Modified quantum delayed-choice experiment without quantum control
Qi Guo, Wen-Jie Zhang, Gang Li, Tiancai Zhang, Hong-Fu Wang, Shou Zhang
aa r X i v : . [ qu a n t - ph ] F e b Modified quantum delayed-choice experiment without quantumcontrol
Qi Guo ∗ ,
1, 2, 3
Wen-Jie Zhang,
1, 2, 3
Gang Li,
2, 3
TiancaiZhang † ,
2, 3
Hong-Fu Wang, and Shou Zhang College of Physics and Electronic Engineering, Shanxi University,Taiyuan, Shanxi 030006, People’s Republic of China State Key Laboratory of Quantum Optics and Quantum Optics Devices,and Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China Collaborative Innovation Center of Extreme Optics,Shanxi University, Taiyuan 030006, China Department of Physics, College of Science, Yanbian University,Yanji, Jilin 133002, People’s Republic of China
Wheeler’s delayed-choice experiment delays the decision to observe either the waveor particle behavior of a photon until after it has entered the interferometer, andthe quantum delayed-choice experiment provides the possibility of observing thewave and particle behavior simultaneously by introducing quantum control device.We here propose a modified quantum delayed-choice experiment without quantumcontrol or entanglement assistance, in which a photon can be prepared in a wave-particle superposition state and the morphing behavior of wave-to-particle transitioncan be observed easily. It is demonstrated that the presented scheme can allow usto rule out classical hidden variable models in a device-independent manner viaviolating dimension witness. We also extend the scheme to the situation of twodegrees of freedom, first constructing a hybrid quantum delayed-choice experimentwhich enables simultaneous observation of a photon’s wave and particle behaviorsin different degrees of freedom, and then proposing a scheme to prepare the single-photon wave-particle entanglement. This study is not only meaningful to explore thewave and particle properties of photons, but also provides potential for the research ∗ E-mail: [email protected] † E-mail: [email protected] of the single-particle nonlocality from the perspective of the wave-particle degree offreedom.
I. INTRODUCTION
Wave-particle duality, one of the most fascinating characters of quantum physics, meansa quantum object has both wave-like and particle-like properties that are two distinct andmutually exclusive natures from the perspective of classical physics. Bohr’s complemen-tarity principle shows that the wave behavior and particle behavior cannot be observedsimultaneously, and which behavior a quantum object will exhibit depends on the measure-ment arrangement [1]. Mach-Zehnder interferometer (MZI) provides an effective platformfor testing the wave-particle duality of a single photon. The fist beam splitter (BS) of aMZI splits the input photon into two paths, and the second BS of the MZI recombine thetwo paths. Therefore, if the second BS is inserted into the MZI, the interference betweenthe two paths can be observed at the two output ports, and the input photon shows wavebehavior; if the second BS is removed, the which-path information will be revealed at theoutput ports, and the photon shows particle behavior.However, one objection is that maybe a hidden variable in the initial state can tell theinput photon in advance the second BS is inserted or not, so the photon can adjust itsbehavior to the corresponding measurement apparatus. To rule out the hidden variabletheory, Wheeler proposed the famous delay-choice experiment [2, 3], in which the secondBS is decided to be inserted or removed after the photon has entered the interferometer,so the photon cannot know which measurement apparatus lie ahead in advance. Withthe development of experiment technology, this Gedanken experiment has been realized inactual laboratory by using different system, such as photons [4–6], and atoms [7–9], evenbeen implemented between satellite and ground stations [10]. Moreover, the delay-choiceexperiment was also extended to other domains of quantum physics [9, 11–17], such asdelay-choice quantum erase [12, 13], delayed-choice entanglement swapping [14–16], delayed-choice decoherence suppression [17], entanglement-separation duality [16], and so on. Theseexperiments exhibit profound and amazing quantum effects. On the other hand, the wave-particle duality has also been studied quantitatively by the complementarity inequality [18–20].The reason that the wave behavior and particle behavior of a photon cannot be observedsimultaneously is the two measurement apparatuses (removing the second BS or not) aremutually exclusive. However, a quantum version delayed-choice experiment was proposedby replacing the second BS in MZI with a quantum-controlled BS [21], in which the secondBS can be prepared in a quantum superposition state of presence and absence by using anancilla qubit. Thus, the quantum delayed-choice (QDC) experiment enables the simulta-neous observation of a photon’s wave and particle behaviors. The key module of the QDCexperiment is the quantum control device of the second BS, therefore, many researchers havedesigned the quantum BS for either massless photons or massive particles to implement theQDC experiment in different systems [22–26], and observed the wave-particle superpositionbehavior. The QDC experiment enriches the basic contents of wave-particle duality andBohr’s complementarity principle, so people’s interest in the foundations of quantum me-chanics has been further stimulated, and a variety of works exploring quantum phenomenaunder the frame of the QDC experiment has been proposed in theory [27–30] and experiment[31–38].The original intention of Wheeler’s delay-choice (WDC) experiment is to exclude clas-sical hidden variable models. However, very recently, Chaves et al. considered the WDCexperiment and its quantum version from the perspective of device-independent causal mod-els [39], and proposed that a causal two-dimensional hidden-variable theory can reproducethe quantum mechanical predictions of the WDC experiment and the QDC experiment,which means the original WDC experiment cannot rule out the classical hidden variablemodel. In that work, the authors treated the WDC experiment as a device-independentprepare-and-measure (PAM) scenario [40], and suggested a slight modification for the WDCexperiment can exclude any two-dimensional nonretrocausal hidden classical model in adevice-independent manner by violating the dimension witness inequality [40–42]. Subse-quently, this causal-modeled WDC experiment was carried out independently in a seriesof experiments [43–45]. The causal modeling approach [46] provides an effective way toanalyze the nonclassical nature of an experiment by classical causal assumptions [47–49].These theoretical and experimental works gave further evidences for the nonclassicality ofwave-particle duality.The essential advance of the QDC experiment over the WDC experiment is that the wavebehavior and particle behavior of a quantum object can be observed simultaneously due tothe quantum-controlled BS, i.e., the wave-particle superposition state can be prepared. Inthe existing schemes for the QDC experiment, the quantum control device were achievedwith the help of ancilla qubits [21–26] or entanglement [27, 35]. Here, we first proposean alternative proposal for QDC experiment with classical strategies, that is, the photon’swave-particle superposition state can be generated without any quantum control or entan-glement assistance, and the same statistical result as the existing schemes can be obtained.We demonstrate the theoretical results of the presented scheme can violate the dimensionwitness, which means it can rule out classical hidden variable models in a device-independentmanner. Moreover, we generalize this proposal to the case of two degrees of freedom, whichshows that a single photon can behave as wave and particle simultaneously in the differentdegrees of freedom, and the single-photon wave-particle entanglement can also be prepared.
II. THE QUANTUM DELAYED-CHOICE EXPERIMENT WITH TUNABLEBEAM SPLITTER
We now introduce how to realize the QDC experiment with classical strategies. Theschematic depiction of the presented proposal is shown in Fig. 1(a). Note that, actually, thecomplete setup diagram of the presented proposal is Fig. 1(c). Fig. 1(a) can only generatethe same statistics results as Fig. 1(c), but cannot rule out the two-dimensional hidden-variable model in Ref. [39], which can be ruled out in Fig. 1(c) by inserting an additionalphase shifter as shown in the following section. Here, to exhibit the photon’s statisticsclearly, we explain the procedure using Fig. 1(a). The second BS in the MZI is replaced bya tunable beam splitter (TBS) with reflectivity cos θ and transmissivity sin θ , where θ iscontinuously tunable between 0 and π . The TBS is always placed in the interferometer inthe proposed scheme (the idea of delayed choice will reflect in the choice of φ in Fig. 1(c)).We denote the two paths of the MZI with quantum states | i and | i . Let a photon enterthe MZI initially from the path 0, i.e., the initial state is | i . The BS transforms the stateas {| i → √ ( | i + | i ) , | i → √ ( | i − | i ) } , and the phase shifter ϕ induces a phase shift ϕ for the photon in the path 1. Thus, after the photon passing through BS and ϕ , the stateevolves to | ψ i → √ | i + e iϕ | i ) . (1) MR BS MR TBS D D (a) X(cid:13)
Y(cid:13) preparer(cid:13) measurer(cid:13)
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Fig. 1: (a) The QDC experiment without quantum control device. MR: normal mirror. BS: 50:50beam splitter. ϕ : phase shifter. TBS: tunable beam splitter. The two paths of the MZI are labeledby 0 and 1. D : conventional photon detector. (b) The device-independent PAM scenarioconsists of a preparer (the first black box) with buttons X , a measurer (the second black box)with buttons Y , and a detection D . (c) The complete schematic of the proposed QDC experimentwith an additional phase shifter φ that can be used to exclude the two-dimensional hidden-variablemodel. The gray dashed boxes correspond to the preparer and the measurer in (b). Then the photon is reflected by mirrors (MR) and reaches the TBS, whose action is equiv-alent to the rotation {| i → cos θ | i + sin θ | i , | i → sin θ | i − cos θ | i} . When the photonleaves the MZI, the state becomes | ψ i f = 1 √ θ + e iϕ sin θ ) | i + (sin θ − e iϕ cos θ ) | i ] . (2)Obviously, if θ = 0, | ψ i f = √ ( | i − e iϕ | i ), detectors D can reveal the which-pathinformation of the photon in the MZI, and the photon behaves as a particle; if θ = π , | ψ i f = e iϕ/ (cos ϕ | i − i sin ϕ | i ) the photon behaves as a wave. Therefore, following theoperational description of the wave and particle behavior of a photon in Ref. [21], we canintroduce the definition of the particle state | particle i = √ ( | i − e iϕ | i ) and the wave state | wave i = e iϕ/ (cos ϕ | i − i sin ϕ | i ). This operational definition not only provides suitableexpressions for the capacity and incapacity of the photon to produce interference in thecontext of quantum mechanics, but also can be conveniently used to study the intermediatebehavior and the transition behavior between wave and particle nature [21–27, 35]. In thewave-particle representation, the final state in Eq. (2) can be rewritten as | ψ i f = α | particle i + β | wave i , (3) θ ϕ π π π /16 2 π π /8 π I ( ϕ , θ ) π /160.6 0 π /40.81 FIG. 2: The morphing behavior between wave ( θ = π/
4) and particle ( θ = 0) by continuouslytuning θ . where the coefficients α = cos θ − sin θ and β = √ θ . It can be seen from Eq. (3),for θ = 0 and π , the photon is in the particle and wave state respectively. While θ is anarbitrary value between 0 and π , Eq. (3) is a wave-particle superposition state, which meansthe photon will exhibit wave property and particle property simultaneously. In order toshow the intermediate morphing behavior between wave and particle nature visually, weshould explore the interference pattern at the output ports of the MZI, which, for a singlephoton, can be reflected by the probabilities that the detector D clicks. Take the outputport 0 for example, the probability of D clicking is I ( ϕ, θ ) = Tr( ρ f | ih | ) = 12 (1 + sin 2 θ cos ϕ ) , (4)where ρ f = | ψ i ff h ψ | is the density matrix of the photon’s final state. For an arbitrary θ ,the visibility of the interference pattern at the output port can be obtained V = ( I max − I min ) / ( I max + I min ) = sin 2 θ . We plot the probability distribution I ( ϕ, θ ) versus θ and ϕ inFig. 2, from which one can see the continuously morphing behavior between wave ( θ = π/ θ = 0). Thus, by varying θ we can observe the photon’s behavior of wave-to-particle transition, that is to say, both wave and particle properties can be measured in asingle experiment by classical strategies and without quantum control. III. RULING OUT THE CLASSICAL HIDDEN VARIABLE MODELS VIADIMENSION WITNESS METHOD
The complementary properties can be observed in a single experimental setup with thepresented scheme, but the setup above cannot rule out the classical hidden variable thatmaybe tell the input photon in advance about the value of θ . One possible way is to tunethe parameter θ of the TBS when the photon has entered the MZI and before it reachesthe TBS, so that the photon cannot adjust itself beforehand to the specific superpositionstate corresponding to θ . However, the same as the WDC experiment, this method canbe explained by the classical two-dimensional hidden variable causal model proposed inRef. [39]. To exclude the the causal model, we here adopt the method similar to Ref. [39],which can be demonstrated with the device-independent PAM scenario by the violation ofdimension witness. The PAM scenario as shown in Fig. 1(b) consists of a preparer (thefirst black box), a measurer (the second black box), a detector D [40, 41]. The preparercan prepare a physical system in the state ρ ( x ) by pressing one of the buttons X , andthen the system is sent to the measurer. By choosing one of the buttons Y , the system ismeasured and an outcome D will be produced. For the PAM scenario, to produce the samestatistical distribution, a classical system is required higher dimensionality than a quantumsystem, which is the theoretical basis for testing classical and quantum systems by usingthe dimension witness. For example, consider a scenario with 2 k preparations and k binarymeasurements, the dimension witness can be achieved with the help of the k × k matrix [40]W k ( i, j ) = p (2 j, i ) − p (2 j + 1 , i ) , (5)with 0 ≤ i, j ≤ k −
1, and p ( x, y ) = p ( D = 0 | x, y ). The dimension witness | Det(W k ) | equals0 for any classical system of dimension d ≤ k , but for any quantum system of dimension d ≤ √ k . Hence we can test classical and quantum systems by using the dimension witness | Det(W k ) | .In order to rule out the classical causal model, we should insert an extra phase shifter φ in the path 0 as shown in Fig. 1(c), which has the same experimental results and photonstatistical behaviors as Fig. 1(a) by absorbing φ into ϕ , so it can also measure the photon’swave property and particle property at the same time. As demonstrated by Chaves etal. [39], the delayed-choice experiment is equivalent to the PAM scenario. The first dottedrectangular box and the second one in Fig. 1(c) correspond to the preparer and the measurerin the PAM scenario. The value of φ should be chosen after the preparation process to ensurethere is no correlation between the preparer and the measurer. We also sent the photon frompath 0 initially, and after passing through the BS and ϕ , the quantum state correspondingto ρ ( x ) is √ ( | i + e iϕ | i ). Then the photon enters the measurer, and passes through φ andTBS. The state evolves to | ψ ′ i f = 1 √ e iφ cos θ + e iϕ sin θ ) | i + ( e iφ sin θ − e iϕ cos θ ) | i ] . (6)Because there are two detection results or the photon has two spatial modes in theexperiment, the dimension of system to be test is 2. Therefore, we should set four preparationchoices X (i . e .ϕ ) ∈ { ϕ , ϕ , ϕ , ϕ } , and two measurement choices Y (i . e .φ ) ∈ { φ , φ } . Thematrix used for dimension witness is given byW = p (0 , − p (1 , p (2 , − p (3 , p (0 , − p (1 , p (2 , − p (3 , , (7)where p ( x, y ) = p ( D = 0 | ϕ x , φ y ) is the probability the photon is detected by D for the choice ϕ x and φ y . For the state in Eq. (6), it can be obtained that p ( x, y ) = [1+sin 2 θ cos( ϕ x − φ y )].We can getW = 14 sin θ { [cos( ϕ − φ ) − cos( ϕ − φ )][cos( ϕ − φ ) − cos( ϕ − φ )] − [cos( ϕ − φ ) − cos( ϕ − φ )][cos( ϕ − φ ) − cos( ϕ − φ ) } , (8)Without loss of generality, we choose ϕ = ϕ = ϕ = ϕ = ϕ , φ = 0, and φ = π forevaluating | Det(W k ) | , then,W = 14 sin θ (2 sin 2 ϕ − sin ϕ − sin 3 ϕ ) . (9)Now we plot the change of | Det(W ) | versus ϕ and θ in Fig. 3(a). We can see | Det(W ) | > | Det(W ) | = 0 .
29 for ϕ = 3 π/ θ = π/ |Det(W )| π /16 π /8 3 π /16 π /4 θ π /2 π π /22 π ϕ (a) I DW π /16 π /8 3 π /16 π /4 θ π /4 π /23 π /4 π ϕ -1-0.500.511.522.533.54 (b) Fig. 3: (a) The dimension witness | Det(W ) | versus ϕ and θ . (b) The linear dimension witnessversus ϕ and θ , and the dimension witness inequality I DW ≤ and the hidden variable is also independent of any noise term. However, in the causal modelproposed in Ref. [39], noise terms might affect the hidden variable, and thus influence thestatistical behaviors of the photon at the output ports. In order to rule out the hiddenvariable model with such correlation, we should test the presented scheme with the lineardimension witness, which has been given by an inequality in Ref. [41] I DW = h D i + h D i + h D i − h D i − h D i ≤ , (10)where h D xy i = p ( D = 0 | ϕ x , φ y ) − p ( D = 1 | ϕ x , φ y ), i.e. the probability difference between D and D to detect the photon. To employ the dimension witness inequality, we should setthree preparation choices ϕ ∈ { ϕ , ϕ , ϕ } , and two measurement choices φ ∈ { φ , φ } . Byusing Eq. (6), we can obtain I DW = sin 2 θ [cos( ϕ − φ ) + cos( ϕ − φ ) + cos( ϕ − φ ) − cos( ϕ − φ ) − cos( ϕ − φ )] . (11)To evaluate I DW , we here also choose ϕ = − ϕ = ϕ , φ = 0, φ = π , and ϕ = π formaximizing I DW , then I DW = sin 2 θ [2(cos ϕ + sin ϕ ) + 1]. From Fig. 3(b), it can be seenthat the dimension witness inequality is violated in the area marked by the dotted line, i.e., I DW > θ, ϕ ). Specially, the maximum violation [39, 41] in quantumsystems can be obtained I DW = 1 + 2 √ ϕ = π/ θ = π/
4. Thus, the hiddenvariable correlated with noise terms can also be ruled out in the presented scheme.0
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FIG. 4: The schematic of the hybrid QDC experiment in path degree of freedom and polarizationdegree of freedom. PBS: polarizing beam splitter. PR: polarization rotator. Other optical elementsare the same as Fig. 1.
IV. HYBRID QUANTUM DELAYED-CHOICE EXPERIMENT IN DIFFERENTDEGREES OF FREEDOM
Now we extend the scheme to two degrees of freedom, i.e., implement the QDC experimentsimultaneously in path degree of freedom and polarization degree of freedom. The basicsetup diagram of the scheme is shown in Fig. 4. The photon is initially prepared in thesuperposition state of horizontal polarization | H i and vertical polarization | V i , and entersthe setup from the path 0, that is, the initial state of the photon can be given by | Φ i = √ ( | H i + | V i ) | i . The polarization beam splitter (PBS) transmits | H i component andreflects | V i component. The phase shifters ϕ and φ induces phase shifts ϕ and φ for | V i component and | i component, respectively. So after passing through BS, PBSs, and phaseshifters, the photon is in the state | Φ i = 12 ( | H i + e iφ | V i )( | i + e iϕ | i ) . (12)The polarization rotator (PR) is used to rotate the photon by an angle ϑ , i.e., | H i → cos ϑ | H i + sin ϑ | V i and | V i → sin ϑ | H i − cos ϑ | V i . The action of TBS with parameter θ isthe same as that in Fig. 1. Therefore, the state after TBS evolves to | Φ i f = 12 [(cos ϑ + e iφ sin ϑ ) | H i + (sin ϑ − e iφ cos ϑ ) | V i ] ⊗ [(cos θ + e iϕ sin θ ) | i + (sin θ − e iϕ cos θ ) | i ] . (13)1 MR(cid:13)
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FIG. 5: The schematic for generating single photon wave-particle entangled state. HWP: half-waveplate oriented at 22 . ◦ . σ z : π -phase shifter Other. optical elements are the same as Fig. 4. The particle state and the wave state in the polarization degree of freedom can be defined as | particle i = √ ( | H i − e iφ | V i ) and | wave i = e iφ/ (cos φ | H i − i sin φ | V i ). Thus, the state inEq. (13) is a superposition state of wave and particle in two degrees of freedom. Especially, | Φ i f = | particle i pol | wave i path for ϑ = 0 and θ = π/
4, and | Φ i f = | wave i pol | particle i path for ϑ = π/ θ = 0, where the subscripts pol and path respectively indicate polarization andpath degree of freedom. That is, the hybrid QDC experiment allows a single photon to be inparticle state in a degree of freedom but in wave state in the other degree of freedom at thesame time. Note that we here have used only one phase shifter in each degree of freedomfor simplicity, which cannot rule out the causal model. To achieve this, every phase shiftershould be divided into two parts and placed in the preparer and the measurer similar toFig. 1(c), then the causal model can be excluded by using the same way as above section.The hybrid QDC experiment above can be straightforward used to generate single photonwave-particle entangled state, whose schematic diagram is shown in Fig. 5. Two hybrid QDCexperiment setups are combined by one PBS, two half-wave plates (HWP), and a π -phaseshifter σ z . The photon is initially prepared in the state | Ψ i = √ ( | H i + | V i ) | i . Throughdirect calculation, the state of the photon passing through the whole setup becomes | Ψ i f = 12 √ { [(cos ϑ | H i + sin ϑ | V i ) + e iφ (sin ϑ | H i − cos ϑ | V i )]2[(cos θ | i + sin θ | i ) + e iϕ (sin θ | i − cos θ | i )]+[(cos ϑ | H i + sin ϑ | V i ) + e iφ (sin ϑ | H i − cos ϑ | V i )][(cos θ | i + sin θ | i ) + e iϕ (sin θ | i − cos θ | i )] } , (14)where ϑ and ϑ respectively denote the rotated angles by PR and PR , and θ isthe transmission parameter of TBS . When we choose ϑ = θ = 0 and θ = ϑ = π/
4, inthe wave-particle representation, the state above can be written as | Ψ i f = 1 √ | particle i pol | wave i path + | wave i pol | particle i path ) , (15)which is a wave-particle entangled state of a single photon. The concurrence of the stateabove equals 1, i.e., the state is a maximally entangled state. By choosing proper parame-ters, other Bell-like states can also be obtained. The entanglement generation scheme canbe regarded as a simple application of the presented hybrid QDC experiment. Comparedwith the two-photon wave-particle entanglement in Ref. [33], the single-photon wave-particleentanglement proposed here maybe more counterintuitive for exhibiting the photon’s dualwave-particle behavior. V. DISCUSSION AND CONCLUSIONS
The wave-particle duality is a fundamental topic of quantum mechanics. The emergenceof the QDC experiment has enriched people’s understanding of Bohr’s complementarity prin-ciple. The relevant QDC schemes presented here requires only the most ordinary opticalelements in optical laboratory [50]. Compared to existing schemes, a crucial optical elementhere is TBS that plays key role in observing the behavior of wave-to-particle transition. For-tunately, such TBS has been realized experimentally [51]. Moreover, the device-independentmanner used here is robust to arbitrarily losses inside the interferometer and the inefficiencyof detectors as pointed out in Ref. [39], which has been demonstrated in current experi-ments [43–45]. Therefore, the presented scheme is feasible under the current experimentalcondition.In conclusion, we have proposed an alternative scheme for the QDC experiment withoutquantum control or entanglement assistance, which means the wave-particle superpositionstate of a photon can be obtained with classical strategies and provides a compact wayto observing the morphing behavior of wave-to-particle transition. By violating nonlinear3dimension witness and linear dimension witness inequality, it has been demonstrated thatthe presented scheme can exclude classical two-dimensional hidden variable causal modelsin a device-independent manner. We have also constructed a hybrid QDC experiment intwo degrees of freedom that makes it possible for a photon to exhibit particle behaviorin one degree of freedom but exhibit wave behavior in the other one. The single-photonwave-particle entanglement between two degrees of freedom can be prepared by using thehybrid QDC experiment. Therefore, these works may be meaningful for the research of thesingle-particle nonlocality and quantum information protocols from the perspective of thewave-particle representation.
Acknowledgments
This work is supported by the National Natural Science Foundation of China underGrant No. 11604190 and No. 11974223, the Natural Science Foundation of Shanxi ProvinceNo. 201901D211167, Scientific and Technological Innovation Programs of Higher EducationInstitutions in Shanxi No. 2019L0043, and the Fund for Shanxi “1331 Project” Key SubjectsConstruction. [1] N. Bohr, The quantum postulate and the recent development of atomic theory, Nature ,580C590 (1928).[2] J. A. Wheeler, in Mathematical Foundations of Quantum Theory (ed. Marlow, A. R.) 9C48(Academic Press, New York, 1978).[3] J. A. Wheeler and W. H. Zurek, Quantum Theory and Measurement (Princeton UniversityPress, 1984).[4] V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, Ex-perimental realization of wheelers delayed-choice gedanken experiment. Science , 966C968(2007).[5] Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. , 1 (2000).[6] T. Hellmuth, H. Walther, A. 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