, Sigl and Stodolsky[28]. The chirality states |± of Harris and Stodolsky[26] are analogous to our macroscopic states |W1,2 of the previous paragraph, and their "medium" is the ion in the trap in our case. Parallels b etween the chirality and macroscopic states were already p ointed out by Joos and Zeh[29]. It seems strange at first sight that a single ion in a given "simple" state plays the role of the "medium". With "simple" I mean that the state of the ion in its trap has only few degrees of freedom which are completely determined e.g. by a harmonic oscillator wavefunctions, whereas a "medium" typically has a very large numb er of degrees of freedom and is thus able to exert random influences on a system. Take into account however that in quantum field theory the wave field always has an infinite numb er of degrees of freedom[30]. In the MWI it is this field which represents all systems and the "simplicity" of the state |A of the ion b efore decoherence at time t0 exists only relative to the subsystem S1 =(|P1 + |P2 ) |f ilter | in eq.(1) (Everett called the MWI "relative-state interpretation" [13]). If this subsystem decohered into two orthogonal states |W1,2 at time t2 the ion |A can no longer b e in a "simple" state relative to b oth of them, and additional degrees of freedom of the wave function |A b ecome dynamically imp ortant. After interaction of |A with the environment, at time t3 there will b e two orthogonal comp onents |A1,2 . Each one has an overall centre of mass wavefunction describ ed e.g. by a "simple" harmonic oscillator state relative to one of the worlds |W1,2 . It is wrong to conclude from that that they are identical, however: |A1 and |A2 are different for the same reason that the "copies" produced by branching from a given macroscopic ob ject are not identical: their "fine structure" in phase space is different. It is clear that our treatment is a gross simplification of the real world. An exact treatment has b een p ossible only for idealized models of the environment, e.g.: toy systems with few particles [18], ensembles of noninteracting harmonic oscillators[31] and scalar fields[32]. For the gravitational field an exact treatment is not p ossible even in principle at the moment, b ecause we lack a quantum theory of gravity. It has b een shown exp erimentally though that gravitational fields decohere if the MWI is correct[33]. The purp ose of this pap er is not to improve on the treatment of the very difficult theoretical problem of decoherence, but to suggest a new exp erimental approach on the quantum mechanical measurement problem. Our treatment gives roughly the correct order of magnitude for the decoherence time scale. Let us now define the relative states in the sense of Everett[13] of |A with resp ect to |W1 1 1 and |W2 at time t2 as |A1 = 2 |A and |A2 = 2 |A , resp ectively. At time t2 |A1 and |A2 1 are still the same or "parallel" in Hilb ert space[27]. We also have |A = 2 (|A1 + |A2 ) a decomp osition which is always p ossible for a pure state according to the sup erp osition 8


principle. The total wavefunction at time t2 can then b e written as: (7) |
t2

= |A1 |W

1

+ |A2 |W

2

This equation is analogous to equation (3) in Ref.[27]. Further following Harris and Stodolsky[26] we now write this wavefunction in the form of a density matrix in a basis of the Hilb ert space spanned by |W1,2 to represent the role of the phases in a b etter way: (8) (t2 ) = A1 |A1 A2 |A1 A1 |A2 A2 |A2

In the initial state t2 the ion and its environment are uncorrelated and all elements of this matrix have the value 1/2 in our case. In our approximation decoherence now leads to an exp onential damping of the off-diagonal elements of this density matrix, while the diagonal elements remain unaffected. At time t3 the matrix is given to a very good approximation by 1/2 the identity matrix. The decoherence time scale in the transition from t2 to t3 is then given as the inverse of the exp onential damping time constant. If there was no internal excitation during the process of decoherence, |A1 and |A2 are identical yet distinguishable in the classical sense (i.e. by way of their structure in phase space) at time t3 . I approximate the temp oral evolution of the off-diagonal elements of as an effect of rep eated scatterings of particles from |W1 and |W2 [26]. If the particles in |W1,2 are atoms (e.g. rest-gas atoms, see b elow section 4) their de Broglie wavelength (< 0.1 ° at A room temp erature) is much smaller than the typical spatial extension of the wavefunction |A of the ion in the trap (typically 0.1-1 µm in current technical setups[34]). It is then a good approximation for the treatment of the scattering to assume that the initial state of the ion is approximated by a plane wave front, and that the elastically scattered wave of the trapp ed ion is approximated by a radially outgoing wave front. I will always use this approximation in the following even in cases where it is less well justified b ecause the wavelength of the scattering particles in |W1,2 is equal to or larger than the spatial extension of |A (e.g. for microwave photons scattering on the ion). In this case the decoherence time scale will b e larger than my estimate (the scattering is less "effective"). To demonstrate that the decoherence timescale can b e large enough to allow interworld communication, my approach is sufficient. Also we will see b elow in section 4 that in our situation the most effective mechanism for decoherence is elastic scattering with rest gas atoms, for which my assumption holds well. The diagonal element A1 |A2 has to b e multiplied by a damping factor D for each scattering of the ion with a particle of |W1,2 as a target. If |AS is the wavefunction of the ion after scattering one can write: (9)
S AS |A2 = D A1 |A2 1

The damping factor after n collisions is given as: (10) Dn = D n . 9


In the sp ecial case of elastic and isotropic scattering and integrating over time one has for the final state after one scattering: (11) A1 S = o(eikz + f · eikr /r )
r + )

(12) A2 S = o(eikz + f · e(ik

/r )

where k is the wave numb er and z the direction of relative motion b etween the particle and the trapp ed ion. f is the scattering amplitude and r the radial distance from the ion. is a relative phase angle which takes random values over rep eated scatterings b ecause |W1 and |W1 are not in phase. The normalization factor o is given by: (13) o = 1 1 + f 2 /r
2

Inserting eqs.(11,12,13) into eq.(9) and integrating over the spatial volume one obtains: (14) D = o2 = 1 1 - f 2 /r 1 + f 2 /r 2
2

The neglect of higher order terms is justified in the dilute gas approximation. For n consecutive scatterings ones gets: (15) Dn (1 - f 2 /(r 2 ))n exp(-f 2 n/(r 2 )) Let us set f2 = /(4 ), where is the total elastic cross section, and n=4 r2 t, where is numb er of particles p er unit area and time on which the ion scatters and t the time span over which interactions b etween |A and |W1,2 takes place. The time evolution of the diagonal elements of the ion-environment density function is then obtained as: (16) Dt exp(- t) The decoherence time is then defined as: (17) t
dec

= 1/( )

This result for the decoherence time agrees with a different and more general calculation by Tegmark[25] for the sp ecial case of a system that is spatially much larger than the effective wavelength of the scattering particles. It was exactly this case that I assumed ab ove. Note that Tegmark calls "coherence time" what I call "decoherence time".

4

Practical realization of communication b etween parallelworlds

I will show that it is technically p ossible to realize a system which approximates the situation outlined in section 2. and which has macroscopic decoherence timescales. For my discussion I will assume the setup which Itano et al.[34] used for a measurement of quantum 10


pro jection noise. This is not in order to suggest that this is an optimal setup for inter-world communication; I only wanted to show that the technical capabilities to test the MWI exist in one concrete case. Itano et al.[34] trap single ions in radio frequency and Penning traps. The ion (I consider 199 Hg+ ) can b e stored for hours in a vacuum of ab out 10-9 atmospheres. They observe transitions b etween the 6s2 S1/2 F=0 and F=1 hyp erfine structure levels by applying rf fields of well-controlled frequency, amplitude and duration. The transition is in the microwave region (40.5 GHz). UV Lasers op erated at 194 nm are used to cool the ion, prepare its state and to measure whether the ion is in F=0 or F=1 state after an application of microwaves. In our example Silvia traps an ion and prepares it in the ground state. If Silvia1 now detects a photon after the p olarization filter she applies the rf field resonant with the F=0 F=1 transition, for a time long enough to excite the ion completely from the ground state to the F=1 state according to the Rabi flopping formula[35]. According to the orthodox interpretation she has to apply a so called " pulse" pulse of length tp and field strength E so that (18) E = ( ¯ )/(tp ) h is the magnetic dip ole transition element b etween the F=0 and F=1 states (the transition is forbidden for electric dip ole radiation) which is given in good aproximation by the Bohr magneton b ecause the wavefunctions of the two states are quite similar. Let us assume that Silvia1 applies a pulse which is a factor 2 longer to comp ensate for the fact that Silvia2 does not apply any pulse ("MWI pulse"). This whole action will take something like a second at least (for a mechanical "Silvia" it could b e p erformed faster, certainly within a µsec). Silvia2 waits for a certain time to allow Silvia1 to apply the microwave field. After this she applies a Laser field to determine the state of the ion. If the MWI interpretation is correct, Silvia2 will find it in a fraction p of the exp eriments in the F=1 case prepared by Silvia1. If the inelastic microwave excitation is the only interaction of ion with the environment (i.e. the ion is completely isolated) we get for the damping factor due to excitation according to eq.(16): (19) Dt exp(-
exc

t)

here exc is the cross section of the ion for excitation from the F=0 to the F=1 level with resonant microwave radiation. t is the time p eriod for which the rf field was applied, and is the flux of the exciting radiation. The excitation probability is given as: (20) p = sin2 ( t) where =E /(h2 2). For a "MWI pulse" p is 1 and Dt can b e easily evaluated as ¯ 1/e. Intuitively one can say, that in this situation only one full interaction took place (the absorption of one microwave photon). Complete decoherence needs more than one interaction so Dn is much larger than zero. Normally Silvia2 will completely decohere the 11


ion when determining its state with the method decrib ed by Itano et al.[34], b ecause the detection of the fluorescence radiation is very inefficient, and many inelastic collisions of 194 nm photons take place for a state determination. This calculation is only correct in the "one-and-only-one interaction" approximation of Stodolsky[27] in which the different collisions of the ion on other particles are treated as completely indep endent. It is unavoidable in our situation that there is "feedback", i.e. a given collision acts on a wavefunction of the ion which is already decohered to some degree by the previous collisions. As a result the excitation of |A2 will b e less effective and p will b e somewhat smaller than 1. Its exact value dep ends on the detailed geometry of the exp erimental setup but is clearly never much smaller than 1, b ecause in the absence of other mechanisms the correlation has its origin in the inelastic scattering of the ion. I find with a numerical calculation that e.g. a "MWI pulse" applied in world 1 would lead to p=0.163 in the "feedback" case, versus p=1.0 in the "one-and-only-one interaction" approximation. In this calculation I made the simplifying assumption that D develops strictly according to eq.(16). Itano et al.[34] rep eated the cycle "preparation-rf field application-measurement" for hundreds of times in their exp eriment so also values of p much smaller than 1 would b e measurable. We have to check if decoherence by other sources can b e avoided for at least a few seconds so that the assumption of complete isolation of the ion made in the previous paragraph is justified. These sources are: a. scattering of remnant gas atoms in the trap on the ions b. elastic scattering of the microwave field on the ion c. interaction with the constraining fields holding the ion Note that only b. is in principle unavoidable, the others could b e avoided with a more advanced technology. For contribution a. I get, inserting typical op erating parameters of the set up used by Itano et al.[34] into eq.(17): (21) t
dec

=8

2.4·10-18 m c

2

T 300K

o

nbar sec p

here c is the elastic cross section; its size (for room temp erature) has b een taken for H2 -Hg collisions (at room temp erature) from the calculation of Bernstein [36]. T is the temp erature, its dep endence here does not take into account the change of c with energy (which is however very small around room temp erature). p is the rest-gas pressure. It is p ossible to achieve vacua much b etter than a nbar in ion traps (see e.g.Ref. [37]). For b. one gets in the same way the decoherence time scale of elastic scattering of a microwave field with a frequency and an intensity that effects a pulse in tp seconds. For the flux in eq.(17) I set: (22) =
2 0 cE h ¯

12


where E is the electric field strength of a MWI- pulse(eq.(18). Inserting this relation gives: (23) t
dec

2.8 · 1022

µB /c

tp sec

5.2·10-40 m

2

40.5Ghz

sec

The cross section is the Thompson cross section which I averaged over scattering angle. The Rayleigh cross section is negligible in our situation. Case c. is treated in a similar way b ecause it is well known that only time dep endent fields can cause decoherence [6]. Even for Penning traps with static fields it is imp ossible to prevent residual time variability with a fraction fv of the total field strength. Without load (as in our case) fv 10-10 is achievable for static confining fields Ec with a strength of ab out 1000 V/m typical for the traps used by Itano et al. (their ion-trap setup is describ ed in Gilb ert et al.[38]). The "worst" case (leading to the shortest decoherence time) is a variability on a time scale similar to the duration of the exp eriment. For this case one then obtains: (24) t
dec

76

5.2·10-40 m

2

1H z

1000V /m Ec

2

f

-2 v

sec

Though it is not of critical imp ortance for our problem, it is easy to show that the decoherence time scale induced by UV Lasers used by Itano et al.[34] via Rayleigh scattering is on a time scale of many years. This surprising ineffectiveness of light to decohere wave functions was already noticed by Joos and Zeh in the connection with chiral eigenstates of molecules [29]. As p ointed out in the previous paragraph eqs.(23,24) are exp ected to underestimate the true decoherence time, b ecause I assumed in their derivation that the wavelength of the particles on which the ion scatters is much smaller than the spatial extension of the ion wavefunction, which does not hold in typical setups. The reader might ob ject that something has to b e wrong with my prop osal b ecause it violates energy conservation in a given world (Silvia2 could receive energy from a parallel world). Fundamental principles (like invariance to time translations [39]) require energy conservation only for the whole universe however, and not for single branches which are very sp ecial entities singled out by individual humans. Because the energy Silvia2 receives is always lost by Silvia1 there is no violation of energy conservation in the universe. Dicke found some time ago that energy conservation is violated in certain quantum mechanical measurement setups for arbitrarily long times[24]. He holds that this p oses no serious problem b ecause the exp ectation value for the amount of energy violation turns out to b e zero (i.e. rep eating the measurement many times, energy is lost as often as it is gained). In the conventional interpretation of quantum mechanics there seems to b e a problem however, b ecause Dicke's result means that e.g. the fundamental principle of time tanslation invariance would b e violated on macroscopic time scales. In the MWI Dicke's situation corresp onds to worlds which have a different energy exp ectation value of the system immediately after they were created due to branching (one is higher and the other lower than the one b efore 13


branching[24]). The average of their energy exp ectation values is the energy exp ectation value b efore the branching, and energy conservation holds at all times. This "restoration of conservation laws" in the MWI, which arises when all branches of the quantum state are considered together was already p ointed out by Elitzur and Vaidman[40].

5

Conclusion

The prediction of the orthodox interpretation [5] is that the ion in our example exp eriment is never observed in an excited state by Silvia2: the measurement is surely finished after the photon from the p olarization filter has not b een detected by Silvia2 and thereafter only Silvia2 exists. The "logical" and "many histories" interpretations [12] probably lead to a similar exp ectation, though it is not completely clear to me what their quantitative prediction would b e. Hidden variable models are devised in order to "destroy" Silvia1; their predicition is therefore the same as in the orthodox interpretation by definition. For the MWI it has b een shown in the previous sections that inter-world communication on a time scale of minutes should b e p ossible with state of the art quantum-optical equipment. The exp erimental verification of this p ossibility would thus rule out the ab ove mentioned alternatives to the MWI. The limiting factor in extending tdec even further (i.e. in "keeping the communication channel op en for longer") seems to b e the rest gas in the vacuum of the ion trap at the moment. The fascinating problem of how to optimize the communication in order to transfer large amounts of data (e.g. TV pictures) would b e b eyond the scop e of this pap er. The detection of parallel worlds would finally clarify the fundamentals of nonrelativistic quantum mechanics: nature would have an ob jective deterministic reality completely indep endent of human consciousness and fully describ ed by the Schrodinger equation together ¨ with the second quantization conditions for the wave field. To communicate with parallel worlds goes of course completely against "common sense", but it does not lead to any inconsistencies or violations of known physical principles. A similar opinion was voiced by Polchinski[41] who showed that interworld communication is p ossible within Weinb erg's nonlinear quantum mechanics. The recent sp eculation of Gell-Mann and Hartle[42] ab out a p ossible communication with "goblin worlds" has also certain parallels with the prop osal of this pap er. The applications of this effect in physics would b e manifold e.g. in the investigation of Chaos or for improving statistics in the study of rare processes. Outside physics interworld communication would lead to truly mind-b oggling p ossibilities, e.g. in psychological research or for the extension of computing capabilities in computers and humans.

References

14


[1] R.Omn`s, Rev.Mod.Phys. 64, 339 (1992), is an exhaustive review of the field; e W.H.Zurek, Physics Today 44(10),36 (1991), is a p edagocical introduction, see also letters ab out this article and Zurek's reply in: Physics Today 46(4),13 (1993). [2] W.Heisenb erg, The physicists conception of nature (Hutchinson,London,1958). [3] L.E.Ballentine, Rev.Mod.Phys. 42,358 (1970). [4] N.Bohr, Atomic Theory and the Description of Human Know ledge (Cambridge University Press, Cambridge, 1934) p.19. [5] J. von Neumann, Mathematical Foundations of quantum mechanics (Princeton University Press,Princeton,1955) chapt.VI. [6] W.H.Zurek, Prog.Theor.Phys. 89,281 (1993), also available in xxx.lanl.gov e-Print archive under gr-qc 9402011; is an exhaustive review on the sub ject of decoherence, I followed Zurek's standp oint in the present pap er. [7] R.Omn`s, Ann.Phys.(N.Y.) 201,354 (1990) (citation from p.361). e [8] P.R.Holland and J.P.Vigier, Found.Phys. 18,741 (1988); F.J.Belinfante, A Survey of Hidden-Variable Theories (Pergamon, Oxford, 1973). [9] a review on the issue of EPR correlations is: D.N.Mermin, Physics Today 39(4),38 (1985); the most recent exp eriments are: P.R.Tapster,J.G.Rarity and P.C.M.Owens, Phys.Rev.Lett. 73, 1923 (1994); P.G.Kwiat,A.M.Steinb erg and R.Y.Chiao, Phys.Rev. A47, R2427 (1993); T.E.Kiess,Y.H.Shih,A.V.Sergienko and C.O.Alley, Phys.Rev.Lett. 71,3893 (1993). [10] R.T.Jones and E.G.Adelb erger, Phys.Rev.Lett. 72,2675 (1994). [11] M. Gell-Mann and J.B.Hartle,in Proc. Intern. symposium Foundations of Quantum Mechanics, S.Kobayashi et al.,ed. (The Physical Society of Japan, Tokyo, 1989) p.321;the remark can b e found in the discussion section in resp onse to a question by P.Mittelstaedt. [12] R.B.Griffiths, Phys.Rev.Lett. 70,2201 (1993); M.Gell-Mann and J.B.Hartle, Phys.Rev. D47,3345 (1993); exhaustive and p edagogic lectures can b e found in: J.B.Hartle, in Quantum Cosmology and Baby Universes, S.Coleman et al.,ed. (World Scientific, Singap ore, 1991), p.67. [13] H.Everett I I I, Rev.Mod.Phys. 29,454 (1957).

15


[14] B.S. DeWitt, Physics Today 23(9),30 (1970); B.S. DeWitt, in Fondamenti di meccanica quantisitica, B.D'Espagnat,ed. (Academic, New York, 1971), p.211; (here DeWitt uses the expression "many-universe interpretation" which can give rise to misunderstandings in my opinion). [15] occasionally the MWI is interpreted in a way in which the "splitting" requires some new mechanism outside of known physics, see e.g. M.A.B. Whitaker, J.Phys. A 18,253 (1985); the assumption of such a mechanism leads to various problems with the MWI as discussed in this reference. I hold the view that one is lead inevitably (and without further mechanisms) to the MWI if one assumes that the Schrodinger equation is a ¨ complete and ob jective description of reality and takes into account decoherence. A similar view is voiced by Zurek who finds the MWI "unsatisfying" however, see the discussion section to his article: W.H.Zurek, in Conceptual Problems of Quantum Gravity, A.Ashtekar and J.Stachel,ed. (Birkh¨user, Boston, 1991), p.43. a [16] H.D.Zeh, Found.Phys. 3,109 (1973). [17] H.D.Zeh, Phys.Lett.A 172,189 (1993). [18] A.Albrecht, Phys.Rev. D48,3768 (1993). [19] D.Deutsch, Int.J.theor.Phys. 24,1 (1985). [20] J.Clarke et al., Science 239,992 (1988). [21] H.Dehmelt, Am.J.Phys. 58,17 (1990). [22] W.H.Zurek, S.Habib, J.P.Paz, Phys.Rev.Lett. 70,1187 (1993). [23] M.O.Scully and H.Walther, Phys.Rev. A39,5229 (1989). [24] R.H.Dicke, Am.J.Phys. 49,925 (1981); R.H.Dicke, Found.Phys. 16,107 (1986). [25] M.Tegmark, Found.Phys.Lett. 6,571 (1993). [26] R.A.Harris and L.Stodolsky, Phys.Lett. 116B,464 (1982). [27] L.Stodolsky, in Quantum coherence,J.S.Anandan,ed. (World Scientific, Singap ore, 1990), p.320. [28] G.Raffelt,G.Sigl and L.Stodolsky, Phys.Rev.Lett. 70,2363 (1993). [29] E.Joos and H.D.Zeh, Z.Phys. B59,223 (1985). 16


[30] L.I.Schiff, Quantum Mechanics, (Mc Graw Hill, Singap ore, 1985), 3rd ed.,Chap.14. [31] A.O.Caldeira and A.J.Legget, Phys.Rev. A31,1059 (1985). [32] W.G.Unruh and W.H.Zurek, Phys.Rev. D40,1071 (1989). [33] D.N.Page and C.D.Geilker, Phys.Rev.Lett. 47,979 (1981). [34] W.M.Itano et al., Phys.Rev. A47,3354 (1993). [35] M.SargentI I I,M.O.Scully and W.E.Lamb, Laser physics (Addison-Wesley, Reading, 1974), p.27. [36] R.B.Bernstein, J.Chem.Phys. 34,361 (1961). [37] F.Diedrich and H.Walther, Phys.Rev.Lett. 58,203 (1987). [38] S.L.Gilb ert et al., Phys.Rev.Lett. 60,2022 (1988). [39] E.L.Hill, Rev.Mod.Phys. 23,253 (1951). [40] A.C.Elitzur and L.Vaidmann, Found.Phys. 23,987 (1993). [41] J.Polchinski, Phys.Rev.Lett. 66,397 (1991). [42] M.Gell-Mann and J.B.Hartle, Equivalent Sets of Histories and Multiple Quasiclassical Domains, preprint University of California at Santa Barbara UCSBTH-94-09 (1994).

Acknowledgements
I thank M.Jarnot, R.Mirzoyan and S.Pezzoni for discussions and helpful remarks on the manuscript and L.Stodolsky for explanations ab out the nature of decoherence. My sister Fritzi Plaga encouraged me in an imp ortant way to work on the sub ject of the present pap er.

17