Time from quantum entanglement: An experimental illustration

Ekaterina Moreva, Giorgio Brida, Marco Gramegna, Vittorio Giovannetti, Lorenzo Maccone, and Marco Genovese

Phys. Rev. A 89, 052122 – Published 20 May 2014

Abstract

In previous years several theoretical papers discussed if time can be an emergent property deriving from quantum correlations. Here, to provide insight into how this phenomenon can occur, we present an experiment that illustrates Page and Wootters' mechanism of “static” time [D. N. Page and W. K. Wootters, Phys. Rev. D 27, 2885 (1983)], and Gambini et al. for subsequent refinements [R. Gambini et al., Phys. Rev. D 79, 041501(R) (2009)]. A static, entangled state between a clock system and the rest of the universe is perceived as evolving by internal observers that test the correlations between the two subsystems. We implement this mechanism using an entangled state of the polarization of two photons, one of which is used as a clock to gauge the evolution of the second: An “internal” observer that becomes correlated with the clock photon sees the other system evolve, while an “external” observer that only observes global properties of the two photons can prove it is static.

Received 7 November 2013

DOI:https://doi.org/10.1103/PhysRevA.89.052122

Authors & Affiliations

Ekaterina Moreva^1,2, Giorgio Brida¹, Marco Gramegna¹, Vittorio Giovannetti³, Lorenzo Maccone⁴, and Marco Genovese¹

¹INRIM, Strada delle Cacce 91, I-10135 Torino, Italy
²International Laser Center of M.V. Lomonosov Moscow State University, 119991, Moscow, Russia
³NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
⁴Dipartimento di Fisica “A. Volta”, INFN Sez. Pavia, Universitàà di Pavia, Via Bassi 6, I-27100 Pavia, Italy

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Vol. 89, Iss. 5 — May 2014

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Images

Figure 1
Gate array representation of the PaW mechanisms for a CR noninteracting model. Here $U_{r} (t) = e^{- i H_{r} t / ℏ}$ and $U_{c} (t) = e^{- i H_{c} t / ℏ}$ are the unitary time evolution operators of the clock $C$ and of the rest of the universe $R$ , respectively. $| Ψ 〉$ is the global state of the system which is assumed to be an eigenstate with null eigenvalue of the global Hamiltonian $H = H_{c} + H_{r}$ (see text).
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Figure 2
Scheme for the two experiments. (a) PaW experiment in “Observer” mode [left, pink (light gray) box] and “Super-observer” mode [right, orange(dark gray) box]. The experimenter in observer mode can prove the time evolution of the first photon (upper path) using only correlation measurements between it and the clock photon (lower path) without access to an external clock. The experimenter in super-observer mode [orange(dark gray) box] proves that the global state of the system is static through state tomography. (b) Two-time measurements in the GPPT mechanism. The two time measurements are represented by the two polarizing beam splitters PBS $_{1}$ and PBS $_{2}$ , respectively. The blue (gray dashed) boxes (A) represent different thicknesses of birefringent plates which evolve the photons by rotating their polarization; different thicknesses represent different time evolutions. The PaW mechanism (a) is completely independent of the thickness, whereas the GPPT mechanism (b) allows it to be measured by the experimenter only through the clock photon (the abstract coordinate time is unaccessible and averaged away); the dashed box (B) represents a (known) phase delay of the clock photon only. PBS stands for polarizing beam splitter in the $H / V$ basis; BS stands for beam splitter.
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Figure 3
PaW experimental results. (a) Observer mode. Plot of the clock-time-dependent probabilities of measurement outcomes as a function of the optical plate thickness (corresponding to abstract coordinate time $T$ ). Downward triangles and squares represent $p (t_{1}) = P_{3 | 1}$ and $p (t_{2}) = P_{3 | 2}$ , respectively, namely the probabilities of measuring $V$ on the subsystem 1 as a function of the clock time $t_{1}$ , $t_{2}$ ; upward triangles and circles represent $P_{4 | 1}$ and $P_{4 | 2}$ , the probabilities of measuring $H$ on the subsystem 1 as a function of the clock time. As expected from the PaW mechanism, these probabilities are independent of the abstract coordinate time $T$ (dimensionless unit), represented by different phase plate A thicknesses (here we used a 957- $μ$ m-thick quartz plate rotated by 15 different equiseparated angles). The inset shows the graph that the observer himself would plot as a function of clock time; squares representing the probabilities of finding the system photon $V$ at the two times $t_{1}$ , $t_{2}$ , the circles of finding it $H$ . (b) Super-observer mode. Plot of the conditional fidelity between the tomographic reconstructed state and the theoretical initial state $| Ψ 〉$ of Eq. (2) as a function of the abstract coordinate time $T$ . The fact that the fidelity is constant and close to one (up to experimental imperfections) proves that the global entangled state is static.
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Figure 4
GPPT experimental results. Probability $P_{v}$ that the upper photon is $V$ as a function of the time $τ$ (dimensionless unit) recovered from the lower photon. The points with matching colors and forms represent $p (t_{1} + τ_{i})$ and $p (t_{2} + τ_{i})$ : blue pentagon, red circle, violet square, etc., for $i = 0, 1, 2, ...$ , respectively. Here nine different values of $τ_{i}$ are obtained from a thick quartz plate rotated by nine different angles. The dashed line is the theoretical value from Eq. (12). Since $t_{2} = t_{1} + π / 2 ω$ , we have plotted the points relative to $p (3 | t_{2})$ as displaced by $π / 2$ with respect to the points relative to $p (3 | t_{1})$ , so that the two curves (11) and (12) are superimposed in the graph.
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