A blueprint for a simultaneous test of quantum mechanics and general relativity in a space-based quantum optics experiment

Abstract

In this paper we propose an experiment designed to observe a general-relativistic effect on single photon interference. The experiment consists of a folded Mach-Zehnder interferometer, with the arms distributed between a single Earth orbiter and a ground station. By compensating for other degrees of freedom and the motion of the orbiter, this setup aims to detect the influence of general relativistic time dilation on a spatially superposed single photon. The proposal details a payload to measure the required effect, along with an extensive feasibility analysis given current technological capabilities.

Appendices

Appendix 1: Derivation of the relativistic time delay

Here we show how to derive Eq. (1), generalising the calculation described in [10].

First of all, we identify the reference frame of a distant observer with the ECI (Earth-Centered Inertial) frame. In order to take into account the velocity of the satellite and of the ground station rotating with the Earth’s surface, Eq. (7) in [10] must be modified. Let $\tau_{r}$ and t be the proper time recorded by a clock at the radial distance r and the coordinate time recorded by a distant observer, respectively. Then we have:

$$ d\tau_{r}^{2}= \biggl(1+\frac{2V(r)}{c^{2}}- \frac{v_{r}^{2}}{c^{2}} \biggr)\, dt^{2}, $$

(9)

where $V(r)=-\frac{GM}{r}$ is the Earth’s gravitational potential and $v_{r}$ is the speed of the clock at the radial distance r, as measured in the ECI frame.

If the fibre refractive index is n and c is the speed of light in vacuum, the proper time spent by a photon in a fibre loop of proper length l is $nl/c$. Hence, using Eq. (9), we can find the time spent by a photon in the two fibres, as measured by a distant observer:

$$ t_{r}=\frac{nl_{r}}{c\sqrt{1+\frac{2V(r)}{c^{2}}-\frac{v_{r}^{2}}{c^{2}}}}, $$

(10)

where $l_{r}$ is the proper length of the fibre at the radial distance r. Henceforth, the quantities related to the satellite and to the ground station will be indicated by the subscripts s and g, respectively. We assume $l_{s}=l$ and $l_{g}=l+dl$. At this stage, we also assume the radial paths towards the Earth to be the same, for the two parts of the photon wave function. We will return on this issue at the end of this appendix.

The difference in photon arrival times (referred to as the ‘time delay’) measured by a distant observer is $t_{s}-t_{g}$. Hence, using Eq. (9) and (10), the time delay measured by the local observer at the ground station is:

$$ \begin{aligned}[b] \Delta\tau&=\sqrt{1+\frac{2V_{g}}{c^{2}}- \frac{v_{g}^{2}}{c^{2}}}(t_{s}-t_{g}) \\ &=\frac{nl}{c} \biggl(\frac{\sqrt{1+\frac{V_{g}}{c^{2}}-\frac{v_{g}^{2}}{c^{2}}}}{ \sqrt{1+\frac{V_{s}}{c^{2}}-\frac{v_{s}^{2}}{c^{2}}}}-1 \biggr)-\frac {ndl}{c} . \end{aligned} $$

(11)

Taking the first order approximation to the potential and the second order approximation to the velocity, Eq. (11) becomes:

$$ \Delta\tau=\frac{nl}{c^{3}} \biggl(V_{g}-\frac{v_{g}^{2}}{2}-V_{s}+ \frac {v_{s}^{2}}{2} \biggr)-\frac{ndl}{c} . $$

(12)

Assuming that the ground station is at rest on the geoid, we can use the conventional geoidal potential $W_{0}$ to write the time delay of Eq. (12) as:

$$ \Delta\tau=\frac{nl}{c^{3}} \biggl(-W_{0}-V_{s}+ \frac{v_{s}^{2}}{2} \biggr) -\frac{ndl}{c} . $$

(13)

Finally, for an elliptical orbit with semi-major axis a, indicating with h the altitude of the satellite on the Earth’s surface, the time delay of Eq. (13) becomes:

$$ \Delta\tau=\frac{nl}{c^{3}} \biggl[-W_{0}+GM \biggl( \frac{2}{R_{\oplus}+h}- \frac{1}{2a} \biggr) \biggr]-\frac{ndl}{c} $$

(14)

which is exactly Eq. (1) in the text.

This time delay (excluding dl) is only due to relativistic effects, assuming the radial paths are the same and neglecting all the possible additional effects changing the total path length. These effects are carefully discussed in the paper, and either compensated for in transit or taken into account in post-processing. Among these effects, the one mentioned at the end of Section 3.3 deserves more attention: we know that the radial motion of the satellite gives a contribution to the time delay. Indeed, the radial paths cannot be the same due to the satellite displacement during the time spent by the photon in the satellite fibre loop. In Section 3.3 we suggest how to compute this additional delay, but only at a classical level. Since we are interested in the relativistic time delay, we must take into account also the relativistic corrections related to this path change. For example, following two different radial paths, two photons would cross two different gravitational fields, being affected by a different Shapiro delay. However, the relativistic corrections related to the difference in the radial paths, are orders of magnitude smaller than the time delay in Eq. (14) and so can be safely neglected.

Appendix 2: Polarisation in the interferometer

The core task in experiment is to implement a Mach-Zehnder interferometer in which one path of the interferometer is in space, whereas the other other is situated on the ground (see [10]). If a traditional Mach-Zehnder interferometer was used to do this, then two different telescope and receiver pairs would have to be used, which would significantly complicate the requirement of keeping the two interferometer paths identical. Instead, we propose a ‘folded’ Mach-Zehnder interferometer: the simplified schematic of our experiment is shown in Figure 1, and the more detailed schematic is in Figure 4. In this appendix we show how the optical elements in this folded Mach-Zehnder interferometer allow us to perform the experiment in [10].

We begin by sending a single linearly polarised photon through a half wave plate (HWP), which creates a superposition of horizontal and vertical polarisations, i.e. the state

$$ {1 \over \sqrt{2}} \bigl( \vert H \rangle + \vert V \rangle \bigr). $$

(15)

Whilst there is clearly only one photon, which is in a superposition, for ease of explanation we refer to the $\vert H \rangle$ part of the superposition as the ‘horizontally polarised photon’ and the $\vert V \rangle$ part of the superposition as the ‘vertically polarised photon’. The first polarising beam splitter (PBS) sends the horizontally polarised photon around a fibre loop on the satellite, and subsequently this photon is sent, via the telescope, to the ground station. In contrast, the vertically polarised photon is reflected by this PBS, and travels straight to the ground station. The two photons therefore arrive at the ground station at different times. This can be formulated by introducing a time degree of freedom. It can be readily checked by Fresnel equations that reflection of the photons by the atmosphere is negligible from any choice of polarisation unless the transmission occurs at very oblique angles. A more pertinent effect is the rotation of polarisation due to the atmosphere. However, as demonstrated in [38] this effect can be compensated after reception by making use of the classical reference laser. Given this compensation, the state arriving at the ground station is then

$$ {1 \over \sqrt{2}} \bigl( \vert H \rangle \vert \tau_{1}+t \rangle + \vert V \rangle \vert t \rangle \bigr), $$

(16)

where $\tau_{1}$ is the same taken to traverse the fibre loop on the satellite, and t is the time taken to travel from the satellite to the ground station. On arriving at the ground station a HWP performs the transformation $\vert H \rangle \rightarrow \vert V \rangle$ and $\vert V \rangle \rightarrow \vert H \rangle$, resulting in the state

$$ {1 \over \sqrt{2}} \bigl( \vert V \rangle \vert \tau_{1}+t \rangle + \vert H \rangle \vert t \rangle \bigr). $$

(17)

This HWP is necessary so that the correct photon traverses the fibre loop on the ground station.

Arriving at the final HWP we see that both photons have traversed a fibre loop, one on the satellite and one on the ground, resulting in state

$$ {1 \over \sqrt{2}} \bigl( \vert V \rangle \vert \tau_{1}+t \rangle + \vert H \rangle \vert t+\tau_{2} \rangle \bigr), $$

(18)

where $\tau_{2}$ is the time taken to traverse the fibre loop on the ground station. We now let $t_{s}=\tau_{1}+t$ and $t_{g}=\tau_{2}+t$, which are the total times spent by the two photons which traverse different paths, in accordance with the notation of Appendix 1. Next, a final HWP is used to performs the transformation

$$\begin{aligned}& \vert H \rangle \longrightarrow{1 \over \sqrt{2}} \bigl( \vert H \rangle + \vert V \rangle \bigr), \end{aligned}$$

(19)

$$\begin{aligned}& \vert V \rangle \longrightarrow{1 \over \sqrt{2}} \bigl( \vert H \rangle - \vert V \rangle \bigr). \end{aligned}$$

(20)

This HWP therefore acts on our state to give

$$\begin{aligned} &{1 \over 2} \bigl[ \bigl(\vert H \rangle - \vert V \rangle \bigr)\vert t_{s} \rangle + \bigl(\vert H \rangle + \vert V \rangle\bigr)\vert t_{g} \rangle \bigr] \end{aligned}$$

(21)

$$\begin{aligned} &\quad = {1 \over 2} \bigl[ \vert H \rangle \bigl(\vert t_{g} \rangle+\vert t_{s} \rangle\bigr) + \vert V \rangle \bigl(\vert t_{g} \rangle - \vert t_{s} \rangle \bigr) \bigr]. \end{aligned}$$

(22)

The final PBS and SNSPDs constitute a measurement in the H and V polarisation basis. We can see that if there was no time dilation then $\vert t_{g} \rangle=\vert t_{s} \rangle$ and the photon is always in the state $\vert H \rangle$, whereas if $\vert t_{g} \rangle=-\vert t_{s} \rangle$ the photon is always $\vert V \rangle$. In between these extremes we will see the interference fringes in Figure 3 - the calculation of these interference fringes in shown in Appendix 3 and in [10].

Appendix 3: Derivation of the expected interferometric signal

Here we show how to obtain Eq. (2) in the text.

The commutation relation for continuous mode photon states can be written [17]:

$$ \bigl[a_{\nu'},a_{\nu}^{\dagger}\bigr] = \delta\bigl(\nu- \nu'\bigr). $$

(23)

Say we have a Mach-Zehnder interferometer with equal arm lengths, but with one arm subject to time-dilation; then the state at detector D± with a single input photon is:

$$ | 1 \rangle_{\nu\pm} \propto \int d \nu f(\nu) \bigl( e ^{i \frac{\nu}{c}(x_{r} - c \tau_{r})} \pm e ^{i \frac{\nu}{c}(x_{r} - c (\tau_{r} + \Delta\tau))} \bigr) a^{\dagger}_{\nu}|0\rangle. $$

Where $x_{r}$ is the local Cartesian coordinate (perpendicular to the radial coordinate r), $\tau_{r}$ is the local time coordinate, $f(\nu)$ is the spectrum of the light source, ν is angular frequency, c is the speed of light, and Δτ is as given in Eq. (1).

The probability of a photon arriving at this detector is then

$$\begin{aligned} P _{\pm} =& {}_{v' \pm}\langle1 | 1 \rangle_{\nu\pm} \\ \propto&\biggl\langle 0 | a_{\nu'} \int d \nu \int d \nu' f(\nu) f\bigl(\nu'\bigr) \\ &{}\times \bigl( e^{i \frac{\nu}{c}(x_{r} - c \tau_{r})} \pm e^{i \frac{\nu}{c}(x_{r} - c ( \tau_{r} + \Delta\tau))} \bigr) \\ &{}\times \bigl(e^{-i \frac{\nu'}{c}(x_{r} - c \tau_{r})} \pm e ^{-i \frac{\nu'}{c}(x_{r} - c ( \tau_{r} + \Delta\tau))} \bigr) a^{\dagger}_{\nu}|0\biggr\rangle . \end{aligned}$$

(24)

From commutation relation Eq. (23), it is shown that

$$ {}_{\nu'} \langle1 | 1 \rangle_{\nu} = \delta\bigl(\nu- \nu'\bigr). $$

(25)

Thus

$$\begin{aligned} P _{\pm} \propto& \int d \nu \int d \nu' \bigl( e ^{i \frac{\nu}{c}(x_{r} - c \tau_{r})} \pm e ^{i \frac{\nu}{c}(x_{r} - c ( \tau_{r} + \Delta\tau))} \bigr) \\ &{} \times f(\nu) f\bigl(\nu'\bigr) \bigl( e ^{-i \frac{\nu'}{c}(x_{r} - c \tau_{r})} \pm e ^{-i \frac{\nu'}{c}(x_{r} - c ( \tau_{r} + \Delta\tau))} \bigr) \\ &{}\times\delta\bigl(\nu- \nu'\bigr) \\ =& \int d\nu\bigl\vert f(\nu) \bigl(e^{i \frac{\nu}{c}(x_{r} - c \tau_{r})} \pm e ^{i \frac{\nu}{c}(x_{r} - c ( \tau_{r} + \Delta\tau))} \bigr)\bigr\vert ^{2} \\ =& \frac{1}{2} \biggl( 1 \pm \int d\nu\bigl\vert f(\nu)\bigr\vert ^{2} \cos(\nu \Delta \tau) \biggr). \end{aligned}$$

(26)