A field theoretic causal model of a Mach-Zehnder Wheeler delayed-choice experiment

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by  P. N. Kaloyerou

A field theoretic causal model of a Mach-Zehnder Wheeler
delayed-choice experiment
P.N. Kaloyerou
6 The University of Oxford, Wolfson College, Linton Road,
0 Oxford OX2 6UD, UK∗
0
2
February 1, 2008

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Abstract

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WeconsideraWheelerdelayed-choiceexperimentbasedontheMach-ZehnderInterferometer.
v
Sincethedevelopmentofthecausalinterpretationofrelativisticbosonfieldstherehavenotbeen
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anyapplicationsforwhichtheequationsofmotionforthefieldhavebeensolvedexplicitly. Here,
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weprovideperhapsthefirstapplication ofthecausalinterpretation ofboson fieldsforwhichthe
0
equations of motion are solved. Specifically, we consider the electromagnetic field. Solving the
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1 equationsofmotionallowsustodeveloparelativisticcausalmodeloftheWheelerdelayed-choice
3 Mach-Zehnder Interferometer. We show explicitly that a photon splits at a beam splitter. We
0 alsodemonstratetheinherentnonlocalnatureofarelativisticquantumfield. Thisisparticularly
/ revealed in a which-path measurement where a quantumis nonlocally absorbed from both arms
h
of the interferometer. This feature explains how when a photon is split by a beam splitter it
p
- neverthelessregisters onadetectorinonearm oftheinterferometer. Bohmet al[1]haveargued
t that a causal model of a Wheeler delayed-choice experiment avoids the paradox of creating or
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changing history, but they did not provide the details of such a model. The relativistic causal
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u model we develop here serves as a detailed example which demonstrates this point, though our
q model is in terms of a field picture rather than the particle picture of the Bohm-de Broglie
: nonrelativistic causal interpretation.
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i
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1 INTRODUCTION
r
a
In 1978 Wheeler [2] described seven delayed-choice experiments. The experiments are such that the
choice of which complementary variable to measure is left to the last instant, long after the relevant
interaction has taken place. Of the seven experiments the delayed-choice experiment based on the
Mach-Zehnder interferometer is the simplest for detailed mathematical analysis. Here we present
a detailed model of this experiment based on the causal interpretation of the electromagnetic field
(hereafter referred to as CIEM) [3]. CIEM is a specific case of the causal interpretation of boson
fields. The experimentalarrangementofthe delayed-choiceMach-Zehnderinterferometeris shownin
figure 1.
A single photon1 enters the interferometer at the first beam splitter BS . The two beams that
1
emergearerecombinedatthesecondbeamsplitterBS byuseofthetwomirrorsM andM . C and
2 1 2
D are two detectors which can be swung either behind or in frontof BS . The detectors in positions
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C and D in front of BS measure which path the photon traveledand hence a particle description
1 1 2
is appropriate according to the orthodox interpretation. With the counters in positions C and D
2 2
after BS interference is observed and a wave picture is appropriate. A phase shifter P producing a
2
phase shift φ is placed in the β-beam. For φ=0 and a perfectly symmetrical alignment of the beam
splitters and mirrors, the d-beam is extinguished by interference and only the c-beam emerges.
∗emailaddress: [email protected]
1Hereweusetheterm‘photon’ tomeanasinglequantum of energyoftheelectromagnetic field. Wedonotimply
anyparticle-likepropertiesbythisterm. Wehaveshowninreference[4]thatasinglephotonstateisanonlocalplane
wavespreadoverspace.
1
Fig.1. Delayed-choice Mach-Zehnder interferometer
Ifweattributephysicalrealitytocomplementaryconceptssuchaswaveandparticleconcepts,then
we are forcedto conclude either that (1) the history ofthe micro-systemleading to the measurement
is alteredby the choice of measurement, or (2) the history of the micro-systemis created at the time
of measurement.
Wheeler [2] [5], following Heisenberg [6], in some sense attributed reality to complementary con-
cepts following measurementand adoptedview (2) above,namely, that historyis createdatthe time
of measurement. Thus he states, ‘No phenomenon is a phenomenon until it is an observed phe-
nomenon,’[7]. Headdsthat‘Registeringequipmentoperatinginthe hereandnowhasanundeniable
part in bringing about that which appears to have happened’ [8]. Wheeler concludes, ‘There is a
strange sense in which this is a “participatory universe” ’ [8].
In Wheeler’s description, the question of the possibility of creating a causal paradox is raised.
One canargue,however,in the spirit ofBohr,thatWheeler-delayedchoice experimentsare mutually
exclusive in the sense that if the history of a system is fixed by one experiment, this history cannot
be affected by another Wheeler delayed-choice experiment. But, it is not obvious that the paradox
can be avoided in this way.
Bohr and Wheeler share the view that ‘no phenomenon is a phenomenon until it is an observed
phenomenon’ but Bohr differs from Wheeler (and Heisenberg) in that he denies the reality of com-
plementary concepts such as the wave concept and the particle concept. We summerize the features
of Bohr’s principle of complementarity [9][10][11] as follows: (1) Pairs of complementary concepts
require mutually exclusive experimental configurations for their definition, (2) Classicalconcepts are
essential as abstractions to aid thought and to communicate the results of experiment, but, physical
reality cannot be attributed to such classical concepts, and (3) The experimental arrangement must
be viewed as a whole, not further analyzable. Indeed, Bohr defines “phenomenon” to include the
experimental arrangement. Hence, according to Bohr a description of underlying physical reality is
impossible. It follows from this that the complementary histories leading to a measurement have no
more reality than the complementary concepts to which the histories are associated. According to
Bohr, then, complementary histories, like complementary concepts, are abstractions to aid thought.
Infact,Bohrhadanticipateddelayed-choiceexperimentsandwrites[12],‘...itobviouslycanmake
nodifferenceasregardsobservableeffectsobtainablebyadefiniteexperimentalarrangement,whether
our plans of constructing or handling the instruments are fixed beforehand or whether we prefer to
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postpone the completion of our planing until a later moment when the particle is already on its way
from one instrument to another’. Bohr also considers a Mach-Zehnder arrangement [13], but not in
the delayed-choice configuration.
Complementarity is not tied to the mathematical formalism. Jammer writes [14], ‘That comple-
mentarity and Heisenberg-indeterminacy are certainly not synonymous follows from the simple fact
that the latter... is animmediate mathematical consequence of the formalism of quantum mechanics
or, more precisely, of the Dirac-Jordan transformation theory, whereas complementarity is an extra-
neous interpretative addition to it’. Indeed, the whole process from the photon entering BS to the
1
finalactofmeasurementisdescribeduniquelybythewavefunction(orthewavefunctionalifquantum
field theory is used, as we shall see). The mathematical description leading up to the measurement
is completely independent of the last instant choice of what to measure. The wave function or wave
functionaldevelopscausally. Indeed,itisbecauseintheCausalInterpretationmathematicalelements
associatedwiththewavefunctionorwavefunctionalareinterpreteddirectlythatacausaldescription
is possible. Wheeler’s assertion that a present measurement can affect the past is seen not to be a
consequence of the quantum formalism, but rather, rests on an extraneous interpretative addition.
The Bohr view can also be criticized. The denial of the possibility of a description of underlying
physical reality seems a high price to pay to achieve consistency.
Clearly, in a causal model of the delayed-choice experiment the issue of changing or creating
history is avoided. The history leading to measurement is unique and completely independent of the
last instant choice of what to measure. There is no question of a present measurement affecting the
past. Bohm et al [1] provided just such a causal description of the Mach-Zehnder Wheeler delayed-
choice experiment based on the Bohm-de Broglie causal interpretation [15] [16], though in general
terms without solving the equations of motion. In this nonrelativistic model, electrons, protons etc.
are viewed as particles guided by two real fields that codetermine each other. These are the R and
S-fields determined by the wave function ψ(x,t)=R(x,t)exp[iS(x,t)/¯h]. The particle travels along
one path which is revealed by a which-path measurement (detectors in front of BS ). The R and
2
S-fields explain interference when the detectors are positioned after BS .
2
Attempts to extend the Bohm-de Broglie causal interpretation to include relativity led to the
causal interpretation of Boson fields [17][18][19][3] of which CIEM is a particular example. In CIEM
the beable is a field; there are no particles. Here, we apply CIEM to the Wheeler delayed-choice
Mach-Zehnderinterferometer. Inparticular,wesetupandsolvetheequationsofmotionforthefield.
Thesolutionsallowustobuildadetailedcausalmodeloftheexperiment. InCIEMthebasicontology
is that of a field, not of a particle as in the Bohm-de Broglie nonrelativistic causal interpretation.
We will see that a quantum field behaves much like a classical field in many respects but not in
all. The essential difference is that a quantum field is inherently nonlocal. We will show explicitly
that a photon is split by a beam splitter. In this case, we will have to show how in a which-path
measurement,despitebeingsplitbythebeamsplitter,aphotonregistersinonlyoneofthedetectors.
Wewilldothisbymodelingthedetectorsashydrogenatomsundergoingthephotoelectriceffect,and
show, using standard perturbation theory, that a photon is absorbednonlocally from both beams by
only one of the atoms. Since we have a wave model interference is explained in the obvious way. In
the next section we will briefly summerize CIEM, and in section 3 we apply CIEM to the Wheeler
delayed-choice Mach-Zehnder interferometer.
2 OUTLINE OF CIEM
In what follows we use the radiation gauge in which the divergence of the vector potential is zero
.A(x,t) = 0, and the scalar potential is zero φ(x,t) = 0. In this gauge the electromagnetic field
∇
has only two transverse components. Heavyside-lorentz units are used throughout.
SecondquantizationiseffectedbytreatingthefieldA(x,t)anditsconjugatemomentum Π(x,t)
as operators satisfying the equal-time commutation relations. This procedure is equivalent to intro-
ducing a field Schr¨odinger equation
∂Φ[A,t]
(A′, Π′)Φ[A,t]dx′ =i¯h , (1)
H ∂t
Z
3
where the Hamiltonian density operator is obtained from the classical Hamiltonian density of the
H
electromagnetic field,
1 1
= (E2+B2)= [c2 Π2+( A)2], (2)
H 2 2 ∇×
by the operator replacement Π i¯hδ/δA. A′ is shorthand for A(x′,t) and δ denotes the
→ −
variational derivative2. The solution of the field Schr¨odinger equation is the wave functional Φ[A,t].
The square of the modulus of the wave functional Φ[A,t]2 gives the probability density for a given
| |
field configurationA(x,t). This suggests that we take A(x,t) as a beable. Thus, as we have already
said, the basic ontology is that of a field; there are no photon particles.
WesubstituteΦ=R[A,t]exp(iS[A,t]/¯h),whereR[A,t]andS[A,t]aretworealfunctionalswhich
codetermine one another, into the field Schr¨odinger equation. Then, differentiating, rearranging and
equating imaginary terms gives a continuity equation:
∂R2 δ δS
+c2 R2 dx′ =0.
∂t δA′ δA′
Z (cid:18) (cid:19)
The continuity equation is interpreted as expressing conservation of probability in function space.
Equating real terms gives a Hamilton-Jacobi type equation:
∂S 1 δS 2 ¯h2c2 δ2R
+ c2+( A′)2+ dx′ =0. (3)
∂t 2 δA′ ∇× − R δA′2
Z (cid:18) (cid:19) (cid:18) (cid:19)
This Hamilton-Jacobi equation differs from its classical counterpart by the extra classical term
1 ¯h2c2 δ2R
Q= dx′,
−2 R δA′2
Z
which we call the field quantum potential.
By analogy with classical Hamilton-Jacobi theory we define the total energy and momentum
conjugate to the field as
∂S[A] δS[A]
E = , Π = .
− ∂t δA
InadditiontothebeablesA(x,t)and Π(x,t)wecandefineotherfieldbeables: theelectricfield,
the magnetic induction, the energy and energy density, the momentum and momentum density, the
intensity, etc. Formulae for these beables are obtained by replacing Π by δS/δA in the classical
formula.
Thus, we can picture an electromagnetic field as a field in the classical sense, but with the ad-
ditional property of nonlocality. That the field is inherently nonlocal, meaning that an interaction
at one point in the field instantaneously influences the field at all other points, can be seen in two
ways: First, by using Euler’s method of finite differences a functional can be approximated as a
function of infinitely many variables: Φ[A,t] Φ(A ,A ,...,t). Comparison with a many-body
1 2
→
wavefunction ψ(x ,x ,...,t) reveals the nonlocality. The second way is from the equation of motion
1 2
of A(x,t), i.e., the free field wave equation. This is obtained by taking the functional derivative of
the Hamilton-Jacobi equation, (3):
1 ∂2A δQ
2A = .
∇ − c2 ∂t2 δA
In general δQ/δA will involve an integral over space in which the integrand contains A(x,t). This
means that the way that A(x,t) changes with time at one point depends on A(x,t) at all other
points, hence the inherent nonlocality.
2Forascalarfunctionφthe variationalorfunctional derivativeisdefinedas δδφ = ∂∂φ −Σi ∂ ∂∂φ [20]. Fora
(cid:18) ∂xi (cid:19)
vector functionAwehavedefinedittobe δδA = δAδxi+ δAδyj+ δAδzk. (cid:0) (cid:1)
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2.1 Normal mode coordinates
To proceed it is mathematically easier to expand A(x,t) and Π(x,t) as a Fourier series
1 1
A(x,t)= εˆ q (t)eik.x, Π(x,t)= εˆ π (t)e−ik.x, (4)
1 kµ kµ 1 kµ kµ
V 2 V 2
kµ kµ
X X
where the field is assumed to be enclosed in a large volume V = L3. The wavenumber k runs from
to + and µ=1,2 is the polarization index. For A(x,t) to be a real function we must have
−∞ ∞
εˆ q =εˆ q∗ . (5)
−kµ −kµ kµ kµ
Substituting eq.’s (2) and (4) into eq. (1) gives the Schr¨odingier equation in terms of the normal
modes q :
kµ
1 ∂2Φ ∂Φ
¯h2c2 +κ2q∗ q Φ =i¯h . (6)
2 kµ − ∂qk∗µ∂qkµ kµ kµ ! ∂t
X
The solution Φ(q ,t) is an ordinary function of all the normal mode coordinates and this simplifies
kµ
proceedings. We substitute Φ = R(q ,t)exp[iS(q ,t)/¯h], where R(q ,t) and S(q ,t) are real
kµ kµ kµ kµ
functionswhichcodetermineoneanother,intoeq. (6). Then,differentiating,rearrangingandequating
real terms gives the continuity equation in terms of normal modes:
∂R2 c2 ∂ ∂S c2 ∂ ∂S
+ R2 + R2 =0.
∂t 2 ∂q ∂q∗ 2 ∂q∗ ∂q
kµ " kµ kµ! kµ (cid:18) kµ(cid:19)#
X
Equating imaginary terms gives the Hamilton-Jacobi equation in terms of normal modes:
∂S c2 ∂S ∂S κ2 ¯h2c2 ∂2R
+ + q∗ q + =0. (7)
∂t kµ " 2 ∂qk∗µ∂qkµ 2 kµ kµ − 2R ∂qk∗µ∂qkµ!#
X
The term
¯h2c2 ∂2R
Q= (8)
− 2R ∂q∗ ∂q
kµ kµ kµ
X
is the field quantum potential. Again, by analogy with classical Hamilton-Jacobi theory we define
the total energy and the conjugate momenta as
∂S ∂S ∂S
E = , π = , π∗ = .
−∂t kµ ∂q kµ ∂q∗
kµ kµ
The square of the modulus of the wave function Φ(q ,t)2 is the probability density for each q (t)
kµ kµ
| |
to take a particular value at time t. Substituting a particularset ofvalues of q (t) at time t into eq.
kµ
(4) gives a particular field configuration at time t, as before. Substituting the initial values of q (t)
kµ
gives the initial field configuration.
The normalized ground state solution of the Schr¨odinger equation is given by
Φ0 =Ne− kµ(κ/2h¯c)qk∗µqkµe− kiκct/2,
withN = ∞ (k/¯hcπ)12 3. Higher excitePdstatesareobtainePdby the actionofthe creationoperator
k=1
a† :
kµ Q
(a† )nkµ
Φ = kµ Φ e−inkµκct.
nkµ n ! 0
kµ
3The normalization factor N is found by substitupting qk∗µ = fkµ+igkµ and its conjugate into Φ0 and using the
normalizationcondition −∞∞|Φ0|2dfkµdgkµ=1,withdfkµ≡dfk11dfk12dfk21...,andsimilarlyfordgkµ.
R
5
For a normalized ground state, the higher excited states remain normalized. For ease of writing we
will not include the normalization factor N in most expressions, but normalization of states will be
assumed when calculating expectation values.
Again, the formula for the field beables are obtained by replacing the conjugate momenta π
kµ
and π∗ by ∂S/∂q and ∂S/∂q∗ in the corresponding classical formula. The following is a list of
kµ kµ kµ
formulae for the beables:
The vector potential A(x,t) is given in eq. (4). The electric field is
1∂A c ∂S
E(x,t)= c Π(x,t)= = εˆ e−ik.x. (9)
− −c ∂t −V 12 kµ∂qkµ
kµ
X
The magnetic induction is
i
B(x,t)= A(x,t)= (k εˆ )q (t)eik.x. (10)
∇× V 12 × kµ kµ
kµ
X
We may also define the energy density, which includes the quantum potential density (see reference
[21]), but we will not write these here as we will not need them. The total energy is found by
integrating the energy density over V to get,
∂S c2 ∂S ∂S κ2 ¯h2c2 ∂2R
E = = + q∗ q + .
−∂t kµ " 2 ∂qk∗µ∂qkµ 2 kµ kµ − 2R ∂qk∗µ∂qkµ!#
X
The intensity is equal to momentum density multiplied by c2
I(x,t)=c2G = −ic2 εˆk′µ′ (k εˆkµ) ∂S qkµei(k−k′).x . (11)
V kµ k′µ′(cid:20) × × ∂qk′µ′ (cid:21)
XX
We have adopted the classical definition of intensity in which the intensity is equal to the Poynting
vector (in heavyside-lorentz units), i.e., I = c(E B). The definition leads to a moderately simple
×
formula for the intensity beable. We note that the definition above contains a zero point intensity.
But, because I is a vector (whereas energy is not) the contributions to the zero point intensity from
individual waves with wave vector k cancel each other because of symmetry; for each k there is
anotherk pointing inthe opposite direction. The above,however,is notthe definitionnormallyused
in quantum optics. This is probably because although it leads to a simple formula for the intensity
beable it leads to a very cumbersome expression for the intensity operator in terms of the creation
and annihilation operators:
Iˆ= −4¯hVc2 kk′εˆkµ×(k′×εˆk′µ′)− kk′(k×εˆkµ)×εˆk′µ′
kµ k′µ′(cid:20) (cid:21)
XX
× aˆkµaˆk′µ′ei(k+k′).x−aˆkµaˆ†k′µ′ei(k−k′).x−aˆ†kµaˆk′µ′e−i(k−k′).x+aˆ†kµaˆ†k′µ′e−i(k+k′).x . (12)
h i
In quantum optics the intensity operator is defined instead as Iˆ=c( Eˆ+ Bˆ− Bˆ− Eˆ+), and
× − ×
leads to a much simpler expression in terms of creation and annihilation operators
Iˆ= ¯hVc2 kˆ√kk′aˆ†kµaˆk′µ′ei(k′−k).x. (13)
kµ k′µ′
XX
This definition is justified because it is proportional to the dominant term in the interaction Hamil-
tonian for the photoelectric effect upon which instruments to measure intensity are based. We note
that the two forms of the intensity operator lead to identical expectation values and perhaps further
justifies the simpler definition of the intensity operator.
Fromthe abovewe seethatobjectssuchasq ,π ,etc. regardedastime independent operators
kµ kµ
in the Schr¨odinger picture of the usual interpretation become functions of time in CIEM.
6
For a given state Φ(q ,t) of the field we determine the beables by first finding ∂S/∂q and its
kµ kµ
complex conjugate using the formula
¯h Φ
S = ln . (14)
2i Φ∗
(cid:18) (cid:19) (cid:18) (cid:19)
This gives the beables as functions of the q (t) and q∗ (t). The beables can then be obtained in
kµ kµ
terms of the initial values by solving the equations of motion for q∗ (t). There are two alternative
kµ
but equivalent forms of the equations of motion. The first follows from the classical formula
∂ 1 dq∗
kµ
π = L = ,
kµ ∂ dqkµ c2 dt
dt
(cid:16) (cid:17)
where is the Lagrangian density of the electromagnetic field, by replacing π by ∂S/∂q . This
kµ kµ
L
gives the equation of motion
1 dq∗ (t) ∂S
kµ
= . (15)
c2 dt ∂q (t)
kµ
The second form of the equations of motion for q is obtained by differentiating the Hamilton
kµ
Jacobi equation (7) by q∗ . This gives the wave equations
kµ
1 d2q∗ ∂Q
kµ +κ2q∗ = . (16)
c2 dt2 kµ −∂q
kµ
The corresponding equations for q are the complex conjugates of the above. These equations of
kµ
motion differ from the classical free field wave equation by the derivative of the quantum potential.
Fromthis it followsthat wherethe quantumpotentialis zero orsmallthe quantum fieldbehaveslike
a classical field. In applications we will obviously choose to solve the simpler eq. (15).
In the next section we apply CIEM to the Mach-Zehnder Wheeler delayed-choice experiment.
3 A CAUSAL MODEL OF THE MACH-ZEHNDER WHEELER
DELAYED-CHOICE EXPERIMENT
Consider the Mach-Zehnder arrangement shown in figure 1. BS and BS are beam splitters, M
1 2 1
andM aremirrorsandP is a phaseshifter thatshifts the phase ofawaveby anamountφ. Inwhat
2
follows we will assume for simplicity that the beam suffers a π/2 phase shift at each reflection and a
zero phase shift upon transmission through a beam splitter. In general, phase shifts upon reflection
and transmission may be more complicated than this. The requirement is that the commutation
relationsmustbepreserved. Thelatterisamorestringentrequirementthanenergyconservation(or,
equivalently, of photon number conservation) alone [22]. The polarization unit vector is unchanged
by either reflection or transmission.
3.1 Input Region
The input regionis the regionbefore the first beamsplitter BS . The incoming beamis a Fock state
1
containing one quantum:
1
2κ 2
Φ (q ,t)= 0 q∗ (t)Φ e−iκ0ct.
i kµ ¯hc k0µ0 0
(cid:18) (cid:19)
From the formula (14) we can find the S corresponding to the state Φ (q ,t). Using this result in
i kµ
eq. (15) gives the equations of motion:
dq∗ ∂S i¯hc2 dq∗ ∂S
k0µ0 =c2 = , kµ =c2 =0 for k = k . (17)
0
dt ∂q 2q dt ∂q 6 ±
k0,µ0 k0µ0 kµ
7
The solutions are easily found:
q∗ (t)=q ei(ω0t+θ0), q∗ (t)=q eiζkµ0 for k = k . (18)
k0µ0 0 kµ kµ0 6 ± 0
q and q are constant amplitudes, and θ and ζ are constant phases all fixed at t = 0. These
0 kµ0 0 kµ0
initialconditions cannotofcoursebe precisely determinedbut aregivenwith someprobabilityfound
from Φ (...q∗ ...,...q ,t)2.
| i kµ kµ |
Wefindexpressionsforthebeablesintheinputregionbysubstitutingthe∂S/∂q and∂S/∂q
k0,µ0 kµ
found from eq. (17) together with solutions (18) into the formulae for the beables. In finding the
expression for the beables we use formula (5). Defining Θ =k .x ω t θ the beables are:
0 0 0 0
− −
2 g (x)
A(x,t) = εˆ q cosΘ + A ,
1 k0µ0 0 0 1
V 2 V 2
2ω
E(x,t) = − 0q εˆ sinΘ , (19)
1 0 k0µ0 0
V 2c
2 g (x)
B(x,t) = − q (k εˆ )sinΘ + B ,
V 12 0 0× k0µ0 0 V 12
with
g (x)= εˆ q eik.x, g (x)= g (x)=i (k εˆ )q eik.x.
A kµ kµ B ∇× A × kµ kµ
Xkµ Xkµ
k6=±k0 k6=±k0
3.2 Region I
Weconsiderthe stateΦ inregionIanddeterminefromthisstatethecorrespondingbeables. Region
I
I is the region after the phase shifter P and the mirror M but before BS .
1 2
The incoming beam Φ is split at BS into two beams: the α and β-beams4. The α-beam
i 1
undergoes a π/2 phase shift at M and becomes Φ eiπ/2 = iΦ . The β-beam undergoes two π/2
1 α α
phaseshiftsfollowedbyaφphaseshiftandbecomesΦ eiφeiπ = Φ eiφ. Alsomultiplyingbya1/√2
β β
−
normalization factor the state Φ in region I becomes
I
1
Φ = iΦ Φ eiφ , (20)
I α β
√2 −
(cid:0) (cid:1)
where Φ and Φ are solutions of the normal mode Schr¨odinger equation (6) and are given by
α β
1 1
2κ 2 2κ 2
Φ (q ,t)= α α∗ Φ e−iκαct, Φ (q ,t)= β β∗ Φ e−iκβct.
α kµ ¯hc kαµα 0 β kµ ¯hc kβµβ 0
(cid:18) (cid:19) (cid:18) (cid:19)
The magnitudes of the k-vectors are equal, i.e., k =k =k .
α β 0
By using formula (14) to first determine S,
¯h
S = ikct 2ik ct+ln iα∗ β∗ eiφ ln iα β e−iφ , (21)
2i"− − 0 kαµα − kβµβ − − kαµα − kβµβ #
Xk (cid:16) (cid:17) (cid:0) (cid:1)
we can determine ∂S/∂α , ∂S/∂β , and ∂S/∂q . Substituting the latter into eq. (15) gives
kαµα kβµβ kµ
the equations of motion for α∗ , β∗ and q∗ :
kαµα kβµβ kµ
dα∗ ∂S ¯hc2 1
kαµα = c2 = − , (22)
dt ∂α 2 iα +β e−iφ
kαµα kαµα kβµβ
4Some workers insist that two inputs into the beam splitter must be used even when one of the inputs is the
(cid:2) (cid:3)
vacuum [23], while other workers use a single input [24] [25]. In passing, we mention that Caves [26], in connection
withthesearchforgravitationalwavesusingaMichelsoninterferometer,suggests,asoneoftwopossibleexplanations,
that vacuum fluctuations are responsible for the ‘standard quantum limit’ which places a limit on the accuracy of
any measurement of the position of a free mass. We will use the single input approach for BS1 as it simplifies the
mathematical analysiswhileallessentialresultsremainthesame.
8
dβ∗ ∂S ¯hc2 e−iφ
kβµβ = c2 = − , (23)
dt ∂β 2i iα +β e−iφ
kβµβ kαµα kβµβ
dq∗ ∂S
kµ = c2 =0, fork(cid:2) = k , k . (cid:3) (24)
α β
dt ∂q 6 ± ±
kµ
Eq.’s (22) and (23) are coupled differential equations. The coupling shows that the α and β-beams
depend nonlocally on each other. Taking the ratio of Eq.’s (22) and (23) gives the relation
α∗ (t)=ieiφβ∗ (t), (25)
kαµα kβµβ
which can be used to decouple the two differential equations. The decoupled differential equations
and eq. (24) are easily solved to give
α∗ (t)=α ei(ωαt+σ0), β∗ (t)=β ei(ωβt+τ0), q∗ (t)=q eiζkµ0 fork = k , k , (26)
kαµα 0 kβµβ 0 kµ kµ0 6 ± α ± β
where σ and τ are integration constants corresponding to the initial phases, and α and β are
0 0 0 0
constant amplitudes. Different values of the constants q and ζ in different regions of the
kµ0 kµ0
interferometer would correspondto amplitude and phase changes of the normal mode coordinates of
the ground state. These changes to the normal mode coordinates of the ground state, if indeed they
occur, do not lead to any observable differences in measured physical quantities so that there is no
way (at present) to detect them. For this reasonwe choose q and ζ to have the same values in
kµ0 kµ0
allthe regionsof the interferometer. The omega’s,ω =h¯c2/4α2 and ω =h¯c2/4β2, arenonclassical
α 0 β 0
frequencies which depend on the amplitudes α and β .
0 0
Substitution eq.’s (26) into eq. (25) gives the following relations among the constants associated
with α (t) and β (t):
kαµα kβµβ
π
α =β , σ =τ +φ+ . (27)
0 0 0 0
2
Substituting eq.’s (27) into ω or ω shows that ω =ω .
α β α β
3.3 The beables in region I
In this section we obtain explicite expressions for the beables A(x,t), E(x,t), B(x,t) and I(x,t) as
functions of time and the initial values. The expression for the energy density is very cumbersome
andisnotasusefulinthepresentcontextasthe intensity. Forthisreasonwewillnotgivetheenergy
density here. For the same reason we will also leave out the quantum potential density, though we
will need the quantum potential. We note that in what follows εˆ = εˆ = εˆ , where εˆ
k0µ0 kαµα kβµβ k0µ0
is the polarization of the incoming wave.
Tofindthebeablesasexplicitefunctionsofpositionandtimewesubstitutethepartialderivatives
ofS withrespecttothenormalmodecoordinatesfoundineq.’s(22),(23),and(24)togetherwiththe
solutions for α (t), β (t) and q (t) given in eq.’s (26) into the formulae for the beables given
kαµα kβµβ kµ
in eq.’s (4), (9), (10), and (11). Eq. (5) is used in deriving the beable expressions. After lengthy
manipulation and simplification, and defining Θ = k .x ω t σ and Θ = k .x ω t τ we
α α α 0 β β β 0
− − − −
get:
2 u (x)
A (x,t) = εˆ α cosΘ +εˆ β cosΘ + I ,
I V 21 kαµα 0 α kβµβ 0 β V 12
E (x,t) = −¯hc(cid:0) εˆkαµα sinΘ + εˆkβµβ sinΘ ,(cid:1) (28)
I 2V 21 (cid:18) α0 α β0 β(cid:19)
2 v (x)
B (x,t) = − (k εˆ )α sinΘ +(k εˆ )β sinΘ + I ,
I V 21 α× kαµα 0 α β × kβµβ 0 β V 12
¯hc2 (cid:2) f (x)g(cid:3)(x,t)
I (x,t) = (k +k k cos2Θ k cos2Θ ) I I ,
I α β α α β β
2V − − − V
9
with
u (x) = εˆ q eik.x, v (x)= u (x)=i (k εˆ )q eik.x, (29)
I kµ kµ I I kµ kµ
∇× ×
Xkµ Xkµ
k6=±kα,±kβ k6=±kα,±kβ
f (x) = i¯hc2 εˆ (k εˆ )q eik.x, g (x,t)=sinΘ +sinΘ .
I kαµα × × kµ kµ I α β
Xkµ
k6=±kα,±kβ
Whattheabovebeablesshow,andthepointwewanttoemphasize,isthatthesingleinputphoton
is split by the beam splitter BS into two beams. Each beam carries half the total momentum,
1
I ¯hk ¯hk
G = G dV = I dV = α + β =h¯k ,
I I c2 2 2 0
ZV ZV
and energy. The energy distribution given by the Hamilton-Jacobi equation (7) turns out to be very
cumbersome to calculate, but the total energy is easily found from eq. (21):
∂S ¯hck
E = =h¯ck + .
0
−∂t 2
k
X
There is no question of the whole photon choosing a single path through the beam splitter. We note
thatsinceΦ isaneigenstateoftheenergyandmomentumoperators,the expectationvaluesofthese
I
operators are equal to the energy and momentum beables.
We see that the field beable behaves much like a classical electromagnetic field, but there are
two differences: The first difference is that frequencies ω = ω have a nonclassical dependence on
α β
the amplitude of the field. The second and more significant difference is that each beam depends
nonlocally on the other beam. This is shown, as we have already mentioned, by the fact that the
equationsofmotionofthe twobeams, eq. (22)andeq. (23), arecoupleddifferentialequations. That
is to say, the behaviour of each beam depends nonlocally on the behaviour of the other beam.
The nonlocal dependence of each beam on the other can also be seen from the wave equations of
α and β . These can be found by inserting the total quantum potential in regionI (found by
kαµα kβµβ
using formula (8)),
1 ¯hck ¯h2c2
Q = k2q∗ q +h¯ck + ,
I −2 kµ kµ 0 2 − 2h∗h
kµ k I I
X X
into the wave equation (16) and differentiating. This gives
1 d2α ¯h2c2 α iβ e−iφ
kαµα = − kαµα − kβµβ ,
c2 dt2 2(h∗h )2
(cid:2) I I (cid:3)
1 d2βkβµβ = −¯h2c2 βkβµβ +iαkαµαeiφ ,
c2 dt2 2(h∗h )2
(cid:2) I I (cid:3)
withh = iα β exp( iφ). Ineachwaveequationtherighthandsidedependsonfunctions
I − kαµα− kβµβ −
from both beams and therefore indicates a nonlocal time dependence of each beam on the other.
Usingeq. (29)itiseasytoshowthattheaboveexpressionsfortheA (x,t),E (x,t),andB (x,t)
I I I
beables satisfy the usual classical relations E = (1/c)∂A /∂t and B = A .
I I I
− ∇×
There are a number of ways to establish a connection between the initial (constant) amplitudes
and phases of the input region and region I. One convenient way is to compare the electric field
beablesintheαandβ-beams,eq. (28),withthoseobtainedbybeginningwiththeinputelectricfield
beable, eq. (19), and inserting the appropriate amplitude and phase changes as it splits at BS and
1
thenpassesthroughM ,M andthe phaseshifterP intoregionI.Thiscomparisongivesthe relation
1 2
between the initial constants in the input region and in region I:
q π
0
α =β = , σ =θ , τ =θ φ π.
0 0 0 0 0 0
√2 − 2 − −
10