kroeger-vink=true

# Microwave-based quantum control and coherence protection of tin-vacancy spin
qubits in a strain-tuned diamond membrane heterostructure

Xinghan Guo1,∗ Alexander M. Stramma2,∗ Zixi Li1 William G. Roth2 Benchen
Huang3 Yu Jin3 Ryan A. Parker2 Jesús Arjona Martínez2 Noah Shofer2 Cathryn P.
Michaels2 Carola P. Purser2 Martin H. Appel2 Evgeny M. Alexeev2,4 Tianle Liu5
Andrea C. Ferrari4 David D. Awschalom1,5,6 Nazar Delegan1,6 Benjamin
Pingault6,7 Giulia Galli1,3,6 F. Joseph Heremans1,6 Mete Atatüre2,† Alexander
A. High1,6,† 1Pritzker School of Molecular Engineering, University of Chicago,
Chicago, IL 60637, USA  2Cavendish Laboratory, University of Cambridge,
Cambridge CB3 0HE, United Kingdom  3Department of Chemistry, University of
Chicago, Chicago, IL 60637, USA  4Cambridge Graphene Centre, University of
Cambridge,Cambridge CB3 0FA, United Kingdom  5Department of Physics,
University of Chicago, Chicago, IL 60637, USA  6Center for Molecular
Engineering and Materials Science Division, Argonne National Laboratory,
Lemont, IL 60439, USA  7QuTech, Delft University of Technology, 2600 GA Delft,
The Netherlands

###### Abstract

Robust spin-photon interfaces in solids are essential components in quantum
networking and sensing technologies. Ideally, these interfaces combine a long-
lived spin memory, coherent optical transitions, fast and high-fidelity spin
manipulation, and straightforward device integration and scaling. The tin-
vacancy center (SnV) in diamond is a promising spin-photon interface with
desirable optical and spin properties at $1.7\text{\,}\mathrm{K}$. However,
the SnV spin lacks efficient microwave control and its spin coherence degrades
with higher temperature. In this work, we introduce a new platform that
overcomes these challenges – SnV centers in uniformly strained thin diamond
membranes. The controlled generation of crystal strain introduces orbital
mixing that allows microwave control of the spin state with
$99.36(9)\text{\,}\mathrm{\%}$ gate fidelity and spin coherence protection
beyond a millisecond. Moreover, the presence of crystal strain suppresses
temperature dependent dephasing processes, leading to a considerable
improvement of the coherence time up to
$223(10)\text{\,}\mathrm{\SIUnitSymbolMicro s}$ at $4\text{\,}\mathrm{K}$, a
widely accessible temperature in common cryogenic systems. Critically, the
coherence of optical transitions is unaffected by the elevated temperature,
exhibiting nearly lifetime-limited optical linewidths. Combined with the
compatibility of diamond membranes with device integration, the demonstrated
platform is an ideal spin-photon interface for future quantum technologies.

††preprint: APS/123-QED

## Introduction

Color centers in diamond are a leading platform in quantum technologies, key
achievements such as the demonstration of a quantum register [1, 2, 3],
distant entanglement generation between three nodes [4], quantum teleportation
[5], along with myriad landmarks in quantum sensing [6, 7]. In recent years,
group IV centers have gained much attention due to their excellent optical
properties [8, 9, 10, 11, 12, 13, 14, 15]. Their $D_{3d}$ symmetry renders
optical transitions insensitive to first-order charge noise [16, 17, 18].
Additionally, a favorable Debye Waller factor leads to the majority of photons
being emitted into the zero-phonon line, critical for spin-photon entanglement
[19]. However, the electronic structure of group IV centers – a spin 1/2
system with two ground state orbital branches – renders the electron spin
susceptible to phonon-driven transitions between the two branches [20]. This
temperature-dependent spin dephasing can be mitigated by operating at
millikelvin temperatures [21, 22] or by engineering the local phonon density
of states through nanostructuring [23, 24]. Alternatively, dephasing can be
mitigated by qubit engineering such as working with group IV centers with high
spin-orbit coupling and thus large orbital splitting [25], or by leveraging
spin-strain interaction in randomly-, or controllably strained group IV
centers[3, 24]. With a spin-orbit coupling significantly higher than those of
the silicon vacancy (SiV) and the germanium vacancy (GeV) centers, the SnV
center has the highest reported spin coherence time at
$1.7\text{\,}\mathrm{K}$ [26]. However, efficient microwave (MW) control of
group IV spins requires the magnitude of spin-strain interaction to be
comparable with the spin-orbit interaction, which for SnV necessitates strain
approaching $0.1\text{\,}\mathrm{\%}$. This degree of strain is challenging to
achieve in microelectrical mechanical structures (MEMS) such as diamond
cantilevers, with reported values on the order of $0.015\text{\,}\mathrm{\%}$
[23]. Therefore, a controlled process to generate $\approx
0.1\text{\,}\mathrm{\%}$ strain in diamond is desired to improve SnV qubit
performance by both increasing the operational temperature and enabling
efficient MW driving.

Figure 1: Strained SnV in diamond membrane heterostructures. (a) Schematics of
the diamond-fused silica heterostructure. The static, tensile strain inside
the membrane is generated from the disparity of thermal expansion ratios of
diamond and fused silica. (b) The microscope image of the diamond membrane
(dashed cyan region) bonded to the fused silica substrate. A trench (dashed
green region) was fabricated prior to bonding. The gold coplanar waveguide is
fabricated post bonding to introduce microwave signals. The location of the
SnV center used in this study is highlighted by a red star. (c) Energy level
of strained SnVs. Unstrained centers, strained centers and strained centers in
the presence of a magnetic field are colored in purple, blue and green,
respectively. (d) The PL spectrum of a strained SnV center (orange), showing a
red-shifted zero-phonon line (ZPL) wavelength with a much larger ground-state
splitting compared with the values in bulk diamond (purple). (e) The
statistics of the SnV ground-state splitting. Two different devices with
identical layout were measured. Device 1 (orange) was used for all-optical
spin control (discussed in the SI) and device 2 (purple) was used for
microwave spin control.

In this work, we utilize heterogeneous integration of diamond membranes to to
generate strain-tuned SnVs. By bonding SnV incorporated pristine diamond
membranes to a glass substrate, we leverage the heterogeneous thermal
expansion coefficients of the two materials to generate a uniform, in-plane
strain in the diamond to the order of $0.1\text{\,}\mathrm{\%}$. This strain
greatly increases the energy splitting between the two orbital levels of the
SnV and induces orbital mixing in the spin ground state. We demonstrate MW
manipulation of the spin with $99.36(9)\text{\,}\mathrm{\%}$ Rabi fidelity at
$4.50(2)\text{\,}\mathrm{MHz}$ for 24 dBm MW input power. At
$1.7\text{\,}\mathrm{K}$, the implementation of dynamical decoupling allows
the SnV to reach millisecond coherence time, which is largely preserved even
at $4\text{\,}\mathrm{K}$, owing to the strain-induced increased ground state
orbital splitting. In combination with near lifetime-limited optical
linewidths up to $7\text{\,}\mathrm{K}$, our spin-photon interface is
compatible with broadly utilized low-infrastructure and cost-effective
portable cryogenic systems. Additionally, the demonstrated strained-membrane
heterostructure maintains robustness and flexibility for additional photonic,
electronic, and micro-electromechanical systems (MEMS) integration. Our SnV-
based diamond membrane platform greatly reduces the technological barrier for
establishing quantum nodes for networking.

### SnVs in strained diamond

This work relies on strain engineering to improve SnV qubit performance.
First, we demonstrate that heterogeneous thermal expansion disparities between
diamond and glass in a diamond-membrane heterostructure are sufficient to
generate uniform strain of the magnitude necessary to beneficially impact SnV.
The diamond membranes used in this work were generated via the “smart-cut”
method combined with isotopically purified (${}^{12}C$) overgrowth. The
membrane thickness is nominally 150 nm, with pristine crystal quality and
atomically smooth surfaces [27]. To introduce a positive tensile strain inside
the diamond membrane, we bond them onto
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$-thick fused silica substrates—a
material with a low thermal expansion coefficient ($<1\times
10^{-6}$$\text{\,}{\mathrm{K}}^{-1}$) – using a layer of hydrogen
silsesquioxane (HSQ). The schematic of this strain generation method is shown
in Figure 1 (a). The device is then annealed at
$600\text{\,}\mathrm{\SIUnitSymbolCelsius}$, beyond the temperature at which
the HSQ solidifies to glass, bonding the heterostructure in a ”zero-strain”
condition [28]. Due to the mismatch in thermal contraction between diamond and
fused silica and the negligible thickness of the diamond membrane compared to
that of the fused silica substrate, cooling down the device to cryogenic
temperature regime generates a positive (tensile), static strain profile in
the diamond membrane with an estimated magnitude of
$0.05\text{\,}\mathrm{\%}0.1\text{\,}\mathrm{\%}$ (see section 1.3 and 1.4 in
SI for details). This passive, uniform, and membrane-compatible strain
generation is complimentary to recent demonstrations of electromechanically-
induced strain on suspended diamond beams [24, 29].

Figure 1 (b) is the microscope image showing the layout of our diamond-
membrane heterostructure device. Prior to the membrane bonding, we patterned
and etched a $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ deep trench on the
fused silica to suspend part of the membrane and mitigate background
fluorescence from the HSQ resist. To study MW control of the SnV centers, we
patterned and deposited gold coplanar waveguides following membrane bonding.

The strain monotonically increases the orbital splitting of the SnV centers in
the membranes, which can be directly verified in the photoluminescence (PL)
spectra at $1.7\text{\,}\mathrm{K}$. The energy level diagram of the strained
SnV is shown in Figure 1 (c), highlighting the ground state orbital splitting
($\Delta_{gs}$) and the respective contributions of spin-orbit coupling,
strain, and magnetic Zeeman interaction in purple, blue, and green boxes.
Figure 1 (d) compares the spectra of a strained (unstrained) SnV center in a
diamond membrane (bulk diamond) with $\Delta_{gs}=$ $\approx
1300(850)\text{\,}\mathrm{GHz}$. This particular strained center is used in
further optical, microwave and spin characterizations in this work.
Remarkably, we note that all color centers in the membrane are comparably
strained. As shown in Figure 1 (e), we observed a distribution of the orbital
branches splitting centered around $1500\text{\,}\mathrm{GHz}$ across
different devices with a minimum (maximum) value of
$1200(1800)\text{\,}\mathrm{GHz}$. We carried out density functional theory
(DFT) calculations to compute strain-susceptibilities and characterize the SnV
spin-strain interaction (see SI); our results show that the increase of the
splitting between orbital branches from $850\text{\,}\mathrm{GHz}$ to $\approx
1500\text{\,}\mathrm{GHz}$ due to strain, corresponds to a diamond membrane
strain magnitude of $0.075\text{\,}\mathrm{\%}$(see section 1.2 in the SI for
details). The consistent strain generation, in combination with our ability to
perform additional integration and nanofabrication following membrane bonding
[30, 31], highlights the robustness and versatility of our platform.

### Optical properties of SnV under strain

To investigate the potential of strained SnV as a spin-photon interface, we
first verify that the symmetry of the defect is preserved even under
considerable strain by characterizing the optical transitions as a function of
the magnetic ($B$) field orientation. Using the $\langle 111\rangle$
crystallographic axis – the high symmetry axis of the SnV as the reference, we
rotate the $B$ field in both polar ($\theta$) and azimuthal ($\phi$) angles at
the same magnitude ($0.2\text{\,}\mathrm{T}$). The absolute energy splitting
between the two spin-conserving transitions (A1-B2) with respect to $\theta$
and $\phi$ is shown in Figure 2 (a), indicating that large splittings at
moderate values of magnetic field are achievable which is ideal for later SnV
spin initialization and control. Similarly to the unstrained case, we observe
a $\phi$ rotational symmetry of the splitting with respect to $\langle
111\rangle$, which corresponds to the intrinsic spin quantization axis. We
further verify that the polarization of the SnV transitions (i.e. dipole
operator matrix elements) remain along the $\langle 111\rangle$ direction (see
section 3.1 of the SI), as in the unstrained case [18].

Figure 2: Optical properties of the strained SnV center under applied magnetic
fields at $1.7\text{\,}\mathrm{K}$. (a) The energy splitting rate between the
A1-B2 spin conserving transitions with respect to the polar angle $\theta$ of
the applied magnetic field at different azimuthal angle $\phi$. The aligned
field is highlighted with a black arrow. (b) PLE scan, averaged over
$20\text{\,}\mathrm{s}$, of the {A1, B2} transitions at an aligned $B$-field
with a magnitude of $81.5\text{\,}\mathrm{mT}$. The average linewidth for both
transitions are below $48\text{\,}\mathrm{MHz}$, which is less than 1.5 times
of the lifetime limited value ($32.26(19)\text{\,}\mathrm{MHz}$). (c) The
initialization curve of the A1 transition, showing a time constant of
$24.2(3)\text{\,}\mathrm{\SIUnitSymbolMicro s}$ and an initialization fidelity
of $98.82\text{\,}\mathrm{\%}$.

From the $B$-field scan of the strained SnV, we note that besides the normal
A1-B2 splitting maximum along the quantization axis, an additional local
maximum at $\theta=$$90\text{\,}\mathrm{\SIUnitSymbolDegree}$ – the equator
plane perpendicular to the quantization axis – is observed, with the relative
A1-B2 position being inverted, as verified by coherent population trapping
measurements (see SI). This differs from the unstrained case. The novel
feature arises from the moderate crystal strain (comparable in magnitude to
the spin-orbit coupling) which increases the difference in effective Zeeman
shift between ground and excited states, mostly visible for a magnetic field
orthogonal to the spin-orbit-dictated quantization axis. As is the case for
moderately strained SiV centers [22] for MW-based control, we roughly align
the $B$-field towards the quantization axis to achieve highly cycling optical
transitions with cyclicity reaching $\eta\approx 2500$ (see section 4.2 of
SI). We note that $\eta$ can be as low as 6 when the $B$ field is
perpendicular to the quantization axis, which is ideal for Raman-based all-
optical control of strained SnV (see section 4.3 of SI). Moreover, by
comparing the dependence on $\theta$ of the A1-B2 splitting with calculated
results, we are able to determine the Stevens reduction factor $g_{L}$ for
ground and excited states mentioned in [32]. This model is then used to
explain the optically detected magnetic resonance (ODMR) frequency of the
strained SnV discussed below.

Additionally, our measurements reveal near-transform limited optical
linewidths, thereby showing that the application of strain does not alter the
excellent coherence properties of the optical transitions, as previously
demonstrated with unstrained centers [25, 11]. As shown in Figure 2 (b), the
$20\text{\,}\mathrm{s}$ average scan returns a mean linewidth of
$47.4(16)\text{\,}\mathrm{MHz}$, only $40\text{\,}\mathrm{\%}$ more than the
lifetime-limited value of $32.26(19)\text{\,}\mathrm{MHz}$
($4.933(190)\text{\,}\mathrm{ns}$ optical lifetime, see section 3.2 of SI).
The long term frequency stability of the {A1, B2} transitions returns a center
frequency standard deviation of $\sigma_{c}=$$23.8(1)\text{\,}\mathrm{MHz}$
and a A1-B2 splitting standard deviation of
$\sigma_{s}=$$13.28(6)\text{\,}\mathrm{MHz}$ (see section 3.4 of SI). This
linewidth and peak stability is comparable to that of other measurements of
group IV color centers in nanostructures [13, 3, 33] and thus confirms the
excellent potential of these defects for quantum photonic applications.

The resolvable splitting and narrow optical transitions are crucial for the
spin initialization and readout of the SnV qubit. The spin initialization
curve with subtracted background is shown in Figure 2 (c), indicating a fitted
exponential decay constant of $24.2(3)\text{\,}\mathrm{\SIUnitSymbolMicro s}$.
The initialization pulse duration was set to
$200\text{\,}\mathrm{\SIUnitSymbolMicro s}$ allowing us to reach a fidelity of
$98.8\text{\,}\mathrm{\%}$. We note that with a cyclicity of over
$2500\text{\,}$, this platform is a prime candidate for single shot readout if
the signal counts can be improved via on-chip structures (nanophotonics, fiber
couplers or grating couplers, solid immersion lenses) [34, 35, 33, 36, 37, 38]
or external methods (microcavities) [39, 40, 41].

### Efficient MW control of the SnV spin

Figure 3: MW control of the strained SnV center at $1.7\text{\,}\mathrm{K}$.
(a) Pulsed ODMR spectrum with scanned MW frequency. The data (purple dots) is
fitted with two Lorentzian functions (dashed line) split by
$628(182)\text{\,}\mathrm{kHz}$ and with a linewidth of
$1047(208)\text{\,}\mathrm{kHz}$ and $891(197)\text{\,}\mathrm{kHz}$,
respectively. (b) Rabi oscillation of the SnV at zero detuning, indicating a
Rabi frequency of $4.50(2)\text{\,}\mathrm{MHz}$ with a fidelity of
$99.36(9)\text{\,}\mathrm{\%}$. (c) Rabi oscillation as a function of the MW
driving frequency. (d) Randomized benchmarking at $1.7\text{\,}\mathrm{K}$,
showing an average gate fidelity of $97.7(1)\text{\,}\mathrm{\%}$. The Rabi
frequency is set to $2.8\text{\,}\mathrm{MHz}$ to avoid excess heating
effects.

A critical component of a spin-photon interface is high-fidelity spin control,
commonly achieved through MW driving of the electron spin. In the case of
group IV centers, a MW field can only drive the spin transition in the
presence of strain [42, 23]. This arises due to the orthogonality of orbital
states associated with the electron spin qubit of group IV centers [18].
Strain that is comparable in strength to spin-orbit coupling relaxes this
orthogonality, enabling microwave control. SnV, with larger spin-orbit
coupling ($850\text{\,}\mathrm{GHz}$) and smaller strain susceptibility than
SiV and GeV, requires large crystal strain to meet this criteria. This strain
requirement goes beyond the achievable magnitude demonstrated via active
strain tuning [23] or implantation-induced strain [3].

To demonstrate efficient MW control, we utilize the nominal
$0.1\text{\,}\mathrm{\%}$ crystal strain in the diamond membrane. We estimate
an effective Landé factor $g$ of $1.62\text{\,}$ for the transverse microwave
field with the external magnetic field roughly aligned to the SnV quantization
axis (see section 2.1 in SI). This value is relatively high compared with
spin-orbit-dominated regime for unstrained centers ($\leq 0.3\text{\,}$) and
is close to the free electron value ($g=2$). In addition, we tapered the MW
waveguide around the measurement area by shrinking its width to
$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to enhance the microwave amplitude,
as shown in Figure 1 (b). The distance between the target SnV and the
waveguide is $\approx 4\text{\,}\mathrm{\SIUnitSymbolMicro m}$, ensuring an
efficient exposure to the MW driving field (see section 2.1 - 2.3 in SI for
details).

We begin the MW control characterization by initializing the spin via optical
pumping and scan the frequency of a MW field across the expected spin
resonance while monitoring the fluorescence intensity of the spin readout at
$1.7\text{\,}\mathrm{K}$. In Figure 3 (a) we observe clear signature of
optically detected magnetic resonance (ODMR) for the target SnV center. The
$81.5\text{\,}\mathrm{mT}$ external magnetic field is aligned to the
quantization axis by polarisation measurements and 3D field scan. The ODMR
shows a profile with two overlapping peaks separated by
$628(182)\text{\,}\mathrm{kHz}$, indicating an interaction between the
electronic spin of the SnV with another system in the vicinity, likely a
[^13C] nuclear spin or the electron spin of a P1 center. Further investigation
is needed to understand the nature of this interaction. By driving both power-
broadened ODMR transitions, we are able to resonantly manipulate the spin
state of the SnV with a Rabi frequency of $4.50(2)\text{\,}\mathrm{MHz}$. The
Rabi oscillation curve and the chevrons (Rabi oscillations with varied driving
frequency) are shown in Figure 3 (b) and (c). We observe a long-time averaged
Rabi $\pi$-gate fidelity of $99.36(9)\text{\,}\mathrm{\%}$, improving
significantly from previously demonstrated optical Raman-based spin control
value [26]. We note that the MW power delivered to the device is approximately
24 dBm ($250\text{\,}\mathrm{mW}$) which is comparable to previous
demonstrations on strained SiV [3]. We also characterized the power dependence
of the Rabi rate. Starting from a linear dependence, the Rabi rate deviates to
sub-linear when the power surpasses 24 dBm due to excessive heating (see
section 2.4 in SI), which could be optimized by replacing gold with
superconducting metals (such as niobium or NbTiN) to deliver the MW signal.

We further characterize the single qubit gate fidelity of MW control via
randomized benchmarking. For this, we use the following set of Clifford gates:
{$I$, $\pi_{x}$, $\pi_{y}$, $\pi_{x}/2$, $-\pi_{x}/2$, $\pi_{y}/2$,
$-\pi_{y}/2$} (see section 5.1 in SI). To prevent excessive heating effect
during benchmarking which would lead to undesired spin decoherence, we apply a
slightly slower Rabi rate ($2.8\text{\,}\mathrm{MHz}$, 18 dBm) which requires
no time buffer between gates. The benchmarking result is shown in Figure 3
(d). We extract an average Clifford gate fidelity of
$97.7(1)\text{\,}\mathrm{\%}$, indicating power efficient MW control with high
fidelity under stringent randomized benchmarking.

### SnV spin coherence properties

We next utilize microwave control to characterize the SnV coherence at
$1.7\text{\,}\mathrm{K}$. We perform a Ramsey measurement as shown in Figure 4
(a). The Gaussian envelope of the Ramsey oscillations corresponds to a spin
dephasing time $T_{2}^{*}$ of $2.5(1)\text{\,}\mathrm{\SIUnitSymbolMicro s}$.
Similar to ODMR, we observe interaction with a proximal spin in the Ramsey
measurement, and we verify that this does not originate from the detuning of
the MW signal via phase dependent readout (see section 5.2 in SI). Possible
decoherence sources could be nearby vacancies and defects in the diamond
membrane, as well as surface spins from both sides of the membrane [43].

Figure 4: Spin coherence of the strained SnV at $1.7\text{\,}\mathrm{K}$. (a)
$T_{2}^{*}$ Ramsey of the SnV center, showing a dephasing time of
$2.5(1)\text{\,}\mathrm{\SIUnitSymbolMicro s}$. The extra beating pattern of
$554(5)\text{\,}\mathrm{kHz}$ is estimated to be an interaction with the
electron or nuclear spin in the vicinity. (b) Dynamical decoupling of the SnV
via CPMG pulses. The CPMG-1 (spin-echo) returns a $T_{2,echo}$ of
$100(1)\text{\,}\mathrm{\SIUnitSymbolMicro s}$, while the CPMG-128 reaches a
$T_{2,CPMG128}$ of $1.57(8)\text{\,}\mathrm{ms}$. (c) The scaling of $T_{2}$
with the number of CPMG and XY pulses, showing a sub-linear dependence.

Advanced pulse sequences, such as dynamical decoupling via CPMG (Carr-Purcell-
Meiboom-Gill) and XY pulse sequences [44, 45], allow us to extend the spin
coherence to millisecond timescales. The CPMG results are shown in Figure 4
(b). The $T_{2,echo}$ returns a value of
$100(1)\text{\,}\mathrm{\SIUnitSymbolMicro s}$, which is already longer than
$35.5(30)\text{\,}\mathrm{\SIUnitSymbolMicro s}$ measured using all-optical
spin echo process (see section 4.3 and 4.4 in SI), in the absence of optically
induced dephasing mechanisms. The $T_{2,CPMG128}$, comprising 128 refocusing
microwave pulses, prolongs the SnV spin coherence to
$1.57(8)\text{\,}\mathrm{ms}$. We note that with no signal normalization being
applied, the CPMG figure indicates a high signal fidelity of $\approx
80\text{\,}\mathrm{\%}$ for up to 128 pulses. Future developments on the MW
driving fidelity including superconducting metals and faster Rabi pulses can
further improve the signal fidelity to higher numbers of pulses. We plot the
relationship between the $T_{2}$ and the number of CPMG or XY pulses $N$ in
Figure 4 (c) and fit it with $T_{2}\sim N^{\beta}$. The fitting curve returns
a sub-linear dependence with a $\beta$ factor of $0.593(8)\text{\,}$. We
observed minimal $T_{2}$ differences between CPMG and XY sequences. XY
sequences are more resilient to control pulse errors compared to CPMG [45],
verifying that the observed coherence is not limited by our control (see
section 5.4 in SI).

### Spin-photon interface at $4\text{\,}\mathrm{K}$

Finally, we demonstrate that our strained SnV platform shows state-of-the-art
spin coherence for Group IV color centers at $4\text{\,}\mathrm{K}$. For Group
IVs, the dominant decoherence source of the electronic spin is the electron-
phonon interaction (phonon-mediated decay) between orbital branches [20, 42].
The electron-phonon interaction rate depends on the temperature-dependent
phonon population and the energy splitting $\Delta_{gs}$ between orbital
branches. Therefore, enhanced coherence of the group IV centers can be
achieved via either cooling down to millikelvin temperature [22, 21],
increased energy splitting by using heavier group IV elements [25],
engineering of the phonon density of states [46], or strain engineering [24].
Here we utilize both a heavy element (Sn as compared to Si and Ge) and crystal
strain in diamond to improve electron spin coherence at elevated temperatures.

The Rabi oscillation of the SnV at $4\text{\,}\mathrm{K}$ is shown in Figure 5
(a). The fidelity is characterized to be $97.7(5)\text{\,}\mathrm{\%}$, only
slightly lower than the value at $1.7\text{\,}\mathrm{K}$ due to background
heating limitations. We characterize the average gate fidelity via randomized
benchmarking at $4\text{\,}\mathrm{K}$ using the same
$2.8\text{\,}\mathrm{MHz}$ Rabi rate, returning a gate fidelity of
$95.7(3)\text{\,}\mathrm{\%}$, confirming the maintained high performance spin
manipulation of the strained SnV at $4\text{\,}\mathrm{K}$.

Figure 5: Performance of the strained SnV center at $4\text{\,}\mathrm{K}$.
(a) Rabi oscillation of the SnV center, showing a gate fidelity of
$97.7(5)\text{\,}\mathrm{\%}$ (b) Randomized benchmarking at
$4\text{\,}\mathrm{K}$, showing an average gate fidelity of
$95.7(3)\text{\,}\mathrm{\%}$. (c) Temperature dependence of the spin decay
time $T_{1}^{\text{spin}}$, dephasing times $T_{2}^{*}$, $T_{2,\text{echo}}$,
and $T_{2,\text{2XY8}}$. (d) ZPL linewidths of the two spin conserving
transitions (A1, B2) with respect to the temperature, showing negligible
broadening with the maximum linewidth below $52.0(8)\text{\,}\mathrm{MHz}$.
The transform-limited linewidth is shown with a dashed line.

Equipped with high fidelity Rabi control, we investigate the spin coherence of
the SnV centers at elevated temperatures. Due to the much larger splitting
$\Delta_{gs}$ of the strained SnV ($\approx 1300\text{\,}\mathrm{GHz}$)
compared with bulk SnV ($\approx 850\text{\,}\mathrm{GHz}$), electron-phonon
dephasing onsets at higher temperatures. Figure 5 (c) shows the
$T_{1}^{\text{spin}}$, $T_{2}^{*}$, $T_{2,echo}$ and $T_{2,2XY8}$ versus
temperature. Fitting the same $\beta$ factor in $T_{2}\sim N^{\beta}$ using
Hahn-echo and XY4 coherence times returns a value of $0.391(8)\text{\,}$ at
$4\text{\,}\mathrm{K}$ and $0.014()\text{\,}$ at $4.5\text{\,}\mathrm{K}$,
indicating that the dominant decoherence mechanism becomes phonon-induced
orbital transitions.

From Figure 5 (c) we notice a much lower dephasing time compared with the
decay time $T_{1}^{\text{spin}}$ [47]. This feature originates from the fact
that only spin-flipping transitions between the lower and upper orbital branch
drive $T_{1}^{\text{spin}}$, whereas $T_{2}$ is sensitive to dephasing by the
spin-conserving transitions due to different precession frequencies in the
orbital branches [23]. In our case, the phonon transitions are highly cycling
due to the aligned magnetic field. Nevertheless, $T_{2}^{*}$ at
$4\text{\,}\mathrm{K}$ remains at $2.7(1)\text{\,}\mathrm{\SIUnitSymbolMicro
s}$ – comparable to the $1.7\text{\,}\mathrm{K}$ value, and $T_{2,echo}$ only
decreases slightly to $74(2)\text{\,}\mathrm{\SIUnitSymbolMicro s}$, with
$T_{2,2XY8}$ reaching the depolarization-limited $T_{2}$ –
$223(10)\text{\,}\mathrm{\SIUnitSymbolMicro s}$. It is worth emphasizing that
all of these are record high values for all group IV spin qubits at
$4\text{\,}\mathrm{K}$ to date.

To demonstrate the potential of the strained SnV center as a promising spin-
photon interface at elevated temperature, we investigate the temperature
dependence of the SnV optical coherence. As shown in Figure 5 (d), we observe
that the ZPL linewidth remains unchanged for both A1 and B2 transitions up to
$7\text{\,}\mathrm{K}$ with the maximum linewidth remaining below
$52.0(8)\text{\,}\mathrm{MHz}$—only $60\text{\,}\mathrm{\%}$ higher than
lifetime-limited values. In the future, modest Purcell enhancement of SnV
emission rates with on-chip nanophotonics or microcavities can generate fully
lifetime-limited photons suitable for efficient entanglement generation.

## Conclusions

In this work, we demonstrate that SnV in strained diamond membranes is a
promising platform for quantum technologies. We create simple heterostructures
that leverage differences in thermal expansion to passively generate
significant strain of $0.05\text{\,}\mathrm{\%}$ to $0.1\text{\,}\mathrm{\%}$
in diamond, enabling efficient, high fidelity microwave control of the SnV
spin. The presence of the strain also suppresses the phonon-mediated decay and
improves the spin coherence of the SnV at $4\text{\,}\mathrm{K}$, which
greatly reduces the technological barrier for quantum networking applications.
We reach a Rabi $\pi$ gate fidelity of $99.36(9)\text{\,}\mathrm{\%}$
($97.7(5)\text{\,}\mathrm{\%}$) with a randomized single qubit gate fidelity
of $97.7(1)\text{\,}\mathrm{\%}$ ($95.7(3)\text{\,}\mathrm{\%}$) at
$1.7\text{\,}\mathrm{K}$ ($4\text{\,}\mathrm{K}$). Dynamical decoupling
sequences allow the SnV spin coherence to reach $1.57(8)\text{\,}\mathrm{ms}$
at $1.7\text{\,}\mathrm{K}$ and $223(10)\text{\,}\mathrm{\SIUnitSymbolMicro
s}$ at $4\text{\,}\mathrm{K}$. In the future this value can be further
enhanced by generating higher strain through heterostructure optimization
and/or additional active tuning. Our platform, derived from scalable diamond
membrane generation, is compatible with further on-chip integration, such as
microwave coplanar waveguides, integrated photonics [31], and MEMS. Finally,
$4\text{\,}\mathrm{K}$ cryostats are relatively affordable and less
infrastructure-intensive in comparison to cryogen-free
$1.7\text{\,}\mathrm{K}$ and $\text{\,}\mathrm{mK}$ dilution-fridge systems.
Therefore, the demonstrated spin-photon interface at $4\text{\,}\mathrm{K}$
can reduce barriers to widespread utilization and deployment of solid-state
quantum technologies.

## Acknowledgements

This work on strain engineering of Group IV color centers is supported by the
Air Force Office of Scientific Research under award number FA9550-22-1-0518.
This work acknowledges funding through Q-NEXT, supported by the U.S.
Department of Energy, Office of Science, National Quantum Information Science
Research Centers. The experiment receives support from the ERC Advanced Grant
PEDESTAL (884745) and the EU Quantum Flagship 2D-SIPC. Membrane integration
research is supported by NSF award AM-2240399. Diamond growth related efforts
were supported by the U.S. Department of Energy, Office of Basic Energy
Sciences, Materials Science and Engineering Division (N.D.). The membrane
bonding work is supported by NSF award AM-2240399. This work made use of the
Pritzker Nanofabrication Facility (Soft and Hybrid Nanotechnology Experimental
Resource, NSF ECCS-2025633) and the Materials Research Science and Engineering
Center (NSF DMR-2011854) at the University of Chicago. A.M.S. acknowledges
support from EPSRC/NQIT, R.A.P. from the General Sir John Monash Foundation
and G-research, J.A.M. from the Winton Programme and EPSRC DTP, C.P.M. from
the EPSRC DTP. B. P. acknowledges funding from the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie Grant
Agreement No. 840968. The authors thank Srujan Meesala, Dorian Gangloff for
insightful discussions, Haoxiong Yan and Ming-Han Chou for experimental help.

∗ These authors contributed equally to this work.

† Correspondence should be addressed to<EMAIL_ADDRESS><EMAIL_ADDRESS>
## Competing interest

A. A. H., X. G., Z. L., T. L., N. D., and F. J. H. filed a provisional patent
for the strain generation of bonded membranes.

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