Delayed-Choice Gedanken Experiments and Their Realizations

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Delayed-choice gedanken experiments and their realizations

Xiao-song Ma∗ Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria Department of Electrical Engineering, Yale University, 15 Prospect Street, New Haven, CT 06520, USA National Laboratory of Solid State Microstructures, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China Johannes Koﬂer† Max Planck Institute of Quantum Optics (MPQ), Hans-Kopfermann-Strasse 1, 85748 Garching, Germany Anton Zeilinger‡ Vienna Center of Quantum Science and Technology (VCQ), University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria

The wave-particle duality dates back to Einstein’s explanation of the photoelectric effect through quanta of light and de Broglie’s hypothesis of matter waves. Quantum mechanics uses an abstract description for the behavior of physical systems such as photons, electrons, or atoms. Whether quantum predictions for single systems in an interferometric experiment allow an intuitive understanding in terms of the particle or wave picture, depends on the specific configuration which is being used. In principle, this leaves open the possibility that quantum systems always behave either definitely as a particle or definitely as a wave in every experimental run by a priori adapting to the specific experimental situation. This is precisely what is tried to be excluded by delayed-choice experiments, in which the observer chooses to reveal the particle or wave character of a quantum system – or even a continuous transformation between the two – at a late stage of the experiment. The history of delayed-choice gedanken experiments, which can be traced back to the early days of quantum mechanics, is reviewed. Their experimental realizations, in particular Wheeler’s delayed choice in interferometric setups as well as delayed-choice quantum erasure and entanglement swapping are discussed. The latter is particularly interesting, because it elevates the wave-particle duality of a single quantum arXiv:1407.2930v3 [quant-ph] 19 Mar 2016 system to an entanglement-separability duality of multiple systems.

CONTENTS B. von Weizs¨acker, Einstein, Hermann 4 C. Bohr’s account 5 I. Introduction 2 D. Wheeler’s delayed-choice wave-particle duality gedanken experiment 5 II. Delayed-choice gedanken experiments 3 E. Delayed-choice quantum erasure 7 A. Heisenberg’s microscope 3 F. Delayed-choice entanglement swapping 9 G. Quantum delayed-choice 10

III. Realizations of delayed-choice Wave-particle duality ∗ [email protected] experiments 11 † johannes.koﬂ[email protected] A. First realizations of Wheeler’s delayed-choice ‡ [email protected] experiment 11 2

B. Wheeler’s delayed-choice experiment with single nickel crystal (Davisson and Germer, 1927) and through particles: Photons and atoms 13 diffraction of helium atoms at a crystal face of lithium C. Wheeler’s delayed-choice experiment with single fluoride (Estermann and Stern, 1930). In 1961, the photons and spacelike separation 14 first nonphotonic double-slit-type experiment was per- IV. Realizations of delayed-choice quantum-eraser formed using electrons (Joensson, 1961). A good decade experiments 15 later, neutron interference (H. Rauch, W. Treimer and A. Photonic quantum erasure 15 U. Bonse, 1974) allowed one to measure the quantum- B. Matter-wave quantum erasure 16 C. Quantum erasure with delayed choice 17 mechanical phase shift caused by the Earth’s gravita- D. Quantum erasure with active and causally tional field (Colella et al., 1975). In modern interfero- disconnected choice 18 metric experiments, the wave nature of molecules of ap- E. Quantum delayed-choice 20 proximately 7000 atomic mass units and 1 pm de Broglie F. Delayed-choice quantum random walk 22 wavelength has been demonstrated (Gerlich et al., 2011). V. Realizations of delayed-choice entanglement-swapping In the language of quantum mechanics, the wave- experiments 22 particle duality is reflected by the superposition princi- A. Delayed entanglement swapping 22 B. Delayed-choice entanglement swapping 23 ple, i.e. the fact that individual systems are described by quantum states, which can be superpositions of dif- VI. Conclusion and outlook 24 ferent states with complex amplitudes. In a Young-type double-slit experiment, every quantum system is at one Acknowledgments 25 point in time in an equal-weight superposition of being References 26 at the left and the right slit. When detectors are placed directly at the slits, the system is found only at one of the slits, reflecting its particle character. At which slit I. INTRODUCTION an individual system is found is completely random. If, however, detectors are not placed at the slits but at a In the 17th century, two different theories of light were larger distance, the superposition state will evolve into developed. While Huygens explained optical phenomena a state which gives rise to an interference pattern, re- by a theory of waves, Newton put forward a corpuscu- flecting the wave character of the system. This pattern lar description where light consists of a stream of fast cannot emerge when the state at the slit would have been particles. At first, the large authority of Newton led to a mere classical mixture of systems being at the left or the general acceptance of the corpuscular theory. How- the right slit. ever, at the beginning of the 19th century, Young demon- To make things more precise, we consider a situa- strated the wave character of light, in particular by show- tion almost equivalent to the double-slit experiment, ing interference fringes in the shadow of a “slip of card, namely quantum systems, e.g. photons, electrons, or about one-thirtieth of an inch in breadth,” formed by the neutrons, which enter a Mach-Zehnder interferometer “portions of light passing on each side” (Young, 1804). (MZI) (Mach, 1892; Zehnder, 1891) via a semitranspar- Many other subsequent experiments further established ent mirror (beam splitter). We denote the transmitted the wave nature of light, in particular the discovery of and reflected arm by b and a, respectively (Fig. 1). Let electromagnetic waves with light being a special case. there be a phase shift ϕ in the reflected arm a, addition- The picture changed again in 1905, when Einstein ex- π ally to a 2 shift due to the reflection. Then the quantum plained the photoelectric effect with his hypothesis that state of the system is a superposition of the two path light consists of “energy quanta which move without states with in general complex amplitudes: splitting and can only be absorbed or produced as a whole” (Einstein, 1905). These massless corpuscles of |ψi = √1 (|bi + i eiϕ |ai). (1) light, called photons, carry a specific amount of energy 2 E = hν with h being Planck’s constant and ν the light’s frequency. In 1909, Taylor performed a low-intensity Whenever one decides to measure through which path the Young-type experiment, measuring the shadow of a nee- system is traveling by putting detectors into the arms a dle with an exposure time of the photographic plate of 3 and b, one will find it in one and only one arm, in agree- months (Taylor, 1909). Despite the feeble light with on ment with its particle character. Until the measurement average less than one photon at a time, the interference the system is considered to be in a superposition of both pattern was observed. paths. The state |ψi determines only the probabilities In 1924, de Broglie postulated that also all massive par- for the respective outcomes a and b. They are given ticles behave as waves (de Broglie, 1924). The wavelength by the squared modulus of the amplitudes and are thus 1 associated with a particle with momentum p is given by pa = pb = 2 . If, however, the two paths are recom- λ = h/p. This wave-particle duality was confirmed ex- bined on a second beam splitter with outgoing paths a0 perimentally through diffraction of an electron beam at a and b0, the quantum state will (up to a global phase) be 3

a mathematical tool for predicting measurement re- b' a sults. Delayed-choice experiments have particularly high- lighted certain peculiarities and non-classical features. j a' In interferometric delayed-choice experiments, the choice b whether to observe the particle or wave character of a quantum system can be delayed with respect to the system entering the interferometer. Moreover, it is possi- FIG. 1 Schematic of a Mach-Zehnder interferometer. A quan- ble to observe a continuous transformation between these tum system enters from the left via a semitransparent beam splitter. When detectors are placed in the paths a and b in- two extreme cases. This rules out the naive classical inside the interferometer, the system is found in one and only terpretation that every quantum system behaves either 1 definitely as a particle or definitely as a wave by adapt- one of the arms with probability 2 each. This reflects the picture that it traveled one of the two paths as a particle. ing a priori to the specific experimental situation. Using If, however, detectors are not placed inside the interferometer multi-partite states, one can decide a posteriori whether 0 0 but at the exit ports a and b after the second beam splitter, two systems were entangled or separable, showing that, the probability of detection depends on the phase ϕ. This re- just as “particle” and “wave”, also “entanglement” and flects the view that the system traveled both paths a and b as a wave, leading to constructive and destructive interference. “separability” are not realistic physical properties carried by the systems. This review is structured as follows: In chapter II, we discuss the history of delayed-choice gedanken exper- transformed into iments, regarding both single (wave-particle duality) and multi-partite (entanglement) scenarios such as delayed- 0 ϕ 0 ϕ 0 |ψ i = cos 2 |a i − sin 2 |b i. (2) choice quantum erasure and entanglement swapping. In chapters III, IV, and V, we review their experimental re- This state gives rise to detection probabilities pa0 = alizations. Chapter VI contains conclusions and an out- 2 ϕ 2 ϕ 0 cos 2 and pb = sin 2 . The ϕ-dependent interference look. Some fractions of this review are based on Ref. (Ma, fringes indicate that the system traveled through the in- 2010) terferometer through both arms, reflecting its wave character. Particle and wave behavior are complimentary (Bohr, 1928) in the sense that they can only be revealed in different experimental contexts and not simultaneously II. DELAYED-CHOICE GEDANKEN EXPERIMENTS (see section II.C). When two physical systems 1 and 2 interact with each A. Heisenberg’s microscope other, they will in general end up in an entangled state, i.e., a (non-separable) superposition of joint states. An The history of delayed-choice gedanken experiments example would be two particles, each of them in a sepa- can be traced back to the year 1927, when Heisenberg rate interferometer: put forward a rudimentary and semiclassical version of the uncertainty relation (Heisenberg, 1927). He visual- iϕ ¯ ized a microscope which is used to determine the position |Ψi12 = cos α |ai1|¯ai2 + sin α e |bi1 b 2 . (3) of an electron. Due to the Abbe limit the accuracy x Here, with probability cos2 α the first system is in path of the position measurement is essentially given by the a in interferometer 1, and the second system is in path wavelength λ of the light used, and since the resolution ¯aof interferometer 2. With probability sin2 α they are gets better with shorter wavelengths, one also often talks in paths b and b,¯ respectively. Again, the superposi- about the “gamma ray microscope”. Considering the mi- tion state “a¯aand bb”¯ is distinctly different in char- croscope’s opening angle ε (see Fig. 2), the laws of optics acter from a classical mixture “a¯aor bb”.¯ Entangle- yield the approximate relation x ∼ λ/ sin ε for the ac- ment can be studied for multi-partite systems, arbi- curacy. For a position measurement, at least one photon trary high-dimensional state spaces, and for mixed states needs to (Compton) scatter from the electron and reach (Horodecki et al., 2009; Pan et al., 2012). Entanglement the observer through the microscope. The momentum also plays a crucial role in Bell tests of local realism transfer depends on the angle of the outgoing photon, (Bell, 1964; Brunner et al., 2013) and it is an essential which is uncertain within ε. The uncertainty of the mo- resource for modern quantum information applications mentum transfer in x direction is thus ηp = sin ε · h/λ (Horodecki et al., 2009; Nielsen and Chuang, 2000). and implies the same uncertainty of the electron momen- Due to the many counter-intuitive features of quantum. The product of position accuracy and momentum tum mechanics, a still heavily debated question is which disturbance reads meaning the quantum state has, in particular whether it is a real physical property or whether it is only x ηp ∼ h. (4) 4

Busch et al., 2013; Ozawa, 2004). In particular, it is now known that Heisenberg’s uncertainty appears in three manifestations, namely (i) for the widths of the position and momentum distributions in any quantum state, (ii) for the inaccuracies of any unsharp joint measurement of both quantities, and (iii) for the inaccuracy of a measurement of one of the quantities and the resulting disturbance in the distribution of the other one (Busch et al., 2007). Note that these manifestations are in close connection to wave-particle duality and complementarity, as they provide partial information about complementary observables.

B. von Weizsäcker, Einstein, Hermann FIG. 2 Heisenberg’s microscope drawing from the notes of his 1929 lectures, printed in 1930. A photon with energy hν In 1931, von Weizsäcker gave a detailed account of is scattered at an electron (represented by a dot) and reaches Heisenberg’s thought experiment (Weizsäcker, 1931). He the observer via a microscope with opening angle ε. The remarked that one can place the observer not in the im- product of uncertainty of the position measurement and the momentum disturbance is of the order of Planck’s constant age plane, as originally intended, but in the focal plane h. Figure taken from Ref. (Heisenberg, 1991). of the microscope. This constitutes a measurement not of the electron’s position but of its momentum. A small but conceptually very important step was made by Ein- stein (for a similar type of experiment) and Hermann (for For Heisenberg, this mathematical relation was a “direct the Heisenberg microscope), who made explicit the pos- illustrative explanation” (Heisenberg, 1927) of the quan- sibility to delay the choice of measurement after the rel- tum mechanical commutation relation [ˆx, pˆx] = i~ for the evant physical interaction had already taken place (Ein- position and momentum operator. He noted (Heisenberg, stein, 1931; Hermann, 1935). This paved the path for the 1991) (translated from German) paradigm of delayed-choice experiments. In an article on the interpretation of quantum mechanics, von Weizsäcker that every experiment, which for instance al- wrote (Weizsäcker, 1941) (translated from German, ital- lows a measurement of the position, necessar- ics in the original): ily disturbs the knowledge of the velocity to a certain degree It is not at all the act of physical interaction In the subsequent years, Heisenberg’s uncertainty prin- between object and measuring device that de- ciple was derived accurately within the formalism of fines which quantity is determined and which quantum mechanics (Kennard, 1927; Robertson, 1929; remains undetermined, but the act of notic- Schrödinger,1930; Weyl, 1928). However, the resulting ing. If, for example, we observe an electron famous inequality with initially known momentum by means of a single photon, then we are in principle able, ∆x ∆p ≥ ~/2 (5) after the photon has traversed the lens, therefore certainly not interacting with the elec- acquired a different meaning, as it did not involve the tron any more, to decide, whether we move a notion of disturbance any longer: ∆x and ∆p are the photographic plate into the focal plane or the standard deviations of position and momentum for an en- image plane of the lens and thus determine semble of identically prepared quantum systems. These the momentum of the electron after the ob- quantities can be inferred by measuring either the posi- servation or its position. For here the physical tion or the momentum of every system. No sequential or “disturbance” of the photon determines the joint measurements are made. Every quantum state pre- description of the state of the electron, which dicts intrinsic uncertainties, which cannot be overcome, is related to it not any more physically but that is, which necessarily fulfill Heisenberg’s uncertainty only via the connection of the state probabil- relation (5). Experimental observations of ensembles of ities given in the wave function, the physical identically prepared systems confirm these predictions. influence is apparently merely important as More recently, Heisenberg’s original derivation in the technical auxiliary means of the intellectual sense of an error-disturbance relation has been revis- act of constituting a well-defined observation ited in fully quantum mechanical terms (Branciard, 2013; context. 5

C. Bohr’s account

We briefly review Bohr’s viewpoint on complementarity, measurement and temporal order in quantum experiments. Already in 1928, Bohr said about the requirement of using both a corpuscular and a wave description for electrons that “we are not dealing with contradictory but complementary pictures of the phenomena” (Bohr, 1928). In his “Discussion with Einstein on epistemologi- cal problems in atom physics” (Bohr, 1949) he wrote: Consequently, evidence obtained under different experimental conditions cannot be com- prehended within a single picture, but must be regarded as complementary in the sense that only the totality of phenomena exhausts FIG. 3 The quantum “phenomenon” can be viewed as a the possible information about the objects. “great smoky dragon”. Figure taken from Ref. (Miller and Wheeler, 1983). In other words, ”in quantum theory the information provided by different experimental procedures that in principle cannot, because of the physical character of the fects obtainable by a definite experimental ar- needed apparatus, be performed simultaneously, cannot rangement, whether our plans for construct- be represented by any mathematically allowed quantum ing or handling the instruments are fixed be- state of the system being examined. The elements of forehand or whether we prefer to postpone information obtainable from incompatible measurements the completion of our planning until a later are said to be complementary” (Stapp, 2009). moment when the particle is already on its The term “phenomenon” is defined by Bohr as follows way from one instrument to another. In the (Bohr, 1949): quantum-mechanical description our freedom of constructing or handling the experimental As a more appropriate way of expression I arrangement finds its proper expression in the advocated the application of the word phe- possibility of choosing the classically defined nomenon exclusively to refer to the observa- parameters entering in any proper application tions obtained under specified circumstances, of the formalism. including an account of the whole experimental arrangement. Therefore, in the language of Heisenberg, von Miller and Wheeler vividly illustrated the concept of Weizsäcker, and Bohr – the main proponents of the “elementary quantum phenomenon” in a cartoon shown Copenhagen interpretation of quantum mechanics – the in Fig. 3. The sharp tail and head of a dragon corre- observer is free to choose at any point in time, even af- spond to Bohr’s “specified circumstances” (the experi- ter physical interactions have been completed, the fur- mental preparation and arrangement) and the result of ther classical conditions of the experiment. This decision, the observation (the outcome of the experiment), respec- e.g., the positioning of the detector in the focal or image tively. The body of the dragon, between its head and plane in the Heisenberg microscope experiment, defines tail, is unknown and smoky: “But about what the dragon which particular of the complementary observables is de- does or looks like in between we have no right to speak, termined. either in this or any delayed-choice experiment. We get a counter reading but we neither know nor have the right to say how it came. The elementary quantum phenomenon D. Wheeler’s delayed-choice wave-particle duality is the strangest thing in this strange world.”(Miller and gedanken experiment Wheeler, 1983) Already in his response to the Einstein-Podolsky- The paradigm of delayed-choice experiments was re- Rosen (EPR) argument (Einstein et al., 1935), Bohr vived by Wheeler in Ref. (Wheeler, 1978) and a series stresses the experimenter’s “freedom of handling the of works between 1979 and 1981 which were merged in measuring instruments” (Bohr, 1935). Later, he wrote Ref. (Wheeler, 1984). To highlight the inherently non- (Bohr, 1949): classical principle behind wave-particle complementarity, he proposed a scheme shown at the top in Fig. 4, where It may also be added that it obviously can one has a Mach-Zehnder interferometer and a single- make no difference as regards observable ef- photon wave packet as input. After the first half-silvered 6

FIG. 4 Wheeler’s delayed-choice gedanken experiment with a single-photon wave packet in a Mach-Zehnder interferometer. 1 FIG. 5 Delayed-choice gedanken experiment at the cosmolog- Top: The second half-silvered mirror ( 2 S) of the interferometer can be inserted or removed at will. Bottom left: When ical scale. Left: Due to the gravitational lens action of galaxy 1 G-1, light generated from a quasar (Q) has two possible paths 2 S is removed, the detectors allow one to determine through which path the photon propagated. Which detector fires for to reach the receptor. This mimics the setup in Fig. 4. Center: an individual photon is absolutely random. Bottom right: The receptor setup. Filters are used to increase the coherence 1 length of the light, thus allowing to perform the interference When 2 S is inserted, detection probabilities of the two detectors depend on the length difference between the two arms. experiment. A fiber optics delay loop adjusts the phase of the Figure taken from Ref. (Wheeler, 1984). interferometer. Right: The choice to not insert (top) or insert (bottom) the half-silvered mirror at the final stage of the experiment, allows to either measure which particular route the light traveled or what the relative phase of the two routes mirror (beam splitter) on the left, there are two possi- was when it traveled both of them. Given the distance be- ble paths, indicated by ‘2a’ and ‘2b’. Depending on the tween the quasar and the receptor (billions of light years), the choice made by the observer, different properties of the choice can be made long after the light’s entry into the inter- photon can be demonstrated. If the observer chooses to ferometer, an extreme example of the delayed-choice gedanken experiment. Figure taken from Ref. (Wheeler, 1984). reveal the its particle nature, he should not insert the 1 second half-silvered mirror ( 2 S), as shown at the bottom left in Fig. 4. With perfect mirrors (A and B) and 100% detection efficiency, both detectors will fire with equal on what we have a right to say about the already past probabilities but only one will fire for every individual history of that photon.” And: “Thus one decides whether photon and that event will be completely random. As the photon ‘shall have come by one route or by both Wheeler pointed out, “[...] one counter goes off, or the routes’ after it has already done its travel” (Wheeler, other. Thus the photon has traveled only one route” 1984). Very much along the line of the reasoning of Bohr, (Wheeler, 1984). one can talk only about a property of the quantum sys- On the other hand, if the observer chooses to demon- tem, for example, wave or particle, after the quantum strate the photon’s wave nature, he inserts the beam phenomenon has come to a conclusion. In the situation 1 splitter 2 S as shown at the bottom right in Fig. 4. For just discussed, this is only the case after the photon has identical beam splitters and zero path difference (or an completely finished its travel and has been registered. integer multiple of the photon wavelength), only the de- Illustrated in Fig. 5, Wheeler proposed a most dra- tector on the bottom right will fire. As Wheeler pointed matic “delayed-choice gedanken experiment at the cos- out: “This is evidence of interference [...], evidence that mological scale” (Wheeler, 1984). He explained it as fol- each arriving light quantum has arrived by both routes” lows: (Wheeler, 1984). One might argue that whether the single-photon wave We get up in the morning and spend the packet traveled both routes or one route depends on day in meditation whether to observe by whether the second half-silvered mirror is inserted or not. “which route” or to observe interference be- In order to rule out naive interpretations of that kind, tween “both routes.” When night comes and Wheeler proposed a “delayed-choice” version of this ex- the telescope is at last usable we leave the periment, where the choice of which property will be ob- half-silvered mirror out or put it in, according served is made after the photon has passed the first beam to our choice. The monochromatizing filter splitter. In Wheeler’s words: “In this sense, we have a placed over the telescope makes the counting strange inversion of the normal order of time. We, now, rate low. We may have to wait an hour for by moving the mirror in or out have an unavoidable effect the first photon. When it triggers a counter, 7

we discover “by which route” it came with delayed-choice wave-particle duality experiment are (1) one arrangement; or by the other, what the a free or random choice of measurement with space-like relative phase is of the waves associated with separation between the choice and the entry of the quan- the passage of the photon from source to re- tum system into the interferometer, and (2) using single- ceptor “by both routes”–perhaps 50,000 light particle quantum states. years apart as they pass the lensing galaxy G-1. But the photon has already passed that galaxy billions of years before we made our E. Delayed-choice quantum erasure decision. This is the sense in which, in a loose way of speaking, we decide what the Scully and collaborators proposed the so-called quan- photon “shall have done” after it has “al- tum eraser (Scully and Drühl, 1982; Scully et al., 1991), ready” done it. In actuality it is wrong to talk in which an entangled atom-photon system was studied. of the “route” of the photon. For a proper They considered the scattering of light from two atoms way of speaking we recall once more that it located at sites 1 and 2 and analyzed three different cases makes no sense to talk of the phenomenon (Fig. 6): until it has been brought to a close by an A. A resonant light pulse l1 impinges on two two-level irreversible act of amplification: “No elemen- atoms (Fig. 6A) located at sites 1 and 2. One of the tary phenomenon is a phenomenon until it is two atoms is excited to level a and emits a photon la- a registered (observed) phenomenon.” beled γ, bringing it back to state b. As it is impossible to know which atom emits γ, because both atoms are Given the distance between the quasar and the recep- finally in the state b, one obtains interference of these tor (billions of light years), the choice is made by the ex- photons at the detector. This is an analog of Young’s perimenter long after the photon’s entry into the cosmic double-slit experiment. interferometer (i.e. emission by the quasar). The speed of light of intergalactic space is not exactly the vacuum B. In the case of three atomic levels (Fig. 6B), the reso- speed of light. Therefore, whether the experimenter’s nant light l1 excites the atoms from the ground state choice is in the time-like future of the emission event or c to the excited state a. The atom in state a can space-like separated therefrom depends on the size of the then emit a photon γ and end up in state b. The interferometer and the amount of time between the choice other atom remains in level c. This distinguishabil- event and the photon arrival at the second beam splitter. ity of the atoms’ internal states provides which-path Depending on the specific parameters, Wheeler’s delayed information of the photon and no interference can be choice can thus be thought of being in the time-like future observed. of, or space-like separated from, the photon emission. C. An additional light pulse l2 takes the atom from level While Wheeler did not specifically discuss the latter b to b’ (Fig. 6C). Then a photon labeled φ is emitted case, it is particularly appealing because it rules out any and the atom ends up in level c. Now the final state of causal influence from the emission to the choice which both atoms is c, and thus the atoms’ internal states might instruct the photon to behave as a particle or cannot provide any which-path information. If one as a wave. Note that this resembles the freedom-of- can detect photon φ in a way that its spatial origin choice loophole (Bell, 2004; Gallicchio et al., 2014; Scheidl (thus which-path information of γ) is erased, inter- et al., 2010) discussed in the context of Bell tests for the ference is recovered. Note that in this case, there falsification of hidden variable theories using entangled are two photons: One is γ for interference, the other states of at least two systems. The question in Wheeler’s one is φ, acting as a which-path information carrier. gedanken experiment is about if and when a single quan- (This resembles closely von Weizsäcker’s account of tum system decides to behave as a particle or as a wave. Heisenberg’s microscope.) Space-like separation excludes unknown communication from this decision to the choice of the experimenter. Scully and Drühldesigned a device based on an electro- Although Wheeler suggested employing (thermal) light optical shutter, a photon detector, and two elliptical cavi- from a quasar, it is conceptually important to use true ties to implement the above described experimental con- single photons rather than thermal light. This is because figuration C in a delayed-choice arrangement (Fig. 7). the indivisible particle nature of single photons guaran- There, one can choose to reveal or erase the which-path tees that the two detectors will never click at the same information after the photon γ has been generated. time. Otherwise, one could explain the results by what In another proposal (Scully et al., 1991), the interfer- is often called a semi-classical theory of light, where light ing system is an atomic beam propagating through two propagates as a classical wave and is quantized only at cavities coherently. The atomic state is the quantum superposition √1 (|ei + |ei ), where |ei denotes the ex- the detection itself (Paul, 1982). 2 1 2 i Therefore, important requirements for an ideal cited state of the atom passing through cavity i = 1, 2 2210 MARLAN 0. SCULLY AND KAI DRUHL 25 are sure I c ) states so that they no longer orthogonal. that the ith atom has decayed to the I c ) state If we could do this, then we would have an in- via the emission of a photon which we designate as teresting situation. That is, the photons could be ((i; The exact specification of the state is y I ). I P; ) the e. far removed the same is well on their way to detector (i. , as that of the I y) photon state, i.e., from atoms 1 and 2) and the fringes made to ap- given by Eq. {4) with the obvious changes in wave pear or not depending on what we do with the vector, and decay rates, etc. The state vector atoms long after the y emission has taken place. describing the experimental arrangement after P However, upon carrying out the appropriate calcu- emlsslon now 1eads lation (including effects of the second laser pulse c2&( . I 6&= I di& I 42& I)'2&) l2) we find that the field correlation function [Eq. lci I)'i&+ (5b)] is changed only in that the atomic state inter- Consider next an experimental arrangement product now becomes which, in effect, allows us to "reduce" the photon states and to the vaccum with the exci- 8 ) I &i U'U I Pi $2) i c2 lci b2& (bl c2 I lci»2& tation of a common photodetector. ' In order to The time evolution operator U describes the in- accomplish this we place the scattering atoms in a teraction of the second pulse as it mixes states b elliptical cavity' as in The cavi- I ) particular Fig. 2. and c Thus if our time development matrix U is taken be the radiation asso- I ). ty to transparent to is unitary, U U =1, and we see that we have not ciated with the I&, I2, and y radiation, but to be succeeded in producing fringes by applying the highly reflecting in the case of the P photons. second pulse. Yet one wonders if some other This is possible since the frequency of the (() radia- scheme designed to retrieve the interference fringes tion is different from that of the l&, l2, and y light. might not work. After all the presence of the in- Since atoms 1 and 2 are located at the foci of the formation contained in our three level atom is very two ellipses all the P radiation leaving atoms 1 and analogous to having information stored in the form 2 is focused to their common foci, where we place of an observation, and we know that the process of FIG.a photodetector, 6 (Color online).see TheFig. delayed-choice2. quantum eraser following Ref. (Scully and Drühl,1982). In A, two two-level atoms observation changes the state vector in a nonuni- are initiallyThe photodetection in the state b. Theof iI'i incidentphotons pulse(at p,l1r)excitesfol- one of the two atoms to state a from where it decays to state b, emitting a photon labeled γ. Because the final states of both atoms are identical, one can observe interference of the photons at the tary fashion. More pictorally the question may lowed by detection of y radiation (at r, t) is detector D. In B, two atoms are initially in the ground state c and one of them is excited by the pulse l to state a from where well be asked "can we erase the information described by the intensity correlation function 1 it decays to state b. Since the final states of both atoms are different, one cannot observe interference of the photons. In C, (memory) locked in our atoms and thus recover a fourth level is added. A pulse l2 excites the atom from state b to b’. The atom in b’ emits a photon labeled φ and ends up fr1nges? in state c. If one can detect φ in a way that the which-path information is erased, interference can be recovered for photon γ. Motivated by these considerations let us consider Figure taken from Ref. (Aharonov and Zubairy, 2005). the following information eraser: allo~ our atoms 1 and 2 to take on a slightly more involved level structure involving four relevent levels as depicted in Fig. 1(d). The second laser pulse l2 is tuned so ETECTOR as to be resonant with the b~b' transition and ELECTRO-OPTI tailored such that it transfers 100% of the popula- SHUTTER tion from b to b'& That is, in .the of I ) I jargon quantum optics, let the second laser pulse be a m pulse. Such a pulse is defined' by the requirement that the integrated amplitude of the laser pulse en- velope be such that 4 DETECTOR Pbb I Ch'8'(r')lA=~, (7) wherep~b is the dipole matrix element connecting b b') The the I ) and I states. point is that such a will take . atom it encounters in b FIG. 2. Laser pulses I~ and lq incident on atoms at appulse every I ) FIG. 7 Proposed delayed-choice quantum-eraser setup in to b'). Hence, the state of the system after in- Ref.sites (Scully1 and and2. Drühl,1982);Scattered photons figureyl and takeny& result therefrom.from Laser I a~b transition. Decay of atoms from b'~e results in FIG. 8 Proposed quantum-eraser setup in Ref. (Scully et al., teracting with the l2 pulse is pulses l1 and l2 are incident on atoms at sites 1 and 2. A photon emission. E11ipticai cavities reflect photons scatteredP photon, γ1 or γ2, is generated by a →Pb atomic tran- 1991); figure taken therefrom. A detector wall, separating two Electro-optic0 shutter = & & onto common photodetector. cavities for microwave photons, is sandwiched by two electro- I 6& I bi, c2 I)'i &+ I ci,b2 I r~& sition. The atom’s decay from b → c produces a photon φ. Thistransmits correspondsP photons to theonly situationwhen switch depictedis open. in Fig.Choice 6C. In or- optic shutters. a. By always opening only one of the shut- But, as indicated in Fig. 1(d), b') is strongly cou- I derof toswitch operateposition this experimentdetermines inwhether a delayed-choicewe emphasize mode, two ters, the photon detections reveal the cavity where the photon pled to c so that after a short time we may be particle or wave nature of photons. I ), elliptical cavities and an electro-opticaly shutter are employed. was emitted and thus, which-path information for the atoms. The cavities reflect φ onto a common detector. The electro- Consequently, no interference pattern emerges. When, open- optical shutter transmits φ only when the switch is open. The ing both shutters the photon detections will erase which-path choice of open or closed shutter determines whether or not the information of the atoms and interference shows up.© 19b.9 Both1 Nature Publishing Group information which atom (1 or 2) emitted the photon is erased. shutters are open. It is assumed that the detector wall can This determines whether one can observe the wave or particle only be excited by the symmetric photon state |+i12. Hence, nature of γ. The choice can be delayed with respect to the if a photon is being emitted in one of the cavities but not generation of γ. detected, it was in the antisymmetric state |−i12. The detections of the symmetric and antisymmetric photon state give rise to oppositely-modulated interference fringes of the atoms (solid and dashed curve), respectively. (see Fig. 8). The excited atom can decay to its ground state |gii and emit a photon in state |γii. In conjunction with the perfectly reflective shutters, the two cavities, separated by a photon detector wall, are used to trap If shutter 1 is open and shutter 2 is closed, detection of a the photon. Conditional on the emission of one photon photon (in cavity 1) reveals the atom’s position in cavity γ from the atom in one of the cavities, the state of atom 1, and vice versa if shutter 2 is open while 1 is closed. (a) and photon (p) becomes: Repeating experiments with these two configurations (i.e. only one shutter open) will not lead to an interference |Φi = √1 (|gi |γi + |gi |γi ). (6) ap 2 1 1 2 2 pattern of the atom detections (dashed curve in Fig. 8a). 9

The same pattern will emerge when both shutters remain authors considered the quantum (system-apparatus) cor- closed at all times. The lack of interference in both cases relations which are at the root of decoherence rather than is because the which-path information is still present in the recoil or collision. This topic will be further discussed the universe, independent of whether an observer takes in Chapters III and IV. note of it or not. Ignoring the photon state, which carries which-path information about the atom, leads to a mixed 1 F. Delayed-choice entanglement swapping state of the atom of the from 2 (|gi1hg| + |gi2hg|) which cannot show an interference pattern. However, the state (6) can also be written as When two systems are in an entangled quantum state, the correlations of the joint system are well defined but 1 1 not the properties of the individual systems (Einstein |Φiap = 2 (|gi1 + |gi2)|+i12 + 2 (|gi1 − |gi2)|−i12, (7) et al., 1935; Schrödinger,1935). Peres raised the question with the symmetric and antisymmetric photon states of whether it is possible to produce entanglement between |+i = √1 (|γi + |γi ) and |−i = √1 (|γi − |γi ). If 12 2 1 2 12 2 1 2 two systems even after they have been registered by de- one opens both shutters and detects the symmetric pho- tectors (Peres, 2000). Remarkably, quantum mechanics ton state |+i12, one cannot in principle distinguish which allows this via entanglement swapping (Zukowski et al., cavity the atom propagated through as its state is the 1993). We note that Cohen had previously analyzed a coherent superposition √1 (|gi + |gi ). Detection of the similar situation in the context of counterfactual entan- 2 1 2 glement generation in separable states (Cohen, 1999). photon in the state |+i12 has erased the which-path information of the atom. Therefore, interference in the atom In a photonic implementation of entanglement swap- detections shows up again (solid curve in Fig. 8a and b). ping, two pairs of polarization-entangled photons, 1&2 and 3&4, are produced from two different EPR sources If one detects the antisymmetric photon state |−i12, the atomic state becomes a superposition with a different rel- (Fig. 9). The initial four-photon entangled state is, e.g., ative phase between the two paths, √1 (|gi − |gi ), lead- of the form 2 1 2 ing to a shift of the interference pattern (dashed curve in |Ψi = |Ψ−i |Ψ−i , (8) Fig. 8b). In Ref. (Scully et al., 1991) it was assumed that 1234 12 34 the detector has perfect detection efficiency but cannot − 1 where |Ψ iij = √ (|Hii|Vij − |Vii|Hij) are the entan- be excited by the antisymmetric photon state, which is 2 gled antisymmetric Bell (singlet) states of photons i and why the shifted interference pattern emerges in the case j. H and V denote horizontal and vertical polarization, of both shutters being open and no eraser photon being respectively. While photon 1 is sent to Alice and photon detected. The detector wall used here is sufficiently thin 4 is sent to Bob, photons 2 and 3 propagate to Victor. such that it cannot distinguish which side the photon has Alice and Bob perform polarization measurements on impinged on, and hence is able to collapse the photons’ photons 1 and 4, choosing freely between the three mutu- superposition states into the symmetric or antisymmetric ally unbiased bases (Wootters and Fields, 1989) |Hi/|Vi, state. It is important to note that the interference pat- |Ri/|Li, and |+i/|−i, with |Ri = √1 (|Hi + i|Vi), |Li = terns of the atoms can only be seen in coincidence with 2 √1 (|Hi − i|Vi), and |±i = √1 (|Hi ± |Vi). If Victor the corresponding photon projections into the symmetric 2 2 or antisymmetric states. chooses to measure his two photons 2 and 3 separately This gedanken experiment triggered a controversial in the H/V basis, i.e. in the basis of separable (product) discussion on whether complementarity is more funda- states |Hi2|Hi3, |Hi2|Vi3, |Vi2|Hi3, and |Vi2|Vi3, then mental than the uncertainty principle (Englert et al., the answer of the experiment is one of the four random 1995; Storey et al., 1994, 1995). Wiseman and colleagues results. Upon Victor’s measurement, also photons 1 and reconciled divergent opinions and recognized the novelty 4 will remain separable and be projected into the cor- of the quantum eraser concept (Wiseman and Harrison, responding product state |Vi1|Vi4, |Vi1|Hi4, |Hi1|Vi4, 1995; Wiseman et al., 1997). Experimental demonstra- or |Hi1|Hi4, respectively. Alice’s and Bob’s polarization tions of quantum erasure for atomic systems have been measurements are thus only correlated in the |Hi/|Vi ba- realized in Refs. (Eichmann et al., 1993) and (Dürr et al., sis. 1998), and will be reviewed in Section IV B. However, the state (8) can also be written in the basis A delayed-choice configuration can be arranged in this of Bell states of photons 2 and 3: experiment: one can choose to reveal or erase the which- 1 + + − − path information of the atoms (by not opening or opening |Ψi1234 = 2 (|Ψ i14|Ψ i23 − |Ψ i14|Ψ i23 + + − − both shutters) after the atom finishes the propagation − |Φ i14|Φ i23 − |Φ i14|Φ i23, (9) through the two cavities. A detailed analysis of the fundamental aspects of with the entangled symmetric Bell (triplet) states |Ψ+i = √1 (|Hi |Vi +|Vi |Hi , |Φ±i = √1 (|Hi |Hi ± single-particle interference experiments facing decoher- ij 2 i j i j ij 2 i j ence has been reported in Ref. (Scully et al., 1989). The |Vii|Vij. When Victor decides to perform a Bell-state 10

Alice Victor Bob Peres suggested an addition to the entanglement- swapping protocol, thereby combining it with Wheeler’s |-ñ |ñV |ñV |ñR delayed-choice paradigm. He proposed that the correla- - |ñV |ñF |ñV tions of photons 1 and 4 can be deﬁned even after they |ñL |ñH |ñH |+ñ |-ñ |ñH |ñV |ñH have been detected via a later projection of photons 2 - |ñL |ñY |ñR |-ñ |ñF+ |-ñ and 3 into an entangled state. According to Victor’s |ñR |ñV |ñH |ñV - choice and his results, Alice and Bob can sort their al- |-ñ |ñF |ñL |ñ+ |ñH |ñH |ñ- ready recorded data into subsets and can verify that each |ñH |ñV |ñV |ñH subset behaves as if it consisted of either entangled or |ñR |ñF+ |ñL |-ñ |ñY+ |-ñ separable pairs of distant photons, which have neither |ñV |ñH |ñV |ñR communicated nor interacted in the past. Such an experiment leads to the seemingly paradoxical situation, that “entanglement is produced a posteriori, after the Entangled- or Separable-state entangled particles have been measured and may even Measurement no longer exist” (Peres, 2000). Since the property whether the quantum state of photons 1 and 4 is separable or entangled, can be freely decided by Victor’s choice of applying a separable-state or Polarization Polarization Measurement Measurement Bell-state measurement on photons 2 and 3 after photons 1 and 4 have been already measured, the delayed-choice 1 2 3 4 wave-particle duality of a single particle is brought to an

EPR Source I EPR Source II entanglement-separability duality of two particles.

FIG. 9 The concept of delayed-choice entanglement swapping. Two entangled pairs of photons 1&2 and 3&4 are pro- − − duced, e.g., in the joint state |Ψ i12|Ψ i34 from the EPR G. Quantum delayed-choice sources I and II, respectively. Alice and Bob perform polarization measurements on photons 1 and 4 in any of the three mutually unbiased bases and record the outcomes. Vic- Wheeler’s delayed choice experiment [redrawn in Fig. tor has the freedom of either performing an entangled- or 10(a)] of a photon in an interferometer with phase ϕ separable-state measurement on photons 2 and 3. If Victor can be translated into the language of quantum circuits decides to perform a separable-state measurement in the four- (Nielsen and Chuang, 2000), where Hadamard gates rep- dimensional two-particle basis {|Hi2|Hi3, |Hi2|Vi3, |Vi2|Hi3, resent the beam splitters and an ancilla is used in a quan- |Vi |Vi }, then the outcome is random and one of these four 2 3 tum random number generator (QRNG) for making the product states. Photons 1 and 4 are projected into the corresponding product state and remain separable. On the other choice [Fig. 10(b)]. A quantum version of this experiment hand, if Victor chooses to perform an entangled-state mea- was suggested (Ionicioiu and Terno, 2011), where the an- + surement on photons 2 and 3 in the Bell-state basis {|Ψ i23, cilla can coherently control the second beam splitter of − + − |Ψ i23, |Φ i23, |Φ i23}, then the random result is one of the interferometer [Fig. 10(c)]. Bias can be achieved by the four Bell states. Consequently, photons 1 and 4 are also more general ancilla states cos α |0i + sin α |1i with am- projected into the corresponding Bell state. Therefore, en- plitudes depending on a parameter α [Fig. 10(d)]. By tanglement is swapped from pairs 1&2 and 3&4 to pairs 2&3 this, the second beam splitter can be in a superposition and 1&4. Figure adapted from Ref. (Ma et al., 2012). of being present and absent. Following the language of Wheeler, the photon must consequently be in a superposition of particle and wave at the same time. Moreover, measurement, i.e. when he measures in the basis of en- + − + − one can arbitrarily choose the temporal order of the mea- tangled states |Ψ i23, |Ψ i23, |Φ i23, and |Φ i23, then surements. In particular, if one measures the ancilla af- the answer of the experiment is one of the four random ter the photon, the latter can be described as having results. Alice’s and Bob’s photons are then projected + − behaved as a particle or as a wave after it has been al- into the corresponding entangled state |Ψ i14, |Ψ i14, + − ready detected. From the experimental point of view, it |Φ i14, or |Φ i14, respectively. Alice’s and Bob’s polar- is advantageous that no fast switching of any devices is ization measurements are thus correlated in all possible required. bases. This implies that Victor can establish entanglement between photons 1 and 4, although they have never With an appropriate alignment of the interferometer, interacted nor share any common past. After entangle- before the detectors in Fig. 2(d) the state of the photon ment swapping, pairs 1&2 and 3&4 are no longer entan- and the ancilla reads gled, obeying the monogamy of entanglement (Coﬀman et al., 2000). |Ψi = cos α |particlei |0i + sin α |wavei |1i , (10) week ending PRL 107, 230406 (2011) PHYSICAL REVIEW LETTERS 2 DECEMBER 2011

Proposal for a Quantum Delayed-Choice Experiment

Radu Ionicioiu1,2,3 and Daniel R. Terno4,5 1Centre for Quantum Science and Technology, Macquarie University, Sydney New South Wales 2109, Australia 2Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada 3Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada 4Department of Physics & Astronomy, Macquarie University, Sydney New South Wales 2109, Australia 5Centre for Quantum Technologies, National University of Singapore, Singapore 117543 (Received 29 March 2011; published 2 December 2011) Gedanken experiments help to reconcile our classical intuition with quantum mechanics and nowadays are routinely performed in the laboratory. An important open question is the quantum behavior of the controlling devices in such experiments. We propose a framework to analyze quantum-controlled experiments and illustrate it by discussing a quantum version of Wheeler’s delayed-choice experiment. Using a quantum control has several consequences. First, it enables us to measure complementary phenomena with a single experimental setup, pointing to a redeﬁnition of complementarity principle. Second, it allows us to prove there are no consistent hidden-variable theories having ‘‘particle’’ and ‘‘wave’’ as realistic properties. Finally, it shows that a photon can have a morphing behavior between particle and wave. The framework can be extended to other experiments (e.g., Bell inequality).

DOI: 10.1103/PhysRevLett.107.230406 PACS numbers: 03.65.Ta, 03.65.Ud, 03.67.a

Wave-particle duality, a quintessential property of quan- setup and photon’s behavior: the photon should not tum systems, defies our classical intuition. In the context of ‘‘know’’ beforehand if it has to behave like a particle or the double-slit experiment, duality played a central role in like a wave. The choice of inserting or removing BS2 is the famous Bohr—Einstein debate and prompted Bohr to classically controlled by a random number generator. formulate the complementarity principle [1]: ‘‘the study of In this article we examine what happens if we replace complementary phenomena demands mutually exclusive this classical control with a quantum device. This enables experimental arrangements.’’ Classical concepts like par- us to extend Wheeler’s gedanken experiment to a quantum ticle or wave (as in ‘‘wave-particle duality’’) do not trans- delayed choice. Quantum elements in various experimental 11 late perfectly into the quantum language. For example, although we observe interference (a definite wavelike be- (a) 0 (b) D A hidden-variable based analysis of quantum delayed- havior), the pattern is produced click-by-click, in a dis- ϕ ϕ 1 〉 b D choice experiments needs to describe the entire (entan- crete, particlelike manner [2]. Notwithstanding this |0 H H 1 BS gled) system of photon and ancilla. It was argued that ambiguity, and with this proviso, we adopt as operational 2 b=0,1 QRNG 〉 quantum delayed-choice experiments without space-like definition of ‘‘wave’’ or ‘‘particle’’ to stand for ‘‘ability’’ or |0 H QRNG separation between system photon and ancilla are equiv- ‘‘inability’’ to produce interference [3]. BS 0 1 alent to classical delayed-choice experiments with space- A good illustration of wave-particle complementarity is (c) (d) like separation (Céleri et al., 2014). The continuous mor- given by a Mach-Zehnder interferometer (MZI), Fig. 1.A ϕ ϕ D D phing behavior predicted by quantum mechanics in quan- photon is first split by beam splitter BS1, travels inside an |0〉 |0> H H H H tum delayed-choice experiments cannot be described by interferometer with a tunable phase shifter ’, and is finally 〉 D’ α|0〉 + α|1〉 D’ hidden-variable theories for the system photon and the recombined (or not) at a second beam splitter BS2 before |0 H cos sin ancilla, which obey objectivity (“particle” and “wave” detection. If the second beam splitter is present we observe are intrinsic attributes of the system photon during its interference fringes, indicating the photon behaved as a FIG. 1 (color online). (a) In the classical delayed-choice ex- FIG. 10 (Color online). (a) The ‘classical’ delayed-choice ex- lifetime), determinism (the hidden variables determine wave, traveling both arms of the MZI. If BS2 is absent, we periment the second beam splitter is inserted or removed ran- 1 the individual outcomes), and independence (the hidden randomly register, with probability , a click in only one of domlyperiment: after The the second photon beam is already splitter inside BS2 theis inserted interferometer. or not 2 (b)after The the equivalent photon has quantum already network.entered the An interferometer. ancilla (red line), (b) variables do not depend on the experimental setting, i.e. the two detectors, concluding that the photon travelled 1 initiallyAn equivalent prepared quantum in the state network:jþi ¼ Anpffiffi ðj ancilla0iþj1 (loweriÞ then input, measured, red) the choice of α) (Ionicioiu et al., 2014). Moreover, these along a single arm, showing particle properties. acts as a quantum random number2 generator (QRNG). Its acts as a quantum random number generator (QRNG). three assumptions are indeed incompatible with any the- This contradictory behavior prompted Wheeler to for- initial state |0i is transformed by a Hadamard gate H into (c) Delayed√ choice with a quantum beam splitter. The classical ory, not only quantum mechanics (Ionicioiu et al., 2015). mulate the delayed-choice experiment [4–8]. In Wheeler’s (|0i + |1i)/ 2. A measurement in the computational |0i / |1i control (red double line) after the measurement of the ancilla in delayed-choice experiment one randomly chooses whether basis gives a random outcome, which determines whether or (b)not is the equivalent second toHadamard a quantum gate control is applied before the with measurement; the system or not to insert the second beam splitter when the photon is the second beam splitter BS2 is now in superposition of present qubit (equivalent to BS2) is applied. (c) Delayed choice with III. REALIZATIONS OF DELAYED-CHOICE already inside the interferometer and before it reaches BS2 and absent, equivalent to a controlled-Hadamard CðHÞ gate. a quantum beam splitter: The second beam splitter BS2 (rep- WAVE-PARTICLE DUALITY EXPERIMENTS [Fig. 1(a)]. The rationale behind the delayed choice is to (d)resented We bias by the a controlled QRNG by Hadamard preparing the gate) ancilla is coherently in an arbitrary con- avoid a possible causal link between the experimental statetrolledcos byj the0iþ statesin ofj1i the. ancilla qubit. It is now in superpo- A. First realizations of Wheeler’s delayed-choice sition of present and absent. (d) The QRNG can be biased by experiment = = = preparing the ancilla inÓ the state cos α |0i + sin α |1i. Figure 0031-9007 11 107(23) 230406(5) 230406-1taken from Ref. (Ionicioiu2011 and Terno, American 2011). Physical Society Inspired by Wheeler’s gedanken experiment, there have been several concrete experimental proposals and with the photon states analyses for different physical systems, including neutron interferometers (Greenberger et al., 1983; Miller, |particlei = √1 (|0i + eiϕ |1i), (11) 2 1983; Miller and Wheeler, 1983) and photon interferom- iϕ/2 ϕ ϕ eters (Alley et al., 1983; Mittelstaedt, 1986). Pioneering |wavei = e (cos 2 |0i + i sin 2 |1i). (12) endeavors in realizing these experiments have been re- The overlap between the latter states is hparticle|wavei = ported in Refs. (Alley et al., 1986; Hellmuth et al., 1987; 2−1/2 cos ϕ. As ϕ varies, the probability to find the pho- Schleich and Walther, 1986). 1 ton in state 0 is Ip(ϕ) = 2 (visibility V = 0) for the par- Hellmuth and collaborators performed delayed-choice 2 ϕ ticle state and Iw(ϕ) = cos 2 (visibility V = 1) for the experiments with a low-intensity Mach-Zehnder interfer- wave state. Eq. (10) is a quantitative expression of com- ometer (MZI) in the spatial domain as well as time- plementarity, and the question whether a system behaves resolved atomic fluorescence in the time domain (Hell- as a wave can now be seen in the language of mutually muth et al., 1987). The layout of the delayed-choice ex- unbiased bases. If the photon data is analyzed in the re- periment in the spatial domain is shown in Fig. 11. An spective subensembles of the ancilla outcomes, it shows attenuated picosecond laser (on average less than 0.2 pho- either perfect particle-like (ancilla in |0i, photon visibil- tons per pulse) was used as the light source for the MZI. ity V = 0) or wave-like behavior (ancilla in |1i, photon Two 5 m (20 ns) glass fibers were used to delay the in- visibility V = 1). put photon. The transit time of the photon through the π For an equal-weight superposition (α = 4 ), analyzing whole interferometer was about 24 ns. The combination only the photon data itself as a function of ϕ leads to an of a Pockels cell (PC) and a polarizer (POL) was placed 1 interference pattern with a reduced visibility of V = 2 . in the upper arm of the MZI as a shutter. Changing α from 0 (photon certainly in state |particlei) When a half-wave voltage was applied on the Pockels to π (|wavei) allows to continuously morph into parti- 2 cell, it rotated the polarization of the photons propa- cle and wave properties. Ignoring the ancilla outcome, gating through it, such that they were reflected out of the detector for the photon state 0 fires with probability the interferometer. In this case the shutter was closed 2 2 Ip(ϕ) cos α + Iw(ϕ) sin α, i.e. and interference vanished as the upper path of the in- 1 cos2 α + cos2 ϕ sin2 α, (13) terferometer was interrupted and only photons from the 2 2 lower arm could reach the photomultipliers (PM 1 and corresponding to a visibility V = sin2 α. PM 2). This provided which-path information, as the 2536 T. HELLMUTH, H. WALTHER, A. ZAJONC, AND W. SCHLEICH 35 12 with complex coefficients a and P. Since initially no photons are present, the electric field is in the vacuum mode ps-Laser Due to spontaneous emission the two states decay back 'Beam Splitter'! ppL to the ground state c which yields the two paths I ) c ~ a ~ c Microscope Objectives ~(F'M/2 and Beam Splitter2 . ic)~ fb)~ic) Glass Fiber Interference between these "routes" results in a modulation of the time-resolved fluorescence intensity I. More FIG. 3. Setup of the spatial-interference experiment with FIG. 11 Setup of the delayed-choice experiment reported in FIG. 13 Schematic diagram of the device generating the ran- precisely, the intensity is given by Pockels cell (PC) and Gian prism polarizer (POL). Ref. (Hellmuth et al., 1987); figure taken therefrom. The dom choices proposed in (Alley et al., 1983) and used in (Al- combination of a Pockels cell (PC) and a polarizer (POL) in ley et al., 1986); figure taken therefrom. A weak light pulse the upper arm of the interferometer was used as a shutter. emitted from a light emitting diode has a pulse duration of 0.67 ns. The detection event of this light pulse makes the 2538 T. HELLMUTH, H. WALTHER, A. ZAJONC, AND W. SCHLEICH 3S A. Delayed-choice interference experiment random choice which determines the setting of the Pockels cell. To realize that, a photocathode with 50% probability N PulsesN, ,N+ 2pwith a pulse duration of 150 ps were produced of producing~ 14— a photo-electron within 1 ns is amplified by a 180 + (a) N+ (a) by an actively mode-locked krypton ion laser (wavelength fast amplifier1.2 within~ 2~ ns. This electric pulse then triggers "I60 ~ ~ ~ ~ ~ 10- ~ o ~ ~ O ~ ~ 647 nm) 'I with a repetition rate of 81 MHz. An acousto- the avalanche transistor chain switch and hence the Pockels where r = r and denotes the Heaviside step function. 4Q ~ ~ ~ I I e 0.8- ~ optical switch120 selected one pulse+ out of 8000. This reduc- cell. The time of the choice can be tuned with~ respect to the For the sake of simplicity we have assumed one decay ~ 0.6— 100 ~ photon’s entry into the MZI. constant for the two states. The quantities 8'&, 8'2 cor- tion in the pulse-repetition rate was necessary+ ~ since the 04- y 80 ~ Pockels cell used to block one of the +tarms of the inter- 0.2— respond to electric fields associated with the two transi- 6Q ferometer could Fur- I I I I I I I t I i I i I 4Q not be switched more frequently. tions and 5 =1. 8 16 24 32 40 48 56 64 72 80 88 96 104112 120 c ha nn e l thermore, the reduced The analogy to the spatial-interference phenomenon 20 pulse frequency guaranteed that the time between0 two pulses was much longer than the transit discussed in Sec. II is obvious. In addition, we emphasize 8 'l6 24 32 40 48 56 64 72 80 88 96 1041&2120 c ha ~ nel beam splitter 1) to the delayed-choice mode (opening the time of the light through the interferometer which was the single-photon character of quantum beats; even when shutter after the photon reaches beam splitter 1) for each about 24 ns. Between the laser and acousto-optic modula- FIG. 12 Experimental results of the delayed-choice experi- N ~ many atoms are in the interaction zone the interference is +200) successive1.6 light pulse, while they kept all the other exper- tor an optical attenuator (T =10 ) was ~inserted(b) into the N+ due to a single-photon scattering via two indistinguishable ment in Ref.'1 80 (Hellmuth— et al., 1987); figure taken therefrom. 1.4 laser beam. ~This ensured that +the average +number pho- imental configurations to be the same, in particular the 16O -++ + of 1.2- channels. Care must be taken in the measurement Interference patterns for normal mode (dots) and delayed- ' process ~ ~ ~ O ~ + ~ ~ — ~ ~ tons per 14Qpulse- was less than 0.2. phase of the1.0 MZI.~ ~ The~ o photono counts detected by PM to insure that both channels remain indistinguishable. If choice mode (crosses)~ measured by PM 1 are similar and con- O 120— + 08 — o oo ~ ~ one tries to obtain information as to which channel actu- sistentThe withincident quantumlight mechanicalpasses through predictions.the first beam splitter 1 as a function of the phase variation are presented in (Fig. 3) and100 - the ttwo+ beams are then directed and focused Q6- ally participates, using, for example, a filter which 80— ++ Fig. 12. The04— results measured by PM 2 showed comple- into two separate ~single-modeOf + optical fibers of 5 m in transmits only photons of the transition 60- 4 +O mentary behavior,0.2— i.e. the pattern was shifted by a phase length (core diameter 4 The+ principal axis of the I I I I I I I I I I I I I I I the modulation (interference) disap- 4p pm). Q ctianne( I c)~ I b)~ I c), π with respect8 to16 24 the32 one40 48 recorded'56 64 72 80 88 by96104112120 PM 1. and fibers was20 aligned such that the polarization of the light pears =0. I I I I I I I S I I I 1 I I I I photonleaving arrivedthe0 fiber atwas beamlinear. splitterAfter 2 becauserecombining it could~QQnnelthe onlytwo ThisFIG. experiment7. Ratio was oneusing ofthe theresults pioneeringfrom realizations6. Again In this case the observable 8 16 24 32 40 48 56 64 72 80 88 96 104 1'l2 120 N/N+ Fig. havebeams comeby viathe thesecond other,beam open,splitter path.the Oninterferences the other hand,are ofthe Wheeler’shorizontal gedankenaxis is equivalent experiment,to time. althoughThe Copenhagen no trueinter- sin- FIG. 6. Comparison of interference patterns for normal and ifdetected the shutterby photomultipliers was open upon1 theand photon’s2 (PM1 arrival,and PM2) one glepretation photonsof quantum were usedmechanics and nopredicts real activeN/N+ choices——1. were delayed-choice configurations. Dots represent the data taken couldwhich observewere cooled the interferenceto reduce the pattern,dark count becauserate. thenThe noin- implemented. The switch-on time of the Pockels cell was is measured where denotes the eigenstate of the cir- with the interferometer in its normal configuration, and crosses I + ) tensity recorded in the experiment by each one of the two o. informationare data for wasdelayed-choice present aboutoperation. the path(a) is thefor photonphotomultiplier took. delayed, but eventually it was turned on such that always cularly polarized photon. Therefore, the observable + photomultipliers changed with the path difference in a The1, while temporalthe phase-inverted structuresignal in thedetected “delayed-choiceby photomultiplier mode” 2 the light’s waveB. characterQuantum-beat was tested.experiment corresponds to I„given by Eq. (4). complementary way having opposite turning points for an If the superposition of both paths is observed the ob- ofis thisshown experimentin (b). The waspoints as follows.are four-channel The inputaverages photonof the metraw Alley and co-workers have put forward a concrete optical path difference of A, /2. Since the path difference We start by first showing results for the quantum-beat servable beamdata. splitterThe horizontal 1 first, whereaxis is itsequivalent amplitudeto wastime splitwith be-30 ofs/channel.the two arms is strongly influenced by temperature- schemeexperiment for realizingin normal Wheeler’smode. In gedankenFig. 8(a) experimenta quantum-beat with tweeninduced tworefractive-index paths throughvariations the interferometer.in the fibers Itand thenin wasthe asignal delayedis andobtained randomwithout choicevoltage for theapplied configurationto the Pockels (open kept in a fiber, one in each path, for 20 ns. During the orcell; closed)in this ofcase a MZIa superposition (Alley et al.,of 1983).o+ and Threea. yearslight in later,the ' the normal (denoted dots) and delayed-choice modes = —,()(+cr„) photon propagation inby the fiber, the shutter opened af- theyfluorescence reportedis successfulobserved experimentaland the exponential demonstrationdecay inis (indicated crosses). The resulting counts are stored in a ter 4 ns PCby rise time. Then the photon exited from the modulated. Applying the quarter-wave voltage to the multichannel in the rnultiscaling mode. Ref. (Alley et al., 1986). The full details of this work are is measured which corresponds to J» defined by Eq. (11). analyzer operating Pockels cell results in the detection of a single polarized fibers, and metwas the openedI) Atomics/channel. shutterBeam and beamresults splittershown 2in described in (Jakubowicz, 1984). The experiment was In the delayed-choice version the filter is removed long The counting for 30 The component and thus in the exponential time dependence but when the sequentially.6 are a Therefore,four-channel in thisaverage case,of openingthe raw ofdata. the shut-The conceptually similar to that in (Hellmuth et al., 1987) after each fluorescence photon is emitted, Fig. X of the signal as shown in Fig. 8(b). Note that in Figs. 8, 9, photon has not reached the filter. tertime wasaxis delayedis untildetermined after theby inputthe temperature-induced photonPockels Cell met beam with some important differences. It was realized with a yet = B Field and 10 time zero corresponds to the arrival time of the splitterrefractive-index 1 and wasvariation well insidewithin of thethe interferometer.interferometer.(') WithThe i@~ 4 m by 0.3 m free space interferometer, where delayed IV. EXPERIMENTAL SETUP visibilityps-Laserof theBeaminterferencer patterns in() this experimenti this experimental(linearly arrangement, the photon’sPolarizer entry into random choices were implemented. An additional pho- polarized) was reduced from its ideal value— of 100%. The origin of the MZI was clearly located in8rn the past light cone of tomultiplier was used to detect the photons which were In this section we discuss the experimental setups for this be related imperfections of the beam splitters openingcan the shutter. to the delayed-choice interference and delayed-choice and the detection scheme of the interferences. The light reflected out ofIFL the interferometer by the combination 3 quantum-beat experiments summarized in Figs. and 4, leavingInFIG. the4. “normaltheSchematicfibers mode”,wasarrangementcollimated openingof thebythemicroscope shutterquantum-beat wasobjectives priorexperi- to of the Pockels cell and the polarizer. The random choice respectively. thement.as inputshown photonin Fig. meeting3. Due to beamthe splitterremaining 1.divergence The authorsthe was made at a photocathode which had a 50% probability alternatedinterference thepattern experimentalat the output arrangementport of the frominterferome- the nor- of producing a photo-electron upon the strike of a laser malter modewas a (openingring system. the shutterOnly thebeforezeroth-orderthe photonmaximum reaches pulse. This photo-electron was then amplified and used

was detected by the photomultipliers. Due to the finite I I I i I ~ ~ i~ I I I aperture of the photomultipliers the visibility was re- 10 20 30 40 50 60 duced. time (ns) A more quantitative comparison between the data for I I I delayed and normal modes is achieved by taking the ratio I 1 I I I / of the corresponding channel counts. These ratios for IFL photomultipliers 1 and 2 are presented in Figs. 7(a) and 7(b), respectively, and yield for the average value N~/N+ —1.00+0.02 and 0 I i 0 10 20 30 40 50 60 N~/N+ —0.99+0.02 . time(nsj FIG. 8. Time-resolved fluorescence from pulse-excited bari- This result is in very good agreement with N/N+ —1 um in normal quantum-beat configuration when (a) Pockels cell predicted by the Copenhagen interpretation of quantum voltage is zero, (b) a quarter-wave voltage is applied to the Pock- mechanics. els cell. 13

FIG. 15 Setup of the delayed-choice experiment reported in Ref. (Baldzuhn et al., 1989); figure taken therefrom. Pho- ton pairs were produced by parametric down-conversion in the LiIO3 crystal. Detection of the (trigger) photon 1 in D1 heralded (signal) photon 2 propagating through fiber F to a Sagnac interferometer. The detection at D1 triggered a Pockels cell P in the interferometer through which the signal photons propagated in a clockwise or anti-clockwise path before reaching detector D2. The signal photons showed wave behavior, if the Pockels cell was continuously left on or off. Particle behavior was revealed if the Pockels cell was switched on at the moment when the signal photons reached the refer- FIG. 14 Space-time diagram reported in Ref. (Alley et al., ence point I in the interferometer. 1986); figure taken therefrom. It shows the locations of the random-choice events for different runs with respect to the photons meeting the beam splitter and hence entering into the MZI in the laboratory reference frame. In runs R, T, U, and Y, the choice events were space-like separated from the alded single photons generated from spontaneous para- photon’s entry into the interferometer (origin of the diagram). metric down-conversion (SPDC) (Friberg et al., 1985) to perform a delayed-choice wave-particle experiment (Baldzuhn et al., 1989). The layout of the setup is shown in Fig. 15A. The detection of one (trigger) pho- to switch the EOMs in the MZI. Detailed information of ton was used to trigger a Pockels cell (P) in a Sagnac the random-choice generation is shown in Fig. 13. interferometer (Sagnac, 1913) through which the other Five experimental runs with different space-time con- (signal) photon propagated. figurations were implemented. The equivalent one-space- In the clockwise path, the signal photon first passed dimensional space-time diagram is shown in Fig. 14. In the Pockels cell P and then the reference point I. In the runs R, T and U the choice events were not only 3.5 ns anti-clockwise path, however, the situation is reverse. (a) delayed with respect to the entry of the photons into the If the Pockels cell was off during the photon’s propaga- MZI in the laboratory reference frame (event E, the origin tion through the whole interferometer, the polarization point in Fig. 13) but also space-like separated from E. In of the signal photon was not rotated and remained the run Y, the choices were also space-like separated from E same for both the clockwise and the anti-clockwise path. but took place 5 ns earlier. In run W, the choices were in (b) Similarly, if the Pockels cell was continuously on, the the time-like past of E. About 90% interference visibility polarization was rotated in both paths. In both cases (a) was obtained when the wave property of the input pho- and (b) the final polarization state was the same for both tons was measured, and no observable interference was paths, leading to interference. If, however, the Pockels obtained when the particle property was measured. The cell was switched on at the time when the signal pho- authors conclude: “The predictions of quantum mechan- ton arrives at the reference point I and was kept on until ics are confirmed even with the choice of the final con- after the photon met the beam splitter again, no inter- figuration being made randomly during the course of the ference was observed. This is because the polarization of ‘elementary quantum phenomenon” (Alley et al., 1986). the clockwise path remained unchanged, while the polarization of the counter-clockwise path was rotated. The polarization degree of freedom introduced a distinguisha- B. Wheeler’s delayed-choice experiment with single bility between the two paths and hence destroyed the particles: Photons and atoms possibility of interference. The experimental results are presented in Fig. 16. If To meet the requirement of using a single-particle the Pockels cell was continuously on or off, one observed quantum state, Baldzuhn and collaborators used her- an interference pattern (Fig. 16A). This corresponds to 14

B FIG. 16 Experimental results of the delayed-choice experiment in Ref. (Baldzuhn et al., 1989); ﬁgures taken therefrom. A: If the Pockels cell was continuously on or oﬀ, an interference pattern was observed. B: If the Pockels cell was switched on when the signal photon reached the reference point I, indicated in Fig. 15, no interference showed up.

the photon’s wave-like behavior. On the other hand, if the Pockels cell was switched on at the time when photon passed the reference point I, no interference pattern was observed (Fig. 16B). This corresponds to the particle-like behavior of the photon. CD The delayed-choice aspect of this experiment was realized by delaying the signal photon by an optical fiber (labeled ‘F’ in Fig. 16A) and varying the time of the application of the voltage on the Pockels cell via electronic delays. This allowed to switch the Pockels cell at the time when the photon was at the reference point, i.e. already within the interferometer. Space-like separation FIG. 17 (Color online). The delayed-choice experiment re- between the choice of the performed measurement and alized in Ref. (Jacques et al., 2007); figures taken therefrom. the entering of the photon into the interferometer was A: Layout of the setup. Single photons were generated by not implemented in this experiment. NV color centers in diamond. A 48-meter-long polarization Very recently, a realization of Wheelers delayed-choice interferometer and a fast electro-optical modulator (EOM), controlled by a quantum random number generator (QRNG), gedanken experiment with single atoms has been re- were used to fulfil the relativistic space-like separation con- ported (Manning et al., 2012). The physical beam split- dition. The space-time diagram is shown in B. The choice ters and mirrors were replaced with optical Bragg pulses. whether to open or close the interferometer was space-like The choice of either applying the last beam splitting pulse separated from the entry of the photon into the interferom- or not was controlled by an external quantum random eter. If the EOM was on, the polarization distinguishability number generator. This choice event occurred after the of the two paths was erased and thus an interference pattern entry of the atoms into the interferometer. emerged (C). If, however, the EOM was switched off, no interference showed up due to the polarization distinguishability of the two paths (D). C. Wheeler’s delayed-choice experiment with single photons and spacelike separation

Two important requirements of an ideal realization space-like separation. The random numbers were gener- of delayed-choice wave-particle duality gedanken exper- ated from the amplified shot noise of a white light beam. iment – namely, use of single-particle quantum states as well as space-like separation between the choice of The space-time diagram of this experiment is illus- the measurement and the entry of the particle into the trated in Fig. 17B. The sequence for the measurement interferometer – have been fulfilled simultaneously in applied to the n-th photon constituted of three steps. Refs. (Jacques et al., 2007, 2008). NV color centers in First, the choice was made by the QRNG, creating a bi- diamonds were employed as single-photon sources (Kurt- nary random number (blue), which determined the inter- siefer et al., 2000). As shown in Fig. 17A, a 48-meter- ferometer configuration. This choice happened simulta- long polarization interferometer and a fast electro-optical neously with the trigger pulse of the n-th photon’s emis- modulator (EOM) controlled by a quantum random num- sion. Second, the random number (bit values 1, 0, 1 for ber generator (QRNG) were used to fulfill relativistic photons n−1, n, n+1 in Fig. 17B) drove the EOM volt- 15 PHYSICAL REVIEW LETTERS week ending PRL 100, 220402 (2008) 6 JUNE 2008

(a) 1. 0 of entangledscanning particles, through where a the measurement phase on of one the interferometer. The 1.0 particle allows one to obtain which-path information on the otherdistinguishability particle. (or which-path information) is deﬁned 2 V [4] A. Tonomuraas Det= al.,D Am.1 + J. Phys.D2 with57, 117D (1989).i = |p(i, 1) − p(i, 2)| and p(i, j) 0.5 0. 5 2 D [5] J. Summhammer et al., Phys. Rev. A 27, 2523 (1983). [6] O. Carnalthe and probability J. Mlynek, Phys. Rev. that Lett. the66, 2689 photon (1991). traveled path i = 1, 2 [7] D. W. Keithandet is al. recorded, Phys. Rev. Lett. by66 detector, 2693 (1991).j = 1, 2. The quantity D1 is 0.0 0. 0 [8] M. Arndt et al., Nature (London) 401, 680 (1999). 0 40 80 120 160 measured by blocking path 2, and vice versa. EOM voltage (V) [9] T. Pfau et al., Phys. Rev. Lett. 73, 1223 (1994). [10] M. S. Chapman et al., Phys. Rev. Lett. 75, 3783 (1995). 2.0 (b) [11] E. Buks et al., Nature (London) 391, 871 (1998). et al. 395 2 1.5 [12] S. Durr , Nature (London) , 33 (1998).

D [13] P. BertetIV.et al. REALIZATIONS, Nature (London) 411, OF 166 (2001). DELAYED-CHOICE 1.0 [14] P. Grangier et al., Europhys. Lett. 1, 173 (1986). + QUANTUM-ERASER EXPERIMENTS

2 [15] C. Braig et al., Appl. Phys. B 76, 113 (2003).

V 0.5 [16] V. Jacques et al., Eur. Phys. J. D 35, 561 (2005). [17] G. Greenstein and A. G. Zajonc, The Quantum Challenge 0.0 Delayed-choice experiments with two particles offer 0 40 80 120 160 (Jones and Bartlett Publishers, Boston, 2005), 2nd ed. EOM voltage (V) [18] J. A. Wheeler,morepossibilities in Quantum Theory than and Measurementthose with, single particles. Es- editedpecially by J. A. Wheeler in the and experiments W. H. Zurek (Princeton performed with entangled FIG. 4 (color online). Delayed choice test of complementarity2 University Press, Princeton, NJ, 1984), pp. 182-213. FIG. 18with (Color single-photon online). pulses. Experimental (a) Wavelike informationvisibilityV (2V and, starting[19] T. Hellmuthparticleset al., Phys. in theRev. A context35, 2532 (1987). of quantum erasure, the choice 2 at 0) andwhich-path distinguishability information D2 as (D a function, starting of the at EOM 1) voltage results from[20] J. Baldzuhnof measurementet al., Z. Phys. B 77, setting 347 (1989). for one particle can be made Ref. (Jacquescorrespondinget al. to, a 2008); given value figureR of the taken VBS reflectivity. therefrom. The (a)[21]V 2 B. J. Lawson-Daku et al., Phys. Rev. A 54, 5042 (1996). 2 solid lines are the theoretical expectations, with 24 and [22] V. Jacquesevenet al. after, Science the315, other 966 (2007). particle has been registered. This and D as functions of the EOM voltage (corresponding2 2 to V 217 V, using Eqs. (2), (3), and (7). (b) V D as a [23] W. K. Woottershas been and W.shown H. Zurek, in Phys. delayed-choice Rev. D 19, 473 quantum eraser exper- the reflectivityfunction of of the the EOM second voltage. beam splitter). Solid lines are(1979). the theoretical expectations. (b) V 2 + D2 as a function of[24] the D. M. Greenbergeriments, where and A. Yasin, the Phys. which-path Lett. A 128, information 391 of one particle EOM voltage, in agreement with Ineq. (14). (1988).was erased by a later suitable measurement on the other out [34], ‘‘It obviously can make no difference as regards [25] G. Jaeger et al., Phys. Rev. A 51, 54 (1995). observable effects obtainable by a definite experimental [26] B.-G. Englert,particle. Phys. Rev. This Lett. allowed77, 2154 (1996). to a posteriori decide a single- arrangement, whether our plans of constructing or handling [27] A similarparticle inequality characteristic, is found in P. Grangier, namely The`se d’E´ whethertat the already mea- the instrument are fixed beforehand or whether we prefer to (1986),sured Institut d’Optique photon et Universite behaved´ Paris as 11; a available wave or as a particle. We postpone the completion of our planning until a later mo- online at http://tel.ccsd.cnrs.fr/tel-00009436. age to V = 0 (bit value 0) or V = Vπ (bit value 1) within ment when the particle is already on its way from one [28] S. Durrwillet al., discuss Phys. Rev. the Lett. 81 experimental, 5705 (1998). realizations along this line rise timeinstrument 40 ns (red), to another.’’ which Such determined an intriguing property the state of of[29] the X. Peng et al., J. Phys. A 36, 2555 (2003). [30] P. Mittelstaedtin theet following al., Found. Phys. sections.17, 891 (1987); P.D. D. quantum mechanics forces one to renounce some common- second beam splitter (BSoutput). Finally, the photon wasSchwindt et al., Phys. Rev. A 60, 4285 (1999). As stressed recordedsense by representations detectors D1 of the or physical D2, afterreality. its time of flightin [28], these experiments can actually be fully interpreted We warmly thank A. Clouqueur and A. Villing for the τ in the interferometer. The blue zone in Fig. 17Bin the framework of classical electrodynamics since they interf realization of the electronics, and J.-P. Madrange for all do notA. correspond Photonic to the case quantum of true single-photon erasure pulses representsmechanical the future realizations light of the cone interferometer. of the choice. This work The is event(see Ref. [14]). “entry ofsupported photon by Institut into the Universitaire interferometer” de France. was space-like[31] M. O. Scully et al., Nature (London) 351, 111 (1991). [32] A. BeveratosEnergy-timeet al., Eur. Phys. (Friberg J. D 18, 191et (2002). al., 1985; Joobeur et al., 1994), separated from the respective choice. If the EOM[33] was The nonidealmomentum value of the (Rarity parameter and is due Tapster,to residual 1990) and polariza- on with voltage V = V , one erased the polarization dis-background photoluminescence of the diamond sample *[email protected]π and totion its two-phonon (Kwiat Ramanet scattering al., 1995; line, which Shih both and Alley, 1988) entan- tinguishability[1] N. Bohr, of Naturwissenschaften the two paths16, and245 (1928). thus observed anproduceglement uncorrelated of photons photon associated pairs generatedto Poissonian from SPDC have been interference[2] R. pattern P. Feynman, when R. B. Leighton, tuning and the M. L. phase Sands, Lectures of the inter-statistics. on Physics (Addison Wesley, Reading, MA, 1963). [34] N. Bohr,widely in Albert used Einstein: in Philosopher experiments Scientist, realizing edited photonic quantum ferometer[3] by We tilting stick to the BS originalintput discussions(Fig. 17 onC complementarity). If, however, theby P.A.erasure. Schilpp Herzog (Library of and Living co-workers Philosophers, used photon pairs gener- for single particles and do not consider schemes with pairs Evanston, IL, 1951), 2nd ed., p. 230. EOM was switched off, due to the polarization distin- ated from type-I SPDC and demonstrated the quantum guishability of the two paths, no interference showed up eraser concept via various experiments (Herzog et al., (Fig. 17D). 1995). Polarization as well as time delay were used as Furthermore, Jacques and co-workers varied the driv- quantum markers, and wave plates as well as narrow- ing voltages applied to the EOM and thus realized a fast bandwidth interference filters as quantum erasers. They switchable beam splitter with an adjustable reflection co- harnessed the momentum entanglement and polarization efficient R (Jacques et al., 2008). The QRNG switched correlation between photon pairs, and performed remote this beam splitter on and off randomly. Each randomly 220402-4 measurements on one photon either revealing or erasing set value of R allowed them to obtain partial interfer- which-path information of the other one. ence with visibility V and partial which-path informa- An arrangement consisting of a double slit and two tion. The which-path information was parameterized by entangled particles allows a combination of the gedanken the distinguishability D. The authors confirmed that experiments of Heisenberg’s microscope and the quantum V and D fulfilled the complementary relation (Englert, eraser. Dopfer and collaborators employed photon pairs 1996; Greenberger and Yasin, 1988; Jaeger et al., 1995; generated from type-I SPDC (Dopfer, 1998; Zeilinger, Wootters and Zurek, 1979) 1999, 2005). Due to the phase matching condition, pho- V 2 + D2 ≤ 1, (14) tons 1 and 2 were entangled in their linear momentum states. Fig. 19A shows one of their experimental config- where equality holds for pure states (see Fig. 18). The urations. Photon 2 passed a double-slit and a lens and visibility is defined as V = (pmax − pmin)/(pmax + pmin), was measured by a static detector D2 in the focal plane. with pmax and pmin the maximal and minimal proba- Photon 1 was sent through another lens with focal length bility for recording a photon in a chosen detector when f and was measured by detector D1, which was mounted 16

A Therefore, when coincidence counts between D1 and D2 D1 Photon 1 were measured as a function of D1’s position along the x- NLC axis, an interference pattern showed up with a visibility Photon 2 Lens as high as 97.22% (Fig. 19B). Double slits On the other hand, when D1 was placed in the image plane (i.e. distance 2f from the lens), the detection events D2 of photon 1 revealed the path photon 2 took through the B 140 double slit. In Fig. 19C, two prominent peaks indicate the proﬁle of the double-slit assembly with no interfer- 120 ence pattern. 100 In the experiment of Walborn and collaborators (Wal- 80 born et al., 2002), one photon of a polarization-entangled

60 pair impinged on a special double-slit device, where two quarter-wave plates, oriented such that their fast axes 40 are orthogonal, were placed in front of each slit to serve

Coincidence counts in 60s 20 as which-path markers. The quarter-wave plates rotated

0 the polarization states of the photons passing through -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 them and hence the subsequent slits. This rotation in- Position of D1 [um] (D2 is fixed.) C troduced a distinguishability of the two possible paths and thus destroyed the interference pattern. To recover 3500 Fit: d=253.0µm interference, polarization entanglement was used and the 3000 polarization of the other entangled photon was measured 2500 in a proper basis. This experiment was also performed 2000 under delayed erasure conditions, in which the interfer-

1500 ing photon is detected before its entangled twin. The experimental data was in agreement with the predictions 1000 Fit: a=73.2µm a

Coincidence counts in 30s of quantum mechanics. 500

0 -300 -200 -100 0 100 200 300 400 B. Matter-wave quantum erasure Position of D1 [um] (D2 is fixed.)

FIG. 19 (Color online). A: Experimental scheme of the ex- Light scattered from laser-cooled atoms provides infor- periment, using a momentum entangled state of two photons. mation on the localization of atoms and can be used to See text for details. B: A high-visibility interference pattern realize quantum eraser experiments, if the atomic separa- in the conditional photon counts was obtained when D1 was tion is large enough and the wave length of the scattered positioned in the focal plane of the lens, thus erasing all path light short enough to allow in principle identification of information. C: The profile of the double slit was resolved the atom’s position by imaging conditions. In Ref. (Eich- when D1 was positioned in the image plane of the lens, revealing path information and therefore no interference pattern mann et al., 1993), an experiment with light scattered arose. Figures taken from (Dopfer, 1998). from two ions was performed. By employing the polarization of the scattered light, the authors realized the above mentioned cases A and B in Section II.E and observed polarization-detection dependent interference patterns. on translation stages capable of moving along both axes x Dürrand collaborators carried out an atomic interfero- and z. This allowed an implementation of the idea of von metric experiment showing that the disturbance of path Weizsäcker switching from the focal plane to the image detection on an atom’s momentum is too small to de- plane. stroy the interference pattern (Dürr et al., 1998). The If D1 was placed in the focal plane of the lens (i.e. principle of this experiment is shown in Fig. 20a. By us- at distance f from the lens), one measured photon 1’s ing a standing-wave grating formed by off-resonant laser momentum state and hence lost the information about light, the collimated atomic beam A was split into two its position. Due to the momentum entanglement, the beams: beam B was reflected and beam C was transmit- measurement of photon 1’s momentum state projected ted. After free propagation for a time duration of tsep, the state of photon 2 into a momentum eigenstate which they were separated by a lateral distance d. The beams could not reveal any position information. One therefore B and C were then split again by a second standing light had no information whatsoever about which slit photon 2 wave grating. In the far field, complementary spatial went through. When both photons were detected, neither interference patterns were observed in two regions. Ex- photon 1 nor photon 2 revealed any path information. perimentally, the authors varied the phase of the atomic 17

The internal electronic states |2i and |3i were used as a which-path detector for the paths B and C (shown in Fig. 20d). These two states were addressed and manip- ulated with microwave pulses. Fig. 20e shows how the atomic internal electronic states were employed in controlling the paths the atoms took. The authors converted the input state |2i to a superposition state |2i + |3i by a π/2 microwave pulse with frequency ωmw = ω3 − ω2, where ω2 and ω3 are the frequencies of states |2i and |3i. (We omit the normalization to be consistent with the original notation). Then a standing-wave grating was formed by a laser with frequency ωlight, which was tuned to be halfway between the |2i → |ei and |3i → |ei tran- ω3−ω2 sitions to the excited state |ei, i.e. ωlight = ωe − 2 . Due to these detunings an internal-state dependent phase shift was implemented. In the reflected arm (B), the light grating induced a π phase shift on state |2i with respect to |3i resulting in state |3i − |2i. In the transmitted arm (C), no phase shift was induced and hence the state remained |3i + |2i. A subsequent π/2 microwave pulse converted the superposition states in the reflected and transmitted arm to |2i and |3i, respectively. Therefore, FIG. 20 Quantum-eraser experiment from Ref. (Dürr et al., the atom path in the interferometer was correlated with 1998) based on an atomic interferometer; figures taken there- its internal electronic states. Consequently, no interfer- from. a: The atomic beam A was split into two beams; beam B was reflected by the first Bragg grating formed by a stand- ence patterns did arise, as shown in Fig. 20f. ing wave, and beam C was transmitted. The atomic beams In this experiment, the disturbance of the path, which freely propagated for a time duration of tsep and acquired a was induced by using microwave pulses, was four orders of lateral separation d. The beams B and C were then split magnitude smaller than the fringe period and hence was again by a second standing light wave grating. In the far not able to explain the disappearance of the interference field, complementary spatial interference patterns were ob- patterns. Instead, “the mere fact that which-path infor- served in two regions. These interference patterns were due mation is stored in the detector and could be read out to superpositions of beams D and E (F and G). b and c show the spatial fringe patterns in the far field of the interferom- already destroys the interference pattern” (Dürr et al., eter for tsep = 105 µs with d = 1.3 µm and tsep = 255 µs 1998). with d = 3.1 µm, respectively. The left and right comple- Recently, an experimental realization of quantum era- mentary interference patterns were respectively generated by sure in a mesoscopic electronic device has been re- the atomic beams D and E, and F and G (shown in a). The ported in Ref. (Weisz et al., 2014). Interacting electrons dashed lines indicate the sum of the intensities of two in- have been used to extract which-path information and a terference patterns obtained with a relative phase shift of smooth variation of the degree of quantum erasure has π. d illustrates the simplified scheme of the internal atomic states, which were addressed using microwave (mw) radia- been demonstrated. tion and light. e illustrates the principle of correlating the We also remark here that neutral kaon systems have path the atoms took with their internal electronic states. The been theoretically suggested to be suitable for a demon- standing-wave grating produced a relative π phase shift of stration of quantum erasure as shown in Ref. (Bramon state |2i relative to |3i conditional on its path. A Ramsey in- et al., 2004). There, strangeness oscillations would repre- terferometer employed two microwave π pulses and converted 2 sent the interference pattern linked to wave-like behavior. different relative phases into different final internal states |2i and |3i, respectively. f: When the which-path information was stored in the internal atomic state, the interference patterns vanished. C. Quantum erasure with delayed choice

In (Kim et al., 2000), pairs of entangled photons were used to mimic the entangled atom-photon system pro- interferometer by setting diﬀerent separation durations posed in (Scully et al., 1991). The layout of the experi- tsep between the ﬁrst and the second standing-wave grat- mental setup is shown in Figure 21a. Photon pairs were ings. Interference patterns with visibilities of (75 ± 1)% generated noncollinearly either from region A or region and (44 ± 1)% for tsep = 105 µs and 255 µs, shown in B of a β-Barium borate (BBO) crystal via type-I SPDC. Fig. 20b and c, have been observed which were in good From each pair, photon 1, simulating the atom, propa- agreement with the theoretical expectations. gated to the right and was focused by a lens. It was then 18

firmed the theoretical prediction. The “choice” of observing interference or not was made randomly by photon 2 by being either reflected or transmitted at BSA or BSB. In the actual experiment, the photons traveled almost collinearly, but the distance from the BBO to BSA and BSB was about 2.3 m (7.7 ns) longer than the distance from the BBO to D0. Thus, after D0 was triggered by photon 1, photon 2 was still be on its way to BSA or BSB, i.e., the which-path or the both- path choice was “delayed” compared to the detection of photon 1. As an extension, a delayed-choice quantum eraser experiment based on a two-photon imaging scheme using entangled photon pairs (signal and idler photons) was reported in Ref. (Scarcelli et al., 2007). The complete which-path information of the signal photon was trans- ferred to the distant idler photon through a “ghost” image. By setting different sizes of the apertures, the au- FIG. 21 Delayed-choice quantum-eraser experiment realized thors could either obtain or erase which-path informa- in Ref. (Kim et al., 2000); figures taken therefrom. a: Experi- tion. In the case of which-path information erasure, in- mental scheme. Pairs of entangled photons were emitted from terference with a visibility of about 95% was obtained. either region A or region B of a BBO crystal via spontaneous When not erasing which-path information, no interfer- parametric down-conversion. These two emission processes ence was observed. were coherent. Detections at D3 or D4 provided which-path information and detections at D1 or D2 erased it. b: Coinci- dence counts between D0 and D3, as a function of the lateral D. Quantum erasure with active and causally disconnected position x0 of D0. Absence of interference was demonstrated. choice c: Coincidence counts between D0 and D1 as well as between D0 and D2 are plotted as a function of x0. Interference fringes were obtained. See text for details. Quantum erasure with an active and causally disconnected choice was experimentally demonstrated in Ref. (Ma et al., 2013). To this end, the erasure event of which-path information had to be space-like separated detected by D0, which was mounted on a step motor ca- from the passage of the interfering system through the in- pable of changing the lateral position x0. terferometer as well as its detection event. Based on the Photon 2, propagating to the left, passed through one special theory of relativity, the event of quantum erasure or two of the three beam splitters. If the pair was gener- was therefore causally disconnected from all relevant in- ated in region A, photon 2 would follow path a and meet terference events. beam splitter BSA, where it had a 50% chance of being The concept of the experiment is illustrated in Fig. reflected or transmitted. If the pair was generated in re- 22A. Hybrid entangled photon pairs (Ma et al., 2009) gion B, photon 2 would propagate path b and meet beam were produced, with entanglement between the path a splitter BSB, again with a 50% chance of being reflected or b of one photon (the system photon s) in an interfer- or transmitted. ometer, and the polarization H or V of the other photon (the environment photon e): In the case that photon 2 was transmitted at BSA or BSB, it would be detected by detector D3 or D4, respec- |Ψi = √1 (|ai |Hi + |bi |Vi ). (15) se 2 s e s e tively. The detection of D3 or D4 provided which-path information (path a or path b) for photon 2, thus also Analogous to the original proposal of the quantum eraser, providing the which-path information for photon 1 due to the environment photon’s polarization carried which- the linear momentum entanglement of the photon pair. path information of the system photon due to the entan- Therefore, there was no interference, as verified by the glement between the two photons. Depending on which results shown in Figure 21b. polarization basis the environment photon was measured On the other hand, given a reflection at BSA or BSB, in, one was able to either acquire which-path informa- photon 2 continued its path to meet another 50:50 beam tion of the system photon and observe no interference, or splitter BS and was then detected by either D1 or D2. erase which-path information and observe interference. The detection by D1 or D2 erased the which-path infor- In the latter case, it depended on the specific outcome of mation carried by photon 2 and therefore an interference the environment photon which one out of two different pattern showed up for photon 1 (Figure 21c). This con- interference patterns the system photon was showing. 19

A Mirror Choice Path b Det 2 “0” BS S Interferometer “1” Path a “0”: Welcher-weg Det 1 information Mirror “1”: Interference B Lab 2 Lab 1 Interferometer Polarization 27 m 50 m Lab 3 projection Hybrid path b Det 2 “0” entanglement Coaxial “1” PBS2 source FPC BS QRNG cables, 28 m Fiber, 55 m S PBS1 path a Det 3 Det 1 EOM Fiber, 28 m Det 4

C P e Future light Transmission of environmentcone of Is photon in optical fiber Labeling of space-time events:

Ese ... Emission of system and Is enviroment photon C FIG. 23 Experimental test of the complementarity inequal- e .... Choice of setting for ity under Einstein locality, manifested by a trade-oﬀ of the in coaxial cable Optical enviroment photon Transmission of choice fiber Pe ..... Polarization projection of Time / ns Time delay of which-path information parameter D and the interference system enviroment photon photon visibility V . The dotted line is the ideal curve from the Is ...... Interferometer-related events Ce Ese of system photon saturation of the complementary inequality. The solid line, 2 1/2 Past light cone of Is V = 0.95 [1 − (D/0.97) ] , is the estimation from experi- Lab 3 Lab 2 Lab 1 mental imperfections. Figure taken from (Ma et al., 2013).

Space / m

FIG. 22 (Color online). Quantum erasure with causally dis- partite complementarity inequality of the form (14) (En- connected choice. A: Principle: The source S emitted path- polarization entangled photon pairs. The system photons glert, 1996; Greenberger and Yasin, 1988; Jaeger et al., propagated through an interferometer (right side), and the en- 1995; Wootters and Zurek, 1979), in which D and V vironment photons were subject to polarization measurements stand for conditional which-path information (distin- (left). B: Scheme of the Vienna experiment: In Lab 1, the po- guishability) and interference visibilities respectively. It larization entangled state generated via type-II spontaneous is an extension of the single-particle complementarity in- parametric down-conversion, was converted into a hybrid en- equality (experimentally verified in Ref. (Jacques et al., tangled state with a polarizing beam splitter (PBS1) and two 2008) and discussed in Section III.C). Under Einstein lo- fiber polarization controllers (FPC). In Lab 2, the polarization projection setup of the environment photon consisted of cality, D and V were measured in sequential experimen- an electro-optical modulator (EOM) and another polarizing tal runs as a function of the applied voltage of the EOM, beam splitter (PBS2). In Lab 3, the choice was made with which changed the polarization projection basis of the a quantum random number generator (QRNG) (Jennewein environment photon. Hence, a continuous transition be- et al., 2000). C: Space-time diagram. The choice-related tween measurements of particle nature and wave nature events Ce and the polarization projection of the environment was acquired. The results are shown in Fig. 23. photon Pe were space-like separated from all events of the Note that similar setups have been proposed in Refs. interferometric measurement of the system photon Is. Addi- tionally, the events Ce were also space-like separated from the (Ballentine, 1998; Grangier, 1986; Kwiat and Englert, emission of the entangled photon pair from the source Ese. 2004). Another successful experiment along this line was The shaded areas are the past and the future light cones of reported in (Kaiser et al., 2012). Kaiser and collaborators events Is. This ensured that Einstein locality was fulfilled. used polarization-entangled photon pairs at the telecom Figures taken from (Ma et al., 2013). wavelength. Every ‘test’ photon was sent into an interferometer (with phase θ ≡ ϕ), while the corresponding ‘corroborative’ photon was subject to a polarization mea- The quantum eraser concept under Einstein locality surement. While no active random choices were imple- was tested on two different length scales. In a first ex- mented in their experiment, the detection events of the periment performed in Vienna in 2007, the environment corroborative and test photon were space-like separated photon was sent away from the system photon via a 55 m (Fig. 24). long optical fiber (Fig. 22B and C). In a second experi- The interferometer employed a polarization-dependent ment performed on the Canary Islands in 2008, the pho- beam splitter (PDBS) with bulk optics which was able tons were separated by 144 km via a free-space link. See to reflect horizontally polarized test photons with close the caption of Fig. 22 for details on the first experiment to 100% probability and reflect/transmit vertically polar- and its space-time diagram. ized photons with 50%/50% probability. Then polarizing In order to quantitatively demonstrate quantum era- beam splitters oriented at 45 to the H/V basis erased all sure under Einstein locality, the authors employed a bi- polarization information that potentially existed at the

20 NATURE PHOTONICS DOI: 10.1038/NPHOTON.2012.179 LETTERS

Single photon Time (ns) BD1 BD2 BD5 α Particle layer

APD 45° φ Forward light Forward light cone cone Wave layer Quartz Corroborative 250 ns 230 ns photon detected Test photon 45° Polarizer detected BD3 BD4 Particle layer: HWP1 (θ = 0°) θ Wave layer: HWP2 (θ = 22.5°)

: Path 0 : Path 1 HWP1 Pair (HWP2) : Polarization generation Position (m) : 3 ns delay -17 m 3 m (0/0) Layer

Figure 2 | Experimental set-up. The set-up includes four parts: single photons generated by the SAQD (not shown), the opened (closed) MZI (quartz, BD1, FIG. 24 (Color online). Space-time diagram of the experi-BD3, HWP1 (HWP2), BD4), the quantum-control apparatus (a, BD2, particle layer, wave layer, BD5), and the detection apparatus (movable polarizer, APDs and the counter and time analyser, which are not shown). Note: the arrows are used to represent the photon polarizations, and the HWPs are not shown; ment reported in Ref. (Kaiser et al., 2012); figure taken there-‘Layer’ in the lowerFIG. figure represents 26 (Colora detailed sketch online). of the particle andExperimental wave layers. quantum delayed- from. The detections of the corroborative photon and the test choice with single photons. Single photons entered the in- photon were space-like separated. D ¼ 0.973+0.003,terferometer which is defined as at beam displacer BD1‘quantum’ which here is the split use of thea quantum light detecting into device (pc-BS). When the single photon reaches BD2 (beam displacer) it has com- horizontal| and− vertical| polarization. The phase ϕ was scanned = N01 N11 pletely passed BD1, because the 3 ns delay is somewhat larger than D + by the quartzN01 N plates11 before BD1.theThe 0.563 ns “second time duration beam of this photon. splitter” The photon makes a choice, but because the pc-BS remains at a quantum superposition in the closed (open) interferometer was provided by the com- | where Nij (i,j ¼ 0,1) is the detected photon number of path j of the eigenstates (presence and absence, corresponding to Hl and when path i binationis unblocked (and of BD3, the other BD4 path blocked). and Thehalf-wave|Vl, respectively), plates the photon HWP2 can never (HWP1). know which choice it has 1 which-path information is almost fully extracted. Further details made, even after it has passed the pc-BS; that choice is only revealed of the definitionFor and HWP2 distinguishability in the for other wave caseslayer, have been thewhen optical-axis the state of the pc-BS direction is detected on theθ eigenstates.was Therefore, addressed elsewhereset22 to. 22.5◦ and interference appearsin this scheme, (closed the photon interferometer). cannot know the state of the detecting device before it passes BD1 in order to adjust itself to exhibit the We see a mixing of Fig. 3a and Fig. 3e in the intermediate cases. ◦ The visibilitiesFor become HWP1 small, but the in centres the (that particle is, the average layer of correspondingθ was set property. to 0 The(open hidden-information interfer- theory is denied the maximum and minimum) and the curve shapes do not change. once again. 0.5 The differencesometer), between the experimental showing and theoretical the particle values are properties.By tracing out the Depending pc-BS states, the results on of the collapse for both caused by thepolarization counting statistics, the (parameter dark and backgroundα counts,), BD2thecontrolled presence state and thewhether absence state the are summed. pho- Because of this

Probability and the tiny instability of the MZIs. summation, it is convenient for us to detect directly the number of However, thetons situation passed is different through for the quantum the wave–particle particlewhole or wave photons in layer. the same path,The without two splitting layers them with a hori- 6 superposition, as shown by the blue symbols in Fig. 3. The blue lines◦ zontally placed polarization beamsplitter and then summing them 0 show the correspondingwere combined theoretical fit, derived by BD5. from equation A 45 (3) again.polarizer This process could produces be the√ inserted classically mixed to results that we 80 4 (see Methods). The fitting parameters d and d change very little, presented previously, that is, the morphing between oscillation 60 post-select on the0 polarization1 state (|Hi + |Vi)/ 2. Finally, 40 2 which further illustrates the stability of our experimental set-up. and non-oscillation states. 20 Except in thetwo cases detectorsof a ¼ 0andp/ counted2, in which the the photons photonsIt is interestingin paths to consider 0 and what 1. will happen Figure if the pc-BS is not Angle α (deg) 0 0 Phase θ (rad) all go through the particle layer or the wave layer, the curves collapsedp to the eigenstates, but to a superposition state, such as are strikinglytaken different between from the Ref. quantum-superposition (Tang et al. and, 2012).(1/ 2)(|presencel þ |absencel) (corresponding to | þ l). This is a FIG. 25 (Color online). Experimental results reportedclassical-mixture in cases. novel scenario that can be realized with a quantum beamsplitter To show these differences more comprehensively, we extracted experiment. equation (3) (see Methods) shows that the result is a Ref. (Kaiser et al., 2012); figure taken therefrom. Whenthree the quantities from Fig. 3: the centre of the curve, the visibility quantum superposition of the wave and particle states of the and the ratio of the rise period to the total period (‘centre’, ‘visibility’ photon, as opposed to a classical statistical mixture of the two corroborative photon was found to be horizontally polarized,and ‘ratio’ forton short). itself The results as are shownthe in ancilla. Fig. 4. The red Onlystates. The the experimental horizontally results give a completely polar- different result the test photons produced an intensity pattern followingsymbols ex- showized the classical photons wave–particle|H mixture,i passed and the through blue from the a classical second mixture, beam because there splitter, is quantum interference symbols the quantum superposition. The larger symbols show the between the wave and particle properties. This phenomenon has pression (13), with θ ≡ ϕ. The same pattern emerged whenexperimental resultswhile taken for from vertical Fig. 3, and the polarization smaller symbols not|V previouslyi the been interferometer revealed, and brings a whole was new meaning to the theoretical simulation results. More analyses are found in the concept of wave–particle duality. the corroborative photon was measured in the |+i/|−i basis,Supplementaryopen. Section S3. Similar to Eq. (10), withThe initial application polarization of a quantum detecting state device leads to a rein- verifying the entanglement. We should note that, although we use terminology that refers to terpretation of the complementarity principle10, because the the ‘classical mixture’sin α and|V thei+cos ‘quantumα superposition’|Hi the of total wave– one-photondetecting device should state be classical was according trans- to the regular particle properties due to equation (2) and equation (3) (see concept. The same problem also appears in many experiments Methods), bothformed terms reflect into: the results of the quantum delayed- that are the foundations of quantum mechanics, such as the choice experiment because the primary character of the Bell inequality test23, the Kochen–Specker inequality test24, and

PDBS output. The corroborative photon passed an EOMNATURE PHOTONICS | VOL 6 | SEPTEMBERψi = 2012 sin | www.nature.com/naturephotonicsα |particlei |Vi + cos α |wavei |Hi , (17) 601 which rotated its polarization state by an angle α before √ where |particlei = (©| 2010i2 Macmillan+ eiϕ Publishers|1i) Limited./ 2 All rights and reserved.|wave i = eiϕ/2× it was measured. The total quantum state of the test (t) (cos ϕ |0i − i sin ϕ |1i), similar to (11) and (12), and |0i and corroborative photon (c) was 2 2 and |1i are the path states in the interferometer. (Note 1 the different conventions for the ancilla bias parameter α |Ψi = √ [(cos α |particleit − sin α |waveit) |Hi tc 2 c in states (10) and (17).) + (cos α |waveit + sin α |particleit) |Vic]. (16) The experimental setup is shown and explained in Fig. 26. If the final path measurement was not sensitive to the Here, |particlei and |wavei are defined similar to (11) and polarization (i.e. no polarizer at the end), the detection (12). This allowed for a continuous transition between results were described by a mixed state (density matrix) wave and particle properties, verifying the predicted in- of the form tensity pattern of Eq. (13). sin2 α |particleihparticle| + cos2 α |waveihwave|. (18) This corresponded to ignoring the ancilla outcome in E. Quantum delayed-choice chapter II.G and lead to a visibility pattern of the form cos2 α. If, however, the photon was post-selected in the Quantum delayed choice shares a few features with √ polarization state (|Hi + |Vi)/ 2 (i.e. polarizer at 45◦), quantum erasure. An experiment following the proposal its path state was left in the “wave-particle superposi- described in chapter II.G has been realized using single tion” photons in an interferometer (Tang et al., 2012). This was achieved by taking the polarization state of the pho- sin α |particlei + cos α |wavei. (19) NATURE PHOTONICS

REPORTS

2 splitter is represented by a Hadamard operation took. The measured intensities are ID′ =cos(ϕ/2) data shows interference fringes. Our results are 2 (22), which transforms the initial photonpffiffiffi state and ID″ =sin(ϕ/2). in excellent agreement with theoretical predic- |0〉s into the superposition ðj0〉s þj1〉sÞ= 2.A The main feature of this quantum controlled tions (Fig. 3). phase shifter then modifies the relative phase beamsplitter is that it can be put in a superpo- To achieve our main goal—to refute models between the two modes,pffiffiffi resulting in the state sition of being present and absent. Indeed, if the in which the photon knows in advance with iϕ jy〉s ¼ðj0〉s þ e j1〉sÞ= 2. Both modes are ancilla photon is initially in a superposition—for which setup it will be confronted—we must go 21 then recombined on a second beamsplitter before instance, in the state |y〉a =cosa|0〉a +sina|1〉a— one step further. Indeed, the result of Fig. 3 does a final measurement in the logical ({|0〉s,|1〉s}) then the global state of the system evolves into not refute such models. Although we have in- basis. In the standard delayed-choice experiment, serted the ancilla photon in a superposition, hence the presence of this second beamsplitter is con- testing both wave and particle aspects at the same 1.00 |Yf (a,ϕ)〉 = cosa|y〉s |0〉a + trolled by the observer (see Fig. 1A). For a closed ,particle time, we have in fact not checked the quantum (3) interferometer, the statistics of the measurements sina| y〉s,wave|1〉a nature of this superposition. This is because the at detectors D′ and D″ will depend on the phase final measurement of the ancilla photon was made ϕ, revealing the wave nature of the0.75 photon. For The system and ancilla photons now become en- in the logical ({|0〉a,|1〉a}) basis. Therefore, we can- an open interferometer, both detectors will click tangled, when 0 < a < p/2. not exclude the fact that the ancilla may have been 2 with equal probability, revealing the particle na- The measured intensity at detector D′ is then in a statistical mixture of the form cos a|0〉〈0|a + 2a 〉〈 ture of the photon. Visibility 0.50 given by sin |1 1|a, which would lead to the same mea- Here, on the contrary, the presence of the sec- sured statistics. Hence, the data can be explained 2 2 ond beamsplitter depends on the state of an an- ID′ (ϕ,a) ¼ Iparticle(ϕ)cos a þ Iwave(ϕ)sin a by a classical model, in which the state of the cillary photon. If the ancilla photon0.25 is prepared ancilla represents a classical variable (a classical in the state |0〉a, no beamsplitter is present; hence, 1 ϕ bit) indicating which measurement, particle or ¼ cos2a þ cos2 sin2 a ð4Þ the interferometer is left open. Formally, this cor- 2 2 wave, will be performed. Because the state of the responds to the identity operator acting on |y〉s, ancilla may have been known to the system pho- resulting in the state 0.00 whereas intensity at D″ is ID″(ϕ, a)=1− ID′(ϕ, a). toninadvance—indeed, here no delayed choice We fabricated the quantum circuit shown in is performed by the observer—no conclusion can p1ffiffiffi iϕ Fig. 2 in a silica-on-silicon photonic chip (18). be drawn from this experiment. This loophole jy〉s,particle ¼ ðj0〉s þ e j1〉sÞð1Þ 2 0.0The0 Hadamard operation0.25 is implemented0.5 by0 a also plagues0.7 the5 recent theoretical1.00 proposal of directional coupler of reflectivity 1/2, which is (17), as well as two of its NMR implementations The final measurement (in the {|0〉s,|1〉s}basis) equivalent to a 50/50 beamsplitter. The controlled-α/ π (19, 20). indicates which path the photon took, revealing Hadamard (CH) is based on a nondeterministic In order to show that the measurement choice the particle nature of the photon. The measured control-phase gate (23, 24). The system and an- could not have been known in advance, we must intensities in both output modes are equal and cilla photon pairs are generated at 808 nm via ensure that our quantum controlled beamsplitter phase-independent, ID′ = ID″ =1/2. parametric down conversion and detected with behaves in a genuine quantum way. In particular, If, however, the ancillaFIG. photon is 27prepared (Color in silicononline). avalanche photodiodes Visibility at the circuit’ ass we a must function ensure that it createsof α entanglementfor the be- the state |1〉a, the beamsplittermixed is present, state and the (18)output. in red (filled circles) andtween thefor system the andsuperposition ancilla photons, which is the interferometer is therefore closed. Formally, this We first characterized the behavior of our clear signature of a quantum process. The global FIG. 29 (Color online). Continuous transition between wave corresponds to applyingstate the Hadamard (19) opera- insetup blue for various (unfilled quantum states diamonds). of the an- state of the The system andred ancilla curve photons, given has in tionto|y〉s, resulting in the state cilla photon. We measured the output intensities Eq. 3, is entangled for all values 0 < a < p/2. and particle behavior. The experimental data are shown by 2I ϕ a I ϕ a a ∈ p ϕ ∈ 〈y y 〉 ∼ ϕ the form cos D′(α,,)and whileD″( , the)for blue[0, /2], and one alsoBecause reflectsparticle| wave interferencecos , the degree of ϕ ϕ [−p/2, 3 p/2]. In particular, by increasing the entanglement depends on ϕ and a; in particular, white dots and were fitted (colored surface) based on Eq. (13). jy〉s,wave ¼ cos j0〉s − i sin j1〉s ð2Þ 2 between2 wavevalue and of a we particle observe the morphing properties. between for Thea = p/4 larger and ϕ = p symbols/2 the state in Eq. are 3 is a particle measurement (a = 0) and a wave mea- maximally entangled. Figure taken from Ref. (Peruzzo et al., 2012). The final measurement givesexperimental information about surement data, (a = whilep/2). For a = the 0 (no beamsplitter), smaller symbolsIn order to certify are the presence theoretical of this entan- the phase ϕ that was applied in the interferome- the measured intensities are independent of ϕ. glement, we tested the Clauser-Horne-Shimony- ter, but indeed not aboutsimulation which path the photon results.For a = p/2, the Figure beamsplitter taken is present, fromand the Ref.Holt (CHSH) (Tang Bell inequalityet al. (25),, the 2012). violation

Fig. 2. Implementation of the quantum delayed- choice experiment on a reconfigurable integrated photonic device. Non- was finally measured in its computational basis, the sys- entangled photon pairs are generated by using tem photon data could be explained by a classical model type I parametric down- conversion and injected in which the ancilla photon was prepared in a mixture into the chip by using 2 2 polarization maintaining of the form cos α |0ih0| + sin α |1ih1| . The particular fibers (not shown). The system photon (s), in the state in every run would be known to the system pho- lower part of the circuit, enters the interferometer at the Hadamard gate (H). A relative phase ϕ is state of the second beamsplitter—a superposition of present and absent. tons beforehand, deciding whether their particle or wave applied between the two modes of the interferometer. Then, the controlled- Last,thelocalmeasurementsfortheBell test are performed through single- U U Hadamard (CH) is implemented by a nondeterministic CZ gate with two qubit rotations ( A and B) followed by APDs. The circuit is composed of behavior is measured. additional MZ interferometers. The ancilla photon (a), in the top part of the directional couplers of reflectivity 1/2 (dc1−5 and dc9−13)and1/3(dc6−8)and circuit, is controlled by theFIG. phase shifter 28a, which (Color determines online). the quantum resistive Two-photon heaters (orange rectangle experimentals) that implement the phase quantum shifters (25). delayed-choice experiment. Non-entangled photon pairs were injectedwww.sciencemag.org into an integratedSCIENCE VOL photonic 338 2 NOVEMBER device. 2012 The system pho-635 To ensure that the choice cannot have been a clas- ton (s, black optical path) passed a Hadamard gate (H) and a sical variable known in advance, the entanglement of phase shifter ϕ. The “second beam splitter” was a controlled- state (10) needed to be shown. This was done using Hadamard gate (CH), implemented with additional Mach- unitary transformations at the final stage of the setup Zehnder interferometers. The ancilla photon (a, red optical (Fig. 28) and performing a test of the Clauser-Horne- path) passed a phase shifter α, allowing a superposition of Shimony-Holt (Clauser et al., 1969) inequality. Maximal present and absent beam splitter for the system photon. For π entanglement of the state (10) is reached for α = 4 (an- the Bell test, single qubit rotations (UAlice and UBob) were π performed before the photon detectors. Directional couplers cilla initially in equal weight superposition) and ϕ = 2 are abbreviated by ‘dc’, and resistive heaters are shown by (for which hparticle|wavei = 0). For this parameter orange rectangles. Figure taken from Ref. (Peruzzo et al., choice, a Bell value of S = 2.45 ± 0.03 was reported, a 2012). significant violation of the local realistic bound 2 (Pe- ruzzo et al., 2012). However, the authors acknowl- edged correctly that the claim to have ruled out a classi- The experimental results for these two states were very cal description of the wave-particle duality without fur- different. Fig. 27 shows the visibility as a function of ther assumptions would require a loophole-free Bell test, α for state (18) in red and for state (19) in blue. The which has been demonstrated recently by three groups red curve follows the expected form cos2 α, as only the (Hensen2015; Giustina2015; Shalm2015). wave-part in Eq. (18) leads to fringes. The blue curve is more complicated and reflects the fact that there was Two other successful realizations of the quantum also quantum interference between the wave and particle delayed-choice scenario were achieved in nuclear mag- 13 properties. netic resonance (NMR) experiments with CHCl3 Also a two-photon experiment was performed realizing molecules. In Ref. (Roy et al., 2012) the system qubits the proposal of Ref. (Ionicioiu and Terno, 2011) has been (i.e. path in the interferometer) were encoded in the hy- performed (Peruzzo et al., 2012). The setup, which used drogen nuclear spins, while the ancilla qubits (control of an integrated photonic device, is explained in Fig. 28. the interferometer) were encoded in carbon nuclear spins. The measured intensity at detector D0 was in excel- In Ref. (Auccaise et al., 2012) it was exactly the oppo- lent agreement with the theoretical prediction given by site. Both experiments showed excellent agreement with Eq. (13), as shown in Fig. 29. Since the ancilla photon the quantum predictions. 22

mental simulation of a Grover walk, a model that can be used to implement the Grover quantum search algo- rithm (Grover, 1997). The similarities between the theoretical and experimental probability distributions in the Grover walk were above 0.95.

V. REALIZATIONS OF DELAYED-CHOICE ENTANGLEMENT-SWAPPING EXPERIMENTS

Entanglement swapping (Zukowski et al., 1993) is a generalization of quantum teleportation (Bennett et al., 1993) and can teleport entangled states. It is of crucial importance in quantum information processing because it is one of the basic building blocks of quantum repeaters (Briegel et al., 1998; Duan et al., 2001), third-man quantum cryptography (Chen et al., 2005) and other protocols. On the other hand, entanglement swapping also allows experiments on the foundations of quantum physics, including loophole-free Bell tests (Simon and Irvine, 2003) and other fundamental tests of quan- FIG. 30 (Color online). Experimental scheme of the delayed- tum mechanics (Greenberger et al., 2008a,b). The en- choice quantum walk reported in Ref. (Jeong et al., 2013); tanglement swapping protocol itself has been experimen- figure taken therefrom. Entangled photon pairs were gener- tally demonstrated with various physical systems (Bar- ated in a PPKTP crystal. One photon of every pair delayed rett et al., 2004; Halder et al., 2007; Kaltenbaek et al., in a 340 m optical fiber and then sent to Alice, who was 2009; Matsukevich et al., 2008; Pan et al., 1998; Riebe able to perform polarization measurements in any basis. This et al., 2004; Yuan et al., 2008). constituted a delayed-choice projection of the initial coin (polarization) state of the other photon, which was already sent In the light of finding which kind of physical inter- to Bob without any fiber delay. In an optical loop, Bob’s actions and processes are needed for the production of photons performed a 2D (x and y steps) quantum walk in the quantum entanglement, Peres has put forward the radi- time domain. Before each step operation is taken, the coin cal idea of delayed-choice entanglement swapping (Peres, operation (‘Coin 1’ and ‘Coin 2’ are Hadamard gates) was ap- 2000). Realizations of this proposal are discussed in the plied. In order to map the 2D quantum walk lattice uniquely following. onto the photon arrival times, the lengths of the optical fibers (L1-L4) were chosen appropriately.

A. Delayed entanglement swapping

F. Delayed-choice quantum random walk In (Jennewein et al., 2001), a delayed entanglement swapping experiment was performed. For the conceptual An experimental realization of a delayed-choice two- setup see Fig. 9. Detection of photons 2 and 3 by Victor dimensional (2D) quantum walk has been reported was delayed by two 10 m (about 50 ns) optical fiber de- in (Jeong et al., 2013). There, the standard single-photon lays after the outputs of the Bell-state analyzer. Alice’s interferometer was replaced by a 2D quantum walk lat- and Bob’s detectors were located next to each other. The tice, which was mapped to a temporal grid for the arrival traveling time of photons 1 and 4 from the source to these times of a single photon by using polarization optical el- detectors was about 20 ns. Victor was separated from ements and fibers. In a quantum walk, a coin and a Alice and Bob by about 2.5 m, corresponding to lumi- shift operator are applied repeatedly. The experimental nal traveling time of approximately 8 ns between them. scheme is shown in Fig. 30. Therefore, Victor’s measurements were in the time-like The essence of the experiment is similar to the quan- future of Alice’s and Bob’s measurements. The observed tum eraser concept. The way in which a photon inter- fidelity of the measured state ρ14 of photons 1 and 4 − − fered in the 2D quantum walk circuitry depended on its with the ideal singlet state, defined as 14hΨ |ρ14|Ψ i14 polarization, which was determined by the (delayed) po- was around 0.84, both above the classical limit of 2/3 larization measurement of its distant twin. This was and the limit of approximately 0.78 necessary to violate the first experiment realizing a 2D quantum walk with Bell’s inequality, as shown in Fig. 31. This was the first a single photon source and in a delayed-choice fashion. attempt of the realization of delayed-choice entanglement Additionally, the authors also showed the first experi- swapping, although a switchable Bell-state analyzer has VOLUME 88, NUMBER 1 PHYSICAL REVIEW LETTERS 7JANUARY 2002 23

The nondeterministic nature of the photon pair produc- Victor b’’ c’’ tion implies an equal probability for producing two photon BS 2 pairs in separate modes (one photon each in modes 0, 1, 2, 3) or two pairs in the same mode (two photons each in modes 0 and 1 or in modes 2 and 3). The latter can Piezo QRNG Diode laser lead to coincidences in Alice’s detectors behind her beam c’ b’ splitter. We exclude these cases by accepting events only EOM 2 BS 1 where Bob registers a photon each in mode 0 and mode 3. EOM 1 It was shown by Zukowski [12] that despite these effects PID of the nondeterministic photon source experiments of our bc kind still constitute valid demonstrations of nonlocality in 104 m fibre quantum teleportation. BS BBOs IF Mirror PD The entanglement of the teleported state was char- FIG.FIG. 3. 31 Observed Experimental entanglement results offi delayeddelity obtained entanglement through swap- cor- 23 acterized by several correlation measurements between relationping reported measurements in Ref. (Jenneweinbetween photonset al., 0 2001); and 3, figure which taken is a l/2 l/4l/8 PBS SPCM lower bound for the fidelity of the teleportation procedure. f photons 0 and 3 to estimate the fidelity of the entan- therefrom. Data points show the entanglement fidelity ob-0 (f ) is the setting of the polarization analyzer for photon 0 (pho- 1 4 ෇ ͗ 2 2͘ tained3 through correlation measurements between photons 1 glement. As is customary the fidelity F C jrjC ton 3) and f0 ෇ f3. The minimum fidelity of 0.84 is well above and 4. Data shown with͞ white open squares (a black filled cir- measures the quality of the observed state r compared thecle) classical was obtained limit of when2 3 and Victor’s also above Bell state the limit measurement of 0.79 neces- was j 2͘ sary for violating Bell’s inequality. The fidelity is maximal for to the ideal quantum case C . The experimental space-like separated± ± from (in the time-like future of) Alice’s f0 ෇ f3 ෇ 0 , 90 since this is the original basis in which the Alice Bob correlation coefficient E is related to the ideal one and Bob’s measurements. The angles φ0/φ3 are the setting exp photon pairs are produced (jHV͘ or jVH͘). For f ෇ f ෇ 45± ෇ ͓͑ ͒͞ ͔ of the polarization analyzer for photons 1/4 (Fig.0 9),3 which EQM via Eexp 4F 2 1 3 EQM [13]. The correla- the two processes must interfere (jHV͘ 2 jVH͘) which is non- were aligned to be equal. The minimum fidelity is above the FIG. 32 (Color online). Experimental setup of delayed-choice tion coefficientsP are defined as E ෇ ͑N11 2 N12 2 perfect due effects such as mismatched photon collection or limit achievable with classical swapping protocols as well as entanglement swapping reported in Ref. (Ma et al., 2012); ͒͞ ͑ ͒ beam walk-off in the crystals. This leads to a fidelity varia- N21 1 N22 Nij, where Nij f0,f3 are the coinci- above the limit necessary for violating a Bell inequality with figure taken therefrom. Two polarization-entangled photon dences between the i channel of the polarizer of photon 0 tion for the initially entangled pairs, which fully explains the pairs (photons 1&2 and photons 3&4) were generated from observedthe swapped variation entangled of the state. shown fidelity. Thus we conclude that set at angle f0, and the j channel of the polarizer of BBO crystals. Alice and Bob measured the polarization of the fidelity of our Bell-state analysis procedure is about 0.92, photons 1 and 4 in whatever basis they chose. Photons 2 and photon 3 set at angle f3. The results (Fig. 3) show the independent of the polarizations measured. The square dots rep- 3 were each delayed with 104 m fiber and then overlapped on high fidelity of the teleported entanglement. not been implemented. resent the fidelity for the case that Alice’s and Bob’s events are the tunable bipartite state analyzer (BiSA) (purple block). The Clauser-Horne-Shimony-Holt (CHSH) inequality spacelikeWe note separated; that in Ref. thus (Sciarrino no classicalet information al., 2002) a transfersuccessful be- The BiSA either performed a Bell-state measurement (BSM) tween Alice and Bob can influence the results. The circular dot [14] is a variant of Bell’s inequality, which overcomes the experiment on delayed entanglement swapping was per- or a separable-state measurement (SSM), depending on the is the fidelity for the case that Alice’s detections are delayed by inherent limits of a lossy system using a fair sampling formed with two singlet entangled states comprised by outcome of a QRNG. An active phase stabilization system 50 ns with respect to Bob’s detections. This means that Alice’s was employed in order to compensate the phase noise in the hypothesis. It requires four correlation measurements measurementthe vacuum projects and the photons one-photon 0 and states. 3 in an entangledThis allowed state, to at tunable BiSA. performed with different analyzer settings. The CHSH ause time a pairafter of they entangled have already photons been rather registered. than four photons. inequality has the following form: The obtained correlation visibility was (91 ± 2)%.

෇ ͑ 0 0 ͒ ͑ 0 00͒ The travel time from the source to the detectors was S jE f0, f3 2 E f0, f3 j equalB. Delayed-choice within 2 ns for entanglement all photons. swapping Both Alice’s and Bob’s ͑ 00 0 ͒ ͑ 00 00͒ 1 jE f0 , f3 1 E f0 , f3 j # 2, (4) detectors were located next to each other, but Alice and erator (QRNG) was used to make the random choice. BobA were refined separated and conclusive by about realization 2.5 m, ofcorresponding Peres’ gedanken to a Both the choice and the measurement of photons 2 and 3 were in the time-like future of the registration of pho- where S is the “Bell parameter,” E͑f0, f3͒ is the cor- luminalexperiment signaling was reported time of (Ma8 nset between al., 2012). them. The Since layout the relation coefficient for polarization measurements where timeof this resolution experiment of the is illustrated detectors in is Fig.,1 ns, 32. Alice The essen-’s and tons 1 and 4. This projected the state of the two already registered photons, 1 and 4, onto either an entangled or f is the polarizer setting for photon 0, and f is the Bobtial’s point detection was the events implementation were spacelike of bipartite separated state for pro- all 0 3 a separable state. setting for photon 3 [15]. The quantum mechanical pre- measurements.jections based on the random and delayed choice. The 2 QM diction for photon pairs in a C state is E ͑f0, f3͒ ෇ choiceA seemingly was to paradoxicaleither perform situation a Bell-state arises — measurementas suggested ͓ ͑ ͔͒ ͑ 0 0 00 00͒ ෇ (BSM) or a separable-state measurement (SSM) on pho- The diagram of the temporal order of the relevant 2 cos 2 f0 2f3 . The settings f0, f3, fp0 , f3 by Peres [4]—when Alice’s Bell-state analysis is delayed ͑0±, 22.5±,45±, 67.5±͒ maximize S to SQM ෇ 2 2, which longtons after 2 and Bob 3.’s In measurements. order to realize This this, seems a bipartite paradoxical, state events is shown in Fig. 33. For each successful run (a 4- clearly violates the limit of 2 and leads to a contra- becauseanalyzer Alice (BiSA)’s measurement with two-photon projects interference photons 0 on and a high- 3 into fold coincidence count), both Victor’s measurement event diction between local realistic theories and quantum anspeed entangled tunable state beam after splitter they havecombined been with measured. photon Nev- de- and his choice were in the time-like future of Alice’s and Bob’s measurements. mechanics [6]. In our experiment, the four correlation ertheless,tections was quantum used. mechanics predicts the same correla- coefficients between photons 0 and 3 gave the following tions.The Remarkably, initial four-photon Alice entangledis even free state to choosewas of the the form kind results: E͑0±, 22.5±͒ ෇ 20.628 6 0.046, E͑0±, 67.5±͒ ෇ of(8). measurement Alice and she Bob wants measured to perform the polarization on photons of 1 and pho- 2. In that experiment, the existence of entanglement was 10.677 6 0.042, E͑45±,22.5±͒ ෇ 20.541 6 0.045, Insteadtons 1 ofand a 4 Bell-state without anymeasurement delay. Photons she could 2 and also 3 were mea- verified by measuring the state fidelities and the expec- and E͑45±, 67.5±͒ ෇ 20.575 6 0.047. Hence, S ෇ suresent the through polarizations 104 m single-mode of these photons fibers, individually. corresponding Thus to tation values of entanglement witness operators (Gühne 2.421 6 0.091 which clearly violates the classical limit of dependinga delay time on Alice of 520’s laterns. Victor measurement, actively Bob chose’s earlier and im- re- and Toth, 2009). It was found that whether photons 1 2 by 4.6 standard deviations as measured by the statistical sultsplemented indicate the either measurements that photons on photons 0 and 3 2 were and entangled3 (either and 4 were entangled or separable only depended on the BSM or SSM) by using a high-speed tunable bipartite type of the measurements Victor implement, not on the error. The differences in the correlation coefficients come or photons 0 and 1 and photons 2 and 3. This means that state analyzer (BiSA). A quantum random number gen- temporal order (Fig. 34). from the higher correlation fidelity for analyzer settings the physical interpretation of his results depends on Alice’s closer to 0± and 90±, as explained in Fig. 3. later decision.

017903-3 017903-3

24 ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS2294

Time 12.5 ns / unit ab0.8 0.8 VICTOR 520 ns 0.6 0.6 MV 0.4 0.4 BSM or SSM 0.2 0.2 |+〉/|¬〉 0 0

¬0.2 |H〉/|V〉|R〉/|L〉|¬0.2 H〉/|V〉|R〉/|L〉|+〉/|¬〉

¬0.4 ¬0.4 Choice ¬0.6 ¬0.6 Correlation function for photons 1 and 4 Correlation function for photons 1 and 4 ¬0.8 ¬0.8

Figure 3 | Experimental results. Correlation function between photons 1 and 4 for the three mutually unbiased bases (|Hi/|Vi,|Ri/|Li,|+i/|−i). a,b, Victor subjects photonsFIG. 2 and 3 to 34 either Correlation a BSM (a) or an SSM (b functions). These results are from obtainedthe from coincidence experiment counts of photons (Ma 1 andet 4, al.conditioned, on the 00 00 coincidence of same polarization and different spatial output√ modes of photons 2 and 3 (b and c in Fig. 2). a, When Victor performs a BSM and finds − photons 2 and 32012); in the state |φ figurei23 = (|HHi taken23 −|VVi23) therefrom./ 2, entanglement is swappeda: Victor to photons subjected 1 and 4. This is confirmed photons by all three 2 correlation functions being of equal magnitude (within statistical error) and their absolute sum exceeding 1. b, When Victor performs an SSM and finds photons 2 and and| i 3| toi a Bell-state measurement and observed the| i result| i 3 in either the state HH− 23 or VV 23, entanglement is not swapped. This is confirmed by only the correlation function in the H / V basis being significant whereas the others|Φ vanish.i23 The. experimentally Alice’s obtained and correlation Bob’s functions photons of photons 1 1and and4 in the (| 4Hi/| wereVi,|Ri/|Li, projected|+i/|−i) bases are 0.511 ± 0.089, 0.603 ± 0.071, −0.611 ± 0.074 respectively for case a and 0.632 ± 0.059, 0.01 ± 0.072, −0.045 ± 0.070− respectively for case b. Whereas entangled states can showinto maximal correlationsthe corresponding in all three bases (the magnitude entangled of all correlation state functions|Φ equals14i 1, ideally), showing separable states corre- can be maximally C(49 ns, 348correlated ns ) (ideal correlation function 1) only in one basis, the others being 0. The uncertainties represent plus/minus one standard deviation deduced from V ∈ lations in all three mutually unbiased bases |Hi/|Vi, |Ri/|Li, propagated Poissonian statistics. and |+i/|−i. Entanglement between photons 1 and 4 is wit- optical delay splitter. Therefore, the two photons interfere and are projected was performed, and to |HHi23 or |VV i23 when the SSM was onto a Bell statenessed by polarization-resolving by the absolute single-photon sum detections. of theperformed. correlation The normalized values correlation exceeding function E(j) between two The separable-state1. b measurement: When (SSM) Victor corresponds performed to turning aphotons separable-state is defined as: measurement ALICE BOB off the switchable quarter-wave plates. Then the interferometer ⊥ ⊥ ⊥ ⊥ Polarization Polarization acts as a 0/100in beam the splitter,|Hi that/|V is,i abasis, fully reflective also mirror. photons 1 andC 4(j,j) ended+C(j ,j ) up−C(j in,j)− theC(j,j ) measurement measurement E(j) = (3) Therefore, the two photons do not interfere and are projected onto a C(j,j)+C(j⊥,j⊥)+C(j⊥,j)+C(j,j⊥) separable statecorresponding by polarization-resolving separable single-photon detections. state and hence showed correlations M, M35 ns ⊥ | i | i 12 34 AB For detailed informationonly in on that the tunable basis BiSA, but see the not caption the of otherwhere j/j two.stands for horizontal/vertical√ ( H / V ) or plus/minus Fig. 2, the Supplementary Information and ref. 41. (|+i/|−i, with |±i = (|√Hi ± |V i)/ 2), or right/left√ (|Ri/|Li, EPR Source I EPR Source II G,I GII 0 ns For each successful run (a fourfold coincidence count), not only with |Ri = (|Hi + i|V i)/ 2 and |Li = (|Hi − i|V i)/ 2) circular does Victor’s measurement event happen 485 ns later than Alice’s polarization. In equation (3), C(j,j⊥) is the number of coincidences and Bob’s measurement events, but Victor’s choice happens in an under the setting (j,j⊥). In Fig. 3, we show the correlation functions FIG. 33 (Color online). Time diagram of the delayed-choiceinterval of 14 ns to 313 ns later than Alice’s and Bob’s measurement of photons 1 and 4 in these three mutually unbiased bases derived events. Therefore,made independent even ofafter the reference the frame, former Victor’s systemfrom the measurement has already results. Note been that the de- reason why we use entanglement swapping experiment reported in Ref. (Machoice and measurementtected. are in the future light cones of Alice’s one specific entangled state but both separable states to compute and Bob’s measurements. Given the causal structure of special the correlation function is that the measurement solely depends on et al., 2012); figure taken therefrom. Two entangled photonrelativity, that pastIn events delayed-choice can influence (time-like) entanglement future events the settings swapping of the EOMs in experiments, the BiSA. Then the same coincidence pairs (1&2 and 3&4) were generated by EPR sources I andbut II not vice versa, we explicitly implemented the delayed-choice counts (HH and VV combinations of Victor’s detectors) are taken scenario as describedone bycan Peres. demonstrate Only after Victor’s measurement, that whetherfor the computation two quantum of the correlation systems function of photons 1 and (events GI and GII) at 0 ns. Alice and Bob measured thewe po- can assert the quantum states shared by Alice and Bob. Our 4. These counts can belong to Victor obtaining the entangled state − experiment reliesare on the entangled assumption of the or statistical separable independence can|φ bei23 in decided a BSM or the states even|HHi23 afterand |VV theyi23 in an SSM. larization of photons 1 and 4 at 35 ns (events MA and MofB the). QRNG from other events, in particular Alice’s and Bob’s We quantified the quality of the experimentally obtained | i = Photons 2 and 3 were delayed and sent to Victor who chosemeasurement results.have Note been that in a measured. conspiratorial fashion, This Victor’s generalizesstates ρêxp using the the fidelity wave-particle defined as F(ρêxp, out id) choice might not be free but always such that he chooses an SSM Tr(ρêxp|outiidhout|), which is the overlap of ρêxp with the ideally (event CV) to perform a Bell-state measurement (BSM)whenever or Alice’sduality and Bob’s for pair issingle in a separable systems state, and to he anexpected entanglement-separability output state |outiid. The state fidelity of the Bell state can chooses a BSM whenever their pair is in an entangled state. This be decomposed into averages of local measurements in terms of a separable-state measurement (SSM) (event MV). Victor’swould preserveduality the viewpoint for that twoin every (and single run more) Alice and systems.Pauli σ matrices42,43, such as choice and measurement were made after Alice’s and Bob’sBob do receive a particle pair in a definite separable or a definite It is a general feature of delayed-choice− experiments− − entangled state. A possible improvement of our set-up would be F(ρêxp,|φ i) = Tr(ρêxp|φ ihφ |) polarization measurements. space-like separation of Victor’s choice event and the measurement 1 that quantum effects can mimic an influence= Tr[ρˆ (Iˆ of+σˆ σ futureˆ +σˆ σˆ −σˆ σˆ )] (4) events of Alice and Bob to further strengthen the assumption of the 4 exp z z y y x x mutual independenceactions of these onevents. past events. However, there never emerges For each pair of photons 1 and 4, we record the chosen mea- where Iˆ is the identity operator for both photons. An entanglement surement configurationsany paradox and the fourfold if the coincidence quantum detection statewitness is is also viewed employed to only characterize as whether‘cat- entanglement 42,44 ˆ | i = ˆ − VI. CONCLUSION AND OUTLOOK events. All raw data are sorted into four subensembles in real existed between the photons. It is defined as W ( out id) I/2 time accordingalogue to Victor’s choiceof our and measurement knowledge’ results. After(Schrödinger,1935)|outiidhout|. A negative expectation without value of any this entanglement all of the data had been taken, we calculated the polarization witness operator, W (ρêxp,|outiid) = Tr[ρêxpWˆ (|outiid)] = 1/2 − correlation functionunderlying of photons 1 and hidden 4. It is derived variable from their description.F(ρêxp,|outiid), is a sufficient Then condition thestate for entanglement. is coincidence counts of photons 1 and 4 conditional√ on projecting Figure 3a shows that when Victor performs the BSM and projects Delayed-choice gedanken experiments and their real- − − photons 2 anda 3 to probability|φ i23 = (|HHi23 − |VV listi23) for2 when all the possible BSM photons measurement 2 and 3 onto |φ i23, this swaps outcomes the entanglement, which is izations play an important role in the foundations of and not a real physical object. The relative temporal or- 482 NATURE PHYSICS | VOL 8 | JUNE 2012 | www.nature.com/naturephysics quantum physics, because they serve as striking illustra- der of measurement© 2012 events Macmillan Publishers is not Limited. relevant, All rights reserved. and no physi- tions of the counter-intuitive and inherently non-classical cal interactions or signals, let alone into the past, are nec- features of quantum mechanics. A summary of the pho- essary to explain the experimental results. To interpret tonic delayed-choice experiments discussed in this review quantum experiments, any attempt in explaining what is presented in Table I. happens in an individual observation of one system has Wheeler’s delayed-choice experiments challenge a re- to include the whole experimental configuration and also alistic explanation of the wave-particle duality. In such the complete quantum state, potentially describing joint an explanation every photon is assumed to behave either properties with other systems. According to Bohr and definitely as a wave (traveling both paths in an interfer- Wheeler, no elementary phenomenon is a phenomenon ometer) or definitely as a particle (traveling only one of until it is a registered phenomenon (Bohr, 1949; Wheeler, the paths), by adapting a priori on the experimental sit- 1984). In light of quantum erasure and entanglement uation. Especially when the choice of whether or not to swapping, one might like to even say that some regis- insert the second beam splitter into an interferometer is tered phenomena do not have a meaning unless they are made space-like separated from the photon’s entry into put in relationship with other registered phenomena (Ma the interferometer, this picture becomes untenable. et al., 2012). In delayed-choice experiments with two entangled Delayed-choice gedanken experiments and their real- quantum systems such as the delayed-choice quantum izations have played important roles in the development eraser, one can choose that one system exhibits wave or of quantum physics. The applicability of the delayed- particle behavior by choosing different measurements for choice paradigm for practical quantum information pro- the other one. These choices and measurements can be cessing is yet to be explored. For example, the au- 25

TABLE I A summary of delayed-choice experiments realized with photons. ‘C and I’, and ‘M and I’ stand for the space- time relations between events C (choice), I (entry into interferometer), and M (measurement of the photon). Note that in the experiments involving more than one photon, M stands for the measurement of ancillary photon(s). Other abbreviations: ‘sep.’ stands for ‘space-like separated’, ‘after’ and ‘before’ stand for ‘time-like after’ and ‘time-like before’, ‘ext.’ and ‘int.’ stand for ‘external’ and ‘internal’, ‘QRNG’ stands for ‘quantum random number generator’, ‘BS’ for ‘beam splitter’. For example, the entry ‘before’ in the ‘C and I’ column means that C happens time-like before I. Experiment / Ref. Number of photons Nature of the Choice C and IM and I (Alley et al., 1983) 1 ext. choice, photon detection sep. after (Hellmuth et al., 1987) 1 fixed setting before after (Baldzuhn et al., 1989) 1 fixed setting before after (Jacques et al., 2007, 2008) 2 ext. choice, shot noise sep. after (Dopfer, 1998) 2 fixed setting before after (Walborn et al., 2002) 2 fixed setting before after (Kim et al., 2000) 2 int. choice, 50/50 BS after after (Ma et al., 2013) 2 ext. choice, QRNG with 50/50 BS sep. & after sep. & after (Tang et al., 2012) 1 quantum delayed choice, fixed setting before after (Kaiser et al., 2012) 2 quantum delayed choice, fixed setting before sep. (Peruzzo et al., 2012) 2 quantum delayed choice, fixed setting before after (Jeong et al., 2013) 1 fixed setting before after (Jennewein et al., 2001) 4 fixed setting before after (Sciarrino et al., 2002) 2 fixed setting before n/a (Ma et al., 2012) 4 ext. choice, QRNG with 50/50 BS after after thors in (Lee et al., 2014) introduced and experimentally stantaneous quantum computation is intrinsically prob- demonstrated a delayed-choice decoherence suppression abilistic because the BSM results in all four Bell states protocol. In their experiment, the decision to suppress with equal probability of 1/4. decoherence on an entangled two-qubit state is delayed Finally, we observe that the development of quantum until after the decoherence and even after the detection mechanics has been accompanied initially by a series of of the qubit. This result suggests a new way to tackle ingenious gedanken experiments, which have – with the Markovian decoherence in a delayed-choice way, which advance of technology – found more and more realiza- could be useful for practical entanglement distribution tions over time. This again has opened up avenues for over a dissipative channel. new experiments and even applications. Likewise, while The concept of delayed-choice entanglement swapping the history of the delayed-choice paradigm dates back is of importance for the security of quantum commu- to the early days of quantum mechanics, only in the nication schemes such as third-man quantum cryptog- past decades have many remarkable experiments demon- raphy (Chen et al., 2005) and could also be employed strated its counter-intuitive aspects in different scenarios in probabilistic instantaneous quantum computing. In and with different physical systems. It can be expected the latter case, quantum state teleportation and entan- that delayed-choice gedanken experiments will continue glement swapping imply a computational speed-up in to lead to novel foundational tests as well as further prac- time over classical procedures (Brukner et al., 2003; Jen- tical implementations. newein, 2002). This can be realized by sending one photon of a Bell state into the input of a quantum computer and performing a quantum computation with it. Since this photon is a part of a Bell state, its individual prop- ACKNOWLEDGMENTS erty is not well defined. Therefore, also the output of the quantum computation will not be defined. However, as We thank Guido Bacciagaluppi, Daniel Greenberger, soon as the required input is known, it can be teleported Radu Ionicioiu, Thomas Jennewein, Florian Kaiser, Mar- onto the state of the photon which had been fed into lan Scully, Sébastien Tanzilli, Daniel R. Terno, and the quantum computer. If the Bell-state measurement Marek Zukowski for helpful comments and remarks. (BSM) results in one specific Bell state which requires X.S.M. is supported by a Marie Curie International Out- no corrective unitary transformation, then immediately going Fellowship within the 7th European Community the output of the quantum computer will be projected Framework Programme. A.Z. acknowledges support by into the correct result. By this means the computation the Austrian Science Fund through the SFB program and is performed quasi instantaneously. Note that this inby the European Commission. 26

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