# Quantum satellites and general relativity

The race towards quantum communication satellites is under way. Various collaborations around the world are designing the first prototype experiments and the Institute for Quantum Computing's (IQC) Quantum Encryption and Science Satellite (QEYSSat) is at the forefront. The challenge is motivated by the potential of building a world-wide Quantum Key Distribution (QKD) network. It involves ground-breaking science and engineering: from building detectors that survive space to understanding the security of relevant QKD protocols in realistic conditions. Now, as we plan to launch state-of-the-art quantum optics equipment into space we ask what other uses can we have for these satellites? Can we use this space-based equipment to test fundamental physics?

One of the advantages of experiments with satellites is that they can be used to probe general relativity. Satellites are at a different gravitational potential from Earth and the experiments can be carried out over long distances. As a result, we can expect general relativity to kick in and affect our quantum systems. While there have been a number of suggestions regarding possible experiments with quantum communications satellites, two of the most likely candidates for experiments that will include general relativistic effects are the rotation of photon polarization (Lens Thirring effect) and gravitationally induced phase shift in an interferometer (the optical COW experiment).

## The optical Lens Thirring effect

We all know that massive objects create a gravitational field that distorts space-time. Around 1918, two Austrian physicists predicted that if the massive object rotates, it further distorts the space-time around it, i.e., the angular momentum of the mass is a parameter of the induced space-time metric. There are a number of ways to understand this intuitively, the simplest is in terms of symmetry.

If $M$ is some spherically symmetric massive object, we expect it to induce a spherically symmetric metric such as the Schwarzschild metric. If the object starts to spin it introduces a new preferred direction, the spin axis. As a result the spherical symmetry of the metric may be (and indeed is) broken. That is, a metric induced by a spinning object is only symmetric around the rotation axis.

The original formulation of the Lens Thirring effect was in terms of gyroscopes precession, and in the previous decade a gyroscope was launched into space in order to test this effect. The mission, named gravity probe B, detected a precession of about 40 mili arc seconds per year (a mili arc second is about $5\times10^{-9}$ radians). The effect is, however, not limited to gyroscopes. For photons, the change in the metric induces a rotation of polarization.

The problem with polarization is that photons do not carry around a reference frame. In empty space there is no way to set up polarization detectors in a consistent way without communication. A photon arriving at a detector will only carry information about its current polarization $P$ and its current direction $k$. While the plane of polarization is always perpendicular to $k$, there is no canonical way to decide on the directions of and for horizontal and vertical polarization, i.e., to break $P$ into $H,V$ components $P=\alpha H +\beta V$. Remember, on an optical table we always take $V$ to be aligned with the gravitational pull $w$ and $H$ to be perpendicular to it. This is only possible when $k\cdot w=0$. Let us say we start by sending light with polarization $P_0$ and detect it to be $P_f$ at the final point. The rotation would be the angle between $P_0$ and $P_f$, however we don't have any a-priori way to align $P_f$ and compare it with $P_0$ (note: parallel transport requires us to know the trajectory).

Intuition from a spherically symmetric space-time can be used to define a standard way to set $H$ and $V$ using only local information. Assume we are at some point $A$ in space and that the only contribution to the metric is some spherical mass $M$. Let $w$ be the direction of the gravitational pull at $A$, i.e., a unit vector pointing from $A$ to $M$. If b is the direction of a photon arriving at b we can say that the horizontal polarization at $A$ (labeled $H_A$) is perpendicular to $k$ and $w$, likewise $V_A$ is perpendicular to $k$ and $H_A$. We call this setup the `Newton gauge'.

This setup has a neat feature, it will detect no rotation for a free photon in a spherically symmetric space-time. The trajectory of such a photon is confined to the plane that includes $k$ and $w$. Since $H$ is always perpendicular to this plane (by construction), it must remain invariant under the rotations. An interesting fact is that Helicity states are not affected by the gravitational field, consequently, since the overall effect must be unitary, the polarization state of the photon remains invariant in a spherically symmetric gravitational field.

The Newton Gauge: To set the polarization $H,V$ basis for a photon with momentum $k$ we use the direction of the gravitational pull $w$ and set $H \propto w\times k$ and $V=H\times K$

As expected, when the symmetry is broken, the detectors set up in the Newton gauge will detect some rotation for polarization. Calculations using the metric of a spinning mass can be used to predict the angle of rotation. Previous attempts at calculating the optical Lens Thirring effect were seemingly contradictory (sometimes disagreeing by an order of magnitude), however, the reason for the apparent mismatch was due to different assumptions about how detectors would be set up. For example, if one assumes the detectors are parallel transported along the photon's trajectory the effect is much smaller.

### So what is the effect near Earth?

If we send a photon from earth and detect it on a satellite in the same (low earth) orbit as QEYSSat we would get a rotation which is about 50 mili arc seconds (the value will not change significantly for higher orbits). The main drawback of this type of measurement is the requirement to align the detectors according to the Newton gauge. While this is a simple issue in theory, it may prove difficult in practice.

The need to align detectors can be removed if we consider an interferometery experiment which allows us to compare the rotation of polarization on the two arms of the interferometer. If the interferometer is balanced, the phase shift will cause dechoerence that can be observed experimentally. Unfortunately in this case the Lens Thirring effect is a few orders of magnitude smaller and would not be observable in Earth's gravitational field. But that can play to our advantage since we don't need to take polarization into account for an interferometry experiment.

## The optical Colella-Overhauser-Werner effect

In the 1970s Roberto Colella, Albert Overhauser and Samuel Werner (COW) performed the first experiment to couple a quantum particle with gravity. They set up a Neutron interferometer so that the different arms of the interferometers experience slightly different gravitational potentials. As a result, particles following each path arrive at the interference point with different phases, effectively changing the interference pattern. This effect can be used as a test of the weak equivalence principle, however it is not, strictly speaking, a general relativistic effect. In other words it can be fully explained using Newtonian gravity.

Photons don't have mass and do not couple to Newtonian gravity, so by observing the same effect on photons we can be certain that the effect is post-Newtonian. The problem with performing such an experiment is that the coupling between photons and gravity is very small and consequently we need a very large interferometer. Satellite experiments provide the perfect solution.

The photons can originate from Earth where the first beam splitter is located. One arm of the interferometer gets delayed and is later sent to the satellite while the other goes directly to the satellite and gets delayed there. Since the two arms experience different gravitational potentials during the delay, they will have a relative phase. This relative phase can be calculated from the interference pattern.

A calculation based on realistic parameters gives a large phase difference to the order of a few radians for an Earth to satellite experiment. However, other issues such as the Doppler effect must be taken into account. As a result the required accuracy to verify the COW phase difference is about 10-7rad.

While this result is somewhat optimistic in terms of experimental feasibility there are still some open problems that relate to realistic experiments. For example, it is unclear if the atmosphere will cause the phase to fluctuate at an amplitude which is too large for the experiment to be realistic. These realistic issues are now being investigated.

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