Delayed-choice gedanken experiments and their realizations

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 system to an entanglement-separability duality of multiple systems.

Figure 1

Schematic of a Mach-Zehnder interferometer. A quantum system enters from the left via a semitransparent beam splitter. When detectors are placed in the paths $a$ and $b$ inside the interferometer, the system is found in one and only one of the arms with probability $1/2$ each. This reflects the picture that it traveled one of the two paths as a particle. If, however, detectors are not placed inside the interferometer but at the exit ports $a^{\prime }$ and $b^{\prime }$ after the second beam splitter, the probability of detection depends on the phase $\phi$. This reflects the view that the system traveled both paths $a$ and $b$ as a wave, leading to constructive and destructive interference.

Figure 2

Heisenberg’s microscope drawing from the notes of his 1929 lectures, printed in 1930. A photon with energy $h\nu$ is scattered at an electron (represented by a dot) and reaches the observer via a microscope with opening angle $ϵ$. The product of uncertainty of the position measurement and the momentum disturbance is of the order of Planck’s constant $h$. From [52].

Figure 3

The quantum “phenomenon” can be viewed as a “great smoky dragon.” From [87].

Figure 4

Wheeler’s delayed-choice gedanken experiment with a single-photon wave packet in a Mach-Zehnder interferometer. Top: The second half-silvered mirror ($\frac{1}{2}S$) of the interferometer can be inserted or removed at will. Bottom left: When $\frac{1}{2}S$ is removed, the detectors allow one to determine through which path the photon propagated. Which detector fires for an individual photon is absolutely random. Bottom right: When $\frac{1}{2}S$ is inserted, detection probabilities of the two detectors depend on the length difference between the two arms. From [125].

Figure 5

Delayed-choice gedanken experiment at the cosmological scale. Left: Because of the gravitational lens action of galaxy $G\text{−}1$, light generated from a quasar ($Q$) has two possible paths to reach the receptor. This mimics the setup in Fig. 4. Center: The receptor setup. Filters are used to increase the coherence length of the light, thus allowing one to perform the interference experiment. A fiber optics delay loop adjusts the phase of the interferometer. Right: The choice to not insert (top) or insert (bottom) the half-silvered mirror at the final stage of the experiment allows one to either measure which particular route the light traveled or what the relative phase of the two routes was when it traveled both of them. Given the distance between the quasar and the receptor (billions of light years), the choice can be made long after the light’s entry into the interferometer, an extreme example of the delayed-choice gedanken experiment. From [125].

Figure 6

The delayed-choice quantum eraser following [108]. (A) Two two-level atoms are initially in the state $b$. The incident pulse $l_1$ excites one of the two atoms to state $a$ from where it decays to state $b$, emitting a photon labeled $\gamma$. Because the final states of both atoms are identical, one can observe interference of the photons at the detector $D$. (B) Two atoms are initially in the ground state $c$ and one of them is excited by the pulse $l_1$ to state $a$ from where it decays to state $b$. Since the final states of both atoms are different, one cannot observe interference of the photons. (C) A fourth level is added. A pulse $l_2$ excites the atom from state $b$ to $b^{\prime }$. The atom in $b^{\prime }$ emits a photon labeled $\varphi$ and ends up in state $c$. If one can detect $\varphi$ in a way that the which-path information is erased, interference can be recovered for photon $\gamma$. From [1].

Figure 7

Proposed delayed-choice quantum-eraser setup. Laser pulses $l_1$ and $l_2$ are incident on atoms at sites 1 and 2. A scattered photon, ${\gamma }_1$ or ${\gamma }_2$, is generated by $a\to b$ atomic transition. The atom’s decay from $b^{\prime }\to c$ produces a photon $\varphi$. This corresponds to the situation depicted in Fig. 6. In order to operate this experiment in a delayed-choice mode, two elliptical cavities and an electro-optical shutter are employed. The cavities reflect $\varphi$ onto a common detector. The electro-optical shutter transmits $\varphi$ only when the switch is open. By opening or closing the shutter, one can either erase the information which atom (1 or 2) emitted the photon or not. This determines whether one can observe the wave or particle nature of $\gamma$. The choice can be delayed with respect to the generation of $\gamma$. From [108].

Figure 8

Proposed quantum-eraser setup. A detector wall, separating two cavities for microwave photons, is sandwiched by two electro-optic shutters. (a) By always opening only one of the shutters, the photon detections reveal the cavity where the photon was emitted and, thus, which-path information for the atoms. Consequently, no interference pattern emerges. When opening both shutters the photon detections will erase which-path information of the atoms and interference shows up. (b) Both shutters are open. It is assumed that the detector wall can be excited only by the symmetric photon state $|+{⟩}_{12}$. Hence, if a photon is being emitted in one of the cavities but not detected, it was in the antisymmetric state $|-{⟩}_{12}$. The detections of the symmetric and antisymmetric photon state give rise to oppositely modulated interference fringes of the atoms (solid and dashed curves), respectively. From [110].

Figure 9

The concept of delayed-choice entanglement swapping. Two entangled pairs of photons 1 and 2, and 3 and 4 are produced in the joint state $|{\mathrm{\Psi }}^{-}{⟩}_{12}|{\mathrm{\Psi }}^{-}{⟩}_{34}$ from the EPR 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. Victor has the freedom of performing either an entangled- or a separable-state measurement on photons 2 and 3. If Victor decides to perform a separable-state measurement in the four-dimensional two-particle basis $\{|H{⟩}_2|H{⟩}_3,|H{⟩}_2|V{⟩}_3,|V{⟩}_2|H{⟩}_3,|V{⟩}_2|V{⟩}_3\}$, then the outcome is random and one of these four product states. Photons 1 and 4 are projected into the corresponding product state and remain separable. On the other hand, if Victor chooses to perform an entangled-state measurement on photons 2 and 3 in the Bell-state basis $\{|{\mathrm{\Psi }}^{+}{⟩}_{23},|{\mathrm{\Psi }}^{-}{⟩}_{23},|{\mathrm{\Phi }}^{+}{⟩}_{23},|{\mathrm{\Phi }}^{-}{⟩}_{23}\}$, then the random result is one of the four Bell states. Consequently, photons 1 and 4 are also projected into the corresponding Bell state. Therefore, entanglement is swapped from pairs 1 and 2, and 3 and 4 to pairs 2 and 3, and 1 and 4. Adapted from [81].

Figure 10

(a) The “classical” delayed-choice experiment: The second beam splitter ${\mathrm{BS}}_2$ is inserted or not after the photon has already entered the interferometer. (b) An equivalent quantum network: An ancilla (lower input) acts as a quantum random number generator (QRNG). Its initial state $|0⟩$ is transformed by a Hadamard gate $H$ into $(|0⟩+|1⟩)/\sqrt{2}$. A measurement in the computational $|0⟩/|1⟩$ basis gives a random outcome, which determines whether or not the second Hadamard gate is applied with the system qubit (equivalent to ${\mathrm{BS}}_2$). (c) Delayed choice with a quantum beam splitter: The second beam splitter ${\mathrm{BS}}_2$ (represented by a controlled Hadamard gate) is coherently controlled by the state of the ancilla qubit. It is now in the superposition of present and absent. (d) The QRNG can be biased by preparing the ancilla in the state $\cos \alpha |0⟩+\sin \alpha |1⟩$. From [60].

Figure 11

Setup of the delayed-choice experiment. The combination of a Pockels cell (PC) and a polarizer (POL) in the upper arm of the interferometer was used as a shutter. From [53].

Figure 12

Experimental results of the delayed-choice experiment. Interference patterns for normal mode (dots) and delayed-choice mode (crosses) measured by PM 1 are similar and consistent with quantum-mechanical predictions. From [53].

Figure 13

Schematic diagram of the device generating the random choices proposed by [2] and used by [3]. A weak light pulse emitted from a light emitting diode has a pulse duration of 0.67 ns. The detection event of this light pulse makes the random choice which determines the setting of the Pockels cell. To realize that, a photocathode with 50% probability of producing a photoelectron within 1 ns is amplified by a fast amplifier within 2 ns. This electric pulse then triggers the avalanche transistor chain switch and hence the Pockels cell. The time of the choice can be tuned with respect to the photon’s entry into the MZI. From [2], and [3].

Figure 14

Space-time diagram. 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 spacelike separated from the photon’s entry into the interferometer (origin of the diagram). From [3].

Figure 15

Setup of the delayed-choice experiment reported by [5]. Photon pairs were produced by parametric downconversion in the ${\mathrm{LiIO}}_3$ 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 anticlockwise 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 reference point $I$ in the interferometer. From [5].

Figure 16

Experimental results of the delayed-choice experiment by [5]. (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. From [5].

Figure 17

The delayed-choice experiment realized by [61]. (a) Layout of the setup. Single photons were generated by NV color centers in diamond. A 48-m-long polarization interferometer and a fast EOM, controlled by a QRNG, were used to fulfill the relativistic spacelike separation condition. (b) The space-time diagram. The choice whether to open or close the interferometer was spacelike separated from the entry of the photon into the interferometer. (c) If the EOM was on, the polarization distinguishability of the two paths was erased and thus an interference pattern emerged. (d) If, however, the EOM was switched off, no interference showed up due to the polarization distinguishability of the two paths. From [61].

Figure 18

Experimental visibility ($V^2$, starting at 0) and distinguishability ($D^2$, starting at 1) results from [62]. (a) $V^2$ and $D^2$ as functions of the EOM voltage (corresponding to the reflectivity of the second beam splitter). Solid lines are the theoretical expectations. (b) $V^2+D^2$ as a function of the EOM voltage, in agreement with Eq. (14). From [62].

Figure 19

(a) Experimental scheme of the experiment, using a momentum entangled state of two photons. See text for details. (b) A high-visibility interference pattern in the conditional photon counts was obtained when $D1$ was positioned in the focal plane of the lens, thus erasing all path information. (c) The profile of the double slit was resolved when $D1$ was positioned in the image plane of the lens, revealing path information and therefore no interference pattern arose. From [29].

Figure 20

Quantum-eraser experiment from [31]) based on an atomic interferometer. (a) The atomic beam $A$ was split into two beams: beam $B$ was reflected by the first Bragg grating formed by a standing wave, and beam $C$ was transmitted. The atomic beams freely propagated for a time duration of $t_{\text{sep}}$ and acquired a lateral separation $d$. The beams $B$ and $C$ were then split again by a second standing light wave grating. In the far field, complementary spatial interference patterns were observed in two regions. These interference patterns were due to superpositions of beams $D$ and $E$ ($F$ and $G$). (b), (c) The spatial fringe patterns in the far field of the interferometer for $t_{\text{sep}}=105\mu \mathrm{s}$ with $d=1.3\mu \mathrm{m}$ and $t_{\text{sep}}=255\mu \mathrm{s}$ with $d=3.1\mu \mathrm{m}$, respectively. The left and right complementary interference patterns were, respectively, generated by the atomic beams $D$ and $E$, and $F$ and $G$ [shown in (a)]. The dashed lines indicate the sum of the intensities of two interference patterns obtained with a relative phase shift of $\pi$. (d) The simplified scheme of the internal atomic states, which were addressed using microwave (mw) radiation and light. (e) The principle of correlating the path the atoms took with their internal electronic states. The standing-wave grating produced a relative $\pi$ phase shift of state $|2⟩$ relative to $|3⟩$ conditional on its path. A Ramsey interferometer employed two microwave $\pi /2$ pulses and converted different relative phases into different final internal states $|2⟩$ and $|3⟩$, respectively. (f) When the which-path information was stored in the internal atomic state, the interference patterns vanished. From [31].

Figure 21

Delayed-choice quantum-eraser experiment realized by [74]. (a) Experimental scheme. Pairs of entangled photons were emitted from either region $A$ or region $B$ of a BBO crystal via spontaneous parametric downconversion. These two emission processes were coherent. Detections at $D_3$ or $D_4$ provided which-path information and detections at $D_1$ or $D_2$ erased it. (b) Coincidence counts between $D_0$ and $D_3$ as a function of the lateral position $x_0$ of $D_0$. Absence of interference was demonstrated. (c) Coincidence counts between $D_0$ and $D_1$ as well as between $D_0$ and $D_2$ are plotted as a function of $x_0$. Interference fringes were obtained. See text for details. From [74].

Figure 22

Quantum erasure with causally disconnected choice. (a) Principle: The source $S$ emitted path-polarization-entangled photon pairs. The system photons propagated through an interferometer (right side), and the environment photons were subject to polarization measurements (left). (b) Scheme of the Vienna experiment: In Lab 1, the polarization-entangled state generated via type-II spontaneous parametric downconversion was converted into a hybrid entangled state with a polarizing beam splitter (PBS1) and two fiber polarization controllers (FPC). In Lab 2, the polarization projection setup of the environment photon consisted of an electro-optical modulator (EOM) and another polarizing beam splitter (PBS2). In Lab 3, the choice was made with a quantum random number generator (QRNG) ([66]). (c) Space-time diagram. The choice-related events ${\mathbf{C}}_e$ and the polarization projection of the environment photon ${\mathbf{P}}_e$ were spacelike separated from all events of the interferometric measurement of the system photon ${\mathbf{I}}_s$. Additionally, the events ${\mathbf{C}}_e$ were also spacelike separated from the emission of the entangled photon pair from the source ${\mathbf{E}}_{se}$. The shaded areas are the past and the future light cones of events ${\mathbf{I}}_s$. This ensured that Einstein locality was fulfilled. From [82].

Figure 23

Experimental test of the complementarity inequality under Einstein locality, manifested by a trade-off of the which-path information parameter $D$ and the interference visibility $V$. The dotted line is the ideal curve from the saturation of the complementary inequality. The solid line $V=0.95[1-{(D/0.97)}^2{]}^{1/2}$ is the estimation from experimental imperfections. From [82].

Figure 24

Space-time diagram of the experiment reported by [71]. The detections of the corroborative photon and the test photon were spacelike separated. From [71].

Figure 25

Experimental results reported by [71]. When the corroborative photon was found to be horizontally polarized, the test photons produced an intensity pattern following Eq. (13), with $\theta \equiv \phi$. The same pattern emerged when the corroborative photon was measured in the $|+⟩/|-⟩$ basis, verifying the entanglement. From [71].

Figure 26

Experimental quantum delayed choice with single photons. Single photons entered the interferometer at beam displacer BD1 which split the light into horizontal and vertical polarization. The phase $\phi$ was scanned by the quartz plates before BD1. The “second beam splitter” in the closed (open) interferometer was provided by the combination of BD3, BD4, and half-wave plates HWP2 (HWP1). For HWP2 in the wave layer, the optical-axis direction $\theta$ was set to 22.5° and interference appears (closed interferometer). For HWP1 in the particle layer $\theta$ was set to 0° (open interferometer), showing the particle properties. Depending on the polarization (parameter $\alpha$), BD2 controlled whether the photons passed through the particle or wave layer. The two layers were combined by BD5. A 45° polarizer could be inserted to postselect on the polarization state $(|H⟩+|V⟩)/\sqrt{2}$. Finally, two detectors counted the photons in paths 0 and 1. From [117].

Figure 27

Visibility as a function of $\alpha$ for the mixed state (18) (solid circles) and for the superposition state (19) (diamonds). The solid circle curve has the form ${\cos }^2\alpha$, while the diamond one also reflects interference between wave and particle properties. The larger symbols are experimental data, while the smaller symbols are theoretical simulation results. From [117].

Figure 28

Two-photon experimental quantum delayed-choice experiment. Nonentangled photon pairs were injected into an integrated photonic device. The system photon ($s$, black optical path) passed a Hadamard gate ($H$) and a phase shifter $\phi$. The “second beam splitter” was a controlled Hadamard gate ($CH$), implemented with additional Mach-Zehnder interferometers. The ancilla photon ($a$, top optical path) passed a phase shifter $\alpha$, allowing a superposition of present and absent beam splitter for the system photon. For the Bell test, single qubit rotations ($U_{\text{Alice}}$ and $U_{\text{Bob}}$) were performed before the photon detectors. Directional couplers are abbreviated by “dc” and resistive heaters are shown by rectangles. From [95].

Figure 29

Continuous transition between wave and particle behavior. The experimental data are shown by white dots and were fitted (colored surface) based on Eq. (13). From [95].

Figure 30

Experimental scheme of the delayed-choice quantum walk reported by [68]. Entangled photon pairs were generated in a periodically poled ${\mathrm{KTiOPO}}_4$ (PPKTP) crystal. One photon of every pair delayed in a 340 m optical fiber and then sent to Alice, who was able to perform polarization measurements in any basis. This constituted a delayed-choice projection of the initial coin (polarization) state of the other photon, which was already sent to Bob without any fiber delay. In an optical loop, Bob’s photons performed a 2D ($x$ and $y$ steps) quantum walk in the time domain. Before each step operation is taken, the coin operation (“Coin 1” and “Coin 2” are Hadamard gates) was applied. In order to map the 2D quantum walk lattice uniquely onto the photon arrival times, the lengths of the optical fibers ($L1$–$L4$) were chosen appropriately. From [68].

Figure 31

Experimental results of delayed entanglement swapping. Data points show the entanglement fidelity obtained through correlation measurements between photons 1 and 4. Data shown with white open squares (a black filled circle) was obtained when Victor’s Bell-state measurement was spacelike separated from (in the timelike future of) Alice’s and Bob’s measurements. The angles ${\varphi }_0/{\varphi }_3$ are the setting of the polarization analyzer for photons $1/4$ (Fig. 9), which were aligned to be equal. The minimum fidelity is above the limit achievable with classical swapping protocols as well as above the limit necessary for violating a Bell inequality with the swapped entangled state. From [67].

Figure 32

Experimental setup of delayed-choice entanglement swapping. Two polarization-entangled photon pairs (photons 1 and 2, and photons 3 and 4) were generated from BBO crystals. Alice and Bob measured the polarization of photons 1 and 4 in whatever basis they chose. Photons 2 and 3 were each delayed with 104 m fiber and then overlapped on the tunable bipartite state analyzer (BiSA) (top block). The BiSA performed either a Bell-state measurement (BSM) or a separable-state measurement (SSM), depending on the outcome of a QRNG. An active phase stabilization system was employed in order to compensate the phase noise in the tunable BiSA. From [81].

Figure 33

Time diagram of the delayed-choice entanglement-swapping experiment. Two entangled photon pairs (1 and 2, and 3 and 4) were generated by EPR sources I and II (events ${\mathbf{G}}_{\mathrm{I}}$ and ${\mathbf{G}}_{\mathrm{II}}$) at 0 ns. Alice and Bob measured the polarization of photons 1 and 4 at 35 ns (events ${\mathbf{M}}_{\mathrm{A}}$ and ${\mathbf{M}}_{\mathrm{B}}$). Photons 2 and 3 were delayed and sent to Victor who chose (event ${\mathbf{C}}_{\mathrm{V}}$) to perform a Bell-state measurement (BSM) or a separable-state measurement (SSM) (event ${\mathbf{M}}_{\mathrm{V}}$). Victor’s choice and measurement were made after Alice’s and Bob’s polarization measurements. From [81].

Figure 34

Correlation functions from the experiment of [81]. (a) Victor subjected photons 2 and 3 to a Bell-state measurement and observed the result $|{\mathrm{\Phi }}^{-}{⟩}_{23}$. Alice’s and Bob’s photons 1 and 4 were projected into the corresponding entangled state $|{\mathrm{\Phi }}_{14}^{-}⟩$, showing correlations in all three mutually unbiased bases $|H⟩/|V⟩$, $|R⟩/|L⟩$, and $|+⟩/|-⟩$. Entanglement between photons 1 and 4 is witnessed by the absolute sum of the correlation values exceeding 1. (b) When Victor performed a separable-state measurement in the $|H⟩/|V⟩$ basis, photons 1 and 4 also ended up in the corresponding separable state and hence showed correlations only in that basis but not the other two. From [81].