# A chip-scale second-harmonic source via injection-locked all-optical poling

Marco Clementi<EMAIL_ADDRESS>Photonic Systems Laboratory (PHOSL),
École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland  Edgars
Nitiss  Photonic Systems Laboratory (PHOSL), École Polytechnique Fédérale de
Lausanne, Lausanne, 1015, Switzerland  Elena Durán-Valdeiglesias  Almae
Technologies, Route de Nozay, 91460 Marcoussis, France  Sofiane Belahsene
Almae Technologies, Route de Nozay, 91460 Marcoussis, France  Junqiu Liu
Laboratory of Photonics and Quantum Measurements (LPQM), École Polytechnique
Fédérale de Lausanne, Lausanne, 1015, Switzerland
Tobias J. Kippenberg  Laboratory of Photonics and Quantum Measurements (LPQM),
École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland  Hélène
Debrégeas  Almae Technologies, Route de Nozay, 91460 Marcoussis, France
Camille-Sophie Brès<EMAIL_ADDRESS>Photonic Systems Laboratory (PHOSL),
École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland

###### Abstract

Second-harmonic generation allows for coherently bridging distant regions of
the optical spectrum, with applications ranging from laser technology to self-
referencing of frequency combs. However, accessing the nonlinear response of a
medium typically requires high-power bulk sources, specific nonlinear
crystals, and complex optical setups, hindering the path toward large-scale
integration. Here we address all of these issues by engineering a chip-scale
second-harmonic (SH) source based on the frequency doubling of a semiconductor
laser self-injection-locked to a silicon nitride microresonator. The
injection-locking mechanism, combined with a high-Q microresonator, results in
an ultra-narrow intrinsic linewidth at the fundamental harmonic frequency as
small as 57 Hz. Owing to the extreme resonant field enhancement, quasi-phase-
matched second-order nonlinearity is photoinduced through the coherent
photogalvanic effect and the high coherence is mapped on the generated SH
field. We show how such optical poling technique can be engineered to provide
efficient SH generation across the whole C and L telecom bands, in a
reconfigurable fashion, overcoming the need for poling electrodes. Our device
operates with milliwatt-level pumping and outputs SH power exceeding 2 mW, for
an efficiency as high as 280 %/W under electrical driving. Our findings
suggest that standalone, highly-coherent, and efficient SH sources can be
integrated in current silicon nitride photonics, unlocking the potential of
$\chi^{(2)}$ processes in the next generation of integrated photonic devices.

## Introduction

Second-harmonic generation (SHG)Franken  _et al._ (1961) plays a fundamental
role in the realm of nonlinear optics, as it enables linking octave-spaced
regions of the spectrum while preserving the coherence of the optical field.
Applications range from laser physics and technologySvelto (2010); Armstrong
(1967), to imagingCampagnola (2011), material scienceShen (1989), and self-
referencing of frequency combsReichert  _et al._ (1999), to name a few. Since
its inception in 1961Franken  _et al._ (1961), SHG has widely been applied in
bulk optics, whereas the nonlinear nature of the process requires an
appropriate combination of i) a high-intensity coherent source, ii) a material
endowed with a second-order nonlinearity ($\chi^{(2)}$) and iii) carefully
engineered phase-matching conditions. Such hurdles have stimulated a great
research effort in the domain of integrated optics, whereas the integration of
frequency doubling on-chip bears the promise of realizing novel devices in a
compact, power-efficient and scalable fashion, while eliminating the need for
bulky and complex experimental apparatuses.

On-chip integration has indeed proven advantageous, as it enables to enhance
the interaction strength thanks to the transverse confinement in
waveguidesWang  _et al._ (2017), and also via the use of resonant
structuresWang  _et al._ (2020). Moreover, the development of nano-fabrication
techniques in several $\chi^{(2)}$ materials has yielded the demonstration of
highly-efficient SHG in platforms such as lithium niobate on insulator
(LNOI)Lu  _et al._ (2020) and III-V semiconductors, such as AlNBruch  _et al._
(2018), GaNWang  _et al._ (2020), AlGaAsKuo  _et al._ (2014) and GaPLake  _et
al._ (2016), especially in combination with quasi-phase-matching (QPM)
techniques. However, despite great progress, such emerging platforms struggle
to find employ in practical devices, owing to the lack of compatibility with
established fabrication processes, in particular those of the silicon-based
complementary metal-oxide semiconductor (CMOS) technology, widely adopted by
the electronics market.

In contrast, silicon nitride (Si3N4) photonicsBlumenthal  _et al._ (2018);
Bucio  _et al._ (2020); Xiang  _et al._ (2022) has emerged as a mature
integrated photonics platform, thanks to its compatibility with CMOS
fabrication, that favors scalability while allowing co-integration with
microelectronics. Silicon nitride photonic devices benefit notably from ultra-
low propagation losses, a wide transparency window, ranging from the mid-
infrared to the near-UV, large bandgap and negligible Raman effect, making
them an ideal choice for high-power and specifically nonlinear applications,
such as Kerr microcombsKippenberg  _et al._ (2018); Gaeta  _et al._ (2019),
supercontinuum generationGrassani  _et al._ (2019), and parametric quantum
light sourcesKues  _et al._ (2019); Arrazola  _et al._ (2021). While these
advantages are usually ascribed to third-order ($\chi^{(3)}$) nonlinear
processes, owing to the centrosymmetric nature of the amorphous material, it
was recently shown that Si3N4 waveguidesBillat  _et al._ (2017); Porcel  _et
al._ (2017); Hickstein  _et al._ (2019) and resonatorsLu  _et al._ (2021);
Nitiss  _et al._ (2022) can be endowed with a photoinduced second-order
nonlinearity by the coherent photogalvanic effect, whereas high conversion
efficiency ($\mathrm{CE}=P_{\rm SH}/P_{\rm FH}^{2}$) of second harmonic (SH)
light, exceeding 2,500%/W, can be reached particularly in resonant structures,
owing to the high values of field enhancement achievable.

Furthermore, the lack of active sources in group IV semiconductors can be
overcome by heterogeneous integration with III-V sourcesXiang  _et al._
(2021a). As shown by recent findings, the combination of such sources with
silicon nitride chips is not only technologically accessible, but can also be
exploited to improve the coherence properties of the latter through a self-
injection-locking (SIL) mechanism to a microring resonatorDahmani  _et al._
(1987); Kondratiev  _et al._ (2017); Raja  _et al._ (2019); Xiang  _et al._
(2021b); Lihachev  _et al._ (2022); Voloshin  _et al._ (2021); Jin  _et al._
(2021); Corato-Zanarella  _et al._ (2023). In these schemes, the
backscattering from a high quality factor (Q) microring resonator is injected
into the cavity of a semiconductor laser. Under the appropriate conditions,
the ring resonator acts as an effective narrowband filter, resulting in the
locking of the source to its resonance frequency and a narrowing of the laser
linewidth proportional to Q2. Recent experimental evidence has shown a
reduction of the intrinsic laser linewidth below the hertz-levelJin  _et al._
(2021), while the combination of this technique with Kerr microcombs has led
to the realization of heterogeneously integrated turnkey soliton sourcesShen
_et al._ (2020); Xiang  _et al._ (2021b).

Figure 1:  Self-injection-locked second-harmonic source. a. Schematic of the
SIL mechanism. The DFB laser inject light at the FH wavelength (solid red
arrow) to the ring resonator bus waveguide. A small fraction of the light
circulating inside the ring is reflected by Rayleigh backscattering (dashed
arrows) and injected to the DFB cavity, yielding a dramatic narrowing of the
emission linewidth compared to the free-running regime (lower panel). Such
high-coherence laser field displays a high intracavity intensity, which is
used to trigger the coherent photogalvanic effect and generate SH light. b.
Schematic of the experimental device. The DFB laser (detail in inset) is
mounted and wire-bonded on a temperature stabilized control board, whose
position is finely tuned by micro-actuators. The temperature of the Si3N4 chip
is independently controlled to tune the resonant conditions. Light is
collected at the output using a collimation lens or a lensed fiber (not shown
here). The Si3N4 chip size is $5\times 5$ mm2. The DFB chip length is
$400\text{\,}\mathrm{\SIUnitSymbolMicro m}$. c. Transmission spectrum at the
the FH wavelength. d. Detail of the resonance used for SIL-SHG. The inset
shows a SEM cross-section of one of the fabricated waveguides. e. Finite
element simulation of the transverse TE modes involved at the fundamental and
SH frequencies. The horizontal axis refers to the pump wavelength.

In this work, we show how SIL and photoinduced second-order nonlinearity can
occur concurrently in a Si3N4 microring resonator to create a standalone dual-
wavelength source emitting highly-coherent light at both the fundamental and
SH frequencies (Fig. 1a). The resonator’s high Q factor yields a narrowing of
the intrinsic laser linewidth near the hertz-level, and its high finesse
results in the efficient generation of milliwatt-level light at the SH,
despite the use of a non-intrinsic $\chi^{(2)}$ material. We show how the
generated SH wavelength can be tuned by simply adjusting the device operating
conditions (current and temperature), and we provide a full mapping of the
suitable operating points over the C and L bands, showing abundance of
suitable doubly-resonant conditions. Remarkably, the generated light shares
the same properties of the pump field, including its coherence. This
establishes our chip-scale source as a potentially powerful tool for
applications that benefit not only from the ultra-narrow linewidth, such as
RbNewman  _et al._ (2019) and SrOriglia  _et al._ (2016) based chip-scale
atomic clocks and integrated quantum photonicsMehta  _et al._ (2020); Moody
_et al._ (2022), but also from mutual coherence of the two beams, such as the
self-referencing of optical frequency combsReichert  _et al._ (1999).

## Results

A prototype realization of our device is shown in Fig. 1b. The layout consists
of an electrically pumped distributed feedback laser diode (DFB), edge-coupled
to the Si3N4 photonic chip. The DFB is realized in an InGaAsP multi-quantum
well, buried waveguide geometry, and it is packaged on a wire-bonded and
thermally stabilized stage. To prove the generality of our results, in this
study we used two batches of DFB lasers, operating respectively in the C and L
telecom bands and characterized by similar performance and output power,
respectively up to 60 mW and 90 mW at room temperature. By varying the driving
current and device temperature, the emission wavelength can be tuned within a
range of approximately $5\text{\,}\mathrm{nm}$. The Si3N4 chip is fabricated
at wafer-scale through the photonic Damascene process Liu  _et al._ (2021) and
contains microring resonators with a radius of
$896\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The waveguide cross-section is
$2\times 0.55$ ${\mathrm{\SIUnitSymbolMicro m}}^{2}$, and adiabatic mode
converters are used for input- and output-coupling. Also in this case, several
chips from the same wafer were tested in order to assess repeatability, with
the same nominal geometric parameters. In this demonstration, the DFB position
is finely adjusted to inject light efficiently to the bus waveguide. Light is
then coupled to the microring resonator through a single point coupler with a
gap of 550 nm. We measure an insertion loss of approximately 5 dB at the
DFB/Si3N4 chip interface. The temperature of the Si3N4 chip is separately
controlled to finely tune the resonant frequencies through thermo-optic
effect. Finally, the light at the output facet is collected either in free-
space or by means of a lensed fiber.

Before operating in SIL regime, we characterized both the linear and nonlinear
properties of the resonator in the telecom band (1500-1600 nm) using a table-
top external cavity tunable laser source. Transmission spectroscopy (Fig. 1c)
at low power reveals a set of slightly overcoupled resonances with an average
free spectral range (FSR) of 25.6 GHz for the fundamental transverse electric
(TE00) mode, a loaded quality factor of $Q=2\times 10^{6}$ (Fig. 1d), and an
estimated intrinsic quality factor of $Q_{0}=7\times 10^{6}$. In the SH band
(750-800 nm), both the ring and the bus waveguides display a multimode
behavior, with up to 5 TE modes supported, which we hereby label from SH1
(TE00) to SH5 (TE40). The associated azimuthal resonances are undercoupled to
the fundamental mode of the bus waveguide, and therefore not accessible
through standard transmission spectroscopy. However, their FSR can be
accurately estimated via numerical simulations (Fig. 1e).

Figure 2:  All-optical poling and second harmonic generation. a. Ring
resonator all-optical poling using an external tunable laser. Light is in-
coupled using a lensed fiber. The generated SH light is visible to the camera
sensor showing the intense circulating power inside the resonator. b-c. Two
dimensional maps of the output FH and SH obtained by scanning the pump power
at different temperatures of the sample. Panel b shows the results for a Si3N4
sample designed to operate in the C band, while panel c corresponds to a
different sample operated in the L band. The approximate tunability region of
the DFB is shaded in white. The dashed lines are guides to the eye
highlighting the variation of the doubly-resonant condition for the
corresponding families of modes. d. Map of the CE as a function of the pump
wavelength and sample temperature, highlighting the best detuning condition.
e. Scaling trend of the generated peak SH power as a function of the input
pump level. f. Two-photon microscope image of the inscribed $\chi^{(2)}$
grating.

The same tunable laser was then amplified and used to pump the resonant modes
in order to investigate and map the second-order nonlinear response (Fig. 2a).
Owing to the high field enhancement provided by resonant modes at the SH
frequency, the CE is maximized whenever a nearly doubly-resonant condition is
met, that is, as the pump and SH frequencies are both tuned closely to a
resonanceNitiss  _et al._ (2022). When this condition is satisfied and the
circulating pump intensity is high enough, the coherent photogalvanic effect
is triggered: a static electric field is established inside the waveguide due
to the displacement of heavy charges, resulting in the breaking of the centro-
symmetry condition, and in the consequent establishment of a permanently
photoinduced $\chi^{(2)}$ responseDianov and Starodubov (1995); Yakar  _et
al._ (2022). Moreover, the local sign and amplitude of the $\chi^{(2)}$ is
such that the QPM condition is automatically fulfilled, resulting in the
inscription of a $\chi^{(2)}$ grating inside the waveguideNitiss  _et al._
(2020, 2022). Such all-optical poling (AOP) phenomenon manifests in the sudden
increase of the generated SH signal, which reaches its equilibrium state in
the millisecond timescale, as soon as the appropriate pump detuning and power
conditions are met. Since the occurrence of a doubly-resonant condition is
strongly dependent on the fabrication tolerance, we implemented a technique to
map the AOP-SHG configurations displaying the highest CE as a function of the
sample temperature and pump wavelength. The results are shown in Figs. 2b-c,
where two different samples – targeting respectively the C and L bands – were
measured by slowly scanning the pump laser (scan speed: 50 pm/s) at varying
values of the sample temperature. Such two-dimensional maps reveal the
presence of families of doubly-resonant configurations, that can be visually
identified as linear patterns in the generated SH plots (highlighted by the
white dashed lines). The slope of such patterns depends on both the FSR
difference between the two modes involved and on their thermo-optic
coefficientNitiss  _et al._ (2023), while their horizontal spacing depends
only on the FSR difference (see Methods). From comparison between such
estimated slopes and the calculated group index ad different wavelength, we
are able to retrieve the resonant modes involved, in this case the pair FH-
SH1. Notably, the presence of hotspots associated with particularly high
generated power – exceeding 20 mW – was observed, which we attribute to
fluctuations in the resonances Q factor and coupling conditions. These
hotspots – some of which, falling within the DFBs tunability range, we labeled
from A to C for illustrating purpose – represent the optimal operating points
for injection-locked SHG, and we therefore investigated further their
properties. To confirm the validity of our picture, we applied our mapping
technique in a narrow range after optical poling with significantly lower
input power (14 mW) and higher scan speed (1 nm/s), in order not to alter the
properties of the inscribed grating. The result at hotspot A, shown in Fig.
2d, confirms the existence of an optimal combination of parameters in the
temperature/wavelength space. Furthermore, remarkably, it points to a
significant increase of the CE compared to the high-power case, which we
registered to be as high as 250%/W. This increase in the CE at low power is
due to the absence of parasitic effects such as pump depletion and the
generation of free-carriers associated to the nonlinear photoconductivityYakar
_et al._ (2022). A similar value of generated power and CE can be assessed for
several of the hotspots identified, both in the C and L bands, that were also
observed to preserve a high CE up to several tens of milliwatts of pump power.
This was confirmed by power scaling measurements, illustrated for hotspot B in
Fig. 2e, which display a nearly quadratic trend as a function of the input
pump power up to about 50 mW. Finally, to confirm the QPM nature of SHG, we
performed two-photon imaging of the $\chi^{(2)}$ gratingNitiss  _et al._
(2020) after poling the sample at hotspot C. The result, shown in Fig.2f
reveals a periodic pattern, with a period of approximately
$2.47\text{\,}\mathrm{\SIUnitSymbolMicro m}$. From comparison with the
simulated values of the effective index, we infer a QPM condition between the
FH and SH1 mode, in excellent agreement with our deductions drawn from the
linear pattern observed in Fig. 2c.

Figure 3:  Injection-locked second-harmonic generation. a. The SIL-SHG source
in operation. The generated SH is visible as scattering from the silicon
nitride ring and at the chip output. b. Current scans of the DFB laser
targeting several operating points (hotspots) among the two samples studied.
The SIL events display are identified as strongly asymmetric dips
characterized by hysteresis, whose width and depth depends on the amplitude
and phase of the backscattered signal by the microresonator, as well as the
dip visibility. SIL-SHG events are marked by spikes in the generated SH power,
that reaches a peak value of 2.3 mW. All powers are rated at the output of the
bus waveguide. c. Optical heterodyne spectrum of the output FH as a function
of the driving current. In the proximity of the resonance, the emission
frequency deviates from the otherwise linear shifting trends, and the
linewidth is narrowed significantly. Once the current if further increased (or
decreased), the trend recovers its linear character. The width of the gap in
the frequency range spanned by the laser emission represents the locking
bandwidth. d. Device emission at the FH and SH wavelength recorded by an
optical spectrum analyzer (resolution: 20 pm), showing the absence of side
modes within the instrument’s dynamic range (65 dB).

From the measurements shown in Fig. 2, we identified several operating points
compatible with the tunability bandwidth of our DFBs, that could be probed for
injection-locked SHG. The tunable laser was thus replaced by the DFB diode,
realizing the SIL-SHG source as described above (Fig. 3a). By fixing the DFB
temperature and sweeping the driving current, we were able to observe several
SIL events, marked by strongly asymmetric dips in the transmitted spectrum
(Figs. 3b), owing to the locking of the lasing frequency to the bottom of the
resonance dip. The width of such dips reflects the locking bandwidth, which
depends on both the magnitude of the back-reflected signal and on its phase,
two quantities that are controlled by finely adjusting the position of the DFB
with respect to the chip facet using a piezoelectric positioner. By retrieving
the DFB spectrum as a function of the driving current through optical
heterodyne measurements (Fig. 3c), we estimated a locking bandwidth in the GHz
range, visualized by stark deviation in the otherwise linear frequency
shifting trend, and characterized by a pronounced hysteresisKondratiev  _et
al._ (2017) between the increasing current (decreasing frequency) and the
decreasing current (increasing frequency) scans. When the doubly-resonant
condition is fulfilled in correspondence of a SIL event, the AOP effect is
triggered, resulting in the emission of light at the SH frequency. The
phenomenon was investigated for several doubly-resonant configurations
(hotspots) shown in Figs. 2b-c and the result was found consistently
repeatable both for pumping in the C and L bands, with a maximum emission
power as high as 2.3 mW in the bus waveguide, and a peak CE as high as 280%/W.
Remarkably, the former value is comparable, if not greater, to the best
results obtained in for injection-locking in LNOI technologyLing  _et al._
(2023). This result highlights the high potential of silicon nitride for the
engineering of second-order nonlinear processes, despite relying on a
photoinduced, rather than intrinsic, nonlinearity. By fixing the current in
correspondence of a SIL-SHG event, a constant CW emission is observed (Fig.
3a). When visualized on an optical spectrum analyzer (3d), such emission shows
a monochromatic spectrum, characterized by strong side-mode suppression ratio
(SMSR) exceeding 60 dB, limited by the sensitivity of our instrument.

Figure 4:  Emission linewidth. a. Heterodyne spectrum of the emission
linewidth at FH in the free running (red dots) and SIL (orange line) regimes.
The linewidth of the FH emission in the SIL regime is estimated to be well-
below the resolution bandwidth (RBW) of the instrument. b. Schematic of the
frequency discriminator setup used for the frequency noise measurements. Light
from the SIL-SHG source is collected at the output of the chip, and the de-
multiplexed FH is routed to an unbalanced Mach-Zehnder interferometer. On one
arm, the phase is regulated using a fiber phase shifter. The output signal is
detected at one of the outputs with a fast photodiode and visualized on an
electrical spectrum analyzer. c. Frequency noise spectra measured by the
frequency discriminator. The technical noise pattern, scaling as $\nu^{-1}$
and associated with a Gaussian contribution to the broadening of the emission
line, is marked by a dashed grey line. The white noise plateaus, associated
with the Lorentzian contribution to the broadening of the emission line, are
highlighted by dotted grey lines. From the value of such plateaus we estimated
a narrowing of the intrinsic linewidth of 37dB, close to the limit of
sensitivity of our technique
($6\text{\,}{\mathrm{Hz}}^{2}\text{\,}{\mathrm{Hz}}^{-1}$).

Finally, we investigated the coherence properties of our dual-wavelength
source. A first analysis of the emission linewidth was performed at the
fundamental wavelength by optical heterodyne with a reference tunable laser
(Fig. 4a). A significant narrowing can be immediately appreciated when passing
from the free-running to the injection-locked regime, with a decrease in
linewidth from $\delta\nu\approx$1\text{\,}\mathrm{MHz}$$ in the former case
(3 dB bandwidth from Voigt fit), to $\delta\nu<$50\text{\,}\mathrm{kHz}$$ when
locked, limited by the noise of the reference laser used as a local
oscillator. To get more insight on the emitted light properties, we
implemented a frequency discriminator apparatus to assess the frequency noise
of the emitted lightLlopis  _et al._ (2011); Tran  _et al._ (2019), as shown
in Fig. 4b. The retrieved power spectral density (PSD) is shown in Fig. 4c,
where a dramatic reduction in both the technical ($f^{-1}$) and white noise,
exceeding 37 dB, is observed, with a noise floor lower plateau as small as
$S_{\nu}^{0}=$18\text{\,}{\mathrm{Hz}}^{2}\text{\,}{\mathrm{Hz}}^{-1}$$,
corresponding to an intrinsic (Schawlow-Townes) linewidth $\delta\nu_{\rm
ST}=\pi S_{\nu}^{0}=$57\text{\,}\mathrm{Hz}$$. As a comparison, the SIL source
outperforms the commercial tunable laser used for characterization in terms of
intrinsic linewidth.

## Discussion

The device developed here represents a novel approach to the engineering of
on-chip SH sources, whereas the resonant element not only enhances the CE, but
first and foremost improves the coherence properties of the FH and SH fields
through the SIL mechanism. While this effect has been widely investigated
recently for perspective application to other nonlinear processesRaja  _et
al._ (2019); Shen  _et al._ (2020); Xiang  _et al._ (2021b); Lihachev  _et
al._ (2022); Voloshin  _et al._ (2021), the results presented here represent
one of the first demonstrations of injection-locked SHG on a chip.

Only a single similar result has been reported so far, to the best of our
knowledge, by Ling and co-workersLing  _et al._ (2023). In that work, the
authors exploit LNOI technology to realize a SIL-SHG source similar to the one
presented here, by leveraging the high intrinsic $\chi^{(2)}$ and electric
field poling of lithium niobate. Despite promising results, their device still
suffers from some limitations inherent to the LNOI platform, most notably i)
the need for electrodes used to inscribe a QPM grating, that limits the
operation to a fixed design wavelength, and ii) a relatively low quality
factor ($Q\approx 4\times 10^{5}$), that sets the best narrowing performance
reported to an estimated intrinsic linewidth of $4.7\text{\,}\mathrm{kHz}$ at
the SH. In contrast, our device displays wide tunability across the whole
telecom spectrum, only requiring control over the pump laser wavelength and on
the sample temperature. The AOP mechanism allows indeed to erase and re-write
the QPM grating by solely changing these two parameters, as long as a doubly-
resonant condition is satisfiedNitiss  _et al._ (2022). As a result, the same
microresonator can be dynamically reconfigured in order to match a different
pump wavelength and/or family of modes, with ample choice provided by the
abundance of doubly-resonant configurations, thus eliminating the need for
poling electrodes and enhancing the flexibility of the final device. Our
solution also excels in terms of coherence, as it displays an intrinsic
linewidth almost reaching the hertz-level, owing to the high Q of the
resonators used. It is worth stressing that such short-term linewidth is
mapped on the generated SH field, with a predicted intrinsic SH linewidth as
small as 228 Hz (note that a 4-fold increase in the frequency noise is
expected as a result of frequency doubling), thus implying the mutual
coherence between the output fields at the FH and SH wavelengths. Our device
also performs well in terms of generated power, being capable to reach and
exceed a milliwatt-level SH output (up to 2.3 mW) with a pump power of about
33 mW, corresponding to a net (i.e. non-normalized) conversion efficiency
$\eta=P_{\rm SH}/P_{\rm FH}\approx 7\%$ at continuous-wave regime, and
consistently displaying a normalized CE exceeding 100 %/W across all the
hotspots tested, with a peak value recorded as high as 280 %/W. This result is
particularly remarkable given the relatively low value of the photoinduced
nonlinearity in silicon nitride – up to $\chi^{(2)}\approx 0.3$ pm/VBillat
_et al._ (2017), compared to $\chi^{(2)}\approx 54$ pm/V in the case of
LNOIWang  _et al._ (2017) – and highlights the maturity of the silicon nitride
photonics platform, as well as its suitability for applications in nonlinear
optics. Our device performance proves superior also compared with existing
single-wavelength SIL sources in the visible and near-infrared rangesSiddharth
_et al._ (2022); Corato-Zanarella  _et al._ (2023), with an order-of-magnitude
narrower intrinsic linewidth and significantly higher SMSR. This advantage can
be attributed to the more difficult realization of the laser and
microresonator components, which requires an obvious increase in the
fabrication accuracy to efficiently operate at a shorter wavelength.

In the perspective of commercial applications, our proof-of-concept
realization could be further improved, both in terms of device engineering and
figures of merit. In particular, the output power can be significantly
increased by engineering an optimal coupling between the DFB facet and the bus
waveguide, for example through the use of optimized adiabatic mode converters.
Ultimately, one could also foresee a full heterogeneous integration, which has
been shown to be within reach of state-of-art fabrication technologyXiang  _et
al._ (2021a), thus enabling a wafer-scale integration of this type of sources.
Finally, further improvements in the microresonator Q factor may enable
increased conversion efficiencies and an even lower laser linewidths,
potentially unlocking access to hertz-level dual-wavelength coherence on a
photonic chip. This result has indeed already proven possible in single-
wavelength SIL sourcesJin  _et al._ (2021), whereas the use of very long ring
resonators in a folded spiral geometry also showed promising advantage in
reducing the thermo-refractive contribution to technical noise, the latter
being effectively averaged over the whole device length. However, this
approach may not be suitable for the purpose of frequency doubling, as the use
of low-confinement waveguides increases the transverse mode area, ultimately
weakening the nonlinear interaction. In this perspective, the use of high-
confinement waveguides based on thick silicon nitride layersLiu  _et al._
(2021) is more advantageous, as it maximizes the nonlinear interaction.
Moreover, the use of long resonators reduces the field enhancement, ultimately
setting a trade-off between coherence and conversion efficiency. Finally, the
combination of SIL and AOP can be potentially extended to further processes,
such as the cascaded sum-frequency generation of the optical third-
harmonicYakar  _et al._ (2022); Hu  _et al._ (2022). Not least, one could
foresee, through fine-tuned dispersion engineering, to employ the same
microring resonator for the generation of a self-starting soliton
microcombShen  _et al._ (2020), whose frequency doubling could potentially
allow access to $f-2f$ interferometry on-chipBrasch  _et al._ (2017). Despite
bearing high technical difficulties, this target would come with great
benefit, allowing to realize a self-referenced microcomb on a single
integrated photonic chip. Such achievement would unravel the potential of
optical atomic clocks in a fully integrated chip-scale photonic device, and
potentially be exploited to bring hertz-level coherence over the whole near-
infrared spectrum and beyond.

In conclusion, we have demonstrated a chip-scale dual-wavelength source based
on the injection-locking of a DFB laser to a high-Q Si3N4 microresonator. The
device displays a near-hertz intrinsic linewidth of 57 Hz, milliwatt-level SH
output power and high side-modes suppression exceeding 60 dB, over a locking
bandwidth of several gigahertz. By exploiting an all-optical poling technique,
our system can be reconfigured to operate across the whole C and L telecom
bands by solely tuning the sample temperature and pump wavelength. Our
findings confirm the suitability of silicon nitride photonics for the
integration of highly-efficient second-order nonlinear processes, and open a
pathway towards the realization of novel chip-scale devices such as
miniaturized atomic clocks and fully integrated self-referenced microcombs.

_Note._ Preliminary results about this work have been presented by the authors
in the form of a conference proceedingClementi  _et al._ (2023). During the
preparation of this manuscript, a similar example of SIL-SHG in silicon
nitride has been reported onlineLi  _et al._ (2023) in the form of a preprint.

Methods

Device information. The Si3N4 microresonator we used in the experiment was
fabricated by the photonic Damascene process that features ultralow loss
operationLiu  _et al._ (2021). It is a ring structure (radius $R=896~{}{\rm\mu
m}$) coupled with a bus waveguide, both buried in SiO2 cladding. The waveguide
nominal cross-section (width $\times$ height) is $2000\times 550~{}{\rm
nm^{2}}$, whichs support multiple spatial modes at the SH wavelength. A
scanning electron microscope image of the cleaved sample reveals a fabricated
cross-section of $2150\times 572~{}{\rm nm^{2}}$, which was used for
simulation of the resonant modes. The microresonator is characterized to
exhibit normal dispersion at the pump wavelength.

AOP mapping. The formula used to calculate the slope of the doubly-resonant
trends in Fig. 2b-c isNitiss  _et al._ (2023):

$\frac{dT}{d\lambda_{\rm p}}\approx-\frac{\Delta\nu_{\rm FH}-\Delta\nu_{\rm
SH}}{\Delta\nu_{\rm FH}\left(\frac{d\lambda_{\rm FH}}{dT}-2\frac{d\lambda_{\rm
SH}}{dT}\right)}$ (1)

where $T$ is the sample temperature, $\lambda_{\rm p}$ is the pump wavelength,
$\Delta\nu_{\rm FH(SH)}$ is the FSR at the FH (SH) wavelength expressed in
hertz and $d\lambda_{\rm FH(SH)}/dT$ is the thermo-optic coefficient at the FH
(SH) wavelength. The approximation is valid as long as the ratio between the
two FSRs is close to 1. The horizontal spacing between similar trends is
calculated as:

$\Delta\lambda_{\rm spacing}\approx\frac{\lambda_{\rm p}^{2}\Delta\nu_{\rm
FH}^{2}}{2c\lvert\Delta\nu_{\rm FH}-\Delta\nu_{\rm SH}\rvert}$ (2)

where $c$ is the speed of light.

TPM imaging. For characterization of the inscribed $\chi^{(2)}$ gratings, a
high power femtosecond Ti:Sapphire laser is focused at the grating plane of
the microresonator in an upright configuration. The focal spot is then raster-
scanned across the plane while, in the meantime, its generated SH signal is
monitored so that the $\chi^{(2)}$ response is probed. From the periodicity
retrieved, the original phase mismatch between the modes involved is inferred.

Optical heterodyne. To obtain the data shown in Figs.3c and 4a, an external
cavity laser, serving as a local oscillator (LO) is tuned at frequency close
to the emission under test. The two fields are mixed at a 50:50 fiber beam-
splitter and routed to a fast photodiode, in order to retrieve the optical
beat-note. The resulting electrical signal is visualized on an electrical
spectrum analyzer, retrieving the narrowband spectrum of the emitted light.
The resolution of the technique is approximately 50 kHz, and it is limited by
the resolution bandwidth of the spectrum analyzer and by the finite linewidth
of the LO laser.

Acknowledgements

This work was funded by ERC grant PISSARRO (ERC-2017-CoG 771647).

Competing interests

The authors declare no competing interests.

Data availability

The data and code that support the plots within this paper and other findings
of this study are available from the corresponding authors upon reasonable
request.

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# Linear cooling of a levitated micromagnetic cylinder by vibration

Chris Timberlake<EMAIL_ADDRESS>School of Physics and Astronomy,
University of Southampton, Southampton SO17 1BJ, UK  Elliot Simcox  School of
Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK
Hendrik Ulbricht<EMAIL_ADDRESS>School of Physics and Astronomy,
University of Southampton, Southampton SO17 1BJ, UK

###### Abstract

We report feedback cooling of translational and librational degrees of freedom
of a levitated micromagnet cylinder, utilizing a piezoelectric actuator to
apply linear feedback to high-Q mechanical modes. The normal modes are
measured with a superconducting pick-up coil coupled to a DC SQUID, and phase
information is fed back to the piezoelectric actuator to feedback cool a
center-of-mass mode to $1.8\pm 1$ K, and a librational mode to $830\pm 200$
mK. Q-factors of $1.0\times 10^{7}$ are evaluated for the center-of-mass mode.
We find that ground state cooling of the center-of-mass mode is plausible by
introducing vibration isolation and optimizing the geometry of the pick-up
coil to focus on the specific mode of interest.

Cooling the center-of-mass motion of macroscopic objects to their quantum
ground state has long been a goal within the physics community as it is
regarded as a crucial first step towards not only observing quantum mechanical
effects on the macroscale - for example by generating spatial quantum
superpositions of single trapped large-mass particles, aka matterwave
interferometry [1, 2, 3, 4] \- but also for searching for new physics in form
of deviations from known interactions and by checking postulates of new
particles [5, 6, 7, 8, 9]. The study of gravitational effects of massive
particles in quantum states is of much interest [10, 11] as it might be a way
to shine light on the interplay between quantum mechanics and gravity via
experiments. It is understood that a larger macrosopicity of quantum states
[12] can be achieved by decoupling mechanical oscillators from their
environment by different ways of levitation. Trapping and cooling the motion
of large (larger than $\mu m$ length scale) particles to the quantum ground
state is extremely challenging. Optical trapping techniques are suited to
trapping sub-micron sized particles, and linear feedback techniques have
already been utilized in levitated optomechanics to cool nanoparticles to
their motional ground state [13, 14]. Recently, simultaneous ground state
cooling of two mechanical modes was achieved [15] and even the motion of the
large LIGO mirrors have been cooled close to the quantum ground state by
feedback [16], beside many other clamped mechanical systems [17]. However, the
absorption and recoil of photons from the trapping field act as dissipation
limit which scales with the sixth power of the radius of the trapped particle
[18], and there is a hard decoherence limit for quantum states in optical
levitation by interaction with black body and trapping laser radiation [19].

Trapping of charged particles in ion traps [20, 21, 22, 23, 22] offers more
flexibility on particle radius, but the active electric fields required for
trapping inherently induces noise on the oscillator. This ultimately limits
the center-of-mass motional quantum states of charged particles because of
anticipated charge-induced decoherence effects [24]. While ground state
cooling might be very possible, it has not been achieved for particles beyond
atomic ions. In general, the ability to manipulate and control the position
and motion of trapped particles by external fields inherently comes with the
introduction of noise and decoherence as the particle’s motion is coupling to
the environment more or less.

Instead, magnetically levitated oscillators, in particular Meissner levitated
ones, have the potential to not only trap and cool the mechanical motion of
macroscopic objects to the quantum ground state, but due to the trapping
mechanisms being entirely passive, it offers the possibility of extended
periods of coherent state evolution for subsequent preparation of motional
quantum states [25]. Furthermore, trapping magnets with a size range from
micrometer [26, 27, 28] to millimeter sized magnets [29, 30, 31] and beyond
[32] is possible in contrast to nanoscale trapping in optical and electrical
systems. The passive trapping can be regarded as a disadvantage when it comes
to controlling and manipulating the system, but we show in this paper that by
modulating vibrations, which affect all forms of particle traps, one can
selectively act on the center-of-mass motion of large-mass systems and cool
all the way to the quantum regime.

A further advantage of magnetic traps using superconductivity is the known
fact that all dominant decoherence effects are dramatically reduced in the
cryogenic environment [33]. Several schemes for ground state cooling of super-
micron sized objects have been proposed, for levitated magnets [34, 35, 36],
levitated superconductors [37, 38], and using ground state cooled
superconducting spheres for matterwave interferometry [39], but cooling near
the ground state has not yet been achieved experimentally in levitated
systems, despite efforts toward this in cooling levitated superconducting
spheres [40]. A challenge of feedback cooling macroscopic particles to the
quantum ground state is that the zero-point motion
$X_{\text{ZPM}}\sim\sqrt{\hbar/2m\omega_{0}}$ decreases with increasing mass
$m$. The ability to cool a particle depends on the strength of the control
fields that can be exerted on the particle, which typically scales as $g=\eta
X_{\text{ZPM}}$, where $\eta$ is the coupling strength to the particle’s
position. This scaling makes cooling to the ground state harder as mass
increases [35]. Additionally, macroscopic magnets tend to levitate with low
mechanical frequencies, meaning their motion is often vastly overwhelmed by
low frequency vibrational noise. Similar experiments in dry dilution
refrigerators suffer from extreme vibrations due to the pulse tube, that is
required to cool to cryogenic temperatures [40, 31]. Here we show that only
moderate vibration isolation (factor 100 in amplitude) is required to reach
the thermal noise limit in our system - which is extremely important for
ground state cooling capabilities.

In this letter, we demonstrate linear feedback cooling of both translational
and librational modes of a levitated micromagnet in a Meissner trap by
modulating vibrations with a piezoelectric actuator, and analyse in detail, by
considering leading effects, that cooling to the quantum ground state with the
very same technique is within scope in the near future.

The equation of motion for a damped, driven harmonic oscillator with an
applied feedback force can be written as

$\displaystyle\ddot{x}(t)+\Gamma_{0}\dot{x}(t)+\omega_{0}^{2}x(t)=\frac{F_{\text{th}}(t)+F_{\text{FB}(t)}}{m},$
(1)

where where $\Gamma_{0}$ is the background damping, as measured without
feedback cooling on, $F_{\text{th}}(t)$ is the driving force due to thermal
stochastic noise and $F_{\text{FB}}$ is the feedback force. The PSD of a
single oscillator mode, in thermal equilibrium with a thermal bath $T$, with
an applied feedback force with dissipation rate $\Gamma_{\text{FB}}$ can be
written as

$S_{x}(\omega)=\frac{4k_{B}T\Gamma_{0}}{m}\frac{1}{(\omega_{0}^{2}-\omega^{2})^{2}+\omega^{2}(\Gamma_{0}+\Gamma_{\text{FB}})^{2}},$
(2)

The thermal force noise of a mechanical oscillator is given by

$S_{F}=4k_{B}Tm\omega_{0}/Q,$ (3)

where $k_{B}$ is the Boltzmann constant, $T$ the temperature of the thermal
bath, $m$ is the mass of the oscillator, $\omega_{0}$ is the resonance
frequency and $Q$ is the quality factor, defined as as $Q=\omega_{0}/\Gamma$,
with $\Gamma$ being the energy dissipation rate, or damping factor. An
analogous torque noise can be expressed as $S_{\tau}=4k_{B}TI\omega_{0}/Q,$
where I is the moment of inertia. It can be seen that in order to lower the
thermal noise, high Q-factors and low thermal bath temperatures are required.

Figure 1: (a) Schematic of the experimental setup. A cylinder magnet is
levitated in a lead superconducting trap. The position is measured using a
pick-up coil and a SQUID, with the position information fed back to a
piezoelectric actuator for cold damping. (b) The coordinate system for
describing the normal modes. (c) A photograph of the NdFeB cylinder magnet, as
viewed from the circular face. The magnet has a diameter of 200 $\mu$m and a
thickness of 100 $\mu$m. A plastic rule is shown in the shot with 1 mm spacing
for reference. Figure 2: (a) An example spectrum of the levitated magnet,
showing three translational ($x$, $y$ and $z$) along with two librational
modes ($\alpha$ and $\beta$). (b) The power spectral density (PSD) of the 42.4
Hz translational mode. The orange represents the mode at a temperature of $T$
= 4.4 K and a pressure of $P$ = $3\times 10^{-1}$ mbar, which is when the
resonator is thermal noise limited and acts as a calibration. The blue
represents the same mode at $T$ = 410 mK and $P=1\times 10^{-8}$ mbar. In
these conditions the magnet is no longer thermal noise limited. The inset
shows the amplitude of the modes. (c) Shows the same as (b) but for the
178.7Hz librational mode. This save was taken at $P=4\times 10^{-8}$ mbar. At
this higher frequency, less vibrational noise is coupled into the system which
results in the motion of the mode being closer to the thermal noise limit.

The experimental setup consists of a 100 $\mu$m length x 200 $\mu$m diameter
cylindrical ferromagnet levitated in a lead superconducting box, which is
cooled to cryogenic temperatures. Below the critical temperature of lead (7.2
K), the magnet levitates due to the Meissner effect. The magnet oscillates
around a stable position, and its motion is coupled to a superconducting pick-
up coil, which itself is coupled to a DC SQUID for detection, as shown in Fig.
2, where three translational ($x$, $y$ and $z$) and two librational ($\alpha$
and $\beta$) mechanical modes are identified. The lead trap is mounted on a
spring loaded piezoelectric actuator, which is used for linear feedback
cooling. More details on the experimental setup can be found in the
Supplemental Material.

In order to calibrate the motion of our normal modes, we take a reference save
at a known temperature $T$ and a high background gas pressure $P$, such that
the motion is known to be thermal noise limited. Data taken at lower
pressures, where no longer thermal noise limited, is measured relative to this
reference data. All pressure measurements in this letter have been corrected
for helium gas pressure, and the cryogenic temperatures (see Supplemental).

Figs. 2 (b) and (c) show the reference power spectral density (PSD) compared
to the equilibrium data taken at $T$ = 410 mK and low pressure for the $z$ and
$\beta$ mode respectively. An equilibrium effective temperature of the
$T_{\text{eq}}^{z}$ = 3400 K and and $T_{\text{eq}}^{\beta}$ = 97 K is
evaluated for the $z$ and $\beta$ mode respectively. Considering the lack of
vibration isolation, these effective temperatures are remarkably low; similar
levitated systems, with comparable mass and frequency, have observed effective
temperatures of $\sim 10^{10}$ K in dry dilution refrigerators [40] without
any isolation, and temperatures of $\sim 3$ K [31] with extensive multi-stage
vibration isolation. Here, we are less than a factor of 100 in amplitude above
the thermal noise floor, meaning relatively moderate vibration isolation could
be implemented in our system. By using the piezo to excite the levitated
magnet on resonance for the $z$ and $\beta$ mode independently, the quality
factors were evaluated by ring down measurement. For the $z$ mode
$Q_{z}=1.0\times 10^{7}$ is measured, whereas $Q_{\beta}=2.1\times 10^{6}$ is
measured for the $\beta$ mode. More information can be found in the
Supplemental Material.

By using eq. (3) and the values measured for the $z$ mode, we can infer a
thermal force noise of $S_{F}^{1/2}=3.4\times 10^{-16}$ N/$\sqrt{\text{Hz}}$
at the equilibrium temperature of $T_{\text{eq}}^{z}$ = 3400 K. By introducing
sufficient vibration isolation, such that the mechanical motion is in thermal
equilibrium with the 410 mK bath, force noises of $S_{F}^{1/2}=3.7\times
10^{-18}$ N/$\sqrt{\text{Hz}}$ could be achieved. For the $\beta$ mode, we
estimate a torque noise of $S_{\tau}^{1/2}=1.5\times 10^{-20}$
Nm/$\sqrt{\text{Hz}}$ at $T_{\text{eq}}^{\beta}$ = 97 K, which with adequate
vibrational damping could be lowered to $S_{\tau}^{1/2}=9.8\times 10^{-22}$
Nm/$\sqrt{\text{Hz}}$ at at bath temperature of 410 mK.

Figure 3: PSDs comparing the reference data to uncooled data and feedback
cooled data. (a) Shows the 42.4 Hz translational mode. The reference save was
taken at $T=4.4$ K and $P=3\times 10^{-1}$ mbar. Here the feedback cooled data
was taken at $P=6\times 10^{-7}$ mbar, with a final motional temperature of
$T_{\text{eff}}^{z}$ = 1.8$\pm$1 K calculated. This temperature is extracted
by fitting to the PSD. (b) Shows the 178.8 Hz librational mode. A final
motional temperature of $T_{\text{eff}}^{\beta}$ = 830$\pm$200 mK is reached
at $P=2\times 10^{-7}$ mbar. Here the reference save was taken at $T=4.2$ K
and $P=1\times 10^{-1}$ mbar

.

By fitting to the PSD of our cooled mode, we can compare terms of fitting
parameters to a save that is thermalized to its environment to calculate the
cooled effective temperature. Our reference save for the center-of-mass mode
was taken at a pressure of $P$ = $3\times 10^{-1}$ mbar, where the magnet was
well thermalized with the background environment of $T$ = 4.4 K. For the
librational mode, the reference save was taken at $T$ = 4.2 K and $P=1\times
10^{-1}$ mbar. Assuming equipartition theorem we can write the effective
temperature of the particle mode as

$T_{\text{eff}}=T\frac{\Gamma_{0}}{\Gamma_{0}+\Gamma_{\text{FB}}},$ (4)

More details of this derivation can be found in [41, 42]. In order to apply a
feedback force, the mode of interest is tracked with a phase-locked loop. This
signal is appropriately phase shifted, such that the feedback signal is
applied as a direct force proportional to the particle’s velocity, and fed to
a piezoelectric actuator. This increases the effective damping on the motion.
The voltage gain is manually adjusted such that the cooling is maximized. This
sort of feedback cooling is referred to in the literature as cold damping,
velocity damping or linear feedback cooling [43, 44, 45]. For the $z$ mode,
which is aligned with gravity, the feedback cooling data was taken at a
measured background gas pressure of $P=2\times 10^{-7}$ mbar. Here a cooled
effective temperature of $T_{\text{eff}}^{z}=1.8\pm 1$ K is reached. For the
$\beta$ mode, which is librational, the background gas pressure was measured
as $P=6\times 10^{-7}$ mbar and an effective temperature of
$T_{\text{eff}}^{\beta}=830\pm 200$ mK was evaluated (see Fig. 3).

A natural question after achieving these temperatures is what is the limit of
feedback cooling that could be achieved with this style of detection and
feedback? One can write the minimum achievable temperature $T_{\text{min}}$
from feedback cooling, in terms of minimum phonon number $N_{\text{min}}$ as
[46, 44]

$N_{\text{min}}=\frac{k_{B}T_{\text{min}}}{\hbar\omega_{0}}=\frac{1}{2\hbar}\sqrt{S_{F}S_{x_{d}}},$
(5)

where $S_{x_{d}}$ is the detector displacement noise and $\hbar$ is Planck’s
constant. In terms of minimum temperature, we have
$T_{\text{min}}=\frac{{\omega_{0}}}{2k_{B}}\sqrt{S_{F}S_{x_{d}}}$
($T_{\text{min}}=\frac{{\omega_{0}}}{2k_{B}}\sqrt{S_{\tau}S_{\theta_{d}}}$ for
librational degrees of freedom). By reducing the vibrational noise to the
below the thermal noise limit, and by measuring to the SQUID nosie floor, a
minimum possible temperature achieved by feedback cooling is predicted for
both the z and $\beta$-mode of $T_{\text{min}}^{z}\sim$ 5 $\mu$K and
$T_{\text{min}}^{\beta}\sim$ 80 $\mu$K respectively. In terms of minimum
phonon number, this corresponds to $N_{\text{min}}^{z}\sim 2500$ and
$N_{\text{min}}^{\beta}\sim 9400$. A more thorough description of experimental
parameters can be seen in the Supplemental Material.

In order to further reduce the minimum achievable temperature, or phonon
number, either the thermal noise of the oscillator or the detector noise must
be reduced. Thermal noise reduction could be achieved by reducing the
temperature further, with a dilution refrigerator. However, given the
extensive costs involved in purchasing a new system, plus the extensive
vibrations introduced by having a pulse tube in dry systems, it isn’t a
practical solution. In our current experiment, our pick-up coil is not
optimized for detecting any particular mode. Therefore, tailoring the geometry
and orientation of the pick-up coil to be sensitive to the $z$-mode can
substantially increase the coupling of motion into the pick-up coil, and
therefore the SQUID. The magnetic flux through a loop of area $\bm{A}$ given
by the surface integral $\Phi_{B}=\iint_{S}\bm{B}\cdot d\bm{A}$. By
considering our magnet to be a magnetic dipole $\bm{\mu}$ loop to be made of
$N$ circular loops of radius $R$, we have that

$\Phi_{B}=N\bm{B}\cdot\bm{A},$ (6)

where $\bm{B}$ is the magnetic field produced by the magnet at the pick-up
coil and $\bm{A}$ is the vector area of the pick-up coil. The magnetic field
produced by a dipole is given by

$\bm{B(r)}=\frac{\mu_{0}}{4\pi r^{3}}(3\bm{(\mu\cdot\hat{r})\hat{r}-\mu}),$
(7)

where $\bm{r}$ is a position vector with unit vector $\bm{\hat{r}}$. Here, we
consider two coil orientations. One, as in the experiments presented here, has
the normal area vector of the coil perpendicular to $\bm{\mu}$, as shown in
Fig. 4 (a), and one where the coil is positioned such that the normal area
vector is parallel to $\bm{\mu}$ (Fig.4 (b)). In these orientations, the
magnetic flux through the pick-up loop is given by

Figure 4: Schematic showing the pick-up coil and magnet geometry. A pick-up
coil of $N$ turns is placed in the vicinity of a magnetic dipole $\bm{\mu}$.
(a) Shows the geometry currently used for detection. The pick-up coil is
placed at a vertical distance $z$ from the magnet, and misaligned by a small
amount $x$, to ensure all normal modes couple into the pick-up coil. (b) Shows
a proposed coil geometry where the coil is oriented at 90 degrees to the
current geometry. Such a geometry will be more strongly coupled to the
$z$-mode, while being less coupled to $x$ and $y$ modes.

$\displaystyle\Phi_{B\perp}$ $\displaystyle=\frac{3N\mu_{0}R^{2}\mu
xz}{4(x^{2}+z^{2})^{2}},$ (8) $\displaystyle\Phi_{B\parallel}$
$\displaystyle=\frac{N\mu_{0}R^{2}\mu}{4}\left(\frac{3x^{2}}{(x^{2}+z^{2})^{2}}-\frac{1}{(x^{2}+z^{2})^{5/2}}\right),$
(9)

where $\Phi_{B\perp}$ and $\Phi_{B\parallel}$ represent the cases in Fig. 4
(a) and (b) respectively, and $x$ and $z$ are the horizontal and vertical
components of the position $r$. In order to compare how the couplings of
either orientation compare for the $z$-mode, we differentiate $\Phi_{B\perp}$
and $\Phi_{B\parallel}$ with respect to $z$, resulting in

$\displaystyle\frac{\partial\Phi_{B\perp}}{\partial z}$
$\displaystyle=\frac{3N\mu_{0}R^{2}\mu x(x^{2}-3z^{2})}{4(x^{2}+z^{2})^{3}},$
(10) $\displaystyle\frac{\partial\Phi_{B\parallel}}{\partial z}$
$\displaystyle=\frac{N\mu_{0}R^{2}\mu}{4}\left(\frac{3z}{(x^{2}+z^{2})^{5/2}}-\frac{12x^{2}z}{(x^{2}+z^{2})^{3}}\right).$
(11)

Experimentally, in the current setup, we have values of $N=15$, $x=0.3$ mm,
$z=2.5$ mm, $R=1.0$ mm and $\mu=\frac{1}{\mu_{0}}B_{r}V=3.6\times 10^{-6}\
\text{A}\cdot\text{m}^{2}$, where $B_{r}$ is the residual flux density of the
magnet, and $V$ the volume of the magnet. Plugging these numbers into eq. (10)
and (11) we find that $\lvert\frac{\partial\Phi_{B\perp}}{\partial
z}\rvert=1.11\times 10^{-9}$ Wb/m and
$\lvert\frac{\partial\Phi_{B\parallel}}{\partial z}\rvert=1.25\times 10^{-6}$
Wb/m. The ratio of these couplings is
$\lvert\frac{\partial\Phi_{B\parallel}}{\partial
z}\rvert/\lvert\frac{\partial\Phi_{B\perp}}{\partial z}\rvert=1100$, meaning
that rotating the pick-up coil by 90 degrees would result in a detection noise
of $S_{x_{d}}^{1/2}=1.3\times 10^{-16}$ m/$\sqrt{\text{Hz}}$. By using eq.
(5), we find that this translates to a minimum phonon number
$N_{\text{min}}^{z}\sim 2$, or 5 nK in temperature. Given the $1/z^{4}$
dependence in eq. (11), reducing $z$ from $z=2.5$ mm to $z=2.0$ mm would
result in a coupling strong enough to reach the quantum ground state. As we
approach the quantum ground state, other limiting noise sources may become
dominant, such as SQUID backaction and current noise within the SQUID, as
considered in Ref. [40].

In conclusion, we have demonstrated feedback cooling, utilizing cold damping
with a piezoelectric actuator, of two normal modes of a levitated
micromagnetic cylinder within a superconducting trap. By driving the piezo
with an AC voltage on resonance, proportional to the velocity of the
resonator, we have reduced the effective temperture of one center-of-mass
(CoM) mode to $1.8\pm 1$ K, while a librational degree of freedom was cooled
to $830\pm 200$ mK. Such effective temperatures have yet to be reached in the
literature for levitated magnets. Our equilibrium temperatures are 3400 K and
97 K for the $z$ and $\beta$ mode respectively are within a factor of 100 of
the thermal noise floor, in amplitude. This is remarkable low compared to
similar levitated systems, where effective temperatures of up to $10^{10}$ K
are reported in dry dilution refrigerator systems. By introducing vibrational
isolation, and measuring to the SQUID noise floor, we predict temperatures of
$T_{\text{min}}^{z}\sim$ 5 $\mu$K and $T_{\text{min}}^{\beta}\sim$ 80 $\mu$K
could be obtained via feedback cooling of the z and $\beta$-mode respectively,
corresponding to minimum phonon numbers of $N_{\text{min}}^{z}\sim$ 2500 and
$N_{\text{min}}^{\beta}\sim$ 9400\. By reorienting the detection pick-up coil
to maximize magnetic coupling in the $z$-direction, it will be possible to
have couplings strong enough to feedback cool to the quantum ground state.
Such an achievement would open up a toolbox for generating macroscopic quantum
states of motion, probing fundamental physics questions and for precision
sensing applications. Our Meissner-levitation scheme strikes with
exceptionally low noise and low decoherence features, giving rise to the hope
of generation of marcoscopic quantum superposition states of single trapped
ferromagnets in the near future [4].

We would like to thank M. Paternostro for helpful comments on the manuscript,
as well as P. Connell, J. Chalk and D. Grimsey for machining the lead traps
and piezoelectric actuator mount used in this study. We acknowledge support
from the QuantERA grant LEMAQUME, funded by the QuantERA II ERA-NET Cofund in
Quantum Technologies implemented within the EU Horizon 2020 Programme, and
funding from the UK funding agency EPSRC grants (EP/W007444/1, EP/V035975/1,
EP/V000624/1, EP/X009491/1), the Leverhulme Trust project MONDMag
(RPG-2022-57), the EU Horizon 2020 FET-Open project TeQ (766900), the EU
Horizon Europe EIC Pathfinder project QuCoM (10032223).

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Supplemental Material: Linear cooling of a levitated micromagnetic cylinder by
vibration

This is the Supplemental Material for the manuscript titled ”Linear cooling of
a levitated micromagnetic cylinder by vibration”, where more details on the
experimental setup and detection system, as well as a description on how the
pressure measurements presented in the manuscript are calibrated for cryogenic
temperature, before decribing how the Q-factor is evaluated and a full list of
relevant experimental parameters are tabulated.

##  I Description of Experimental Setup

The magnetic trap consists of a superconducting well, made of the type-I
superconductor lead, with a flat elliptical base (long axis = 5 mm, short axis
= 3 mm), with a lead lid attached to fully contain the magnet inside the
superconductor. A neodymium (NdFeB) cylindrical ferromagnetic particle (100
$\mu$m length x 200 $\mu$m diameter) is placed at the bottom of the trap, and
the setup is cooled inside a helium-3 sorption refrigerator to $\approx$ 410
mK in temperature. To dampen the effects of external vibrations the apparatus
is mounted on an 850 kg granite block, which is suspended on optical table
supports with compressed air. When the superconducting transition temperature
is reached, the particle is lifted off of the lead surface and levitates due
to the Meissner currents induced in the superconducter. The walls of the well
provide lateral stability while the base of the trap, combined with gravity,
provide vertical confinement. The particle acts as a harmonic oscillator with
three translational and two librational modes.

These oscillations induce a change in magnetic flux $\Phi$ in the pick-up
coil, due to the magnetic dipole $\mu$ oscillating. The pick-up coil, of
inductance $L$, is connected to the input coil of the SQUID. Therefore the
oscillating magnet produces a flux $\Phi_{S}=MI=\frac{M}{L}\Phi$ in the SQUID,
where $I$ is the induced current in the pick-up coil and $M$ is the mutual
inductance between the SQUID and the input coil. For these experiments the
pick-up coil consists of 15 loops of 75 $\mu$m diameter NbTi wire, wrapped
around a polyether ether ketone (PEEK) holder of radius of 1 mm. This pick-up
coil holder is located above the magnet, inside the magnetic trap, as shown in
Fig. 5. For electromagnetic shielding purposes, the NbTi wires connecting the
SQUID to the pick-up coil are in twisted pairs and fed through a lead
superconducting sleeve, and the SQUID itself is housed in a Nb can.

Figure 5: Technical drawing of the lead magnetic trap used for the levitation
experiments. The main body of the trap is represented in dark gray (3), with
the PEEK pick-up coil holder in yellow (2) and the lead lid in light gray (1).
The pick-up coil holder is positioned 0.3 mm off axis to break symmetry, and
allow detection of five mechanical modes.

The entire lead trap and is housed within a cryoperm shield. The motion of the
particle is detected with a pick-up coil which is inductively coupled to a DC
SQUID. The SQUID signal output is connected to a lock-in amplifier and an
oscilloscope. The trap is spring loaded onto a piezoelectric actuator, which
is used to modulate the particle motion in the vertical direction. Each mode
is tracked using a lick-in amplifier, which tracks the phase of the mechanical
mode, which can be fed back to the piezoelectric actuator at a phase to oppose
the motion, resulting in linear feedback cooling.

##  II Correcting Pressure Measurements

The inner vacuum chamber (IVC) of our cryostat has a cold side, where
experiments are undertaken, and a warm side, where pumps and pressure gauges
are attached. All pressure measurements are taken at the warm side of the
vacuum chamber with a Pirani-Penning vacuum gauge. Such gauges measure
nitrogen standard pressure, meaning a correction factor is needed to account
for the helium gas. Additionally, the pressure at the warm side of the vacuum
chamber $P_{w}$, at a temperature $T_{w},$ will differ to the pressure at the
cold side of the chamber $P_{c}$ at temperature $T_{c}$ according to the
Weber-Schmidt model [47, 26], which states that

$\frac{P_{c}}{P_{w}}=\left(\frac{T_{c}}{T_{w}}\right)^{1/2}.$ (12)

All listed pressure measurements in the manuscript have been corrected for
helium gas and temperature.

##  III Q-factor

An important parameter used to characterise the mechanical modes of our magnet
is the quality factor, or Q-factor. This Q-factor is evaluated by resonantly
exciting the normal modes and recording how the amplitude decays once the
excitation field is no longer present, in a ring-down measurement. The
Q-factor of the $z$ and $\beta$ modes can be seen in Fig. 6 (a) and (b)
respectively.

Figure 6: Ring-down measurements to evaluate the Q-factor of the $z$ (a) and
$\beta$ (b) mechanical modes.

The Q-factors which are listed in the manuscript are evaluated via ringdown
measurement, as shown here.

##  IV Experimental Parameters

This table lists the physical parameters of the magnet, as well as measured
and predicted experimental parameters of the $z$ and $\beta$ mechanical mode.

|  Parameter | Symbol | Value | Unit
---|---|---|---|---
Properties of magnet |  Mass | m | 23 |  $\mu$g
|  Radius | r | 100 |  $\mu$m
|  Thickness | d | 100 |  $\mu$m
|  Density | $\rho$ | 7430 | $kg/m^{3}$
|  Perpendicular moment of inertia | I | $7.8\times 10^{-17}$ | $kg\cdot m^{2}$
|  Magnetization | M | $1.15\times 10^{6}$ | A/m
z-mode parameters |  Frequency | $\omega_{z}/2\pi$ | 42.4 | Hz
|  Quality factor | $Q_{z}$ | $1.0\times 10^{7}$ | -
|  Equilibrium effective temperature | $T_{\text{eq}}^{z}$ | 3400 | K
|  Force noise | $S_{F}^{1/2}$ | $3.4\times 10^{-16}$ | N/$\sqrt{\text{Hz}}$
|  Detection noise | $S_{x_{d}}^{1/2}$ | $2.1\times 10^{-11}$ | m/$\sqrt{\text{Hz}}$
|  Feedback cooled effective temperature | $T_{\text{eff}}^{z}$ | $1.8\pm 1$ | K
|  Minimum possible feedback cooled temperature | $T_{min}^{z}$ | 70 | mK
|  SQUID noise floor at 42.4 Hz | $S_{\Phi_{0}}^{1/2}$ | $\sim 0.6$ | $\mu\Phi_{0}/\sqrt{\text{Hz}}$
$\bm{\beta}$-mode parameters |  Frequency | $\omega_{z}/2\pi$ | 178.8 | Hz
|  Quality factor | $Q_{\beta}$ | $2.1\times 10^{6}$ | -
|  Equilibrium effective temperature | $T_{\text{eq}}^{\beta}$ | 97 | K
|  Torque noise | $S_{\tau}^{1/2}$ | $1.5\times 10^{-20}$ | N${\cdot}$m/$\sqrt{\text{Hz}}$
|  Detection noise | $S_{\theta_{d}}^{1/2}$ | $3.0\times 10^{-7}$ | rad/$\sqrt{\text{Hz}}$
|  Feedback cooled effective temperature | $T_{\text{eff}}^{\beta}$ | $830\pm 200$ | mK
|  Minimum possible feedback cooled temperature | $T_{min}^{\beta}$ | 180 | mK
|  SQUID noise floor at 178.8 Hz | $S_{\Phi_{0}}^{1/2}$ | $\sim 0.5$ | $\mu\Phi_{0}/\sqrt{\text{Hz}}$
z-mode expected future parameters |  Force noise at thermal equilibrium | $S_{F}^{1/2}$ | $3.7\times 10^{-18}$ | N/$\sqrt{\text{Hz}}$
|  Detection noise at SQUID noise floor | $S_{x_{d}}^{1/2}$ | $1.4\times 10^{-13}$ | m/$\sqrt{\text{Hz}}$
|  Minimum possible feedback temperature at thermal noise and SQUID noise limit | $T_{min}^{z}$ |  $\sim$ 5 |  $\mu$K
|  Minimum phonon number | $N_{min}^{z}$ |  $\sim$ 2500 | -
$\bm{\beta}$-mode expected future parameters |  Torque noise at thermal equilibrium | $S_{\tau}^{1/2}$ | $9.8\times 10^{-22}$ | Nm/$\sqrt{\text{Hz}}$
|  Detection noise at SQUID noise floor | $S_{\theta_{d}}^{1/2}$ | $2.1\times 10^{-9}$ | rad/$\sqrt{\text{Hz}}$
|  Minimum possible feedback temperature at thermal noise and SQUID noise limit | $T_{min}^{\beta}$ |  $\sim$ 80 |  $\mu$K
|  Minimum phonon number | $N_{min}^{\beta}$ |  $\sim$ 9400 | -
Table 1: Table showing the relevant experimental parameters of the magnet and
normal modes of interest. In the top row basic parameters of the magnet itself
are listed, followed by the measured parameters of the z-mode and $\beta$-mode
respectively. Finally, predicted experimental values for the z and
$\beta$-mode are listed. These predictions are based on introducing sufficient
vibration isolation to reach the thermal noise limit, and by measuring to the
full capability of the SQUID used for readout. Currently the detection noise
is limited by the data acquisition, rather than SQUID noise.