# The non-explosive stellar merging origin of the ultra-massive carbon-rich
white dwarfs

Adela Kawka1, Lilia Ferrario1,2 and Stéphane Vennes1,2
1 International Centre for Radio Astronomy Research - Curtin University, GPO
Box U1987, Perth, WA 6845, Australia
2 Mathematical Sciences Institute, The Australian National University,
Canberra, ACT 0200, Australia Contact e-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We have investigated the origin of a sub-class of carbon-polluted white dwarfs
(DQ) originally identified as the “hot DQ" white dwarfs. These objects are
relatively hot ($10\,000\lesssim T_{\rm eff}\lesssim 25\,000$ K), have
markedly higher carbon abundance (C-enriched), are more massive ($M\gtrsim
0.8$ M⊙) than ordinary DQs ($M\sim 0.6$ M⊙), and display high space
velocities. Hence, despite their young appearance their kinematic properties
are those of an old white dwarf population. The way out of this dilemma is to
assume that they formed via the merging of two white dwarfs. In this paper we
examine the observed characteristics of this population of “C-enriched" DQ
white dwarfs and confirm that nearly half of the 63 known objects have
kinematic properties consistent with those of the Galactic thick disc or halo.
We have also conducted population synthesis studies and found that the merging
hypothesis is indeed compatible with observations. Studies of this sub-class
of white dwarfs have important implications for our understanding of Type Ia
Supernovae (SNeIa), commonly used to determine the expansion history of the
universe, since the same formation channel applies to both kind of objects.
Hence probing the properties of these white dwarfs that failed to explode may
yield important constraints to the modelling of the mechanisms leading to a
thermonuclear runaway.

###### keywords:

white dwarfs – stars: evolution – stars: kinematics and dynamics – stars:
atmospheres – supernovae: general

††pubyear: 2022††pagerange: The non-explosive stellar merging origin of the
ultra-massive carbon-rich white dwarfs–4

##  1 Introduction

White dwarfs are the final stage of stellar evolution for the majority of
stars and about a quarter of white dwarfs are found in binary systems
(Hollands et al., 2018). However, about half of intermediate mass main-
sequence stars (the progenitors of the white dwarfs) are in binaries
(Ferrario, 2012; Duchêne & Kraus, 2013). This discrepancy provides evidence
that some white dwarfs formed in stellar mergers (Briggs et al., 2015; Toonen
et al., 2017).

The merger of double degenerates (DDs) provides one path toward Type Ia
supernova explosions. The study of white dwarfs that lead to such events, but
failed to explode, yield important constraints to the modelling of the
mechanisms leading to a thermonuclear runaway (Ruiter, 2020). The expectation
is that white dwarfs formed via DD mergers are more massive than white dwarfs
descending from single stars and this was indeed shown to be the case by the
population synthesis calculations of Briggs et al. (2015, 2018). Temmink et
al. (2020) further investigated the impact of binary evolution on apparently
single white dwarfs and found that about a quarter of them should have formed
in a stellar merger, but that only about 3 per cent formed from a DD merger,
which is consistent with the findings of Briggs et al. (2015). Schwab (2021)
calculated evolutionary models for these white dwarf merger remnants and
showed that they should be massive and fast rotating. Overall, the products of
DD mergers should be rare.

The entire sub-class of hot, carbon-rich white dwarfs are candidate merger
products (Dunlap & Clemens, 2015). White dwarfs with carbon lines or molecular
carbon bands are generally classified as DQs (Dufour et al., 2005). The
presence of carbon in most of these stars can be explained by the dredge-up of
core carbon by a deep helium convection zone (Pelletier et al., 1986; Fontaine
& Brassard, 2005). This model explains the observed increase in the carbon
abundance with increasing temperatures up to $T_{\rm eff}\lesssim 10\,000$ K
but fails to explain the surge in carbon observed in hotter DQ white dwarfs
(Dufour et al., 2007, 2008b; Coutu et al., 2019). These hot DQ white dwarfs
display neutral and ionized carbon line spectra in a carbon dominated
atmosphere at effective temperatures ranging from $\sim 18\,000$ to $25\,000$
K (Dufour et al., 2008b). Cooler objects display a blend of neutral carbon
lines and molecular carbon bands at temperature ranging from $\sim 10\,000$ to
$16\,000$ K (Coutu et al., 2019) in a mixed carbon/helium atmosphere. Using
Gaia parallaxes, Coutu et al. (2019) suggested that these hot DQs should also
be very massive ($\gtrsim 0.8$ M⊙). Furthermore, Dunlap & Clemens (2015)
showed that these hot DQs have high transverse velocities. This means that
despite their high temperatures and high masses, characteristic of a young
white dwarf population, they would in fact belong to an old one. These
properties, along with the presence of a magnetic field and high rotation rate
(Dunlap & Clemens, 2015), strongly suggest that these objects formed through
the merger of two white dwarfs.

This population of massive carbon-polluted merger products should extend below
$\sim 10\,000$ K and could be distinguished from canonical DQ white dwarfs by
having a higher carbon abundance at a given temperature. Therefore, we are
dealing with an unusual white dwarf population that has a different origin
from the general white dwarf population. To avoid confusion, we shall refer to
the combined population of hot DQ white dwarfs and their cooler counterparts
as “C-enriched DQs”.

Strong independent evidence in support of the merger hypothesis for the origin
of the C-enriched DQs is provided by a kinematical study of the DQ white dwarf
LP 93-21. Kawka et al. (2020) showed that the total age of LP 93-21, assuming
single star evolution, is too short and inconsistent with its Galactic
kinematics which place LP 93-21 in the halo hence bolstering a merger origin
for the star.

In this paper we extend the kinematical approach to the now much larger sample
of massive DQ white dwarfs and we investigate, through a population synthesis
study, whether the stellar merger hypothesis for the origin of the C-enriched
DQs is consistent with observed population properties (age, mass).

##  2 Massive DQ white dwarfs

We have gathered all known DQ white dwarfs from several sources including
those from the recent studies by Coutu et al. (2019), Blouin & Dufour (2019)
and Koester & Kepler (2019) and we have adopted the properties (effective
temperature, surface gravity, mass and carbon abundance) from these sources.
For stars that have not been analysed using Gaia measurements, we used
available spectroscopy, photometric measurements and Gaia distance
measurements to determine their properties. If the effective temperature and
carbon abundance are known, we combined this information with the Gaia
parallax and photometry and our model photometry to estimate the surface
gravity and mass. We used the evolutionary mass-radius relations for helium
atmospheres of Benvenuto & Althaus (1999) combined with our model DQ spectra
(Kawka et al., 2020) to calculate absolute magnitudes.

From this sample we extracted white dwarfs with masses greater than 0.8 M⊙
that we consider representative of potential merger products. We checked all
massive candidates for their carbon abundance to ensure they lie in the
enhanced carbon sequence and removed those that appear to lie in the lower
carbon abundance sequence. To this sample we added two new massive DQs from
the SkyMapper survey of high proper motion white dwarfs (Vennes et al., in
preparation, see Appendix A). The spectra of these two stars were obtained
with the Focal Reducer and low dispersion Spectrograph (FORS) at the European
Southern Observatory and the Wide-Field Spectrograph (WiFeS) at Siding Spring
Observatory (SSO). These massive and C-enriched DQs are listed in Table 2 with
their atmospheric parameters and they are shown as the green circles in Fig.
1.

Figure 1: Carbon abundance measurements as a function of the effective
temperature of the known population of DQ white dwarfs. The massive C-enriched
DQs are shown in green whilst the others are shown in blue. The size of the
circles is proportional to the mass of the white dwarf.

###  2.1 Kinematics

We calculated the Galactic velocity components using the distance and proper
motion from Gaia, and, wherever possible, a measurement of the radial
velocity. For most stars we used spectra from the Sloan Digital Sky Survey
(SDSS). In a few cases we extracted archival spectra from the European
Southern Observatory (ESO) Archive and the Keck Observatory Archive: A UVES
spectrum of J0045$-$2336 obtained on 2016 Sep 4 (Programme ID 097.D-0063), a
series of PMOS (FORS2) spectra of J0106$+$1513, J0236$-$0734, and J0818$+$0102
obtained on 2012 Nov 17-18 (Programme 090.D-0536), and Low Resolution Imaging
Spectrometer (LRIS) spectrum of J0205$+$2057 obtained on 1996 Dec 12. We also
obtained a FORS2 spectrum of J2255$-$2826 on 2016 Jun as part of our ESO
Programme 097.D-0694, and spectra of J1225$+$0959 and J2140$-$3637 obtained at
Siding Spring Observatory (SSO) with the 2.3-m telescope and the Wide-Field
Spectrograph (WiFeS) on 2020 Mar 19 and 2015 Sep 25, respectively.

Interestingly, the high resolution UVES spectra of J0045-2336 revealed that
some lines show strong red asymmetry, similar to Si ii lines observed in the
heavily polluted GALEX J193156.8+011745 (Vennes et al., 2011). At low
resolution, these asymmetric profiles appear as redshifted profiles as noted
in the analysis presented by Kawka et al. (2020). In the high electronic
density of massive DQs, the carbon line positions are altered by the Stark
effect which induces large radial velocity shifts (see a discussion in Kawka
et al., 2020). In cooler objects dominated by neutral carbon, the extent of
this shift ($d_{e}$) depends on the electronic density in the atmosphere and,
given the range of temperatures and masses among these objects, the Stark
shift was estimated at $d_{e}\approx-15$ km s-1 for measurements based on the
spectral line C i$\,\lambda 5181.83$Å which is only minimally Stark shifted.
In hotter objects, dominated by singly ionized carbon, the total Stark shift
is estimated at $d_{e}+d_{\text{C\,{ii}}}\approx 45$ km s-1 for measurements
based on C ii$\,\lambda 4270$Å (Larbi-Terzi et al., 2012). These velocity
measurements were corrected for the Stark shift as well as for the
gravitational redshift. All measurements are barycentric corrected. In the
presence of a magnetic field, the central ($\pi$) component was employed for
the measurement. For seventeen stars it was not possible to measure a radial
velocity because of a low signal-to-noise ratio or a lack of publicly
available spectra. For these stars we assumed a radial velocity of zero.

The Galactic space velocity components were computed using Johnson & Soderblom
(1987) and assuming that the Solar motion relative to the local standard of
rest is ($U_{\odot},V_{\odot},W_{\odot}$) = (11.1, 12.2, 7.3) km s-1
(Schönrich et al., 2010). The resulting velocity vectors for the C-enriched
DQs are listed in Table 4. We have also derived the $z-$component of their
angular momenta, $J_{z}$, and the eccentricity of their orbit, $e$ using galpy
(Bovy, 2015). We assumed that the rotational speed of the Galactic disc is 220
km s-1 and the Sun’s distance from the Galactic centre is 8 kpc. These
kinematical properties provide an additional constraint to help distinguish
between thin disc, thick disc and halo stars (Pauli et al., 2003, 2006).

###  2.2 Space density of C-enriched DQ white dwarfs

Hollands et al. (2018) estimated a white dwarf space density of $4.49\pm
0.38\times 10^{-3}$ pc-3 using the 20 pc sample. In this sample 75 per cent
are isolated white dwarfs, that is 104 white dwarfs. Using population
syntheses Temmink et al. (2020) showed that 10 to 30 per cent of observed
single white dwarfs would have formed through binary mergers, and of these 15
per cent would have formed via DD mergers. Therefore within 20 pc, about 10 to
31 single white dwarfs would have formed in binary mergers and of these, 1 to
5 would have formed in DD mergers for a space density of 7-$35\times 10^{-5}$
pc-3 . The C-enriched white dwarfs would only be a subset of all merger
products alongside other merger candidates such as ultra-massive magnetic
white dwarfs (e.g., RE/EUVE J0317$-$855, Ferrario et al., 1997; Vennes et al.,
2003). We can estimate the space density of C-enriched DQs by bracketing its
value between a maximum value based on a complete, local sample, and a minimum
value based on a list of all known objects. There are no C-enriched DQs within
20 pc allowing us to place an upper limit to their space density at $\la
3\times 10^{-5}$ pc-3. Two C-enriched DQs (G47-18 and SMSSJ2140-3637) are
found within 40 pc. In the same volume McCleery et al. (2020) reported 1233
white dwarf candidates from the Gaia DR2 (Gentile Fusillo et al., 2019), out
of which a minimum of 521 ($\approx 40$ percent) were spectroscopically
confirmed white dwarfs, most of them in the Northern hemisphere. Allowing for
a minimum of two current C-enriched DQ identifications to be part of that
selection, for a possible total of 4 identifications within the complete 40 pc
sample, we estimate a space density of C-enriched DQs of at least
$0.75-1.5\times 10^{-5}$ pc-3, hence bracketing the C-enriched DQ space
density in the range $\rho\approx 0.8-3.0\times 10^{-5}$ pc-3. At the upper
range of space densities, C-enriched DQs could represent nearly half of all DD
mergers, or at the lower range no more than a few percent.

Figure 2: Left panel: Cumulative distance distribution function (histogram) of
C-enriched DQs identified in the SDSS with a sky coverage of 9380 degree2. The
long-dash, short-dash, and full line curves represent cumulative distributions
assuming space density in the Galactic plane of $3\times 10^{-5}$, $1.5\times
10^{-5}$, and $7.5\times 10^{-6}$ pc-3, respectively, assuming a Galactic
scale-height of 250 pc and a fractional sky coverage of 0.227 (9380 degree2).
The full red line curve represents a cumulative distribution at
$\rho_{0}=7.5\times 10^{-6}$ pc-3 and $Z_{0}=350$ pc. Right panel: The
height,$|Z|$, above the plane of all known massive C-enriched DQs.

Fig. 2 shows the cumulative distance distribution of C-enriched DQs in the
SDSS. Most of these objects were identified in the SDSS which has a current
(DR17) sky coverage of 23 percent (9380 degree2), and only five objects out of
63 are not included in the SDSS. The observed SDSS cumulative distribution is
compared to calculated distance distributions $N_{<d}$ assuming a space
density in the Galactic plane of $\rho_{0}=3.0$, $1.5$, and $0.75\times
10^{-5}$ pc-3, a scale-height $Z_{0}=250$ pc, and a fractional sky coverage
$\Phi=0.227$:

$N_{<d}=\Phi\,4\pi\rho_{0}Z_{0}^{3}\Big{[}\frac{1}{2}\Big{(}\frac{d}{Z_{0}}\Big{)}^{2}+\Big{(}\frac{d}{Z_{0}}+1\Big{)}e^{-d/Z_{0}}-1\Big{]}$
(1)

The SDSS distribution appears to follow the calculated distribution at
$\rho_{0}=1.5\times 10^{-5}$ pc-3 up to a distance of 80 pc. Assuming that all
C-enriched DQs were correctly identified over the whole sky and within a
distance of 40 pc, i.e., $\rho_{0}=1.5\times 10^{-5}$ pc-3, we find that only
one fifth of C-enriched DQs have been identified within a distance of 100 pc
leaving $\approx 40$ bright objects to be discovered, mostly in the Southern
hemisphere and outside the SDSS footprint. Gentile Fusillo et al. (2019)
showed that the white dwarf space density decreases with distance using the
100 pc sample of white dwarfs that were selected from the Gaia 2nd Data
Release and that this decrease can be modelled with a scale height of 230 pc.
Based on the 2DF sample of distant blue white dwarfs Vennes et al. (2002)
showed that the scale height can be as high as 350 pc thereby minimizing the
effect of Galactic scale-height of the expected number count of C-enriched
DQs. Application of Equation (1) at $Z_{0}=350$ pc shows that Galactic scale-
heights in the 230-350 pc range have a negligible effect on the number count
in local surveys ($d\la 100$ pc). We calculated the height above the Galactic
plane for all the massive C-enriched DQs and found that they drop off at
higher scale heights most likely because of the SDSS survey limit. The
majority (87 percent) of stars are found below Z=200 pc (Fig. 2).

##  3 Origin of C-enriched DQ white dwarfs

The characteristics of the C-enriched DQs are different from those of ordinary
white dwarfs as summarised below.

  1. 1.

Their optical spectra show notable absorption lines of C i at 4270.2, 4933.4,
5053.6, 5181.8, 5381.8Å and/or C ii at 4267, 4300, 4370, 4860, 6578, and 6583Å
and weak He i lines. Weak Swan bands are detected at the cooler end of the
population ($\approx 10\,000$K). The detection of hydrogen in C-enriched DQ
white dwarfs is relatively rare, i.e., in less than $\approx 20$ percent of
the population (Dufour et al., 2007, 2008a; Coutu et al., 2019), such as in
the two massive DQ white dwarfs G35-26 (Thejll et al., 1990) and G227-5
(Wegner & Koester, 1985). An upper limit of $\log{\rm(H/C})\approx-2.7$ is
otherwise achieved among the hottest objects from the absence of hydrogen
lines in the spectra (Dufour et al., 2008a). Upper limits to the hydrogen
abundance in cooler objects have yet to be determined relative to the dominant
species, helium or carbon (Coutu et al., 2019).

  2. 2.

Their masses are substantially higher. The average mass of H-rich white dwarfs
with effective temperatures below 60 000 K is $0.540$ M⊙ while that of He-rich
white dwarfs below 50 000 K is $0.575$ M⊙ (Bédard et al., 2020). The average
mass of the C-enriched DQs is $\langle
M\rangle=1.027\,M_{\odot},\sigma_{M}=0.116\,M_{\odot}$ (see section 6), thus
substantially larger. In fact, their average mass surpasses even that of the
magnetic white dwarfs ($\langle M\rangle=0.87\,M_{\odot},\
\sigma_{M}=0.22\,M_{\odot}$; Kawka, 2020).

This suggests that either the main sequence progenitors of the C-enriched DQs
were substantially more massive or that they are the result of mergers.

  3. 3.

About 70 percent of hot DQs are magnetic (Dufour et al., 2013; Dunlap &
Clemens, 2015) exhibiting field strengths from 0.3 MG up to 18 MG but with the
majority of fields between 1 and 4 MG. This is a much higher percentage than
among the general white dwarf population (14 - 20 percent in volume-limited
samples, Kawka et al., 2007). This is again a strong indicator that stellar
merger occurred and was responsible for the generation of their magnetic
fields (Tout et al., 2008; García-Berro et al., 2012; Wickramasinghe et al.,
2014; Briggs et al., 2015).

  4. 4.

The kinematics characteristics of the C-enriched DQs indicate that they belong
to an older population (see Section 5 and Dunlap & Clemens, 2015). Thus these
C-enriched DQs are highly unlikely to be the progenies of single massive stars
since white dwarfs with masses of $0.8-1.2$ M⊙ are the descendants of main
sequence stars of $3.5-8$ M⊙ (e.g., Romero et al., 2015) whose evolution to
the compact phase is very fast ($\leq 0.5$ Gyrs). Such warm/hot white dwarfs
would be confined to the thin Galactic disd. If, instead, they evolved from
single stars and became white dwarfs billions of years ago, they would be very
cool and dim, unlike the population of the currently observed C-enriched DQs.
Therefore, it is much more likely that these DQs are the outcome of merging
events many of which occurred with long delay times from the formation of the
progenitor binary.

The points highlighted above strongly support a merger hypothesis for the
C-enriched DQ white dwarfs (Dunlap & Clemens, 2015). However, not all mergers
are expected to produce C-enriched DQ white dwarfs. The studies of Briggs et
al. (2015) have shown that high field magnetic white dwarfs are the result of
merger events during common envelope evolution with the major contributors
coming from low-mass main-sequence stars merging with the degenerate cores of
red giant branch (RGB) and asymptotic giant branch (AGB) stars. The DD mergers
only represented a very small fraction of the population of magnetic white
dwarfs and populated the high-mass tail of the magnetic white dwarf mass
distribution. The latter are the most likely progenitor’s candidates of the
C-enriched DQs. A second possible channel may consist in the merger of a white
dwarf with a naked helium star or in the merger of two naked helium stars. The
common property that characterises these two channels is the absence of
hydrogen in their envelopes.

##  4 Population synthesis calculations

Figure 3: Star formation rate (SFR) of the Galactic disc (dashed line) and
bulge (dotted line) with the total SFR shown as a solid curve. The disc’s SFR
has been used in the present studies. These SFRs yield the observed stellar
masses and age distributions of the given components. The SFR reached a
maximum about 12.5 Gyr ago but it is only the Galactic disc that has continued
to contribute to star formation up to the present day.

In order to test the stellar merger hypothesis as an explanation for the
origin of the C-enriched DQs, we have used the rapid Binary Stellar Evolution
(BSE) code of Hurley et al. (2002) which includes the updates of Kiel & Hurley
(2006). Common envelope (CE) evolution, as first proposed by (Paczynski,
1976), is necessary to explain compact binaries whose size is smaller than the
initial radius of the primary star. The outcome of CE evolution is either a
merged object or a close binary. Since the processes that govern the CE phase
are still not fully understood and the ejection of the envelope may or may not
be complete, two parameters are usually introduced. The first is the CE
efficiency parameter $\alpha_{\rm CE}$ which was introduced to parametrise the
efficiency of the injection of orbital energy into the envelope (Livio &
Soker, 1988) and the other is $\lambda_{\rm b}$ which depends on the structure
of the donor star and on how tightly bound to the core the envelope is. Here
we use the BSE option whereby $\lambda_{\rm b}$ is calculated from the
detailed stellar evolution models of Pols et al. (1995) obtained with the
Cambridge STARS code (Eggleton, 1971). An extensive explanation of the nature
of this parameter and the range of values it can attain are in Loveridge et
al. (2011).

Since the envelope clearance efficiency is low at small $\alpha_{\rm CE}$’s,
the envelope has a longer time to exert a drag on the orbit and consequently
the number of stars that merge during CE increases as $\alpha_{\rm CE}$
decreases. Most of these merger events occur during the RGB or AGB phases of
the primary star. Those systems that do not coalesce emerge from CE evolution
at smaller orbital separations. Ivanova et al. (2013) have shown that the
post-CE orbital separation is directly proportional to $\alpha_{\rm CE}$.
Ruiter et al. (2011) found that in order to obtain a number of events that is
consistent with the predicted rate of SNe Ia from the DD merger channel one
has to assume complete CE efficiency, that is, $\alpha_{\rm CE}\lambda_{\rm
b}=1$. Similarly, Ruiter et al. (2019) investigated the various pathways to
neutron star formation via the accretion induced collapse (AIC) of oxygen-neon
white dwarfs in interacting binaries or via merger induced collapse. They
explored their results using two different approaches for CE evolution. In one
they have $\alpha_{\rm CE}\lambda_{\rm b}=1$. In the other they keep
$\alpha_{\rm CE}=1$ but with the donor binding energy parameter based on the
stellar evolution calculations of Xu & Li (2010) and on the evolutionary state
of the donor at the onset of CE evolution. They find that the AIC birthrates
are similar in both cases. In the present work, we also require $\alpha_{\rm
CE}=1$, since low $\alpha_{\rm CE}$’s yield too many merged objects with
$M\lesssim 0.95$. Because of our choice of $\lambda_{b}$, our simulations are
closer to those labelled Model 2 in Ruiter et al. (2019).

The masses of the stars, $M_{1}$ for the primary and $M_{2}$ for the
secondary, are assigned values in the range $0.8-10$ M⊙ while the initial
orbital period at the ZAMS, $P_{\rm i}$, varies in the range $10-10\,000$
days. The masses of the primary are randomly chosen according to Kroupa (2001)
mass function and those of the secondary stars according to a flat mass
distribution with $q=M_{2}/M_{1}\leq 1$ (e.g. Ferrario, 2012). The initial
period distribution is assumed to be uniform in the logarithm (Kouwenhoven et
al., 2007). The metallicity is near solar taking into consideration that we
may slightly underestimate the number of merger events since their rates are
higher at lower metallicity due to lower wind-mass-loss rates (Côté et al.,
2018).

In our population synthesis calculations we have followed the evolution of
$10^{7}$ binaries up to an age $t_{Gal}=13$ Gyrs. From this evolved population
we then extracted all single white dwarfs that were the result of either the
merger of two white dwarfs or of a white dwarf with a naked helium star or two
naked helium stars. All such mergers yield white dwarfs with no hydrogen in
their atmospheres. The evolutionary path that leads to these events requires
one or two common envelope phases. If the two stars do not coalesce during
common envelope evolution, they both evolve to the compact star stage and form
a close binary system consisting of two white dwarfs. Because the merger of
two white dwarfs is driven by gravitational wave radiation, their merging can
be delayed substantially (see section 6). To gain more familiarity with these
complex processes, we refer the reader to the thorough review of Ivanova et
al. (2013) on binary evolution, on the role that common envelope evolution
plays in bringing stars together, and on possible mergers or explosions.

Our population synthesis calculations correspond to a single starburst (one
generation of stars). In order to model the currently observed C-enriched DQ
population, which is the result of binaries that were born at different
Galactic times, we have assigned various birth epochs to the starburst, in
agreement with the Galactic disc star formation history (SFH) of Crocker et
al. (2017) and shown in Fig 3. Briefly, this SFH is given by

$\log_{10}[SFR+D]={\rm max}[Az^{2}+Bz+C,0],$ (2)

where $z$ is the cosmological redshift as first proposed by van Dokkum et al.
(2013) and Snaith et al. (2014). This form was then renormalized by Crocker et
al. (2017) so that the integrated stellar mass of the disc is $(3.7\pm
0.5)\times 10^{10}$ M⊙, in agreement with Bland-Hawthorn & Gerhard (2016). In
this context we would like to remark that the star formation history that we
have adopted does not take into account a possible inside-out assemblage
history of our Galaxy (Xiang et al., 2018) and that the effect on the present
day merger population remains to be investigated.

This SFH allowed us to scale up the progenitors of the DQs to a number (and
thus mass) that makes this subgroup of binaries consistent with the total mass
of the Galactic disc. The method consisted in producing many generations of
DQs whose progenitor binaries were born at times that were randomly sampled
from the SFH of equation (2) under the simplifying assumption that there is an
equal number of single stars as number of binaries. Each of our merged white
dwarf was then assigned a location in the Galaxy in the cylindrical coordinate
system ($R$, $\Phi$, $z$) with origin in the Galactic centre. The distribution
of stars in the $R$ and $z$ directions were taken to follow exponential laws
(e.g. van der Kruit & Searle, 1982, and references therein) with radial and
vertical scale-lengths of $r_{0}=2.6\pm 0.5$ kpc and $z_{0}=800\pm 180$ pc
respectively at $R=8$ kpc (distance of the Sun from the Galactic centre Bland-
Hawthorn & Gerhard, 2016). A multiplicative factor $(t/t_{Gal})^{1/2}$, where
$t$ is the total age of the star (from the birth of the binary on the main
sequence to the present time, Eggleton et al., 1989), has been applied to the
distribution in the $z$ direction. This was done to take into consideration
the vertical age gradient in the Milky Way disc. Effective temperature and
magnitude were assigned to each synthetic object using the tables available at
http://www.astro.umontreal.ca/~bergeron/CoolingModels (Bergeron et al., 2011;
Bédard et al., 2020).

We have limited our population analysis to objects that have a Gaia
G-magnitude, $G$, less than 20. This choice was determined by the study of
Boubert & Everall (2020) on the completeness of Gaia DR2. More specifically,
these authors found that over $3<G<20$, Gaia is essentially complete and falls
from 100 to 0 per cent over $20.0<G<21.5$. Thus, all our model data represent
a magnitude-limited sample of mergers with $G\leq 20$.

The 2nd data release of Gaia revealed an enhancement of massive white dwarfs
in a narrow temperature range (Q-branch; Gaia Collaboration et al., 2018) that
cannot be explained with the current cooling models. Cheng et al. (2019)
showed that this enhancement could be explained by a delay in the cooling of
massive ($>1.08$ M⊙) white dwarfs by 22Ne settling in C/O white dwarfs. This
delay could potentially increase the cooling age of the cooler white dwarfs by
8 Gyr. Recently, it was shown that this cooling delay can only occur in white
dwarfs with C/O cores and not in O/Ne core white dwarfs because
crystallization occurs much earlier in the evolution of O/Ne white dwarfs as
compared to C/O core white dwarfs (Camisassa et al., 2021; Blouin et al.,
2021). Schwab (2021) showed the massive white dwarfs that form from the merger
of two C/O white dwarfs end up being O/Ne white dwarfs, and therefore these
would not experience this additional delay. The properties of the C-enriched
DQs indicate that they may fall into this category.

Before examining the implications of the population syntheses, we first
establish the kinematic properties of the population, and in particular its
age distribution.

##  5 Galactic orbits and stellar populations

Figure 4: Top panel: $\sqrt{U^{2}+W^{2}}$ versus $V$ diagram of the C-enriched
DQs. The dotted and short-dashed curves correspond to
$v_{t}=(U^{2}+V^{2}+W^{2})^{1/2}$ km s-1 at $70$, and $180$ km s-1,
respectively, and with respect to the local standard of rest. Bottom panel:
Regions $A$, $B$ and $C$ denote thin disc, thick disc, and halo C-enriched
DQs.

We now examine the kinematical properties of the observed sample of C-enriched
DQs to establish to which Galactic population they belong. We plot in the top
panel of Fig. 4 the Galactic space velocity components as $\sqrt{U^{2}+W^{2}}$
versus $V$, where $U$ is positive in the direction of the Galactic centre, $V$
is positive in the direction of the Galactic rotation and $W$ is positive
toward the North Galactic pole. The velocities are relative to the local
standard of rest.

To a first approximation, stars with a total velocity
$v_{t}=(U^{2}+V^{2}+W^{2})^{1/2}\lesssim 70$ km s-1 belong to the thin disc,
stars with $70\lesssim v_{t}\lesssim 180$ km s-1 belong to the thick disc
(Venn et al., 2004), while stars with $v_{t}\gtrsim 180$ km s-1 are likely to
be halo objects. There is a likely overlap of thin and thick disc white dwarfs
between about $50$ and $70$ km s-1.

We present the plot of $J_{z}$ against $e$ in the bottom panel of Fig. 4.
According to Pauli et al. (2006) thin disc stars occupy region $A$ which is
characterised by low eccentricities and $J_{z}$ in the range $1\,600-2\,000$
kpc km s-1. In region $B$, stars have larger eccentricities and lower $J_{z}$
and are likely to belong to the thick disc population. Region $C$ is generally
populated by halo stars. We can see that the location of the C-enriched DQs in
the $J_{z}$ against eccentricity plot is consistent with that of the
$\sqrt{U^{2}+W^{2}}$ versus $V$ diagram. In particular, we note that there are
two DQs that belong to the Galactic halo, one of which is LP93-21 (Kawka et
al., 2020) which is on a retro-grade orbit and the other, SDSS J0918+4843,
appears to have a very eccentric orbit with near-zero $J_{z}$. The two other
halo candidates identified in the $\sqrt{U^{2}+W^{2}}$ versus $V$ diagram fall
in the thick-disc region in the $J_{z}$ versus $e$ diagram, however they have
a much higher $J_{z}$ than the other thick-disc candidates. These are likely
halo stars since they also have a maximum vertical amplitude $z_{\rm max}>1.5$
kpc (Martin et al., 2017).

Figure 5: Same as Fig. 4, however here we compare the kinematics of white
dwarfs for which we measured a radial velocity (red points) to the same white
dwarfs assuming, instead, a zero radial velocity (blue points).

We have found that about half of the observed sample of DQs have kinematic
properties that are consistent with those attributed to the Galactic thick
disc or halo. However this a lower limit because for about one quarter of the
sample we had to assume radial velocities of $0$ km s-1 (see section 2.1).
Pauli et al. (2003) suggest that as much as 23 per cent of thick disc white
dwarfs can be misclassified as thin disc when assuming a radial velocity of
$0$ km s-1. We revisit this problem by recalculating the kinematics of the
sample for which we have radial velocity measurements and assuming a zero
velocity instead. Fig. 5 shows the shifts in the Galactic velocity components
$\sqrt{U^{2}+W^{2}}$ versus $V$ and $J_{z}$ versus $e$ which confirms that the
inclusion of the radial velocity measurements increases the kinematical age,
i.e., it pushes some thin disc stars to the thick disc and thick disc stars to
the halo.

We calculated kinematics for all known C-enriched DQs using Gaia parallaxes
and proper motions, and radial velocities for about two thirds of the sample.
Radial velocities for the remaining third should be acquired to confirm the
population kinematics.

##  6 Discussion and conclusions

Figure 6: The distribution of delay times of the simulated types of mergers
within 1 kpc and for one starburst.

We extracted two samples from the population synthesis calculations. The first
contains white dwarfs that formed via DD mergers. The second consists of He
star mergers. We show in Fig. 6 the type of merger versus delay time, noting
that these are simulated data that only pertain to a single starburst. It is
obvious that white dwarf-white dwarf progenitors have the longest delay times
ranging from a few hundred Myrs to a Hubble time. The delay times of the
second sample, instead, are shorter and mostly confined to below a couple of
Gyrs. After performing integration over time (see section 4), we applied the
following selection criteria to compare theory to observations. We took all
relevant merger products, the DD mergers and He star mergers, within a
distance of 200 pc which is the distance encompassing the majority of known
C-enriched DQs (see Fig. 2). We also assembled the mergers products brighter
than (Gaia magnitude) $G<20$ noting that neither observed sample following
such criteria is complete since a number of very dim objects would escape
detection even within a distance of 200 pc. From these selections we extracted
the number of objects with a temperature in the range $8\,000\leq T_{\rm
eff}\leq 25\,000$ K, and, separately, in the range $T_{\rm eff}\geq 25\,000$
K. Table 1 shows the average and dispersion of the age and mass distributions
for the two different samples as well as the number of objects selected under
these criteria. Although the simulated mass distributions appear similar both
in their average and dispersion, the projected age distribution of DD mergers
corresponds to a much older population than that of He star mergers. This is
not surprising since it is entirely consistent with the simulated delay time
data portrayed in Fig. 6.

Table 1: Statistics of synthetic and observed populations.

|  | DD |  | He-star
---|---|---|---|---
| $T_{\rm eff}$ | age/$\sigma_{a}$ | $M/\sigma_{M}$ | $N$ |  | age/$\sigma_{a}$ | $M/\sigma_{M}$ | $N$
| ($10^{3}$ K) | (Gyr) | ($M_{\odot}$) |  |  | (Gyr) | ($M_{\odot}$) |
$d<200$ pc | $8-25$ | 6.45/3.43 | 1.131/0.108 | 150 |  | 2.36/1.20 | 1.108/0.100 | 291
$d<200$ pc | $25-\ \ \ $ | 4.55/3.99 | 1.217/0.109 | 10 |  | 0.70/0.44 | 1.205/0.132 | 26
$G<20$ | $8-25$ | 5.80/3.38 | 1.098/0.098 | 277 |  | 1.84/0.99 | 1.088/0.097 | 512
$G<20$ | $25-\ \ \ $ | 5.75/3.79 | 1.159/0.124 | 126 |  | 0.63/0.31 | 1.156/0.108 | 172

| $T_{\rm eff}$ | age/$\sigma_{a}$ | $M/\sigma_{M}$ | $N$
---|---|---|---|---
| ($10^{3}$ K) | (Gyr) | ($M_{\odot}$) |
observed | $8-25$ | 8.51/2.14 | 1.026/0.116 | 63

Fig. 7 (top panel) shows the observed mass distribution of the C-enriched DQs.
The mean average of the sample is $1.026$ M⊙ with a dispersion of $0.116$ M⊙.
This sample is compared to the mass distribution from the DD merger and He
star merger population syntheses. The DD mergers produce more massive white
dwarfs, whereas He star mergers can produce a higher fraction of lower mass
white dwarfs when compared to DD mergers. Both simulated mass distributions
peak at slightly higher mass than the observed distribution with relatively
fewer objects than observed at the low end ($0.8\,M_{\odot}$). The mass
distribution of DD mergers simulation peaks at a slightly lower mass than that
of He star mergers. The He star channel appears to form twice as many objects
as the DD channel in the $d<200$ pc sample.

Figure 7: Observed mass distribution of massive and C-enriched DQs (top)
compared to the mass distribution derived from the $d<200$ pc population
synthesis (bottom). The red and blue histograms in the bottom panel show the
mass distributions from DD mergers and He star mergers, respectively. Figure
8: The top panel shows the measured cooling age (+200 Myr) distribution (blue)
and eccentricity-determined age distribution (green) of C-enriched DQs. The
latter has been smoothed with a 4 Gyr boxcar function to simulate the
intrinsic dispersion in the Age-eccentricity relation. The middle panel shows
the simulated age distributions for the DD mergers and the bottom panel shows
the simulated age distributions for the He star mergers. The simulated data
apply to the $G<20$ synthetic sample.

Our population synthesis study does not provide information on space velocity,
however we can use the observed kinematics, such as the eccentricity of the
Galactic orbit, to estimate a total age for each white dwarf. Kordopatis et
al. (2011) determined the average eccentricity for the thin disc, thick disc
and halo, and showed that the eccentricity increases with age. We assigned an
age of 8 Gyr for the thin disc, an age of 10 Gyr for the thick disc (Sharma et
al., 2019) and an age of 11 Gyr for the halo (Kilic et al., 2019). We fitted a
function (Age $=13.5-2.05e^{-0.5}$, where $e$ is the eccentricity of the
Galactic orbit) to these three points that we can apply to the C-enriched DQ
white dwarf sample. Fig. 8 (top panel) compares ages of the observed
C-enriched DQs assuming single star evolution (blue histogram) to the age
determined from the eccentricity (green histogram). The age assuming single
star evolution includes the cooling age of the white dwarf that is based on
its mass and temperature added to an average pre-white dwarf lifetime of 200
Myr. In the lower two panels we show the synthetic populations ($G<20$) that
emerged from DD mergers (middle panel) and He star mergers (bottom panel). The
age distribution of objects emerging from the DD channel is in qualitative
agreement with the kinematic age of the C-enriched DQs, although our
simulations suggest that there should be a larger number of C-enriched DQs
than currently observed. As mentioned earlier, the C-enriched white dwarfs
would only be a subset of all merger products alongside other merger
candidates such as ultra-massive magnetic white dwarfs. It demonstrates that
the much longer lifespan of C-enriched DQs relative to their apparent
(cooling) age is the result of binary evolution and interaction in the form of
DD mergers. The age of the He star channel products is much lower and
corresponds to a thin Galactic disc population rather than the observed thick
disc or halo populations.

Results obtained from the $G<20$ sample in the population synthesis show a
large excess of He star merger products relative to the DD merger products.
Again the age distribution of these objects does not match the observed
distribution.

Neither observed samples, $d<200$ pc and $G<20$, are complete but the
simulations show that many more objects should be identifiable at current and
future survey limits and that He star mergers should dominate in the thin
Galactic disc. However, only DD merger products take the appearance of the
kinematically old population of C-enriched DQs.

Because our population synthesis study does not provide information on space
velocity we have not addressed the excess of C-enriched DQs with a transverse
velocity $>70$ km s-1 in the Q-branch region of the H-R diagram that was
attributed by Cheng et al. (2019) to some additional cooling delay mechanisms.
However, we have concluded that the products of mergers are likely to produce
O/Ne core white dwarfs which do not experience additional cooling since
crystallization occurs at higher temperatures than those of white dwarfs on
the Q-branch. Therefore, if white dwarfs produced from mergers do experience
additional cooling as they pass through the Q-branch, it cannot be through
22Ne settling and another mechanism is required.

We know that one of the evolutionary channels leading to Type Ia SNe, used as
standard candles to measure cosmological distances, consists in the merger of
two white dwarfs. We can therefore state that these massive C-enriched DQs are
failed Type Ia SNe, as first noted by Dunlap & Clemens (2015). We have also
shown that these mergers are rare events and that only a few C-enriched DQs
are observed with an estimated space density that is between 0.2 to 0.7
percent of the local space density of white dwarfs. Nonetheless, they may
constitute from a few to about $50$ percent of all DD merger products.

Our study shows that this population of white dwarfs is old, with nearly half
of the observed objects having kinematic properties consistent with those of
stars belonging to the Galactic thick disc and halo. Our population synthesis
results largely support these findings and are compatible with a population of
white dwarfs descending from the merger of two white dwarfs. We found that the
merger of stars whose envelope was stripped of hydrogen during common envelope
evolution (He star mergers) would leave remnants much younger than actually
observed. Note that the simulated distributions were not actually fitted to
observed distributions; the predicted and observed population numbers may
differ by more than a factor of two but are generally of the same order of
magnitude. The population synthesis predicts a large number of very hot white
dwarfs ($25\,000<T_{\rm eff}\lesssim 110\,000$) that resulted from DD mergers.
They would represent 15 percent of DD mergers in a volume-limited survey
($d<200$ pc), and up to 75 percent in a magnitude-limited survey ($G<20$).
These white dwarfs will most likely have carbon rich atmospheres not unlike
the hottest known objects in the C-enriched population. Few such objects are
known, but the ultra-hot, massive DZQ H1504+65 and cooler siblings (Werner &
Rauch, 2015) which show a mixed carbon-oxygen atmosphere are emerging as
possible candidates. The observed trend in carbon abundance with temperature
in these likely merger products remains to be explained.

To conclude, we note that binary white dwarfs are sources of low-frequency
gravitational waves. Therefore, some of the progenitors of these merging
binaries will be detectable with the space-based gravitational wave
observatory LISA, which is an European Space Agency-led mission, scheduled to
launch in the early 2030’s. Whilst most binary white dwarfs are invisible in
the electromagnetic spectrum, LISA will be able to ‘hear’ thousands of them
millions of years before they merge.

## Acknowledgements

LF and SV would like to express their gratitude for the hospitality of the
staff at the International Centre for Radio Astronomy Research. We thank D.T.
Wickramasinghe for useful discussions. This study is partly based on
observations made with ESO telescope at the La Silla Paranal Observatory under
programmes 097.D-0694 and 097.D-0063 and 090.D-0536. We thank M.S. Bessell for
sharing with us the spectrum of J2140$-$3637\. Funding for the SDSS IV has
been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy
Office of Science, and the Participating Institutions. SDSS-IV acknowledges
support and resources from the Center for High-Performance Computing at the
University of Utah. The SDSS web site is www.sdss.org. This work presents
results from the European Space Agency (ESA) space mission Gaia. Gaia data are
being processed by the Gaia Data Processing and Analysis Consortium (DPAC).
Funding for the DPAC is provided by national institutions, in particular the
institutions participating in the Gaia Multi-Lateral Agreement (MLA). The Gaia
mission website is https://www.cosmos.esa.int/gaia. The Gaia archive website
is https://archives.esac.esa.int/gaia.

## Data Availability

The SDSS spectra are available publicly from the SDSS Archive
(https://www.sdss.org/). The FORS2 and UVES spectra are from the author (AK).

## References

  * Bédard et al. (2020) Bédard A., Bergeron P., Brassard P., Fontaine G., 2020, ApJ, 901, 93
  * Benvenuto & Althaus (1999) Benvenuto O. G., Althaus L. G., 1999, MNRAS, 303, 30
  * Bergeron et al. (2011) Bergeron P., et al., 2011, ApJ, 737, 28
  * Bland-Hawthorn & Gerhard (2016) Bland-Hawthorn J., Gerhard O., 2016, ARA&A, 54, 529
  * Blouin & Dufour (2019) Blouin S., Dufour P., 2019, MNRAS, 490, 4166
  * Blouin et al. (2021) Blouin S., Daligault J., Saumon D., 2021, ApJ, 911, L5
  * Boubert & Everall (2020) Boubert D., Everall A., 2020, MNRAS, 497, 4246
  * Bovy (2015) Bovy J., 2015, ApJS, 216, 29
  * Briggs et al. (2015) Briggs G. P., Ferrario L., Tout C. A., Wickramasinghe D. T., Hurley J. R., 2015, MNRAS, 447, 1713
  * Briggs et al. (2018) Briggs G. P., Ferrario L., Tout C. A., Wickramasinghe D. T., 2018, MNRAS, 478, 899
  * Camisassa et al. (2021) Camisassa M. E., Althaus L. G., Torres S., Córsico A. H., Rebassa-Mansergas A., Tremblay P.-E., Cheng S., Raddi R., 2021, A&A, 649, L7
  * Cheng et al. (2019) Cheng S., Cummings J. D., Ménard B., 2019, ApJ, 886, 100
  * Côté et al. (2018) Côté B., Denissenkov P., Herwig F., Ruiter A. J., Ritter C., Pignatari M., Belczynski K., 2018, ApJ, 854, 105
  * Coutu et al. (2019) Coutu S., Dufour P., Bergeron P., Blouin S., Loranger E., Allard N. F., Dunlap B. H., 2019, ApJ, 885, 74
  * Crocker et al. (2017) Crocker R. M., et al., 2017, Nature Astronomy, 1, 0135
  * de Martino et al. (2007) de Martino D., Koester D., Treves A., Sbarufatti B., Falomo R., 2007, in Napiwotzki R., Burleigh M. R., eds, Astronomical Society of the Pacific Conference Series Vol. 372, 15th European Workshop on White Dwarfs. p. 273
  * Duchêne & Kraus (2013) Duchêne G., Kraus A., 2013, ARA&A, 51, 269
  * Dufour et al. (2005) Dufour P., Bergeron P., Fontaine G., 2005, ApJ, 627, 404
  * Dufour et al. (2007) Dufour P., Liebert J., Fontaine G., Behara N., 2007, Nature, 450, 522
  * Dufour et al. (2008a) Dufour P., Fontaine G., Liebert J., Schmidt G. D., Behara N., 2008a, ApJ, 683, 978
  * Dufour et al. (2008b) Dufour P., Fontaine G., Liebert J., Williams K., Lai D. K., 2008b, ApJ, 683, L167
  * Dufour et al. (2013) Dufour P., Vornanen T., Bergeron P., Fontaine G., Berdyugin A., 2013, in Krzesiń ski J., Stachowski G., Moskalik P., Bajan K., eds, Astronomical Society of the Pacific Conference Series Vol. 469, 18th European White Dwarf Workshop.. pp 167–172
  * Dunlap & Clemens (2015) Dunlap B. H., Clemens J. C., 2015, in Dufour P., Bergeron P., Fontaine G., eds, Astronomical Society of the Pacific Conference Series Vol. 493, 19th European Workshop on White Dwarfs. pp 573–577
  * Eggleton (1971) Eggleton P. P., 1971, MNRAS, 151, 351
  * Eggleton et al. (1989) Eggleton P. P., Fitchett M. J., Tout C. A., 1989, ApJ, 347, 998
  * Ferrario (2012) Ferrario L., 2012, MNRAS, 426, 2500
  * Ferrario et al. (1997) Ferrario L., Vennes S., Wickramasinghe D. T., Bailey J. A., Christian D. J., 1997, MNRAS, 292, 205
  * Fontaine & Brassard (2005) Fontaine G., Brassard P., 2005, in Koester D., Moehler S., eds, Astronomical Society of the Pacific Conference Series Vol. 334, 14th European Workshop on White Dwarfs. p. 49
  * Gaia Collaboration et al. (2018) Gaia Collaboration et al., 2018, A&A, 616, A10
  * García-Berro et al. (2012) García-Berro E., et al., 2012, ApJ, 749, 25
  * Gentile Fusillo et al. (2019) Gentile Fusillo N. P., et al., 2019, MNRAS, 482, 4570
  * Hardy et al. (2018) Hardy F., Dufour P., Jordan S., 2018, A New Look at Magnetic White Dwarfs, https://repositories.lib.utexas.edu/handle/2152/71592
  * Hollands et al. (2018) Hollands M. A., Tremblay P. E., Gänsicke B. T., Gentile-Fusillo N. P., Toonen S., 2018, MNRAS, 480, 3942
  * Hurley et al. (2002) Hurley J. R., Tout C. A., Pols O. R., 2002, MNRAS, 329, 897
  * Ivanova et al. (2013) Ivanova N., et al., 2013, A&ARv, 21, 59
  * Johnson & Soderblom (1987) Johnson D. R. H., Soderblom D. R., 1987, AJ, 93, 864
  * Kawka (2020) Kawka A., 2020, IAU Symposium, 357, 60
  * Kawka et al. (2007) Kawka A., Vennes S., Schmidt G. D., Wickramasinghe D. T., Koch R., 2007, ApJ, 654, 499
  * Kawka et al. (2020) Kawka A., Vennes S., Ferrario L., 2020, MNRAS, 491, L40
  * Kiel & Hurley (2006) Kiel P. D., Hurley J. R., 2006, MNRAS, 369, 1152
  * Kilic et al. (2019) Kilic M., Bergeron P., Dame K., Hambly N. C., Rowell N., Crawford C. L., 2019, MNRAS, 482, 965
  * Koester & Kepler (2019) Koester D., Kepler S. O., 2019, A&A, 628, A102
  * Koester et al. (1982) Koester D., Weidemann V., Zeidler E.-M., 1982, A&A, 116, 147
  * Kordopatis et al. (2011) Kordopatis G., et al., 2011, A&A, 535, A107
  * Kouwenhoven et al. (2007) Kouwenhoven M. B. N., Brown A. G. A., Portegies Zwart S. F., Kaper L., 2007, A&A, 474, 77
  * Kroupa (2001) Kroupa P., 2001, MNRAS, 322, 231
  * Larbi-Terzi et al. (2012) Larbi-Terzi N., Sahal-Bréchot S., Ben Nessib N., Dimitrijević M. S., 2012, MNRAS, 423, 766
  * Leggett et al. (2018) Leggett S. K., et al., 2018, ApJS, 239, 26
  * Livio & Soker (1988) Livio M., Soker N., 1988, ApJ, 329, 764
  * Loveridge et al. (2011) Loveridge A. J., van der Sluys M. V., Kalogera V., 2011, ApJ, 743, 49
  * Martin et al. (2017) Martin P., Jeffery C. S., Naslim N., Woolf V. M., 2017, MNRAS, 467, 68
  * McCleery et al. (2020) McCleery J., et al., 2020, MNRAS, 499, 1890
  * Paczynski (1976) Paczynski B., 1976, in Eggleton P., Mitton S., Whelan J., eds, IAUS Vol. 73, Structure and Evolution of Close Binary Systems. p. 75
  * Pauli et al. (2003) Pauli E. M., Napiwotzki R., Altmann M., Heber U., Odenkirchen M., Kerber F., 2003, A&A, 400, 877
  * Pauli et al. (2006) Pauli E. M., Napiwotzki R., Heber U., Altmann M., Odenkirchen M., 2006, A&A, 447, 173
  * Pelletier et al. (1986) Pelletier C., Fontaine G., Wesemael F., Michaud G., Wegner G., 1986, ApJ, 307, 242
  * Pols et al. (1995) Pols O. R., Tout C. A., Eggleton P. P., Han Z., 1995, MNRAS, 274, 964
  * Romero et al. (2015) Romero A. D., Campos F., Kepler S. O., 2015, MNRAS, 450, 3708
  * Ruiter (2020) Ruiter A. J., 2020, IAU Symposium, 357, 1
  * Ruiter et al. (2011) Ruiter A. J., Belczynski K., Sim S. A., Hillebrandt W., Fryer C. L., Fink M., Kromer M., 2011, MNRAS, 417, 408
  * Ruiter et al. (2019) Ruiter A. J., Ferrario L., Belczynski K., Seitenzahl I. R., Crocker R. M., Karakas A. I., 2019, MNRAS, 484, 698
  * Schönrich et al. (2010) Schönrich R., Binney J., Dehnen W., 2010, MNRAS, 403, 1829
  * Schwab (2021) Schwab J., 2021, ApJ, 906, 53
  * Sharma et al. (2019) Sharma S., et al., 2019, MNRAS, 490, 5335
  * Snaith et al. (2014) Snaith O. N., Haywood M., Di Matteo P., Lehnert M. D., Combes F., Katz D., Gómez A., 2014, ApJ, 781, L31
  * Temmink et al. (2020) Temmink K. D., Toonen S., Zapartas E., Justham S., Gänsicke B. T., 2020, A&A, 636, A31
  * Thejll et al. (1990) Thejll P., Shipman H. L., MacDonald J., Macfarland W. M., 1990, ApJ, 361, 197
  * Toonen et al. (2017) Toonen S., Hollands M., Gänsicke B. T., Boekholt T., 2017, A&A, 602, A16
  * Tout et al. (2008) Tout C. A., Wickramasinghe D. T., Liebert J., Ferrario L., Pringle J. E., 2008, MNRAS, 387, 897
  * van der Kruit & Searle (1982) van der Kruit P. C., Searle L., 1982, A&A, 110, 79
  * van Dokkum et al. (2013) van Dokkum P. G., et al., 2013, ApJ, 771, L35
  * Venn et al. (2004) Venn K. A., Irwin M., Shetrone M. D., Tout C. A., Hill V., Tolstoy E., 2004, AJ, 128, 1177
  * Vennes et al. (2002) Vennes S., Smith R. J., Boyle B. J., Croom S. M., Kawka A., Shanks T., Miller L., Loaring N., 2002, MNRAS, 335, 673
  * Vennes et al. (2003) Vennes S., Schmidt G. D., Ferrario L., Christian D. J., Wickramasinghe D. T., Kawka A., 2003, ApJ, 593, 1040
  * Vennes et al. (2011) Vennes S., Kawka A., Németh P., 2011, MNRAS, 413, 2545
  * Wegner & Koester (1985) Wegner G., Koester D., 1985, ApJ, 288, 746
  * Werner & Rauch (2015) Werner K., Rauch T., 2015, A&A, 584, A19
  * Wickramasinghe et al. (2014) Wickramasinghe D. T., Tout C. A., Ferrario L., 2014, MNRAS, 437, 675
  * Williams et al. (2013) Williams K. A., et al., 2013, ApJ, 769, 123
  * Xiang et al. (2018) Xiang M., et al., 2018, ApJS, 237, 33
  * Xu & Li (2010) Xu X.-J., Li X.-D., 2010, ApJ, 716, 114

##  Appendix A Two new carbon-enriched DQ white dwarfs

Figure 9: Optical spectra (in grey) and best-fit spectral syntheses (in red)
of the newly identified C-enriched DQs WD J2140$-$3637. Figure 10: Optical
spectra (in grey) and best-fit spectral syntheses (in red) of the newly
identified C-enriched DQs WD J2255$-$2836.

We included in this work two newly identified carbon-enriched white dwarfs.
The objects were identified in the SkyMapper survey of white dwarfs (Vennes et
al., in preparation). The two objects, WD J2140$-$3637 and WD J2255$-$2836,
lie along the abundance versus temperature relation (Fig. 1) at $T_{\rm
eff}=11\,800\pm 1\,000$ K and $\log{\rm C/He}\approx-2$, and $T_{\rm
eff}=16\,800\pm 1\,800$ K and $\log{\rm C/He}\approx-0.6$, respectively. The
cooler object, WD J2140$-$3637, shows C2 Swan bands and atomic carbon lines.
Also, along with approximately one in five C-enriched DQs in the Coutu et al.
(2019) sample, WD J2255$-$2836 shows contamination by hydrogen ($\log{\rm
H/He}\approx-2.3$). These objects are among new DQ identifications in the
Southern hemisphere complementing the SDSS Northern hemisphere coverage. Fig.
9 shows the SSO2.3m/WiFeS optical spectrum of WD J2140$-$3637 and Fig. 10
shows the VLT/FORS spectrum of WD J2255$-$2836, along with their best-fit
spectral syntheses. We employed mixed H/He/C convective model atmospheres. A
complete description of these two objects and models will be presented
elsewhere (Vennes et al., in preparation).

##  Appendix B Atmospheric and kinematic parameters

Table 2 lists all known DQ white dwarfs with a surface composition enriched in
carbon relative to normal DQ white dwarfs; its members have higher than
average mass and effective temperatures extending up to 25 000 K. We list the
distance calculated from the Gaia parallax and the stellar parameters
(effective temperature, surface gravity, mass, and carbon abundance) obtained
from the literature or determined in this work (see Section 2).

Table 4 lists the radial velocities measured in this work along with the
Galactic velocity components $U,V,W$, the Galactic orbital eccentricity $e$,
and the angular momentum along the Z-axis $J_{Z}$ (see Sections 2.1 and 5).

Table 2: Atmospheric parameters of C-enriched DQ white dwarfs. | Name | Distance | $T_{\rm eff}$ | $\log{g}$ | $M$ | $\log{{\rm C/He}}$ | Reference
---|---|---|---|---|---|---|---
|  | (pc) | (K) | (c.g.s.) | ($M_{\odot}$) |  |
J0005$-$1002a | SDSSJ000555.90$-$100213.3, PHL 657 | 150.7 | 19018 | 8.80 | 1.10 |  2.00 | 1
J0019$+$1847 | SDSSJ001908.63$+$184706.0 | 150.3 | 10280 | 8.55 | 0.93 |  $-$2.84 | 2
J0045$-$2336 | G268-40 | 47.2 | 10500 | 8.65 | 1.00 |  $-$2.70 | 3,4
J0106$+$1513a | SDSSJ010647.92$+$151327.8 | 372.0 | 23430 | 8.50 | 0.93 |  1.00 | 4,5
J0205$+$2057 | G35-26 | 85.4 | 16150 | 9.04 | 1.20 |  3.00 | 6
J0236$+$2503 | SDSSJ023633.74$+$250348.9 | 177.7 | 13376 | 8.70 | 1.03 |  $-$1.58 | 7
J0236$-$0734a | SDSSJ023637.42$-$073429.5 | 663.0 | 24400 | 9.07 | 1.22 |  2.00 | 4
J0243$+$0101 | SDSSJ024332.77$+$010111.1, WD0240$+$008 | 182.4 | 8225 | 8.63 | 0.99 |  $-$4.23 | 7
J0807$+$1949 | SDSSJ080708.48$+$194950.7 | 171.3 | 13501 | 8.78 | 1.08 |  $-$1.24 | 7
J0818$+$0102 | SDSSJ081839.23$+$010227.5 | 266.8 | 24483 | 8.33 | 0.81 |  2.00 | 2
J0852$+$2316 | SDSSJ085235.43$+$231644.3 | 185.9 | 11099 | 8.61 | 0.97 |  $-$3.18 | 7
J0856$+$4513 | SDSSJ085626.94$+$451336.9 | 205.9 | 9484 | 8.51 | 0.91 |  $-$3.27 | 7
J0859$+$3257 | SDSSJ085914.63$+$325712.1, G47-18 | 23.1 | 9486 | 8.45 | 0.87 |  $-$3.52 | 7
J0901$+$5751 | SDSSJ090157.93$+$575135.9, WD0858+580 | 153.8 | 13576 | 8.76 | 1.07 |  $-$1.99 | 7
J0918$+$4843 | SDSSJ091830.27$+$484323.0 | 184.4 | 9203 | 8.80 | 1.09 |  $-$3.72 | 7
J0919$+$0236 | SDSSJ091922.22$+$023604.5, WD0916$+$028 | 159.9 | 11319 | 8.61 | 0.98 |  $-$2.85 | 7
J0936$+$0607 | SDSSJ093638.07$+$060710.0 | 161.7 | 11013 | 8.61 | 0.97 |  $-$3.07 | 7
J0958$+$5853 | SDSSJ095837.00$+$585303.0 | 175.1 | 15444 | 8.95 | 1.16 |  $-$0.50 | 2
J1036$+$6522a | SDSSJ103655.38$+$652252.0, WD1033+656 | 175.3 | 15500 | 8.83 | 1.12 |  $-$1.00 | 8,4
J1040$+$0635 | SDSSJ104052.40$+$063519.7 | 283.7 | 13882 | 8.40 | 0.84 |  $-$2.08 | 7
J1045$+$5904 | SDSSJ104559.14$+$590448.2, LP93-21 | 57.7 | 9730 | 8.90 | 1.14 |  $-$2.73 | 9
J1049$+$1659 | SDSSJ104906.61$+$165923.6 | 194.1 | 12799 | 8.92 | 1.15 |  $-$1.64 | 7
J1058$+$2846 | SDSSJ105817.66$+$284609.3 | 162.6 | 9422 | 8.49 | 0.89 |  $-$3.60 | 7
J1100$+$1758 | SDSSJ110058.03$+$175806.9 | 150.9 | 12367 | 8.76 | 1.07 |  $-$1.28 | 7
J1104$+$2035a | SDSSJ110406.68$+$203528.7 | 173.4 | 23476 | 8.60 | 0.99 |  2.00 | 1
J1113$+$0146 | SDSSJ111341.33$+$014641.7 | 43.1 | 5961 | 8.71 | 1.03 |  $-$5.14 | 10
J1133$+$6331 | SDSSJ113359.94$+$633113.3, WD1131$+$637 | 194.1 | 11517 | 8.57 | 0.95 |  $-$2.68 | 7
J1140$+$0735 | SDSSJ114059.85$+$073530.1 | 160.8 | 10651 | 8.56 | 0.94 |  $-$3.36 | 7
J1140$+$1824 | SDSSJ114006.35$+$182402.3 | 94.2 | 9656 | 8.35 | 0.81 |  $-$3.54 | 7
J1148$-$0126 | SDSSJ114851.68$-$012612.7, WD1146$-$011 | 68.2 | 9680 | 8.46 | 0.88 |  $-$3.48 | 7
J1153$+$0056 | SDSSJ115305.54$+$005646.2 | 165.3 | 21650 | 9.40 | 1.39 |  2.00 | 4,5
J1200$+$2252 | SDSSJ120027.73$+$225212.9 | 378.6 | 21880 | 8.50 | 0.92 |  2.00 | 2
J1203$+$6451 | SDSSJ120331.89$+$645101.4, WD1200$+$651 | 87.2 | 12359 | 8.77 | 1.07 |  $-$1.59 | 7
J1209$+$5355 | SDSSJ120936.50$+$535525.7 | 236.0 | 11721 | 8.53 | 0.92 |  $-$2.71 | 7
J1215$+$4700 | SDSSJ121510.64$+$470011.0 | 160.6 | 13230 | 8.87 | 1.13 |  $-$2.00 | 7
J1225$+$0959 | LP495-79 | 84.4 | 11100 | 8.60 | 0.97 |  $-$2.60 | 11,4
J1328$+$5908a | SDSSJ132858.19$+$590851.0, WD1327$+$594 | 147.7 | 18755 | 9.01 | 1.19 |  3.00 | 6
J1331$+$3727 | SDSSJ133151.38$+$372754.8 | 134.9 | 16741 | 9.03 | 1.16 |  0.41 | 2
J1332$+$2355 | SDSSJ133221.56$+$235502.1 | 210.6 | 14205 | 8.70 | 1.03 |  $-$1.76 | 7
J1337$-$0026a | SDSSJ133710.19$-$002643.7 | 304.5 | 22711 | 8.66 | 1.02 |  1.50 | 1
J1339$+$5036 | SDSSJ133940.53$+$503612.8 | 182.4 | 11680 | 8.62 | 0.98 |  $-$2.20 | 2
J1341$+$0346 | SDSSJ134124.28$+$034628.7 | 196.1 | 13765 | 8.76 | 1.07 |  $-$2.18 | 7
J1400$-$0154 | SDSSJ140051.57$-$015414.4, 2QZJ140051.6$-$015413 | 142.3 | 9394 | 8.69 | 1.02 |  $-$3.56 | 7
J1402$+$3818a | SDSSJ140222.26$+$381848.9 | 320.5 | 17232 | 8.61 | 0.99 |  $-$0.50 | 1
J1426$+$5752a | SDSSJ142625.70$+$575218.4 | 306.3 | 18809 | 8.72 | 1.04 |  2.00 | 2
J1428$+$3238 | SDSSJ142812.54$+$323817.7 | 161.8 | 10718 | 8.56 | 0.94 |  $-$3.20 | 7
J1434$+$2258 | SDSSJ143437.82$+$225859.5 | 191.9 | 14575 | 8.75 | 1.06 |  $-$1.14 | 7
J1435$+$5318 | SDSSJ143534.01$+$531815.0 | 196.0 | 15167 | 8.85 | 1.12 |  $-$1.91 | 7
J1444$+$0434 | SDSSJ144407.25$+$043446.7, WD1441$+$047 | 180.5 | 9813 | 8.44 | 0.87 |  $-$3.47 | 7
J1448$+$0519 | SDSSJ144854.80$+$051903.5 | 120.3 | 15966 | 8.94 | 1.16 |  0.30 | 2
J1452$+$6020 | SDSSJ145236.57$+$602036.3, WD1451$+$605 | 224.9 | 12572 | 8.65 | 1.00 |  $-$1.73 | 7
J1455$+$4209 | SDSSJ145524.89$+$420910.8 | 266.4 | 14288 | 8.78 | 1.08 |  $-$1.48 | 7
J1542$+$4329 | SDSSJ154248.67$+$432902.4 | 184.4 | 9799 | 8.49 | 0.89 |  $-$3.61 | 7
J1555$+$3219 | SDSSJ155539.51$+$321914.1 | 189.1 | 9195 | 8.59 | 0.96 |  $-$3.74 | 7
J1615$+$4543 | SDSSJ161531.71$+$454322.4 | 455.5 | 20940 | 8.62 | 1.00 |  1.74 | 10,4
J1622$+$1849 | SDSSJ162205.12$+$184956.7 | 187.7 | 16693 | 9.13 | 1.16 |  $-$0.08 | 2
J1622$+$3004 | SDSSJ162236.13$+$300454.5 | 74.7 | 16131 | 8.93 | 1.15 |  $-$0.10 | 2
J1728$+$5558 | SDSSJ172856.19$+$555823.0, G227-5 | 47.1 | 14453 | 8.90 | 1.14 |  $-$1.37 | 7

Table 3: continued

Atmospheric parameters of massive DQ white dwarfs.  Name Distance $T_{\rm
eff}$ $\log{g}$ $M$ $\log{{\rm C/He}}$ Reference (pc) (K) (c.g.s.)
($M_{\odot}$) J2140$-$3637 SMSSJ214023.58$-$363757.5 39.8 11800 8.70 1.02
$-$2.00 4 J2200$-$0741a SDSSJ220029.09$-$074121.5 207.2 21271 8.64 1.01 2.00 1
J2250$+$1240 SDSSJ225000.22$+$124019.8 182.5 9801 8.50 0.90 $-$3.66 7
J2255$-$2836 SMSSJ225523.30$-$283649.6 153.7 16800 9.07 1.22 $-$0.60 4
J2348$-$0942 SDSSJ234843.30$-$094245.2 375.6 21550 8.54 0.93 2.00 4,5
References: (1) Hardy et al. (2018) (2) Koester & Kepler (2019) (3) Koester et
al. (1982) (4) This work (5) Dufour et al. (2008b) (6) Leggett et al. (2018)
(7) Coutu et al. (2019) (8) Williams et al. (2013) (9) Kawka et al. (2020)
(10) Blouin & Dufour (2019) (11) de Martino et al. (2007)
a Confirmed to be magnetic.

Table 4: Kinematics of C-enriched DQs. | $v_{r}$ | $U$ | $V$ | $W$ | $e$ | $J_{z}$
---|---|---|---|---|---|---
| (km s-1) | (km s-1) | (km s-1) | (km s-1) |  | (kpc km s-1)
J0005$-$1002 | $-10$ | $-49$ | $-6$ | $11$ | $0.174$ | $1648$
J0019$+$1847 | $-38$ | $109$ | $-16$ | $-6$ | $0.346$ | $1575$
J0045$-$2336 | $76$ | $-25$ | $-45$ | $-72$ | $0.194$ | $1341$
J0106$+$1513 | $228$ | $-45$ | $120$ | $-193$ | $0.631$ | $2703$
J0205$+$2057 | $-195$ | $227$ | $-79$ | $31$ | $0.720$ | $1044$
J0236$+$2503 | $-1$ | $14$ | $4$ | $0$ | $0.038$ | $1757$
J0236$-$0734 | $28$ | $20$ | $13$ | $-31$ | $0.063$ | $1876$
J0243$+$0101 | ($0$) | $29$ | $-96$ | $-19$ | $0.496$ | $938$
J0807$+$1949 | $93$ | $-41$ | $-113$ | $19$ | $0.578$ | $814$
J0818$+$0102 | $137$ | $-109$ | $-61$ | $29$ | $0.440$ | $1269$
J0852$+$2316 | $53$ | $-9$ | $-22$ | $53$ | $0.109$ | $1549$
J0856$+$4513 | ($0$) | $-15$ | $15$ | $-23$ | $0.062$ | $1852$
J0859$+$3257 | $6$ | $-17$ | $12$ | $-16$ | $0.060$ | $1796$
J0901$+$5751 | $-113$ | $126$ | $-12$ | $-30$ | $0.393$ | $1606$
J0918$+$4843 | ($0$) | $-29$ | $-208$ | $1$ | $0.989$ | $33$
J0919$+$0236 | $47$ | $68$ | $-59$ | $65$ | $0.334$ | $1221$
J0936$+$0607 | $-7$ | $5$ | $-17$ | $-36$ | $0.102$ | $1576$
J0958$+$5853 | $-24$ | $-49$ | $-16$ | $-66$ | $0.172$ | $1590$
J1036$+$6522 | $-103$ | $83$ | $-28$ | $-57$ | $0.287$ | $1479$
J1040$+$0635 | ($0$) | $-16$ | $-1$ | $-12$ | $0.057$ | $1709$
J1045$+$5904 | $15$ | $-203$ | $-423$ | $57$ | $0.614$ | $-1672$
J1049$+$1659 | $-12$ | $14$ | $29$ | $2$ | $0.105$ | $1940$
J1058$+$2846 | $26$ | $81$ | $-74$ | $50$ | $0.424$ | $1097$
J1100$+$1758 | $-18$ | $23$ | $-6$ | $-16$ | $0.098$ | $1652$
J1104$+$2035 | $36$ | $-16$ | $-19$ | $28$ | $0.125$ | $1559$
J1113$+$0146 | ($0$) | $16$ | $-57$ | $-39$ | $0.291$ | $1235$
J1133$+$6331 | ($0$) | $-11$ | $-137$ | $74$ | $0.553$ | $606$
J1140$+$0735 | $37$ | $44$ | $-17$ | $37$ | $0.173$ | $1554$
J1140$+$1824 | $67$ | $-43$ | $-45$ | $52$ | $0.251$ | $1346$
J1148$-$0126 | $69$ | $15$ | $-66$ | $40$ | $0.329$ | $1167$
J1153$+$0056 | $-90$ | $-70$ | $-19$ | $-111$ | $0.216$ | $1559$
J1200$+$2252 | ($0$) | $-38$ | $-10$ | $-3$ | $0.145$ | $1630$
J1203$+$6451 | $-58$ | $29$ | $-70$ | $-8$ | $0.380$ | $1136$
J1209$+$5355 | ($0$) | $-11$ | $-30$ | $14$ | $0.182$ | $1468$
J1215$+$4700 | $-79$ | $55$ | $-44$ | $-50$ | $0.271$ | $1344$
J1225$+$0959 | $5$ | $-44$ | $-29$ | $3$ | $0.222$ | $1466$
J1328$+$5908 | ($0$) | $-37$ | $-30$ | $19$ | $0.215$ | $1464$
J1331$+$3727 | $61$ | $19$ | $-5$ | $74$ | $0.050$ | $1655$
J1332$+$2355 | $-17$ | $17$ | $0$ | $-10$ | $0.064$ | $1685$
J1337$-$0026 | $57$ | $40$ | $8$ | $58$ | $0.129$ | $1721$
J1339$+$5036 | ($0$) | $-59$ | $35$ | $-12$ | $0.231$ | $1981$
J1341$+$0346 | $45$ | $-61$ | $-36$ | $78$ | $0.238$ | $1402$
J1400$-$0154 | ($0$) | $-8$ | $-83$ | $-6$ | $0.431$ | $1026$
J1402$+$3818 | ($0$) | $-12$ | $14$ | $9$ | $0.057$ | $1798$
J1426$+$5752 | $2$ | $2$ | $24$ | $1$ | $0.071$ | $1893$
J1428$+$3238 | $-70$ | $54$ | $-3$ | $-73$ | $0.168$ | $1655$
J1434$+$2258 | $13$ | $17$ | $-32$ | $29$ | $0.184$ | $1423$
J1435$+$5318 | $78$ | $-16$ | $60$ | $68$ | $0.266$ | $2175$
J1444$+$0434 | ($0$) | $38$ | $8$ | $-11$ | $0.118$ | $1734$
J1448$+$0519 | $-57$ | $-63$ | $-46$ | $-9$ | $0.317$ | $1328$
J1452$+$6020 | $32$ | $-59$ | $-3$ | $50$ | $0.210$ | $1673$
J1455$+$4209 | ($0$) | $24$ | $-8$ | $16$ | $0.094$ | $1620$
J1542$+$4329 | ($0$) | $53$ | $9$ | $-1$ | $0.156$ | $1759$
J1555$+$3219 | ($0$) | $-23$ | $-3$ | $35$ | $0.095$ | $1658$
J1615$+$4543 | $-27$ | $-37$ | $0$ | $-6$ | $0.152$ | $1672$
J1622$+$1849 | $-23$ | $59$ | $-39$ | $-38$ | $0.266$ | $1364$
J1622$+$3004 | $-55$ | $-8$ | $-61$ | $0$ | $0.329$ | $1201$
J1728$+$5558 | $-38$ | $-44$ | $-28$ | $6$ | $0.219$ | $1478$
J2140$-$3637 | $-3$ | $51$ | $15$ | $46$ | $0.166$ | $1799$
J2200$-$0741 | $82$ | $62$ | $64$ | $-37$ | $0.307$ | $2177$
J2250$+$1240 | $99$ | $-68$ | $49$ | $-113$ | $0.358$ | $2081$
J2255$-$2826 | $-90$ | $-14$ | $-62$ | $82$ | $0.234$ | $1193$
J2348$-$0942 | $18$ | $29$ | $4$ | $-14$ | $0.076$ | $1725$


# Free to Harmonic Unitary Transformations in Quantum and Koopman Dynamics

Gerard McCaul<EMAIL_ADDRESS>Tulane University, New Orleans, LA 70118,
USA  Denys I. Bondar  Tulane University, New Orleans, LA 70118, USA

(August 30, 2024)

###### Abstract

In the context of quantum dynamics there exists a coordinate transformation
which maps the free particle to the harmonic oscillator. Here we extend this
result by reformulating it as a unitary operation followed by a time
coordinate transformation. We demonstrate that an equivalent transformation
can be performed for classical systems in the context of Koopman-von Neumann
(KvN) dynamics. We further extend this mapping both to dissipative evolutions
as well as for a quantum-classical hybrid, and show that this mapping imparts
an identical time-dependent scaling on the dissipation parameters for both
types of dynamics. The derived classical procedure presents a number of
opportunities to import squeezing dependent quantum procedures (such as
Hamiltonian amplification) into the classical regime.

##  I Introduction

It is a little known fact that there exists an mapping between a quantum free
particle and a harmonic oscillator. This formal connection between two
distinct systems may seem surprising initially, particularly given that a free
particle has a continuous spectrum, while the oscillator’s is discrete.
Nevertheless, there exists a simple set of coordinate transforms which will
transform the wavefunction which solves for the free particle into one which
is a solution for a harmonic oscillator Solov’ev (1982); Niederer (1973). This
mapping allows one to transition between free and bound representations of
dynamics Steuernagel (2005), which can be useful for obtaining otherwise
inaccessible analytic results Steuernagel  _et al._ (2013). Moreover, being
able to map a free particle to a trapping potential avoids the unbounded
spreading that can make numerical simulations of free systems challenging.

The origin of this mapping lies in quantum optics Takagi (1990); Steuernagel
(2005), but has also found use in the analysis of freely moving waves
Andrzejewski  _et al._ (2014), inverted oscillators Barton (1986); Yuce  _et
al._ (2006), instantaneous transitions Steuernagel (2014), as well as to the
equivalence principle Sen  _et al._ (2022); Dhasmana  _et al._ (2021). The
mapping itself can be compactly stated - if one has a wave function
$\phi(x,t)$ which obeys (setting the mass $m=\hbar$=1 for convenience)

$i\frac{\partial\phi(x,t)}{\partial
t}=-\frac{1}{2}\frac{\partial^{2}\phi(x,t)}{\partial x^{2}},$ (1)

this may be brought to the form of a harmonic oscillator using an invertible
coordinate transformation given by:

$t(\tau)=\tan(\omega\tau),\ \ \ \ \ \ \
x(\xi,\tau)=\frac{\sqrt{\omega}}{\cos(\omega\tau)}\xi.$ (2)

This then leads to a harmonic type Schödinger equation:

$i\frac{\partial\psi(\xi,\tau)}{\partial\tau}=-\frac{1}{2}\frac{\partial^{2}\psi(\xi,\tau)}{\partial\xi^{2}}+\frac{1}{2}\omega^{2}\xi^{2},$
(3)

where

$\psi(\xi,\tau)=\frac{{\rm
e}^{\frac{itx^{2}}{2(t^{2}+1)}}}{(1+t^{2})^{\frac{1}{4}}}\phi(x(\xi,\tau),t(\tau)).$
(4)

This mapping may be checked by direct substitution Steuernagel (2014), and can
be extended to both an arbitrary number of spatial dimensions and a time
dependent frequency $\omega\to\omega(\tau)$ Takagi (1990).

Significantly, this mapping is quite distinct from either the manner in which
quantum harmonic Hamiltonians are diagonalised, or the canonical action-angle
transformations which convert free to harmonic systems in classical dynamics
McCaul  _et al._ (2019). The existence of a quantum coordinate transformation
of this type naturally raises the question of whether an equivalent exists in
the classical case. To investigate this question, we first ask whether the
coordinate transform for the Schrödinger equation mapping can be performed in
a representation-free manner - i.e. whether there exists a unitary operator
which is equivalent to the coordinate mapping shown above. In Sec.II, we
derive the free to harmonic mapping as consisting of a unitary transformation
followed by a time coordinate transform. An equivalent procedure is found for
classical systems in Sec.III, where an analogous unitary transforming the
Koopman operator is constructed. We further investigate the effect of this
mapping on both dissipative and quantum-classical hybrid evolutions in Sec.IV
and close with a discussion of the results in Sec.V.

##  II Quantum Case

We wish to first investigate whether the coordinate map between a free
particle and the harmonic oscillator can be expressed as a unitary
transformation, as the existence of such a unitary should provide a
transparent method to replicate the free to harmonic map in a classical
context. To find this unitary, it is worth considering first both how the
coordinates and the wavefunction are transformed. $x\to\xi$ is a time
dependent scaling, while the $\phi\to\psi$ map is strongly reminiscent of a
gauge transformation. This immediately suggests that any overall unitary
$\hat{\mathcal{U}}$ mapping the free to harmonic system will consist of the
composition of two unitary operators.

We begin by considering a scaling unitary $\hat{U}(t)$ of the form:

$\displaystyle\hat{U}(t)$ $\displaystyle={\rm
e}^{-i\frac{\ln(c(t))}{2}\left(\hat{x}\hat{p}+\hat{p}\hat{x}\right)},$ (5)
$\displaystyle\hat{U}^{\dagger}(t)\hat{x}\hat{U}(t)$
$\displaystyle=c(t)\hat{x},$ (6)
$\displaystyle\hat{U}^{\dagger}(t)\hat{p}\hat{U}(t)$
$\displaystyle=c^{-1}(t)\hat{p},$ (7) $\displaystyle
i\hat{U}^{\dagger}(t)\frac{\partial\hat{U}(t)}{\partial t}$
$\displaystyle=\frac{\dot{c}(t)}{2c(t)}\left(\hat{x}\hat{p}+\hat{p}\hat{x}\right).$
(8)

This is nothing more than the squeezing operator, common in quantum optics but
here represented in first quantisation Schleich (2001). As a time dependent
unitary transformation it will transform the system Hamiltonian as

$\displaystyle\hat{H}^{\prime}=\hat{U}^{\dagger}(t)\hat{H}\hat{U}(t)-i\hat{U}^{\dagger}(t)\frac{\partial\hat{U}(t)}{\partial
t}.$ (9)

If we initially have a free-particle system where
$\hat{H}=\frac{1}{2}\hat{p}^{2}$, then the transformed Hamiltonian
$H^{\prime}$ will be given by

$\displaystyle\hat{H}^{\prime}$
$\displaystyle=\frac{1}{2c^{2}(t)}\hat{p}^{2}-\frac{\dot{c}(t)}{2c(t)}\left(\hat{x}\hat{p}+\hat{p}\hat{x}\right)$
$\displaystyle=\frac{1}{2c^{2}(t)}\left(\hat{p}-c(t)\dot{c}(t)\hat{x}\right)^{2}-\frac{\dot{c}^{2}(t)}{2}\hat{x}^{2}.$
(10)

This scaling unitary has introduced a harmonic potential term, at the cost of
a longitudinal vector potential. This can be removed via a unitary gauge
transformation $\hat{R}(t)$:

$\displaystyle\hat{R}(t)$ $\displaystyle={\rm
e}^{i\frac{1}{2}c(t)\dot{c}(t)\hat{x}^{2}},$ (11)
$\displaystyle\hat{R}^{\dagger}(t)\hat{p}\hat{R}(t)$
$\displaystyle=\hat{p}+c(t)\dot{c}(t)\hat{x},$ (12)
$\displaystyle-i\hat{R}^{\dagger}(t)\frac{\partial\hat{R}(t)}{\partial t}$
$\displaystyle=\frac{1}{2}(\dot{c}^{2}(t)+c(t)\ddot{c}(t))\hat{x}^{2}.$ (13)

Applying this to $\hat{H}^{\prime}$, we obtain the Hamiltonian:

$\hat{H}^{\prime\prime}=\frac{1}{c^{2}(t)}\left(\frac{1}{2}\hat{p}^{2}+\frac{1}{2}\ddot{c}(t)c^{3}(t)\hat{x}^{2}\right).$
(14)

It is interesting to note that while $\hat{H}^{\prime\prime}$ is constructed
from unitaries $\hat{R}(t)$ and $\hat{U}(t)$ which clearly do not commute, it
is in fact possible to derive an expression for
$\hat{\mathcal{U}}=\hat{U}(t)\hat{R}(t)$ as a single unitary. This is
principally due to the fact that we have the commutation relation:

$\left[\hat{x}\hat{p}+\hat{p}\hat{x},\hat{x}^{2}\right]=-4i\hat{x}^{2}.$ (15)

This is relevant as the operator $\hat{Z}$ in ${\rm e}^{\hat{Z}}={\rm
e}^{\hat{X}}{\rm e}^{\hat{Y}}$ has a closed form when
$[\hat{X},\hat{Y}]=u\hat{Y}$ Van-Brunt and Visser (2015). Explicitly this is
given by

$\hat{Z}=\hat{X}+f(u)\hat{Y}$ (16)

where

$f(u)=\frac{u}{2}(1+\coth(\frac{u}{2})).$ (17)

Taking
$\hat{X}=-i\frac{\ln(c(t))}{2}\left(\hat{x}\hat{p}+\hat{p}\hat{x}\right)$ and
$\hat{Y}=i\frac{1}{2}c(t)\dot{c}(t)\hat{x}^{2}$ results in

$[\hat{X},\hat{Y}]=-2\ln(c(t))\hat{Y}.$ (18)

From which we obtain

$\hat{\mathcal{U}}={\rm
e}^{-i\frac{\ln(c(t))}{2}\left(c(t)\dot{c}(t)\left(1-\frac{c^{2}(t)+1}{c^{2}(t)-1}\right)\hat{x}^{2}+\hat{x}\hat{p}+\hat{p}\hat{x}\right)}.$
(19)

To make contact with the harmonic oscillator, it is necessary to specify the
form of $c(t)$ such that it satisfies $\ddot{c}(t)=c^{-3}(t)\omega^{2}$.
Solving this differential equation gives an expression for $c(t)$:

$c(t)=\sqrt{1+\omega^{2}t^{2}}.$ (20)

The final step is also to transform the time coordinate, using:

$\displaystyle\tau$ $\displaystyle=\int^{t}_{0}{\rm d}t^{\prime}\
\frac{1}{1+\omega^{2}t^{2}}=\frac{1}{\omega}\arctan(\omega t),$ (21)
$\displaystyle c(\tau)$ $\displaystyle=\frac{1}{\cos(\omega\tau)},$ (22)

such that ${\rm d}t=c^{2}(t){\rm d}\tau$. This then completes the mapping,
which can be summarised as:

$\displaystyle i\frac{\partial\ket{\psi}}{\partial t}$
$\displaystyle=\frac{1}{2}\hat{p}^{2}\ket{\psi}$ (23) $\displaystyle\implies
i\frac{\partial\ket{\bar{\psi}}}{\partial\tau}$
$\displaystyle=\frac{1}{2}\left(\hat{p}^{2}+\omega^{2}\hat{x}^{2}\right)\ket{\bar{\psi}}$
(24)

provided that

$\omega t=\tan(\omega\tau),\ \ \ \
\ket{\bar{\psi}}=\hat{R}^{\dagger}(t)\hat{U}^{\dagger}(t)\ket{\psi}.$ (26)

Critically, we find that this mapping has the form of a unitary transformation
in addition to a time coordinate transformation. As has been noted in Ref.
Steuernagel (2014), this latter transform plays a particularly important role
as the discontinuities of the $\tan$ function are what effect the transition
from the continuous free particle energetic spectrum to the discrete spectrum
of the oscillator. This crucial additional step also means that this mapping
cannot be related to the usual free-to-harmonic canonical transformation in
classical dynamics Goldstein (2014).

Additionally, the transformation may have a significant impact on boundary
conditions, depending on what is chosen. For example, if the free particle is
embedded in a box of length $L$ with periodic boundaries, the condition
$\phi(L,t)=\phi(0,t)$ transforms via Eq. (4) to produce

$\psi\left(\frac{\cos(\omega\tau)}{\sqrt{\omega}}L,\tau\right)={\rm
e}^{\frac{1}{4}i\sin(2\omega t)L^{2}}\psi(0,\tau),$ (27)

i.e. even the relatively simple case of periodic boundary conditions in the
free case corresponds in the harmonic system to a non-trivial phase relation
between the origin and a point in space that oscillates with time. It is
interesting to observe that the boundary condition becomes tautological
$\psi(0,\tau)=\psi(0,\tau)$, but only at the asymptotic times $t=\pm\infty$ in
the original free particle picture.

##  III Classical Case

An important advantage of the preceding mapping is that it can be immediately
replicated within the Koopman-von Neumann (KvN) framework. The KvN approach to
classical dynamics is one which adopts the formal trappings of a Hilbert space
theory Wigner (1932); Baker (1958); Curtright  _et al._ (2014); Groenewold
(1946); Cabrera  _et al._ (2015), employing four dynamical variables with the
fundamental commutation relations:

$\displaystyle\left[\hat{x},\hat{\lambda}\right]=\left[\hat{p},\hat{\theta}\right]=i,\quad\left[\hat{\lambda},\hat{\theta}\right]=\left[\hat{\lambda},\hat{p}\right]=\left[\hat{\theta},\hat{x}\right]=0.$
(28)

The Bopp operators Bopp (1956); Cohen (1966); Bondar  _et al._ (2013)
$\hat{\lambda}$ and $\hat{\theta}$ are introduced such that time evolutions
are described by the Poisson bracket in phase space. The generator of time
translations is then the _Koopman_ (or equivalently the Liouvillian)
generating operator $\hat{K}$:

$\displaystyle\hat{K}=\hat{p}\hat{\lambda}-\hat{V}^{\prime}\left(\hat{x}\right)\hat{\theta}+f(\hat{x},\hat{p}),$
(29)

where $f(\hat{x},\hat{p})$ is an arbitrary function of the phase space
variables Bondar  _et al._ (2012), which here we set to zero. This generator
evolves the classical wavefunction $\left|\psi_{\rm cl}\right\rangle$
according to

$\displaystyle i\frac{{\rm d}}{{\rm d}t}\left|\psi_{\rm
cl}\right\rangle=\hat{K}\left|\psi_{\rm cl}\right\rangle.$ (30)

For a free particle, we have $\hat{K}=\hat{p}\hat{\lambda}$. Proceeding
entirely in analogy with the quantum case, we first consider a squeezing
procedure, using $\hat{U}_{\rm cl}(t)$:

$\displaystyle\hat{U}_{\rm cl}(t)$ $\displaystyle={\rm
e}^{i\frac{\ln(c(t))}{2}\left(\hat{p}\hat{\theta}+\hat{\theta}\hat{p}-\left(\hat{x}\hat{\lambda}+\hat{\lambda}\hat{x}\right)\right)},$
(31) $\displaystyle\hat{U}_{\rm cl}^{\dagger}(t)\hat{x}\hat{U}_{\rm cl}(t)$
$\displaystyle=c(t)\hat{x},$ (32) $\displaystyle\hat{U}_{\rm
cl}^{\dagger}(t)\hat{p}\hat{U}_{\rm cl}(t)$ $\displaystyle=c^{-1}(t)\hat{p},$
(33) $\displaystyle\hat{U}_{\rm cl}^{\dagger}(t)\hat{\lambda}\hat{U}_{\rm
cl}(t)$ $\displaystyle=c^{-1}(t)\hat{\lambda},$ (34)
$\displaystyle\hat{U}_{\rm cl}^{\dagger}(t)\hat{\theta}\hat{U}_{\rm cl}(t)$
$\displaystyle=c(t)\hat{\theta},$ (35) $\displaystyle-i\hat{U}_{\rm
cl}^{\dagger}(t)\frac{\partial\hat{U}_{\rm cl}(t)}{\partial t}$
$\displaystyle=\frac{\dot{c}(t)}{2c(t)}\left(\hat{p}\hat{\theta}+\hat{\theta}\hat{p}-\left(\hat{x}\hat{\lambda}+\hat{\lambda}\hat{x}\right)\right).$
(36)

Physically, for both free and harmonic oscillators this squeezing operator has
a simple interpretation, namely that it scales the mass as $m\to c(t)m$. Such
mass transformations are of some interest, owing to the fact that in a
relativistic context, mass and proper time can be represented as conjugate
variables. This means that the dynamics of the proper time can be represented
in the Hamiltonian with a varying mass Greenberger (2010), which can itself be
implemented via $\hat{U}_{\rm cl}$. The ability to construct a squeezing
operator in a classical context also presents a number of opportunities to
mimic quantum procedures. One example is the amplification of quantum
Hamiltonians, which is achieved via squeezing Arenz  _et al._ (2020), and
could be replicated classically using $\hat{U}_{\rm cl}$.

If the squeezing operator is applied to $\hat{K}$, we obtain the transformed
$\hat{K}^{\prime}$,

$\displaystyle\hat{K}^{\prime}=\frac{1}{c^{2}(t)}\left[\hat{p}\hat{\lambda}+\frac{c(t)\dot{c}(t)}{2}\left(\hat{p}\hat{\theta}+\hat{\theta}\hat{p}-\left(\hat{x}\hat{\lambda}+\hat{\lambda}\hat{x}\right)\right)\right].$
(37)

Application of squeezing to the Koopman operator has in this case resulted in
harmonic potential terms of the form
$\hat{x}\hat{\lambda}+\hat{\lambda}\hat{x}$, but also additional
$\hat{p}\hat{\theta}+\hat{\theta}\hat{p}$ terms . These in fact represent a
minimal coupling to a vector potential, entirely in analogy with Eq.(10).
While in the quantum case the minimal coupling is obtained via
$\hat{p}\to\hat{p}-\hat{A}$, for KvN dynamics the coupling substitution is
given by Gozzi and Mauro (2002):

$\displaystyle\hat{p}\hat{\lambda}\to$
$\displaystyle\frac{1}{2}\left[\left(\hat{p}-\hat{A}\right)\hat{\lambda}+\hat{\lambda}\left(\hat{p}-\hat{A}\right)\right]$
$\displaystyle+$
$\displaystyle\frac{1}{2}\frac{\partial\hat{A}}{\partial\hat{x}}\left[\left(\hat{p}-\hat{A}\right)\hat{\theta}+\hat{\theta}\left(\hat{p}-\hat{A}\right)\right],$
(38)

which for $\hat{A}=c(t)\dot{c}(t)\hat{x}$ yields $c^{2}(t)\hat{K}^{\prime}$.
This suggests we should continue in an identical fashion as the previous
quantum case, and apply a further transformation $\hat{R}_{\rm cl}$ designed
to eliminate this vector potential:

$\displaystyle\hat{R}_{\rm cl}(t)$ $\displaystyle={\rm
e}^{-ic(t)\dot{c}(t)\hat{x}\hat{\theta}},$ (39) $\displaystyle\hat{R}_{\rm
cl}^{\dagger}(t)\hat{p}\hat{R}_{\rm cl}(t)$
$\displaystyle=\hat{p}+c(t)\dot{c}(t)\hat{x},$ (40) $\displaystyle\hat{R}_{\rm
cl}^{\dagger}(t)\hat{\lambda}\hat{R}_{\rm cl}(t)$
$\displaystyle=\hat{\lambda}-c(t)\dot{c}(t)\hat{\theta},$ (41) $\displaystyle
i\hat{R}_{\rm cl}^{\dagger}(t)\frac{\partial\hat{R}_{\rm cl}(t)}{\partial t}$
$\displaystyle=(\dot{c}^{2}(t)+c(t)\ddot{c}(t))\hat{x}\hat{\theta}.$ (42)

After some fairly tedious algebra, we obtain the sought for Koopman operator
$\hat{K}^{\prime\prime}$:

$\hat{K}^{\prime\prime}=\frac{1}{c^{2}(t)}\left(\hat{p}\hat{\lambda}-\ddot{c}(t)c^{3}(t)\hat{x}\hat{\theta}\right).$
(43)

Once again, if one specifies $c(t)$ to have the form of Eq.(22) and makes the
time transformation given by Eq.(21) then $\hat{K}^{\prime\prime}$ becomes the
generator for simple harmonic motion. Furthermore it is again possible to find
a closed form for the combined unitary $\hat{\mathcal{U}}_{\rm
cl}=\hat{U}_{\rm cl}(t)\hat{R}_{\rm cl}(t)$ due to the fact that

$\left[\hat{p}\hat{\theta}+\hat{\theta}\hat{p}-\left(\hat{x}\hat{\lambda}+\hat{\lambda}\hat{x}\right),\hat{x}\hat{\theta}\right]=-4i\hat{x}\hat{\theta},$
(44)

meaning that Eq.(16) can once again be applied to yield:

$\hat{\mathcal{U}}_{\rm cl}={\rm
e}^{-i\ln(c(t))\left(c(t)\dot{c}(t)\left(1+\frac{c^{2}(t)+1}{c^{2}(t)-1}\right)\hat{x}\hat{\theta}+\frac{1}{2}\left(\hat{x}\hat{\lambda}+\hat{\lambda}\hat{x}-\hat{p}\hat{\theta}-\hat{\theta}\hat{p}\right)\right)}.$
(45)

One rather remarkable fact worth highlighting is that transformations of the
form of $\hat{U}_{\rm cl}(t)\hat{R}_{\rm cl}(t)$ already exist in the KvN
literature. Ref. Ramos-Prieto  _et al._ (2018) demonstrates that this unitary
(with suitably chosen $c(t)$) can be used to map a simple harmonic oscillator
to one with a time dependent frequency. As has been made explicit here, this
mapping can be extended even to the free particle provided an additional time
coordinate transformation is performed. This of course raises the question of
invariants, as it is easy to see that after the unitary transformations not
only does the Koopman operator become explicitly time dependent, but so too
does the Hamiltonian as the coordinate transformation is itself non-canonical.
That is, while the transformation yields the same Koopman operator and
dynamical equations of motion as the simple harmonic oscillator, it does not
have the same Hamiltonian, and that Hamiltonian is not invariant. It is
possible however to show that in this case the time invariant operator
$\hat{I}$ is given by Ramos-Prieto  _et al._ (2018); Sen and Silagadze (2020):

$\displaystyle\hat{I}=$
$\displaystyle\frac{\hat{x}^{2}}{2c^{2}(t)}+\frac{1}{2}\left(\dot{c}(t)\hat{x}-c(t)\hat{p}\right)^{2}$
$\displaystyle+$
$\displaystyle\frac{\hat{\theta}^{2}}{2c^{2}(t)}+\frac{1}{2}\left(\dot{c}(t)\hat{\theta}+c(t)\hat{\lambda}\right)^{2}.$
(46)

##  IV Mapping Open and hybrid System Evolutions

Having demonstrated an almost identical procedure can be used to map a free
particle to a harmonic oscillator in both quantum and classical dynamics, we
now ask what effect this mapping will have when the evolution in question is
no longer unitary, but has a dissipative component. In particular, we will
consider a process which will cause decoherence in the position basis Zurek
(2003). For quantum dynamics this is described by a Lindblad master equation
Breuer and Petruccione (2007), given by

$\displaystyle\frac{d\hat{\rho}(t)}{dt}$
$\displaystyle=-\frac{i}{\hbar}[\hat{H},\hat{\rho}(t)]+\frac{\gamma}{2}\left(2\hat{x}\hat{\rho}(t)\hat{x}-\left\\{\hat{\rho}(t),\hat{x}^{2}\right\\}\right).$
(47)

where $\hat{\rho}$ is the density matrix. The easiest way to interrogate the
effect of the mapping on the additional dissipative terms is to exploit the
fact that an infinitesimal evolution can be represented as a sum of two
unitary maps McCaul  _et al._ (2021):

$\displaystyle\hat{\rho}(t+{\rm d}t)=$ $\displaystyle\frac{1}{2}\left({\rm
e}^{-i\hat{H}_{+}{\rm d}t}\hat{\rho}(t){\rm e}^{i\hat{H}_{+}{\rm d}t}+{\rm
e}^{-i\hat{H}_{-}{\rm d}t}\hat{\rho}(t)\ {\rm e}^{-i\hat{H}_{+}{\rm
d}t}\right)$ $\displaystyle+$ $\displaystyle O\left({\rm d}t^{2}\right),$ (48)

where

$\hat{H}_{\pm}=\frac{1}{2}\hat{p}^{2}\pm\sqrt{\frac{\gamma}{{\rm
d}t}}\hat{x}.$ (49)

We can therefore assess the action of the mapping purely on $\hat{H}_{\pm}$ in
order to ascertain its effect on the dissipative part of the action. Applying
$\hat{\mathcal{U}}$, we obtain:

$\displaystyle\hat{\mathcal{U}}^{\dagger}\hat{H}_{\pm}\hat{\mathcal{U}}$
$\displaystyle-i\hat{\mathcal{U}}^{\dagger}\frac{\partial\hat{\mathcal{U}}}{\partial
t}=$
$\displaystyle\frac{1}{c^{2}(t)}\left(\frac{1}{2}\hat{p}^{2}+\frac{1}{2}\ddot{c}(t)c^{3}(t)\hat{x}^{2}\pm\sqrt{\frac{\gamma}{{\rm
d}t}}c^{3}(t)\hat{x}\right).$ (50)

If the time coordinate transformation ${\rm d}t=c^{2}(t){\rm d}\tau$ is also
made, we find that the effect on the mapping can be expressed in terms of its
effect on the dissipation parameter $\gamma\to c^{4}(\tau)\gamma$. Hence, a
free system undergoing continuous measurement with strength $\gamma$ maps to a
harmonic oscillator where the measurement has a time dependent strength
$\cos^{-4}(\omega\tau)\gamma$.

It is relatively easy to show that the classical evolution that corresponds to
the preceding quantum evolution is Cabrera  _et al._ (2015):

$\displaystyle i\frac{{\rm d}}{{\rm d}t}\left|\psi_{\rm cl}\right\rangle$
$\displaystyle=\hat{K}_{\rm FP}\left|\psi_{\rm cl}\right\rangle.$ (51)
$\displaystyle\hat{K}_{\rm FP}$
$\displaystyle=\hat{\lambda}\hat{p}-iD\hat{\theta}^{2}$ (52)

which is itself nothing but a Fokker-Planck evolution with a constant
dissipation parameter $D$. Application of $\hat{\mathcal{U}}_{\rm cl}$ yields

$\displaystyle\hat{\mathcal{U}}_{\rm cl}^{\dagger}\hat{K}_{\rm
FP}\hat{\mathcal{U}}_{\rm cl}$ $\displaystyle-i\hat{\mathcal{U}}_{\rm
cl}^{\dagger}\frac{\partial\hat{\mathcal{U}}_{\rm cl}}{\partial t}=$
$\displaystyle\frac{1}{c^{2}(t)}\left(\hat{p}\hat{\lambda}-\ddot{c}(t)c^{3}(t)\hat{x}\hat{\theta}-ic^{4}(t)D\hat{\theta}^{2}\right).$
(53)

Meaning that $D\to c^{4}(\tau)D$. Significantly, this means that the effect of
the mapping on the dissipative part of the evolution is identical in both the
classical and quantum scenarios, namely a scaling of the dissipation parameter
by $c^{4}(\tau)$.

Finally, it is worth considering the free-harmonic mapping in the context of a
quantum-classical hybrid system Bondar  _et al._ (2019). A typical example
would be a quantum system with non-commuting variables
$\hat{z}=(\hat{x},\hat{p})$ coupled bilinearly to a classical one with
commuting phase space classical variables $\hat{Z}=(\hat{X},\hat{P})$. An
example hybrid Hamiltonian describing this scenario would be $\hat{H}_{h}$

$\hat{H}_{h}=\frac{1}{2}\hat{P}^{2}+\frac{1}{2}\hat{p}^{2}+k\hat{x}\hat{X}.$
(54)

The hybrid Liouvillian describing the evolution of this system is given by
Bondar  _et al._ (2019)

$\displaystyle\hat{\mathcal{L}}_{h}$
$\displaystyle=\hat{H}_{h}+\frac{1}{2}\left(\frac{\partial\hat{H}_{h}}{\partial\hat{P}}\hat{\lambda}-\frac{\partial\hat{H}_{h}}{\partial\hat{X}}\hat{\theta}-\hat{Z}\cdot\nabla_{Z}\hat{H}_{h}\right)$
(55)
$\displaystyle=\hat{\lambda}\hat{P}+\frac{1}{2}\hat{p}^{2}-\frac{k}{2}\hat{x}\left(\hat{\theta}+\hat{X}\right),$
(56)

where $\hat{\lambda}$ and $\hat{\theta}$ are the Bopp operators associated
with the classical variables $\hat{Z}$. Applying the appropriate unitary
transformations to both the quantum and classical variables followed by the
time coordinate transformation, we obtain a classical and quantum harmonic
oscillator, together with the additional coupling term
$\frac{k}{2}c^{4}(\tau)\hat{x}\left(\hat{\theta}+\hat{X}\right)$. As might be
expected, the effect of the unitary mapping on the bilinear coupling is
therefore the introduction of a time dependence into the coupling via $k\to
c^{4}(\tau)k$.

##  V Discussion

Taking as inspiration the existence of the coordinate mapping between the free
particle and harmonic oscillator Schrödinger equation, we have demonstrated
that an equivalent transformation can be achieved in both quantum and
classical dynamics via a combination of unitary operators and a time
coordinate transform. This mapping can be further extended to time dependent
frequency oscillators and open systems. The discontinuities in the time
coordinate transformation also sheds some insight into how the continuous
Hamiltonian (or Koopman) spectrum of a free particle can become the discrete
oscillator spectrum. Moreover, it is also possible to map the free particle to
an inverted oscillator with $\omega\to i\omega$. Significantly, an imaginary
frequency removes the discontinuities in the time transformation, and
immediately implies that both the Hamiltonian and Koopman spectra for the
inverted oscillator are continuous.

Given the harmonic oscillator is perhaps the paradigmatic model for any
physical system, the existence of a novel mapping to it may be of some
interest to a number of fields. A relevant example here would be that such a
mapping would allow one to incorporate time dependent frequencies into both
quantum and classical environment models, and the path integral methods used
to solve them McCaul and Bondar (2021). Additionally, the methods presented
here can also be used to map a free particle to a linear potential, which can
itself be mapped to a Lindblad-type model for entropic gravity Schimmoller
_et al._ (2021). Connecting these two would therefore allow one to identify
the unitary operators corresponding to the Kraus map for this model of
entropic gravity. Even divorced from application however, the existence of two
almost identical mappings in both quantum and classical dynamics highlights
the centrality of the harmonic oscillator model, and the unexpected
commonalities it possesses between classical and quantum dynamics.

###### Acknowledgements.

This work was supported by Army Research Office (ARO) (grant W911NF-19-1-0377;
program manager Dr. James Joseph). The views and conclusions contained in this
document are those of the authors and should not be interpreted as
representing the official policies, either expressed or implied, of ARO or the
U.S. Government. The U.S. Government is authorized to reproduce and distribute
reprints for Government purposes notwithstanding any copyright notation
herein.

## References

  * Solov’ev (1982) E. A. Solov’ev, Sov. Phys. - Nucl. Phys 35, 136 (1982).
  * Niederer (1973) U. Niederer, Helv. Phys. Acta, v. 46, no. 2, pp. 191-200 46, 191 (1973).
  * Steuernagel (2005) O. Steuernagel, American Journal of Physics 73, 625 (2005), https://doi.org/10.1119/1.1900099 .
  * Steuernagel  _et al._ (2013) O. Steuernagel, D. Kakofengitis, and G. Ritter, Phys. Rev. Lett. 110, 030401 (2013).
  * Takagi (1990) S. Takagi, Progress of Theoretical Physics 84, 1019 (1990), https://academic.oup.com/ptp/article-pdf/84/6/1019/5406385/84-6-1019.pdf .
  * Andrzejewski  _et al._ (2014) K. Andrzejewski, J. Gonera, and P. Kosiński, Phys. Rev. A 89, 014103 (2014).
  * Barton (1986) G. Barton, Annals of Physics 166, 322 (1986).
  * Yuce  _et al._ (2006) C. Yuce, A. Kilic, and A. Coruh, Physica Scripta 74, 114 (2006).
  * Steuernagel (2014) O. Steuernagel, The European Physical Journal Plus 129 (2014), 10.1140/epjp/i2014-14114-3.
  * Sen  _et al._ (2022) A. Sen, B. K. Parida, S. Dhasmana, and Z. K. Silagadze, “Eisenhart lift of koopman-von neumann mechanics,” (2022), arXiv:2207.05073 .
  * Dhasmana  _et al._ (2021) S. Dhasmana, A. Sen, and Z. K. Silagadze, Annals of Physics 434, 168623 (2021).
  * McCaul  _et al._ (2019) G. McCaul, A. Pechen, and D. I. Bondar, Phys. Rev. E 99, 062121 (2019).
  * Schleich (2001) W. P. Schleich,  _Quantum Optics in Phase Space_ (Wiley-VCH Verlag, Weinheim, Germany, 2001).
  * Van-Brunt and Visser (2015) A. Van-Brunt and M. Visser, Journal of Physics A: Mathematical and Theoretical 48, 225207 (2015).
  * Goldstein (2014) H. Goldstein,  _Classical Mechanics_ , 3rd ed. (Pearson, Essex, 2014).
  * Wigner (1932) E. Wigner, Phys. Rev. 40, 749 (1932).
  * Baker (1958) G. Baker, Phys. Rev. 109, 2198 (1958).
  * Curtright  _et al._ (2014) T. L. Curtright, D. B. Fairlie, and C. K. Zachos,  _a Concise Treatise On Quantum Mechanics In Phase Space_ (World Scientific, 2014).
  * Groenewold (1946) H. J. Groenewold, Physica 12, 405 (1946).
  * Cabrera  _et al._ (2015) R. Cabrera, D. I. Bondar, K. Jacobs, and H. A. Rabitz, Phys. Rev. A 92, 042122 (2015).
  * Bopp (1956) F. Bopp, Ann. Henri Poincaré 15, 81 (1956).
  * Cohen (1966) L. Cohen, Philos. Sci. 33, 317 (1966).
  * Bondar  _et al._ (2013) D. I. Bondar, R. Cabrera, D. V. Zhdanov, and H. A. Rabitz, Phys. Rev. A 88 (2013), 10.1103/PhysRevA.88.052108.
  * Bondar  _et al._ (2012) D. I. Bondar, R. Cabrera, R. R. Lompay, M. Y. Ivanov, and H. A. Rabitz, Physical Review Letters 109 (2012), 10.1103/PhysRevLett.109.190403.
  * Greenberger (2010) D. M. Greenberger, “Conceptual problems related to time and mass in quantum theory,” (2010), arXiv:1011.3709 .
  * Arenz  _et al._ (2020) C. Arenz, D. I. Bondar, D. Burgarth, C. Cormick, and H. Rabitz, Quantum 4, 271 (2020).
  * Gozzi and Mauro (2002) E. Gozzi and D. Mauro, Annals of Physics 296, 152 (2002).
  * Ramos-Prieto  _et al._ (2018) I. Ramos-Prieto, A. R. Urzúa-Pineda, F. Soto-Eguibar, and H. M. Moya-Cessa, Scientific Reports 8, 8401 (2018).
  * Sen and Silagadze (2020) A. Sen and Z. Silagadze, International Journal of Theoretical Physics 59, 2187 (2020).
  * Zurek (2003) W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).
  * Breuer and Petruccione (2007) H.-P. Breuer and F. Petruccione,  _The Theory of Open Quantum Systems_ (OUP, Oxford, 2007).
  * McCaul  _et al._ (2021) G. McCaul, K. Jacobs, and D. I. Bondar, Phys. Rev. Research 3, 013017 (2021).
  * Bondar  _et al._ (2019) D. I. Bondar, F. Gay-Balmaz, and C. Tronci, Proc. R. Soc. A. 475, 20180879 (2019).
  * McCaul and Bondar (2021) G. McCaul and D. I. Bondar, The European Physical Journal Special Topics 230, 733 (2021).
  * Schimmoller  _et al._ (2021) A. J. Schimmoller, G. McCaul, H. Abele, and D. I. Bondar, Phys. Rev. Research 3, 033065 (2021).