Measuring Which-Path Information with Coupled Electronic Mach-Zehnder Interferometers

1 MB·English

by  J. Dressel

Measuring Which-Path Information with
Coupled Electronic Mach-Zehnder Interferometers
J. Dressel, Y. Choi, A. N. Jordan
Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
(Dated: January 20, 2013)
We theoretically investigate a generalized “which-path” measurement on an electronic Mach-
Zehnder Interferometer (MZI) implemented via Coulomb coupling to a second electronic MZI act-
ing as a detector. The use of contextual values, or generalized eigenvalues, enables the precise
constructionofwhich-pathoperatoraveragesthatarevalidforanymeasurementstrengthfrom the
available drain currents. The form of the contextual values provides direct physical insight about
the measurement being performed, providing information about the correlation strength between
2 system and detector, the measurement inefficiency, and the proper background removal. We find
1 that the detector interferometer must display maximal wave-like behavior to optimally measure
0 the particle-like which-path information in the system interferometer, demonstrating wave-particle
2
complementaritybetweenthesystemanddetector. Wealsofindthatthedegreeofquantumerasure

n thatcanbeachievedbyconditioningonaspecificdetectordrainisdirectlyrelatedtotheambiguity
a of the measurement. Finally, conditioning the which-path averages on a particular system drain
J using thezero frequency cross-correlations produces conditioned averages that can become anoma-

1 louslylarge duetoquantuminterference;theweak couplinglimit oftheseconditionedaveragescan
3 produceboth weak values and detector-dependent semi-weak values.

]
h I. INTRODUCTION the Coulomb interaction, as shown in Figure 1. This ge-
p ometry has similarities to Hardy’s paradox34–37 and has
-
t been considered previously at various levels of detail by
n The construction of electronic Mach-Zehnder interfer-
severalauthors38–41 toexploresuchphenomenaasquan-
a ometers (MZIs) in the solid state is a recent innovation
u inthefabricationandcontrolofcoherentmesoscopicsys- tum erasure42–47 and Bell inequality violations. In our
q tems. The first experiment of this kind, published by treatment, we include interactionsbetween the MZIs via
[ the Heiblum group,1 used the edge states2 of an integer a minimal phenomenological model that adds a relative
interaction-induced phase shift between a pair of elec-
2 quantumHallCorbinogeometryasthe electronicanalog
v of light beams and quantum point contacts (QPCs)3–5 trons that occupy adjacent edge states simultaneously.
7 astheelectronicanalogsofopticalbeamsplitterstocon- The relative phase shift has the effect of entangling48–50
8 the states,mixing thepathinformationofthe twoMZIs.
struct an interferometer with a visibility of 62%. Other
5
interferometer designs have since been similarly imple-
2
mentedaselectronicinterferometersinthe integerquan-
.
5 tum Hall regime.6–9
0
TheelectronicMZIdiffersfromitsopticalcounterpart
1
1 in several respects. The arms of the MZI accumulate
: relative phase differences not only due to kinetic prop-
v
agation of electrons along the arms, but also because
i
X the electrons are charged particles and can thus acquire
r a geometric Aharanov-Bohm phase10,11 when the arms
a enclose a magnetic flux. This charge can also lead to
strongelectron-electroninteractions,givingrisetoavari-
etyofeffectsthathavenocounterpartinanopticalMZI.
For example, the interactions can produce differences FIG. 1. (color online) Schematic of coupled electronic MZIs.
in the counting statistics,12,13 can induce temperature- An ohmic source in a quantum Hall system at filling factor
dependentdecoherence,14–19 canbeusedtodetectexter- ν = 2 injects chiral excitation pairs biased at energy eV rel-
nal charges,20,21 and can even lead to lobe structure in ative to the ohmic reference drains D1, D2, S1, and S2 into
the visibility at high voltage bias.22–25 Such lobe struc- independent edge channels. The bias is kept low enough to
allow only one excitation per channel on average. The outer
ture was unexpected and has generated numerous the-
oretical explanations,26–33 some hypothesizing Luttinger (red)channelistransmittedentirelythroughtheQPCQd1and
liquid physics as the cause. partiallytransmittedthroughQs1 andQs2,formingthesystem
MZI. The inner (blue) channel is reflected entirely from Qs1
Here we take a more modest theoretical approach for andQs2andpartiallytransmittedthroughQd1andQd2,forming
describingelectronicMZIsinaquantumHallsystemthat a separate detector MZI. The Coulomb interaction between
focuses on the low bias regime within a single-particle the copropagating arms Ld and Us induces an average rela-
edge-state model. We consider a configuration of two tive phase shift γ between each excitation pair that couples
such single-particle electronic MZIs coupled together by theinterferometers.
2
This kind of interaction has been experimentally shown the Fermi energy E injects chiral electron-like edge ex-
F
to be capable of producing a π-phase shift on a single citations of chargee into the sample that propagateuni-
electron—perhapsthe mostdramaticdifferencefromthe directionallyalongtwoindependentedgechannels.2Each
optical analog.16 Thus, all elements of our theoretical edgechannelformsaninterferometerfromtwoappropri-
analysis are based on currently available technology. ately tuned quantum point contacts (QPCs) that coher-
Our work considers the task of detecting which-path ently split and then recombine the possible paths. The
information in one MZI by using the second MZI as the relative phase between the arms of each interferometer
detector. Since the system and the detector are identi- is determined not only by a local dynamical phase accu-
cal devices, this arrangement has several appealing fea- mulated during kinetic propagationalong each arm, but
tures. First, the symmetry of the geometry indicates also by a global geometric phase11 (in the form of the
that there should be a duality between “which-path” Aharonov-Bohm (AB) effect)10 arising from the closed
information in one MZI versus “which-fringe” informa- paths. After the paths interfere, the charges are col-
tion in the other. We will show that this is indeed lected at ohmic reference drains held at the Fermi en-
the case, which relates this work to earlier “controlled ergy, producing fluctuating output currents that can be
dephasing” experiments.51–54 We apply the contextual temporally averaged.
values formalism55,56 for generalized measurements57–59 The two interferometers accrue an additional relative
to show how even with inefficient detection60,61 and low phase shift due to the Coulomb interaction where the
visibility, the which-path information may be extracted charges copropagate. Intuitively, the mutual repulsion
fromthe detector currents systematically. Next, the fact affects the dynamical phases by effectively warping the
that both the system and the detector have their own propagation paths, which also affects the geometrical
inputs, outputs and coherence allows the effects of mea- phases by changing the areas enclosed by the paths. A
surementto be exploredin detail. In particular,the cor- morecarefulanalysisofthejointinteractionphaseispro-
relations between them can be experimentally measured vided in the Appendix. Such additional relative phase
andanalyzed,takingvariousformssuchasjointcounting hastheeffectofentangling thejointstateofthetwointer-
statistics,orevenconditionedmeasurements(seeSukho- ferometers, mixing the which-path information. Due to
rukov et al.62 for an example with incoherent electrons). theentanglement,extractinginformationfromthedrains
The ability to condition (or post-select) measurements ofoneinterferometerallowsonetoinfercorrelatedwhich-
performed on a system with quantum coherence also pathinformationabouttheotherinterferometer. Thatis,
allows the possibility of measuring weak values.55,63–66 oneinterferometercanbeusedasadetectortoindirectly
Weak values, in addition to being of interest in their measure55,57–59 the which-pathinformationof the other.
ownright,have been shownto be useful as anamplifica- As we shall see, the characteristics of the measurement
tiontechniqueformeasuringsmallvariationsofasystem will depend on the tuning of the detector interferometer
parameter67–71, aswellas for tests ofbona fide quantum as well as on the coupling phase.
behavior.35,56,72–74 We model the coupled MZI system using the elastic
Thepaperisorganizedasfollows. InSectionII,wede- scattering approach of Landauer and Bu¨ttiker3–5 for co-
scribe the considered geometry and our physical model- herent charge transport. As the transport is largely bal-
ing principles before reviewing and applying the scatter- listic in the integer quantum Hall regime, the formalism
ing theory of mesoscopic transport to the two edge state directly relates the average currents I collected at each
l
physics. In Section III, we take the joint predictions of ohmicleadl D ,D ,S ,S tothetransmission prob-
1 2 1 2
∈{ }
the scattering model and interpret them as a measure- abilities P (E) [0,1] for plane waves of fixed energy E
l
∈
ment by the detector electron that extracts information to traverse the sample successfully. Treating the ohmic
aboutthepathstateofthesystemelectron. We describe leads as thermal reservoirs, the average currents from
a principled way to construct this information from the spinless single-channel transport are,
data in a variety of regimes. In Section IV we intro-
duce the conditioning procedure and use it to clarify the eV dE
I =e (f(E+eV) f(E))P (E) (1a)
phenomenon of quantum erasure, as well as to calculate l h − l
conditioned averages of the which-path measurements. Z0
e2V
Weshallseethatbothweakvaluesandsemi-weakvalues P (E ). (1b)
l F
≈ h
willcompeteasweakcouplinglimitsoftheseconditioned
averages. In Section V we describe our conclusions.
Here, f(E) = (exp((E E )/k T)+1)−1 is the equi-
F B
−
librium Fermi distribution relative to the Fermi energy
E at a temperature T; h is Planck’s constant; and, k
F B
II. COUPLED MZIS is Boltzmann’s constant.
The approximate equality (1b) holds in the low-bias
We consider a pair of electronic MZIs embedded in regime when E eV k T and the transmission
F B
≫ ≫
a two-dimensional electron gas in the integer quantum probabilities P are constant with respect to the small
l
Hall regime at filling factor ν = 2 as illustrated in Fig- variations in energy. We assume that the source oper-
ure 1. An ohmic source with a small DC-bias eV above ates in such a regime. Due to the small spectral width
3
of the source, the fermionic excitations will then be well
approximated as plane waves at a fixed energy on the
scale of the sample; hence, on average only one excita-
tionperchannelwilloccupythesampleandintrachannel
interactions can be ignored. In particular, we avoid the
anomalouslobestructureintheinterferencethatappears
at higher bias.22–25
We also assume for simplicity of discussion that the
source only injects spinless excitation pairs with one ex-
citation per channel so that the coupling interaction be- FIG.2. (coloronline)ComplementaryQPCbalanceparame-
ters (4) for m∈{d,s} and i∈{1,2} as parametrized by the
tween the channels is fixed; the results will be averaged
balance angle θm. Left: transmission probability Tm (solid,
overa more realistic source distribution in Section IIIG. i i
blue) and reflection probability Rm (dashed, red). Right:
With these approximations, the initial joint scattering particle-likeparameterδm (solid, bliue)andwave-likeparam-
i
state for an excitation pair can be written in second- eter ǫm (dashed, red).
i
quantized notation as,
Ψ =aˆd†aˆs† 0 , (2) mentary QPC balance parameters,
| i | i
δm =Tm Rm [ 1,1], (4a)
where 0 is the filled Fermisea of the edge channels and i i − i ∈ −
| i
aˆd† and aˆs† are creation operators for plane waves of a ǫm =2 TmRm [0,1], (4b)
i i i ∈
fixed energy injected into the inner and outer channels,
respectively. Operators corresponding to different edge form d,s andi p1,2 thatsatisfy(ǫm)2+(δm)2 =
∈{ } ∈{ } i i
channels commute due to the independence of the chan- 1. All such QPC parameters can be related by a QPC
nels. balance angle θm [0,π/2] such that, Tm = cos2θm,
The inner channel will form a Mach-Zehnder interfer- Rm = sin2θm, δim∈= cos2θm, and ǫm = isin2θm asiil-
i i i i i | i |
ometerasshowninFigure1,whichwerefertoastheup- lustrated in Figure 2. We shall see that the parameters
per MZI, or the detector MZI. Similarly, the outer chan- δm control the particle-like path-bias of the excitation
i
nel will form an identical interferometer, which we refer after a QPC, while the parameters ǫm control the com-
i
to as the lowerMZI, or the system MZI. We will use the plementary wave-like interference visibility.
lowercase superscripts d and s throughout to distinguish The joint state (2) can be scattered through Qd and
1
quantities specific to the detector and the system, re- Qs using(3)intothebasisoftheMZIpaths,yieldingthe
1
spectively, and to avoid confusion with the detector and replacements,
systemdrainsthatwedenotewithcapitallettersD ,D ,
S1 and S2. 1 2 aˆd† =eiχd1td1aˆ†Ld +eiξ1dr1daˆ†Ud, (5a)
The QPCs Qd1, Qd2, Qs1, and Qs2 shown in Figure 1 aˆs† =eiχs1ts1aˆ†Ls +eiξ1sr1saˆ†Us. (5b)
eachelasticallyscattertheplanewaves,affectingonlythe
complex amplitudes of the joint scattering state. Hence, During propagation to the second pair of QPCs, each
for m d,s , i 1,2 we can represent the effect of path p Ld,Ud,Ls,Us accumulates an additional dy-
∈ { } ∈ { } ∈{ }
each QPC as a unitary scattering matrix, namical phase φ that depends on the excitation energy
p
andthepath-length. Whenthepathsrecombine,thedif-
Uˆim = eeiiχχmimi trmim eeiiξξimimrtimm , (3) finerteenrfceerebnectew.eeCnlotshinegdythneampaictahlspfhoarsMesZcIonmtributse,sdtoatlhsoe
(cid:18) i i (cid:19) contributes a relative geometric Aharonov-∈Bo{hm}(AB)
phase φm that depends on the magnetic flux enclosed
wheretm = Tm andrm =i Rm aregivenintermsof AB
i i i i by the path.
the transmission and reflection probabilities Tm [0,1]
andRm =1pTm thoughQmp. Theadditionalsica∈ttering Wecancompactlyaccountforthevariousphaseeffects
phasesiχm a−nd ξim may arisei from QPC asymmetry. contributingtotheinterferencebydefiningtuningphases
i i for each MZI,
The QPCs are kept tunable subject to the constraints
tQahnsadtanQthdd2eQaosnudtteortchcreheaaitnnennetelhreiscthwfauonllnyseelptraiasrnasftmuelilityntteerdreffleterhcirntoegudgphafrtoQhmsd1. φφds ==φφsAdABB ++φφLLsd−−φφUUsd++χχs1d1 −−ξξ1s1d., ((66ba))
1 2
There is an additional QPC near drain S1 not shown in Finally, the joint scattering amplitude correspond-
Figure 1 that is kept fixed to allow full transmission of ing to co-occupation of Ld and Us acquires a effective
the outer channeland full reflection of the inner channel Coulomb interaction phase γ that couples the two in-
in order to divert the outer channel for collection at the terferometers. (See the Appendix for discussion about
drain S1. how the Coulombeffect can produce such a phase shift.)
For later convenience we also introduce the comple- Thisinteractionphasecompactlyencodesopposingshifts
4
inthe combineddynamicalandgeometricphasesofeach III. MEASUREMENT INTERPRETATION
MZI due to the Coulomb repulsion of the charge pair.
Forsimplicity,weassumefornowthattherelativephase The joint scattering model is useful for computing
is constant; we will allow it to fluctuate for consecu- probabilities and average currents, but it does not pro-
tive pairs in Section IIIG. We also note that any ad- videdirectinsightintothemeasurementbeingperformed
ditional Coulomb phase acquired during copropagation by one interferometer on the other. To make the con-
after QPC Qd1 and before QPC Qs1 will only contribute nection to measurement more apparent, we will use the
to the tuning phase φd and can therefore be ignored. contextual values formalism55,56 that links the detector
After adding the phenomenological phases, the scat- drainprobabilitiesdirectly tothe “which-path”operator
tered joint state just before the second pair of QPCs is, for the system. We will see that we can understand the
varioussubtletiesofthemeasurementquitetransparently
Ψ′ = tdtsei(φd+φs)aˆ† aˆ† +rdrsaˆ† aˆ† (7)
| i 1 1 Ld Ls 1 1 Ud Us using this technique.
+(cid:16)rdtseiφsaˆ† aˆ† +tdrsei(φd+γ)aˆ† aˆ† 0 ,
1 1 Ud Ls 1 1 Ld Us | i
(cid:17) A. POVM
up to a global phase of exp(i(φUd +φUs +ξ1d+ξ1s)) not
written.
To facilitate the interpretation of the distinguishable
Theinteractionphaseγ hastheeffectofentangling the
detector drains as the outcomes of a measurement being
twointerferometers,whichwecanshowbycomputingthe
performed on the system, we define the single particle
concurrence,48
state kets,
γ
Ψ′ =ǫdǫs sin [0,1]. (8)
C | i 1 1 2 ∈ D =aˆ† 0 , D =aˆ† 0 , (12a)
h i (cid:12) (cid:12) | 1i D1| i | 2i D2| i
Wthee speheatsheatγthe eπntaanngdlemvaenni(cid:12)(cid:12)tshreesaca(cid:12)(cid:12)hsesγa max0i.muFmurwthheern- |Lsi=aˆ†Ls|0i, |Usi=aˆ†Us|0i, (12b)
→ →
more, the entanglement directly depends on the QPCs definethereducedsystemstateinabsenceofinteraction,
Qd and Qs preparing interfering wave-like excitations
in1each M1ZI, which is measured by ǫdǫs; maximum ψs =eiφsts Ls +rs Us , (13)
1 1 | i 1| i 1| i
entanglement can only occur for balanced QPCs with
Td =Ts =1/2, or ǫdǫs =1. and write (10) in the form,
1 1 1 1
Atthispoint,weconceptuallybreakthesymmetrybe-
tween the two interferometers to treat one as a detector Ψ′′ = D Mˆ ψs + D Mˆ ψs . (14)
| i | 1i⊗ D1| i | 2i⊗ D2| i
for informationaboutthe other. We willtreatthe upper
MZIasthedetector andthelowerMZIasthesystem be- The interaction with the detector in (14) is entirely
ingmeasured,thoughobviouslywecouldexchangethose represented by operators acting on the reduced system
roles by the symmetry of the geometry. To do this we state (13) that contain all the scattering information of
finish scattering the detector MZI through Qd into the the detector,
2
basis of the ohmic detector drains D ,D using (3),
{ 1 2} MˆD1 =CD1,Ls|LsihLs|+CD1,Us|UsihUs|, (15a)
(cid:18)aˆaˆ†U†Ldd(cid:19)=Uˆ2d aaˆˆ†D†D12! (9) MˆD2 =CD2,Ls|LsihLs|+CD2,Us|UsihUs|. (15b)
The operator Mˆ encodes the interaction followed by
yielding, D1
the absorption of the detector excitation at the drain
|Ψ′′i= aˆ†D1 CD1,Lseiφsts1aˆ†Ls +CD1,Usr1saˆ†Us (10) Dfol1l.owSeimdiblayrlayb,stohrepotipoenraattorDMˆ.DW2 eenrceofderestotheMˆinter,aMˆction
+(cid:16)aˆ†D2h[CD2,Lseiφsts1aˆ†Ls +CD2,Usr1saˆ†Us]i |0i, asAmseatshuerecmouepnltinogpeprahtaosres.5γ7–259 0 the meas{ureDm1entDo2p}-
(cid:17) →
up to the same global phase as in (7). For later con- erators (15) become nearly proportional to the identity.
venience, we have defined the detector scattering ampli- We call γ 0 the weak coupling limit since the reduced
→
tudes, system state is only weakly perturbed for small γ. Con-
verselythe limitγ π iscalledthestrong coupling limit
CD1,Ls =eiχd2 td1td2eiφd +r1dr2d , (11a) sincethemeasurem→entoperatorsaremaximallydifferent
fromtheidentityandmaximallyperturbthereducedsys-
CD1,Us =eiχd2 (cid:16)td1td2ei(φd+γ)+r1d(cid:17)r2d , (11b) tem state.
The measurement operators also form a Positive
CD2,Ls =eiξ2d (cid:16)td1r2deiφd +r1dtd2 , (cid:17) (11c) Operator-Valued Measure (POVM) on the system,
CD2,Us =eiξ2d (cid:16)td1r2dei(φd+γ)+r(cid:17)1dtd2 . (11d) EˆD1 =MˆD†1MˆD1, EˆD2 =MˆD†2MˆD2, (16)
(cid:16) (cid:17)
5
such that Eˆ + Eˆ = ˆ1. The POVM elements under which the operator basis is orthonormal,
D1 D2
Eˆ ,Eˆ act as probability operators for the measure-
{meDnt1ouDtc2o}mes. σˆµs,σˆνs =δµν. (20)
Hence, the probability of absorbing the detector exci-
(cid:10) (cid:11)
tation at a drain D D ,D can be expressed either Hereµ,ν 0,1,2,3andδ istheKroneckerdeltathatis
∈ { 1 2} ∈ µν
as an expectation of the projection operator of the de- 1 if µ=ν and 0 otherwise. Using this basis, any system
tector drain under the joint state (10) or, equivalently, observable can be written,
as an expectation of the probability operator (16) under
the unperturbed system state (13), Aˆ= a σˆs, (21a)
µ µ
µ
P = D Ψ′′ 2 = ψs Eˆ ψs (17) X
D |h | i| h | D| i a = Aˆ,σˆs =Tr Aˆσˆs /2, (21b)
=|CD,Lstd2|2+|CD,Usr2d|2. µ D µE h µi
where a arereal-valuedcomponentsoftheobservable.
By working with the reduced state (13), the measure- µ
{ }
ment operators (15), and the probability operators (16), Using (21), we can expand the probability operators
we treat the detector as an abstract entity whose sole on the system (16) in the basis (18) to determine their
purpose is to measure the system. Such abstraction al- structure,
lows us to more clearly examine the measurement being
1 1
made upon the system. Eˆ = βd Vd∆d σˆs VdΓdσˆs, (22a)
D1 2 +− 0− 2 3
Eˆ = 1(cid:0)βd +Vd∆d(cid:1)σˆs+ 1VdΓdσˆs, (22b)
B. Contextual Values D2 2 − 0 2 3
(cid:0) (cid:1)
where we see that the measurement is characterized by
In order to relate the measurement on the system to
the detector parameters,
observableinformationthatwecaninterpret,wewilluse
contextual values55,56 to formally construct system ob-
βd =2 TdTd +RdRd =1+δdδd, (23a)
servables from the probability operators (16). This for- + 1 2 1 2 1 2
malism acknowledges that the only quantities to which β−d =2(cid:0)T1dR2d +R1dT2d(cid:1)=1−δ1dδ2d, (23b)
wehaveexperimentalaccessarethedetectordrainprob-
Vd =4(cid:0) TdRdTdRd =(cid:1)ǫdǫd, (23c)
abilities, so all observations we wish to make about the 1 1 2 2 1 2
system must be contained somehow in those probabili- qγ γ
Γd =sin sin +φd , (23d)
ties. Generally,the correspondencebetweenthe detector 2 2
drains and a particular system observable will be imper- ∆d =cosφd (cid:16)Γd. (cid:17) (23e)
fect, but we can compensate for such ambiguity of the −
detection by weighting the drain probabilities with ap-
definedin terms ofthe QPC balanceparameters(4), the
propriate values for the particular measurement setup.
tuning phases (6), and the coupling phase γ. We will
Generally,wecannotconstructinformationaboutjust
describe these parameters in detail in the next section.
any system observable from a particular measurement.
The probability operators only contain components in
To find which observables we can measure, it is useful thesubspacespannedby σˆs,σˆs ;therefore,we can only
to decompose the probability operators (16) into an or- { 0 3}
construct observables that are contained within that sub-
thonormalbasisfortheobservablespace. Inourcase,the
space. That is, we can construct any observable of the
system state space is two-dimensional,so any Hermitian formAˆ=a σˆs+a σˆs. Wedenoteobservablesofthisform
operator can be spanned by the four basis operators, 0 0 3 3
as being compatible with the measurement (22). Other
observables are incompatible with the measurement.
σˆs =ˆ1= Ls Ls + Us Us , (18a)
0 | ih | | ih | Toconstructsuchacompatiblesystemobservablefrom
σˆs =σˆs = Ls Us + Us Ls , (18b)
1 x | ih | | ih | the measurement, we expand its operator directly in
σˆs =σˆs = i(Ls Us Us Ls ), (18c) terms of the probability operators (16),
2 y − | ih |−| ih |
σˆs =σˆs = Ls Ls Us Us , (18d)
3 z | ih |−| ih | Aˆ=a σˆs+a σˆs =α Eˆ +α Eˆ . (24)
0 0 3 3 D1 D1 D2 D2
which are equivalent to the identity operator and the
Pauli spin operators. To find the real components of Therequiredexpansioncoefficientsα andα aregen-
D1 D2
an observable in this basis we introduce the normalized eralized eigenvalues, or contextual values55,56, of the op-
Hilbert-Schmidt inner product between operators, erator. Usingthisexpansion,wecanrecoverthesamein-
formationon average asaprojectivemeasurementbyus-
Tr Aˆ†Bˆ ingonlythedrainprobabilities,hAˆi=αD1PD1+αD2PD2.
Aˆ,Bˆ = , (19) To determine the appropriate contextual values to as-
Thr ˆ1 i signinorderto constructAˆ,we insert(22)into (24)and
D E
(cid:2) (cid:3)
6
solveitasastandardmatrixequationusingtheorthonor-
mal basis, which yields the unique contextual values,
a βd
α =a 3 − +∆d , (25a)
D1 0− Γd Vd
(cid:18) (cid:19)
a βd
α =a + 3 + ∆d . (25b)
D2 0 Γd Vd −
(cid:18) (cid:19)
Aslongasthecontextualvaluesdonotdiverge,theex-
pansion(24)ofthecompatibleoperatorAˆiswelldefined,
and we can perfectly recover its average,
Aˆ = ψs Aˆψs =a +a δs. (26)
h i h | | i 0 3 1
The observable parameter a0 sets the reference point for FIG. 3. (color online) The contextual values (25) of the
theaverage,socontributesnoinformationaboutthesys- which-path operator σˆ3s: αD1 (solid, blue) and αD2 (dashed,
tem; we will set it to zero in what follows without loss red), as a function of the coupling phase γ. The curves are
of generality. Similarly, the remaining parameter a sets shown for efficient detection Vd = 1 and detector tunings
3
the scale of the average; we will set it to one in what φd ={0,π/2,3π/4,π}. Thetuningstrongly affects theambi-
follows. guityofthemeasurement;moreover,therolesofthedetector
drains flip as the tuningvaries from φd =0 toφd =π.
Weformallyconcludethatthedetectordrainsperform
ageneralizedmeasurementofthewhich-pathoperator σˆs,
3
asmightbeintuitivelyexpectedfromthepath-dependent
interaction. Moreover,the QPCparameterδs definedin
1
(4a) determines the particle-like which-path behavior on
average. No other information about the system can be
inferred from the measurement.
The contextual values (25) are shown in Figure 3 for
a few parameter choices. If they are equal to the eigen-
values of the which-path operator, α ,α = 1, then
D1 D2 ±
the measurement is unambiguous: one obtains perfect
knowledge about the path information with every drain
detection, and the system state is projected to a pure
path state. If the contextual values diverge, α ,α
D1 D2 →
, then the measurementis completely ambiguous: no
±∞
knowledge about the path information can be obtained,
and the system state is unprojected; however, will shall
see in Section IIID that the system state may still be
unitarily perturbed by the coupling. In between these
extremes the measurement is partially ambiguous: par-
tial knowledge is obtained about the path information
with each drain detection, and the system state is par-
tially projected toward a particular path state.
FIG. 4. (color online) The drain probability PD1 (27) as a
C. Parameters function of the detector tuning φd and the which-path infor-
mation δ1s, shown for efficient detection Vd =1 and coupling
phases γ = {0,π/4,π/2,π}. For zero coupling the interfer-
Tobetterunderstandtheparameters(23)wewritethe
enceisindependentofthewhich-pathinformation;forstrong
drain probabilities (17) explicitly,
couplingγ =π theinterferencemaximallycorrespondstothe
which-path information.
1
P = βd Vd ∆d+δsΓd , (27a)
D1 2 +− 1
1(cid:0) (cid:0) (cid:1)(cid:1)
PD2 = 2 β−d +Vd ∆d+δ1sΓd . (27b) they indicate the average background signal of each de-
tector drain and satisfy (βd + βd)/2 = 1. The wave-
(cid:0) (cid:0) (cid:1)(cid:1) + −
The probability PD1 is illustrated in Figure 4 for several like parameter Vd [0,1] is determined entirely by the
values of the coupling strength. path-uncertainty pa∈rameters ǫd and ǫd; it indicates the
1 2
The particle-like parameters βd,βd [0,1] are deter- visibility of the interference. The parameter Γd [ 1,1]
mined entirely by the path-bias+para−m∈eters δd and δd; indicates the deviation in the interference cause∈d b−y the
1 2
7
coupling phase γ, which is the only effect of the charge terference encodes the measurement result: Γd cosφd
coupling. Theparameter∆d [ 1,1]indicatestheinter- and ∆d 0. →
∈ − →
ference unrelated to the path information of the system. Practically speaking, the detector must be calibrated
As the coupling γ 0, then Γd 0 and ∆d cosφd, in the laboratory before it can be used to probe an un-
→ → →
which recoversthe signalfor an isolatedinterferometer.1 known system state. That is, the detector parameters
As the coupling γ π, then Γd cosφd and ∆d 0, (23)mustbe predeterminedbyexaminingthe drainout-
→ → →
and the interference maximally corresponds to the path puts of the detector under known system configurations.
information. Forexample,pinchingoffQPCQstopreventanyinterac-
1
Theparameters(23)alsogiveinsightintothenatureof tions allows most of the parametersto be set directly by
the measurement by the role they play in the contextual tuningthedetectorQPCsandthemagneticfield. There-
values (25). The parameter Γd indicates the correlation maininginteractionparameterγ canbe inferredfroman
betweenthedetectordrainsandthewhich-pathinforma- additionalreferencesystemstate. Therefore,the process
tion. Its magnitude Γd [0,1] denotes the correlation of detector calibration can be viewed as the experimen-
strength, with 1 indic|ati|n∈g perfect correlation and 0 in- taldeterminationofthe appropriatecontextual values to
dicating nocorrelation;due to the inversedependence in assign to the detection apparatus.
(25), any imperfect correlation will amplify the contex-
tualvalues to compensate forthe resultingmeasurement
ambiguity. ThesignofΓdindicatesthecorrespondenceof D. Measurement Disturbance
the detector drains to the which path information, with
denoting the mapping D ,D Ls,Us and + Themeasurementnecessarilydisturbsthesystemstate
1 2
−denoting the mapping D{,D } ↔Ls,{Us . N}ote that by extracting information. We can see the effect of such
2 1
{ } ↔ { }
thecorrelationstrengthdependsnotonlyonthecoupling disturbance by characterizing the system interferometer
phaseγ,butalsoonthetuningphaseφd;hence,itispos- with analogous parameters to (23),
sible for the detector drains to be uncorrelated with the
system paths even under strong coupling (e.g. examine β+s =2(T1sT2s +R1sR2s)=1+δ1sδ2s, (28a)
φd =π/2 in Figure 4 when γ =π). βs =2(TsRs +RsTs)=1 δsδs, (28b)
− 1 2 1 2 − 1 2
The parametersβ+d andβ−d in(25) counterbalancethe Vs =4 TsRsTsRs =ǫsǫs, (28c)
bias in the average drain background caused by a pre- 1 1 2 2 1 2
γ γ
ferred particle-like path. For instance, if βd > βd then Γs =sipn sin φs , (28d)
+ − 2 2 −
the signal at drain D is stronger on average in (27);
1 ∆s =cosφs Γ(cid:16)s. (cid:17) (28e)
hence, the contextual value (25a) assigned to D is pro-
1 −
portional to the smaller value βd to compensate.
− Using these parameters the absorption probabilities for
The visibility parameter Vd controls the wave-like in- the system drain take the simple form similar to (27),
terference produced by Qd and Qd. The transmission of
1 2
eachQPCshouldbe balancedinorderto providethe in- 1
P = βs Vs ∆s δdΓs , (29a)
teraction phase with an equal-amplitude reference phase S1 2 +− − 1
forlaterinterference. Anyimbalanceleadstoinefficiency 1(cid:0) (cid:0) (cid:1)(cid:1)
P = βs +Vs ∆s δdΓs . (29b)
ofthemeasurement60byreducingtheinterferencevisibil- S2 2 − − 1
ity, effectively hiding the correlations. Such inefficiency (cid:0) (cid:0) (cid:1)(cid:1)
increases the measurement ambiguity and results in an With efficient detection Vd = 1 and strong coupling
γ π, then δd 0 and ∆s 0, so the system drain
oaarmreQphld2iifidicdsaeftnuiollanytotzrfeartnohseminctiseosrnifvteeerxeontrucaerelvflviesacilbtuiiveliset.,yTAw1dlhl,eTcn2odreri∈tehl{ae0tr,io1Qn}d1s, pPparSo→t2hb→ambieβlai−stsi/ue2sr;edmthi1seapn→tltaiwys,ilnalofsotirrncotenegrpflaey→rrtecinocucleep-,lleikPdeS,1setffia→tciisetβni+scts/w2inhaitcnhhde-
whichleadstodivergentcontextualvalues. Maximumin-
system.16,38
terferencevisibilityoccursforbalancedtransmissionwith
Vd = 1. We see that to optimally measure the particle- The measurement disturbance may be analyzed more
explicitly by rewriting the measurement operators (15)
like which-path information for the system, the detector
must itself exhibit maximal wave-like interference; the for the case of efficient detection Vd =1,
detector and system behaviors are therefore complemen-
tary. MˆD1 =ieiχd2eiφd/2UˆγEˆD1/12, (30a)
Theparameter∆distheportionoftheinterferencenot Mˆ =ieiξ2deiφd/2Uˆ Eˆ1/2, (30b)
affected by the coupling, meaning ∆d +Γd = cosφd. It D2 γ D2
φd φd+γ
indicates an additional bias in the drain correspondence Eˆ1/2 =sin Ls Ls +sin Us Us , (30c)
caused by the interference not pertinent to the which- D1 2 | ih | 2 | ih |
pathmeasurement. Thecontextualvaluesnaturallysub- Eˆ1/2 =cosφd Ls Ls +cosφd+γ Us Us , (30d)
tract the contribution from this irrelevant background D2 2 | ih | 2 | ih |
interferenceto retrievethe measurementinformation. In Uˆ =exp iγ Us Us . (30e)
the limit of strong coupling γ π, all the detector in- γ 2| ih |
→ (cid:16) (cid:17)
8
Thedisturbancemanifestsitselfastwodistinctprocesses. ambiguity in the measurement, however,the system will
First, the positive roots of the POVM (16) Eˆ1/2,Eˆ1/2 always be perturbed by the additional unitary evolution
perform the information extraction necess{arDy1forDt2h}e (30e) that induces a relative phase shift of π/2 between
measurement, partially projecting the reduced system the arms. Since the system state will be appreciably al-
state toward a particular path. Second, the coupling- tered by the strong coupling, the measurement will not
dependent unitary factor Uˆ contributes an additional beweakevenwhencompletelyambiguous. Hence, ambi-
γ
evolutionofthesystemthatisunrelatedtotheextraction guity of the measurement need not indicate weakness of
of information. The remaining phase factors contribute the measurement.
only to the global phase of the measured state and do The measurement becomes unambiguous when the
not alter the subsequent measurement statistics. tuningis heldfixedatcosφd = 1. Inthis situation,the
±
Unambiguousmeasurementsextractmaximalinforma- detector drains are perfect “bright” and “dark” ports:
tion from the system and thus project the system state detection at the dark port will occur deterministically
to a definite path; they are frequently known as projec- whenthe systemexcitationis inthe upper arm. The de-
tive or strong measurements. Ambiguous measurements tector drains are perfectly correlated to the which-path
extractpartialinformationfromthesystemandthuspar- information of the system, so the system state is pro-
tially project the system state toward a particular path. jected to a definite path, and the measurementis strong.
Completely ambiguous measurements extract no infor-
mationfromthesystemandthusarecompletelyunitary.
F. Weak Coupling Limit
When the system state is nearly unperturbed up to a
globalphase,the measurementis calledweak, whichcor-
responds to the case of a nearly completely ambiguous The weak coupling limit is the limit as the coupling
measurement with a negligible unitary evolution. phaseγ 0 andthe systemanddetectorbecome nearly
→
uncoupled. Since at zero coupling the measurement op-
erators(30)musteitherbezeroorbeproportionaltothe
E. Strong Coupling identity, the weak coupling limit of a measurement must
haveoutcomesthatareinherentlyambiguous. Hence,we
An unambiguous measurement can only be obtained expect the contextual values (25) to diverge. However,
in the limits of efficient detection Vd 1 and strong sinceΓd =sinφd(γ/2)+cosφd(γ/2)2+O(γ3) the nature
coupling γ π. In this situation, the a→mbiguity will be of the divergence will also depend upon the tuning.
determined→only by the tuning phase of the detector φd, Ifthe tuning is notanintegermultiple ofπ, then both
andthePOVMwillhavethemostsymmetricdependence measurement operators (30) will approach the identity
on the which-path operator, as γ 0 and the measurement will be weak for all out-
→
comes. That is, the system state will be nearly unper-
1 turbedforanyoutcomeofthemeasurement. Inthiscase,
α − , (31a)
D1 → cosφd Γd =sinφd(γ/2)+O(γ2) and the divergence of the con-
1 textual values (25) will be linear in γ. For an efficient
αD2 → cosφd, (31b) detector with Vd =1, we find to O(γ2),
Eˆ 1 ˆ1 σˆscosφd , (31c) 21+cosφd
D1 → 2 − 3 αD1 →1− γ sinφd , (32a)
Eˆ 1(cid:0)ˆ1+σˆscosφd(cid:1). (31d) 21 cosφd
D2 → 2 3 α 1+ − , (32b)
D2 → γ sinφd
(cid:0) (cid:1)
As the tuning phase φd varies, the POVM elements 1 cosφd γ
oscillate between pure path projections and the identity, EˆD1 → − 2 ˆ1+ 2 sinφd|UsihUs|, (32c)
despitethestrongcoupling. Thetuning-dependentdrain
1+cosφd γ
ambiguity contributes to the inefficiency ofthe measure- Eˆ ˆ1 sinφd Us Us . (32d)
D2 → 2 − 2 | ih |
ment by erasing the potentially extractable which-path
informationfromthe detectorstate. Indeed,we shallsee The POVM has simple dependence on the projection to
in Section IVB that suchambiguity in the measurement theupperpath,whichcanalsobewrittenintermsofthe
allows the system interference to be recovered by condi- which-path operator, Us Us = (ˆ1 σˆs)/2. The most
| ih | − 3
tioning the system results on specific detector outcomes: symmetric case of φd = π/2 is shown in the upper-right
Such a phenomenon is known as quantum erasure.42–47 of Figure 3.
InalaboratoryquantumHallsystemtheABphasewill However, if the tuning φd = nπ with integer n, then
precessduetoslowdecayofthetransversemagneticfield, onlyone ofthemeasurementoperatorswillapproachthe
so the tuning phase φd will also precess slowly. Hence, identity as γ 0. The remaining outcome remains pro-
→
theambiguityofthemeasurementwillgenerallyoscillate portional to a projector with a vanishing coefficient and
betweenextremes,whilealsoflippingthecorrespondence will thus strongly perturb the system state. Hence, only
ofthedrainstothewhich-pathinformation. Despiteany one contextual value diverges while the other remains a
9
constant eigenvalue. In this case, Γd = ( 1)nsin2(γ/2) PHΓ’L ΗHΣL
−
so the divergence will be quadratic in γ. We call such Γ 1
a measurement a semi-weak measurement56 since only a
1
subset of outcomes are weak. For an efficient detector 0.5
with Vd =1, we find, Σ
0 Γ’ Σ
1 γ 0 Π Π 3Π Π 0 Π Π 3Π Π
αD1 → si−n2 γ (−1)n+cos2 2 , (33a) 4 2 4 4 2 4
2 (cid:16) (cid:17)
1 γ
α ( 1)n cos2 , (33b)
D2 → sin2 γ − − 2 FIG. 5. (color online) Left: The raised cosine distribution
2 (cid:16) (cid:17) showing a spread in the coupling phase centered at γ = π/2
1 γ
Eˆ (1 ( 1)n)ˆ1+( 1)nsin2 Us Us , (33c) by a half-width σ =π/4. Right: The inefficiency factor η(σ)
D1 → 2 − − − 2| ih |
definedin (36) as a function of thehalf-width σ.
1 γ
Eˆ (1+( 1)n)ˆ1 ( 1)nsin2 Us Us . (33d)
D2 → 2 − − − 2| ih |
hascompactsupport. We centerthe distributionaround
ThePOVMretainsthesimpledependenceontheprojec-
γ [0,2π], and give it the half-width σ [0,π]. The
tion to the upper path. The cases for n = 0 and n = 1 ∈ ∈
densityforthedistributionisnonzerointhedomainγ′
are shown in the upper-left and lower-right of Figure 3, ∈
[γ σ,γ+σ] and has the form,
respectively. −
For the semi-weak measurement, the effect of absorp- 1 π
P(γ′)= 1+cos (γ′ γ) . (34)
tion at one of the drains is projective. The projective
2σ σ −
drain outcome unambiguously indicates that the system (cid:16) (cid:16) (cid:17)(cid:17)
excitation took the upper Us path; therefore, the con- An example of the distributionis showninFigure 5 cen-
textual value assigned to the complementary drain is tered at γ =π/2 and with half-width σ =π/4.
an eigenvalue. In contrast, the effect of absorption at Averaging the probability operators (16) only affects
the complementary drain only weakly perturbs the sys- the constant Γd, which is replaced by the averaged ver-
tem state. Its outcome only ambiguously corresponds to sion,
which-path information; therefore, the contextual value
γ+σ
assignedto the projective drain must be amplified. Such Γd(γ) dγ′P(γ′)Γd(γ′)=η(σ)Γd(γ), (35)
complementary behavior of the contextual value ampli- →Zγ−σ
fication can be counter-intuitive, but it emphasizes that π2 sinσ
η(σ)= . (36)
thefunctionoftheamplificationistocompensateforthe π2 σ2 σ
ambiguity of the measurement. (cid:18) − (cid:19)(cid:18) (cid:19)
WeshallseeinSectionIVFthatwhileconditionedav- A plot of the damping factor η(σ) is shown in Figure 5.
erages of the weak measurements (32) will lead to weak We assumed in (2) that the source emits only excita-
values, the conditioned averages of the semi-weak mea- tion pairs. However, any contribution of unpaired exci-
surements (33) have different limiting behavior and lead tations in the initial joint state is equivalent to a contri-
to different values. The two limiting cases will compete bution of joint states with γ = 0. The net effect of the
depending on the relative magnitudes of γ and φd. sourceemittingsuchunpairedexcitationsisthustomod-
ifyΓd byanadditionalprobabilityfactorP thatdenotes
p
the likelihood of pair emission. Hence, the only effect of
G. Fluctuating Coupling animperfect sourceis to introduce a netinefficiency fac-
tor η′ =P η(σ) [0,1] in the parameter Γd.
p
∈
If the coupling between excitation pairs is not a con- Since the contextual values (25) inversely depend on
stantrelativephaseγ,butinsteadcanfluctuate withina Γd, any such inefficiency will introduce an overallampli-
finite uncertaintywidth σ aroundanaverageγ,then the fication factor of 1/η′. In other words, any uncertainty
average measurement will be correspondingly more am- inthecouplingstrengthwillleadtoadditionalambiguity
biguous. We could quantify this effect by averaging the in the averagemeasurement by degrading the portion of
jointstate(10)overarangeofcouplingphasestocreatea the detector interference that is coupled to the system.
mixed state represented by a density operator; the mea-
surementoperators(15)andresulting POVM(16) could
then be generalized to an averaged measurement from H. Observation Time
that density operator. However, that procedure would
be completely equivalent to the simpler procedure of av- Since ambiguous measurements provide less informa-
eraging the probability operators (16) over the coupling tion about an observable per measurement, more mea-
width directly, which we choose to do here. surements will be required to achieve a desired precision
For simplicity, we consider as a coupling distribution for an observable average. We can characterize the nec-
theraised cosine distribution,whichisGaussian-like,but essary increase in observation time as follows. The total
10
detector current is I = (e2VP)/h = e/τ according to der of,
m
(1b), where P is the total probability for excitations to
traverse the sample; hence, we can infer that the aver- α2 +α2
T τ D1 D2. (40)
age time per detector absorption is τm = h/eVP. For ≈ m ǫ2
our single-particle model to apply we wish for the volt-
As the measurement becomes more ambiguous the con-
age bias V to be low enough that the interferometers
textual values become more amplified and so lengthen
contain less than one excitation per channel on average.
the observationtime necessaryto achievethe RMS error
The characteristicmeasurement time τ will then be on
m
of ǫ. For a strong measurement the upper bound on the
theorderofthe time-of-flightτ ℓ/v ofanexcitation
m ≈ F observation time is T 2τ /ǫ2.
pair through the sample, where ℓ is the average path m
≈
IV. CONDITIONED MEASUREMENTS
length of the interferometers and v is the Fermi veloc-
F
ity of the ballistic excitations. An observation time of T
at the drains D and D therefore roughly corresponds To gain further insight into the which-path informa-
1 2
to n T/τ individual measurement events. tion, we can condition the measurement on the subse-
m
≈
quent absorption of the system excitation at a specific
The contextual values can be used to provide an up-
systemdrain. To do this we must obtainthe joint trans-
per bound for the number of measurement events for
mission probabilities for pairs of detector and system
a desired root-mean-square (RMS) error in the estima-
drains. Conditional probabilities can then be defined in
tion of the average. Specifically, to estimate the aver-
age σˆs from a sequence of n random drain absorptions terms of the joint and single transmission probabilities.
h 3i As pointed out by Kang38 these probabilities are ex-
(d ,d , ,d ), where d D ,D , one can use an
1 2 n i 1 2
··· ∈ { } perimentally accessible in the low-bias regime through
unbiased estimator for the average,
the zero-frequencycross-correlationnoisepowerbetween
a detector drain D D ,D and a system drain
1 n S S ,S , ∈ { 1 2}
E[σˆz]= n αdi, (37) ∈{ 1 2}
Xi e3V
S 2 (P (E ) P (E )P (E )). (41)
D,S S,D F S F D F
≈ h −
that is defined in terms ofthe contextual values assigned
to each measurement realization. As n , the esti- Hence, knowledge of both the average currents (1) and
mator (37) converges to hσˆ3si=αD1PD1 +→α∞D2PD2. The thenoisepower(41)allowsthedeterminationofboththe
mean squared error (MSE) of this estimator is given by joint and single transmission probabilities.
the variance of the contextual values over the number of
measurements,
A. Joint Scattering
α2 P +α2 P σˆs 2
MSE[E[σˆz]]= D1 D1 D2 D2 −h 3i . (38) We can determine the joint probabilities directly in
n
the scatteringmodelby rewriting(10)inthe basisofthe
system drains using (3),
Hence, the RMS error MSE[E] scalesas1/√n andim-
proves with an increasing number of measurements.
peWrbiothuonudtopnreiocranknmowaklpeedfgoertohfethMeSsEtaitse,thaerneaosromn-asbquleaurepd- (cid:18)aˆaˆ†U†Lss(cid:19)=Uˆ2s aaˆˆ†S†S21! (42)
of the contextual values over the number of measure-
yielding,
ments,
Ψ′′′ = C aˆ† aˆ† +C aˆ† aˆ† (43)
α2 +α2 | i D1,S1 D1 S1 D1,S2 D1 S2
MSE[E[σˆ ]] D1 D2. (39) (cid:16)
z ≤ n +CD2,S1aˆ†D2aˆ†S1 +CD2,S2aˆ†D2aˆ†S2 |0i,
(cid:17)
It then follows that to guarantee a maximum desired up to the same global phase as in (10).
RMS error ǫ one needs an observation time on the or- The relevant joint scattering amplitudes are,
C =ei(χd2+χs2) rdrdrsrs +tdtdrsrsei(φd+γ)+rdrdtstseiφs +tdtdtstsei(φd+φs) , (44a)
D1,S1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
C =ei(χd2+ξ2s) hrdrdrsts +tdtdrstsei(φd+γ)+rdrdtsrseiφs +tdtdtsrsei(φd+φs)i, (44b)
D1,S2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
C =ei(ξ2d+χs2)hrdtdrsrs +tdrdrsrsei(φd+γ)+rdtdtstseiφs +tdrdtstsei(φd+φs)i, (44c)
D2,S1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
h i