Experimental Test of Entangled Histories

0.44 MB·

by  Jordan Cotler

MIT-CTP-4749
Experimental test of entangled histories
Jordan Cotler,1 Lu-Ming Duan,2,3,∗ Pan-Yu Hou,2 Frank
Wilczek,4,5 Da Xu,2,6 Zhang-Qi Yin,2,† and Chong Zu2
1Stanford Institute for Theoretical Physics,
Department of Physics, Stanford University
6 2Center for Quantum Information, Institute for Interdisciplinary Information Sciences,
1
0 Tsinghua University, Beijing 100084, China
2
n 3Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
a
J 4Center for Theoretical Physics, MIT, Cambridge MA 02139 USA

0
2 5Origins Project, Arizona State University, Tempe AZ 25287 USA

] 6Department of Physics, Tsinghua University, Beijing 100084, China
h
p
- Abstract
t
n
a We propose and demonstrate experimentally a scheme to create entangled history states of the
u
q Greenberger-Horne-Zeilinger (GHZ) type. In our experiment, the polarization states of a single photon
[

2 at three different times are prepared as a GHZ entangled history state. We define a GHZ functional which
v
3 attainsamaximumvalue1ontheidealGHZentangledhistorystateandisboundedaboveby1/16forany
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9 three-time history state lacking tripartite entanglement. We have measured the GHZ functional on a state
2
0 wehavepreparedexperimentally,yieldingavalueof0.656±0.005,clearlydemonstratingthecontribution
.
1
0 ofentangledhistories.
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1
:
v
i
X
r
a
∗[email protected]
†[email protected]
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I. INTRODUCTION
Thetraditional“observables”ofquantumtheoryareoperatorsinHilbertspacethatactatapar-
(cid:82)2
ticulartime. Butmanyquantitiesofphysicalinterest,suchastheaccumulatedphaseexpi dt(cid:126)v·A(cid:126)
1
ofaparticlemovinginanelectromagneticpotential,oritsaccumulatedpropertime,aremorenat-
urally expressed in terms of histories. We may ask: Having performed a measurement of this
more general, history-dependent sort of observable, what have we learned? For conventional ob-
servables, the answer is that we learn our system is in a particular subspace of Hilbert space, that
is,theeigenspacecorrespondingtotheobservable’smeasuredvalue.
Recently two of us, building on the work of Griffiths and others [1], have formulated a math-
ematical framework which extends many of the concepts and procedures ordinarily used in ana-
lyzingstatesofquantumsystemstotheirhistories[2–4]. Specifically,wehaveconstructed,under
very general assumptions about a quantum dynamical system, a Hilbert space of its possible his-
tories. Theinnerproductreflectsprobabilitiesofhistoriesoccurring.
There is a natural definition of observables on the history Hilbert space. It accommodates
the observables we mentioned initially, and new possibilities which might not have been easy
to imagine otherwise. The result of measuring a history observable is a partial reconstruction
of “what happened” during the evolution of one’s system. Analogously to how measurement
of the value of an ordinary observable on a system establishes the location of the state of the
systemwithinaneigenspaceoftheobservable,measurementofahistoryobservableonasystem’s
evolution establishes the location of its history within an eigenspace (in history space) of the
history observable. Such eigenspaces often contain entangled histories. The defining property of
anentangledhistoryisthatitcannotbeassigned,ateachtime,toadefinitestate.
Aparticularlyinterestingsortofentangledhistorycorrespondstoaparticularstateataninitial
time,andtoanotherparticularstateatafinaltime,andyetcannotbeassignedtoadefinitestateat
intermediate times. Such an entangled history provides a vivid illustration of the “many worlds”
picture of quantum mechanics, for it branches into several incompatible trajectories, which later
cometogether.
Here we describe a detailed protocol for producing an entangled history of that kind. We have
producedhistoriesfollowingthatprotocol,andmeasuredthattheydisplaybehaviorwhichcannot
berealizedbyanyhistorywhichisnotentangled.
Our history state is a temporal analogue of the GHZ state, and our measurement strategy was
2
inspired by the GHZ test. Hence it is appropriate briefly to describe the nature of that test, and its
context.
In1935,Einstein,PodolskyandRosen(EPR)notedapeculiarconsequenceofquantumtheory,
according to which measurement outcomes of distant entangled particles should be perfectly cor-
related – a result they felt to be in tension with relativistic locality (which limits causal influence
by the speed of light) [5]. In 1964, John Bell proved that the predictions of quantum theory differ
quantitativelyfromanythatcanemergefromalargeclassofdeterministic(classical)localmodels
[6]. Following Bell’s suggestion, numerous experiments have been done, and confirmed quantum
theory[7–9]. In 1989,Greenberger, Horne,andZeilinger proposeda relatedtestwhich isslightly
more complicated to set up experimentally, but much simpler to interpret and more striking theo-
retically[10]. TheGHZtestwasthenmadetransparentbyMermin[11],brilliantlyexpoundedby
Coleman[12],andperformedbyPanetal[13].
AparticularmultipartiteentangledstatecalledtheGHZstate,whichinvolvesatleastthreespin-
1/2degreesoffreedom,isessentialfortheGHZtest. Inourproposal,wedealwiththeproperties
of one photon at three different times, rather than three photons at the same time. Below, we
define a functional G which is diagnostic of entangled histories. We will prove that for product
history states G is bounded above by 0, while for history states lacking tripartite entanglement it
isboundedaboveby1/16. ForidealGHZentangledhistories,thefunctionalGisequalto1.
Motivated by these ideas, we performed the GHZ test for entangled histories experimentally.
We generated a candidate GHZ entangled history state for a single photon, and measured its G
functional. WemeasureGtobe0.656±0.005,whichconsiderablyexceedstheboundsmentioned
previously. Therefore, the GHZ test for histories clearly demonstrates the existence of entangled
histories.
We should add that the structure of our temporal analogue, unlike the original GHZ test, does
not preclude its classical modeling. Indeed, in appropriate limits our setup can be understood on
thebasisofclassicaloptics(whichanticipateskeyfeaturesofquantumtheory,i.e. complexwaves
which interfere, and whose absolute square is physically salient). Still, it seems to us noteworthy
that a simple stochastic classical model for the functionalG has an upper bound of 1/16. (See the
supplementary materials.) While that classical model does not map onto our experiment cleanly,
itsanalysisisinstructive.
Weshouldalsomentionthataveryinteresting,butquitedistinctaspectoftemporalcorrelation
in quantum theory has been the subject of previous study [14–18]. These works focus on tem-
3
poral correlations induced by the usual quantum measurement process, whereas we are primarily
concernedwithcorrelationsthatareintrinsictothedynamicalsystem.
II. CONSTRUCTINGANENTANGLEDHISTORYSTATE
In Griffith’s theory of “Consistent Histories” [1], a history state is a sum of tensor-like event
products,intheform
|Ψ) = Pˆin (cid:12)···(cid:12)Pˆi3 (cid:12)Pˆi2 (cid:12)Pˆi1 (1)
tn t3 t2 t1
Here the Pij are projectors at different times in temporal order t < t < t < ··· < t ) where
tj 1 2 3 n
the index i distinguishes orthogonal projectors within a decomposition of the identity. (cid:12) is a
typographical variation on the tensor product symbol ⊗, which we use when the factors in tensor
productrefertodifferenttimes. Aninnerproductonhistoryspaceisdefinedby
(Ψ|Φ) = Tr(K†|Ψ)K|Φ)) (2)
where
K|Ψ) = PˆinT(t ,t )···Pˆi3T(t ,t )Pˆi2T(t ,t )Pˆin (3)
tn n n−1 t3 3 2 t2 2 1 tn
Usingthispositivesemi-definiteinnerproductwecandefinequantumsuperpositionandquantum
interferenceforhistoriesjustaswedoforquantumstates. Foramoredetailedexposition,see[2].
Letusdiscuss,conceptually,howwemightconstructtheGHZhistorystate
|GHZ) := √1 (cid:0)[z+](cid:12)[z+](cid:12)[z+]−[z−](cid:12)[z−](cid:12)[z−](cid:1), (4)
2
where the notation |·) is used to denote the history state, and where [z±] := |z±(cid:105)(cid:104)z±|. Consider a
spin-1/2 particle in the state |x+(cid:105) = √1 (|z+(cid:105)+|z−(cid:105)). We are going to construct an entangled history
2
state via a post-selection procedure [19]. We introduce three auxiliary qubits |0(cid:105) |0(cid:105) |0(cid:105) =: |000(cid:105).
1 2 3
Attimet weperformaCNOToperationbetweenthefirstauxiliaryqubitandthespin-1/2particle,
1
resultingin
1 1
√ |z+(cid:105)|000(cid:105)+ √ |z−(cid:105)|100(cid:105) (5)
2 2
We let this system evolve trivially to time t . Then at time t , we perform a CNOT between the
2 2
secondauxiliaryqubitandthespin-1/2particle,resultingin
1 1
√ |z+(cid:105)|000(cid:105)+ √ |z−(cid:105)|110(cid:105) (6)
2 2
4
The system then evolves trivially to time t , at which time we perform a CNOT between the third
3
auxiliaryqubitandthespin-1/2particle,giving
1 1
√ |z+(cid:105)|000(cid:105)+ √ |z−(cid:105)|111(cid:105) (7)
2 2
If we measure the auxiliary qubits in the {|000(cid:105),|111(cid:105),...} basis, then measuring |000(cid:105) would
indicate that the spin-1/2 particle has been in the history state [z+] (cid:12) [z+] (cid:12) [z+]; and if we
measure |111(cid:105), this would indicate that the spin-1/2 particle has been in the history state [z−] (cid:12)
[z−] (cid:12) [z−]. However we can also choose to measure the auxiliary qubits in the GHZ basis
(cid:110) (cid:111)
√1 (|000(cid:105)±|111(cid:105)),... . Then if we measure √1 (|000(cid:105) − |111(cid:105)), it means that the spin-1/2 parti-
2 2 √
cle has been in the history state [z+] (cid:12) [z+] (cid:12) [z+] with amplitude 1/ 2, and [z−] (cid:12) [z−] (cid:12) [z−]
√
with amplitude −1/ 2. In other words, the particle has been in the entangled history state
√1 ([z+](cid:12)[z+](cid:12)[z+]−[z−](cid:12)[z−](cid:12)[z−]). By changing the basis of the auxiliary qubits, we have
2
erased knowledge about the history of the spin-1/2 particle. As emphasized in [20], selective
erasurecanbeapowerfultoolforexploringquantuminterferencephenomena.
Similartechniqueshavebeenproposedinthecontextof“multiple-timestates”[19]. Inthislan-
guage,wecanwritethetemporalGHZstateas √1 ((cid:104)z+||z+(cid:105)(cid:104)z+||z+(cid:105)(cid:104)z+||z+(cid:105)−(cid:104)z−||z−(cid:105)(cid:104)z−||z−(cid:105)(cid:104)z−||z−(cid:105)).
2
The interpretation of temporal entanglement has been a subject of much debate [19], [21]-[22].
The framework proposed in [2–4], grounded in the consistent histories approach of Griffiths [1],
seemstousclearandunambiguous.
III. TEMPORALGHZTEST
InthissectionwewilldiscusshowtoperformaGHZtestforentangledhistories. Considerthe
operators
σ (cid:12)σ (cid:12)σ , σ (cid:12)σ (cid:12)σ , σ (cid:12)σ (cid:12)σ , σ (cid:12)σ (cid:12)σ (8)
x y y y x y y y x x x x
on a three-time history space of a single spin-1/2 particle with trivial time evolution. The ex-
pectation values of |GHZ) with the history state operators corresponding to the four operators in
Equation 8 are 1, 1, 1 and −1, respectively. The product of these four expectation values is −1.
We represent this procedure of computing the product of the expectation values by the functional
Gwhichsatisfies
G[|Ψ)] = −(cid:104)σ (cid:12)σ (cid:12)σ (cid:105)(cid:104)σ (cid:12)σ (cid:12)σ (cid:105)(cid:104)σ (cid:12)σ (cid:12)σ (cid:105)(cid:104)σ (cid:12)σ (cid:12)σ (cid:105) (9)
x x x y y x y x y x y y
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where|Ψ)isanormalizedhistorystate. FortheidealGHZhistorystate,wehave
G[|GHZ)] = 1 (10)
 
Wecanwriteahistorystatewithtwo-timeentanglementas|ψ)(cid:12)cos(cθo)ss2in(θ(θ))eiφ cos(θs)insi2n((θθ))e−iφ ,
where |ψ) is arbitrary two-time entangled history states. Other history states with two-time entan-
glement take the same form, up to permutations of the tensor product components. Since the G
functional is not sensitive to such permutations, it suffices to consider a single ordering. We have
provedthat(supplementarymaterials)
  
G|ψ)(cid:12)cos(cθo)ss2in(θ(θ))eiφ cos(θs)insi2n((θθ))e−iφ 
1 1
= − sin4(2θ)sin2(2φ)(cid:104)σ (cid:12)σ (cid:105)(cid:104)σ (cid:12)σ (cid:105)(cid:104)σ (cid:12)σ (cid:105)(cid:104)σ (cid:12)σ (cid:105) ≤
x x y y x y y x
4 16
(11)
Andforagenericseparablehistorystate
|ψ ) ≡ P(θ ,φ )(cid:12)P(θ ,φ )(cid:12)P(θ ,φ ) (12)
pure 1 1 2 2 3 3
 
where P(θ,φ) = cos(cθo)ss2in(θ(θ))eiφ cos(θs)insi2n((θθ))e−iφ ,wehave
1
G[|ψ )] = − sin4(2θ )sin4(2θ )sin4(2θ )sin2(2φ )sin2(2φ )sin2(2φ ) ≤ 0 (13)
pure 1 2 3 1 2 3
64
Ourgoalistoconstructanapproximationtothehistorystate|GHZ)experimentally,andtoshow
thatforourconstructedstateG[|GHZ)] (cid:29) 1/16,thusdemonstratingahighdegreeoftemporalen-
tanglement. (InfacttheGfunctionalevendistinguishesaspecificformoftripartiteentanglement.
FortheWentangledhistorystate,|W) = √1 ([z−](cid:12)[z+](cid:12)[z+]+[z+](cid:12)[z−](cid:12)[z+]+[z+](cid:12)[z+](cid:12)[z−]),
3
the G functional vanishes.) This τGHZ test is much simpler than the generalized temporal Bell
testin[3],andrequiresmanyfewermeasurements.
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IV. EXPERIMENTALRESULTS
Wehavephrasedourdiscussioninthelanguageappropriatetothespinstatesofaspin-1 parti-
2
cle. As is well known, we can use the same two-dimensional, complex state space to describe the
polarization states of a photon. In that context, it is known as the Poincare´ sphere. We can adapt
standard optical tools and techniques to create a temporal GHZ state for a photon and measure
the correlations that appear in the GHZ functional. The predicted correlations, as we have seen,
providequantitativeevidenceforthecontributionofhighlyentangledhistories.
Before proceeding, we provide a “dictionary” between the complex state space of a spin-1/2
particleandthepolarizationstateofasinglephoton. Wemaketheidentifications
|z+(cid:105) ←→ |H(cid:105)
|z−(cid:105) ←→ |V(cid:105)
|x+(cid:105) ←→ |D(cid:105)
|x−(cid:105) ←→ |A(cid:105)
|y+(cid:105) ←→ |R(cid:105)
|y−(cid:105) ←→ |L(cid:105)
where “H” stands for “horizontally” polarized light, “V” stands for “vertically” polarized light,
“D” stands for “diagonally” polarized light, “A” stands for “anti-diagonally” polarized light, “R”
stands for “right-circularly” polarized light, and “L” stands for “left-circularly” polarized light.
Wehavethestandardrelations
1
|D(cid:105) = √ (|H(cid:105)+|V(cid:105)) (14)
2
1
|A(cid:105) = √ (|H(cid:105)−|V(cid:105)) (15)
2
1
|R(cid:105) = √ (|H(cid:105)+i|V(cid:105)) (16)
2
1
|L(cid:105) = √ (|H(cid:105)−i|V(cid:105)) (17)
2
To implement the GHZ test for entangled histories experimentally, we prepare a single photon
through spontaneous parametric down conversion (SPDC) shown in Figure 2. The SPDC process
generates photon pairs with perpendicular polarizations, which are then separated by a polarizing
beamsplitter (PBS). Through detection of the reflected photon by an avalanche photodiode single
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photon detector (D1), we get a single photon source on the other outport (Fiber coupler 3). We
prepare this (approximate) single photon as a diagonal polarization state [D] with a fiber-based
polarization controller (PC) and a polarizer, and then send it into a balanced Mach-Zehnder inter-
ferometer (MZI), each arm of which supports a sequence of PBSs and wave-plate sets (WP). The
incoming photon in the [D] state is initially split, by PBS0, into horizontal (H) and vertical (V)
components of equal amplitude, traveling along the two arms. WP2 and PBS2 divide the photon
in the lower arm again, removing one polarization direction, while the other continues along the
arm. We might, for example, remove the [L] polarization, while allowing [R] to continue prop-
agating. WP4 then rotates the propagating photon back to [H] direction. Two more operations
(PBS4, WP6 and PBS6, WP8) of the same type take place, until the surviving photon reaches
PBS7. The surviving photon will have been in the history state [H](cid:12)[H](cid:12)[H], and sampled by
the observable [R](cid:12)∗(cid:12)∗, where the wild cards reflect our choices of polarization for PBS4 and
PBS6. A completely parallel analysis applies to the other arm. Finally, the surviving components
recombine coherently at PBS7, another polarizing beam splitter, and emerge in a direction that
enforcesarelativeminussignbetweenthecontributionsfrom[H](cid:12)[H](cid:12)[H]and[V](cid:12)[V](cid:12)[V].
By post-selecting on the events that trigger D2, and varying the wave-plate sets appropriately, we
canmeasuretheGHZfunctionalforanentangledhistory. Wemustmeasuretheexpectationvalues
(cid:104)σ (cid:12)σ (cid:12)σ (cid:105),(cid:104)σ (cid:12)σ (cid:12)σ (cid:105),(cid:104)σ (cid:12)σ (cid:12)σ (cid:105)and(cid:104)σ (cid:12)σ (cid:12)σ (cid:105)withrespecttotheGHZhistory
x x x y y x y x y x y y
state,andthenmultiplyalloftheexpectationvaluestogether.
In this experiment we often access polarization properties at definite times, through PBS1−6.
Alternatively, in the case where the photon transmits at all of the 6 PBSs and triggers D2, we
access a multi-time observable. Such multi-time observables represent, in Griffiths’ terminology
[1],“contextual”properties. Wecanformafamilybasedonthosecomplementarysingle-timeand
multi-time properties. The character of the history state √1 ([z+](cid:12)[z+](cid:12)[z+]−[z−](cid:12)[z−](cid:12)[z−])
2
emerges clearly only when we measure multi-time observables, which in turn we access when
certainothereventsfailtooccur.
The experimental procedure is divided into 32 trials, which each have separate settings for
the polarizing beam splitters and wave plates. Each trial consists of preparing the settings of the
apparatusandrecordingthenumberofphotonsreceivedbydetectorD2(denotedbyCounts(D2)).
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Wedefine
x = D , x = A , x = R , x = L
1 2 3 4
x = A , x = D , x = L , x = R
1 2 3 4
Let PBS(α,α) denote that the PBS transmits the photon in the |α(cid:105) polarization and reflects the
orthogonal component |α(cid:105), while WP(β,γ) denote that the WP is set up so that the incoming
photon in the |β(cid:105) polarization is transformed into the |γ(cid:105) polarization. Then for a given trial, the
settingshavetheform
{PBS1(x,x), WP3(x,H), PBS2(x,x), WP4(x,V),
i i i i i i
PBS3(x ,x ), WP5(x ,H), PBS4(x ,x ), WP6(x ,V),
j j j j j j
PBS5(x ,x ), WP7(x ,H), PBS6(x ,x ), WP8(x ,V)}
k k k k k k
withfixedvaluesofi, j,k = 1,2,3.
Wedefine
Counts (D2)
C := i,j,k
i,j,k
Counts(total)
where Counts(total) denotes the overall number of photons sent into the MZI, which is a constant
fordifferenttrialsinourexperiment. Bycollectingdatafromallofthetrialswecanevaluate
C −C −C +C −C +C +C −C
(cid:104)σ (cid:12)σ (cid:12)σ (cid:105) = 1,1,1 1,1,2 1,2,1 1,2,2 2,1,1 2,1,2 2,2,1 2,2,2 (18)
x x x C +C +C +C +C +C +C +C
1,1,1 1,1,2 1,2,1 1,2,2 2,1,1 2,1,2 2,2,1 2,2,2
C −C −C +C −C +C +C −C
(cid:104)σ (cid:12)σ (cid:12)σ (cid:105) = 3,3,1 3,3,2 3,4,1 3,4,2 4,3,1 4,3,2 4,4,1 4,4,2 (19)
y y x C +C +C +C +C +C +C +C
3,3,1 3,3,2 3,4,1 3,4,2 4,3,1 4,3,2 4,4,1 4,4,2
C −C −C +C −C +C +C −C
(cid:104)σ (cid:12)σ (cid:12)σ (cid:105) = 3,1,3 3,1,4 3,2,3 3,2,4 4,1,3 4,1,4 4,2,3 4,2,4 (20)
y x y C +C +C +C +C +C +C +C
3,1,3 3,1,4 3,2,3 3,2,4 4,1,3 4,1,4 4,2,3 4,2,4
C −C −C +C −C +C +C −C
(cid:104)σ (cid:12)σ (cid:12)σ (cid:105) = 1,3,3 1,3,4 1,4,3 1,4,4 2,3,3 2,3,4 2,4,3 2,4,4 (21)
x y y C +C +C +C +C +C +C +C
1,3,3 1,3,4 1,4,3 1,4,4 2,3,3 2,3,4 2,4,3 2,4,4
andfinallytocompute
G[|Ψ)] = −(cid:104)σ (cid:12)σ (cid:12)σ (cid:105)(cid:104)σ (cid:12)σ (cid:12)σ (cid:105)(cid:104)σ (cid:12)σ (cid:12)σ (cid:105)(cid:104)σ (cid:12)σ (cid:12)σ (cid:105)
x x x y y x y x y x y y
bytakingtheproduct.
ThefourmeasuredcorrelationsareshowninFig. 2. ThecomputedvalueofGis0.656±0.005,
wheretheerrortakesintoaccountthestatisticsofdetectorphotocounting.
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Detection
Single photon source
D1 Fiber coupler 1
D2 Laser PPKTP Fiber coupler 2
Fiber coupler 4
DM
PBS 7 𝑡3
𝑡3 𝑡2
Polarizer
WP 7 PBS 5 WP 5 PBS 3 Polarization
WP 8 𝑡1 controller
HWP
PBS 6 WP 3
𝑡2
QWP
WP 6 PBS 1
PBS
WP 1
PBS 4
WP 4 WP 2
PBS 2 Fiber coupler 3
Prism
𝑡1
25mm
FIG. 1. Illustration of the experimental setup. A continuous-wave diode laser around 404 nm in wave-
length, after a band-pass filter centered at 404 nm with 3 nm bandwidth, is focused on a type II PPKTP
crystaltogeneratecorrelatedphotonpairsof808nmwavelengthwithperpendicularpolarizationsthrough
spontaneousparametricdownconversion. Adichroicmirror(DM)isusedtofilteroutthepumpbeam. The
photon pairs are split by a polarizing beam splitter (PBS), and then coupled into single mode fibers. With
theregistrationofaphotoncountatafiber-basedsinglephotondetectorD1,wegetaheraldedsinglephoton
sourceintheotherfiberoutport(fibercoupler3). Thepolarizationoftheheraldedphotonissettothestate
√
|D(cid:105) = (|H(cid:105)+|V(cid:105))/ 2withafiber-basedpolarizationcontroller(PC).Afterfilteringbyapolarizeroriented
at45◦,thephotonissentintoaMach-Zehnderinterferometer(MZI)withtwoarmsofequallengths. Inthe
MZI,asetofhalf-waveplates(HWP),quarter-waveplates(QWP),andPBSsareappliedat3differenttimes
(denotedast ,t andt inthefigure)toperformtheprojectivemeasurementsandpolarizationrecoveryop-
1 2 3
erations for the τGHZ test. All of the wave-plates are mounted on motorized precision rotation mounts
that are automatically controlled by a computer. A prism, positioned on a one-axis motorized translation
stage,isusedtopreciselyadjustthelengthofonearmsothatthetwospatialmodesintheMZIcoherently
interferewitheachother atthePBS7beforereadoutby detectorD2. Weregisterthe two-foldcoincidence
counts between D1 and D2 with a 5 ns window through a home-made Field-Programmable Gate Array
(FPGA) board. The GHZ test is repeated with 32 trials, each with different angles of the wave plates (see
supplementary materials). To guarantee that the phase of the MZI is stable during the measurement, we
monitorthecountrateC withafixedwave-platesettingbeforeandaftereachtrial.
ref
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