Conveyor-belt clock synchronization

A protocol for synchronizing distant clocks is proposed that does not rely on the arrival times of the signals which are exchanged, and an optical implementation based on coherent-state pulses is described. This protocol is not limited by any dispersion that may be present in the propagation medium through which the light signals are exchanged. Possible improvements deriving from the use of quantum-mechanical effects are also addressed.

Figure 1

Representation of the conveyor belt synchronization scheme. Alice pours sand on the conveyor belt at positions $A$ and $A^{\prime }$, while Bob scoops away sand at the intermediate position $B$. Measuring the amount of sand at position $D$—once an initial transient has passed—directly reveals the time difference between their two clocks.

Figure 2

Proposal for dispersion-immune synchronization. The laser produces intense ↕-polarized pulses that travel from Alice to Bob, where they are reflected back to Alice. At points $A$ and $A^{\prime }$, Alice delays the 45° polarization with respect to the $-45°$ polarization by an amount proportional to the time shown on her clock. At point $B$, Bob delays the $-45°$ polarization with respect to the 45° polarization by an amount proportional to the time shown on his clock. These delays are in accord with the conveyor belt protocol, i.e., Bob’s proportionality constant is twice Alice’s. The polarizing beam splitter PBS separates the incoming beam into its ↕ and ↔ polarization components. These components are directed to integrating detectors $D_{\updownarrow }$ and $D_{\leftrightarrow }$ respectively, which measure the number of photons impinging on them. As discussed in the text, signal multiplexers allow pulses to travel through the dispersive medium in a common polarization state, thus avoiding polarization-dependent propagation effects.

Figure 3

Model of the time-varying delays introduced by Alice at points $A$ and $A^{\prime }$ in the Fig. 2 system. The left polarizing beam splitter (PBS) separates the two polarization components so that they impinge on opposing faces of a mirror moving at speed $v$ in the direction shown. The right PBS recombines the polarizations. Bob uses a similar setup at point $B$ in the Fig. 2 system, but his mirror moves at speed $2v$ in the opposite direction from what is shown here. Electro-optic modulators would be used, instead of the moving mirror, in an actual system.

Figure 4

Explanation of the delay ${\tau }_D$, from Eq. (11), that is due to the moving mirrors. Prior to the onset of mirror motion, the total optical path length for the ⤢ polarization is $L=2L_1+2L_2$. When the mirrors are moving, by the time the signal reaches point $A$, the first mirror has increased the path length by $2v(L_1∕c-t_0^a)$. This means that the ⤢-polarized signal will incur a propagation delay $(L_1+L_2)∕c+2v(L_1∕c-t_0^a)∕c$ en route to point $B$. However, during this time interval, Bob’s mirror has reduced the path length for the ⤢ polarization by $4v[(L_1+L_2)∕c+2v(L_1∕c-t_0^a)∕c-t_0^b]$. Proceeding in a like manner for the path length increase at $A^{\prime }$, and summing up all the contributions, we can show that the overall delay ${\tau }_D$ is given by Eq. (11) to first order in $v∕c$.

Figure 5

(a) Plot of $J_{\leftrightarrow }(t_0^b-t_0^a)$ versus $t_0^b-t_0^a$ from Eq. (18) for Gaussian pulses. Here the velocity of the phase variation is $8v{\omega }_0∕c={10}^9{\mathrm{s}}^{-1}$ and the bandwidth is $\Delta \omega ={10}^{13}{\mathrm{s}}^{-1}$. (b) Magnification of the box in the previous plot: $J_{\leftrightarrow }(t_0^b-t_0^a)$ has null at $t_0^a=t_0^b$.

Figure 6

Differential conveyor belt scheme: Bob is not required to be at the midpoint of the transmission line.

Figure 7

Two examples of the periodic-ramps protocol. The lines plot the amounts of sand that Alice (solid) and Bob (dashed) must move to $(>0)$ or from $(<0)$ the conveyor belt versus time: (a) Alice and Bob periodically restart the protocol and (b) Alice and Bob periodically reverse their rates.