Testing universal Compton clocks using clock interferometry

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IN

DEGREE PROJECT ENGINEERING PHYSICS, SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2020

Testing Universal Compton

Clocks Using Clock

Interferometry

THOMAS AGRENIUS GUSTAFSSON

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

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KTH Royal Instiute of Technology

M.Sc. Thesis

Testing Universal Compton

Clocks Using Clock

Interferometry

Thomas Agrenius Gustafsson

Supervisor: Dr. Igor Pikovski

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Abstract

It is demonstrated how application of clock interferometry theory to recent atomic interferometry experiments can impose restrictions on speculative theories of dynamical mass. Specifi-cally, we use recently published data to show that the existence of universal Compton clocks, defined as a universal internal dy-namic of particles at their Compton frequency ωC= mc2/~, is

infeasible, as they predict interference patterns which are in-consistent with experimental results.

Sammanfattning

Det demonstreras hur klockinterferometri, en hypotetisk ef-fekt, kan appliceras p˚a nyligen publicerad atominterferometri-data för att erh˚alla begränsningar p˚a spekulativa teorier om dynamisk massa. Specifikt s˚a visar vi fr˚an data att existen-sen av universella Comptonklockor, definierade som en uni-versell intern dynamik i partiklar vid deras Comptonfrekvens ωC= mc2/~, är orimlig, d˚a de skulle ge interferensmönster som

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Preface

This master thesis, and diploma thesis for the Swedish title of civil-ingenj¨or, is the result of six months of independent work under the supervision of Dr. Igor Pikovski at Stockholm University. Dr. Jens Bardarson at KTH Royal Institue of Technology has provided addi-tional supervision and support, and I express my gratitude to both.

Shortly after the start of this thesis work, the sars-cov-2 virus spread globally, causing a pandemic, closing down the buildings of KTH and Stockholm University along with much of the rest of the world. The majority of the work underlying this thesis was therefore completed in quarantine at home. Consequently, I have two groups of people to thank:

First, while universities were open, I wish to thank Dr. Bardarson and the rest of the staff in the KTH Theoretical Physics corridor for their hospitality in letting me work in their thesis office, and the fellow thesis students I shared this office with for our interesting and fun conversations. Next, I wish to express my gratitude to my friends, especially Anna, Hilda, Anders, and Klas, and to my dear mother, all with whom I shared home office for some period of time during the pandemic. Finally, overarching both periods, I wish to thank Dr. Pikovski and his group, Vasilis Fragkos, Dr. Thomas Guff and Dr. Christian K¨ading, for being very kind and including me as a full member of the group, as well as for our many interesting and fun discussions.

Because this thesis concludes my time as a diploma student, or teknolog, at KTH, it seems fitting to use the opportunity to thank my friends and my family for sharing these times with me, the good as well as the bad. Thanks also to the lively community at KTH, which I have very much enjoyed being part of: teachers, staff, fellow students.

This thesis was typeset using the LA_{TEX ecosystem, and made} ex-tensive use of the SciPy ecosystem, including NumPy for the numerical analysis and Matplotlib for the creation of figures. The hard work of all contributors to these projects is gratefully acknowledged.

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1 Introduction

The modern understanding of mass is based on Einstein’s mass-energy equivalence, which arises as a consequence of the transforma-tion of the 4-momentum of particles in Lorentz-invariant relativity [1]. The mass of a particle is by this understanding a result of all the stored interaction energy associated with the particle. By the current understanding of the standard model of particle physics, compos-ite particles, especially hadrons such as protons and neutrons, have large masses due to consisting of strongly interacting constituents, while fundamental particles, such as the quarks and the electron, have masses due to constant interactions with the Higgs boson field, which permeates the universe via its nonzero vacuum expectation value. Gravity is understood to be the result of spacetime curvature due to the presence of any energy, or equivalently, mass.

Usually, the mass of particles is considered to be a static quantity – internal energy states of composite particles are usually assumed to be in or very close to an energy eigenstate, and the Higgs field is assumed to be uniform and non-dynamical. If indications of dy-namical mass were to be observed, this would be a signal of new physics, such as quantum violations of the equivalence principle, or other low-energy signatures predicted by speculative theories of quan-tum gravity [2]. A pertinent question is then: Is it possible for mass to have hidden dynamics which are yet undiscovered?

An experimental window to answer such a general question was recently opened when it was proposed that interference patterns ob-served in matter-wave interferometry should be affected when the matter used in the interferometer contains a dynamical state. [3, 4] The reason is that when interferometer paths are at different heights in a gravitational field, the internal dynamics evolve with different proper times. If the proper time difference is large enough, the inter-nal states associated with each path should become distinguishable, affecting the interference pattern. This clock interferometry effect has so far only been verified in analogue gravity experiments [5], but if true, it opens up a new method by which we can answer the question of whether mass can contain hidden dynamics: By observing that interference fringe patterns in performed experiments so far do not

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Introduction| 6

show signs of internal dynamics, we can put concrete restrictions on dynamical-mass models. We pursue this opportunity in this thesis.

We begin in this Introduction by reviewing the ingredients of this approach: Namely, modern matter-wave interferometry in grav-itational fields, clock interferometry, and a specific example of a dynamical-mass proposal, the universal Compton clock. In the fol-lowing parts of this thesis, we develop the clock interferometry and Compton clock concepts further, before putting Compton clocks to the test against data in the final section. This provides a demon-stration of how the clock interferometry concept can be applied to existing interferometric data to restrict dynamical-mass theories. 1.1 Gravitational interferometry with matter-waves in

Newtonian gravity

To build understanding of modern matter-wave interferometry in gravitational fields, we review two classic experiments and one of their modern descendants, namely the 1975 Colella, Overhauser, Werner [6] (abbreviated as COW) and 1991 Kasevich, Chu [7] experiments, and an experiment by Xu and coworkers in 2019 [8]. In these ex-periments, interference in a two-way matter-wave interferometer is observed due to the presence of a gravitational potential difference between the two interferometer arms.

Interference patterns in atomic interferometry experiments are most easily analyzed using Feynman’s path integral formalism. This is based on the idea that

ψ(xb, tb) = Z

K(xb, tb; xa, ta)ψ(xa, ta) dxa (1.1) where K(xb, tb; xa, ta) is the propagator, defined by

K(xb, tb; xa, ta) =hxb| ˆU (tb, ta)|xai = = Z x(tb)=xb x(ta)=xa Dx(t) exp −i ~ Z L[x(t), ˙x(t), t] dt (1.2) where ˆU (tb, ta) is the time-evolution operator andL is the Lagrangian of the corresponding classical system. The last integral_{R Dx(t) is an}

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Introduction| 7

integral over all possible paths x(t) that the particle may take from xa to xb in the time interval tb − ta, hence the name path integral formalism. The propagator has the composition property

K(xc, tc; xa, ta) = Z

K(xc, tc; xb, tb)K(xb, tb; xa, ta) dxb (1.3) which allows calculating propagation along a longer path by subdivi-sion of the path into parts. A full introduction to the path integral formalism is given in Ref. [9].

The path integral formalism allows for exact calculation of changes in wave packet spreading when passing through slits, and so on, but in gravitational interferometry situations that level of detailed knowl-edge of the position wave function is usually not of interest. The quantity of interest is the difference in wave function phase associ-ated with different paths, since this will determine the fringe pat-tern on a detector. To this end, the path integral formalism may be substantially simplified by employing a semiclassical approxima-tion, where the particle has a characteristic momentum and follows the path of a classical particle with the same momentum, while still carrying a quantum-mechanical amplitude and phase along the clas-sical paths which it traverses. Then, the phase of the particle as it travels along the path can be calculated either exactly by considering the product of propagators along this path, or perturbatively by in-tegrating the perturbation Lagrangian along this path. In this way, quantum-mechanical phase differences between particle paths in in-terferometers are easily calculated, allowing successful prediction of the interference fringe pattern and its dependence on the experimen-tal parameters.

The details of the path integral formalism as applied to quantum interferometry, and also of how the semiclassical formalism follows from the exact formalism, are given in [10] with the Appendix A.1 of this thesis as complement. Presently, we will use it to analyze the COW, Kasevich–Chu, and Xu 2019 experiments. The central results we will need are that the wave function of a particle along the classical paths in an interferometer may be calculated as

ψ(xb, tb) = exp i ~ Z Γcl L(x(t), ˙x(t), t) dt ψ(xa, ta) (1.4)

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Introduction| 8

where Γcl is the classical path connecting (xb, tb) and (xa, ta) and where ψ(xa, ta) has its momentum well-defined to be the correspond-ing classical momentum, and that the composition rule (1.3) reduces to a simple product of propagators along the classical path

e~i R ΓclL(x(t), ˙x(t),t) dt_{= e} i ~ R Γ1 cl L(x(t), ˙x(t),t) dt e i ~ R Γ2 cl L(x(t), ˙x(t),t) dt (1.5) where Γ1

cland Γ2clare subdivisions of the path Γcl.

∗ _{Note that the} inte-gral of_{L along these classical paths is the classical action S}cl(xb, tb; xa, ta), which depends directly on the start and end points, and one can con-veniently use this for calculations instead of explicitly integrating the Lagrangians along the path. This works for Lagrangians which are at most quadratic in x and ˙x. Even simpler, though, and capable of handling more general Lagrangians, is to treat the Lagrangian of interest as a perturbation on top of the free-particle Lagrangian L0(x, ˙x, t) = m ˙x2/2. With L1 = L − L0, the phase of the wave function is then ψ(xb, tb) = ψ(0)(xb, tb) exp ( i ~ Z Γ(0)_cl L 1(x(t), ˙x(t), t) dt ) (1.6)

where Γ(0)_cl is the unperturbed free-particle path from (xa, ta) to (xb, tb) and ψ(0)_(x

b, tb) is the free-particle wave function after propagation along the same path. Since free particles travel in straight rays, this allows for a very convenient analysis.† Interferometers are con-structed so that the particles travel along two paths which are then recombined, and any phase difference of the particle along the paths will cause interference. Typically interferometers are constructed so that the free-propagation phase ψ(0)_(x

b, tb) is the same along the paths. Then, one only needs to consider the phase difference due to the perturbing potential:

∆φ = Z Γ(0)1 L1(x(t), ˙x(t), t) dt− Z Γ(0)2 L1(x(t), ˙x(t), t) dt (1.7) ∗

The normalizing amplitude factors of the full propagator vanish in the semi-classical approximation.

†_{If one uses the non-perturbative approach, one must consider the exact} clas-sical trajectories, e.g. curved parabolas in a gravitational field.

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Introduction| 9

where now Γ(0)_1,2 are the two different paths through the interferom-eter. This quantity is what determines interference fringe patterns observed at detectors.

An interference fringe pattern is the variation of the probability of detecting a particle in each output port of the interferometer with the phase difference accumulated in the interferometer. Two-way in-terferometers have two output ports and hence two probabilities of detecting a particle in each port, which we label P0 and P1, respec-tively. Fringe patterns typically look like

P0,1= u0,1± v0,1cos(∆φ− ϕ) (1.8) where 0, 1 are the labels for the two different output ports of the re-combining beamsplitter, the upper sign is for 0 and the lower sign for 1, u0, u1, v0 and v1 are constants depending on the specific construc-tion of the interferometer, ∆φ is the path phase difference from (1.7), and ϕ is a possible phase shift which may be induced by experimental conditions. Instead of analyzing P0 and P1 separately, it is common to investigate the probability asymmetry,

A = P0− P1 = δu + σv cos(∆φ− ϕ) (1.9) where δu = u0− u1 and σv = v0+ v1. We will switch between P0,1 and A frequently, and collectively refer to them as fringe patterns. A central quantity for this thesis is the visibility of the fringe pattern (also commonly called the contrast). This is defined as

V = maxϕ{P0, P1} − minϕ{P0, P1} maxϕ{P0, P1} + minϕ{P0, P1}

(1.10) The visibility essentially measures how large the interference fringes are: V = 0 implies that the probability of detecting a particle in either detector is identically 1/2 independently of ∆φ, meaning that there are no fringes visible at all. The visibility obeys an inequality known as the quantum complementarity principle

V2+ D2 _{≤ 1} (1.11)

where D is the distinguishability, which is a measure of the informa-tion available about which way a particle took in an interferometer.

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Introduction| 10

The complementarity principle then essentially states that as which-way information becomes available (D increases), the fringe pattern must lose visibility. A well-known example of this principle is the van-ishing of the interference pattern in a double-slit experiment when one places a detector at one of the slits. In this case, perfect which-way information becomes available (D = 1), so the interference pattern vanishes completely (V = 0). However, as (1.11) shows, the general case is not binary. Typically in experiments, experimental imper-fections cause ”leaks” of a small amount of which-way information (1 D > 0), which causes the visibility to be slightly below the theoretical maximum (1 > V _{0). The equal sign in (1.11) holds} whenever the state of the particle in the interferometer is a pure state. [11]

To experimentally control the path of atoms or neutrons in in-terferometers, their momentum state must be controlled. In atomic interferometry, these momentum states coincide with internal energy levels of the atom, typically hyperfine levels (Raman transitions) of the ground state. The momentum state of the atom may then be manipulated by inducing absorption or emission of a photon during transition between the hyperfine levels, since the atom recoils with the absorbed/emitted photon’s momentum. The internal state of the atom may then also be used to label the momentum state.∗ Experi-mentally, the transitions are stimulated by laser pulses, a technique which was pioneered by Kasevich and Chu and is presented in de-tail in [7]. The effect of the laser pulse depends on the interaction time and pulse arrangement. An ideal ”π/2 pulse” acts as a perfect beamsplitter, coherently splitting the internal states of the atom into a superposition of itself and the other internal state. An ideal ”π pulse” interchanges the two states between each other with certainty, which is useful for controlling the path.

We now turn to the experiments themselves and their observed fringe patterns. For the interested, calculations of the fringe pat-terns for each experiment applying the above formalism are shown in Appendix A.2.

∗_{The hyperfine excitations of the ground state of the atoms used} experimen-tally, like sodium and caesium, have very long lifetimes in the controlled conditions of the experiments, so that spontaneous decay of these states can be neglected.

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Introduction| 11 z t 0 T 2T A D π π/2 π/2 B C |1i |0i |1i |0i |1i |0i |0i g

Figure 1: Geometry for analysis of the Kasevich–Chu experiment. An atom in a gravitational field is coherently split at t = 0 by a π/2 pulse into an equal superposition of two momentum states, labeled|0i and _{|1i. At t = T a π pulse flips the momentum states, causing the} paths to meet in a recombining π/2 pulse at t = 2T . The probability of detecting an atom in each respective momentum state post the recombining pulse is subject to interference (Eq. (1.12)). While the paths are parabolas in the exact theory, they may be replaced by straight lines for calculations using perturbation theory.

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Introduction| 12

We start with the Kasevich–Chu experiment. A simplified geom-etry of the experiment is shown in Figure 1. Here, 0 and 1 label the ground state and a hyperfine excitation of the ground state, which in turn also label different momentum states, as explained above. The atom starts in state _{|0i which is coherently split by a π/2 laser} pulse into the state _{|0i + |1i. The two states separate spatially due} to their different momentum and spend time 2T in free fall before being mixed by another π/2 pulse. At time T , a π pulse is applied to interchange the momenta of the states, to make sure they arrive at the same location in the second π/2 laser pulse. The probability of detecting either a_{|0i or |1i state after the final π/2 pulse depends on} the phase shift ∆φ between the two paths, which will be sensitive to the gravitational field as the paths are at different mean height. As shown in Appendix A.2, the fringe pattern on the detectors is

P0,1= 1 2 ± 1 2cos κgT 2 − ϕ (1.12) where the upper sign is for 0 and the lower sign for 1. Note that the pattern is sensitive to product of the acceleration due to gravity g and the laser wavelength 1/κ, but is insensitive to the atomic mass m. The quantity ϕ is a phase shift induced by the phase of the lasers at the time of their interactions with the atoms. Note also that the theoretical visibility for this experiment is 1. Kasevich and Chu fixed T and varied κ to measure g to high accuracy, and their setup has become an important method of g measurement. [7]

Before Kasevich–Chu, the Colella–Overhauser–Werner (COW) ex-periment was the first exex-periment to demonstrate a fringe pattern sensitive to the gravitational field g at all, but it used neutrons in-stead of atoms. The momentum state of neutrons cannot be manipu-lated by lasers as the neutron has no accessible inner structure at the energy scale of lasers. Instead, single-crystal diffraction was used in the pioneering COW experiment. By diffraction at the Bragg angle an incoming neutron beam was coherently split into two paths, and then recombined via successive Bragg reflections: See Figure 2 for a simplified geometry of the experiment. By rotating the diffraction crystal around the axis of the incoming neutron beam, i.e. chang-ing the angle φ in Figure 2, the paths could be placed at different mean heights in the gravitational field, producing a sensitivity to the

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Introduction| 13 a L L θ t a/ ~ mλcos θ L/ ~ mλcos θ L/ ~ mλcos θ φ 2L sin θ a sin θ z g sin φ

Figure 2: Geometry for analysis of the COW [6] experiment. Neu-trons with de Broglie wavelength λ are incident onto a single-crystal interferometer. Crystal slabs are depicted in gray. Reflections and transmissions through the crystal slabs at the Bragg angle θ cause the neutron to coherently split at the first slab and recombine at the final slab. By rotating the crystal by φ around the axis of the incom-ing neutron beam, a gravitational potential difference g sin φ can be generated between the paths. This produces detectable interference in the two output beams after passage through the final crystal slab, Eq. (1.13).

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Introduction| 14

gravitational field. The fringe pattern in this experiment is∗ P1− P2 = β1+ β2cos 2m 2_gλ ~2 L(L + a) sin φ tan θ . (1.13)

The quantities L, a, φ, and θ are geometric quantities defined in Fig-ure 2, λ is the thermal de Broglie wavelength of the neutrons, and β1 and β2are constants from the specifics of the Bragg reflections, which caused the visibility to be less than 1 in the experiment. The obser-vation of these interference fringes as a function of φ was historically the first demonstration of interference due to gravity.

Notably, the COW experiment is sensitive to the product of the mass and gravitational field mg (the second m coming from λ), whereas the Kasevich–Chu experiment was insensitive to m. One pursuit of subsequent atomic-interferometry experiments has been to increase the direct sensitivity of the phase shift to the Newtonian potential mg∆z, where ∆z denotes a height separation of the inter-ferometer paths, in relation to the κg dependence, the latter called the ”laser phase”. The motivation is to provide increasingly accurate tests of the equivalence principle, in turn testing the assumptions underlying the interpretation of gravity as curvature of space-time.

A recent milestone in this direction is the 2019 experiment by V. Xu, H. M¨uller, and coworkers [8], who were able to separate and then hold their atoms at µm height separation in an optical lattice for up to 20 s before recombination. The simplified geometry of their experiment is shown in Figure 3. Two π/2 pulses separated by time tS are used to separate the matter-wave packets by a height ∆z. Then, after a time tA, the wavepackets are adiabatically loaded into the optical lattice, where they are held at the constant separation ∆z for a variable time T , after which they are again released adiabatically and recombined using the reverse pulse sequence. The resulting phase difference is the sum of a laser phase ∆φlaser and a pure gravitational phase difference ∆φgrav, being

∆φlaser = κgtS(tS+ 2tA)− ϕ (1.14) ∆φgrav=

mg∆z

~ T. (1.15)

∗

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Introduction| 15

The experimenters achieved sufficient coherence to observe interfer-ence fringes after hold times as long as T = 20 s at ∆z = 3.9µm sepa-ration while using the sepasepa-ration and load pulse timings tS= 0.516 ms and tA = 11 ms, making the gravitational phase the dominant con-tributor to the total phase difference. Naturally, the visibility of the fringe pattern at T = 20 s was quite low, as which-way information is slowly lost through imperfections in the optical lattice. In some con-trast, the reported visibility for shorter (but not short!) hold times constitute an impressive feat of the experiment as well: the visibility is near-perfect. The calculated fringe pattern of the geometry is∗

P0,1= 1 4 ± 1 4cos mg∆z ~ T + ∆φlaser . (1.16)

The theoretical visibility is then only 1/2, due to only two out of four output ports being resolved (see Figure 3). For hold times T = 0.2 s and T = 1.1 s at the ∆z = 3.9µm separation, the measured visibility is very close to this ideal value. For these hold times, the gravitational phase differences are 5400π and 27 000π, which should be compared to the laser phase at these pulse timings, 530π.

In summary of this section, we have seen three different methods of measuring quantum interference due to the presence of a grav-itational potential, with three different dependencies of the phase shift upon the gravitational field, mass, and experimental tools. The experiment by Xu et al. approaches the geometry of an idealized gravitational Mach–Zehnder interferometer, where a particle is in-stantly separated to follow two paths at constant height separation in a gravitational field, and then instantly recombined, making the phase shift solely on the gravitational potential. This idealized in-terferometer was used as a building block in a proposal by M. Zych et al. on visibility losses in clock interferometry due to time dilation [3], which we will return to in Section 1.3. In light of the Zych et al. proposal, the Xu 2019 experiment data constitutes an excellent theoretical testing ground for proposals of dynamical mass. In this thesis we therefore explore how the Xu 2019 experiment data can be used to restrict dynamical mass-proposals. In the next section, we

∗

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Introduction| 16 z t tS tA T tA tS π/2 π/2 optical lattice π/2 π/2 ∆z |1i |0i |0i |1i |0i |1i |1i |0i |1i g

Figure 3: The 2019 experiment by Xu et al. [8]. An optical lattice is used to hold the paths at constant height separation ∆z for the time T . This contributes a phase difference directly proportional to the Newtonian potential mg∆z. The paths outside the optical lattice contribute a laser phase ∆φlaser similar to the Kasevich–Chu phase. At the second and third π/2 pulses, two of the output beams are discarded, indicated as dashed grey lines. This reduces the theoretical maximum visibility of the experiment to 1/2.

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Introduction| 17

turn to a specific such proposal, which is then developed and put to the test in the remainder of the thesis.

1.2 The Compton frequency and universal Compton clocks

In a 2010 article, H. M¨uller, A. Peters and S. Chu [12] noticed an interesting connection between the path integral formalism for calcu-lating phase differences, and general relativity. According to general relativity, an observer carrying a clock will, by looking at his clock, measure the observer’s proper time, which depends on his particular path (worldline) through space-time. Famously, two observers follow-ing different worldlines will measure different proper times. M¨uller and coworkers applied this idea to a particle traversing two paths in an interferometer in superposition. Introducing a coordinate system with timelike coordinate t and using this to parameterize the world lines in the interferometer, the proper time along each path is

τ = 1 c Z r −gµν dxµ dt dxν dt dt (1.17)

where gµν are the components of the spacetime metric, xµ the com-ponents of the worldline, and c is the speed of light. The central insight of M¨uller et al. was to write the Lagrangian of the particle as

L(t) = mc2dτ_dt, (1.18)

implying that the phase picked up by a particle along an interferome-ter path in the semiclassical approximation may be elegantly written in coordinate-invariant form as φ = Z Γcl mc2 ~ dτ (1.19)

where Γcl is the classical worldline. This gives the correct phase by what is essentially the correspondence principle: Choosing a suitable metric to describe the Earth and expanding for a particle moving with velocity v_{c, the lowest-order terms in c}−2in (1.18) will beautifully yield the exact Newtonian Lagrangian, making (1.19) equivalent with

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Introduction| 18

the Newtonian path integral formalism laid out in the previous section to order _O(c−2).

The quantity under the integral in (1.19) has dimensions of inverse time and is called the Compton frequency

ωC = mc2

~ . (1.20)

The Compton frequency is the frequency of a photon which has the same energy as the rest-mass energy of the particle with mass m. In relativistic quantum field theory, the energy scale set by ωC (or equivalently, mc2_{) is the scale at which particle production} be-comes significant: E.g., illuminating the particle with mass m us-ing photons of frequency ωC will potentially produce a second ticle with mass m, or some other state with multiple massive par-ticles being produced. This energy scale is generally huge, in the GeV range. Specifically, for caesium, which is frequently used in atomic interferometry, mc2 _{= 1.24 GeV.}∗ _{Consequently, expressed in} seconds, the Compton frequency of a caesium atom is gargantuan: ωC,Cs/2π = 3.00× 1025s−1.

Controversially [13–16], M¨uller, Peters, and Chu in their 2010 paper interpreted this finding as indication that there is a physical oscillation at ωC, which they call a “Compton clock”, present in every massive particle. They then argued that if this is the case, measure-ments of the acceleration due to gravity g using atomic interferome-try can be re-interpreted as direct measurements of the gravitational redshift. If correct, this would provide measurements of the gravita-tional redshift several orders of magnitude more precise than other methods, due to the high running frequency of the Compton clocks. It is important to understand that the proposal of “universal Compton clocks”, as we will call the idea of the last paragraph, is conceptually separate from the insight that the interferometric phase can be written in the form (1.19). Once the expansion of (1.19) into the Newtonian limit has been made, this insight corresponds

essen-∗_{For comparison, the current world record for particle collider energy is} 13 TeV, held by the Large Hadron Collider at CERN, Geneva.

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Introduction| 19

tially to the factorization mc2+ mv 2 2 − mgz = mc 2 1 + v 2 2c2 − gz c2 (1.21) where the quantity in parenthesis is the _O(c−2_{) expansion of dτ /dt.} Factorizations do not qualify to be proposals of new physical laws, as they numerically predict exactly identical physics.∗ Rather, the universal Compton clock proposal stands independently besides this insight. In their paper, M¨uller and coworkers do not propose any the-ory of its dynamics. Only the existence of universal Compton clocks is postulated, with the Lagrangian insight as indicative support.

The criticism raised against M¨uller, Peters, and Chu’s 2010 paper concerned two main points: 1) The proper time difference between the paths in the experimental geometry they were using (a slight variant of the Kasevich–Chu geometry in Figure 1) is zero by metric theories of gravity, such that there would be no proper time differ-ence between the two Compton clocks [13]. What we noted earlier, that the Kasevich–Chu phase difference (1.12) depends on the laser phase but not the gravitational phase, is a consequence of this fact. 2) The lack of proposed dynamics for the Compton clocks. Any real quantum-mechanical clock, such as conventional atomic clocks, must employ a superposition of two or more energy eigenstates to mea-sure the passage of time. It is not sufficient to use a single energy eigenstate [15]. Here, we note that point 1) invalidates usage of the particular experiment that M¨uller and coworkers wished to reinter-pret, but that more advanced experimental geometries, especially the 2019 experiment by Xu and coworkers (Figure 3), get around this is-sue. 2) is however a fundamental isis-sue.

Assuming we extend the universal Compton clock postulate to include a proposal of the Compton clock dynamics, is their existence tenable? At first sight, we could claim that the dynamics of the Compton clocks occur in a new abstract Hilbert space, which does not interact with any previously known force or interaction. Such a proposal will be restricted by compatibility issues with known physics, which one would have to solve if one were interested in constructing a

∗_{The presence of the mc}2_{term, not natural to Newtonian physics, can be seen} as a gauge freedom of non-relativistic quantum mechanics [15].

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Introduction| 20

complete theory. More interestingly, restrictions are also imposed by the fact that the dynamics of this Hilbert space may contribute dis-tinguishability by the mechanism of the next section. In the sections following the next, we investigate to what extent universal Compton clocks become restricted by the complementarity principle. The uni-versal Compton clock proposal is an example of a theory where mass is postulated to have some hidden dynamics; thus we will also take some effort to consider how our results in restricting it will impact such proposals more generally.

1.3 Clock interferometry: Time dilation in gravitational interferometry

We now review the proposal on distinguishability from time dilation of quantum clocks by M. Zych and coworkers [3].∗ _{The effect is} essentially the quantum version of the well-known, experimentally established “twin paradox” of classical relativity, where two clocks, initially synchronized to the same time standard, are separated and traverse a gravitational field on two world lines at different height. The prediction of general relativity, experimentally well-established [17, 18], is that the higher-altitude clock runs at a faster rate than the lower clock. Precisely, it runs faster by the amount g∆z/c2_{, where ∆z} is the height separation and g the local acceleration due to gravity, a factor commonly called the redshift factor. Finally, the clocks are brought together again, and will be found to have desynchronized by the accumulated proper time difference.

In the quantum version of this experiment, we consider a clock which travels through an interferometer in superposition of the in-terferometer paths, as illustrated in Figure 4. In the presence of a gravitational field, the upper clock will run faster than the lower. As the clocks are brought back together in the interferometer, we would expect interference due to the phase difference mgzT /~ of the two interferometer paths, as in _{§ 1.1. However, the paths are now} also associated with different states of the clock. By measuring the clock state, we could tell which path the particle took through the

∗

The core idea was also proposed by Sinha and Samuel around the same time [15].

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Introduction| 21 z t 0 T 0 z |τ(0)i |τ(0)i τ ((1 + gz c2)T ) |τ(T )i _g

Figure 4: Clock interferometry in an ideal gravitational Mach– Zehnder interferometer. An incoming particle with an initial internal state |τ(0)i is coherently put in a superposition of two paths sepa-rated by height z at T = 0. The paths are assumed to separate in negligible time and then keep constant height difference until recom-bined, again in negligible time. The internal state at different points on the paths, just after separation and just before recombination, are indicated. Choosing our coordinate time to be the time of the lower path, the upper clock state will evolve at a faster rate (1 + gz/c2_). This potentially makes the internal states distinguishable, affecting fringe visibility.

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Introduction| 22

interferometer without consulting the interferometer detectors. This means that the paths have nonzero distinguishability, D > 0. By the quantum complementarity principle (see_{§ 1.1),}

D > 0⇒ V < 1. (1.22)

Consequently, we expect the presence of the clock to cause loss of interference pattern visibility.

We give a slightly more careful introduction to this effect below, before returning to it in _{§ 2.4, where we will have reason to give a} rigorous derivation of the effect.

In quantum mechanics, the simplest clock mechanism is a system with internal energy levels, in a superposition of at least two such energy levels. Writing the external state (position and momentum state) of the particle as _{|ξi, and the internal state as |τi, typically} one expects the total state of the system to be a product state,

|ψ(t)i = |ξ(t)i ⊗ |τ(t)i , (1.23) implying that there is no entanglement between the internal state_|τi and the external state |ξi – akin to the full separation of centre of mass-degree of freedom and internal degrees of freedom in compos-ite systems. The clock mechanism is provided by having _{|τi be a} superposition of internal energy eigenstates,

|τi =Xci|Eii , ci6= δij, (1.24) where ˆHinternal|Eii = Ei|Eii. Then, |τi will be a dynamically evolv-ing state in time: The projection_{hτ(0)|τ(t)i will be a nontrivial} func-tion of t, as the time-evolving phases of the internal energy states interfere with each other. For the simplest case of an equal mix of two internal levels,

|τ(0)i = √1 2(|E0i + |E1i) (1.25) ⇒ |τ(t)i = √1 2 e−iE0t/~_|E 0i + e−iE1t/~|E1i , (1.26) we will have

| hτ(0)|τ(t)i |2 _{= cos}2 ∆Et

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Introduction| 23

an observable, physical oscillation at the frequency∗ ω = ∆E/~ where ∆E = E1− E0. This periodicity can be used to measure the pas-sage of time: In conventional atomic clocks, an external laser is syn-chronized to ω, providing accurate macroscopic time measurement. But regardless of external measurement, the system with the inter-nal state (1.25) can itself be considered a clock mechanism, becoming orthogonal to itself, i.e. ”ticking”, once every 2π/ω seconds.

In the interferometer of Figure 4, we can as before apply the semiclassical approximation to the external degrees of freedom _|ξi. Then writing the position state along the lower path as |0i, and the upper path as_{|zi, the state just after the coherent separation at t = 0} is

|ψ(0)i = √1 2

|0i + ieiφ1_|zi_{⊗ |τ(0)i} _(1.28)

where φ1 is the relevant laser phase, the details of which will not concern us presently. Even though the position state is in a super-position, the total state is still an unentangled product state of the form (1.23). In quantum mechanics without time dilation, the pre-diction would be that the state remains unentangled forever, because there is no coupling between the centre-of-mass and internal degrees of freedom: The Hamiltonian is

ˆ

H = ˆHcom+ ˆHinternal. (1.29) But, when considering relativistic time dilation, the evolution of the internal system occurs at the position and momentum-dependent proper time, ˆ H = ˆHcom+ dˆτ dtHˆinternal= = ˆHcom+ 1 +gˆz c2 − ˆ p2 2m2_c2 +O(c −4 ) ˆ Hinternal. (1.30) The cross-terms∝ ˆz ˆHinternal(corresponding to gravitational time di-lation) and _{∝ ˆp}2_H_ˆ

internal (corresponding to special-relativistic time dilation) causes entanglement between the center-of-mass degree of

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Introduction| 24

freedom and internal degree of freedom. Consequently, the unentan-gled product state (1.28) evolves into

|ψ(T )i = √1 2

e−iΦ0_{|0i ⊗ |τ(T )i + ie}−i(Φz−φ1)_{|zi ⊗}

τ ((1 + gz c2)T ) E . (1.31) Because the two paths are assumed to have the same momentum p in the semiclassical approximation, we have neglected the special-relativistic contribution −p2_/2m2_c2 _{to the time dilation in this} ex-pression. The phases Φ0 and Φz are the phases corresponding to centre-of-mass propagation along the respective paths. This is an entangled state, with the state of the internal system entangled with the interferometer path, signalling distinguishability of the paths. For the special case of the internal state (1.25), the fringe pattern on the detectors will be [3] P0,1 = 1 2± 1 2cos ∆Egzt 2~c2 cos m +Ecorr c2 + h ˆHinternali c2 ! gzt ~ − ϕ ! . (1.32) There are two effects due to relativity apparent here: 1) The addi-tional cosine factor with frequency ∆E/2~· gz/c2_{, and 2) the} con-tribution of the relativistic energy corrections Ecorr and the internal energyh ˆHinternali to the interferometer phase in the second term. We remind the reader that we will derive this equation, along with the general case, in _{§ 2.4. This fringe pattern is compared to the ideal} Mach–Zehnder fringe pattern without clocks in Figure 5.

The effect 1) is the visibility modulation due to distinguishability of the clocks: If one calculates the visibility of (1.32) by (1.10), it will turn out to be V = | cos∆Egzt_2~c2

|. The visibility is zero at odd multiples of the time

t⊥= π~c 2

∆Egz (1.33)

precisely at which times hτ(0)|τ(t⊥)i = 0, when the states are maxi-mally distinguishable. The effect 2) is an occurrence of the universal coupling of gravity to energy: The mass that the gravitational phase shift couples to is not simply the ”bare mass” m, but also the energy of the internal state, and relativistic corrections to the bare mass m.

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Introduction| 25 P0− P1 t mgz/~ 1 −1 4π 8π 12π 16π 20π (a) V (t) t mgz/~ 1 0 _4π _{8π 12π 16π 20π} (b) no clock clock, ω = ωC/10

Figure 5: The effect of clocks on interferometry. (a) Comparison of interference fringes in an ideal gravitational Mach–Zehnder interfer-ometer with and without an internal clock state. In the presence of a clock, here set to the frequency ω = ωC/10, the fringe pattern is modulated at the clock frequency. (b) Comparison of the visibili-ties in the same cases. The visibility of the clock pattern drops to zero periodically at frequency ω, while the no-clock case has constant visibility.

Relativity teaches us that the mass which any experiment will detect is M = m + Ecorr/c2+h ˆHinternali/c2, the mass equivalent of the total energy of the system, so it will make sense to ”renormalize” the sys-tem’s mass to be M , a point we will return to in_{§ 2.4. However, for} any realistic clock, the energy scale of the clock degrees of freedom, typically hyperfine transitions, is insignificant to the rest mass energy scale mc2_{, so that m}_{h ˆ}_H

internali/c2. Additionally, atoms in inter-ferometers move so slowly that Ecorr is negligible, so that M ≈ m to very good accuracy.

While it is experimentally well-established that quantum clocks measure their proper times along their world path, so that the twin paradox applies to quantum-mechanical clocks [18, 19], clock inter-ferometry as outlined here has not been directly performed to date. An analogue experiment, employing atomic spins for the clocks and using inhomogeneous magnetic fields to simulate time dilation, was performed by Y. Margalit and colleagues in 2015 [5]. Distinguisha-bility from time dilation therefore remains a hypothetical, if highly

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Introduction| 26

plausible, effect. Due to the existing experimental support and theo-retical naturalness of the phenomenon, we will in this thesis assume it to be real.

The reason that the effect is yet to be experimentally verified is that the orthogonalization time (1.33) is very large for currently re-alistic values of ∆E: Larger than achievable holding times in current interferometers. When comparing interferometer performance, it is more useful to consider the product of the separation time and sep-aration distance, z_{· t. Then we expect to be able to detect the effect} only when

zt_{∼ zt}⊥= π~c 2

∆Eg. (1.34)

A table comparing ∆E of available quantum ”clocks”, such as hyper-fine transitions in atoms, spin precessions of fundamental particles, and vibrations of molecules, to the best zt-separation achieved in experiment at the time, is given by Zych et al. in [3]. At the time of writing, the smallest discrepancy between experimentally achieved and required zt is approximately 102 _{[20]. With this understanding,} we see that while the clock technically modulates the visibility peri-odically by (1.32), the expected experimental appearance is rather a slow drop of the visibility after a large number of fringes, so that the effect is sometimes called ”visibility loss due to gravity”.∗

We emphasize that this effect occurs regardless of the nature of the internal system. The only requirement is that the internal state is in a non-eigenstate of the internal Hamiltonian, and the only role of the internal dynamics is to set the timescale of the effect via the inverse energy separation of the superposition (as in (1.33)). Espe-cially, visibility modulations would be expected for any abstract or hidden Hilbert space. This universality opens the door to experimen-tally access proposed ”hidden” dynamics. This new opportunity is the main motivation for this thesis, and we we will put it to use in the next section.

∗

On this note, this effect has also been predicted to cause loss of coherence of superpositions of macroscopic systems in gravitational fields. See Refs. [4, 21].

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Introduction| 27

1.4 Compton clock interferometry

At the end of our discussion on Compton clocks in § 1.2, we asked whether a development of the Compton clock idea, where the clock dynamics are occurring in a Hilbert space postulated to be abstract and practically isolated from the rest of the universe, would be ten-able. If the system starts out in the product state (1.23), the Galilean quantum-mechanical expectation would be that the system remains in a product state. The clock space would simply ”tag along” in calculations, not producing observable consequences because no op-erators would entangle its state with any other state in any other Hilbert space. As we saw in the previous section,§ 1.3, this expecta-tion is incorrect when one takes relativistic correcexpecta-tions into account: the universality of gravity causes entanglement between any dynam-ical state and the centre of mass, even if there is no other known way of accessing the internal state directly.

If we believe that the proposed effect in_{§ 1.3 is real, the fact that} visibility modulations are not observed in current interferometric data tells us something: There is no dynamical process, or superposition of internal states, at an energy scale ∆E _{∼ π~c}2_{/gzt inside these} particles. Clearly then, existing interferometric data should restrict the feasibility of Compton clock-like hypotheses, and other proposals involving dynamics of mass. Indeed, in their original proposal, Zych et al. used the effect to place restrictions on proposals of proper time as a quantum degree of freedom [3]. In this thesis, we will apply the same philosophy, focusing on the universal Compton clock proposal. Specifically, we ask: Does interferometric data rule out the possibility of universal Compton clocks? If so, how restricted are they? If not, is it indeed possible that particles could contain hidden oscillations which are yet undiscovered?

Because we agree with the criticism against the original universal Compton clock proposal that a clock requires dynamics, as raised in [13, 15], to answer these questions, we must first develop the uni-versal Compton clock proposal into a theory containing a dynamical clock, akin to (1.24). Because we are interested in obtaining the most general restrictions possible, we will first consider the effect generally in§ 2.4. Next, we will construct concrete models in § 3–§ 3.2. Again, as our interest is restricting the possibility of universal clocks rather

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Introduction| 28

than proposing them, we will keep the model building simple. We will finally put the models to the test in _{§ 5, obtaining restrictions.} Additionally, as an interlude, we consider links between the universal Compton clock idea and tests of residual symmetries and the equiv-alence principle in_{§ 4.}

To conclude the introduction, we comment on the timing of this thesis. As we discussed in§ 1.1, interferometer geometries similar to the Kasevich–Chu setup, where the atoms are constantly in free fall, are not sensitive to the Newtonian potential mgz. Instead, they are only sensitive to the laser phase κgT2_{. Simply, this is because the} integral of the Newtonian free-fall Lagrangian around a closed loop is zero [10]. As we saw in (1.18) and (1.21),

LNewtonian+ mc2 dt = mc2 dτ dt +O(c −4₎ dt. (1.35)

Consequently, the integral of the differential proper time dτ around the loop formed by the free-falling paths in Figure 1 is also zero to O(c−4_).∗ _{Hence, this geometry does not generate any proper time} difference ∆τ between the paths to first order [13] – so to first order, these experimental geometries are not sensitive to the clock interfer-ometry effect! In contrast, the very recent experiment by Xu et al., where the atoms are held separated at constant height in a gravita-tional field, generate substantial proper time difference ∆τ . Then, at least when considering atomic interferometry data only, it is only in the very recent Xu 2019 experimental data where we can expect to obtain interesting quantitative restrictions on universal Compton clocks.

∗_{We can freely add mc}2_{to the Newtonian integral because its integral around} the closed path is trivially zero.

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General considerations| 29

2 General considerations

Before we develop our simple universal Compton clock models, we consider generally how they could be made consistent with existing theoretical and experimental knowledge.

Because most previous work on clock interferometry has focused on obtaining the visibility for general systems [4, 21], and derived the full fringe pattern only for special cases [3], we begin in§ 2.1 by deriving the fringe pattern of a gravitational Mach–Zehnder inter-ferometer for general clock states. In § 2.2 we specialize to general two-level system clocks. In these sections we will learn generally what requirements clocks must fulfill to be consistent with interferometry data, including restrictions by universal coupling of gravity to energy and fringe pattern considerations.

Because mass has the role of a parameter in standard non-relativistic quantum mechanics, and we will see in _{§ 2.1 and 2.2 that universal} Compton clocks require dynamical (variable) mass, we consider in § 2.4 how particle mass could be generated from energy in an inter-nal Hilbert space of a composite system, and which requirements this places on our Compton clock models.

Finally, we conclude this part of the thesis with a summarized list of requirements on Compton clock models in_{§ 2.5, which we then use} as input for our model building in§ 3.

2.1 General clock interferometry fringe pattern

After a derivation, we present the fringe pattern of general clock states (in fact, general internal states) and discuss its interpretation and implications for Compton clocks.

2.1.1 Derivation

The starting point of obtaining the clock interferometry effect is to obtain a formulation of quantum mechanics which includes general-relativistic corrections. Because we are dealing with atoms whose velocities are a negligible fraction of the speed of light c, and situ-ations where particle number is conserved, we do not need to use a field theory description of the quantum mechanics. Indeed, it is also

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preferable to stay in a first-quantized description because it is far easier to deal with issues of entanglement and internal Hilbert spaces there.

To obtain a first-quantized relativistic description, we can expand the total energy of a particle in the Schwarzschild spacetime in the slow-moving, weak-field limit, discard terms above the desired order in 1/c2_{, and then canonically quantize the resulting energy. This will} provide us with a _O(c−4)-relativistically corrected Hamiltonian for the centre-of-mass degree of freedom of a particle [3],

ˆ Hcom= mc2+ ˆ p2 2m 1 + 3 ˆ p 2mc 2! +mΦ(ˆr) 1 +Φ(ˆr) 2c2 − 3 2 ˆ p mc 2! . (2.1) Here, Φ(ˆr) = −GM/ˆr is the gravitational potential operator. Be-cause the geometry of the interferometer fixes the particle’s paths so that the semiclassical approximation applies, as was outlined in§ 1.1, ˆ

r and ˆp are simultaneously numerical functions of the classical path. We will then not be very interested in the relativistic corrections to the kinetic and gravitational energy. Therefore we summarize these contributions as ˆ Hcom= mc2+ ˆ p2 2m+ mΦ(ˆr) + Ecorr(ˆr, ˆp). (2.2) For systems with internal degrees of freedom, described by a Hamil-tonian ˆHclock, the correct total Hamiltonian is then

ˆ

H = ˆHcom+ ˙τ (ˆr, ˆp) ˆHclock, (2.3) where the dot denotes differentiation with respect to the coordinate time ˙τ = dτ /dt, and ˙τ (r, p) is the rate of proper time passage for a particle of position r and momentum p. At this point, the appearance of ˙τ in the total Hamiltonian (2.3) may best be seen as an assumption based on the strong experimental evidence that quantum systems experience proper time [17–19], but it can also be motivated from quantization of composite systems on metric spacetimes, see Ref. [22]. In the weak-field, slow moving limit of the Schwarzschild metric, to first order in c−2, ˙τ (ˆr, ˆp) = 1 +Φ(ˆr) c2 − 1 2 ˆ p mc 2 , (2.4)

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Figure 6: States in the gravitational Mach–Zehnder interferometer. The position state corresponding to the lower path is _|r1i, and the position state corresponding to the upper path is _|r2i. The verti-cal paths contribute the laser phases φl(0) and φl(T ), but no time-evolution phase. We also allow a phase-shifter which contributes an experimentally variable phase shift ϑ along the lower path, although such a device is not present in the experimental setups we will later compare to. There is an important conceptual difference between the freely variable phase shift ϑ and the laser phases φl, since the latter are interdependent on the path separation z.

so that (2.3) is ˆ H = mc2+pˆ 2 2m+mΦ(ˆr)+ 1 + Φ(ˆr) c2 − 1 2 ˆ p mc 2! ˆ Hclock+Ecorr(ˆr, ˆp). (2.5) Again because we work in the semiclassical approximation, our main interest in this Hamiltonian is to define the system’s time-evolution operator, which will determine the phase picked up by the states as they propagate along the classical paths. We denote this as

ˆ U (ˆr, ˆp, t) = exp −i ~ ˆ Ht . (2.6)

For our clock interferometry setup, we consider a gravitational Mach–Zehnder interferometer as in Figure 6.

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We are careful to account for laser phase contributions of the verti-cal paths, and also allow an experimentally variable phase shift added to the lower path, which adds the controllable phase ϑ to the lower path. The reason for this detailed treatment of the non-gravitational phase is that the laser phases φl(t) are not independently controllable from the height separation z in experiments: rather, they are interde-pendent. This distinction of the phases is relevant for the possibility of detecting universal Compton clocks by clock interferometry, but not for the appearance of the clock interferometry effect in the first place. We will therefore return to this point at the end of the present discussion.

The position state along the lower and upper paths are _|r1i and |r2i, respectively. Within the semiclassical approximation, we will assume that the momentum of the traveling wavepacket along these paths is definite and the same, p. Under these assumptions, the time evolution operator (2.6) acting on the position states, collectively denoted |rii, simplifies into

ˆ U (ˆr, ˆp, t)|rii ⊗ 1 = Ucom(r, p, t)|rii ⊗ ˆUclock(ri, p, t) (2.7) where Ucom(ri, p, t) = exp −i ~Hcom(ri, p)t (2.8) ˆ Uclock(ri, p, t) = exp −i ~ ˆ Hclock˙τ (ri, p)t . (2.9)

Importantly, the total time-evolution operator separates into an ex-ternal phase (2.8) and a time-evolution operator on the inex-ternal sys-tem (2.9) evolving with the proper time of the path ˙τ (ri, p)t. Natu-rally, the corresponding adjoint identities apply tohri| ˆU†. We stress again that this simplification occurs only when the center-of-mass follows a fixed semiclassical path independently of the internal dy-namics.

Then, we are ready to calculate the interference fringe pattern from a general clock state. Consider the initial internal state to be the general density matrix ˆρτ(0), so that the initial state incoming into the interferometer is_|r1i hr1| ⊗ ρτ(0). The beamsplitters are im-plemented as in Appendix A.1, using_|r1i = 1 0

T

, _|r2i = 0 1 T

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so that their action is ˆ V (t) = √1 2 1 ie−iφl(t) ieiφl(t) 1 (2.10) where the laser phase is abbreviated into φl(t). Then, the state post the first beamsplitter is

ˆ

ρ(T ) = ˆU (T )ρ(0) ˆU†(T ) = = 1

!

(2.12) where we introduced the temporary abbreviation

δ = Hcom(r2)− Hcom(r1)

~ T − φl(0), (2.13)

which is the centre-of-mass phase difference between the two paths at this stage. We neglected writing out p-dependence of operators and phases since this is uniform everywhere. The dependence of ˆUclock on T is also implicit for clarity.

Because the internal (clock) state is not measured by the interfer-ometer, we must take the partial trace to obtain the reduced density matrix, which the interferometer measures. The remaining opera-tions, application of the phase shifter∗ and the final beamsplitter, do

∗

The phase shifter operator is ˆP = eiϑ

|r1i hr1| + |r2i hr2|. It commutes with the time evolution operators, so that we can introduce it at any stage between the beamsplitters.

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not affect the internal state. To simplify expressions, we therefore apply the partial trace now. The reduced density matrix is

ˆ ρ0(T ) = Trτρ(T ) =ˆ = 1 2 1 + ie −iδ |r2i hr1|D Û_clock† (r1) Ûclock(r2) E τ − − ieiδ_|r 1i hr2|D Û_clock† (r2) Ûclock(r1) E τ ! (2.14) where D ˆ_U† clock(r1) Ûclock(r2) E τ = Trτ n ˆ_U† clock(r1) Ûclock(r2)ˆρτ(0) o . (2.15) To obtain the final state in the interferometer, from which we can read the detection probabilities, we apply the phase shifter and second beamsplitter operators. We can then read off the detection prob-abilities from the diagonal elements of the final state, as the pro-jection operators corresponding to detection at each output port is

ˆ

P0 =|r1i hr1| and ˆP1=|r2i hr2|, respectively. The result is P0,1= 1 2 ± 1 2Re n

ei(∆φ−ϕ)D ˆU_clock† (r2) ˆUclock(r1) E τ o . (2.16) Here, ∆φ = Hcom(r2, p)− Hcom(r1, p) ~ T (2.17)

is the centre-of-mass phase difference, the quantity which would de-termine the fringe pattern in the absence of internal states, and

ϕ = ϑ + φl(0) + φl(T ) (2.18) is the sum of the phase shifts from the phase shifter and the beam splitter laser pulses.∗

In our geometry, we implicitly assumed that the interferometer is small enough that we only needed to take the radial separation of the interferometer paths into account, not considering any curving of

∗_{So, in terms of our earlier abbreviation δ, ∆φ = δ}

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the gravitational field. For most interferometers on Earth, this is an excellent approximation. In the same vein, we can also approximate

Φ(r2)− Φ(r1)≈ GM

r2 1

z≡ gz (2.19)

where z = r2 − r1 and we identified the local acceleration due to gravity, g. Then, by reference to (2.1), the center-of-mass phase difference to first order in z and c−2 _is

∆φ = mgz(1₋ 3 2 p mc 2 )_·T ~ ≈ mgzT ~ . (2.20)

The relativistic correction to the phase shift is completely negligible for contemporary experiments.

The trace over the clock time evolution operators is more elegantly expressed as D ˆ_U† clock(r2, p, T ) ˆUclock(r1, p, T ) E = exp i ~ ˆ Hclock∆τ (2.21) where ∆τ _{≡ ( ˙τ(r}2, p)− ˙τ(r1, p))T is the proper time difference of the two interferometer paths. Expanded to first order in small z = r2−r1, by reference to (2.4), the proper time difference is∗

∆τ = gz

c2T. (2.22)

Equation (2.21) is the expectation value of the clock time evolution operator evaluated at time ∆τ : It quantifies how much the internal state desynchronized along the two paths due to the time dilation:

exp i ~ ˆ Hclock∆τ

=D ˆU_clock† (∆τ )Efor pure states= _{hτ(∆τ)|τ(0)i} (2.23) where the last equation holds if the initial state of the clock is a pure state_{|τi, ˆ}ρτ =|τi hτ|. Note that, since the time-evolution operator is not a Hermitian operator, its expectation value is generally complex. We apply these simplifications to (2.16) to obtain our final result P0,1= 1 2 ± 1 2Re expimgz ~ T − ϕ exp i ~ ˆ Hclock gz c2T τ . (2.24)

∗_{Note that comparing Eqs. (2.20) and (2.22), we see that ∆φ = ω}

C∆τ , an example of the relation (1.19) of§ 1.2 in action.

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2.1.2 Discussion

If the clock is in an eigenstate of the internal clock Hamiltonian, D

expi ˆHclock∆τ E

= exp_{{iEt/~} constantly, where E is the energy} of the eigenstate. In this case, (2.24) reduces to the Mach–Zehnder fringe pattern for particles without clocks:

P0,1 = 1 2± 1 2cos m + E c2 gzT ~ − ϕ . (2.25)

Eq. (2.25) was plotted as the orange graph in Figure 5. Note the contribution of the mass-equivalent of E to the particle’s mass in this expression. This is fully expected, as by universal coupling of gravity to energy, the gravitational mass of the particle should be its total energy equivalent. As we saw in _{§ 1.2, the gravitational} fringe pattern is connected to the particle’s mass via the Compton frequency, since the phase of the fringe pattern is ∆φ = ωC∆τ . So, we can identify the Compton frequency in (2.25) as the quantity multiplying the factor gzT /c2 _{= ∆τ . Thus by factorization, the} Compton frequency in this case is ωC = (mc2+ E)/~, implying from the definition of ωC (1.20) that the mass of the particle is M = m + E/c2_{. We can then conclude that in clock interferometry, the} quantity m has the character of a ”bare mass”, corresponding to the portion of mass-energy of the particle which is not part of the clock mechanism. Experiments measure the total mass M .

As a bonus, this analysis throws some light on the significance of the Compton frequency ωC. Despite being a large number when expressed in seconds, ωC is in fact the frequency of the interference fringes measured in proper time difference. Eq. (2.24) in terms of ∆τ is P0,1 = 1 2± 1 2Re exp(i(ωC∆τ − ϕ)) exp i ~ ˆ Hclock∆τ τ . (2.26)

For universal Compton clocks, we would expect thatD ˆHclock E ∼ ~ωC. Crudely, then, exp i ~ ˆ Hclock∆τ ∼ exp(iωC∆τ ), (2.27)

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so the visibility modulations due to the Compton clock occurs on the same separation time scale as the Newtonian interference fringe pat-tern. This gives us a better sense of how ”fast” a universal Compton clock would tick.

With the ”renormalization” M = m + E/c2_{, (2.25) should be} compared to the experimentally measured fringe pattern of Xu et al. [8], (1.16) in§ 1.1. We can conclude that the result (2.24) has the cor-rect limit in the case of an internal Hilbert space of the particle in a non-dynamical state. This is expected, as atoms, molecules, neutrons, and other non-fundamental particles presently used in interferometry carry around several internal degrees of freedom, to good approxima-tion in energy eigenstates.∗ _{This is also why visibility modulations are} thought to be useful as a universal probe of internal dynamics: Any internal Hilbert space in a dynamical state (that is, a non-eigenstate of the internal energy operator ˆHclock) would give a signal in the in-terference fringes via (2.24). The only exception would be if there is an excited internal dynamic with a fringe pattern (2.24) that re-produces (2.25) faithfully despite the dynamics. We will attempt to construct precisely such a dynamic for the universal Compton clock proposal in the coming sections.

As mentioned, the effect of (2.24) which has attracted the most interest is its effect on the visibility of the fringe pattern. To find this generally, we write exp i ~ ˆ Hclock gz c2T = exp i ~ ˆ Hclock gz c2T eiα(T ) (2.28) where α(T ) is the complex phase of the expectation value at time T . In this form, P0,1 = 1 2± 1 2 exp i ~ ˆ Hclock gz c2T τ cos mgzT ~ + α(T )− ϕ . (2.29)

∗_{In molecules and atoms, we would naively expect the internal degrees of} freedom to be in thermal states, which could potentially give a signal in (2.24). However, to prevent decoherence from thermal radiation, interferometry experi-ments are carried out at ultracold temperatures, which essentially makes only the internal ground state populated. The effect is then too weak to be measurable.

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Then, looking at the definition of visibility (1.10), which we repeat for convenience:

V = maxϕ{P0, P1} − minϕ{P0, P1} maxϕ{P0, P1} + minϕ{P0, P1}

, (2.30)

it is clear that the maxi/minimizing value of ϕ is ϕ = mgzT /~+α(T ), so that the visibility is easily calculated to be

V (T ) = exp i ~ ˆ Hclock gz c2T . (2.31)

These are the visibility modulations caused by clocks, and was ob-tained in [21] and [4]. But as we already saw, this is not the only effect of clocks predicted by (2.29): The energy of the internal system also affects the fringe pattern via α(T ). In the eigenstate case, which we saw in the beginning of this section, α(T ) = E∆τ (T )/~, so that the energy E contributes to the total mass of the particle. We will see other examples in the next sections. Presciently, we remark that while (2.29) may seem simpler than (2.24) due to the ”visibility times cosine” form of the former, in general α(∆τ ) will be a complicated function of ∆τ , so that factorizing out the phase factor cannot be performed in a simple, or even closed, form.

From the comparison of the general clock interferometer pattern (2.24) and (2.29) to the no-clock interferometer pattern (2.25), can we already say something about the feasibility of universal Compton clocks? Such clocks have to be in a state with tangible dynamics, which bars the non-dynamical or next to non-dynamical cases where V _{≈ 1, α(T ) ≈ const constantly. Then, necessarily, V (T ) → 0 for} some times T = t⊥, because a sufficiently dynamical internal state becomes orthogonal or partially orthogonal to its initial state at some point t⊥ in the dynamics. Essentially, this means that a Compton clock could only be hiding in existing experimental fringe patterns if the visibility V (T ) and the extra phase α(T ) conspicuously con-spired to make the pattern (2.29) look like (2.25). Then, since we can eliminate α(T ) by varying ϕ, the most effective experimental search for Compton clocks would involve experimental control of ϕ. This would allow direct measurement of the visibility V , but infor-mation would be provided already by checking whether the pattern

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behaves as expected when varying ϕ slightly. This is the reason for our conceptual distinction between the controllable phase shift ϑ and the laser phases φlin ϕ = ϑ + φl(0) + φl(T ) (see Figure 6). The fact is that current experimental setups, especially the Xu 2019 setup (Fig-ure 3), does not include a variable phase shift ϑ: i.e., they are not direct measurements of T -dependent visibility. Rather, the only con-tribution to ϕ are the laser phases, whose values are interdependent on the experimental geometry (specifically, the height separation z). So, while far-fetched, a conspiracy of V (T ) and α(T ) is not totally unfeasible, especially considering the timing of this thesis in relation to experimental data, as discussed at the end of_{§ 1.4.}

2.2 General pattern of 2-level clocks

To get a better sense of the predictions of (2.24), we now consider some different states of a two-level clock system as a concrete exam-ple. The two-level Hamiltonian is

ˆ

Hclock= E0|0i h0| + E1|1i h1|

where E0 and E1 are the energies of the two energy eigenstates |0i and |1i. We calculate the predictions of (2.24) for a few different states of this system.

We begin by assuming the clock state to be a pure state, param-eterized on the Bloch sphere as

|τi = cosθ

2|0i + sin θ 2e

iφ_|1i _(2.32)

where θ is the polar angle and φ is the azimuthal angle on the Bloch sphere. Then, we find that (2.24) gives∗

A = cos2 θ 2cos gzT ~c2(mc 2_{+ E} 0)− ϕ + + sin2 θ 2cos gzT ~c2(mc 2_{+ E} 1)− ϕ . (2.33)

The form of this equation is suggestive of the physical picture that the clock behaves in the experiment as having either mass-energy mc2₊

∗

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E0or mc2+E1with their respective weights in the state. Importantly, this is not the only state which produces the interference pattern (2.33). Since the partial trace in (2.24) selects only the diagonal elements of the state operator, the classically mixed state

ˆ ρτ =cos 2 θ 2 0 0 sin2 θ 2 (2.34) also gives (2.33). Then, the azimuthal angle φ on the Bloch sphere does not affect the interference pattern: Indeed, the state need not even be coherent along this angle.

Most earlier investigations of fringe patterns in the literature, in-cluding the original clock interferometry proposals [3, 15], have paid attention only to the special case of an equally mixed 2-level system,

|τ(0)i = √1 2

|0i + eiφ|1i, (2.35) corresponding to θ = π/2. We also used this state as introductory example in § 1.3. Its fringe pattern ((1.32), blue graph of Figure 5) is attractively simple. For convenience, we restate it here:

P0,1= 1 2± 1 2cos gzT ~c2 ∆E 2 cos gzT ~c2(mc 2_{+ ¯}_E) − ϕ (2.36) where ¯E = (E0+ E1)/2 and ∆E = E1− E0. Comparing to (2.29), and as can be verified by calculation, it is clear that for this special case V (∆τ ) = cos ∆E 2~ ∆τ (2.37) α(∆τ ) = E∆τ¯ ~ + πΘ − cos ∆E 2~ ∆τ (2.38) where the last term in α(∆τ ) serves to restore the sign to the V cosine, Θ(t) being Heaviside’s function.

This fringe pattern is simple because of its ”visibility times cosine” form