Quantum time: Experimental multitime correlations

In this paper we provide an experimental illustration of Page and Wootters’ quantum time mechanism that is able to describe two-time quantum correlation functions. This allows us to test a Leggett-Garg inequality, showing a violation from the “internal” observer point of view. The “external” observer sees a time-independent global state. Indeed, the scheme is implemented using a narrow-band single photon where the clock degree of freedom is encoded in the photon’s position. Hence, the internal observer that measures the position can track the flow of time, while the external observer sees a delocalized photon that has no time evolution in the experiment time-scale.

Figure 1

Experimental setup: Two-time measurements in the PW mechanism. The SPS box shows the coherent source attenuated (A) at the photon-counting regime. A system composed by a polarizer (GL) and a half wave plate (HWP) oriented at 22.5° pre-select the single photon state $|\mathrm{\Psi }⟩$. The IF box illustrates the two time measurements, operated by the two polarizing beam splitters BD (selecting the mode $\mathrm{a}=+1,-1$) and PBS (selecting the mode $\mathrm{b}=+1,-1$), respectively. The blue boxes QPs represent different thicknesses of birefringent plates which perform the evolution of the photon by rotating its polarization: different thicknesses represent different time evolutions. At the output of the interferometer 4 SPADs connected to a coincidence electronic chain perform the single-photon detection. The birefringent plate LG in dashed box is used for the Leggett-Garg experiment only.

Figure 2

Experimental setup: Super-Observer Mode. The SPS is an attenuate He-Ne laser operated with a frequency stabilization feed-back control that balances the intensity of two lasing modes under the gain curve ($\sim 1.5\mathrm{GHz}$). The tube length allows only two cavity modes at the output ($\mathrm{FSR}\sim 1\mathrm{GHz}$) with orthogonal polarization. After the suppression of one polarization mode (by a PBS), only one single cavity mode emerges from the source. The line width of this individual axial mode is evaluated by a beat measurement. The spectral analysis returned a bandwidth lower than 3 MHz, corresponding to a coherence length greater than 100 m.

Figure 3

Two-time correlation experiments. Graphs of conditional probabilities of polarization measurement outcomes as a function of the plate thickness (corresponding to the internal time evolution $t$). As expected, these probabilities are in good agreement with the conventional quantum description (7) of the evolution from the internal observer’s point of view.

Figure 4

Leggett-Garg function $K_3$ of Eq. (10) for a two-level system as a function of measurement-time $\omega \mathrm{\Delta }t$. The solid curve shows the theoretical $K_3$ of Eq. (14), the red points are the experimental results.