Evolution of Photon Beams through a Nested Mach-Zehnder Interferometer using Classical States of Light

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by  Anarta Roy

Evolution of Photon Beams through a Nested Mach-Zehnder Interferometer using
Classical States of Light
Anarta Roy1,∗ and Sibasish Ghosh2,†
1Department of Physics, Indian Institute of Science Education and Research, Bhopal Bypass Road, Bhopal 462 066, India
2Optics & Quantum Information Group, The Institute of Mathematical Sciences,
HBNI, C.I.T. Campus, Taramani, Chennai 600 113, India
(Dated: January 12, 2017)
Inthispaper,wepresentacoherentstate-vectormethodwhichcanexplaintheresultsofanested
linearMach-ZehnderInterferometricexperiment. SuchinterferometersareusedwidelyinQuantum
Information and Quantum Optics experiments and also in designing quantum circuits. We have
7 specifically considered the case of an experiment by Danan et al. (Phys. Rev. Lett. 111, 240402
1 (2013))wheretheoutcomeoftheexperimentwasspookybyourintuitiveguesses. Howeverwehave
0 been able to show by our method that theresults of this experiment is indeed expected within the
2 standardformalism ofQuantumMechanicsusinganyclassical stateofasingle-moderadiation field
as theinput intothenested interferometric set-upof theaforesaid experiment and therebylooking
n
a intothe power spectrum of theoutput beam.
J

1
I. INTRODUCTION
1

] Quantum Mechanics can explain various counter-
h
intuitive phenomena of the subatomic world [1]. An in-
p
- terferometerisgenerallyusedtodescribeseveralcounter-
nt intuitive features one gets to see in Quantum Mechanics
a [1].
u In a fairly recent experiment by Danan et al., the re-
q sults were spooky (ref. [2]). The experiment involves a
[
nested Mach-Zehnder Interferometer(MZI) inside which,

1 mirrorsA,B,C,E,andFvibrateatdifferentfrequencies
v (seeFIG.1). Thisinterferometricarrangementallowsthe
4 passageoflight(representedaslineswitharrowsinFIG.
7
1)throughseveralbeamsplittersnamelyBS1,BS2,BS3,
0
andBS4andviathevibratingmirrorsA,B,C,E,andF.
3
Thephotonsoflightrecordthefrequenciesofthevibrat-
0
. ingmirrors(viawhichtheypass). Thefinaloutputbeam
1 oflightfrombeamsplitterBS4remainsdirectedtowards FIG. 1. Mach-Zehnder Interferometer(nested) consisting of
0 a quad-celldetector D, which records the frequencies (in beam splitters BS1, BS2, BS3, BS4 and mirrors A, B, C, E
7 and F. The arrows show the path of light beam from source
1 the form of a power spectrum) of the vibrating mirrors
S to detector D [2].
: via which the photons of light have passed. This exper-
v
iment [2] seems to infer about the path the photon has
i
X taken while passing through the interferometer.
In this paper, we will analyse the outcome of this ex-
r The results were explained by the authors using the
a two-state vector formalism (TSVF) and weak values in periment using a standard coherent-state formulation.
Our results and methods will be able to explain the out-
QM [3–5]. This explanation was followed by a series
come of this experiment and will be able to shed some
of arguments and counter-arguments regarding how the
morelightonthistopic[2]. Giventhatacoherentstateis
outcome of this experiment should be analysed [6–11].
mainly a classicalstate, we can say with confidence that
All these arguments have dealt with the methods which
thisexperimentfallsintheclassicalregime. However,we
might be employed in a search for the path of photons
have followed a purely quantum mechanical way to look
through a nested Mach-Zehnder Interferometer (MZI)
at this problem.
[2]. Two theoretical approaches have been employed for
In section II we will discuss how a quantum mechan-
explaining the results of this experiment. One uses a
ical lossless beam-splitter transforms its input modes to
classical optics formalism [12] and the other employs a
output modes [14]. Insections III andIVwe will discuss
“one-state vector quantum-mechanical” approach [13].
the evolution of the system-states as they pass through
different beam-splitters and mirrors in this experiment
[2]. In section III, we analyze setup 1 and in section IV
∗ [email protected] we analyze setup 2 of this experiment [2]. In section V
† [email protected] we will discuss the power spectrum of the output states
2
for setups 1 and 2. FIG. 1 gives a schematic representa- T and L to denote states entering the Top port and the
tion of setup 1 of this experiment. Finally we conclude Left port of the beam splitter respectively.
in section VI.
ψ = n m =(n!m!)−1/2[(aˆ†)n (aˆ†)m] 0 0
| ini | iT | iL T ⊗ L | i| i
(4)
II. GENERAL LOSSLESS BEAM SPLITTER
TRANSFORMATION Also from (1), we get (noting that Bˆ is invertible and
B† =B∗),
ij ij
aˆ B† B† aˆ′
T = 11 21 T (5)
(cid:18)aˆL(cid:19) (cid:18)B1†2 B2†2(cid:19)(cid:18)aˆ′L(cid:19)
Hence, in terms of output annihilation operators
(modes) aˆ′ & aˆ′ , the input modes can be expressed as,
L T
aˆ =B† aˆ′ +B† aˆ′ (6)
T 11 T 21 L
aˆ =B† aˆ′ +B† aˆ′ (7)
L 12 T 22 L
Putting (6) and (7) in expression (4),
ψ =(n!m!)−1/2[(aˆ†)n (aˆ†)m] 0 0
| ini T ⊗ L | i| i
=(n!m!)−1/2[((B† aˆ′ +B† aˆ′ )†)n (8)
11 T 21 L ⊗
FIG.2. Agenericquantummechanical lossless beam splitter ((B† aˆ′ +B† aˆ′ )†)m] 0 0
with input modes aˆL & aˆT and output modes aˆ′L & aˆ′T. 12 T 22 L | i| i
After some rearrangementwith expression (8), we get
ψ .
There are two input modes and correspondingly two | outi
output modes in a quantum-mechanical lossless beam
splitter. For the beam-splitter representation in Fig. 2, n m n m
ψ =(n!m!)−1/2
the input modes are represented by annihilation oper- | outi j k
ators aˆL & aˆT. The output modes are given by cor- Xj=0kX=0(cid:18) (cid:19)(cid:18) (cid:19) (9)
responding annihilation operators aˆ′ & aˆ′ . They are [(n j+m k)!(j+k)!]1/2Bn−jBj
related by the transformation matrixLBˆ thrTough the fol- Bm−−kBk n− j+m k j1+1 k 21
lowing equation. Please see Ref. [14]: 12 22| − − iT′| iL′
We note that this output state has been obtained af-
ter applying the unitary transformation corresponding
aˆ′ aˆ B B aˆ
T =Bˆ T = 11 12 T (1) to Bˆ on the input state. This gives the most general
aˆ′ aˆ B B aˆ
(cid:18) L(cid:19) (cid:18) L(cid:19) (cid:18) 21 22(cid:19)(cid:18) L(cid:19) transformation of the input state to the output state of
TheunitaryoperatorU ofthetransformationcharac- a quantum mechanical lossless beam splitter. From (3),
B
terizes the beam-splitter. Following the angular momen- we have the values of the matrix elements of Bˆ.
tum theory, U is the two-dimensional representationof
B
the rotation group (SO(3)) [14].
B =cos(Θ/2)ei([Ψ+Φ]/2)
11
B =sin(Θ/2)ei([Ψ−Φ]/2)
U (Φ,Θ,Ψ)=e−iΦLˆ3e−iΘLˆ2e−iΨLˆ3 (2) 12 (10)
B B = sin(Θ/2)e−i([Ψ−Φ]/2)
21
−
From (2), we can derive the expression for matrix Bˆ B =cos(Θ/2)e−i([Ψ+Φ]/2)
22
[14], which is
From the above elements we get the transmittance τ
and reflectance ρ=(1 τ) of the beam splitter [14].
cos(Θ/2)ei([Ψ+Φ]/2) sin(Θ/2)ei([Ψ−Φ]/2) −
Bˆ =
sin(Θ/2)e−i([Ψ−Φ]/2) cos(Θ/2)e−i([Ψ+Φ]/2)
(cid:18)− (cid:19)(3) B11 2 = B22 2 =τ =cos2(Θ/2)
| | | | (11)
where Φ,Θ,Ψ are quantum mechanical analogues of the B 2 = B 2 =ρ=sin2(Θ/2)
12 21
classical Euler angles. | | | |
Now we consider a general input state as ψ which Now we are at a position to begin the analysis of the
in
is a tensor product of two fock states. We use| subiscripts experimental results of Danan et al.
3
III. ANALYSIS OF SETUP 1
(tsaetHtiouenpre,i1n)w.FeTigdhi3iss.cuisWssgeitvhweenillifinbrsetthaesnedatuilaypgzirnoagfmtmehaaecthiecxarpnedeprriemevseeernnyt- |ψoutiBS1 =−(cid:12)(cid:12)(cid:12)αατρeei(−Ψi(1+Ψ2Φ1−12Φ)E1)T8L⊗2 (12)
beam-splitterandmirror,oneaftertheotherasthebeam (cid:12) E
(cid:12)
passes through each of them. a tensor product of two(cid:12) coherent states. Noting that
τ = 1 and ρ = 2 for BS1 (1:2 beam-splitting as
3 3
used iqn ref. [2]), theqoutput state is given by
|ψoutiBS1 =(cid:12)αr13ei(Ψ1+2Φ1)+ ⊗
(cid:12) T8 (13)
(cid:12)
(cid:12)(cid:12) α 2e−i(Ψ1−2Φ1)
(cid:12)− r3 +
(cid:12) L2
(cid:12)
(cid:12)
System T8 now goes t(cid:12)owards mirror C and L2 towards
mirror E.
B. Mirror E
The action of a mirror is the same as that of a beam
splitter which has τ = 0 and ρ = 1. The input state for
mirror E is given by
FIG.3. Setup1of[2]showingstatevectorsofall thephoton
beveoalmutsi.onWoefhtahveesttoatfeosllforwomthSetsotaDte.Tvehcetoproswteorusnpdecetrrsutamndfrtohme |ψiniE =|0iT2⊗(cid:12)−αr23e−i(Ψ1−2Φ1)+ (14)
(cid:12) L2
detectorDrecordsthefrequenciesofoscillationsofallthefive (cid:12)
mirrors A,B,C,E and F. The final state reaching detector D Working on this state, w(cid:12)(cid:12)e obtain the output to be
is |i .
h
ψ = α B α B (15)
| outiE | E 12iT3⊗| E 22ia
A. Beam Splitter 1 where αE =−α 23e−i(Ψ1−2Φ1).
All the mirrorqs in this experiment vibrate with differ-
The action of the first beam splitter (BS1) in FIG. ent frequencies and same amplitudes. Hence the phases
3 is governedby the generalscheme mentioned before in ofthebeamaretime-dependent. Wehavethus(seeequa-
SectionII.Thesameschemealsogovernstheotherbeam tions (2) and (3)),
splitters in the figure. Let α be a coherent state of a
| i B =ei[(ΨE(t)−ΦE(t))/2], B =0
single mode system. We consider the input state as, 12 22
Here Ψ (t) and Φ (t) are the time-dependent Euler an-
E E
ψ = α 0
| iniBS1 | iT1⊗| iL1 gular phase factors for mirror E given by:
∞ αn
=e−|α|2/2 n 0 Ψ (t)=ψ sin(2πf t)
√n!| iT1⊗| iL1 E 0 E (16)
nX=0 ΦE(t)=φ0sin(2πfEt)
=e−|α|2/2 ∞ αn [(n!)−1/2 n n
n=0√n! j=0(cid:18)j(cid:19) where fE (in Hz units) denotes the frequency of oscil-
X X lation for mirror E. ψ and φ are constant amplitudes
0 0
[(n j)!j!]1/2Bn−jBj n j j which are same for all the mirrors in ref. [2]. Thus, the
− 11 21| − iT8| iL2
final state from mirror E is
We have given labels T8 and L2 to the output states of
BS1 (refer to Fig 3). According to eqn (3) of Section II,
ψ = α ei[(ΨE(t)−ΦE(t))/2] 0 (17)
Φ1 andΨ1 arethe constantEulerangles for BS1. Work- | outiE E T3⊗| ia
ing on the above expression, the final state is obtained (cid:12) E
(cid:12)
as System T3 is no(cid:12)w directed towards BS2.
4
C. Beam Splitter 2 E. Mirror B
BS2 is a 1:1 beam splitter as described in the exper- Similar to the case of mirror A, the system T5 now
iment [2]. This means τ = 1/2 & ρ = 1/2. The input enters the Mirror B. We thus take here
state to BS2 is given by:
1
α =α ei[(ΨE(t)−ΦE(t))/2] ei(Ψ2+Φ2)/2
B E
|ψiniBS2 = αEei[(ΨE(t)−ΦE(t))/2] T3⊗|0iL3 (18) r2
(cid:12) E This is according to equation (19), for system T5, where
(cid:12)
If we take α′E =(cid:12) αEei[(ΨE(t)−ΦE(t))/2], the output state αB is coherent. The input state to the mirror B is
from BS2 is found to be
ψ = α 0 (24)
ψ = α′ B α′ B | iniB | BiT5⊗| iL5
| outiBS2 | E 11iT5| E 21iL4
After the action of the mirror B on ψ , the output
= α ei[(ΨE(t)−ΦE(t))/2] 1ei(Ψ2+Φ2)/2 state is given by | iniB
E
(cid:12) r2 + ⊗ (19)
(cid:12) T5
(cid:12) ψ = α B α B (25)
(cid:12) 1 | outiB | B 11ic⊗| B 21iL6
(cid:12) α ei[(ΨE(t)−ΦE(t))/2] e−i(Ψ2−Φ2)/2
E
(cid:12)− r2 + See equations (2) and (3). We take the time dependent
(cid:12) L4
(cid:12) phasesformirrorB asΨB(t)andΦB(t)where(fB inHz
(cid:12)
As in(cid:12)the previous beam splitter, here Φ2 and Ψ2 are units),
the constant Euler angles for the beam splitter BS2. α
E
is known from mirror E. The system T5 goes towards ΨB(t)=ψ0sin(2πfBt)
(26)
the mirror B, while system L4 goes towards mirror A. Φ (t)=φ sin(2πf t)
B 0 B
We note here, that the vibration frequency of mirror E
is stored in both the states of T5 and L4 (see equation This is similar to the previous mirror A. The constant
(18)). amplitudes are the same as in mirror A. The oscillation
amplitudesforallthe mirrorsinref. [2]areconstantand
equal. Henceallamplitudesofoscillationsforthemirrors
D. Mirror A in ref. [2] are given by ψ and φ . f is the oscillation
0 0 B
frequency for mirror B.
SystemL4entersMirrorA.Wethushavehere,accord-
ing to equation (18), |ψoutiB =|0ic⊗ −αBe−i[(ΨB(t)−ΦB(t))/2] L6 (27)
(cid:12) E
(cid:12)
α = α ei[(ΨE(t)−ΦE(t))/2] 1e−i(Ψ2−Φ2)/2 ThesystemL6is also(cid:12)directedtowardsthe beamsplitter
A − E 2 BS3. Now, we will analyse the case of BS3.
r
Thus the input state to the mirror A is given by:
F. Beam Splitter 3
ψ = 0 α (20)
| iniA | iT4 ⊗| AiL4
Before discussing the BS3 (50:50) transformation, we
After the action of the mirror, state is given by:
look at the unitary transformationcorresponding to any
beamsplitter actiononaninputstateoftheform 0
|ψoutiA =|αAB12iT6⊗|αAB22ib (21) γ , where γ is a single mode coherent state: | iT ⊗
| iL | i
WetakethetimedependentphasesformirrorAasΦ (t)
A
& Ψ (t),
A U 0 γ = γB γB (28)
BS| iT ⊗| iL | 12iT′ ⊗| 22iL′
|ψoutiA = αAei[(ΨA(t)−ΦA(t))/2] T6⊗|0ib (22) Thisunitarytransformationfollowsfromequation(9)for
(cid:12) E a coherentstate γ . Hence, if we knowthe RHS in(25),
(cid:12) | i
where (fA in H(cid:12)z units is the oscillation frequency for weareinapositiontofindout|γi. Weapplythismethod
mirrorA; ψ andφ are its constantamplitudes ofoscil- for BS3. Here,
0 0
lation)
α′ =α ei[(ΨA(t)−ΦA(t))/2],
A A
(29)
Ψ (t)=ψ sin(2πf t)
ΦA(t)=φ0sin(2πfAt) (23) α′B =−αBe−i[(ΨB(t)−ΦB(t))/2],
A 0 A
and we get,
System T6 is directed towards beam-splitter 3 or BS3.
Before moving on to BS3, we will first look at mirror B. ψ = α′ α′ (30)
| iniBS3 | AiT6⊗| BiL6
5
The output state is found out to be I. Beam splitter 4
ψ = 0 γ , (31)
| outiBS3 | id⊗| iL7 It is the lastbeam splitter inthis setup which receives
with states of the systems T9 and L9, and produces the final
output beam. The analysis for this transformation has
γ =α 2e−i[(Ψ1−Φ1)/2] ei[(Φ2+Φ3)/2] been done in the same way as in BS3. The input state
3 × × for this is
r
(32)
ei[(ΨE(t)−ΦE(t))/2] ei[(ΨA(t)−ΦA(t))/4]
× × ψ = γei[(ΨF(t)−ΦF(t))/2]
e−i[(ΨB(t)−ΦB(t))/4]. | iniBS4 T9⊗
(cid:12) E (39)
Φ3 istheconstantEulerangleforthe beamsplitterBS3. α 1ei(Ψ1+2Φ1(cid:12)(cid:12))e−i[(ΨC(t)−ΦC(t))/2] .
Now the system L7 enters the mirror F. (cid:12)− r3 +
(cid:12) L9
(cid:12)
(cid:12)
The fi(cid:12)nal state coming out of BS4 is then
G. Mirror F
ψ = ψ = 0 β (40)
| outifinal | outiBS4 | ig⊗| ih
For this mirror (FIG. 3),
where
ψ = 0 γ , (33)
| iniF | iT7⊗| iL7
whereγisgiveninequation(32). Afterthemirroraction, β =αei(Φ1+2Φ4+Φ2+4Φ3+π2)ei(ΨE(t)−4ΦE(t))ei(ΨF(t)−4ΦF(t))
×
the output state is ei(ΨA(t)−8ΦA(t))e−i(ΨB(t)−8ΦB(t))e−i(ΨC(t)−4ΦC(t))
ψ = γei[(ΨF(t)−ΦF(t))/2] 0 (34)
| outiF T9⊗| ie Φ is the Euler angle for the beam splitter BS4. The
where (f in Hz(cid:12)(cid:12) units is the oscillEation frequency for st4ate of the system h is the final state which reaches the
F (cid:12)
detectorD (FIG. 3). We see thatthe state ofthe system
mirror F)
h is a coherent state and in this coherent state β , the
| ih
Ψ (t)=ψ sin(2πf t) oscillation frequencies of all the mirrors A, B, C, E, & F
F 0 F
(35)
are present. Also the phases for mirrors B and C have
Φ (t)=φ sin(2πf t)
F 0 F
signs opposite to that of mirrors A, E and F.
Now the system T9 goes into the beam splitter BS4. Next, we will analyze the spooky results of this exper-
iment [2].
H. Mirror C
IV. ANALYSIS OF SETUP 2
We recall the two states that emerged from BS1. One
ofthose states wasthe state of the systemT8 whichwas
directed towards mirror C. This makes the input state In setup 2 (Fig 4.) of the experiment, everything re-
for mirror C as mains the same as setup 1 until we reach BS3 and then
BS4. So we will analyze the outputs of only these two
|ψiniC =(cid:12)αr13ei(Ψ1+2Φ1)+ ⊗|0iL8 (36) bdoeaesmn-soptlirtetaecrhs.itMainryromroFre.isSnoowteimigpnoorrteanitt.as the beam
(cid:12) T8
(cid:12)
By following our us(cid:12)ual mirror action, the output state,
(cid:12)
in this case, turns out as
A. Beam Splitter 3
|ψoutiC =|0if⊗(cid:12)(cid:12)−αr13ei(Ψ1+2Φ1)e−i[(ΨC(t)−ΦC(t))/2]+L9 byHtehreeawcetiofinrstoflotohkeabteathme supnliittateryr Btˆra,ngsifvoernmiantieoqnugaitvioenn
(cid:12) (37)
(cid:12) (1),
where (f in Hz(cid:12) units is the oscillation frequency for
C
mirror C)
U γ 0 = γB γB (41)
BS| iT | iL | 11iT′| 21iL′
Ψ (t)=ψ sin(2πf t)
C 0 C
(38)
Φ (t)=φ sin(2πf t) In the same way as in the case of BS3 in setup 1, we can
C 0 C
find γ , given the states γB and γB . Proceeding
11 21
| i | i | i
The system L9 is now directed towards BS4, the final in this way, we find the output states here. The input
beam splitter in this setup. states are
6
the final state entering D. This relation is
Ψ (t)+Ψ (t) Φ (t) Φ (t)
A B A B
Ψ +Ψ = − − π+λ
2 3
2 −
(cid:20) (cid:21)
(46)
Equation(46)isobtainedwhilefindingtheexpressionfor
χ. This has been discussed in the Appendix.
B. Beam Splitter 4
The system L9 that enters the beam splitter BS4 in
FIG. 4 has a non-trivial state. This makes the input
state to BS4 as:
ψ = 0
| iniBS4 | iT9⊗
α 1ei(Ψ1+2Φ1)e−i[(ΨC(t)−ΦC(t))/2] . (47)
FIG.4. Setup2[2]showsthestatesofthebeamwhenthereis (cid:12)− r3 +
(cid:12) L9
nophotonbeampassingviamirrorF.AccordingtoDananet (cid:12)
(cid:12)
al. [2] there is complete destructiveinterference of light that Tofind(cid:12) theoutputstateweuserelation(46)inourusual
isdirectedtowardsmirrorF.Thepowerspectrumrecordsthe beamsplittertransformation. Thusthefinaloutputstate
frequencies of only three mirrors A, B and C.
in this setup, from BS4 is
ψ = α α (48)
| outiBS4 | 1ig⊗| finalih
where,
1
α′ =α e−i[(Ψ1−Φ1)/2]ei[(ΨE(t)−ΦE(t))/2]e−i[(Ψ2−Φ2)/2]
A r3 ei[(ΨA(t)−ΦA(t))/2], α1 =−α√32ei(Ψ1+2Φ1)ei(Ψ4−2Φ4)e−i[(ΨC(t)−ΦC(t))/2],
×
1
α′B =αr3e−i[(Ψ1−Φ1)/2]ei[(ΨE(t)−ΦE(t))/2]ei[(Ψ2+Φ2)/2] αfinal = αe−i(Ψ2+Ψ3)ei(Ψ1+2Φ1)e−i(Ψ4+2Φ4)eiλ
e−i[(ΨB(t)−ΦB(t))/2]eiλ 3
× ei[(ΨA(t)−ΦA(t))/2]ei[(ΨB(t)−ΦB(t))/2]
(42)
×
e−i[(ΨC(t)−ΦC(t))/2].
ψ = α′ α′ (43) ×
| iniBS3 | AiT6⊗| BiL6
The coherent state α of the system h reaches the
| finalih
Instate L6,we consideranadditionalphase shift by eiλ. detector D. It corresponds to α which contains the
final
This is taken into account because of the slight shifting vibrational information of only three mirrors A, B and
of mirror B for this setup [2]. The output state is C. Also, the phases Φ ,Φ ,Φ ,Φ ,Ψ ,Ψ ,Ψ ,Ψ are all
1 2 3 4 1 2 3 4
constants. The time dependent phases contain the fre-
ψ = χ 0 (44) quencies of their respective mirrors as below:
| outiBS3 | id⊗| iL7
with ΦA(t)=φ0sin(2πfAt),
Φ (t)=φ sin(2πf t),
B 0 B
2
χ=α e−i[(Ψ1−Φ1)/2]ei[(Φ2−Φ3+π)/2] ΦC(t)=φ0sin(2πfCt), (49)
3 ×
r Φ (t)=φ sin(2πf t),
(45) E 0 E
ei[(ΨE(t)−ΦE(t))/2]ei[(ΨA(t)−ΦA(t))/4]
× ΦF(t)=φ0sin(2πfFt).
e−i[(ΨB(t)−ΦB(t))/4]eiλ
The same relations hold for phases Ψ (t) (where
i
Here Φ is the constant Euler angle associated with the i=A,B,C,E,F). The final state which contains some or
3
beamsplitterBS3. Thesystemshavingthisstatecannot all of these time dependant phases will be measured by
reachthe detector D as it emerges fromthe bottom port thedetectorD,bythecorrespondingfrequenciesofthose
of BS3. This is in accordance to ref. [2] as shown in phases.
FIG. 4. However we take note of one important relation Relation (40) gives the final output for setup 1, and
between the phases, which can be used later for finding (48) gives the final output for setup 2.
7
V. POWER SPECTRUM ANALYSIS OF THE equation(49). This iseasilyseenfromthe expressionfor
FINAL OUTPUT STATES S(1)(ω) where the time dependent phases of all the five
xx
mirrorsarepresent. Thisisinagreementwiththeexper-
Here,webeginwithaquantumPowerSpectralDensity imental result of Danan et al. [2]. Here κ is a constant
(PSD). This is a spectral function which gives the inten- phase.
sity of a time-dependent quantum mechanical operator
aˆ(t) for a given frequency ω. It is defined as [18]:
B. Power Spectrum of Output State from Setup 2
∞
S (ω)= R (t)eiωtdt
aa aa Nowweputαforsetup2intothepowerspectralfunc-
−Z∞ tion:
(50)
∞
= aˆ(t)aˆ(0) eiωtdt S(2)(ω)=2πx2 [(α2ei2κ′ei[ΨA(t)−ΦA(t)]
h i xx zpf 9
Z
−∞ ei[ΨB(t)−ΦB(t)]e−i[ΨC(t)−ΦC(t)]
×
Here R (t)= aˆ(t)aˆ(0) is the auto-correlationfunction δ(ω ω ))
aa 0
h i × −
fwoirllaˆb(te).anTahlyezceodhuerseinngt stthaeteabeonvteerfiunngcitnitoon.the detector D +(α∗2e−i2κ′e−i[ΨA(t)−ΦA(t)]
9
Accordingly, the power spectrum analysis of the final
e−i[ΨB(t)−ΦB(t)]ei[ΨC(t)−ΦC(t)]
statesobtainedfromsetups1and2havebeendoneusing
×
the following power spectral function, given in ref. [15]: δ(ω ω ))
0
× −
∞ +α2δ(ω+ω0)+(1+ α2)δ(ω ω0)].
| | | | −
Sxx(ω)=x2zpf [(α)2e−iω0t+(α∗)2eiω0t+(α∗α)eiω0t+ From this expression for S(2)(ω), we see that the power
xx
Z
−∞ spectrumdependsonthefrequenciesofmirrorsA,Band
(1+α∗α)e−iω0t]eiωtdt Conly(fromequation(49)). Hereκ′ isaconstantphase.
This is exactly what Danan et al. observed in their ex-
=2πx2 [(α)2δ(ω ω )+(α∗)2δ(ω+ω )+
zpf − 0 0 periment [2].
α2δ(ω+ω0)+(1+ α2)δ(ω ω0)] Our quantum mechanical state vector calculations
| | | | −
with coherent states have proved that this observation
Weknowtheαofthefinalcoherentstates α forsetups
| ih is indeed expected. Thus our simple coherent state ap-
1 and 2. Putting those values in, we obtain the final
proach is enough to explain the outcomes of this exper-
power spectra for the two setups separately. Here ω
0 iment. Although we have used a quantum mechanical
and ω are the electromagnetic radiation frequencies
− 0 approach, it is important to note that we have used a
associated with a single quantum state. x = h¯ coherent state and calculated its evolution for both the
zpf 2mω0
is the zero-point fluctuation constant. q setups (1 and 2). This points out the classical nature
of the photon beams used in this experiment. As any
single-mode classical state is a convex mixture of (sin-
A. Power Spectrum of Output State from Setup 1 gle mode) coherent states – via the Glauber-Sudarshan
P-distribution[16,17]–ouranalysisshowsthatthe con-
clusion of the experiment in ref. [2] follows also from
Puttingαforsetup1,intothe powerspectralfunction
any single-mode classical state in the input mode T1 (of
above we have,
FIG.3 as wellas FIG. 4). Hence the observationsofthis
S(1)(ω)=2πx2 [((α)2ei2κei[(ΨE(t)−ΦE(t))/2] experiment can also be expected to be explained using
xx zpf
classical light – as has been done in ref. [12].
ei[(ΨF(t)−ΦF(t))/2]ei[(ΨA(t)−ΦA(t))/4]
×
e−i[(ΨB(t)−ΦB(t))/4]e−i[(ΨC(t)−ΦC(t))/2]
× VI. CONCLUSION
δ(ω ω ))
0
× −
+((α∗)2e−i2κe−i[(ΨE(t)−ΦE(t))/2]
We have provided here (without using the two-state
e−i[(ΨF(t)−ΦF(t))/2]e−i[(ΨA(t)−ΦA(t))/4] vectorformalismofref. [4]), afully quantummechanical
× description of the experiment in ref. [2] using any clas-
ei[(ΨB(t)−ΦB(t))/4]ei[(ΨC(t)−ΦC(t))/2]
× sical (in the sense of Quantum Optics) input state of a
δ(ω+ω0)) single-mode radiationfield, and qualitatively established
×
+α2δ(ω+ω )+(1+ α2)δ(ω ω )]. similartypesoffunctionaldependenceofthepowerspec-
0 0
| | | | − tra on the oscillation frequencies of the mirrors used in
We observe that in this spectrum, we must get the fre- the nested interferometric experiment of ref. [2]. We be-
quencies corresponding to mirrors A,B,C,E and F as in lieve that the power spectra we obtained here do match
8
quantitativelyalsowiththoseofref. [2]. Wedohopethat 1 1
γ e−i[(Ψ3−Φ3)/2] =α e−i[(Ψ1−Φ1)/2]
ouranalysisheredoeshelpinunderstandingthe founda- − 2 3 ×
r r
tional issues related to paths of microscopic systems — ei[(ΨE(t)−ΦE(t))/2]ei[(Ψ2+Φ2)/2] (A.2)
as discussed in ref. [2]. ×
e−i[(ΨB(t)−ΦB(t))/2]eiλ
VII. ACKNOWLEDGEMENTS
Next, we obtain the two different expressions for γ from
equations (A.1) and (A.2). This process is trivial, since
The work of AR has been supported by the Sum-
we know the above equations.
mer Internship Programmein Physics of the Institute of
Mathematical Sciences (IMSc.), Chennai, India and the Now,weneedtoequatethesetwoseparateexpressions
KishoreVaigyanikProtsahanYojana(KVPY)Fellowship for γ. Doing this, we obtain the following relation:
Programmeof the Department of Science & Technology,
Government of India. Most part of the work was done
2
when AR was a final year BS-MS 5-year Dual Degree α e−i[(Ψ1−Φ1)/2]ei[(ΨE(t)−ΦE(t))/2]e−i[(Ψ2−Φ2)/2]
3 ×
coursestudent(inPhysics)atIISER-Bhopal,fromwhere r
he graduated recently. SG would like to thank Sandeep ei[(ΨA(t)−ΦA(t))/2]e−i[(Ψ3+Φ3)/2] =
K. Goyal for useful discussion on ref. [2].
2
α e−i[(Ψ1−Φ1)/2]ei[(ΨE(t)−ΦE(t))/2]ei[(Ψ2+Φ2)/2]
3 ×
r
Appendix: Proof of Eqn. (46) e−i[(ΨB(t)−ΦB(t))/2]ei[(Ψ3−Φ3)/2]eiλeiπ
(A.3)
We use equations (41),(42), and(43), to find initially, On simplifying equation (A.3), and solving for Ψ2+Ψ3,
two separate expressions for γ as shown below: we obtain equation (46):
1 1
γ ei[(Ψ3+Φ3)/2] =α e−i[(Ψ1−Φ1)/2]ei[(ΨE(t)−ΦE(t))/2] ΨA(t)+ΨB(t) ΦA(t) ΦB(t)
r2 r3 Ψ2+Ψ3 = −2 − −π+λ
e−i[(Ψ2−Φ2)/2]ei[(ΨA(t)−ΦA(t))/2] (cid:20) (cid:21)
×
(A.1)
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