Limits to the validity of gravitational redshift as a quantum-optical multimode mixer

Abstract

We analyze the domain of validity of a quantum optical model that describes the effects of gravitational redshift on the quantum state of photons that propagate in curved spacetime. This model assumes that the modes defining the initial state of the photon are mixed with an auxiliary environment mode via an effective multimode mixer. We find that the model, as proposed, is consistent only to first order for small redshift, where the range of validity is conditional not only to the gravitational parameters, but also to those that define the photonic modes. We identify the problem and provide a partial solution in terms of a necessary condition on the transformation matrix representing the process, which requires the use of a number of auxiliary modes that is at least equal to the number of modes that define the photonic state. We conclude by discussing implications for theoretical quantum optics and photonics in curved spacetime, as well as for the development of quantum technologies.

Figures (4)

• Figure 1: General scheme: in this work we consider photons that propagate between two points A and B in a (weak) gravitational field. Photon exchange between an Earth-based user, Alice, and a satellite-based user, Bob, is a paramount example of such general situation.
• Figure 2: Depiction of the two possible scenarios: Alice-to-Bob and Bob-to-Alice photon exchange protocols. The effective transformations as proposed in the literature Bruschi:Ralph:2014 are presented for clarity. Note that the definition of the redshift $\chi^2=\Omega_\textrm{B}/\Omega_\textrm{A}$ is, and must be, the same in both cases.
• Figure 3: Depiction of the two possible scenarios: Alice-to-Bob and Bob-to-Alice photon exchange protocols. The effective transformations as proposed in the literature Bruschi:Ralph:2014 are presented for clarity, together with the changes in average energy.
• Figure 4: Bob-to-Alice scenario: Sketch of the transformation of two modes $F_1$ and $F_2$ into the modes $F_1'$ and $F_2'$, respectively. We have chosen the Bob-to-Alice scenario with $\chi \leq 1$ since the modes move to the right in the frequency domain as $\chi$ increases, which allows for better visualisation. Note that $F_2'$ has moved outside of the figure in the last panel. It is clear from the figure that, with increasing redshift, the overlap of all transformed modes $F_n'$ with any of the initial modes $F_n$ diminishes, and eventually vanishes. All frequencies are to be measured in the local reference frame of Alice.