Atomic Clocks

Ben Roberts (UQ) — 2026 — broberts.io/slides

# About Us

<div style="text-align: left; float: left; width: 50%">
            <br>
            <br>
            <div>
              <h2>High-precision atomic theory</h2>
              <h3> • Heavy atoms for precision physics</h3>
              <h3> • Atomic parity violation</h3>
              <h3> • Probing nuclear and fundamental physics</h3>
              <h3> • Support development of atomic clocks</h3>
            </div>
            <br>
            <div>
              <h2>Low-mass dark matter</h2>
              <h3> • Mass range: $\sim$10 GeV down to $10^{-23}$ eV</h3>
            </div>
            <br>
            <div class="fragment">
              <h3>AMPSCI</h3>
              <h3> • Atomic structure calculations</h3>
              <h3>   • One- and two-valence systems</h3>
              <h3>   • Relativistic scattering</h3>
              <h3> • <a href="https://ampsci.dev">ampsci.dev</a></h3>
            </div>
          </div>

# Precision timing: fundamental physics

<div style="text-align: left; float: left; width: 70%">
            <br>
            <div class="fragment" data-fragment-index="0">
              <h3> 1657: Huygens designs pendulum clock</h3>
              <h4> • Accurate to $\sim15$ s / day</h4>
              <h4> • Works on solid physics principal: $T\propto\sqrt{L/g}$</h4>
            </div>

<br>

<div class="fragment">
              <h3> 1672: Jean Richer expedition to French Guiana </h3>
              <h4> • Observe Mars near equator -- measure scale of solar system</h4>
              <h4> • Failed: pendulum clocks ran slow by 2 min / day</h4>
            </div>

<br>

<div class="fragment">
              <h3> 1687: Newton's Principia (15 years post Richer)</h3>
              <h4> • Explained Richer's measurement: law of gravitation</h4>
            </div>

<br>

<div class="fragment">
              <h3> 1676: Rømer calculates speed of light</h3>
              <h4> • Observe eclipse of Jupiter's moons</h4>
              <h4> • Required accuracy of $\sim30$ s / day</h4>
            </div>

<br>

<div class="fragment">
              <h3> Modern day</h3>
              <h4> • Allows GPS</h4>
              <h4> • Underpins modern geodesy</h4>
            </div>

</div>

<div class="fragment " data-fragment-index="0" style="text-align: center; float: right; width: 30%">
            <br>
            <img src="img/HuygensPendulumBW.png" width="85%"><br>
            <font size="3">[Huygens 1673]</font>
          </div>

# Clock: a thing that ticks

### Predictable, period motation + method to count ticks

<br>
          <div>
            <div class="fragment fade-in" style="text-align: center; float: left; width: 32%">
              <img src="img/StoneHenge.png" width="86%"><br>
              <font size="3">[Getty]</font>
              <br>
              <h3>
                Earth's orbit/rotation
              </h3>
            </div>

<div class="fragment fade-in" style="text-align: center; float: left; width: 32%">
              <img src="img/H1.png" width="96%"><br>
              <font size="3">[H1, Royal Museums Greenwhich]</font>
              <br>
              <h3>
                Swinging pendulum
              </h3>
            </div>

<div class="fragment fade-in" style="text-align: center; float: right; width: 32%">
              <img src="img/laser-JILA.jpg" width="100%">
              <font size="3"> [JILA]</font>
              <br>
              <h3>
                Oscillating electromagnetic wave
              </h3>
            </div>
          </div>

<div class="fragment fade-in" style="text-align: left; float: left; width: 100%">
            <br>
            <br>
            <br>
            <h3>
              • Pendulums (+all kinematic clocks): depend on materials, location, temperature etc.
              <br>
              • Earth rotation: unstable at ~5 ms per day $10^{-7}$
              <br>
              • Lasers/EM wave: depend on cavity, material etc.
              <br>
              <h3 class="fragment fade-in">
              • Atomic clock: universal timekeeping based on fundamental physics? 
              </h3>
            </h3>
          </div>

## Lord Kelvin in 1879 (attributed to Maxwell):

<div style="text-align: left; float: left; width: 70%">
            <div style="text-align: center; float: center; width: 100%">
              <font size="18">
                The recent discoveries ...
                
                indicate to us natural standard <br>pieces of matter such as <b>atoms of hydrogen or sodium,<br>
                  ready made in infinite numbers, <br>all absolutely alike in every physical property</b>.<br><br>
              </font>
            </div>

<div class="fragment fade-in" data-fragment-index="1"
              style="text-align: center; float: center; width: 100%">
              <font size="18">
                The time of vibration of a sodium particle corresponding to <br>any one of its modes of
                vibration is <b>known to be absolutely independent of its position in the Universe</b>...
              </font>
            </div>

<div style="text-align: center; float: center; width: 90%">
              <font size="5">
                Thomson, and Tait, 1879, <i>Elements of Natural Philosophy</i> (Cambridge University Press)</font>
            </div>

</div>
          <div style="text-align: left; float: right; width: 30%">
            <img src="img/Maxwell.png" width="100%"><br>
            <font size="3"> [R. H. Campbell | Institution of Electrical Engineers]</font>
          </div>

# Atomic Clocks 101:

<br>

### Frequency reference: atomic transition

<div class="fragment fade-in" style="text-align: center; float: left; width: 32%">
            <img src="img/Bohr.png" width="70%"><br>
            <font size="3"> JabberWok [Wikimedia Commons]</font>
            <br><br><br>
            <h3>
              Atomic transition: "perfect*" frequency reference
            </h3>
          </div>

<div class="fragment fade-in" style="text-align: center; float: left; width: 32%">
            <img src="img/Wcislo2016.png" width="95%">
            <font size="3">[Wcislo, Nature Astronomy (2016)]</font>
            <br><br><br>
            <h3>
              Compare frequency of oscillator to atomic transition
            </h3>
          </div>

<div class="fragment fade-in" style="text-align: center; float: right; width: 32%">
            <img src="img/Al+Choi2010.png" width="95%">
            <font size="3">[Chou et al., Science 329, 1630 (2010)]</font>
            <br><br><br>
            <h3>
              Lock on: Adjust local oscillator to maximum transition rate
            </h3>
          </div>

# Optical atomic clocks

<div class="fragment fade-in" style="text-align: left; float: left; width: 55%">
            <img src="img/Stability.png" width="95%"><br>
            <font size="3"> [M. Safronova]</font>
            <br>
            <br>
            <br>

* Requires ultra-stable laser ("see" narrow line)
            * Frequency comb ("listen" to high frequency)

</div>

<div class="fragment fade-in" style="text-align: center; float: right; width: 45%">
            <img src="img/basicclock2.png" width="95%">
            <font size="3">[Centre for Quantum Technologies | U. Singapore]</font>
            <br>
            <br>
            <br>
            <img src="img/Diddams-Comb.png" width="95%">
            <font size="3">[Diddams <i>et al.</i>, Science <b>369</b>, 3676 (2020)]</font>
          </div>

* Accurate to parts in $10^{18}$
            * better than 1 s over lifetime of the universe!
          </div>

# Example: gravitational redshift

<br>

$$ \Huge {\color{orange}
          \frac{\Delta f}{f} = -\frac{\Delta U}{c^2} \sim 10^{-16} \frac{\Delta r}{{\rm m}}
          }
          $$

<br>

<div class="fragment fade-in" style="text-align: center; float: left; width: 60%">
            <img src="img/Chou-A.png" width="50%">
            <img src="img/Chou-B.png" width="45%"><br>
            <font size="4">30 cm [Chou et al., Science <b>329</b>, 1630 (2010)]</font><br><br>

* $\sim$ 30 cm
          </div>

<div class="fragment fade-in" style="text-align: center; float: right; width: 40%">
            <img src="img/Ye-Redshift.png" width="85%">
            <br>
            <font size="4">[Bothwell <i>et al.</i>, Nature <b>602</b>, 420 (2022)]
            </font><br><br>

* $\sim$ 1 mm
          </div>

* Gravitational red-shift is one of the biggest systematic which must accounted for in clock comparisons --
            even within the same lab!
          </div>

# The Standard Model + General Relativity

<br>

<div style="text-align: center; float: left; width: 50%">
              <img
                src="img/Standard_Model_of_Elementary_Particles_dark.svg"
                width="95%">
              <font size="3"> Credit: Cush [Wikimedia Commons]</font>
            </div>

<div style="text-align: left; float: right; width: 47%">
              <h2>Tested with extraordinary precision</h2>
              <br>
              <h3>   • e.g., electron magnetic moment, Higgs boson</h3>
              <h3>   • gravitational lensing, gravitational waves</h3>
              <br><br><br>
              <h2>However...</h2>
              <br><br><br>
              <div class="fragment">
                <h2>Several deep inconsistencies</h2>
                <h2>with the observed universe</h2>
              </div>
              
            </div>

# Unexplained mysteries (among others)

<br>
            <div style="text-align: center; float: left; width: 70%">
              <div class="fragment" data-fragment-index="0" style="text-align: left;" data-markdown>
  
                <h2>Matter anti-matter asymmetry</h2>
                <h3> • CP-violation in the Standard Model is too small</h3>
              </div>
  
  
              <br>
              <div class="fragment" data-fragment-index="1" style="text-align: left;" data-markdown>
  
                <h2>Nature of Dark Energy</h2>
                <h3> • Accelerated expansion of universe
                <h4>  • Cosmologists: Why is Λ so large?
                   • Particle theorists: Why is Λ so small?
                </h4>
  
              </div>
  
              <br>
              <div class="fragment" data-fragment-index="2" style="text-align: left;" data-markdown>
  
                <h2>Nature of Dark Matter</h2>
                <h3> • Missing mass from the universe
                  • Matter that interacts <em>only</em> gravitationally? </h3>
  
              </div>

<div class="fragment" data-fragment-index="3" style="text-align: left;" data-markdown>
                <h3> <font color="orange">95% of universe is "dark"; we have to use the 5% to learn about the rest! </font></h3>
              </div>
  
            </div>
  
            <div style="text-align: left; float: right; width: 25%">
              <br>
              <img class="fragment" data-fragment-index="0" src="img/antimatter.png" width="95%">
              <br><br><br><br>
              <img class="fragment" data-fragment-index="1" src="img/dm-pie.jpg" width="95%">
            </div>

# Focus: dark matter

<br>

<div style="text-align: center; float: left; width: 35%">
              <br><br><br>
              <h3>Coma Cluster</h3>
              <br>
              <img
                src="img/Ssc2007-10a1.jpg"
                width="95%">
              <font size="3"> Sloan Digital Sky Survey + Spitzer Space
                Telescope
                [NASA / JPL-Caltech / L. Jenkins]</font>
            </div>

<div style="text-align: left; float: right; width: 47%">
              <h2>First major hints: galaxy clusters</h2>
              <br>
              <h3>Fritz Zwicky: clusters too fast. <br>
                • Termed phrase "dark matter"</h3><br>
              <p>(And some even earlier hints)</p>

<br>
              <div class="fragment">
                <h3>Then: Vera Rubin, along with Kent Ford</h3>
                <br>
                <img src="img/vera-rubin.jpg" width="60%"><br>
                <font size="3">
                  [AIP Emilio Segrè Visual Archives, Rubin Collection]
                </font>

</div>
            </div>

# Galactic Rotation Curves

<div style="text-align: center; float: left; width: 70%">
              <br><br>
              <img
                src="img/Rotation-curve-of-spiral-galaxy-Messier-33-Image-Credit-Mario-De-Leo.png"
                width="95%"><br>
              <font size="3">[Mario De Leo, Wikimedia Commons]</font>
            </div>

<div style="text-align: left; float: right; width: 30%">
              <br><br>
              <h2>Missing Matter?</h2>
              <br>
              <p data-markdown>• Not a small correction</p>
              <br>
              <p data-markdown>• 80 - 95% of mass is "missing"</p>
              <br>
              <div data-markdown class="fragment fade-in">
                <p data-markdown>• Implies <font color="orange">dark matter</font></p><br>
                \[a = \frac{G \, [M(r) + {\color{orange}\delta M(r)}]}{r^2} \]
                <br>
                <p data-markdown>   • Simulations + observations: DM "halo"</p>
              </div>
              <br>
              <p data-markdown class="fragment fade-in">• ... or <font
                  color="green">modified
                  gravity</font><br><br>
                \[a = \frac{[G + {\color{green}\delta G(M,r)}] \, M(r)}{r^2} \]
              </p>
              <br>
              <p data-markdown class="fragment fade-in">Evidence continues to build:</p>
            </div>

# Evidence builds up

<br>
          <div style="text-align: left; float: left; width: 50%">
            <div>
              <h2> Rotation curves</h2>
              <h3> • Missing galactic matter</h3>
            </div>
            <br>
            <div class="fragment"  data-fragment-index="1">
              <br>
              <h2> Gravitational lensing</h2>
              <h3> • Gravitational bending of light</h3>
            </div>
            <br>
            <div class="fragment"  data-fragment-index="2">
              <br>
              <h2> Cosmic Microwave Background</h2>
              <h3> • Gravitational imprint on background radiation</h3>
            </div>
            <br>
            <div class="fragment"  data-fragment-index="3">
              <br>
              <h2> Large-Scale Structure Simulations</h2>
              <h3> • Dark Matter required to match observations</h3>
            </div>

<div class="fragment"  data-fragment-index="4">
                •  Ωc = 0.2589(57) [Plank]
              <br>
                $$
                  \rho_{\rm DM} \approx 0.4 ~ {\rm GeV}/{\rm cm}^3
                $$
            </div>

</div>
          <br>
          <div style="text-align: center; float: right; width: 50%">

<div class="r-stack">
              <img class="fragment fade-out" data-fragment-index="1" src="img/RotCurve.png" width="100%">
              <div class="fragment fade-in-then-out" data-fragment-index="1" >
                <img src="img/1e0657_scale.jpg" width="100%">
                <br><font size="3">[NASA/CXC/M. Weiss - Chandra X-Ray Observatory]</font>
              </div>
              <div class="fragment fade-in-then-out" data-fragment-index="2">
                <img src="img/CMB.png" width="85%">
                <img src="img/CMB-PowerSpectrum-Planck.png" width="75%">
                <br><font size="3">[ESA/Planck Collaboration (2013)]</font>
              </div>
              <div class="fragment fade-in" data-fragment-index="3" >
                <img src="img/Cosmic_web.jpg" width="100%">
                <br><font size="3">[Volker Springel/MPI, Wikimedia Commons]</font>
              </div>
            </div>

</div>

# Dark Matter: What we know

<br>
            <h3> There's lots of it</h3>

<h3> It gravitationally clusters in halos around galaxies</h3>
            <h4> (really, galaxies gravitationally cluster around DM)</h4>

<div style="text-align: left; float: left; width: 50%"
              class="fragment">
              <br><br>
              <h2>Very strong evidence:</h2>
              <br>
              <h3>• Rotation curves, gravitational lensing, CMB </h3>
              <br>
              <h3>• Also: large scale structure, BAO etc. </h3>
              <br>
              <br>
              <div class="fragment">
                <h3>• It's not strange for particles to not interact with
                light (e.g., neutrinos)</h3>
              </div>
            </div>

# Dark Matter: What we don't know

<br>
            <div class="fragment" data-fragment-index="0">
              <h2> ... everything else</h2>
            </div>

<br>

<div class="r-stack">
              <img class="fragment" data-fragment-index="1"
                src="img/DM_masses2.png" width="80%">
              
              <img class="fragment" data-fragment-index="3"
                src="img/DM_masses5.png.svg" width="80%">
              <img class="fragment" data-fragment-index="3"
                src="img/DM_masses6.png.svg" width="80%">
              <img class="fragment" data-fragment-index="3"
                src="img/DM_masses7.png.svg" width="80%">
              <img class="fragment" data-fragment-index="3"
                src="img/DM_masses8.png.svg" width="80%">
              <img class="fragment" data-fragment-index="3"
                src="img/DM_masses9.png.svg" width="80%"> 
            </div>
            <div class="fragment" data-fragment-index="1">
              <font size="3">US "Cosmic Visions" report [arXiv:1707.04591]</font>
            </div>

<div class="fragment" data-fragment-index="1">
              <h3>• Possible mass range: 90 orders of magnitude!</h3>
            </div>
            <div class="fragment" data-fragment-index="2">
              
              <h3>• Fundamental particle: 50 orders (de Broglie to Planck)</h3>
            </div>

<br>

<div class="fragment">
              <h3> • Vast majority of focus on WIMPs, but field is very wide</h3>
            </div>

# WIMP Searches: XENON

<div style="text-align: left; float: left; width: 60%">
              <div class="r-stack">
                <div class="fragment fade-out" data-fragment-index="1">
                  <br>
                  <img src="img/Feynman.png" width="65%">
                  <br><br>
                  <h3 class="fragment highlight-red" data-fragment-index="0">•
                    Direct detection
                    (scattering)</h3>
                  <h3>• Indirect detection (annihilation)</h3>
                  <h3>• Collider searches (production)</h3>
                </div>
                <div>
                  <div class="fragment fade-in" data-fragment-index="1">
                    <br>
                    <img
                      src="img/WIMP-nucleon.png" width="70%">
                  </div>
                  <br>
                  <h3 class="fragment fade-in" data-fragment-index="2">
                     • High mass: fewer particles
                  </h3>
                  <h3 class="fragment fade-in" data-fragment-index="3">
                     • Mass below nuclear mass ( $\sim\,{\rm GeV}$ ): energy threshold
                  </h3>
                  <h3 class="fragment fade-in" data-fragment-index="4">
                     • Plot shuold extend 30 order-of magnetide to the left 
                    <br>
                      (and 15 to the right)
                  </h3>
                </div>
              </div>
            </div>

# En*light*ening the search for Dark Matter?
            ## Atomic Physics!

<br>

<h2> Mass drops below nuclear mass:</h2>
              <h3> • No nuclear recoils</h3>
              <h3> • Instead: electron recoils + ionisation</h3>
              <h4>   • Caddell, Flambaum, BMR, <a href="https://journals.aps.org/prd/abstract/10.1103/PhysRevD.108.083030">Phys. Rev. D <strong>108</strong>, 083030 (2023)</a>, <a href="http://arxiv.org/abs/2305.05125">[2305.05125]</a></h4>
              </div>
              <br>

<h2> Mass drops below electron mass:</h2>
              <h3> • No electron recoils</h3>
              <h3> • Instead: absorption (dark photoelectric effect)</h3>
              </div>
              <br>

<h2> Mass drops below eV:</h2>
              <h3> • Classical DM field</h3>
              <h3> • Quantum sensing: <font color="orange">atomic clocks</font></h3>
            </div>
            <br>

</div>

<div style="text-align: center; float: right; width: 40%">
              <br>
              <img class="fragment" data-fragment-index="0"
                src="img/atom.png" width="35%">
              <br><br>
              <div class="r-stack">
                <div class="fragment" data-fragment-index="2">
                  <img
                    src="img/laser-JILA.jpg" width="85%"><br>
                  <font size="3">[JILA]</font>
                </div>
                <div class="fragment" data-fragment-index="3">
                  <img
                    src="img/Ye_UCBolder-SrLattice.jpg" width="75%"><br>
                  <font size="3">[Ye/UCBolder]</font>
                </div>
              </div>
            </div>

# Light and Ultralight Dark Matter

$$ \Huge
            \rho_{\rm DM}\simeq 0.3 ~ \frac{\rm GeV}{{\rm cm}^3}
            $$

<br>

## Mass decreases $\implies$ number density increases:
            ## Classical boson field (e.g., axions, scalars)

<br>

<div style="text-align: left; float: right; width: 70%">
              <div class="fragment fade-in">
                <h3>
                  1. No interaction: oscillation: $\phi = \phi_0 \cos(m_\phi
                  t)$
                </h3>
                \[\phi_0^2\propto \rho_{\rm DM}\]
              </div>

<h3 class="fragment fade-in">
                2. Interactions: clumps
              </h3>
              <br>
              <h3 class="fragment fade-in">
                3. Also: constant build-up (local over-densities)
              </h3>
            </div>

<h2 class="fragment fade-in">
              Classical field: quantum sensors (clocks)
            </h2>

# "Listen" for DM field using atomic clocks

<br>

<div style="text-align: left; float: left; width: 50%">
              <br> <br>
              <img src="img/NHanacek_NIST.png" width="90%"><br>
              <font size="3">[N Hanacek/NIST]</font>
            </div>

<div style="text-align: left; float: right; width: 50%">
              <br>
              <div>
                <h3>
                   • DM scalar field has small interaction with matter
                </h3>
                <h3>
                   • Shift atomic energy levels and frequencies
                </h3>
              </div>
              <br>
              <div class="fragment fade-in">
                <h3>Observable shift:</h3>
                
                <br>
                \[\huge
                \frac{\delta f}{f} = \kappa \, \phi_{\rm DM}(\vec{r},t)
                \]
              </div>
              <br>
              <div class="fragment fade-in">
                <h4>
                   • $\kappa$ - model dependent parameter
                </h4>
                <h4>
                   • Calculated: high-precision atomic theory
                </h4>
              </div>

<br>
              <div class="fragment fade-in">
                <h3>Example: variation of constants</h3>
                
                <br>
                \[\huge
                \mathcal{L} = \phi F_{\mu\nu}F^{\mu\nu}
                \implies \alpha\to\alpha+\phi_{\rm DM}(\vec{r},t)
                \]
                \[\huge
                \kappa = \frac{{\rm d} f}{{\rm d} \alpha}\frac{\alpha_0}{f_0}
                \]
              </div>

</div>

# GPS.DM

<div style="text-align: left; float: left; width: 35%">
              <br>
              <h3>• 30 Cs, Rb atomic clocks</h3>
              <br>
              <h3>• Over 20 years of high-quality data</h3>
              <br>
              <h3>• Publicly available (JPL)</h3>
              <br>
              <h3>• 50,000 km Dark Matter observatory</h3>
              <br>
              <div class="fragment fade-in">
                <video width="75%" data-autoplay muted>
                  <source src="img/GPSDM.mp4"></video><br>
                <font size="3">[Video: Conner Dailey]</font>
              </div>
            </div>

<div style="text-align: center; float: right; width: 65%">
              <img div class="fragment fade-in" src="img/gps-data.png"
                width="45%">
              <br>
              <br>
              <br>
              <div class="fragment fade-in">
                <img src="img/GPSDM-results.png" width="98%">
                <b>BMR</b>, Blewitt, Dailey, Murphy, Pospelov, Rollings,
                Sherman,
                Williams, Derevianko, <br>
                <a
                  href="http://www.nature.com/articles/s41467-017-01440-4">
                  Nature Comms.
                  (2017)</a>

</div>

# Oscillating signals

<div style="text-align: left; float: left; width: 50%">
              <br>
              <h3>Case: oscillating field:</h3>
              \[{\large
              \phi_{\rm DM} = \phi_0 \cos({m_\phi t})}
              \]
              <br>
              \[\large
              \frac{\delta f}{f} = \kappa \phi_0 \cos({m_\phi t})
              \]
              <br>
              <br>
              <div class="fragment fade-in">
                <h2>Constraints from clock-clock comparisons</h2><br>
                <h3>• Require a <em>pair</em> of clocks</h3>
                <div class="fragment fade-in">
                <h3>• Different systems: probe different couplings</h3>
                <h4>   • Optical-Optical: $d_e$ ($\delta\alpha)$</h4>
                <h4>   • Optical-Microwave: $d_{m_e}-d_{m_p}$ ($\delta(m_e/m_p)$)</h4>
                </div>
                <br>
              </div>
              <div class="fragment fade-in">
                <h3>• Dimensionful couplings always cancel in these ratios</h3>
              </div>
            </div>

<div style="text-align: center; float: right; width: 50%">
              <br> <br>
              <div class="fragment fade-in" data-fragment-index="3">
                <h3>Typical: Compare two local clocks, different $\kappa$</h3>
                <img src="img/scalarphoton2.png" width="100%"><br>
                <p>[O'Hare - GitHub/AxionLimits]</p><br>
              </div>
            </div>

# Space-time separated sensors
      
                * Filzinger, Caddell, Jani, Steinel, Giani, Huntemann, BMR, [Phys. Rev. Lett. **134** 031001 (2025)](https://link.aps.org/doi/10.1103/PhysRevLett.134.031001)
      
                <br>
                <br>
                <br>
      
                <div style="text-align: left; float: left; width: 50%">
      
                  ### Typical local experiment:
      
                  <br>
      
                  $$\large
                    \frac{\delta(\nu_a/\nu_b)}{(\nu_a/\nu_b)} = \phi_0 \\, ( d_{a} - d_b) \\, \cos(\omega t)
                  $$
      
                  <br>

<div class="fragment fade-in">
                  
                  * Non-zero effect from two _different_ transitions only
                  * Certain couplings always cancel 
                  * e.g., Universal couplings

</div>

<br>
      
                  <img src="img/Local.png" width="75%"><br>
                </div>
                
                <div style="text-align: left; float: right; width: 50%" class="fragment fade-in">
      
                  ### Separated (identical) sensors:
      
                  <br>
      
                  <div>
                  \[\large
                    \frac{\delta(\nu_a/\nu_b)}{(\nu_a/\nu_b)} = \phi_0 \, d_{\rm eff} \left(\cos(\omega t) -\cos(\omega t-\delta)\right) \\
                      \delta = \omega\Delta t - k_z D 
                  \]
                  </div>
                  <br>
                  <div class="fragment fade-in">
                  
                  * Now: _field_ is different between clocks
                  * Spatial and temporal component (depends on experiment)
                  * Allows access to unconstrained couplings
  
                  </div>
      
                  <img src="img/Separated.png" width="80%"><br>
                </div>

# Space-time separated sensors: initial results

* Filzinger, Caddell, Jani, Steinel, Giani, Huntemann, BMR, [Phys. Rev. Lett. **134** 031001 (2025)](https://link.aps.org/doi/10.1103/PhysRevLett.134.031001)
        <br>

* Existing data: [Schioppo _et al._, Nat.Comms 13 (2022)](https://www.nature.com/articles/s41467-021-27884-3)
          * Comparison of ultra-stable cavity lasers
          * 750 km apart (2220 km Fibre cable: PTB to NPL via SYRTE)
          * Sensitive to _spatial_ phase shift

<br>
          
          * Also: Data from GPS satellites (Rb clocks)
          * Clocks less accurate, but setup sensitive to _temporal_ term

<br>

* First Constraints on $d_{m_e}$ alone at low-frequency<br>
          * Sensitivity to _spatial_ fluctions of DM, daily oscillation<br>
          
          <br>
          <div class="fragment fade-in" data-fragment-index="2">
          <img src="img/dailymod.png" width="90%"><br>
          </div>

</div>

# Super-massive black hole at the Galactic Centre

<div style="text-align: left; float: left; width: 45%">
            <h3>• with UCLA Galactic Centre Group</h3>
            <h4>   • Observations led by Tuan Do</h4>
            <h4>   • Andrea Ghez: 2020 Nobel
              prize for discovery of black hole</h4>
            <h4>   • Analysis led by Aurelien Hees</h4>
            <br>
            <h3>• Keck telescope in Hawaii</h3>
            <br>
            <h3>• Motion of ∼1000 stars tracked</h3>
            <h4>   • Precise spectroscopy for many stars</h4>
            <br>
            <br>
            <div class="fragment fade-in">
              <h3>• High gravitational potential</h3>
              <h3>• Possibly large concentration of dark
                matter </h3>
              <h3>• (also other "exotic" physics)</h3>
            </div>
          </div>

<div style="text-align: center; float: right; width: 55%">
            <img src="img/Galactic_centre_pillars.jpg" width="100%">
            <font size="4">[ESA / C. Carreau]</font>
            <br>
            <br>
            
            <br>
          </div>

# New physics in high gravity?
          <br>

<div class="fragment fade-in">
              <img src="img/equation.png" width="80%">
              <br>
              <div class="fragment" style="font-size: 1.15em;">
              $$
              \Delta \alpha / \alpha \lesssim 10^{-5}
              $$
              </div>
            </div>
          </div>

<div style="text-align: center; float: right; width: 40%">
            <img src="img/keck-ao.jpg" width="100%">
            <font size="3">[Keck Observatory]</font>
          </div>

<div style="clear: both;"></div>
          <div style="float: left; text-align: left; width: 60%">
            <br>
            <div class="fragment fade-in">
              <h4>• Constrain models for physics beyond GR</h4>
              <h4>• Less sensitive than other searches</h4>
              <h4>• Position-dependent: First measurement near a black hole</h4>
              <h4>• Future dedicated observations: large improvements expected</h4>
            </div>
            <br>
              <font size="6">Hees, Do, <b>BMR</b>, Ghez <i>et al.</i>,
              <a href="https://link.aps.org/doi/10.1103/PhysRevLett.124.081101">Phys. Rev. Lett. <b>124</b>, 081101 (2020)</a></font>
              <br>
              <font size="6">Benergut <i>et al.</i>,
              <a href="10.1103/PhysRevLett.111.010801">Phys. Rev. Lett. <b>111</b> (2013)</a>; Ashby Nat. Phys <b>14</b> (2018)</font>
          </div>
          <div class="fragment" style="float: right; text-align: center; width: 40%">
            <img src="img/Compactness.png" width="75%">
          </div>

# Thorium-229: Nuclear Clock?

<div style="text-align: center; float: left; width: 50%">
              <br>
              <img src="img/Th_PThirolf_LMU.webp" width="85%"><br>
              <font size="4">[P. Thirolf, LMU]</font>
              <br><br>
              <div style="text-align: left; padding-left: 5%">

<div class="fragment" data-fragment-index="2">
              <h2>Timekeeping</h2>
              <br>
              <h3>• No perturbation from external fields</h3>
              <h3>   • Nucleus shielded from environment</h3>
              </div>
              </div>
            </div>

<div style="text-align: left; float: right; width: 48%">
              <br>
              <div class="fragment" data-fragment-index="1">
              <h3>• Unique low-energy nuclear transition (~8 eV)</h3>
              <br>
              <h3>• First laser excitation of a nuclear transition (2024)</h3>
              <h4>   • E. Peik (PTB), T. Schumm (Vienna) <a href="https://link.aps.org/doi/10.1103/PhysRevLett.132.182501">Phys. Rev. Lett. <b>132</b> 182501</a></h4>
              <br>
              <div class="fragment" data-fragment-index="3">
              <h3>• Nuclear energy: balance of <font color="orange">QCD</font> vs. Coulomb</h3>
              <h3>   • Nuclear scale $\sim$ MeV $\rightarrow$ eV</h3>
              <h3>   • Fine tuning: high sensitivity</h3>
              </div>
              </div>
              <br><br>

<div class="fragment" data-fragment-index="4">
              <h2>Fundamental Physics</h2>
              <br>
              <h3>• Probe variation of fundamental constants</h3>
              <h4>    • Flambaum, <a href="https://link.aps.org/doi/10.1103/PhysRevLett.97.092502">Phys. Rev. Lett. <b>97</b>, 092502 (2006)</a>
              <h3>• Highly sensitive <font color="orange">dark matter</font> detector</h3>
              <h3>• Test of local Lorentz invariance and CPT</h3>
              </div>

</div>

# Conclusion

### Dark Matter:

* Extremely strong evidence for existance: no direct detection
          * 90 orders-of-magnitude of mass to search: 
            * requires broad, low-cost experiments

### Atomic Clocks & Ultralight DM

* Exceptionally precise quantum sensors
          * Scalar coupling DM-SM: shift in atomic levels
          * Space-time separated clocks: universal couplings

### UNSW Physics Colloquium, 2026

### Benjamin M. Roberts
  
          _University of Queensland, Australia_

<br>

* Slides [broberts.io/slides](https://broberts.io/slides/)

<!-- <br><br>

* Caddell, Flambaum, BMR, [Phys. Rev. D **108**, 083030 (2023)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.108.083030)
            <br>
            * Savalle _et al._, [Phys. Rev. Lett. **126**, 051301 
            (2021)](https://link.aps.org/doi/10.1103/PhysRevLett.126.051301)
            <br>
            * BMR _et al._, [Nature Comms. **8**, 1195 
            (2017)](http://www.nature.com/articles/s41467-017-01440-4)
            <br>
            * BMR _et al._, [N. J. Phys. **22**, 093010 
            (2020)](https://iopscience.iop.org/article/10.1088/1367-2630/abaace)
            <br>
            * Hees, Do, BMR, Ghez _et al._, [Phys. Rev. Lett. **124** 081101 
            (2020)](https://link.aps.org/doi/10.1103/PhysRevLett.124.081101)
            <br>
            * Filzinger, Caddell _et al._,  [Phys. Rev. Lett. **134** 031001 (2025)](https://link.aps.org/doi/10.1103/PhysRevLett.134.031001)
            <br>
            * Caddell, BMR (coming soon)

# Oscillating signals

<div style="text-align: left; float: left; width: 50%">
              <br>
              <h3>Case: oscillating field:</h3>
              \[\huge
              \phi_{\rm DM} = \phi_0 \cos({m_\phi t})
              \]
              <br>
              \[\huge
              \frac{\delta f}{f} = \kappa \phi_0 \cos({m_\phi t})
              \]
              <br>
              <br>
              <div class="fragment fade-in">
                <h3>• Constraints from clock-clock comparisons</h3>
                <br>
                <h3>• ~ $10^{-2}$ Hz (optical clocks)</h3>
                <h4>
                    • e.g., Wcislo et al
                  <a
                    href="https://advances.sciencemag.org/content/4/12/eaau4869">
                    Science Advances <b>4</b>, 4869 (2018)</a>
                </h4>
                <br>
                <h3>• ~ $10^{-6}-10^{-2}$ Hz (microwave clocks)</h3>
                <h3>• $\implies m = 10^{-21}-10^{-18}$ eV</h3>
                <h4>
                    • e.g., Hees et al
                  <a
                    href="http://link.aps.org/doi/10.1103/PhysRevLett.117.061301">
                    Phys. Rev. Lett. <b>117</b>, 061301 (2016)</a>
                </h4>
              </div>
            </div>

<div style="text-align: left; float: right; width: 50%">
              <br> <br>
              <div class="fragment fade-in">
                <h3>The DAMNED experiment:</h3>
                <img src="img/damned.png" width="90%"><br>
                <p>(DArk Matter search using Non-Equal Delay lines)</p><br>
                <div class="fragment fade-in">
                  <h3>• Modifies length + refractive index</h3>
                  <h3>• 10-100 kHz range ($m=10^{-10}$ eV)</h3>
                  <h3>• Led by Etienne Savalle (SYRTE Paris)</h3>
                  Savalle, Hees, Frank, Cantin, Pottie, <b>BMR</b>, Cros,
                  McAllister,
                  Wolf, <br>
                  <a
                    href="https://link.aps.org/doi/10.1103/PhysRevLett.126.051301">
                    Phys. Rev. Lett. <b>126</b>, 051301
                    (2021)</a>
                </div>
              </div>
            </div>

# Variation of fundamental constants

<br>

<div>
          \[\large
            \omega^{A} = \underbrace{F_A(\alpha)}_{\text{Transition-specific}} 
            \times  \underbrace{m_e c^2\alpha^2}_{\text{Units}} 
          \]
          </div>

<div class="fragment" style="text-align: center; float: left; width: 50%">
            <br><br>
            <img src="img/Bohr.png" width="30%"><br>
            <font size="3">[JabberWok/Wikipedia]</font>
            <br><br><br>
            \[
            \frac{\Delta\omega}{\omega} = \kappa\frac{\Delta\alpha}{\alpha}
            \qquad\qquad\qquad
            \left(
                {\color{red}\kappa \equiv \frac{\partial \omega}{\partial \alpha}\frac{\alpha_0}{\omega}}
            \right)
            \]

<br>

* In general, $\kappa$ calculated from atomic theory
            
            <br>

* Dzuba, Flambaum, Webb [Phys Rev Lett 82, 888 (1999)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.82.888)
          </div>

<div class="fragment" style="text-align: center; float: right; width: 50%">
            <br> <br>
            <img src="img/AlphaSensitivity.png" width="90%"><br>
            <font size="3">[M. Murphy 2012]</font>
            <br>
            <br>
             • Difference transitions: different $\kappa$ sensitivity
          </div>

# Optical atomic clocks

* Requires ultra-stable laser ("see" narrow line)
            * Frequency comb ("listen" to high frequency)

</div>

* Accurate to parts in $10^{18}$
            * better than 1 s over lifetime of the universe!
          </div>

# Bullet Cluster

<div class="r-stack">
                <div class="fragment fade-out" data-fragment-index="0">
                  <img  src="img/Bullet_cluster.jpg" width="100%"><br>
                  <font size="4">Chandra X-ray Observatory [NASA]</font>
                </div>

<div class="fragment fade-in" data-fragment-index="0">
                  <img src="img/1e0657_scale.jpg"  width="90%">
                  <font size="4">NASA/CXC/M. Weiss - Chandra X-Ray Observatory</font>
                </div>
              </div>

</div>

<div style="text-align: left; float: right; width: 35%">
              <br>
              <h3>• Colliding galaxy clusters ~ 4 Gyr</h3>
              <br>
              <h3>• X-ray emission image:<br> distribution of gas (regular
                matter)</h3>
              <br>
              <h3 class="fragment fade-in" data-fragment-index="0">• Also -
                Gravitational lensing:<br>
                distribution of <i>mass</i></h3>
              <br>
              <br>
              <div class="fragment fade-in" data-fragment-index="2">
                <h3>Composite image: X-ray + lensing</h3>
                <br>
                <h3><font color="pink">• Pink = gas (X-ray)</font></h3>
                <h3><font color="sky-blue">• Blue = mass (gravitational
                    lensing)</font></h3>
                <br>
                <h3 class="fragment fade-in" data-fragment-index="3">
                  • Frictionless dark matter: <br>
                  passes straight through
                </h3>
              </div>
            </div>

# Cosmic Microwave Background: ΛCDM model
    
                <div style="text-align: center; float: left; width: 70%">
                  <img src="img/CMB-PowerSpectrum-Planck.png" width="100%"><br>
                  <font size="3">[ESA/Planck Collaboration (2013)]</font>
                </div>
    
                <div style="text-align: center; float: right; width: 30%">

<br>
                  <br>
                  
                  $$\large
                  \Omega_{\rm baryon} = 0.0486(10)
                  $$

$$\large\Omega_{\rm dark~matter} = 0.2589(57)
                  $$

$$\large\Omega_{\rm dark~energy} = 0.6911(1)
                  $$

<h3 class="fragment">• ΛCDM model works extremely well</h3>
                  <br>
                  <br>
                  <h3 class="fragment">• (Just need to work out the Λ and CDM parts)</h3>

</div>

# WIMP miracle

<div style="text-align: left; float: left; width: 30%">
              <br><br>
              <img src="img/Feynman.png" width="95%">
              <br><br>
              <h3> • Early universe: hot and dense</h3>
              <h3> DM + SM in thermal equilibrium</h3>
              <br>
              <h3 class="fragment fade-in" data-fragment-index="0">
                • Universe cools: <br>
                Boltzmann suppression
              </h3>
              <br>
              <h3 class="fragment fade-in" data-fragment-index="1">
                • Universe expands: <br>
                Shuts off annihilation
              </h3>
              
            </div>

<div style="text-align: center; float: right; width: 65%">
              <div class="r-stack">
                <img class="fragment fade-in" data-fragment-index="0"
                  src="img/YDM/Yeq.png" width="90%">
                <img class="fragment fade-in" data-fragment-index="1"
                  src="img/YDM/YDM1.png" width="90%">
                <img class="fragment fade-in" data-fragment-index="2"
                  src="img/YDM/YDM2.png" width="90%">
              </div>
              <h3 class="fragment fade-in" data-fragment-index="3">
                • Weak interaction ($G_F \simeq 10^{-5}\,{\rm
                GeV}^{-2}$): new physics scale<br>
                • $m_{\rm weak} \simeq 100\,{\rm GeV}$ <br>
              </h3>
              <h3 class="fragment fade-in" data-fragment-index="4">
                • $\langle\sigma v\rangle_{\rm weak} \simeq
                3\times10^{-26}\,{\rm
                cm}^3/{\rm s}$ !!!
              </h3>
            </div>