BASE Collaboration

# Image-current mediated sympathetic laser cooling of a single proton in a
Penning trap down to 170 mK axial temperature

C. Will  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117
Heidelberg, Germany  M. Wiesinger  Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, D-69117 Heidelberg, Germany  P. Micke  Present address:
Helmholtz Institute Jena, GSI Helmholtz Centre for Heavy Ion Research,
Planckstraße 1, 64291 Darmstadt, Germany Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, D-69117 Heidelberg, Germany  CERN, 1211 Geneva, Switzerland
H. Yildiz  Institut für Physik, Johannes Gutenberg-Universität, Staudingerweg
7, D-55128 Mainz, Germany  T. Driscoll  Present address: The University of
Oregon, Eugene, Oregon 97403, USA Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, D-69117 Heidelberg, Germany  Department of Physics, The
University of Texas at Austin, Austin, Texas 78712, USA  S. Kommu  Institut
für Physik, Johannes Gutenberg-Universität, Staudingerweg 7, D-55128 Mainz,
Germany  F. Abbass  Institut für Physik, Johannes Gutenberg-Universität,
Staudingerweg 7, D-55128 Mainz, Germany  B. P. Arndt  Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany  RIKEN, Fundamental
Symmetries Laboratory, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan  GSI
Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  B.
B. Bauer  Institut für Physik, Johannes Gutenberg-Universität, Staudingerweg
7, D-55128 Mainz, Germany  S. Erlewein  CERN, 1211 Geneva, Switzerland  RIKEN,
Fundamental Symmetries Laboratory, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
M. Fleck  RIKEN, Fundamental Symmetries Laboratory, 2-1 Hirosawa, Wako,
Saitama 351-0198, Japan  Graduate School of Arts and Sciences, University of
Tokyo, Tokyo 153-8902, Japan  J. I. Jäger  Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, D-69117 Heidelberg, Germany  CERN, 1211 Geneva, Switzerland
RIKEN, Fundamental Symmetries Laboratory, 2-1 Hirosawa, Wako, Saitama
351-0198, Japan  B. M. Latacz  CERN, 1211 Geneva, Switzerland  RIKEN,
Fundamental Symmetries Laboratory, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
A. Mooser  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117
Heidelberg, Germany  D. Schweitzer  Institut für Physik, Johannes Gutenberg-
Universität, Staudingerweg 7, D-55128 Mainz, Germany  G. Umbrazunas  RIKEN,
Fundamental Symmetries Laboratory, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
Eidgenössische Technische Hochschule Zürich, John-von-Neumann-Weg 9, 8093
Zürich, Switzerland  E. Wursten  RIKEN, Fundamental Symmetries Laboratory, 2-1
Hirosawa, Wako, Saitama 351-0198, Japan  K. Blaum  Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany  J. A. Devlin
Centre for Cold Matter, Blackett Laboratory, Imperial College London, Prince
Consort Road, London SW7 2AZ, UK  C. Ospelkaus  Institut für Quantenoptik,
Leibniz Universität Hannover, D-30167 Hannover, Germany  Physikalisch-
Technische Bundesanstalt, D-38116 Braunschweig, Germany  W. Quint  GSI
Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  A.
Soter  Eidgenössische Technische Hochschule Zürich, John-von-Neumann-Weg 9,
8093 Zürich, Switzerland  J. Walz  Institut für Physik, Johannes Gutenberg-
Universität, Staudingerweg 7, D-55128 Mainz, Germany  Helmholtz-Institut
Mainz, D-55099 Mainz, Germany  C. Smorra  Institut für Physik, Johannes
Gutenberg-Universität, Staudingerweg 7, D-55128 Mainz, Germany  S. Ulmer
RIKEN, Fundamental Symmetries Laboratory, 2-1 Hirosawa, Wako, Saitama
351-0198, Japan  Heinrich Heine University, Düsseldorf, Universitätsstrasse 1,
D-40225 Düsseldorf, Germany

###### Abstract

We demonstrate a new temperature record for image-current mediated sympathetic
cooling of a single proton in a cryogenic Penning trap by laser-cooled 9Be+.
An axial mode temperature of 170 mK is reached, which is a 15-fold improvement
compared to the previous best value. Our cooling technique is applicable to
any charged particle, so that the measurements presented here constitute a
milestone towards the next generation of high-precision Penning-trap
measurements with exotic particles.

††preprint: APS/123-QED

Laser cooling of atoms and ions is a widely employed method in the field of
atomic physics [1, 2]. However, only few species offer a suitable optical
transition for laser cooling. For most other particles, sympathetic cooling
techniques have to be employed. In the established sympathetic cooling
schemes, the coupling is realized through direct ion-ion Coulomb interaction,
where the charged particles are either trapped in the same potential well [3,
4, 5] or in two separate ones whose separation distance is a few hundred
micrometers only [6, 7].
Our group has recently demonstrated the sympathetic laser cooling of a single
proton mediated by image currents [8]. Here, the two ion species are placed in
two independent Penning traps that are separated by a distance of 5.5 cm but
are connected to the same parallel RLC circuit [9]. Consequently, this
technique is especially suited for exotic species such as highly charged ions
or charged antimatter [10, 11]. Furthermore, in comparison to resistive
cooling with a cyclotron resonator that only covers a single fixed particle
charge-to-mass ratio $q/m$ [12, 13], the sympathetic cooling technique is
widely tunable and applicable to all relevant $q/m$ simultaneously. The lowest
axial temperature achieved in Ref. [8] was $(2.6\pm 2.5)$ K, measured via the
temperature-induced axial frequency shift of about 100 mHz/K (axial) of the
proton in the presence of an anharmonic trapping potential. The temperature
resolution of this method is fundamentally limited to about 500 mK by the
typical axial frequency stability of about 50 mHz. Furthermore, this
temperature measurement method required that the beryllium ion cloud is laser
cooled comparably strongly, which is detrimental to the achievable proton
temperature [14].
In this Letter, we demonstrate axial temperatures of a single proton in a
Penning trap down to about 170 mK, which constitutes a factor of 15
improvement compared to the previous record. To achieve this, we utilize a
more precise temperature measurement method based on the axial frequency shift
in a quadratic magnetic field inhomogeneity. A new temperature measurement
trap (TMT) has been implemented with an axial frequency shift of 470 Hz/K
(axial) [15].

|  (a) |
---|---
(b) |
---|---
Figure 1: (a) Schematic of the experimental setup. A cloud of beryllium ions
is trapped in the loading trap (LT) and a single proton in the proton trap
(PT). Both traps are connected to a common resonator. The proton is
transported from the PT into the temperature measurement trap (TMT) for
measuring its modified cyclotron energy via a quadratic magnetic field
inhomogeneity. (b) Typical Boltzmann distribution of a temperature
measurement. The red line corresponds to the Boltzmann distribution with a
temperature as determined by the maximum-likelihood method.

The TMT offers a temperature resolution in the mK range as well as a 17-fold
faster temperature readout time compared to a preexisting analysis trap (AT)
with a larger magnetic field inhomogeneity for spin state readout [16]. In
addition, the understanding of the coupling process has been improved through
numerical simulations, which has led to optimized experimental parameters for
the cooling process [17, 14].
Our sympathetic cooling experiments are performed in a cryogenic multi-
Penning-trap system [15, 14] that stores ions by means of a superposition of a
homogeneous magnetic field of $B_{0}\approx 1.9$ T and a quadrupolar electric
potential, shown schematically in Fig. 1(a). The resulting harmonic motion of
a single particle consists of the magnetron and modified cyclotron motion in
the radial plane and an axial motion orthogonal to it with frequencies
$\nu_{-}$, $\nu_{+}$, and $\nu_{\text{z}}$, respectively [18]. The axial
motion of a particle is detected by tuning its axial frequency to the
resonance frequency of a parallel RLC circuit (also called resonator)
consisting of a superconducting coil with inductance $L$, a stray capacitance
$C$, and an effective parallel resistance $R$ [19]. The resonance is generated
by the thermal Johnson-Nyquist noise at a frequency
$\nu_{0}=1/(2\pi\sqrt{LC})$ and has a high quality factor
$Q=R/(2\pi\nu_{0}L)$. For example,
$\nu_{0}^{\text{PTLT}}=$345\,250\text{\,}\mathrm{H}\mathrm{z}$$ and
$Q\approx$10\,000\text{\,}$$ for the resonator that is connected to both the
proton trap (PT) and loading trap (LT) in our case [15]. The voltage signal of
the resonator is amplified by a cryogenic low-noise amplifier at 4 K [20]
followed by another amplification stage at room temperature. Recording the
fast Fourier transform (FFT) of the resonator voltage signal yields the
characteristic spectrum featuring a dip at the ion’s axial frequency [19],
shown schematically in the inset of Fig. 1(a). The coupling of the particles
to the resonator is given by their respective dip width
$\gamma_{\text{z}}=\frac{1}{2\pi}\frac{R}{m}\frac{q^{2}}{D^{2}}N$, where $N$
is the number of ions of the species and $D$ an effective trap size [19]. The
proton dip width in the PT is $2.2$ Hz per ion and the beryllium dip width in
the LT is $0.10$ Hz per ion. The cooling measurements are performed with a
single proton trapped in the PT and a cloud of beryllium ions stored in the
LT. Both traps are connected to the same resonator, which mediates the energy
exchange via image currents between the particles [8]. The proton is
sympathetically cooled by tuning the axial frequency of both species to the
resonator frequency and laser-cooling the beryllium ions via the 313 nm
2S${}_{1/2}\rightarrow^{2}$P3/2 transition with a natural linewidth of about
$\Gamma=2\pi\times$20\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ [21].
In order to measure the temperature of the proton, we employ a two-trap
measurement scheme. Since the resonator constitutes a thermal reservoir, the
axial mode continuously samples a Boltzmann distribution with a correlation
time of $1/\gamma_{\text{z}}$ if $\nu_{\text{z}}\approx\nu_{0}$. The
corresponding time average is, according to the ergodic theorem, equivalent to
the ensemble average which we use as the temperature definition of a single
particle. In contrast, the modified cyclotron mode remains at constant energy.
However, it can be coupled to the axial mode by irradiating a quadrupolar
sideband drive at frequency $\nu_{\text{RF}}=\nu_{+}-\nu_{\text{z}}$ [22]. In
this way, an axial energy is imprinted on the modified cyclotron mode. The
corresponding temperatures obey the relation
$T_{+}=T_{\text{z}}\nu_{+}/\nu_{\text{z}}$, where $T_{+}$ and $T_{\text{z}}$
are the temperatures of the modified cyclotron mode and the axial mode,
respectively. Then, the proton is transported from the PT into a dedicated
temperature measurement trap (TMT), where its modified cyclotron energy
$E_{+}$ is measured. The TMT features a ferromagnetic ring electrode made from
nickel, which introduces a large quadratic magnetic field inhomogeneity, i.e.
$B_{\text{z}}^{\text{TMT}}(z)=B_{0}^{\text{TMT}}+B_{2}^{\text{TMT}}z^{2}$. The
coefficient $B_{2}^{\text{TMT}}$ has been measured to be
$B_{2}^{\text{TMT}}=$27.8(7)\text{\,}\mathrm{k}\mathrm{T}\mathrm{/}\mathrm{m}^{2}$$
[14], which is consistent with the design value [15]. To measure $E_{+}$, we
utilize the fact that the $B_{2}$-coefficient causes an axial frequency shift
in the TMT which is proportional to $E_{+}$ [18, 23], in our case
$\delta\nu_{\text{z}}^{\text{TMT}}/\delta E_{+}=$ 5.8 Hz/($\text{K}\cdot
k_{B}$) for the proton. Thus, the axial temperature of the single proton is
obtained by repeatedly measuring modified cyclotron energies independently
drawn from a Boltzmann distribution. An example of such a Boltzmann
distribution is given in Fig. 1(b).
In order to convert the distribution of axial frequency measurements in the
TMT to an axial temperature in the PT, we employ a maximum-likelihood
approach. The maximum-likelihood estimator for the axial temperature in the
PT, $T_{\text{z}}^{\text{PT}}$, is given by the mean axial frequency shift,

$\displaystyle
T_{\text{z}}^{\text{PT}}=\frac{\nu_{\text{z}}^{\text{PT}}}{\nu_{+}^{\text{PT}}}\frac{B_{0}^{\text{PT}}}{B_{2}^{\text{TMT}}}\frac{4\pi^{2}}{k_{\text{B}}}m\,\nu_{\text{z,0}}^{\text{TMT}}\braket{\nu_{\text{z}}^{\text{i,TMT}}-\nu_{\text{z,0}}^{\text{TMT}}}.$
(1)

Here,
$\nu_{\text{z,0}}^{\text{TMT}}\approx$550\,875\text{\,}\mathrm{H}\mathrm{z}$$
is the unshifted axial frequency in the TMT at $E_{+}=0$ and
$\nu_{\text{z}}^{\text{i,TMT}}$ are the individual axial frequency
measurements. $B_{0}^{\text{PT}}=1.899$ T is the magnetic field in the PT and
$\nu_{+}^{\text{PT}}\approx$28.9\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ and
$\nu_{\text{z}}^{\text{PT}}=$345\,250\text{\,}\mathrm{H}\mathrm{z}$$ are the
modified cyclotron and axial frequencies of the proton in the PT,
respectively, and $k_{\text{B}}$ is Boltzmann’s constant. This formula
incorporates not only the axial frequency shift due to non-zero $E_{+}$, but
also the temperature relation due to sideband coupling and the relative
modified cyclotron energy change during transport into a different magnetic
field, in our case by the magnetic field ratio of the traps,
$B_{0}^{\text{PT}}/B_{0}^{\text{TMT}}$.
In support of the experimental effort, we performed first-principles
simulations of the experimental setting that is shown in Fig. 1(a) [17, 14].
We found that the on-resonance cooling scheme employed in this work, where the
proton and beryllium ion cloud are tuned to the resonator frequency, can be
understood by the formation of a symmetric and antisymmetric normal mode of
the axial motion of the coupled proton-beryllium system. The antisymmetric
mode decouples from the resonator, so that the laser cools it close to the
Doppler limit. In contrast, the symmetric mode couples and thermalizes to the
resonator and the relative proton and 9Be+ component are given by their
respective dip widths. Since image-current coupling relies on the motion of
the particles, for an optimal proton temperature the damping rate of the
cooling laser $\gamma_{\text{L}}$ must be sufficiently weak in order to not
decouple the beryllium ions from the resonator, i.e.
$\gamma_{\text{L}}\ll\gamma_{\text{z,Be}}$, where $\gamma_{\text{z,Be}}$
denotes the dip width of the beryllium cloud. Then, the axial temperature of a
proton with dip width $\gamma_{\text{z,p}}$ is given by [17, 14]

$\displaystyle
T_{\text{z,p}}=\frac{1}{1+\frac{\gamma_{\text{z,Be}}}{\gamma_{\text{z,p}}}}T_{\text{res}},$
(2)

where $T_{\text{res}}$ is the effective resonator temperature. Thus, besides
$T_{\text{res}}$, the ratio of the proton and beryllium dip widths determines
the final proton temperature. In order to compare theory with experiment,
$T_{\text{res}}$ is measured as the first preparatory step to
$T_{\text{res}}=(8.6\pm 0.8)$ K. The cryogenic amplifier is turned off for
this measurement as well as for the sympathetic cooling measurements, since
otherwise the amplifier’s input noise gives rise to a slightly higher
effective axial temperature.
It is crucial to minimize the axial frequency difference of the two species
for efficient sympathetic cooling. Hence, in the following their individual
frequency stabilites are examined. The axial frequency stability of the single
proton in the PT is $\sigma(\nu_{\text{z,p}})\approx 40$ mHz for 60 s
averaging time, which is negligibly small for the sympathetic cooling. In
contrast, the axial frequency stability of the beryllium cloud is adversely
affected by the radial cloud expansion due to the Coulomb interaction [24,
25]. In order to prevent an uncontrolled radial expansion, the radial modes of
a large beryllium cloud must be regularly cooled. However, in our setup the
laser is applied nearly parallel to the trap axis so that the cooling of
radial modes [26] is not efficient. Since large laser powers are not feasible
because the cooling scheme requires low photon scattering rates, a magnetron
sideband drive at $\nu_{\text{RF}}=\nu_{\text{z,Be}}+\nu_{\text{-,Be}}$ [22]
is applied in addition to the cooling laser. We observe axial frequency drifts
of the beryllium ion clouds after turning off the magnetron sideband drive, as
shown exemplarily in Fig. 2 for two ion clouds of 840 ions and 2000 ions [15,
14]. These drifts are related to the harmonicity of the electrical trap
potential, which is adjustable via the tuning ratio (TR) of the trap as
defined in Ref. [18]. We attribute these frequency drifts to the change in the
aspect ratio of the beryllium ion cloud while the cloud expands due to Coulomb
repulsion in an anharmonic trapping potential [25].

---
Figure 2: Axial frequency drift of a cloud of beryllium ions in the LT after
applying the magnetron sideband drive at
$\nu_{\text{RF}}=\nu_{\text{z,Be}}+\nu_{\text{-,Be}}$. The drift is related to
the tuning ratio (TR) of the trap and becomes stronger the larger the cloud
is. The TR design value is 0.442 [15].

The accumulated frequency shift can be reset by applying the magnetron
sideband drive again. Figure 2(a) demonstrates that a TR optimization allows
to control the frequency drifts of clouds of $\lesssim$ 1000 ions to $<1$ Hz
on the relevant time scale, which is sufficiently stable for the sympathetic
cooling process. In contrast, for larger clouds the drifts are not only
stronger, they also exhibit a quadratic component, as shown in Fig. 2(b). For
large clouds above 1000 ions, this requires us to introduce small frequency
offsets and apply the coupling procedure during the times when the change in
detuning between the proton and beryllium ions is minimal.
With a precisely adjusted TR, we were able to sufficiently stabilize beryllium
ion clouds with dip widths of up to 120 Hz or 1200 ions. Assuming the proton
is initially trapped in the PT, one sympathetic cooling cycle consists of the
following steps: First, in order to start the cooling process, the axial
frequencies of the proton and the beryllium ion cloud are matched in between
two iterations of applying the magnetron sideband drive to the beryllium ion
cloud. The cooling laser is kept continuously on with a weak damping rate
during the whole cycle. During the sympathetic cooling, the modified cyclotron
sideband frequency of the proton is continuously applied at
$\nu_{\text{RF}}=\nu_{\text{+,p}}-\nu_{\text{z,p}}$ with about 1 Hz Rabi
frequency. The total cooling time is set to 90 s. The cooling time constant
has not been explicitly measured, however, it was verified that after 90 s the
resulting temperatures have converged and two subsequent values of $E_{+}$ are
uncorrelated. Afterwards, the proton is transported into the TMT to measure
$E_{+}$. Finally, the proton is transported back into the PT and the next
cooling cycle is carried out. A single temperature measurement consists of
$>20$ individual cooling cycles. A more detailed description of the individual
steps can be found in Ref. [14].

---
Figure 3: Axial temperature measurements of a sympathetically cooled proton.
Three differently sized beryllium ion clouds were employed. The red line is
the prediction by Eq. (2) and the shaded red area the associated uncertainty.
In (d) the individual measurements for each cloud size are combined and
plotted as a function of the beryllium ion cloud dip width
$\gamma_{\text{z,Be}}$.

This cooling process has been conducted for three differently sized beryllium
ion clouds, namely with dip widths of 48 Hz, 84 Hz, and 120 Hz, corresponding
to 480, 840, and 1200 ions. For the 48 Hz cloud, the cooling laser was set to
84 MHz red detuning and its power was adjusted to
5–$15\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{W}$, which corresponds to
1.5–4.5% of the saturation power of the cooling transition. The saturation
power of $P_{\text{sat}}=$340\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{W}$$
was measured by in-trap detection of fluorescence photons [27, 15]. For the 84
Hz and 120 Hz clouds, the laser power and red detuning were increased to
$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{W}$ and 200 MHz, respectively.
In contrast to our previous work [8], both parameter sets fulfill the
condition that the beryllium ion cloud is damped only weakly and that no loss
of the SNR of the beryllium dip occurs. Several independent proton temperature
measurements have been conducted for each beryllium cloud with results shown
in Fig. 3, where the uncertainty of each temperature measurement is dominated
by the statistical uncertainty. The horizontal red line is the temperature
predicted by Eq. (2) and the shaded red area is the corresponding uncertainty,
which is dominated by the statistical uncertainty of the measurement of
$T_{\text{res}}$. In Fig. 3(d) the individual measurements are combined and
plotted as a function of the dip width and thus particle number of the
beryllium ion cloud. In general, we observe excellent agreement between the
theoretical prediction and the experimental data. The lowest reproducibly
measured temperature of about 170 mK constitutes a 15-fold improvement
compared to the previous record measurement [8]. We emphasize that not only
the temperatures agree, but also the method to reach them: The prediction by
the simulations that a weak laser damping rate is required [17] has been
confirmed as well.
Moreover, our results demonstrate the capability of the two-trap temperature
measurement technique. By separating the temperature determination from the
cooling process a broad range of temperatures between 5 mK and 10 K can be
measured and resolved. In this regard we also measured a negligible heating
rate of $\lesssim 2$ mK per cycle, where we repeated the cooling sequence but
detuned the modified cyclotron sideband frequency for the proton by 10 kHz
[14].
Further, we study the effect of a relative axial frequency detuning between
the proton and the beryllium ion cloud. To this end, the 120 Hz cloud was
tuned to slightly different axial frequencies than the proton. All other steps
remain the same as before. The resulting proton temperatures are shown in Fig.
4. Notably, these measurements were also recorded with the cryogenic amplifier
turned off, so that the relative frequency detuning is estimated based on the
initial center frequency as well as the TR-related frequency drift. This gives
rise to a small offset of about 2 Hz. As a result, the detunings should only
be considered a coarse estimate. Nevertheless, the existence of a cooling
resonance is evident and the minimum corresponds to the prediction by Eq. (2).

Figure 4: Temperature of the sympathetically cooled proton as a function of
relative axial frequency detuning between the proton and beryllium ions. The
data was recorded with the 120 Hz cloud.

To achieve lower proton temperatures, an alternative to increasing the number
of beryllium ions is to reduce the resonator temperature. Thus, we have also
conducted temperature measurements where the effective resonator temperature
was reduced to $T_{\text{res}}^{\text{fb}}=(2.6\pm 0.3)$ K by the application
of negative feedback [28]. However, here the proton temperature was limited to
$\geq 300$ mK independent of the number of beryllium ions [14], which is a
strong indication for a feedback-induced noise limitation.
In summary, we have demonstrated image-current mediated sympathetic cooling of
a single proton to axial temperatures down to about 170 mK, an improvement by
a factor of 15 compared to the previous record [8]. As such, this work
constitutes a crucial milestone towards the next generation of high-precision
Penning trap measurements with particles that require sympathetic cooling with
separate trapping regions.
Several routes towards lower proton temperatures are conceivable: An optimized
beryllium ion trap for which the dip width per Be+ is maximized and the
anharmonicity-related frequency drifts are minimized would directly enable
lower proton temperatures [29, 15, 14]. Alternatively, with an independent
cooling laser in radial direction it would be possible to control and
stabilize significantly larger beryllium ion clouds via laser cooling only.
Then, the magnetron sideband coupling with its associated frequency drifts
would become obsolete. Another option would be to confine large ion clouds
radially with a rotating wall potential [30], which could enable the use of
significantly larger beryllium ion clouds as well. Besides, the simulation
studies [17] as well as independent work in Ref. [31] predict that a cooling
scheme with several kHz particle-resonator detuning and pulsed laser cooling
can achieve 10 mK axial particle temperatures with about 100 beryllium ions
only. The excellent agreement between experiment and simulation in this work
further corroborates the fundamental feasibility of these cooling methods.
Regardless which cooling scheme turns out to be the most suitable one for
further temperature reduction, the fact that all of them rely on image-current
coupling makes them in principle applicable to any trapped charged particle
and experimental systems beyond Penning traps [32]. Consequently, these
cooling methods are of special interest for charge-, parity-, and time (CPT)
symmetry tests with protons and antiprotons [10], magnetic moment measurements
of light nuclei [33], as well as high-precision mass measurements [34] and
tests of quantum electrodynamics with highly charged ions [31, 35, 36] in
Penning traps. In particular, once even lower temperatures of about 10 mK
(axial) can be reached [37], the sympathetic cooling will significantly boost
the sampling rate and spin state detection fidelity [38, 39] of future
$g$-factor measurements on protons [40], antiprotons [13] and other nuclear
moments, as well as reduce the dominant systematic uncertainties in mass
measurements with the highest precision [41].

##  I Acknowledgements

This study comprises parts of the PhD thesis work of C. Will and M. Wiesinger.
We acknowledge financial support from the Max-Planck-Society, the RIKEN Chief
Scientist Program, the RIKEN Pioneering Project Funding, the RIKEN JRA
Program, the Royal Society, the Helmholtz-Gemeinschaft, the DFG through SFB
1227 ‘DQ-mat’, the cluster of excellence QuantumFrontiers, the CERN Gentner
programme, the Max-Planck IMPRS-PTFS, the European ResearchCouncil (ERC) under
the European Union’s Horizon 2020 research and innovation programme (Grant
agreement Nos. 832848 – FunI, 721559 – AVA, 852818 – STEP), and the Max-
Planck–RIKEN–PTB Center for Time, Constants and Fundamental Symmetries.

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# Two-Sheeted Universe, Analyticity and the Arrow of Time

Latham Boyle1 and Neil Turok1,2 1Perimeter Institute for Theoretical Physics,
Waterloo, Ontario, Canada, N2L 2Y5
2Higgs Centre for Theoretical Physics, University of Edinburgh, Edinburgh,
Scotland, EH8 9YL

(August 2021)

###### Abstract

Our universe seems to be radiation dominated at early times, and vacuum energy
dominated at late times. When we consider the maximal analytic extension of
this spacetime, its symmetries and complex analytic properties suggest a
picture in which spacetime has two sheets, exchanged by an isometry which, in
turn, picks a preferred (CPT-symmetric) vacuum state for quantum fields on the
spacetime. Previously Boyle:2018tzc ; Boyle:2018rgh , we showed how this line
of thought provides new explanations for dark matter, the matter-antimatter
asymmetry, the absence of primordial vector and tensor perturbations, and the
phase of the primordial scalar perturbations; and additional testable
predictions. In this paper, we develop this picture in several respects and,
in particular, point out that it also provides a new explanation for why the
thermodynamic arrow of time points away from the bang.

Introduction. Observations reveal that, seconds after the Big Bang, the
universe was described by a highly ordered state: a spatially-flat radiation-
dominated FRW metric (plus tiny gaussian, adiabatic, purely-growing-mode
scalar perturbations described by a nearly-scale-invariant power spectrum)
Ade:2015xua . What is this profound clue trying to tell us? The conventional
interpretation is that the radiation dominated era was preceded by an earlier
hypothetical epoch of accelerated expansion called inflation. However, for
every property of the universe that inflation claims to explain or predict,
there are numerous published inflationary models that claim to explain or
predict the opposite. We worry that inflation may not be a “good explanation”
in the sense articulated in Ch. 1 of Deutsch , and hope to find an alternative
explanation that is harder to vary.

In Refs. Boyle:2018tzc ; Boyle:2018rgh , we began to explore a different
approach. Instead of imagining that the simple radiation-dominated era we see
was preceded by earlier unseen chaos that must be cleaned up by inflation, we
tried to take the observed simplicity of the early universe at face value and
suppose that it extends all the way back to the bang, to explore where that
line of thought naturally leads. We found that the maximally-extended
spacetime so obtained has a new reflection isometry around the bang, and that
taking this symmetry seriously yields elegant new explanations and predictions
for several observed properties of the early universe, for the dark matter,
and for the neutrino sector.

In this paper, we go further: in addition to being radiation-dominated in the
past, our universe will be vacuum energy dominated in the future. We start by
studying the simplest cosmology of this type – a universe with just radiation
and vacuum energy – paying special attention to the structure (symmetries and
complex-analytical properties) of its maximal analytic extension, which we
regard as key clues. We then add matter, requiring it to respect this
structure. We point out that, in addition to the features and predictions
pointed out in Boyle:2018tzc ; Boyle:2018rgh , this picture provides other
insights, including a new solution to an old problem: why the thermodynamic
arrow of time points away from the bang PenroseWCH ; PenroseR2R .

Figure 1: The real solutions $a(\eta)$. The solid (blue) curve shows the
solution for $3\kappa<\sqrt{4\lambda r}$ (the physically relevant case); and
the dashed (orange) curves show them for $3\kappa>\sqrt{4\lambda r}$.

The background spacetime. Consider the general FRW line element

$ds^{2}=a^{2}(\eta)(-d\eta^{2}+\gamma_{ij}dx^{i}dx^{j})$ (1)

where $\gamma_{ij}$ is the metric on a maximally-symmetric space of curvature
$\kappa$. The Friedmann equation is

$\frac{\dot{a}^{2}}{a^{4}}=\frac{\rho}{3}-\frac{\kappa}{a^{2}}$ (2)

where we have taken $\hbar=c=8\pi G=1$ and a dot denotes $d/d\eta$:
$\dot{a}=da/d\eta$.

Since we would like to study the analytic behavior of the scale factor as
$a\to 0$ (where radiation dominates) and as $a\to\infty$ (where vacuum energy
dominates), we start with the simplest model that has the correct asymptotic
behavior at both extremes – i.e. a universe which only contains radiation and
vacuum energy: $\rho=r/a^{4}+\lambda$, where $r$ and $\lambda$ are positive
constants.

Now the Friedmann equation reads:

$\dot{a}^{2}={1\over 3}(r-3\kappa a^{2}+\lambda a^{4}).$ (3)

Note that in the absence of radiation, the three options for $\kappa$
(positive, negative, and zero) just repesented 3 different ways to slice the
same de Sitter (“dS”) solution; but when we include radiation, these three
options for $\kappa$ represent three physically distinct solutions.

Eq. (3) exhibits an interesting duality: setting $a=1/b$, we find

$\dot{b}^{2}={1\over 3}(rb^{4}-3\kappa b^{2}+\lambda),$ (4)

i.e., (3), with $a\rightarrow b$ and $r\leftrightarrow\lambda$.

Figure 2: The two-sheeted universe. The conformal time $\eta$ is circular: as
$\eta$ increases, we move up the outer ($a>0$) sheet from the $a=0$ to
$a=+\infty$, then back down the inner ($a<0$) sheet from $a=-\infty$ to $a=0$,
and so on in a loop. The metric $g_{\mu\nu}$ and tetrad $e^{a}$ are
meromorphic, with a zero at the bang, and a pole at dS $\infty$, and the map
exchanging each point on the outer sheet with its neighbor on the inner sheet
sends $a\to-a$ and is an isometry. The conformal metric
$\widetilde{g}_{\mu\nu}=a^{-2}g_{\mu\nu}$ and tetrad
$\tilde{e}^{\;\\!a}=a^{-1}e^{a}$ are $\eta$-independent and non-singular.

The solution of (3) is

$a(\eta)=\alpha\,{\rm sn}(\beta\eta,\,m),$ (5)

where ${\rm sn}(z,m)$ is a Jacobi elliptic function AbramowitzAndStegun , $m$
is either root of the quadratic equation

$\frac{(1+m)^{2}}{m}=\frac{(3\kappa)^{2}}{\lambda r},$ (6)

and the constants $\alpha$ and $\beta$ are given by

$\alpha=\sqrt{\frac{r(1+m)}{3\kappa}},\qquad\beta=\sqrt{\frac{\kappa}{(1+m)}}.$
(7)

The limit $(3\kappa)^{2}/(\lambda r)\to 0$ yields the $\kappa=0$ solution.

The choice of roots $m$, $\alpha$ and $\beta$ is just a convention: for
definiteness, when $(3\kappa)^{2}<4\lambda r$ (so $m$ is complex) we choose
the root $m$ in the upper half plane; and when $(3\kappa)^{2}>4\lambda r$ (so
$m$ is real) we choose the root $m<1$ (and imagine it to have an
infinitessimal positive imaginary part). Then $\alpha$ and $\beta$ are fixed
by taking $\sqrt{z}$ to have its standard branch cut along the negative real
axis.

Figure 3: The complex solution $a(\eta)$, in the complex $\eta$ plane, for the
physically relevant case $(3\kappa)^{2}<4\lambda r$ (not too curved). We have
marked the zeros with an “x”, and the poles with a “$+$” or “$-$” depending on
the sign of the residue. The solid curves show the contours of $|a(\eta)|$;
the vertical and horizontal dotted lines show, respectively, where ${\rm
Re}[a(\eta)]$ and ${\rm Im}[a(\eta)]$ vanish; and the arrows indicate the
direction (phase) of $a(\eta)$ in each of the regions enclosed by dotted
lines.

Note that ${\rm sn}(z,m)$ is meromorphic in both $z$ and $m$, and doubly
periodic in $z$: ${\rm sn}(z,m)={\rm
sn}(z+n\omega+n^{\prime}\omega^{\prime},m)$, for $n,n^{\prime}\in\mathbb{Z}$.
The two periods $\omega$ and $\omega^{\prime}$ point in different directions
in the complex plane (i.e. $\omega/\omega^{\prime}\notin\mathbb{R}$) and are
given by $\omega=4K$ and $\omega^{\prime}=2iK^{\prime}$, where $K\equiv K(m)$,
$K^{\prime}\equiv K(1-m)$, with $K(m)$ the complete elliptic integral of the
first kind AbramowitzAndStegun . Thus, we can think of ${\rm sn}(z,m)$ as a
periodic decoration or tiling of the complex $z$ plane, whose fundamental
parallelogram has edges $\omega$ and $\omega^{\prime}$. Alternatively, we can
think of ${\rm sn}(z,m)$ as a function over the toroidal Riemann surface
formed by identifying the opposite edges of this fundamental parallelogram.
The zeros of ${\rm sn}(z,m)$ are at $z=2nK+2n^{\prime}iK^{\prime}$ and the
poles are at $z=2nK+(2n^{\prime}+1)iK^{\prime}$, with
$n,n^{\prime}\in\mathbb{Z}$.

The nature of the solution $a(\eta)$ depends, first, on the sign of the
spatial curvature $\kappa$, and second, on whether the universe is close to
flat, $(3\kappa)^{2}<4\lambda r$, or far from flat, $(3\kappa)^{2}>4\lambda
r$. In all four cases, as we move along the real $\eta$ axis, the solution
$a(\eta)$ is also real, and the universe described by $a>0$ starts at $\eta=0$
and ends at $\eta=\eta_{\ast}$, where $\eta_{\ast}$ is given in the following
table:

| $\quad\;\;\kappa<0\quad\;\;$ | $\kappa>0$
---|---|---
$(3\kappa)^{2}<4\lambda r\;\;$ | $iK^{\prime}/\beta$ | $(2K\\!-\\!iK^{\prime})/\beta$
$(3\kappa)^{2}>4\lambda r\;\;$ | $iK^{\prime}/\beta$ | $2K/\beta$

In the three cases satisfying $(3\kappa)<\sqrt{4\lambda r}$, $a(\eta)$
describes a universe that grows from a simple zero at $\eta=0$ (the bang) to a
simple pole at $\eta=\eta_{\ast}$ (dS asymptopia). In the fourth case
$(3\kappa)>\sqrt{4\lambda r}$, corresponding to the lower right-hand corner of
the table, $a(\tau)$ describes a universe that grows from a simple zero at
$\eta=0$ (the bang) to a finite maximum size, before recollapsing to another
simple zero at $\eta=\eta_{\ast}$ (the crunch), while $a(\tau-
iK^{\prime}/\beta)$ describes a universe that shrinks from a simple pole at
$\eta=0$ (past dS asymptopia) to a minimum size, before re-expanding to
another simple pole at $\eta=\eta_{\ast}$ (future dS asymptopia). These
various solutions are shown in Fig. 1.

In all cases, the universe described by $a(\eta)>0$ extends from $\eta=0$ to
$\eta=\eta_{\ast}$, but the maximal analytic extension of $a(\eta)$ is
periodic, with period $2\eta_{\ast}$, and describes a two-sheeted universe,
with $a>0$ on one sheet and $a<0$ on the other, and with $a(\eta)$ odd under
temporal reflection across either the bang at $\eta=(2n)\eta_{\ast}$ or the dS
boundary at $\eta=(2n+1)\eta_{\ast}$ ($n\in\mathbb{Z}$). We can visualize this
as saying that $\eta$ is curled into a circle, with $\eta$ and
$\eta+2\eta_{\ast}$ identified, so that the universe forms the two-sheeted
cylinder shown in Fig. 2, with an isometry that swaps adjacent points on the
inner and outer cylinder and sends $a\to-a$.

Imaginary time and thermality. The analytic structure of $a(\eta)$ is shown in
Fig. 3, for the physically relevant cases where $(3\kappa)^{2}<4\lambda r$
(“close-to-flat”). As evidence we should take this analytic structure
seriously, note the physical time
$t(\eta)=\int_{0}^{\eta}a(\eta^{\prime})d\eta^{\prime}$ is given by

$t=\frac{\alpha}{\beta}m^{-1/2}{\rm ln}\left(\frac{{\rm
dn}(\beta\eta,m)-m^{1/2}{\rm cn}(\beta\eta,m)}{{\rm dn}(0,m)-m^{1/2}{\rm
cn}(0,m)}\right),$ (8)

where $dn(z,m)$ and $cn(z,m)$ are also Jacobi elliptic functions
AbramowitzAndStegun , and $\eta$ may be complex. The integral
$\int_{0}^{\eta}a(\eta^{\prime})d\eta^{\prime}$ depends on how the contour
from $\eta^{\prime}=0$ to $\eta^{\prime}=\eta$ winds around the poles of
$a(\eta^{\prime})$. This in turn corresponds to the fact that the answer (8)
depends on ${\rm ln}(z)$, which is only defined modulo $2\pi i$, so that $t$
itself is only well defined modulo $\Delta t=2\pi i(\alpha/\beta)m^{-1/2}=2\pi
i\sqrt{3/\lambda}$. But a system whose time coordinate is only defined modulo
an imaginary shift $t\to t+\Delta t$ (or, in other words, whose $t$ is curled
into a circle of circumference $\Delta t$ in the imaginary time direction)
corresponds to a system at finite temperature $T=i/\Delta t$. In this way, we
precisely recover the well-known expression for the de Sitter temperature

$T_{dS}=H/2\pi$ (9)

where $H=\sqrt{\lambda/3}$ is the asymptotic dS Hubble constant (the
reciprocal of the asymptotic dS radius).

Note the generality of this result: $T_{dS}$ is independent of $\eta$,
$\kappa$, or the radiation density, and determined by the residue of $a(\eta)$
at its pole, and hence by $\lambda$.

Matter and the arrow of time. Now let us add matter. For simplicity, we study
a massive scalar field $\varphi$ and leave the details of higher-spin fields
to future work.

The equation of motion for $\varphi$ is

$-\Box\varphi+\xi R\varphi+\mu^{2}\varphi=0$ (10)

where the mass $\mu$ is a constant. For concreteness, we consider the
conformal coupling $\xi=1/6$, which is most natural Penrose ; Callan:1970ze ,
but our conclusions apply for any value of $\xi$, including minimal coupling
$\xi=0$. If we define the $\eta$-independent conformal metric
$\tilde{g}_{\mu\nu}=a^{-2}\,g_{\mu\nu}$ and corresponding field
$\tilde{\varphi}=a\varphi$, Eq. (10) becomes

$-\tilde{\Box}\tilde{\varphi}+\frac{1}{6}\tilde{R}\tilde{\varphi}+\tilde{\mu}^{\;\\!2}\tilde{\varphi}=0$
(11)

where $\tilde{\Box}$ and $\tilde{R}$ are the wave operator and Ricci scalar of
$\tilde{g}_{\mu\nu}$, and $\tilde{\mu}\equiv a\mu$ is time-dependent:
$\tilde{\mu}=\tilde{\mu}(\eta)$.

We have seen that the analytically-extended background metric may be
interpreted as a two-sheeted cylinder, with a reflection symmetry (through the
bang or through the dS boundary) which swaps adjacent points on the two sheets
(see Fig. 2). We regard this as a clue to be taken seriously. We thus require
that the matter fields living on this background share its structure. In
particular, $\tilde{\varphi}$ should satisfy

$\tilde{\varphi}(\eta,{\bf x})=\tilde{\varphi}(\eta+2\eta_{\ast},{\bf x})$
(12)

(i.e. it lives on the cylinder shown in Fig. 2) and

$\displaystyle\tilde{\varphi}(\eta,{\bf x})$ $\displaystyle=$
$\displaystyle\pm\tilde{\varphi}(-\eta,{\bf x})$ (13a)
$\displaystyle\tilde{\varphi}(\eta,{\bf x})$ $\displaystyle=$
$\displaystyle\pm\tilde{\varphi}(2\eta_{\ast}-\eta,{\bf x}),$ (13b)

(i.e. it is even or odd under reflection through either the bang or dS
boundary: we explain how to resolve this sign ambiguity below). Note that
these conditions are not independent: Eqs. (12) and (13a) imply Eq. (13b).

To see what the symmetries (12, 13) imply, let us examine the solutions of
(11). For our purposes, it is sufficent to solve (11) near the bang and the dS
boundary, where the gravitational backreaction of $\varphi$ can be neglected.
111More generally, the solution $\tilde{\varphi}$ is an arbitrary linear
combination of separable solutions $\tilde{\varphi}_{k}(\eta)\psi_{k}({\bf
x})$, where $\psi_{k}({\bf x})$ is an eigenfunction of
$\gamma^{ij}\tilde{\nabla}_{i}\tilde{\nabla}_{j}$; and, if we neglect the
gravitational backreaction of $\varphi$, so that $a(\eta)$ is given by (5),
then (11) becomes the Lamé equation for $\tilde{\varphi}_{k}(\eta)$, which can
be solved in terms of Lamé functions WhittakerWatson .

Near the bang, where radiation dominates, we can neglect the vacuum energy and
spatial curvature of the metric, so $a(\eta)=(r/3)^{1/2}\eta$, and
$\tilde{\varphi}$ is a linear combination of terms of the form
$\tilde{\varphi}_{k}(\eta){\rm exp}(i{\bf k}\cdot{\bf x})$, where

$\ddot{\tilde{\varphi}}_{k}+(k^{2}+\frac{r}{3}\eta^{2}\mu^{2})\tilde{\varphi}_{k}=0,$
(14)

with $k^{2}={\bf k}\cdot{\bf k}$, and the general solution is

$\tilde{\varphi}_{k}=c_{+}D_{p}(\gamma\eta)+c_{-}D_{p}(-\gamma\eta),$ (15)

where $D_{p}(z)$ is the parabolic cylinder function GradshteynAndRyzhik ,
$c_{\pm}$ are constants, $p\equiv-\frac{1}{2}+i\sqrt{3/r}\frac{k^{2}}{2\mu}$,
and $\gamma\equiv{\rm e}^{-i\pi/4}(\frac{4r}{3}\mu^{2})^{1/4}$.

Near the dS boundary, where vacuum energy dominates, we can neglect the
radiation and spatial curvature of the metric, so
$a(\eta)=1/H(\eta_{\ast}-\eta)$, and $\tilde{\varphi}$ is a linear combination
of terms $\tilde{\varphi}_{k}(\eta){\rm exp}(i{\bf k}{\bf x})$, where

$\ddot{\tilde{\varphi}}_{k}+(k^{2}+\frac{\mu^{2}}{H^{2}(\eta_{\ast}-\eta)^{2}})\tilde{\varphi}_{k}=0,$
(16)

with the general solution

$\tilde{\varphi}_{k}=(k(\eta_{\ast}\\!-\eta))^{1/2}[c_{1}J_{\nu}(k(\eta_{\ast}\\!-\eta))\\!+c_{2}Y_{\nu}(k(\eta_{\ast}\\!-\eta))]$
(17)

where $J_{\nu}(z)$ and $Y_{\nu}(z)$ are Bessel functions GradshteynAndRyzhik ,
$c_{1}$ and $c_{2}$ are constants, and
$\nu\equiv\frac{1}{2}(1-4m^{2}/H^{2})^{1/2}$.

From these solutions, we see a crucial difference between the behavior at the
bang and at the dS boundary. At the bang, $\tilde{\varphi}_{k}$ remains
analytic ($\varphi_{k}=a^{-1}\tilde{\varphi}_{k}$ remains meromorphic), and
may be analytically continued across it. For this analytic continuation to be
compatible with the reflection symmetry (13), $\tilde{\varphi}$ must satisfy
the following boundary condition (“b.c.”) at the bang:

$\displaystyle\tilde{\varphi}(0,{\bf x})$ $\displaystyle=$
$\displaystyle\pm\tilde{\varphi}(0,{\bf x}),$ (18a)
$\displaystyle\dot{\tilde{\varphi}}(0,{\bf x})$ $\displaystyle=$
$\displaystyle\mp\dot{\tilde{\varphi}}(0,{\bf x}).$ (18b)

By contrast, $\tilde{\varphi}$ is not analytic at the dS boundary, and cannot
be analytically extended across it, so it does not satisfy a corresponding
b.c. there.

The fact that basic considerations of symmetry and analyticity force all
fields to satisfy a restrictive boundary condition at one end of spacetime
(the bang) but not at the other end (the dS boundary) is striking. It seems to
suggest a fundamental new explanation for why the thermodynamic arrow of time
(i.e. the direction in which entropy increases) points away from the bang.

The upper (lower) sign in (18) corresponds to a Neumann (Dirichlet) b.c. for
$\tilde{\varphi}$ at the bang. This is closely related to the b.c. at the
boundary of anti de Sitter (AdS) spacetime Avis:1977yn ; Breitenlohner:1982jf
: there too, if one conformally maps the spacetime to a portion of the
infinite static (Einstein) cylinder, one finds a simple reflecting (Neumann or
Dirichlet) b.c. for the field $\tilde{\varphi}$. In fact, the AdS b.c.s were
originally derived Avis:1977yn  by mapping AdS to a portion of the static
Einstein cyclinder, and then noting that it could be analytically extended to
form a two-sheeted spacetime, with a natural $\mathbb{Z}_{2}$ symmetry
relating the sheets. Of course, this closely parallels our derivation, but
with some noteable differences: in the AdS case, one analytically continues in
the radial direction, and the resulting b.c.s make the AdS boundary an
ordinary mirror (a timelike surface which reflects radially); while in our
case, one analytically continues in the time direction, and the resulting
b.c.s make the bang an exotic mirror (a spacelike surface which reflects in
the time direction).

For a canonical scalar field, it seems most natural to choose the lower sign
in (18) at the bang, so that $\tilde{\varphi}$ and
$\varphi=a^{-1}\tilde{\varphi}$ satisfy Dirichlet and Neumann boundary
conditions, respectively, since in this case $\tilde{\varphi}$ and $\varphi$
both remain finite and analytic at the bang, consistent with the assumption of
small gravitational back reaction.

The $\pm$ option in (18) also appears in the b.c.s for higher spin fields (as
in the AdS case Breitenlohner:1982jf ; Hawking:1983mx ). Since $a(\eta)$ is
odd under $\eta\to-\eta$, we can infer right choice for the tetrad: the
conformal tetrad $\tilde{e}^{\;\\!a}_{\mu}$ (with
$\eta_{ab}\tilde{e}^{\;\\!a}_{\mu}\tilde{e}^{\;\\!b}_{\nu}=\tilde{g}_{\mu\nu}$)
satisfies a Neumann b.c. (or, equivalently, the symmetric condition
$\tilde{e}^{\;\\!a}_{\mu}(\eta,{\bf x})=\tilde{e}^{\;\\!a}_{\mu}(-\eta,{\bf
x})$) while the physical tetrad $e^{a}_{\mu}=a\tilde{e}^{\;\\!a}_{\mu}$ (with
$\eta_{ab}e^{a}_{\mu}e^{b}_{\nu}=g_{\mu\nu}$) satisfies a Dirichlet b.c. (or,
equivalently, the anti-symmetric condition $e^{a}_{\mu}(\eta,{\bf
x})=-e^{a}_{\mu}(-\eta,{\bf x})$). As shown in Boyle:2018tzc , this condition
successsfully explains: (i) the absence of primordial vector perturbations;
(ii) the absence of primordial “decaying-mode” scalar and tensor perturbations
(which synchronizes the scalar perturbations, explaining the observed
oscillations in the CMB temperature spectrum and their phases); and (iii) the
fact that bang seems to be a particularly mild singularity (a “Weyl
singularity” – a non-singular conformal metric times an overall conformal
factor $a(\eta)$ which momentarily passes through zero).

Remarks. (i) In a forthcoming paper, we present a mechanism for generating the
observed scale-invariant scalar perturbations in this framework perts .

(ii) In our picture, instead of thinking of the universe evolving through the
bang, we can think of the bang as a spacelike mirror – a surface of reflection
symmetry. As with an electromagnetic mirror, we can either think of the bang
as boundary of spacetime, satisfying special boundary conditions, or else (by
the method of images), we can think of a mirror universe on the “other side”
of the bang.

(iii) One might worry that, in deriving (5), we dropped the only term in the
Friedmann equation (the matter density $\propto a^{-3}$) that is odd under
$a\to-a$, while all other terms are even. Would including this term spoil the
symmetry? No: if the matter fields satisfy the condition (18) then, far from
the bang, the matter density is actually $\propto|a|^{-3}$ (which is even
under $a\to-a$); while near the bang it remains perfectly analytic.

(iv) We saw that the solution (17) generally has an essential singularity near
the dS boundary. The exception is when the field is massless ($\mu=0$), in
which case (17) remains analytic at the dS boundary 222The fact that both
massless and massive solutions are analytic at the bang, while only massless
solutions are analytic at the dS boundary is due, physically, to the fact that
both massless and massive particles reach the bang in finite proper time,
while only massless particles reach the dS boundary in finite proper time..
This leaves open the possibility that massless fields might obey a boundary
condition there too, which would quantize the allowed comoving wavenumbers for
massless fields perts .

(v) It is natural to speculate that our classical symmetry conditions
$e^{a}_{\mu}(-\eta,{\bf x})=-e^{a}_{\mu}(\eta,{\bf x})$ and
$\varphi(-\eta,{\bf x})=\varphi(\eta,{\bf x})$ (along with analogous
conditions for the various other fields) follow from a path integral
formulation in which only paths satisfying these conditions are allowed 333In
this picture, the bang may alternatively be thought of as a kind of spacelike
orbifold boundary, with some similarity to the timelike orbifold boundaries
which appear in constructions like those in Horava:1995qa ; Horava:1996ma ..
This suggests a natural wavefunction of the universe: namely, that the
amplitude for a given spacelike 3-configuration of the geometry and matter
fields is the path integral over all appropriately symmetric 4-configurations
ending on the given 3-configuration.

Acknowledgements. Research at Perimeter Institute is supported by the
Government of Canada, through Innovation, Science and Economic Development,
Canada and by the Province of Ontario through the Ministry of Research,
Innovation and Science. The work of NT is supported by the STFC Consolidated
Grant “Particle Physics at the Higgs Centre.”

## References

  * (1) L. Boyle, K. Finn and N. Turok, “CPT-Symmetric Universe,” Phys. Rev. Lett. 121, no.25, 251301 (2018) doi:10.1103/PhysRevLett.121.251301 [arXiv:1803.08928 [hep-ph]].
  * (2) L. Boyle, K. Finn and N. Turok, “The Big Bang, CPT, and neutrino dark matter,” [arXiv:1803.08930 [hep-ph]].
  * (3) P. A. R. Ade et al. [Planck Collaboration], “Planck 2015 results. XIII. Cosmological parameters,” Astron. Astrophys. 594, A13 (2016) doi:10.1051/0004-6361/201525830 [arXiv:1502.01589 [astro-ph.CO]].
  * (4) D. Deutsch, The Beginning of Infinity: Explanations that Transform the World, Penguin Books (2011).
  * (5) R. Penrose, ”Singularities and time-asymmetry,” in General Relativity: An Einstein Centenary Survey, edited by S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979).
  * (6) R. Penrose, The road to reality: A complete guide to the laws of the universe, Random house (2005).
  * (7) M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (1964).
  * (8) R. Penrose, “Conformal Treatment of Infinity,” in Relativity, Groups and Topology, Eds. C. DeWitt and B.S. DeWitt, Gordon and Breach, London (1964); republished as Gen. Rel. and Grav. 43, 901 (2011) doi:10.1007/s10714-010-1110-5.
  * (9) C. G. Callan, Jr., S. R. Coleman and R. Jackiw, “A New improved energy - momentum tensor,” Annals Phys. 59, 42-73 (1970) doi:10.1016/0003-4916(70)90394-5
  * (10) N. D. Birrell and P. C. W. Davies, Quantum fields in curved space, Cambridge University Press (1984).
  * (11) V. Mukhanov and S. Winitzki, Introduction to quantum effects in gravity, Cambridge University Press (2007).
  * (12) E.T. Whittaker, G.N. Watson, A course of modern analysis, Cambridge University Press (1952), Ch. 6
  * (13) M. Abramowitz and I. A. Stegun, Table of Integrals, Series and Products, 7th Edition.
  * (14) S. J. Avis, C. J. Isham and D. Storey, “Quantum Field Theory in anti-De Sitter Space-Time,” Phys. Rev. D 18, 3565 (1978) doi:10.1103/PhysRevD.18.3565
  * (15) P. Breitenlohner and D. Z. Freedman, “Stability in Gauged Extended Supergravity,” Annals Phys. 144, 249 (1982) doi:10.1016/0003-4916(82)90116-6 1497 citations counted in INSPIRE as of 27 Jun 2021
  * (16) S. W. Hawking, “The Boundary Conditions for Gauged Supergravity,” Phys. Lett. B 126, 175-177 (1983) doi:10.1016/0370-2693(83)90585-3
  * (17) L. Boyle and N. Turok, in preparation (2021).
  * (18) P. Horava and E. Witten, Nucl. Phys. B 460, 506-524 (1996) doi:10.1016/0550-3213(95)00621-4 [arXiv:hep-th/9510209 [hep-th]].
  * (19) P. Horava and E. Witten, Nucl. Phys. B 475, 94-114 (1996) doi:10.1016/0550-3213(96)00308-2 [arXiv:hep-th/9603142 [hep-th]].


This paper is in the public domain. It has been peer-reviewed and accepted at
FPGAworld 2012, but ACM refuse to publish public domain papers from non-US
government authors unless paid an unreasonable amount of money. ACM have
decided to ignore the conference’s scientific referee, disrespect their
constitutional purpose of “fostering the open interchange of information”, and
remove this paper from the official conference proceedings on profit grounds.

# A 26 ps RMS time-to-digital converter core for Spartan-6 FPGAs

Sébastien Bourdeauducq

Independent researcher<EMAIL_ADDRESS>
###### Abstract

We have designed, implemented and tested a time-to-digital converter core in a
low-cost Spartan-6 FPGA. Our design exploits the finite propagation speed in
carry chains to realize a delay line in which the propagation distance of the
incoming signal’s edges is measured using hundreds of taps. This technique
enables the core to reach a precision far better than the minimum switching
period of the FPGA flip-flops. To compensate for process, voltage and
temperature (PVT) effects, our design uses a combination of two techniques:
startup calibration and online calibration. The startup calibration uses a
statistical method to estimate the delay between the taps of the delay line
and helps eliminate the effect of process variations. The online calibration,
which takes place without disruption of the core’s operation, uses a ring
oscillator whose frequency instability is measured and used to compensate for
subsequent voltage and temperature effects on the delay line. Our tests show
that our design reaches a precision of $26$ ps RMS over a temperature range of
37°C to 48°C.

###### category:

B.m Hardware Miscellaneous

###### category:

B.7.1 Hardware Integrated circuits

###### keywords:

Types and design styles

###### keywords:

FPGA, TDC, time-to-digital converters

††terms: Experimentation

##  1 Introduction

Several FPGA-based time-to-digital converter (TDC) designs have already been
proposed[1][2]. However, there were many incentives for us to design a new
core.

The PVT compensation mechanism of [1] introduces dead times during which the
core is insensitive to incoming signal transitions, which we found
undesirable. The continuous calibration process of [2] requires that the
statistical distribution of transitions within the reference TDC clock periods
is uniform. This may not be the case in systems meant to be part of a particle
accelerator, where many events are synchronous to a single clock. Therefore,
we devised another technique (online calibration) that does not introduce dead
times and is independent of the statistics of the incoming signal.

We also wanted the design to function on Spartan-6 FPGAs so that it can be
used on the SPEC[3] boards. Previous works are based on Virtex-5 or Cyclone-II
FPGAs.

Finally, no source code is published for any of these designs, which renders
it necessary to develop a new core for all practical purposes (and
incidentally makes it more difficult to reproduce and verify the results). Our
core is available under the LGPL free software license and its full VHDL code
can be freely downloaded from http://www.ohwr.org/projects/tdc-core.

##  2 Design

###  2.1 Overview

The block diagram of the core is given in Figure 1.

Figure 1: Block diagram of the TDC core.

The generated timestamp is based on a cycle count and the arrival time within
a clock cycle. The former needs only a simple counter whereas the latter is
measured with a tapped delay line. The fine time measurement is obtained by
injecting the signal into the tapped delay line which gives a measurement
analogous to a thermometer after the taps are sampled by D flip-flops. The
total delay of the delay line must be greater than the clock period. At each
clock tick, an encoder counts the taps the signal has reached and gives a raw
measurement of the timestamp of the signal within the current clock cycle.
This raw value is fed into a look-up table (LUT) which converts it into a
calibrated value expressed in subdivisions of the clock cycle, called the
fractional value. Finally, in the deskew stage, the fractional value is
combined with the index of the current clock cycle given by the coarse
counter, and the resulting fixed-point value is added a user-defined constant
to enable the TDC core to directly generate timestamps relative to the source
of the system clock.

The main difficulty with this system is that the delay line is subject to
process, temperature and voltage (PVT) induced variations, and it needs to be
calibrated against them.

To generate the LUT contents, the controller switches to the calibration
signal. The key property of the calibration signal is that the probability
density of its transition timestamps within a system clock cycle must be
constant. The controller measures the raw timestamps and books a histogram.
Because of the constant probability density, the heights of the histogram bars
are approximately proportional to the delays between the taps of the delay
line after enough measurements have been taken. Further, the last tap to have
recorded a signal transition corresponds to a delay equal to the system clock
period. This enables the controller to build the initial contents of the LUT.
This process is called startup calibration.

The drawback of the startup calibration is that the system cannot operate
while the calibration is taking place. Therefore, a process of online
calibration has been devised. Each channel contains a ring oscillator that is
placed close to the delay line. The controller periodically measures the
frequency of this ring oscillator, compares it to the frequency that was
measured at the time of startup calibration, linearly interpolates the
fractional timestamps, and updates the LUT. This allows compensation of
temperature and voltage effects while the system keeps running.

The system gives timestamps of both rising and falling edges of the incoming
signal. The rising edges are discerned from the falling edges using the
“polarity” output.

###  2.2 Delay line structure

The delay line uses a carry chain. It is made up of `CARRY4` primitives whose
`CO` outputs are registered by the dedicated D flip flops of the same slices.
The signal is injected at the `CYINIT` pin at the bottom of the carry chain.
The `CARRY4` primitives have their `S` inputs hardwired to 1, which means the
carry chain becomes a delay line with the signal going unchanged through the
`MUXCY` elements (see [4] for reference). Since each `CARRY4` contains four
`MUXCY` elements, the delay line has four times as many taps as there are
`CARRY4` primitives.

Using the Xilinx timing model, a surprising observation is that some delay
differences between consecutive taps are negative. This probably is at the
origin of the “bubbles” mentioned in the EPFL paper [1]. The schematics given
by Xilinx of the `CARRY4` primitive is misleading there, and has probably
little to do with the actual transistor-level implementation. The Xilinx
documentation [4] gives a hint by describing the primitive as “Fast Carry
Logic with Look Ahead”.

To avoid negative differences, we simply reorder the bits at the output of the
delay line to sort the taps by increasing actual delays. We can then think of
the delay line according to Figure 2. The bin widths are uneven, but the
incoming signal reaches the taps in order. This last property simplifies the
encoder design, since it only has to count the number of identical bits at the
beginning of the delay line.

Figure 2: Representation of the delay line.

###  2.3 Calibration details

In the formulas below:

  * •

$T_{sys}$ is the system clock period.

  * •

$H(n)$ is the number of hits in the histogram at output $n$. A hit at output
$n$ means that the signal propagated down to output $n$, without reaching
output $n-1$.

  * •

$W(n)$ is the width of bin $n$.

  * •

$C=\displaystyle\sum\limits_{n=0}^{N-1}H(n)$ is the total number of hits in
the histogram.

  * •

$R(n)$ is the timestamp of an event whose signal propagated down to output $n$
(without reaching output $n-1$), measured backwards from the clock tick.

  * •

$f$ (respectively $f_{0}$) is the current (respectively reference) frequency
of the online calibration ring oscillator.

####  2.3.1 Offline calibration

At startup, the core sends random pulses into the delay line (coming from a
on-chip ring oscillator), builds the histogram, computes the delays (as
explained in [2]), and initializes the LUT.

We take the first output of the delay line to be the origin of the time
measurements, and we define:

$W_{0}(N-1)=0$ (1)

The width of other bins is proportional to their respective number of counts
in the histogram. The widths sum up to a clock period. This leads to the
following equation:

$W_{0}(n)=\frac{H(n+1)}{C}\cdot T_{sys}$ (2)

The timestamp is the sum of the widths of the traversed bins:

$R_{0}(n)=\displaystyle\sum\limits_{i=n}^{N-1}{W_{0}(i)}=\frac{T_{sys}}{C}\cdot\displaystyle\sum\limits_{i=n}^{N-1}{H(i)}$
(3)

In the TDC core, the unit is the clock period, and the output has $F$ base 2
digits after the radix points. The controller also chooses $C=2^{F+P}$, where
$P$ is the number of extra histogram bits. Expressed in units of $2^{-F}$
clock periods (which is the weight of the least significant bit of the fixed-
point output), we have:

$\frac{T_{sys}}{C}=2^{-P}$ (4)

####  2.3.2 Online calibration

Online calibration is performed with a simple linear interpolation of the
delays relative to the ring oscillator frequencies:

$R(n)=\frac{f_{0}}{f}\cdot R_{0}(n)$ (5)

Note that when $f<f_{0}$, some values can go above the maximum fractional part
value of $1-2^{-F}$ and might not fit in the LUT anymore. However, those
correspond to delays that now exceed one clock period, and therefore they
should almost never get used. In case of overflow, the controller saturates
the result by using the maximum value $1-2^{-F}$ in order to give the best
approximation in case those LUT entries still get used.

##  3 Tests and results

###  3.1 General setup

The demonstration design runs on a SPEC board equipped with a FMC DIO
5-channel daughterboard.

Test signals go through the FMC daughterboard. The first LEMO connector on the
daughterboard is configured as output and transmits an oscillating pattern.
The next two LEMO connectors are inputs connected to TDC channels.

For measuring the FPGA temperature, a 1-wire digital thermometer is attached
on top of the FPGA using kapton tape. Thermal paste improves conduction
between the FPGA and the sensor.

The TDC core is configured with 2 channels (to enable differential
measurements, see §4.3) and each delay line has 124 `CARRY4` elements (496
taps).

To minimize variations of the timing properties between runs of the automated
place and route tool and to maximize thermal coupling between each delay line
and its online calibration oscillator, the design is floorplanned.

The two delay lines from each channel are placed close to their respective
IOBs. The ring oscillator components are placed in the `SLICEX` columns just
at the right of the delay lines, and spread evenly along the height of the
delay lines. There is one ring oscillator per channel, which is made of many
LUTs in series. This is illustrated by Figure 3, where the delay lines are
colored green and the ring oscillators are blue (each blue block is a `SLICEX`
component containing a LUT belonging to one of the two ring oscillators).

Figure 3: Floorplan of the delay lines and ring oscillators in FPGA Editor.

In the input signal path, there are one multiplexer and one inverter per
channel. Everything is packed into one FPGA slice, which is also manually
placed to minimize timing variations. The physical input signal path can be
seen in Figure 4. The LVDS IOBs are represented in black, and the routing and
the slice in pink.

A limitation of this TDC design is that it does not compensate for PVT
variations in the input signal path elements.

Figure 4: Input signal path in FPGA Editor.

##  4 Methods and results

###  4.1 Effect of temperature on ring oscillators

The purpose of this experiment is to examine how temperature affects
propagation delays. We slowly heated the FPGA (so it remains in thermal
equilibrium with the sensor) to obtain the plot of Figure 5.

The frequency values are directly reported from the TDC core, and are measured
in cycles per frequency counter period.

As expected, the frequencies decrease linearly with the temperature, and the
two channels follow a near-identical pattern. The variation is small: about
1.3% for the 15°C difference. However, near the end of the delay line, a 1.3%
variation represents about 100ps, so it is important to compensate for the
effects of temperature.

We suspect that the constant difference between the two channels is due to
process variations across the different locations of the FPGA chip where the
two ring oscillators are placed, and/or differences in routes chosen by the
`par` tool to implement the two oscillators.

Figure 5: Dependence of ring oscillator frequencies on temperature.

###  4.2 Startup calibration stability

The startup calibration process relies on an asynchronous clock source which
generates TDC events with a uniform random distribution within the system
clock cycles. We wanted to verify that the process is deterministic enough.

With the FPGA in thermal equilibrium, we ran the startup calibration twice and
compared the resulting LUT contents. The difference is plotted in Figure 6,
and is small enough.

Figure 6: Difference between the LUT contents from two startup calibrations at
the same temperature.

###  4.3 Differential measurements

The purpose of this test is to determine the precision of the system.

We connected the oscillator output of the FMC DIO card to a splitter feeding
two cables of different lengths going to the two TDC channels. Those cables
had propagation delays of approximately 2ns and 4ns. We then observed the
difference between the two TDC timestamps, which is expected to remain
constant (Figure 7). Since the oscillator is asynchronous to the system clock,
the complete delay line can be covered and tested.

Figure 7: Principle of differential measurements.

The advantage of this technique is that it is easy to set up and does not
require expensive equipment. A limitation is that the result is not affected
by common-mode noise of the input path to the delay line (Figure 4).

We made the measurements at thermal equilibrium, with the sensor measuring
36.9375°C. The histogram of the results is shown in Figure 8.

Figure 8: Differential measurement results.

The results can be modeled with a Gaussian distribution having a mean of
2221ps (which is close to the 4ns-2ns difference in propagation times from the
cables) and a standard deviation of 37ps. If we suppose that the jitter in
each channel is independent and also has a Gaussian distribution, we can
estimate that its standard deviation is 26ps. This means that for one channel,
95% of the results are precise to $\pm$52ps.

###  4.4 Temperature compensation

Even though the influence of temperature is small (§4.1), we can still see the
positive action of the online calibration.

After calibrating at 37°C, we brought the temperature to 47.875°C, and ran
startup calibration again. We observed a significant difference between the
LUT contents (figure 9).

Figure 9: Difference between the LUT contents from two startup calibrations at
high and low temperatures.

The new LUT data are very close to what had been extrapolated from the 37°C
data by the online calibration system (Figure 10). In fact, in this sample the
difference is slightly smaller than what we had observed between two startup
calibrations at the same temperature (Figure 6). This shows the good working
of the online calibration system.

Figure 10: Difference between the LUT contents from startup calibration and
the values computed by online calibration.

##  5 Conclusions and future works

The results of this experiment are very encouraging, as they show a precision
better than that of many commercial TDC chips, but using a lower-cost FPGA. We
can also potentially support dozens of channels in the XC6SLX45T of the SPEC
board as the core uses little FPGA resources and also shares the calibration
logic among all channels. Further, the latency of the core is low (6 cycles of
the system clock) and the throughput high (for each channel, the dead time
after an event is 3 cycles of the system clock, which can be brought down to 1
cycle without major architecture changes). This is also better than many
commercial solutions.

There are however several areas of improvement.

First, more testing would be welcome, with many boards and FPGAs, with
deliberate variations of the supply voltage, and within a wider temperature
range.

Examining the startup calibration histograms reveals that almost half of the
bin widths are zero. This is due to the particular propagation characteristics
of carry chains, which are not the best solution for a delay line (their
advantage, however, is that it is relatively easy to keep the exact same
delays between runs of the place-and-route tool). It can make sense to use
regular LUTs and/or general routing to implement the delay line instead, at
the cost of increased design difficulty and reduced portability.

The startup calibration process could be improved (and made almost
deterministic) by using as calibration signal a clock whose frequency is
slightly different from the system clock. This way, the variations shown in
Figure 6 (which peak at almost 17ps) can be reduced or eliminated.

The carry chain is very long and this restricts its possible placements and
compatibility with smaller FPGAs. Using LUTs and/or routing would also
alleviate this problem.

If better precision is needed, multiple delay lines can work in parallel and
their outputs combined, in order to average errors out.

Finally, the influence of the input path (Figure 4) was not thoroughly
studied, even though we expect it to be minor.

##  6 Acknowledgments

This work was prepared under an agreement with and funded by CERN. The content
of this paper does not necessarily reflect the position or the policy of CERN
and no official endorsement should be inferred.

The author wishes to thank Javier Serrano and Tomasz Wlostowski for their
valuable ideas and inputs.

## References

  * [1] C. Favi and E. Charbon.  A 17ps Time-to-Digital Converter Implemented in 65nm FPGA Technology.  Proceedings of the ACM/SIGDA international symposium on Field programmable gate arrays, 2009.
  * [2] J. Wu and Z. Shi.  The 10-ps Wave Union TDC: Improving FPGA TDC Resolution beyond Its Cell Delay.  Nuclear Science Symposium Conference Record, 2008. NSS ’08. IEEE, October 2008.
  * [3] E. van der Bij, J. Serrano, T. Wlostowski, M. Cattin, E. Gousiou, P. Alvarez Sanchez, A. Boccardi, N. Voumard and G. Penacoba.  Open Hardware for CERN’s accelerator control systems.  IOP Journal of Instrumentation, Volume 7., January 2012.
  * [4] Xilinx.  Spartan-6 Libraries Guide for HDL Designs.