Postponing the Past: An Operational Analysis of Delayed-Choice Experiments

Found Phys (2010) 40: 55–92DOI 10.1007/s10701-009-9375-9

Postponing the Past: An Operational Analysisof Delayed-Choice Experiments

M. Bahrami · A. Shafiee

Received: 29 January 2009 / Accepted: 29 October 2009 / Published online: 10 November 2009© Springer Science+Business Media, LLC 2009

Abstract The prominent characteristic of a delayed-choice effect is to make thechoice between complementary types of phenomena after the relevant interactionbetween the system and measuring instrument has already come to an end. In this pa-per, we first represent a detailed comparative analysis of some early delayed-choicepropositions and also most of the experimentally performed delayed-choice proposalsin a coherent and unified quantum mechanical formulation. Taking into the accountthe represented quantum mechanical descriptions and also the rules of probabilitytheory, we discuss that there are two fundamentally different descriptions concerningthe physical mechanism of the delayed-choice between the complementary types ofphenomena in these experiments. In this regard, we finally conclude that the delayed-choice experiments can be classified into two main groups where the temporal aspectof each group is characterized differently within their probabilistic descriptions.

Keywords Delayed-choice · Interference · Complementary phenomena

1 Introduction

The behaviors of quantum systems that are manifested during the measurementprocess are indispensable to the experimental arrangement that the quantum systemsinteract with. On this account, Bohr reckoned that “[t]he quantum mechanical formal-ism permits the well-defined applications referring only to . . . closed phenomenon,1

1Emphasizes are made by authors.

M. Bahrami · A. Shafiee (�)Research Group On Foundations of Quantum Theory and Information, Department of Chemistry,Sharif University of Technology, P.O. Box 11365-9516, Tehran, Irane-mail: [email protected]

M. Bahramie-mail: [email protected]

56 Found Phys (2010) 40: 55–92

[t]he word phenomenon . . . refer[s] only to observations obtained under circum-stances whose description includes an account of the whole experimental arrange-ment” [1].

According to Bohr, it is impossible to subdivide a quantum phenomenon into a se-quence of physically unambiguous well-defined steps and “ascribe customary phys-ical attributes” to them, i.e., the unambiguous use of space-time concepts for de-scription of each step is impossible. “In particular, it must be realized that—besidesin the account of the placing and timing of the instruments forming the experimen-tal arrangement—all unambiguous use of space-time concepts in the description ofatomic phenomena is confined to the recording of observations which refer to markson a photographic plate or to similar practically irreversible amplification effects likethe building of a water drop around an ion in a cloud-chamber” [1]. For ascribing theunambiguous physical attributes to the dynamical evolution of object within the mea-suring instrument, we have to subdivide the phenomenon. “Any attempt of subdivid-ing the phenomena will demand a change in the experimental arrangement”. How-ever, because of “introducing new possibilities of interaction between objects andmeasuring instruments” [1], the change of the experimental arrangement at a propertime and a proper place will result to a new phenomenon. “Consequently evidenceobtained under different experimental conditions cannot be comprehended within asingle picture, but must be regarded as complementary in the sense that only thetotality of the phenomena exhausts the possible information about the objects” [1].Therefore, “we are dealing with individual phenomena and that our possibilities ofhandling the measuring instruments allow us only to make a choice between the dif-ferent complementary types of phenomena we want to study” [1].

In order to “provide new insights” on key ingredients of Bohr interpretation (i.e.,the indivisibility and closure of quantum phenomenon and also refutation of any at-tempt of a pictorial representation of the behavior of quantum systems within mea-suring apparatus), Wheeler asked the following question: Is it possible to make thechoice between complementary modes of observation “after the relevant interactionhas already come to an end” [2], by changing the experimental arrangement at aproper time and a proper place?

To show such possibility, Wheeler suggested some thought experiments that arenow usually called as delayed-choice arrangements. Henceforth, many thought exper-iments called as delayed-choice have been proposed and some of them have been per-formed experimentally, too. In our opinions, the quintessential feature of a delayed-choice scheme is the possibility to choose between different complementary phenom-ena after the relevant interactions between the measuring instrument and the quantumsystem have already come to an end. Of course, the relevant interaction does not meanthe irreversible non-unitary interactions when the quantum system is detected by thedetecting system.

Based on standard formalism of quantum theory, the interpretation of the delayed-choice phenomenon beyond Bohr stand (i.e., subdividing the phenomenon and as-cribing physical attributes to each step) will supposedly result to some paradoxes,for example it seems as if the “[p]resent choice influences past dynamics” [2]. Ac-cordingly, Wheeler concluded that “[g]enerally, we would seem forced to say that nophenomenon is a phenomenon until—by observation, or some proper combination of

Found Phys (2010) 40: 55–92 57

theory and observation—it is an observed phenomenon. The universe does not “exist,out there,” independent of all acts of observation. Instead, it is in some strange sensea participatory universe” [2].

In this paper, however, we neither want to challenge with what has been delin-eated in the above paragraphs by Bohr and Wheeler, nor to represent the possi-ble alternatives for interpreting the delayed-choice experiments (e.g., Ref. [3, 4] orRef. [5, 6]). In effect, here, by representing a unified treatments of various delayed-choice arrangements, we are going to investigate that how and to what extent thedelayed-choice experiments (and suggested proposals) characterize the main featureof a delayed-choice effect mentioned above. On the lines of probability theory, weknow that all probabilities are conditional. On the other hand, quantum theory en-ables us to formulate probabilistic predictions about the outcomes of measurementsin which the conditions are determined by state preparations. That is, the ψ -function(i.e., the quantum state) can be interpreted as a catalog of probability [7, 8]. On thisaccount, we want to examine that how the choice between complementary phenom-ena are represented in relevant quantum mechanical descriptions. Of most intereststo us are the temporal aspect of these experiments (i.e., the effect of time-sequence ofevents on the final results), and also the detailed description of interactions betweenthe system and measuring instrument. Meanwhile, throughout this paper we wouldwork in Schrödinger picture of quantum theory.

The contents of this paper is sketched out as follows. In Sect. 2 we first brieflyreview some original delayed-choice proposals and derive the appropriate probabilis-tic description of the complementary phenomena in different experimental arrange-ments. Then, in Sect. 3 we scrutinize most of the experimentally performed delayed-choice proposals and try to examine their contents and scopes in detail. Finally, inSect. 4, with regard to the formulation of delayed-choice proposals given in Sect. 2,we classify the debated experiments into different categories, investigating that towhat extent each category can satisfy the features of a delayed-choice effect.

2 Some Proposals on Delayed-Choice Idea

In this section, we will examine some delayed-choice proposals where two of themwere represented by Wheeler in his first manuscript on delayed-choice idea in1978 [2]. Each proposal contains some novel features and characteristics about thedelayed-choice effect where they have been used as the measures and criteria for theclassification of delayed-choice proposals introduced in Sect. 4.

2.1 Delayed-Choice Double-Slit Experiment with Swinging Door Detector Plate

Regarding the double-slit arrangement shown in Fig. 1a, the experimental consider-ations are managed such that the beams diffracted from two slits are then directedtoward the plane of interference where the detector plate may or may not be inserted.The plane of interference is the locus in which both of the diffracted beams havenon-zero amplitudes in x-direction. The two diffracted beams could continue their

58 Found Phys (2010) 40: 55–92

Fig. 1 The delayed-choice double-slit experiment with swinging door detector plate. After the passage ofparticle through the double-slit, the observer makes the choice to insert or not the detector plate. The plots(b)–(e) show the probability distribution as a function of x in different times where tb = 0 < tc < td < te

propagations to make clean entry into well separated detectors if the detector plate isnot inserted.2

We shall consider the massive systems (e.g., electrons) as the interfering quantumsystems. The double-slit is fixed at the position z = 0 in xy-plane and the centers ofthe slits are located at + d

2 and − d2 away from the origin in x-direction, both with

width s. The particle3 traverses the double-slit at time t = 0. In this regard, the wavefunction of the particle in x-direction at time t after passing through the double-slitcan be envisaged as two Gaussian wavepacket propagating in different directions

ψ1(x, t) ∝{

exp

[ikx(x − d/2 + Vxt) − (x − d/2 + Vxt)

2

2σ × σt

]

+ exp

[−ikx(x + d/2 − Vxt) − (x + d/2 − Vxt)

2

2σ × σt

]}(1)

2In Wheeler original version of this experiment, this can be achieved by a swinging-door detector plate [2].3The delayed-choice proposals or experiments outlined in this paper are propounded or performed withinthe interferometric arrangements. In this respect, we would call the single quantum mechanical interferingsystems as “interfering particles” or simply as “particles”. Yet, this common notion is also misleadingbecause the interfering systems does not possess the spatial and kinematical properties of a particle in itsclassical sense.

Found Phys (2010) 40: 55–92 59

Fig. 2 The delayed choicesplit-beam experiment. Thesingle-photon radiation enters aMach-Zehnder interferometerwith 50–50 beam splitters. Afterthe radiation passes throughBS1, the observer decides toinsert the detectors into eitherthe position P or the position W

where σ � 2s, σt = σ [1 + i�t/(2mσ 2)], and Vx = �kx/m.In order to introduce the delayed-choice mode of operation, after the particle

passes through the double-slit and before arriving at the interference plane, the ob-server can choose to insert or not the detector plate. If the detector plate is inserted,the corresponding pattern is described by

p(x) = |ψ1(x, t1)|2 (2)

where t1 is the time when the particle is detected on the detector plate. This patternshows interference fringes, as shown in Fig. 1d. Likewise, with no detector platebeing inserted, then the particle is detected on the discrete detectors at the time t2,and the corresponding pattern is described by

p(x) = |ψ1(x, t2)|2 (3)

which shows two well-separated Gaussian wavepackets, as depicted in Fig. 1e.Meanwhile, this experiment is analogous to the one proposed by Bartell [9] where

the interfering particles are photons, but here all calculations are for massive particles(for the sake of simplicity and clarity, of course). In Sect. 4.1, we will discuss thatlike Bartell proposal, this experiment can also include the intermediate cases for themanifestation of both wave and particle aspects, as shown in Fig. 1c, if the detector-plate being inserted at the space interval [L–2L].

2.2 Delayed-Choice Split-Beam Experiment

We would consider the interfering particle as a single quanta of electromagnetic en-ergy. Regarding Mach-Zehnder interferometer with 50–50 beam splitters (Fig. 2),the transformation of the radiation’s state due to the passage through the first beamsplitter BS1 can be described as

|ψ5〉 BS1→ |ψ6〉 (4)

|ψ5〉 = |1in〉 (5)

|ψ6〉 = [|0r ,1t 〉 + i|1r ,0t 〉]/√

2 (6)

where |1in〉 describes the single-photon input radiation in the pulse beam, and |1r ,0t 〉(|0r ,1t 〉) denotes the reflected (transmitted) radiation after BS1.

60 Found Phys (2010) 40: 55–92

After the interaction of radiation with BS1 is already finished, the observer decidesto place the detectors in the position P or the position W . Now, depending on whetherthe radiation will interact with the second beam splitter BS2 or not, the state evolutionof the radiation and the corresponding results of the detectors could be described asfollows:

• PM1 and PM2 are placed at W

|ψ6〉 phase shift→ |ψ8〉 BS2→ |ψ9〉 (7)

|ψ8〉 = [eiφ |0r ,1t 〉 + i|1r ,0t 〉]/√

2 (8)

|ψ9〉 = 1

2[(eiφ − 1)|PM1〉 + i(1 + eiφ)|PM2〉] (9)

p(PM1) = 〈ψ9|(|PM1〉〈PM1|)|ψ9〉= sin2(φ/2) (10)

p(PM2) = 〈ψ9|(|PM2〉〈PM2|)|ψ9〉= cos2(φ/2) (11)

• PM1 and PM2 are placed at P

|ψ6〉 phase shift→ |ψ13〉 (12)

|ψ13〉 = [i|PM1〉 + eiφ |PM2〉]/2 (13)

p(PM1) = 〈ψ13|(|PM1〉〈PM1|)|ψ13〉= 1/2 (14)

p(PM2) = 〈ψ13|(|PM2〉〈PM2|)|ψ13〉= 1/2 (15)

where |PM1〉 (|PM2〉) describes the single-photon radiation that is traveling to-ward the detector PM1 (PM2), and φ is the phase difference between two arms ofinterferometer.

2.3 Delayed-Choice Double-Slit with Cavities as Path Detectors

Let us consider the experimental arrangement of Young’s double-slit in which a sys-tem comprised of two independently identical cavities are introduced in front of adouble-slit arrangement (Fig. 3). The interfering particles are assumed to be the atomswith suitable Zeeman sublevels (e.g., Rydberg atoms). The experimental considera-tion (i.e., the interaction of cavity’s field and particle) are managed such that thepassage of the particle through the cavity will result to the preparation of an initiallyempty cavity into a single-photon Fock state.4 The welcher Weg information is con-

4The preparation of cavity in a Fock state due to the passage of atoms can be achieved via the configurationdescribed by Scully et al. [10]. It can also be prepared by the techniques of adiabatic transition as e.g.discussed by Parkins et al. [11].

Found Phys (2010) 40: 55–92 61

Fig. 3 The delayed choiceEinstein’s recoiling-slitexperiment. (a) Thisexperimental setup is analogousto one proposed by Scully et al.[10]. The cavities are resonantsystems in which a suitableRydberg atom emits a photonwith probability equal to onewhen traveling through it. Thesame probability is zero whenthe atom is outside of thecavities. The atoms are excitedby a laser before entering thecavities. (b) Thick curve iscorresponded to the referentpattern. The dashed curve isrelated to the mutual pattern inwhich the photon has beendetected in the upper cavity. Thedotted curve is related to themutual pattern in which thephoton has been detected in thelower cavity. (c) Thick curve iscorresponded to the referentpattern. The dotted curve isrelated to the mutual patternwhere the photon has beenfound in the state|�x = +1〉.The dashed curve is related tothe mutual pattern in which thephoton has been found in thestate |�x = −1〉

tained in the cavities’ Fock states that are prepared due to the passage of the particlesthrough the cavities. On this account, the state of the system of particle/cavities afterpassage of particle through the double-slit is given by

|ψ16〉 = [|uu(x, t)〉 ⊗ |1u,0l〉 + |ul(x, t)〉 ⊗ |0u,1l〉]/√

2 (16)

62 Found Phys (2010) 40: 55–92

where uu(x, t) (ul(x, t)) describes the propagated wave function5 of the particle dif-fracted from upper (lower) slit, and |1u,0l〉 (|0u,1l〉) represents the Fock states in-dicating that there is one photon in upper (lower) cavity and none in lower (upper)cavity.

In this respect, two types of patterns can be obtained on the detector plate thatshould be carefully distinguished: a referent (or integrated) pattern which is obtainedby the whole set of observed positions of particles, and mutual patterns which areobtained by subsets of the observed positions of particles conditional on a specificresult being obtained for a measurement on the cavities.

The referent pattern denoted by p(x) is given by

p(x) = 〈ψ16|(|x〉〈x| ⊗ I )|ψ16〉= {|u1(x, tp)|2 + |u2(x, tp)|2}/2 (17)

where tp is the time the particle is detected on the detector plate.The observer can also choose to measure some specific observables of the cavities

in order to specify the mutual patterns. Accordingly, he could manage to operatethe experiment in a delayed-choice mode. In order to introduce the delayed-choicemode of operation, the measurement on the cavities is performed after passage ofthe particle through the double-slit, or even after detection of particle on the detectorplate. In this respect, the observer can measure one of the complementary observablesof cavities: either �z or �x (see Appendix A)6:

• Measuring �z: the corresponding mutual patterns p(x,�z = ±1) are given by

p(x,�z = +1) = 〈ψ16|(|x〉〈x| ⊗ |�z = +1〉〈�z = +1|)|ψ16〉= |uu(x, tp)|2/2 (18)

p(x,�z = −1) = 〈ψ16|(|x〉〈x| ⊗ |�z = −1〉〈�z = −1|)|ψ16〉= |ul(x, tp)|2/2 (19)

• Measuring �x : the corresponding mutual patterns p(x,�x = ±1) are given by

p(x,�x = ±1) = 〈ψ16|(|x〉〈x| ⊗ |�x = ±1〉〈�x = ±1|)|ψ16〉= 1

4{|u1(x, tp)|2 + |u2(x, tp)|2

± 2Re[(u1(x, tp))∗u2(x, tp)]} (20)

Our knowledge about which-slit the particle has passed through would be com-pletely destroyed when �x is measured. Thus the mutual patterns described byp(x,�x = ±1) show the interference fringes. However, the mutual patterns describedby p(x,�z = ±1) show no interference fringes because �z = ±1 signifies the truewelcher Weg information of the particles. The far field patterns corresponding to thereferent and mutual patterns described above have been shown in Figs. 3b and 3c.

5〈x|u(x, t)〉 = u(x, t) where |x〉 is the position eigenstate of the particle.6The experimental configuration corresponded to the measurements of these observables are discussed inRef. [10].

Found Phys (2010) 40: 55–92 63

2.4 Delayed-Choice Quantum Marking and Quantum Eraser with Neutral Kaons

Consider the φ-meson resonance decays or S-wave proton–antiproton annihilationinto K0K0 pairs [12]. Immediately after the decay, the state of the products is de-scribed by the following maximally entangled state

|ψ(t = 0)〉 = [|K0〉l ⊗ |K0〉r − |K0〉l ⊗ |K0〉r ]/√

2

= [|KL〉l ⊗ |KS〉r − |KS〉l ⊗ |KL〉r ]/√

2 (21)

where l and r denote the “left” and “right” kaon directions of motion, {|K0〉, |K0〉} isthe strangeness basis with 〈K0|K0〉 = 0, and {|KS〉, |KL〉} is the lifetime basis with〈KS |KL〉 � 0, where

|K0〉 = [|KS〉 + |KL〉]/√2 (22)

|K0〉 = [|KS〉 − |KL〉]/√2 (23)

The lifetime states KS and KL (with well-defined masses mS(L) and decay widths�S(L)) propagate in free space as follows

|KS(L)(t)〉 = exp[−iλS(L)t]|KS(L)〉 (24)

where λS(L) = mS(L) − i�S(L)/2, and �S � 579�L.

With regard to the relation (24), the time evolution of the state (21) is given by

|ψ25(tl, tr )〉 = 1√2[exp[−i(λLtl + λStr )]|KL〉l ⊗ |KS〉r

− exp[−i(λStl + λLtr )]|KS〉l ⊗ |KL〉r ]= 1√

1 + eτ�[|KL〉l ⊗ |KS〉r − eiτmeτ�/2|KS〉l ⊗ |KL〉r ]

= 1

2√

1 + eτ�{(1 − eiτmeτ�/2)[|K0〉l ⊗ |K0〉r − |K0〉l ⊗ |K0〉r ]

+ (1 + eiτmeτ�/2)[|K0〉l ⊗ |K0〉r − |K0〉l ⊗ |K0〉r ]} (25)

where τ = tl − tr , � = �L − �S and m = mL − mS .The individual rate of the results of observation of both the strangeness and life-

time observables for the left and right kaons show no undulatory behavior:

p(K) = 〈ψ25|(|K〉l〈K|l ⊗ I )|ψ25〉 = 〈ψ25|(I ⊗ |K〉r 〈K|r )|ψ25〉= 1/2 (26)

where K = K0, K0,KS or KL.

64 Found Phys (2010) 40: 55–92

However, by correlating the results of measurements on the kaons, the observercan manage to operate the experiment in a delayed-choice mode. The correspondingcoincidence rates are described by the joint probabilities p(Kl,Kr):

p(Kl,Kr) = 〈ψ25|(|K〉l〈K|l ⊗ |K〉r 〈K|r )|ψ25〉 (27)

Since the system in the state (21) is right-left symmetric, then, in order to introducethe delayed-choice mode, the observation on one kaon can be easily postponed to thetimes after the measurement on the other kaon, that is, the observer can considereither the left or the right kaon as the interfering system and the other kaon as thepath detector, and then manage the time considerations of a delayed-choice mode formeasurements on the interfering system and the path detector.

Since “strangeness oscillations play the role of the traditional interference pattern”[12], then, we are interested to those coincidence rates where one of the measure-ments is performed in strangeness basis. In this respect, the observer can measureone of the complementary observables of a kaon: either strangeness or lifetime, to-gether with the strangeness of the other kaon. By correlating the results of measure-ments on strangeness basis of the kaons, the corresponding coincidence rates showthe undulatory behaviors

p(K0l ,K0

r ) = p(K0l , K0

r ) = 1

4

[1 − cos(τm)

cosh(τ�/2)

](28)

p(K0l ,K0

r ) = p(K0l , K0

r ) = 1

4

[1 + cos(τm)

cosh(τ�/2)

](29)

However, the coincidence rates of strangeness measurement of one kaon with theresults of lifetime observation of the other kaon show no oscillating behaviors

p(K0l , (KS)r ) = p(K0

l , (KS)r ) = 1

2[1 + exp(τ�/2)] (30)

p(K0l , (KL)r ) = p(K0

l , (KL)r) = exp(τ�/2)

2[1 + exp(τ�/2)] (31)

Nevertheless, the measurement either in the strangeness or in the lifetime basis canbe decided in two completely different physical mechanisms, either in an active orin a passive process, where their final results are coherently described by the abovequantum mechanical calculations. “For the two possible active measurements oneeither inserts or not a dense piece of matter at a (time-of-flight) distance t along thekaon trajectory”, and then decides to measure the lifetime or strangeness of the kaon.In passive measurement, however, “one allows the entangled kaon pairs to propagatefreely in space and identifies the kaon decay times and modes” [12]. Having the decaytimes and modes, the abovementioned joint probabilities can then be obtained7 [12].In passive measurement, the choice between complementary observables (i.e., theobservation either in the strangeness or in the lifetime basis) is made by the dynamicsof kaon decays and the observer remains passive in this respect.

7The reader can refer to Ref. [12] for the detailed calculations concerning the passive measurement.

Found Phys (2010) 40: 55–92 65

3 Experimentally Performed Delayed-Choice Experiments

In this section, we review most of the experimentally performed delayed-choice pro-posals. We try to present these experiments with enough details and then give a co-herent quantum mechanical description for them.

3.1 Hellmuth et al. (1987): Delayed-Choice Experiments in QuantumInterference [13]

3.1.1 Delayed-Choice Interference Experiment

This experiment was performed as the typical interference of the single quantumof radiation (photon) within Mach-Zehnder interferometer depicted in Fig. 4a. Theexperimental assessments of the source that produces the light pulses were managedin a manner that they ensured there is only one photon in the interferometer at eachrun of experiment. After the incident light passed through the first beam splitter, thetwo resulted beams are directed and focused into two separate single-mode opticalfibers whose principal axis are aligned such that the polarization of the light leavingthe fiber is linear. A Pockels cell (PC) is placed in the upper arm of the interferometer,followed by a Glan polarizing prism (POL). The Pockels cell rotates the polarizationof the incident light by 90◦ if a suitable voltage is applied to it,8 and the POL deflectsthe light whose polarization is rotated. As a result, the upper arm of the interferometerwould be interrupted (closed) if a proper voltage is applied to the Pockels cell at theproper time.

In the normal-mode operation, no voltage is applied to the Pockels during thewhole transition time of the radiation through the interferometer. In the delayed-choice mode, however, the voltage is normally switched on and would be switchedoff 5 ns after the radiation passed through the first beam splitter. Here, the light pulsesare well inside the optical fiber when the voltage is switched off.

Regarding the experimental arrangement and the timing and placing of Pockelscell and POL, in both of the normal and delayed-choice modes of operation, the up-per arm of interferometer is opened when the radiation is passing through it. Conse-quently, the evolution of the radiation within the interferometer in both of the normaland delayed-choice modes of operation is described by relations (4) and (7). Accord-ingly, after recombining the two light pulses by the second beam splitter, the inter-ference fringes are detected by photomultipliers 1 and 2 (PM1 and PM2) as shownin Fig. 4b. The obtained results of these experiments show no discernible differencebetween normal and delayed-choice modes of operation.

3.1.2 Delayed-Choice Quantum Beat Experiment

With the aid of a suitable laser pump and Barium atoms (whose states are schemat-ically shown in Fig. 4c), an experiment is designed such that a signal photon is pro-duced in a specific superposition state of two different modes as

|ψ32〉 ∝ αL(t)|εL〉|1,0〉 + αR(t)|εR〉|0,1〉 (32)

8The angle is positive when the rotation is counterclockwise.

66 Found Phys (2010) 40: 55–92

Fig. 4 Hellmuth et al. (1987):Delayed-choice experiments inquantum interference. (a) Setupof the delayed-choiceinterference experiment withPockels cell (PC) and Glanprism polarizer (POL).(b) Comparison of interferencepatterns for normal anddelayed-choice configurations.Dots represent the data obtainedin the normal mode of operation,and crosses are data fordelayed-choice mode operation.The upper diagram shows theresults of PM1, while the resultsof PM2 is shown in the lowerdiagram. (c) Schematicrepresentation of Bariuminternal states used in thequantum beat experiment.(d) Experimental arrangementof the quantum-beat experiment.(e) The results obtained innormal quantum-beatconfiguration. The upperdiagram is obtained whenPockels cell voltage is zero, andthe lower one is obtained when aquarter-wave voltage is appliedto the Pockels cell.(f) Comparison of the results forthe delayed-choice modes ofoperation in which the appliedvoltage is switched off atdifferent times denoted by s.(g) Quantitative comparison ofthe results of the delayed-choiceand normal modes wheres = 4 ns. The comparison ismade for the time period of10–30 ns. Dots represent thedelayed-choice and crossesdenote the normal mode ofoperation. The pictures,diagrams, and caption (a) aretaken from Ref. [13]

(a)

(b)

(c)

in which |εR〉 (|εL〉) represents right (left) hand circularly polarization,9 |1,0〉 de-notes the photon produced in the path |m = 0〉 → |m = +1〉 → |m = 0〉, |0,1〉 de-

9Henceforth, we will use Jones calculus for the photon polarization and optical elements, introduced inAppendix B.

Found Phys (2010) 40: 55–92 67

(d)

(e) (f)

(g)

Fig. 4 (Continued)

notes the photon produced in the path |m = 0〉 → |m = −1〉 → |m = 0〉, and |αL(t)|2(|αR(t)|2) represents the probability that a photon described by |1,0〉 (|0,1〉) hasbeen emitted. These probabilities are decreased exponentially by time.

After the production of radiation in the state (32), the time dependence of thesignal is measured using a time-to-amplitude converter together with a multichannelanalyzer in the pulse-height analyzing mode. A linear polarizer with a polarizationdirection parallel to y-direction is placed in front of the photomultiplier, as shownin Fig. 4d. The detecting system (polarizer + photomultiplier) is 8 m far from the

68 Found Phys (2010) 40: 55–92

Barium atomic beam (i.e., the source of radiation). So, the time of flight of the photonis about 26 ns to arrive from the source to the detecting system.

Since the position of the photomultiplier is fixed in space, then we calculate thefirst order correlations between two modes of photon field as a function of time. In thecase in which no voltage is applied to the Pockels cell when the light passes throughit, the probability of detecting a photon at time t is proportional to

G(1)(t; t) = 〈ψ32|E(−)(t)E(+)(t)Ay |ψ32〉∝ |αL(t)|2 + |αR(t)|2 − 2Re[αL(t)(αR(t))∗ei(ωL−ωR)t ] (33)

where Ay = |εy〉〈εy | corresponds to the observation of the polarization in y-direction(refer to relation (B.7)), E(−)(t) and E(+)(t) are negative and positive frequencycomponents of photon electric field in a way that

E(+)(t) = a+eiωLt + a−eiωRt (34)

E(−)(t) = a†+e−iωLt + a

†−e−iωRt (35)

where a†+(a+) is creation (annihilation) operator for the photon produced in the path

|m = 0〉 → |m = +1〉 → |m = 0〉, a†−(a−) is creation (annihilation) operator for the

photon produced in the path |m = 0〉 → |m = −1〉 → |m = 0〉, ωL is the frequencyof photon produced in the path |m = 0〉 → |m = +1〉 → |m = 0〉, and ωR is thefrequency of photon produced in the path |m = 0〉 → |m = −1〉 → |m = 0〉.

In the situation where a quarter-wave voltage is applied to the Pockels cell whenradiation passes through it, the evolution of the light’s state is given by

|ψ32〉 Pockels cell→ |ψ37〉 (36)

|ψ37〉 ∝ iαL(t)|εy〉|1,0〉 + αR(t)|εx〉|0,1〉 (37)

and thus, the probability of detecting a photon at time t is calculated as

G(1)(t; t) = 〈ψ37|E(−)(t)E(+)(t)Ay |ψ37〉∝ |αL(t)|2 (38)

The normal mode of this experiment is obtained in two different cases in whichthe voltage was switched on or off within the whole transit time of radiation throughthe experimental setup. In the first case, no voltage is applied to the Pockels cell, andconsequently the results show interference between two modes of radiation (Fig. 4e(above), described by (33)). In the second case, a quarter-wave voltage is applied tothe Pockels cell, leading to the transformation of the radiation state into (37). Hence,no interference fringes appears10 (Fig. 4e (below), described by (38)).

To perform a delayed-choice experiment, the applied voltage to the Pockels cell isswitched off at a very specific moment during the time of flight of photon (i.e., after

10Note that in Fig. 4e the zero time corresponds to the arrival time of the first fluorescence photon at thedetector which is 26 ns after the laser pulse.

Found Phys (2010) 40: 55–92 69

the signal photon has been produced but has not yet reached the detecting system).We denote this specific moment by s. It is self-evident that s ≤ 26 ns. At the mo-ments before s, the photon arrives at the detecting system in the state (26), and thusthe obtained results are given by (38). At the moments after s, the radiation’s stateis described by (32) when the photon arrives at the detecting system, and so the cor-responding results are given by (33). Figure 4f shows the results of delayed-choicemode of operation for different s.

Meanwhile, the obtained results in a particular delayed-choice mode in whichs = 4 ns is compared in details with the results of normal mode. Within the nor-mal mode of operation, the voltage is kept off. Only that part of the signal which liesbetween 10 ns and 30 ns is used in the detailed evaluation in Fig. 4g. These resultsshow no discernible difference between the normal and the delayed-choice modes.

3.2 Baldzuhn et al. (1989): A Wave-Particle Delayed-Choice Experimentwith a Single-Photon State [14]

In this experiment, as shown in Fig. 5a, pairs of colinear and linearly polarized signaland idler photons are produced by parametric fluorescence. One of the photons of apair (photon 1) serves as a trigger photon, and the other one (photon 2) goes througha Mach-Zehnder ring interferometer. The experiment is designed such that one canbe sure that the state of each observed radiation in the interferometer is a one-photonstate in a good approximation. The state of photon 2 is divided into two lightwavesafter passing through the first beam splitter BS1 (Fig. 5a), one of them traverses theinterferometer clockwise, and the other one traverses counter-clockwise, and theyare finally recombined again at BS1. Trigger pulses resulted from the detection ofphoton 1 on D1 apply a halfwave voltage to a Pockels cell P which is positioned in theinterferometer. A coincidence logic unit registers the logic pulses of D1 and D2 afteran additional cable delay for D1. The observation of interference fringes depends onwhether the polarizations of the clockwise and counterclockwise lightwaves are thesame or not when they are recombined again at BS1.

The delayed-choice mode of this experiment was performed in two different cases.In one case, after the photon 2 passed through BS1, the Pockels cell is activated by atrigger pulse of diode D1 at the moment when the lightwave traveling clockwise hasjust left the Pockels cell. In this case, the state of lightwave traveling counterclock-wise is marked by turning its vector of polarization by 90◦, while the polarization ofthe lightwave traveling clockwise remains unchanged. Thus, no interferences appear.The evolution of radiations within the interferometer in this case can be introducedas

|ψ40〉 BS1→ |ψ41〉 (Pockels cell)�→ |ψ42〉 BS1→ |ψ43〉 (39)

|ψ40〉 = [|εx,D1〉 ⊗ |εy,2〉 + |εy,D1〉 ⊗ |εx,2〉]/√2 (40)

|ψ41〉 = 1

2{|εx,D1〉 ⊗ (|εy,ϕ

�2

〉 + eiφ |εy,ϕ�2

〉)

+ |εy,D1〉 ⊗ (|εx,ϕ�2

〉 + eiφ |εx,ϕ�2

〉)} (41)

70 Found Phys (2010) 40: 55–92

Fig. 5 Baldzuhn et al. (1989): A wave-particle delayed-choice experiment with a single-photon state.(a) Experimental setup: BS and BS1 are beam splitters; D1 and D2 are detectors; F stands for the glassfiber, P for the Pockels cell, and M for mirrors. The symbol “I” marks a reference position in the beam.(b) Comparing the results of a delayed-choice experiment (dots) with normal reference mode of operationfor the case in which the polarization of lightwaves traveling the interferometer is not the same whenthey are recombined again at BS1. (c) Comparing the results of a delayed-choice experiment (dots) withnormal reference mode of operation for the case in which the polarization of lightwaves traveling theinterferometer is the same when they are recombined again at BS1. The picture, diagrams, and the caption(a) are taken from Ref. [14]

|ψ42〉 = 1

2{|εx,D1〉 ⊗ (|εy,ϕ

�2

〉 − eiφ |εx,ϕ�2

〉)

+ |εy,D1〉 ⊗ (|εx,ϕ�2

〉 + eiφ |εy,ϕ�2

〉)} (42)

|ψ43〉 = 1

2{|εx,D1〉 ⊗ |εy,D2〉 + |εy,D1〉 ⊗ |εx,D2〉

+ eiφ(−|εx,D1〉 ⊗ |εx,D2〉 + |εy,D1〉 ⊗ |εy,D2〉)} (43)

where |D1〉 represents the photon 1 that is traveling toward D1, |ϕ�2

〉 (|ϕ�2

〉) indi-cates the lightwave traveling clockwise (counterclockwise), φ is the phase differencebetween two paths of interferometer, and |D2〉 denotes the lightwave traveling towardD2 after the recombination of |ϕ�

2〉 and |ϕ�

2〉 at BS1. Accordingly, the coincidence

Found Phys (2010) 40: 55–92 71

rate of D1 and D2 is given by

p(D1,D2) = 〈ψ43|(|D1〉〈D1| ⊗ |D2〉〈D2|)|ψ43〉∝ 1/2 (44)

which shows no interference. The results of this delayed-choice mode of the obser-vation (dots in Fig. 5b, the left ordinate), however, are the same with the results of anormal-mode of operation (drawn line in Fig. 5b, the right ordinate). In the normal-mode, the intensity of a probe light is measured with a Quarter-Wave retardation Plateplaced at position I in the interferometer (Fig. 5a). This plate converts the states ofthe clockwise and counterclockwise lightwaves into the states with different circularpolarization. The Pockels cell is inactive during this reference measurement. Similarto the delayed mode, no interference fringes appear in normal mode, too.

In other case of delayed-choice, the experiment is performed in a manner similarto the first case of delayed-choice described above, but with a quarterwave retardationplate tilted by 45◦ which is placed at position I in the interferometer (Fig. 5a). Thisquarterwave plate (QWP) changes the polarization of light from linear to circular.The evolution of the radiation within this delayed-choice mode can be represented as

|ψ41〉U

�+45,λ/4 and U

�−45,λ/4→ |ψ46〉 (Pockels cell)�→ |ψ47〉 BS1→ |ψ48〉 (45)

|ψ46〉 = 1

2{i|εx,D1〉 ⊗ (|εR,ϕ�

2〉 − eiφ |εL,ϕ�

2〉)

+ |εy,D1〉 ⊗ (|εL,ϕ�2

〉 + eiφ |εR,ϕ�2

〉)} (46)

|ψ47〉 = 1

2{i|εx,D1〉 ⊗ (|εR,ϕ�

2〉 + eiφ |εR,ϕ�

2〉)

+ |εy,D1〉 ⊗ (|εL,ϕ�2

〉 + eiφ |εL,ϕ�2

〉)} (47)

|ψ48〉 = 1

2{i(1 + eiφ)|εx,D1〉 ⊗ |εR,D2〉

+ (1 + eiφ)|εy,D1〉 ⊗ |εL,D2〉} (48)

where U�+45,λ/4 describe the transformation of the polarization of the clockwise light

due to the passage through QWP, U�−45,λ/4 denotes the rotation of the polarization of

counterclockwise light by −45◦ due to the passage of counterclockwise light throughQWP. Here, the lightwaves traveling counterclockwise and clockwise have the samecircular polarization when they are recombined at BS1. Consequently, using the rela-tion (48), the coincidence rate of D1 and D2 is given by

P(D1,D2) = 〈ψ48|(|D1〉〈D1| ⊗ |D2〉〈D2|)|ψ48〉∝ cos2(φ/2) (49)

which shows the interference fringes. The results shown in Fig. 5c exhibit the goodagreement between the delayed-choice results (dots in Fig. 5c, the left ordinate) with

72 Found Phys (2010) 40: 55–92

Fig. 6 Kwait et al. (1994): Delayed choice quantum eraser-theory and experiment. (a) Schematic ofexperiment: A photon pair is created within two indistinguishable possibilities: either directly or via themirror Mp . (b) The upper-left curve (circles, left axis) shows the singles rate DS as a function of mirrorMS position when there is no wave plate or polarizer. The upper-right curve (circles, left axis) showsthe singles rate DS as a function of mirror MS position when there is a wave plate but no polarizer. Thelower curves (triangles, right axis) show the coincidence detections of DS and DI for different polarizingdirection of linear polarizer as a function of mirror MS position. The modified picture (a) is originallytaken from online version of T.J. Herzorg’s Ph.D. thesis. The diagram (b) is taken from Ref. [15]

the reference normal-mode result (drawn line in Fig. 5c, the right ordinate). Thearrangement for the reference (normal-choice) measurement is equivalent to the caseof a typical experiment in interferometer with no Pockels cell and no QWP where theresults show interferences fringes.

3.3 Kwait et al. (1994): Delayed Choice Quantum Eraser-Theoryand Experiment [15]

A photon pair is created within two indistinguishable possibilities: either directly orvia the mirrors (Fig. 6a). Indistinguishability of these paths could lead to interference

Found Phys (2010) 40: 55–92 73

at the detectors. “Because of the type-I phase matching, the photons share the samepolarization”, here vertical [16]. The signal and idler modes associated with a right-going UV pump photon are superposed with the signal and idler modes associatedwith a left-going pump photon:

|ψ50〉 ∝ [eiφ |(εy)r 〉 ⊗ |(εy)

r 〉 + |(εy)l〉 ⊗ |(εy)

l〉]/√2 (50)

where φ is the phase difference of two paths of radiation. The superscripts r andl denote the states corresponding to the radiation produced by right-going and left-going UV pump photon, respectively. In this experiment, φ is set by the positionof MS .

In the absence of the QWP, the single rate at DS (detector of signal photon) andDI (detector of idler photon) displays interference fringes (refer to Fig. 6). Thesefringes are possible because it is not possible in principle to determine whether thesignal and idler pair have been created either from a left-going or from a right-goingUV pump photon. If a QWP is used, the polarization of idler photon associated withright-going UV pump photon is changed from vertical to horizontal.11 Then, thetwo paths are distinguishable, and no interference fringes are observed at DS andDI .

When a Pockels cell followed by a polarizer at ±45◦ is used in the idler arm, thedetection of the idler photon on DI contains no information about whether the idlerphoton and its entangled signal photon were originated either from left-going UVpump photon or from right-going UV pump. Accordingly, the interference fringesappear upon correlating detections at DS and DI .

To introduce the delayed-choice, the experiment is managed such that the idlerphoton enters the Pockels cell after the signal photon is detected. The state evolu-tion of photons where there is no wave-plate and no polarizer in the experimentalarrangement can be described by:

|ψ51〉 = 1√2[eiφ |(εy)

r ,DS〉 ⊗ |(εy)r ,DI 〉

+ |(εy)l,DS〉 ⊗ |(εy)

l,DI 〉] (51)

where |DS〉 (|DI 〉) describes the signal (idler) photon traveling toward DS (DI ).With regard to (51), the single rate DS (left circles, left axis in Fig. 6b) can be

calculated as the following

p(DS) = 〈ψ51|(|DS〉〈DS | ⊗ I )|ψ51〉∝ cos2(φ/2) (52)

With wave-plate in idler arm, the state evolution of photons is given by

|ψ50〉(QWP)2

i→ |ψ54〉 (Pockel cell)i→ |ψ55〉 (53)

11In other word, the welcher Weg information is stored in the polarization of radiation.

74 Found Phys (2010) 40: 55–92

|ψ54〉 = 1

2[ieiφ |(εy)

r ,DS〉 ⊗ |(εx)r ,DI 〉

+ |(εy)l,DS〉 ⊗ |(εy)

l,DI 〉] (54)

|ψ55〉 = 1

2[−eiφ |εy,DS〉 ⊗ |εy,DI 〉

+ i|εy,DS〉 ⊗ |εx,DI 〉] (55)

where (QWP)2i represents the rotation of the vector of polarization of idler photon12

produced by right-going photon pump by 90◦ due to double-passage through QWP.Regarding (55), the single rate DS (right circles, left axis in Fig. 6b) can be calcu-

lated as

p(DS) = 〈ψ55|(|DS〉〈DS | ⊗ I )|ψ55〉∝ 1/2 (56)

Now, by adjusting the orientation of the linear polarizer placed before DI , theobserver can manage to observe one of the complementary observables of the polar-ization of idler photon: either σ1 (given in (B.10)) or σ2 (given in (B.11)). The obser-vations with the polarizer at ±45◦ are corresponded to the measuring the observableσ2, and likewise, the observations with the polarizer at 0◦ and 90◦ are correspondedto the measuring the complementary observable, σ1.

• Observations of σ1– For the orientation of the polarizer at 0◦.

The coincidence rates of detections at detectors DS and DI (triangles, rightaxis PI = 0 in Fig. 6b) can be calculated as

p(DS,DI ) = 〈ψ55|(|DS〉〈DS | ⊗ |εx,DI 〉〈εx,DI |)|ψ55〉∝ 1/2 (57)

– For the orientation of the polarizer at 90◦The coincidence rates of detections at detectors DS and DI (triangles, right

axis PI = 90 in Fig. 6b) is given by

p(DS,DI ) = 〈ψ55|(|DS〉〈DS | ⊗ |εy,DI 〉〈εy,DI |)|ψ55〉∝ 1/2 (58)

• Observations of σ2– For the orientation of the polarizer at +45◦

With regard to (44), the coincidence rates of detections at detectors DS andDI (triangles, right axis PI = 45 in Fig. 6b) is given by

p(DS,DI ) = 〈ψ55|(|DS〉〈DS | ⊗ |ε+,DI 〉〈ε+,DI |)|ψ55〉∝ sin2(φ/2) (59)

12Refer to the Appendix B for more details.

Found Phys (2010) 40: 55–92 75

Fig. 7 Kawai et al. (1998): Development of cold neutron pulser for delayed choice experiment.(a) Schematic experimental arrangement of the delayed choice experiment. The pair of identical com-posite neutron mirrors is arranged like a Jamin type interferometer. (b) The time spectra of the neutronbeam reflected from the second composite mirror at the three relative angles within one period of thefringe in the first “delayed choice” mode. The picture, diagrams, and captions are taken from Ref. [17]

– For the orientation of the polarizer at −45◦Accordingly, the coincidence rates of detections at detectors DS and DI (tri-

angles, right axis PI = −45 in Fig. 6b) can be calculated as the following

p(DS,DI ) = 〈ψ55|(|DS〉〈DS | ⊗ |ε−,DI 〉〈ε−,DI |)|ψ55〉∝ cos2(φ/2) (60)

3.4 Kawai et al. (1998): Development of Cold Neutron Pulser for Delayed-ChoiceExperiment [17]

The cold neutron pulser in a Jamin type interferometer was used to perform a delayed-choice experiment. This Mach-Zehnder type interferometer consists of a combinationof two multi-layer mirrors. Polarized 12.6 Å neutrons are incident on the first multi-layer mirror (CMl) as shown in Fig. 7a, and divided into two coherent partial waves.The multi-layer mirror consists of a Ni/Ti multilayer, a Ge gap layer, and a semi-transparent Permalloy/Ge multilayer (PGM) where the top PGM functions as a wave

76 Found Phys (2010) 40: 55–92

splitter. The induced geometrical phase shift of two interfering beams is set to be iden-tical, and in this respect the phase shift φ between two beams changes by rotating thesecond multi-layer mirror (CM2). The delayed-choice mode of this experiment wasperformed in two different cases.

In the first delayed-choice mode, the neutron optical switch is switched off (thePGM of the second mirror is not placed) when the neutron reaches the first mirror,and it is switched on (the PGM of the second mirror is placed) after the neutronpassed through the first mirror. In this mode, both of mirrors are placed when theneutron is passing through them, and thus the interactions of neutron within the in-terferometer is similar to the typical Mach-Zehnder interferometer where both beamsplitters are placed (analogous to the situation described by (4) and (7)). In this mode,the interference fringe was obtained as shown in Fig. 7b.

In the second mode of delayed choice, the neutron optical switch is switched onwhen the neutron reaches the first mirror, and it is switched off after neutron passedthrough the first mirror. In this mode, the interference fringes disappear. Here, thesecond mirror is not placed when the neutron is passing through it, and thus theinteractions of the neutron within this interferometer is similar to the typical Mach-Zehnder interferometer where the second beam splitter is not inserted (analogous tothe situation described by (4) and (12)).

3.5 Kim et al. (1999): A Delayed Choice Quantum Eraser [18]

As shown in Fig. 8a, a laser beam is first divided by a double-slit and then passesthrough a nonlinear optical crystal BBO. Therefore, a pair of orthogonally polar-ized signal-idler photons is generated either from A region or B region by a spon-taneous parametric down conversion process. The orthogonally polarized signal andidler photons are splitted by a Glen-Thompson prism and then directed toward twodifferent setups.

The signal photon (photon 1, produced from either A or B) passes through a lensLS to meet the detector D0 that can be scanned along its x-direction. The wholeexperimental arrangement for the signal photon is analogous to a hypothetical double-slit setup with path detectors where the slits are located at A and B and also thewelcher Weg information is stored in the idler photon. The idler photon (photon 2,produced from either A or B) is sent to an interferometer with equal-path opticalarms which contains four detectors D1, D2, D3 and D4. The BSB and BSA are50–50 beam splitters. With regard to this interferometer, the detection of the idlerphoton at D1 and D2 contains no information about the region that idler photon wasoriginated, from either A or B, and in turn provides nothing with us about the originof its entangled photon, the signal photon. The triggering of detectors D3 and D4,however, provide the information about the region that these photons were originated.Here we can infer that the signal and idler photons were originated, either from A ifD3 is triggered, or from B if D4 is triggered.

Since the polarization of entangled photons is not involved in interactions with theinterferometers (Fig. 8a), then the state of photons after BBO can be expressed as thefollowing

|ψ62〉 double-slit→ |ψ63〉 BBO→ |ψ64〉 (61)

Found Phys (2010) 40: 55–92 77

Fig. 8 Kim et al. (1999): A delayed choice quantum eraser. (a) A laser pump is first divided by a dou-ble-slit and then passes through a nonlinear optical crystal BBO. Therefore, a pair of orthogonally polarizedsignal and idler photons are split by a Glen-Thompson prism and then directed toward two different setups.The idler photon detection is delayed until about 7.7 ns after the detection of the signal photon. (b) R01and R02 are the joint detection rate of D0 together with D1 and D2, respectively, against the positionof the detector D0 in x-direction. (c) R03 is the joint detection rate of D0 together with D3 against theposition of the detector D0 in x-direction. The picture and the diagrams are taken from Ref. [18]

|ψ62〉 = |p〉 (62)

|ψ63〉 = [|pA〉 + |pB〉]/√2 (63)

78 Found Phys (2010) 40: 55–92

Fig. 8 (Continued)

|ψ64〉 = 1√2[uA

s (x)|s〉 ⊗ uAi (x)|i〉

+ uBs (x)|s〉 ⊗ uB

i (x)|i〉] (64)

where |pB〉 and |pA〉 represent the diffracted pump photon through the upper andlower slits, respectively (Fig. 9a), uA

s (x) (uAi (x)) corresponds to the scalar ampli-

tude13 of the signal (idler) photon produced by pump photon diffracted from lowerslit, and uB

s (x) (uBi (x)) corresponds to the scalar amplitude of the signal (idler) pho-

ton produced by pump photon diffracted from upper slit.Regarding the experimental arrangement depicted in Fig. 8a, the state evolution of

the radiations can be described as the following:

|ψ64〉 BSB and BSA→ |ψ66〉 BS→ |ψ67〉 (65)

|ψ66〉 = 1

2[uA

s (x)|D0〉 ⊗ (i|D4〉 + |χAi 〉)

+ uBs (x)|D0〉 ⊗ (i|D3〉 + |χB

i 〉)] (66)

|ψ67〉 = 1

2

[uA

s (x)|D0〉 ⊗(

i|D4〉 + |D2〉 + i|D1〉√2

)

+ uBs (x)|D0〉 ⊗

(i|D3〉 + i|D2〉 + |D1〉√

2

)]

13In this experiment, x is finally related to the position of the detector Ds where the signal photon isdetected.

Found Phys (2010) 40: 55–92 79

Fig. 9 Walborn et al. (2001): A double-slit quantum eraser. (a) An argon laser is used to pump a BBOcrystal, generating entangled photons (photon p and photon s). Photon s is incident on a double-slitarrangement where the quarter wave plates QWP1 and QWP2 are introduced in front of the upper andlower slit. The polarization of photon p is measured by placing a linear polarizer (POL1) in the path ofphoton p. The experimental setup is designed in a way that photon s is detected before the detection ofphoton p. (b) Coincidence counts vs detector Ds position x with QWP1, QWP2, and POL1 are absent.(c) Coincidence counts vs detector Ds position x with QWP1 and QWP2 in place in front of the doubleslit. (d) Coincidence counts vs detector Ds position x with QPW1, QWP2, and POL1 are in place. POL1was set to the angle of the fast axis of QWP1. (e) Coincidence counts vs detector Ds position x withQPW1, QWP2, and POL1 are in place. POL1 was set to the angle of the fast axis of QWP2. The pictureand the diagrams are taken from Ref. [19]

= 1

2

[iuA

s (x)|D0〉 ⊗ |D4〉 + iuBs (x)|D0〉 ⊗ |D3〉

+ uAs (x) + iuB

s (x)√2

|D0〉 ⊗ |D2〉 + iuAs (x) + uB

s (x)√2

|D0〉 ⊗ |D1〉]

(67)

where |D0〉 represent the signal photon traveling toward Ds , |D4〉 indicates the re-flected light after BSA traveling toward D4, |χA

i 〉 denotes the transmitted light afterBSA traveling toward BS, |D3〉 denotes the reflected light after BSB traveling towardD3, |χB

i 〉 denotes the transmitted light after BSB traveling toward BS, |D1〉 (|D2〉)represents the light after BS traveling toward D1 (D2), and |D0〉 represents the signalphoton that is traveling toward D0.

In order to introduce the delayed-choice mode of operation, the experiment is de-signed such that after detection of photon 1 in D0, the photon 2 would still be on itsway to BSA and BSB. The results obtained in this delayed-choice mode of opera-tion are the “joint detection” counting rate of D0 together with one of the detectorsD1,D2,D3 and D4 against the position of the detector D0 in x-direction. Using thestate (67), we can calculate these coincidence rates as the followings:

p(D0 at x,D1) = 〈ψ67|(|D0〉〈D0| ⊗ |D1〉〈D1|)|ψ67〉= 1

8[|uA

s (x)|2 + |uBs (x)|2

80 Found Phys (2010) 40: 55–92

Fig. 9 (Continued)

− 2Re[i[uAs (x)]∗uB

s (x)]] (68)

p(D0 at x,D2) = 〈ψ67|(|D0〉〈D0| ⊗ |D2〉〈D2|)|ψ67〉= 1

8[|uA

s (x)|2 + |uBs (x)|2

+ 2Re[i[uAs (x)]∗uB

s (x)]] (69)

p(D0 at x,D3) = 〈ψ67|(|D0〉〈D0| ⊗ |D3〉〈D3|)|ψ67〉= |uB

s (x)|2/4 (70)

p(D0 at x,D4) = 〈ψ67|(|D0〉〈D0| ⊗ |D4〉〈D4|)|ψ67〉= |uA

s (x)|2/4 (71)

Found Phys (2010) 40: 55–92 81

The joint detection rate of D0 together with D1 or D2 against the position ofthe detector D0 in x-direction (described by relations (68) and (69)) demonstratesan interference pattern as shown in Fig. 8b. Nevertheless, the joint detection rate ofD0 together with D3 or D4 against the position of the detector D0 in x-direction(described by relations (70) and (71)) demonstrates no interference pattern as shownin Fig. 8c.

3.6 Walborn et al. (2001): A Double-Slit Quantum Eraser [19]

The experimental setup is shown in Fig. 9a. An argon laser is used to pump a BBOcrystal, generating entangled photons (photon p and photon s) by spontaneous para-metric down-conversion. The state of the produced entangled photons can be ex-pressed as

|ψ72〉 = 1√2{us(x)|εx, s〉 ⊗ |εy,p〉

+ eiφus(x)|εy, s〉 ⊗ |εx,p〉} (72)

where us(x) denotes the scalar amplitude14 of the photon s, and φ is a relative phaseshift due to the crystal birefringence. Photon s is incident on a double-slit arrange-ment where the quarter wave plates QWP1 and QWP2 are introduced in front of theupper and lower slit with the fast axes at angle +45◦ and −45◦ respect to x-direction,respectively. Taking into the account that the diffracted wave functions through theupper and lower slits are marked by orthogonal polarizations, then there is no pos-sibility for appearance of interference fringes. However, we can recover interferencepattern for the coincidence detections of photons s and p. In this respect, the polar-ization of photon p is measured by placing a linear polarizer (POL1) in the path ofphoton p at ±45◦ orientations (Fig. 9a).

This experiment has been operated in different experimental setups. In first case,QWP1, QWP2, and POL1 are absent, and the coincidence counts vs detector Ds posi-tion x shows interference fringes, as shown in Fig. 9b. In the second case, QWP1 andQWP2 are in place in front of the double slit and POL1 is absent, and the coincidencecounts vs detector Ds position x shows interference fringes, as shown in Fig. 9c. Intwo other cases, QPW1, QWP2, and POL1 are in place. In the third case, POL1 wasset to the angle of the fast axis of QWP1, and the interference fringes recover for thecoincidence counts vs detector Ds position x, as shown Fig. 9d. In fourth and the lastcase, POL1 was set to the angle of the fast axis of QWP1 plus π/2, and the interfer-ence anti-fringes are recovered for the coincidence counts vs detector Ds position x,as shown Fig. 8e. In order to introduce the delayed-choice mode of operation, the ex-perimental setups are designed in a way that photon s is detected before the detectionof photon p.

The state evolution of the radiations and the corresponding results within differentexperimental arrangements of this experiment can be described as the following. If

14In this experiment, x is finally related to the position of the detector D0 in which the photon s is detected.

82 Found Phys (2010) 40: 55–92

QWP1, QWP2 are absent, then the transformation of the state is given by

|ψ72〉 (double-slit)s→ |ψ74〉 (73)

|ψ74〉 = 1

2[(u1

s (x) + u2s (x))|εx,Ds〉 ⊗ |εy,Dp〉

+ eiφ(u1s (x) + u2

s (x))|εy,Ds〉 ⊗ |εx,Dp〉] (74)

where u1s (x) (u2

s (x)) describes the scalar amplitude of diffracted photon s from theslit 1(2), and |Dp〉 describes the photon traveling toward Dp . Accordingly, the coin-cidence counts vs detector DS position can be given by (Fig. 9b)

p(Ds at x,Dp) = 〈ψ74|(|Ds〉〈Ds | ⊗ |Dp〉〈Dp|)|ψ74〉

= 1

2[|u1

s (x)| + |u2s (x)|

+ 2Re[(u1s (x))∗u2

s (x)]] (75)

If QWP1 and QWP2 are in place in front of the double slit, the state evolution isdescribed by

|ψ74〉(U45◦,λ/4)

1s and (U−45◦,λ/4)

2s→ |ψ77〉 (76)

|ψ77〉 = 1

2

{[u1

s (x)|εL,Ds〉+u2

s (x)|εR,Ds〉

]⊗ |εy,Dp〉

+ieiφ

[u1

s (x)|εR,Ds〉−u2

s (x)|εL,Ds〉

]⊗ |εx,Dp〉

}(77)

where (U45◦,λ/4)1s ((U−45◦,λ/4)

2s ) describes the transformation of the polarization of

photon s diffracted from the slit 1(2) due to the passage through QWP1 (QWP2).With regard to (77), in the absence of POL1, the coincidence counts vs detector

Ds position can be calculated by

P(Ds at x,Dp) = 〈ψ77|(|Ds〉〈Ds | ⊗ |Dp〉〈Dp|)|ψ77〉= [|u1

s (x)|2 + |u2s (x)|2]/2 (78)

which shows no interference fringes as shown in Fig. 9c.If POL1 is in place, then the observer can manage to detect the observable σ2

(given in (B.11)) of the polarization of photon p. The observation of σ2 completelydestroy the welcher Weg knowledge of photons, and thus the coincidence rates showthe interference pattern. On this account, if the angle of POL1 is +45◦, then, thecoincidence counts vs detector Ds position can be given by (Fig. 9d)

p(Ds at x,Dp) = 〈ψ77|(|Ds〉〈Ds | ⊗ |ε+,Dp〉〈ε+,Dp|)|ψ77〉

= 1

4[|u1

s (x)| + |u2s (x)|

Found Phys (2010) 40: 55–92 83

+ 2Re[ieiφu1s (x)(u2

s (x))∗]] (79)

and likewise, if the angle of POL1 is −45◦, then, the coincidence counts vs detectorDs position can be given by (Fig. 9e)

p(Ds at x,Dp) = 〈ψ77|(|Ds〉〈Ds | ⊗ |ε−,Dp〉〈ε−,Dp|)|ψ77〉

= 1

4[|u1

s (x)| + |u2s (x)|

− 2Re[ieiφu1s (x)(u2

s (x))∗]] (80)

3.7 Jacques et al. (2006): Experimental Realization of Wheeler’s Delayed-ChoiceGedankenExperiment [20]

This experiment is performed in a typical Mach-Zehnder polarization interferome-ter (Fig. 10a). A linearly polarized single-photon radiation is sent through the firstpolarization beam splitter (BSinput). The two resulting beams with linear polariza-tions S and P at the output of BSinput are splitted with no spatial overlap and travelalong separated paths. The recombination of these two beams is done in two stepsat BSoutput. BSoutput consists of the combination of a half-wave plate (λ/2), a polar-ization beam splitter BS′, an electro-optical modulator EOM and a Wollaston prismWP. The photons with the linear polarization |εx〉 (|εy〉) are transmitted (reflected)by WP. First, the two beams coming from BSinput (which are spatially separatedand orthogonally polarized) are spatially merged by BS′, but can still be unambigu-ously identified by their polarization. When a suitable voltage is applied to EOM, theEOM is equivalent to a half-wave plate which rotates the vector of polarization by45◦. The EOM switching is randomly decided, in real time, by a Quantum RandomNumber Generator (QRNG) closely located to the output of the interferometer, at 48ms from BSinput (see Fig. 10a). In this respect, the requested space-like separationcondition is satisfied. The phase-shift φ between the two interferometer arms is var-ied by tilting the second polarization beam splitter BS′ with a piezoelectric actuator(PZT).

In order to introduce the delayed-choice mode of operation, a voltage to EOM canbe randomly applied or not, decided by QRNG, after the radiation has passed the firstbeam splitter (BSinput). When no voltage is applied to the EOM, the evolution of thephoton can be described as the following

|ψ82〉 BSinput→ |ψ83〉 λ/2→ |ψ84〉 BS′→ |ψ85〉 WP→ |ψ86〉 (81)

|ψ82〉 = |ε+〉 (82)

|ψ83〉 = [|ϕS, εx〉 + eiφ |ϕP , εy〉]/√

2 (83)

|ψ84〉 = [|ϕS, εy〉 + eiφ |ϕP , εx〉]/√

2 (84)

|ψ85〉 = [|ϕ, εy〉 + eiφ |ϕ, εx〉]/√

2 (85)

|ψ86〉 = [|D2, εx〉 + eiφ |D1, εy〉]/√

2 (86)

84 Found Phys (2010) 40: 55–92

Fig. 10 Jacques et al. (2006): Experimental realization of Wheeler’s delayed-choice GedankenExper-iment. (a) Mach-Zehnder polarization interferometer where BSoutput consists of the combination of ahalf-wave plate (λ/2), a polarization beam splitter BS′ , an electro-optical modulator EOM and a Wollas-ton prism WP. In order to introduce the delayed-choice mode of operation, a voltage to EOM is or notrandomly applied by QRNG after the radiation has passed the first beam splitter (BSintput). (b) The upperdiagram is related to the result of a delayed-choice mode of operation where a half-wave voltage is ran-domly imposed to EOM. The lower diagram shows the results where no voltage is applied to EOM. Thered dots show the results obtained in D1 and the blue dots correspond to the results of D2. The picture andthe diagrams are taken from Ref. [20]

where |ϕS〉 (|ϕP 〉) describes the radiation traveling along the path S (P), |ϕ〉 repre-sents the radiation after recombination of |ϕS〉, |ϕP 〉 at BS′, and |D1〉 (|D2〉) describesthe radiation traveling toward D1 (D2) after WP. Accordingly, the single counts ofD1 and D2 are given by

p(D1) = 〈ψ75|(|D1〉〈D1|)|ψ75〉= 1/2 (87)

Found Phys (2010) 40: 55–92 85

p(D2) = 〈ψ75|(|D2〉〈D2|)|ψ75〉= 1/2 (88)

which show no interference fringes as shown in Fig. 10b (lower diagram). However,when half-wave voltage is applied to the EOM, the evolution of the radiation is de-scribed as:

|ψ85〉 45◦,λ/2→ |ψ90〉 WP→ |ψ91〉 (89)

|ψ90〉 = [|ϕ, ε−〉 + eiφ |ϕ, ε+〉]/√2 (90)

|ψ91〉 = 1

2[(1 + eiφ)|D2, εx〉

+ i(1 − eiφ)|D1, εy〉] (91)

Regarding the relation (91), the single counts of D1 and D2 are given by

p(D1) = 〈ψ91|(|D1〉〈D1|)|ψ91〉= sin2(φ/2) (92)

p(D2) = 〈ψ91|(|D2〉〈D2|)|ψ91〉= cos2(φ/2) (93)

which show the interference fringes as depicted in Fig. 10b (upper diagram).

4 Concluding remarks

Taking into the account the main features of the delayed-choice effect and also theprobabilistic descriptions of the above mentioned proposals and experiments, in thissection we want to discuss that to what extent the delayed-choice effect can be char-acterized in the relevant quantum formulations of the debated experiments.

4.1 Probabilistic Description of Physical Mechanism of the Choice BetweenComplementary Phenomena

In the delayed-choice split-beam experiment, the choice for the locations of the de-tectors is made after the interaction between the photon and BS1 has already fin-ished. Here, the complementary phenomena are described by the probabilities eitherp(PMi |ψ9) or p(PMi |ψ13) (i = 1,2) given in relations (10), (11) and (14), (15).These probabilities are calculated under exclusive conditions described by |ψ9〉 and|ψ13〉. The change of conditions (characterized by unitary transformation of the quan-tum states) is occurred due to the interaction between the photon and BS2. That is,one can choose between exclusive contexts under which the phenomena are recordedby letting whether the photon would interact with BS2 or not.

86 Found Phys (2010) 40: 55–92

Accordingly, in delayed-choice split-beam experiment, after preparing the photonin a superposition of states, the observer makes the choice between the complemen-tary phenomena by deciding whether the photon would interact with BS2 or not. Onthis account, the experiments performed by Hellmuth et al. (1987), Baldzuhn et al.(1989), Kawai et al.(1998), and Jacques et al. (2006) are analogous to the delayed-choice split-beam experiment. In these experiments, the choice between complemen-tary phenomena is delayed to the moments after the particle is prepared in a coherentsuperposition of states. In other word, the complementary results are obtained underexclusive contexts, i.e., the corresponding probabilities are calculated under exclu-sive conditions, and the transformation between different states (which characterizesthe change of conditions) is occurred due to the local interaction between the particleand the measuring instrument. The observer decides to let this interaction occurs bychanging the experimental arrangement at proper time and proper place after prepa-ration of the system in a superposition of states.

In the delayed-choice double-slit experiment with cavities as path detectors, theobserver can manage to observe one of the patterns described by the joint probabil-ities either p(x,�z = ±1|ψ16) or p(x,�x = ±1|ψ16) (given by relations (18), (19)and (20)). These probabilities are calculated under the same given condition char-acterized by |ψ16〉. Here, the observer can conduct the experiment in two differentschemes. That is, in one case, the observer could delay the measurement of the ob-servable of the cavities to the moments after the detection of particle on the detectorplate. In this respect, the corresponding joint probability can be calculated as thefollowing

p(x,β|ψ16) = p(β|ψ16)p(x|ψ16, β) (94)

where β = (�x = ±1), (�z = ±1). On the other hand, the observer could follow theinverted scheme and observe the observable of the cavities before the detection ofparticle. In this regard, the corresponding joint probability can be calculated as

p(x,β|ψ16) = p(x|ψ16)p(β|ψ16, x) (95)

As far as the basic rules of probability theory are concerned, we know that both ofthe above mentioned schemes lead to the same joint probability, i.e., the relation (84)and (85) are identical

p(x,β|ψ16) = p(x|ψ16)p(β|ψ16, x) (96)

= p(β|ψ16)p(x|ψ16, β) (97)

In this situation, the observer can decide to obtain the mutual patterns in differentorders of time in which the delayed choice of the observable to be measured on onesystem is a choice of the condition under which the result of a measurement on theother system is to be calculated. In other word, the probabilities describing mutualpatterns p(x,β|ψ16) are time-symmetric with respect to the time orders of x and β

measurements. Here, the results of measurement on the path detectors enables theobserver to sort the whole set of data obtained on the detector-plate (described byp(x|ψ16)) into different mutually exclusive subsets (described by p(x,β|ψ16)) where

Found Phys (2010) 40: 55–92 87

the description of some subsets show the interference fringes and some others do not.However, the recorded phenomenon which is related to the whole set of data is notchanged by any measurements on the path detectors.

Likewise, the same line of reasoning can be given for the delayed-choice quan-tum marking and quantum eraser with neutral kaons where the joint probabilitiesdescribing the results of measuring the strangeness of a given kaon together with theobservation of the strangeness of the other kaon show the oscillatory behavior, butthe corresponding complementary joint probabilities (measuring the strangeness of agiven kaon together with the observation of the lifetime of the other kaon) show nooscillation. These joint probabilities (which are calculated under the same given con-dition |ψ25〉) are also symmetric with respect to the time-ordering of measurements,and the corresponding marginal probabilities (given in (26)) show no undulatory be-havior, too.

Briefly speaking, the temporal aspect of the delayed-choice double-slit experi-ment with cavities as path detectors and the delayed-choice quantum marking andquantum eraser with neutral kaons has no observable effects since the probabilitiesdescribing the mutual patterns are time-symmetric with respect to the sequence mea-surements. The experiments performed by Kwait et al. (1994), Kim et al. (1999) andWalborn et al. (2001) correspond to this experiment in a way that their outcomes arealso described by a time-symmetric joint probability calculated under the same givencondition (i.e., with a definite state preparation). In these experiments, the results ofmeasurement on the path detector provide the condition under which the result of ameasurement on the interfering system is to be calculated. Thus, regardless of anyfurther interpretation, the time sequence of measurements have no observable effectfor these experiments.

The foregoing experiments mentioned in the previous paragraph are usually calledas Quantum Eraser (QE) proposals15 [10]. Of most intriguing QE proposals are casesin which the welcher Weg information is stored in some external system differentfrom the interfering particles (e.g., Ref. [14, 17, 18]), because these experiments canthen be easily performed in a delayed choice mode of operation. Up to now, the ex-perimental tests of delayed-choice QE have been performed with entangled photons.The proposed experiment by Bramon et al. (Sect. II.D), however, extends QE con-siderations to entangled massive particles where it can also be simply operated in adelayed-choice mode, too16 [12].

15The idea of QE was introduced in a fascinating paper by Scully and Drühl in 1982 [10]. Afterwards, thisproposal has been followed by various theoretical and experimental investigations describing the possibil-ity of restoring the interference fringes for various interfering particles in different experimental arrange-ments [10, 15, 16, 18, 19, 21]. In a typical two-path interferometer, quantum theory seemingly asserts thatan observation of both the interference pattern and the interfering particles’ paths are mutually exclusive.Any attempt for verifying a pictorial representation of the behavior of interfering particles, e.g., by entan-gling them with path detectors (i.e., storing welcher Weg information), makes the interference fringes to bewiped out. However, the interference fringes could be recovered if one examines the joint probability dis-tribution (i.e., coincidence rates) of results in measurements of two observables, including an observableof the interfering particles (e.g., the position) together with an appropriate observable of the path detectors.Measuring the proper observable of path detectors is called the Quantum Eraser process where the resultsof QE measurement contain no welcher Weg information.16The atomic interferometric QE experiment performed by Dürr and Rempe also includes the massiveparticles where the welcher Weg information was stored in the internal states of Rydberg atoms serving as

88 Found Phys (2010) 40: 55–92

In the delayed-choice double-slit experiment with swinging door detector plate,the observer can manage to observe one of the two exclusive patterns described by theprobabilities either p(x|ψ1(x, t1)) or p(x|ψ1(x, t2)), given in relations (2) and (3).These probabilities are also calculated under given different conditions correspond-ing to the states ψ1(x, t1) and ψ1(x, t2). The states characterizing these conditionsare mutually exclusive because they are related to different times and cannot be si-multaneously considered in this experiment. However, the unitary transformation ofψ1(x, t1) into ψ1(x, t2) is occurred due to the free evolution of the particle withinthe time period [t1 − t2]. In other words, in this experiment, there is no interactionbetween the particle and measuring apparatus responsible for the transformation ofstates from ψ1(x, t1) to ψ1(x, t2). Nevertheless, none of experimentally performeddelayed-choice proposals are analogous to this experiments. The discussed situations,described by p(x|ψ1(x, t1)) and p(x|ψ1(x, t2)), however, are the extreme situationsin which only one of the complementary aspects, either wave or particle, are man-ifested. If the detector plate would be inserted at the positions [L–2L] far from thedouble-slit, then both aspects of the wave and particle can be manifested, as shownin Fig. 1c. Concerning these intermediate situations, Complementarity Principle canbe quantified by some sophisticated inequalities [12, 22]. In the case of two-pathinterferometers, the inequality reads as

P 2 + V 2 ≤ 1 (98)

where predictability P quantifies the particle aspects, and visibility V is a measurefor wave aspect [22]. For the extreme situations, one reads (P = 0, V = 1) for thecase described by p(x|ψ1(x, t1)), and (P = 1, V = 0) for the case described byp(x|ψ1(x, t2)).

4.2 Passive and Active Choice Between Complementary Phenomena

As mentioned in Sect. 2.4, the choice between complementary observations can bemade actively or passively with respect to the direct intervention of the observer.Regarding this issue, the choices of the observables to be measured on the path de-tectors in Kwait et al. (1994) and Walborn et al. (2001) experiments were activelymade by the observer, however the choice was made randomly in Kim et al. (1999)by the quantum dynamics of the idler photon serving as path detector. This can becompared with the delayed-choice quantum marking and quantum eraser with kaonswhere the choice between complementary observables of the path detector (i.e., theright or the left kaon) can be made both actively and passively.

This idea can also be extended to the experiments in which the complementary re-sults are obtained under exclusive contexts, i.e., the experiments performed by Hell-muth et al. (1987), Baldzuhn et al. (1989), Kawai et al. (1998), and Jacques et al.(2006). Then, it can be concluded that the choice between complementary contexts(under which the phenomena are manifested) were made actively by the observer in

interfering particles. However, this experiment has not been performed in delayed-choice mode, since theposition and the internal states of atoms were measured “simultaneously” [21].

Found Phys (2010) 40: 55–92 89

the experiments of Hellmuth et al. (1987), Baldzuhn et al. (1989) and Kawai et al.(1998), in comparison with Jacques et al. (2006) where the choice was randomlydecided by a QRNG, of course within a quantum dynamics.

4.3 The Role of Time

Meanwhile, with only two exceptions, in all foregoing experiments and proposals, thetime is merely an external parameter for the chronological scale of the experimentsthat determines the sequence of the system’s interactions (including measurements)for differentiating between the normal and delayed-choice modes of operation. How-ever, in the delayed-choice quantum beam experiment of Hellmuth et al. (1987) andalso the delayed-choice quantum marking and quantum eraser with neutral kaons, thequantum superpositions are temporal and time is a variable by which the undulatorybehaviors are observed too.

4.4 Summarizing Note

The key idea of delayed-choice effect is to make the choice between the complemen-tary phenomena after the preparation of the system in a superposition of states. Sincethe inception of this idea by Wheeler in 1978, many different experimental arrange-ments have been proposed on this idea. All of the delayed-choice experiments char-acterize the delayed-choice effect in a way that the choice between complementarypatterns is made after the system is prepared in a superposition of states. This choicecan be made either actively by a direct intervention of the observer or passively by arandom quantum dynamics. However, the active delayed-choice experiments can alsobe performed passively, in principle, e.g. with a aid of an external quantum dynamicssuch as a QRNG.

Concerning the probabilistic descriptions of the results of measurements, the de-layed choice proposals can be generally categorized into two main groups. The firstgroup includes the experiments in which the complementary phenomena are recordedwithin exclusive contexts, and hence their corresponding probabilities are calculatedunder exclusive conditions. The change of conditions which are characterized bytransformation of the states is occurred due to the local interaction between the par-ticle and measuring instrument. The only expectation in which the transformation isa free evolution, is the delayed-choice double-slit experiment with swinging door de-tector plate which has no experimental counterpart, yet. The second group containsthe experiments in which the coincident results of measuring the entangled degreesof freedom of two entangled systems are considered. Here, the corresponding proba-bilities are joint probabilities that are calculated under the same given condition, andthese joint probabilities are symmetric respect to the time sequence of measurementson two entangled systems.

In the second group of delayed-choice experiment, the results of measurementson a system together with the results of measurement on other spatially-separatedsystems (i.e., the coincidence rates) show the spooky dependency on the parametersof both systems. However, when the outcome of the measurements on a single sys-tem are taken into the account (i.e., the individual rates), no dependency upon the

90 Found Phys (2010) 40: 55–92

measurements of the other spatially-separated system appears. In other word, in thesecond group of delayed-choice experiments, no change in the quantum mechanicalstatistics of a system take place as a consequence of the interactions (e.g., measure-ment) with the other remote systems. However, as described in details, in the firstgroup of delayed-choice experiments, the change of the phenomenon is due to thelocal interactions with the measuring instrument.

Finally, in delayed-choice experiments, the time is merely an external parameterdetermining the sequence of interactions for differentiating between the normal anddelayed-choice modes of operation, and thus it seems that these experiments don’tprovide new insights on the differences between classical and quantum conceptionsof time.

Acknowledgements M. Bahrami wants to thanks Caslav Brukner of Vienna University for his valuablehelps on providing with us the original papers of John Wheeler on delayed-choice.

Appendix A

�z is an observables of cavities defined as:

�z = |1u,0l〉〈1u,0l | − |0u,1l〉〈0u,1l | (A.1)

where its eigenstates are given by

�z|�z = ±1〉 = ±|�z = ±1〉 (A.2)

|�z = +1〉 = |1u,0l〉; |�z = −1〉 = |0u,1l〉 (A.3)

Likewise, for �x we have:

�x = |1u,0l〉〈0u,1l | + |0u,1l〉〈1u,0l | (A.4)

where its eigenstates are given by

�x |�x = ±1〉 = ±|�x = ±1〉 (A.5)

|�x = ±1〉 = |1u,0l〉 ± |0u,1l〉√2

(A.6)

Appendix B

• Linear polarized in x-direction

|εx〉 =(

10

)(B.1)

• Linear polarized in y-direction

∣∣εy

⟩ =(

01

)(B.2)

Found Phys (2010) 40: 55–92 91

• Linear polarized at ±45◦ away from x-axis

|ε±〉 = 1√2

(1

±1

)(B.3)

• Right circular polarized

|εR〉 = 1√2

(1−i

)(B.4)

• Left circular polarized

|εL〉 = 1√2

(1i

)(B.5)

• Linear polarizers

Ai = |εi〉 〈εi | ; (i : x, y, z,±) (B.6)

• Quarter-wave plate, fast axis horizontal

Ux,λ/4 = eiπ4

(1 00 i

)(B.7)

• Quarter-wave plate, fast axis vertical

Uy,λ/4 = eiπ4

(1 00 −i

)(B.8)

• Quarter-wave plate, fast axis at the θ = ±45◦

U±45,λ/4 =(

1/√

2 ±i/√

2±i/

√2 1/

√2

)(B.9)

• We define σ1 with eigenstates |εx〉 and |εy〉

σ1 =(

1 00 −1

)(B.10)

• We define σ2 with eigenstates |ε+〉 and |ε−〉

σ2 =(

0 11 0

)(B.11)

• We define σ3 with eigenstates |εL〉 and |εR〉

σ3 =(

0 −i

i 0

)(B.12)

92 Found Phys (2010) 40: 55–92

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