Gravitational Redshift in Quantum-Clock Interferometry

Abstract

The creation of delocalized coherent superpositions of quantum systems experiencing different relativistic effects is an important milestone in future research at the interface of gravity and quantum mechanics. This could be achieved by generating a superposition of quantum clocks that follow paths with different gravitational time dilation and investigating the consequences on the interference signal when they are eventually recombined. Light-pulse atom interferometry with elements employed in optical atomic clocks is a promising candidate for that purpose, but suffers from major challenges including its insensitivity to the gravitational redshift in a uniform field. All these difficulties can be overcome with a novel scheme presented here which is based on initializing the clock when the spatially separate superposition has already been generated and performing a doubly differential measurement where the differential phase shift between the two internal states is compared for different initialization times. This can be exploited to test the universality of the gravitational redshift (UGR) with delocalized coherent superpositions of quantum clocks and it is argued that its experimental implementation should be feasible with a new generation of 10-meter atomic fountains that will soon become available. Interestingly, the approach also offers significant advantages for more compact set-ups based on guided interferometry or hybrid configurations. Furthermore, in order to provide a solid foundation for the analysis of the various interferometry schemes and the effects that can be measured with them, a general formalism for a relativistic description of atom interferometry in curved spacetime is developed. It can deal with freely falling atoms, but also include the effects of external forces and guiding potentials, and can be applied to a very wide range of situations. As an important ingredient for quantum-clock interferometry, suitable diffraction mechanisms for atoms in internal-state superpositions are investigated too. Finally, the relation of the proposed doubly-differential measurement scheme to other experimental approaches and to tests of the universality of free fall (UFF) is discussed in detail. arXiv:1810.06744v1 [physics.atom-ph]

Post-Newtonian Hamiltonian description of an atom in a weak gravitational field

“…As a result, our work lends justification to current experimental proposals in atom interferometry that so far were based on these heuristic ideas on the basis of which completeness of the relativistic effects could not be reliably judged; e.g. [3][4][5][6][7][8]. Moreover, it also applies to new experimental setups, like that of an ion trap in a gravitational field, currently under investigation [30] in extension of [31].…”

“…'proper time' and 'redshift'); e.g. [3][4][5][6][7][8]. We emphasised in the introduction and also in our discussion of the Equivalence Principle that answers to the fundamental question of gravity-matter coupling in Quantum Mechanics should not be based on a priori restricted states that imply a semi-classical behaviour of some of the (factorising) degrees of freedom.…”

“…As emphasised above, this generalisation serves not only a point of principal interest that deserves clarification, but is also of immediate practical interest, not only in the obvious realm of quantum optics experiments regarding the detection of gravitational waves, but also in atom interferometry; see, e.g. [2][3][4][5][6][7][8]. A calculation using methods very similar to those of [1] including external gravitational fields was performed by Marzlin already in 1995 [9] 2 ; but unlike Sonnleitner and Barnett in [1] or our calculation in this work, Marzlin did not perform a full first-order post-Newtonian expansion and instead focused on the electric dipole coupling only.…”

“…As emphasised before, our central point of concern results from our conviction that any generally valid implementation of the Equivalence Principle into Quantum Mechanics should not be based on a priori assumptions concerning the state of the matter. In particular, it should not rely on semi-classical notions like 'wordline', that often enter discussions in a fundamental way 12 ; see, e.g., [3][4][5][6][7][8]. It should also not rely on other classical formulations, like 'equality of inertial and gravitational mass', for reasons discussed below.…”

“…It is the underlying systematics of producing 'relativistic corrections' that, in our opinion, distinguishes our approach from others, like, e.g., [3][4][5][6][7][8]. These latter attempts make use in an essential way of notions, like 'wordline' and 'redshift', which have no immediate meaning in quantum theory, unless the state of the system is severely restricted in an a priori fashion.…”