Gedankenexperiments have consistently played a major role in the development of quantum theory. A paradigmatic example is Wheeler’s delayed-choice experiment, a wave-particle duality test that cannot be fully understood using only classical concepts. We implement Wheeler’s idea along a satellite-ground interferometer that extends for thousands of kilometers in space. We exploit temporal and polarization degrees of freedom of photons reflected by a fast-moving satellite equipped with retroreflecting mirrors. We observe the complementary wave- or particle-like behaviors at the ground station by choosing the measurement apparatus while the photons are propagating from the satellite to the ground. Our results confirm quantum mechanical predictions, demonstrating the need of the dual wave-particle interpretation at this unprecedented scale. Our work paves the way for novel applications of quantum mechanics in space links involving multiple photon degrees of freedom.

So far, several implementations of Wheeler’s Gedankenexperiment have been realized on the ground [see the study of Jacques et al. ( 27 ) for the realization closest to the original idea and the study of Ma et al. ( 28 ) for a complete review]. An alternative way of interpreting the delayed-choice experiment is within the quantum-erasure framework ( 29 , 30 ). Furthermore, a quantum delayed-choice version of the experiment, where a quantum ancilla controls the second BS, has been recently proposed ( 31 ) and realized ( 32 – 34 ).

Here, we extend the delayed-choice paradigm to space, as sketched in Fig. 1 , by exploiting the temporal degree of freedom of photons reflected by a rapidly moving satellite in orbit. The two paths of the interferometer are represented by two time bins that allow us to both obtain clear which-path information and observe interference modulated by the satellite motion. We also exploit photon polarization as an ancillary degree of freedom to implement the insertion or removal of the BS at the measurement apparatus. We are able to demonstrate the need of the dual wave-particle model for a propagation distance of up to 3500 km, demonstrating the validity of the quantum mechanical description at a much larger scale than all previous experiments.

If the configuration is chosen after the photon enters the interferometer, then a purely classical interpretation of the process in which the photon decides its nature at the first BS would imply a seeming violation of causality. On the other hand, in the quantum mechanical interpretation of the experiment, the photon maintains its dual wave-particle nature until the very end of the experiment, when it is detected.

A photon wave packet enters the first BS of an interferometer, which extends along thousands of kilometers in space. The interferometer can be randomly arranged according to two configurations that correspond to the presence or absence of the second BS (in/out BS) located on Earth. Following Wheeler’s idea, the configuration choice is performed when the photon has already entered the interferometer. In our actual implementation, the interferometer begins and terminates on the ground, extending up to the target satellite, and the measurement choice performed on ground is space-like separated from the photon reflection by the satellite.

John Wheeler pushed this observation to the extreme and conceived his delayed-choice Gedankenexperiment to highlight the contradictory interpretation given by classical physics ( 25 , 26 ). In his idea, a photon emerging from the first beam splitter (BS) of a Mach-Zehnder interferometer (MZI) ( Fig. 1 ) may find two alternative configurations. Given the presence or absence of a second BS at the output of the interferometer, the apparatus measures the wave- or particle-like character of the photon. If the BS is absent, then only one of the two detectors will fire, reflecting the fact that the photon traveled along only one arm of the interferometer and revealing which path it took, as a classical particle would have done. If the BS is present, then interference can be observed, reflecting the fact that the photon traveled both routes, as a classical wave would have done.

These thought experiments played a primary role in the famous debate between Einstein and Bohr ( 21 ), concerning the completeness of quantum mechanics ( 22 , 23 ) and the concept of complementarity ( 24 ). The most disturbing implication of complementarity is the wave-particle duality of quantum matter, which is the impossibility of revealing both the wave- and particle-like properties of a quantum object at the same time. Bohr pointed out that it is necessary to consider the whole apparatus to determine which property is measured, stating that there is no difference “whether our plans of constructing or handling the instruments are fixed beforehand or whether we postpone the completion of our planning until a later moment” ( 21 ).

Quantum communications in space enable the investigation of the basic principles of quantum mechanics in a radically new scenario. As envisioned in theoretical works ( 1 – 6 ) and satellite mission proposals ( 7 – 9 ), quantum information protocols ( 10 , 11 ) have breached the space frontier ( 12 ) in recent experimental demonstrations ( 13 – 20 ). These developments foster the implementation in space of fundamental tests of Physics, such as the Gedankenexperiments that highlight the counterintuitive aspects of quantum theory.

RESULTS

Description of the experiment We realized the experiment at the Matera Laser Ranging Observatory (MLRO) of the Italian Space Agency. At the MLRO, we have already tested the feasibility of receiving qubits encoded in the polarization of single photons (15) and of observing interference between two temporal modes throughout satellite-ground channels in the study of Vallone et al. (17). A pulsed laser [repetition rate, 100 MHz; wavelength λ = 532 nm; energy per pulse, ~1nJ], diagonally polarized and paced by an atomic clock, enters into an unbalanced MZI, as sketched in Fig. 2. The combined action of the first polarizing BS (MZI-PBS) and of the imbalance of the MZI transforms each laser pulse into a superposition of two temporal and polarization modes. The long arm of the MZI is traveled by the vertically polarized component of the beam, whereas the horizontally polarized component travels along the short arm. The separation between the two temporal modes is about Δt ≈ 3.5 ns (see Materials and Methods for more details). Fig. 2 Scheme of the experimental setup and detection histograms. A pulsed laser synchronized with the MLRO atomic clock exits the MZI in two temporal and polarization (pol) modes. The sHWP leaves the pulses unperturbed, and the telescope directs the beam to a target satellite. After the reflection, the photons are collected on the ground by the same telescope and injected into the optical table. The photons pass through the sHWP whose behavior is set according to the bit b extracted from an on-demand QRNG. The QRNG is inquired twice in each 100-ms cycle of the experiment, as detailed in the main text. In the inset, a 1-s sample of the extracted bits is shown. At the MZI output, two wave plates, a PBS, and two single-photon detectors (SPDs) perform a polarization measurement in the {|+⟩, |−⟩} basis. According to the value b of the random bit, interference or which-path measurement is performed, as shown by the detection histograms for a passage of the Starlette satellite. The counts in the central peak on the left histogram are comparable to the sum of the counts associated to the lateral peaks on the right one, as expected. HWP, half–wave plate; QWP, quarter–wave plate. The pulses then pass through two liquid crystal retarders (LCRs) whose combined action is equivalent to a single switchable (on/off) half–wave plate (sHWP) inclined at 45° with respect to the fast axis. During the transmission period, the sHWP is always off, leaving the outgoing beam unperturbed. The light is then directed to a target satellite equipped with polarization-maintaining corner-cube retroreflectors via a telescope (15). The corner cubes of the target satellite redirect the beam back to the ground station. Furthermore, the radial motion of the satellite introduces a kinematic phase shift between the two time bins given by (1)where β(t) = v r (t)/c, with v r (t) as the instantaneous satellite radial velocity with respect to the ground and c as the speed of light in vacuum, as demonstrated by our group in a previous study (17). The photons returning from the satellite are collected by the same telescope and injected into the optical table, where they reencounter the same sHWP and MZI. At an exit port of the MZI-PBS (Fig. 2), we perform a polarization measurement in the diagonal and antidiagonal basis {| + ⟩, | − ⟩} with | ± ⟩ = (|H⟩ ± |V⟩)/√ 2, where |H⟩, |V⟩ are the horizontal and vertical polarization states, respectively. While the photons are propagating back to MLRO, an on-demand quantum random number generator (QRNG) extracts a random bit b ∈ {0, 1} with a 50% probability. The QRNG is based on differences of the arrival time of single photons in attenuated light (35), and its relevant features will be detailed in Materials and Methods. The bit value sets the voltages V b applied to the LCRs, determining the on or off behavior of the sHWP. The latter determines whether we perform a measurement that reveals the particle-like (sHWP on) or wave-like (sHWP off) behavior of the photons returning from the satellite. Because the random bits are generated while the photons are traveling from the satellite to the ground station, we ensure a space-like separation between the measurement choice and the last interaction with the apparatus, that is, the reflection by the satellite (as detailed in the “Implementation of the delayed choice” section). Let us first suppose that the QRNG extracts a b = 0 bit causing the sHWP to remain off, leaving the polarization of the photon unchanged as it reenters the MZI. At the exit port of the MZI-PBS toward the detectors in Fig. 2, only the horizontally polarized component that propagated through the long arm and the vertically polarized component that traveled along the short arm can be detected. Because this is the reverse situation compared to the outward passage through the MZI, the two polarization modes will recombine into a single temporal mode, losing all which-path information and allowing us to observe a ϕ-dependent interference, which is the fingerprint of the wave-like nature of the photon. In this case, the probabilities of a click in the detectors Det ± are given by (2)where is the theoretical visibility as in the study of Vallone et al. (17). Let us now suppose that the QRNG extracts a b = 1 bit, switching the sHWP on and swapping the horizontal and vertical polarizations before the photon reenters the MZI. The polarization transformation causes each component of the state to retravel along the same arm compared to the outwards passage through the MZI. As a result, the photon can be detected at two distinct times separated by 2Δt (with 50% probability for each detector Det ± , that is, ), giving which-path information and evidencing the particle-like nature of the photon.

Implementation of the delayed choice Simultaneous tracking of the target satellite via the satellite laser ranging (SLR) technique allows the determination with few tens of picosecond accuracy of the photon’s time of flight or round trip time (rtt). Furthermore, SLR allows an accurate estimation of the satellite radial velocity, which is crucial for the determination of the kinematic phase ϕ(t). The laser ranging technique exploits a bright laser signal with pulses at a 10-Hz repetition rate, synchronized with the 100-MHz train used in the experiment (see Materials and Methods for more details). We separated each 100-ms cycle between two subsequent SLR pulses in two periods by using two mechanical shutters (Figs. 2 and 3). In the first half of the 100 ms, only the transmitting shutter (TX shutter) is open, whereas the receiving one (RX shutter) is closed to protect the detectors. In the second half of the time slot, the TX shutter is closed, whereas the RX shutter is open, and the detectors can receive the photons coming from the satellite. Furthermore, because the shutters require a certain time to open and close completely, the effective detection time period is limited by the shutters transition time (t trans ~ 5 ms), as sketched in the figure. On that basis, a precise temporal window τ = rtt − t trans exists, where we expect to receive photons from the satellite. The value of τ depends on the actual rtt, which is continuously changing along the satellite orbit. However, the SLR technique described above allows the transmission and reception phases of the protocol to be synchronized in real time by using a fast field-programmable gate array (FPGA) controller. Fig. 3 Minkowski diagram of the experiment. Along the temporal axis (not to scale) a 100-ms cycle between two SLR pulses is represented. The x axis represents the radial coordinate (not to scale) from the detectors, where x 0 is the position of both the sHWP and the QRNG. The dotted line is the satellite worldline. We only considered the detections in the temporal window τ, as detailed in the main text. A fast FPGA controller synchronized in real time with the MLRO tracking system drives the two shutters and the QRNG. For each cycle, we performed two independent measurements via the random bit extracted by the QRNG at times and , causally disconnected from the photon reflection at the satellite. The cycle is repeated for each 100-MHz train between two SLR pulses. A faithful realization of Wheeler’s experiment requires that the entrance of the photon in the interferometer is not in the future light cone of the measurement choice. Moreover, the latter must be realized in a random manner: This prevents any causal influence of the measurement choice on the behavior of the photon. Our implementation is performed over a space channel with a length of the order of thousands of kilometers, corresponding to an rtt of the order of 10 ms. We designed the experiment to guarantee that the choice of the measurement apparatus is space-like separated from the reflection of the photon from the satellite, as shown in the Minkowski diagram in Fig. 3. This guarantees that, in a purely classical interpretation, a photon “should have decided its nature” at most at its reflection from the satellite. For each cycle, we performed two independent choices that will affect the detections in the acceptable temporal window τ by driving the QRNG with the same FPGA controller used for the shutters. The sHWP behavior at the photon return is set according to the bits b 1 and b 2 extracted by the QRNG. The first choice is performed at , corresponding to the middle of the shutter transition phase. The second choice is at , which occurs with a delay rtt/2 with respect to the first choice. The detected photons are divided into two groups, each characterized by a value of the bit choice. In this way, all the photons of a given group were already reflected by the satellite when the corresponding bit choice was performed.