††thanks: H.S and H.B. contributed equally to this work.††thanks: H.S and H.B.
contributed equally to this work.††thanks: Current affiliation: Division of
Quantum Metrology and Nanotechnologies, Istituto Nazionale di Ricerca
Metrologica (INRiM), Strada delle Cacce 91, 10135 Torino, Italy††thanks:
Correspondence<EMAIL_ADDRESS>
# Optically detected magnetic resonance with an open source platform

Hossein Babashah  Institute of Physics, École Polytechnique Fédérale de
Lausanne (EPFL), Lausanne CH-1015, Switzerland  WAINVAM-E, 1 rue Galilée,
56270 Plœmeur, France  Hoda Shirzad  Institute of Physics, École Polytechnique
Fédérale de Lausanne (EPFL), Lausanne CH-1015, Switzerland  Center for Quantum
Science and Engineering, EPFL, Lausanne, Switzerland  Elena Losero  Institute
of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne CH-1015,
Switzerland  Valentin Goblot  Christophe Galland  Institute of Physics, École
Polytechnique Fédérale de Lausanne (EPFL), Lausanne CH-1015, Switzerland
Center for Quantum Science and Engineering, EPFL, Lausanne, Switzerland
Mayeul Chipaux  Institute of Physics, École Polytechnique Fédérale de Lausanne
(EPFL), Lausanne CH-1015, Switzerland  Center for Quantum Science and
Engineering, EPFL, Lausanne, Switzerland

###### Abstract

Localized electronic spins in solid-state environments form versatile and
robust platforms for quantum sensing, metrology and quantum information
processing. With optically detected magnetic resonance (ODMR), it is possible
to prepare and readout highly coherent spin systems, up to room temperature,
with orders of magnitude enhanced sensitivities and spatial resolutions
compared to induction-based techniques, allowing for single spin
manipulations. While ODMR was first observed in organic molecules, many other
systems have since then been identified. Among them is the nitrogen-vacancy
(NV) center in diamond, which is used both as a nanoscale quantum sensor for
external fields and as a spin qubit. Other systems permitting ODMR are rare
earth ions used as quantum memories and many other color centers trapped in
bulk or 2-dimensional host materials. In order to allow the broadest possible
community of researchers and engineers to investigate and develop novel ODMR-
based materials and applications, we review here the setting up of ODMR
experiments using commercially available hardware. We also present in detail
the dedicated collaborative open-source interface named Qudi and describe the
features we added to speed-up data acquisition, relax instrument requirements
and extend its applicability to ensemble measurements. Covering both hardware
and software development, this article aims to overview the setting of ODMR
experiments and provide an efficient, portable and collaborative interface to
implement innovative experiments to optimize the development time of ODMR
experiments for scientists of any backgrounds.

###### Contents

  1. I Optically Detected Magnetic Resonance
    1. I.1 Main principles
    2. I.2 Interactions with external quantities
    3. I.3 Main ODMR systems and their applications
    4. I.4 Main ODMR techniques
      1. I.4.1 Continuous wave mode
      2. I.4.2 Pulse sequences
  2. II Hardware for continuous and pulsed ODMR
    1. II.1 Photoluminescence microscopy
      1. II.1.1 Light detector and imaging mode
      2. II.1.2 Optical components and alignment
      3. II.1.3 Light source
      4. II.1.4 PL digitization and visualization
      5. II.1.5 Scanning in a confocal microscope
    2. II.2 Microwave instrumentation
    3. II.3 Pulse generation
    4. II.4 Bias magnetic field
  3. III Connection and synchronization
    1. III.1 Photon detection
    2. III.2 Confocal scanning
    3. III.3 CW ODMR
    4. III.4 Pulsed measurements
  4. IV Qudi, an open-source interface for ODMR experiments
    1. IV.1 Qudi’s features
    2. IV.2 Main architecture
      1. IV.2.1 Modules
      2. IV.2.2 Hardware interfaces
    3. IV.3 Utilization
  5. V Good practices for ODMR measurements: NV Center characterization
    1. V.1 Procedure
    2. V.2 CW ODMR
    3. V.3 Pulsed experiments

## Introduction

Spin is a quantum property of elementary particles which confers them an
intrinsic magnetic moment that can be associated with a circulating charge
flow in their wave function Ohanian (1998). In atoms and ions, while the
nuclear spins remain essentially shielded, the unpaired electrons result in
magnetic moments that are three order of magnitude stronger – of the order of
the Bohr magneton. It makes electron spins the main contributors to most
materials’ magnetic properties. In the modern context of the second quantum
revolution Dowling and Milburn (2003), electron spins, such as in quantum dots
Vandersypen  _et al._ (2017); Zhang  _et al._ (2018a), color centers in
diamond Chang, Hsiao, and Su (2018) or silicon carbide Castelletto and Boretti
(2020), or rare earth ions Bertaina  _et al._ (2007) have been either studied
as individual quantum systems or as ensembles. While electron spins naturally
interact with various external and internal fields, they can also be isolated
from unwanted perturbations by experimental design or choice of system
Camenzind  _et al._ (2018), making them either great quantum sensors Degen,
Reinhard, and Cappellaro (2017) or highly coherent quantum bits (or qubits)
for quantum information processing Wesenberg  _et al._ (2009); Pezzagna and
Meijer (2021).

Standard inductive radio- and microwave techniques for spin resonance
measurements suffer from poor sensitivity and low spatial resolution. Indeed,
under achievable magnetic fields electronic spin transition energies typically
lie in the microwave (MW) domain, with associated Boltzmann temperatures of
few tens of millikelvin, so that they are very weakly polarized at ambient
conditions. Additionally, sensing MW fields at the single photon level remains
a tremendous challenge Lescanne  _et al._ (2020) and the associated
wavelengths (millimeter to centimeter) is by far larger than the few tens of
nanometers required for directly probing spin-spin interactions Dolde  _et
al._ (2014). Even using magnetic field gradients magnetic resonance imaging
(MRI) cannot achieve spatial resolution below tens of micrometers Blank  _et
al._ (2009). All these limitations affect both the spatial resolution in
quantum sensing and the scaling of entangled qubits’ networks for useful
quantum computing.

Optically detected magnetic resonance (ODRM) Suter (2020) offers a powerful
solution to these problems. It is applicable to a range of material systems
(see Table 1) whose joint optical and magnetic properties render their
electron-spin magnetic transition detectable in the optical domain. The
following ingredients are characteristics of ODMR systems:

  * •

Firstly, the electronic spin can be polarized by optical pumping to a much
higher degree than under thermal equilibrium. Associated with naturally low
spin-lattice coupling Gugler  _et al._ (2018), certain electronic spins can
maintain long polarization and coherence times even at elevated temperature
Balasubramanian  _et al._ (2009). Together these properties can totally relax
the need for cryogenic conditions.

  * •

Secondly, the strength of optical signal (typically absorption, fluorescence
or phosphorescence) must depend on the electronic spin state or its projection
along a quantization axis. Since optical fields can routinely be detected at
the shot-noise limit, the sensitivity of magnetic resonance spectroscopy and
the fidelity of spin state readout can be significantly increased while spin
initialization and readout can be achieved with a much higher spatial
resolution set by the optical wavelength (or below).

  * •

Finally, these advantages can be translated to nuclear spin spectroscopy
through the use of hyperfine coupling Nishimura  _et al._ (2020).

Together, these features are placing ODMR at the core of many emerging spin-
based quantum technologies.

Despite the growing number of researchers and engineers active in this area
and the availability of advanced optical and microwave (MW) equipment,
building and operating an ODMR setup for a specific purpose usually demands
substantial hardware and software developments. Moreover, beyond the apparent
simplicity of ODMR measurements, a certain amount of tacit knowledge is needed
to avoid some pitfalls that can spoil measurement accuracy and
reproducibility.

The object of this article is twofold: (i) to describe a typical ODMR setup,
from its conception and construction to the optimization of specific
measurements and (ii) to present, in details, the ODMR-dedicated open source
platform Qudi Binder  _et al._ (2017) with the features we developed for
speeding-up acquisitions, relaxing instrumental requirements, widening its
applications and lowering entry barriers.

In Sec. I, we first recall ODMR main principles and measurement techniques and
review the most studied material systems with their applications. We then
present the hardware configuration needed for ODMR experiments in Sec. II.
Aspects related to optical setup, generation of MW signals, pulses and bias
magnetic fields are discussed. We show how some ODMR experiments (in
continuous wave and pulsed modes) can be visualized directly with an
oscilloscope and determine when the use of a computer interface is useful or
required. In Sec. III, we describe how these instruments are connected
together and present the most common strategies for synchronizing ODMR
experiments. Both the computer and oscilloscope interfaces are covered. The
section IV is dedicated to the open source platform Qudi, first release in
2017 Binder  _et al._ (2017) and upgraded by us for efficient and versatile
ODMR experiment operations. Finally, in Sec. V we give a few practical advice
to perform reliable and contrasted measurements, in particular accounting for
technical noise and instrument imperfections. We take a specific focus on NV
center ensembles as an illustrative scenario.

We believe that the software experimental developments presented here will
help newcomers achieve faster and better results for their desired application
and provide experts an efficient, portable, and collaborative platform to
explore innovative experiments.

##  I Optically Detected Magnetic Resonance

###  I.1 Main principles

Figure 1: InterSystem Crossing (ISC) based ODMR dynamics when the ground state
is singlet (a) or triplet (b).

ODMR relies on two main features: (i) initialization (i.e. polarization) of
the spin state under optical excitation (spin pumping) and (ii) spin-state-
dependent optical properties, such as fluorescence, phosphorescence,
absorption, or, as more recently demonstrated, photocurrent Bourgeois, Gulka,
and Nesladek (2020), used for spin state readout. While feature (i) boosts the
contrast of the spin resonance signal, feature (ii) allows to advantageously
trade low energy MW photons for much higher energy optical ones in the spin
state measurement. These properties may arise from spin-orbit coupling (see
Fig. 1), a relativistic effect that allows for inter-system crossing (ISC),
e.g. transitions between singlet $S=0$ and triplet $S=1$ state manifolds
Penfold  _et al._ (2018); Marian (2012) (in a spin-1 system). The probability
of ISC may depend on the initial spin state, which in turn may result in spin-
dependent optical properties and spin-dependent relaxation pathways. In other
systems (see Sec. I.3), ODMR can be achieved under resonant optical pumping
such as using $\Lambda$-shaped configurations.

###  I.2 Interactions with external quantities

In a system that is not isotropic, a quantization axis emerges; noting $m_{S}$
the spin quantum number (i.e. the spin projection along that axis), the
dipolar spin-spin interaction lifts the degeneracy between the states
$\left|m_{S}=0\right>$ and $\left|m_{S}=\pm 1\right>$ among the $S=1$ triplet,
by an amount called zero-field splitting (ZFS), often denoted with $D$.
Further degeneracy is lifted between $\left|m_{S}=-1\right>$ and
$\left|m_{S}=+1\right>$ by a parameter $E$ when the symmetry is less than
axial. The eigenstates, potentially made of superpositions of pure spin
states, are then often denoted $x$, $y$ and $z$. $D$ and $E$ depend on
geometrical aspects and therefore on pressure and temperature. In addition,
Stark and Zeeman effects further separate the $\left|m_{S}=-1\right>$ and
$\left|m_{S}=+1\right>$ state which makes ODMR systems naturally sensitive to
magnetic and electric fields. Finally, further interactions may occur with
other electron or nuclear spins, resulting in more level splittings, spin
dephasing and diffusion, etc.

###  I.3 Main ODMR systems and their applications

We can distinguish two main configurations linked to the multiplicity of the
ground state (see Fig. 1). The ground state of organic molecules are typically
spin singlets, where all electrons are paired in covalent bonds or lone pairs
Carbonera (2009). Photoexcitation followed by ISC may promote the molecule
into a spin triplet state and enable ODMR as in Fig. 1 (a). Since 1967 Kwiram,
Alvin L. (1967); Schmidt  _et al._ (1967); Sharnoff (2004), such process has
been exploited to enhance the detection limit of magnetic resonance Carbonera
(2009) with a particular application in photosynthesis research Giacometti
_et al._ (2007); Kwiram and Ross (1982). ODMR from a single molecule was
reported for the first time in 1993 independently by Wrachtrup et al.
Wrachtrup  _et al._ (1993) and Köhler et al. Köhler  _et al._ (1993). Both
experiments relied on the photoexcited metastable triplet state of a pentacene
molecule trapped in a p-terphenyl host crystal. Pentacene’s spin-dependent
optical properties have also been used inside MW resonators Arroo, Alford, and
Breeze (2021), such as for demonstrating the first room temperature maser
Oxborrow, Breeze, and Alford (2012) or conversely for cooling a microwave mode
Wu  _et al._ (2021).

The negatively charged Nitrogen-Vacancy center (NV-) in diamond - hereafter
called NV center - belongs to the second configuration depicted in Fig. 1 (b).
Its ground 3A2 and main excited 3E optical states are spin triplets. The first
single NV center ODMR signal was reported by Gruber et al. in 1997 Gruber  _et
al._ (1997). For the following reasons, NV centers then became the most
studied ODMR system:

  * •

ISC combined with a low Debye-Waller factor allows for an efficient off-
resonance optical pumping and readout over a large range of wavelengths, even
far over room temperature Liu  _et al._ (2019) and under high pressures
Doherty  _et al._ (2014).

  * •

Diamond is bio-compatible making NV-doped nanodiamonds particularly relevant
Chipaux  _et al._ (2018) when used as biomarkers Haziza  _et al._ (2017);
Hemelaar  _et al._ (2017); Miller  _et al._ (2020) or quantum sensors Sigaeva
_et al._ (2022a); Morita  _et al._ (2023); Zhang  _et al._ (2021).

  * •

NV centers can exhibit long spin relaxation and coherence times
Balasubramanian  _et al._ (2009) thanks to the low density of nuclear spins in
diamond and the decoupling of spin states from lattice phonons Gugler  _et
al._ (2018). It makes them versatile quantum sensors Barry  _et al._ (2020)
for detection and imaging of magnetic and electric fields, temperature,
strain, currents and associated noises Degen, Reinhard, and Cappellaro (2017);
Schirhagl  _et al._ (2014); Rondin  _et al._ (2014), as well as for microwave
field imaging Shao  _et al._ (2016) or spectroscopy Chipaux  _et al._ (2015a);
Meinel  _et al._ (2021).

  * •

NV center ODMR properties can be extended to polarize Healey  _et al._ (2021);
Rizzato  _et al._ (2022); Gao  _et al._ (2022) and/or detect Mamin, Sherwood,
and Rugar (2012); Staudacher  _et al._ (2013); Sushkov  _et al._ (2014);
Lovchinsky  _et al._ (2016) other electron or nuclear spins in their vicinity
and perform their magnetic resonance spectroscopyWood  _et al._ (2016); Mignon
_et al._ (2023). In these cases the sensitivity can go down to a single
nuclear spin, orders of magnitude better than conventional electron
paramagnetic resonance (EPR) and nuclear magnetic resonance (NMR) methods.

  * •

Finally, thanks to their excellent coherence properties and couplings to even
longer lived nuclear spins Bradley  _et al._ (2019), NV centers are also
promising qubits for quantum information processing Pezzagna and Meijer (2021)
and fundamental studies of quantum mechanics Hensen  _et al._ (2015).

Other diamond defects from the group IV-vacancy category Bradac  _et al._
(2019) have recently revealed their ODMR signals, such as the negatively
charged silicon-vacancy center (SiV-) Pingault  _et al._ (2017) and its
neutral form (SiV0) Zhang  _et al._ (2020), the Germanium-Vacancy (GeV)
Siyushev  _et al._ (2017) and the Tin-Vacancy (SnV) Iwasaki  _et al._ (2017)
centers. In these cases, ODMR is possible only at cryogenic temperatures since
it requires resonant optical addressing – differing from the scheme presented
in Fig. 1. Single-defect ODMR signals have recently been reported in silicon
carbide Nagy  _et al._ (2018); Castelletto and Boretti (2020); Yan  _et al._
(2020) or hexagonal boron nitride Stern  _et al._ (2022); Murzakhanov  _et
al._ (2022). Notably, the so called V1 center in silicon carbide Nagy  _et
al._ (2018) involves a spin quartet $S=\frac{3}{2}$ ground state. Rare earth
ions in crystal lattices are also intensely studied for long lasting and
broadband optical quantum memories Zhong and Goldner (2019); Grimm  _et al._
(2021). One can cite $Ce^{3+}$ in YAG Xia  _et al._ (2015) or $Yb^{3+}$ Zhou
_et al._ (2020) and $Eu^{3+}$ Ortu  _et al._ (2022) in $Y_{2}SiO_{5}$.
Semiconductor quantum dots have also been used in ODMR experiments Ivanov  _et
al._ (2018); O. Tolmachev  _et al._ (2020). The optical wavelengths and
microwave frequencies associated to those systems are summarized in Table 1.

System |  Optical pumping |  Microwave |  Comments / Applications
---|---|---|---
(host) |  [Off-resonance] |  Zero field splitting $D\textrm{ and }E$ |
|  (On-resonance (ZPL)) |  Gyromagnetic ratio $\gamma$ |
Organic molecules |  Various |  $D=500\textrm{ to }$1500\text{\,}\mathrm{MHz}$$ |  Sensitive EPR spectroscopy Carbonera (2009)
|  |  $E=0\textrm{ to }$500\text{\,}\mathrm{MHz}$$ |  Photosynthesis Agostini  _et al._ (2021)
Pentacene |  [Green - yellow, |  $D=$1395\text{\,}\mathrm{MHz}$$ |  First single molecule ODMR signal Wrachtrup  _et al._ (1993); Köhler  _et al._ (1993)
(p-terphenyl crystal) |  e.g. $585\text{\,}\mathrm{nm}$] |  $E=$53\text{\,}\mathrm{MHz}$$ |  First room temperature maser Oxborrow, Breeze, and Alford (2012); Arroo, Alford, and Breeze (2021)
NV- center |  [Green e.g. $515\textrm{ or }$532\text{\,}\mathrm{n}\mathrm{m}$$] |  $D=$2.87\text{\,}\mathrm{GHz}$$ |  Quantum sensing
(Diamond) |  $($637\text{\,}\mathrm{n}\mathrm{m}$=$1.945\text{\,}\mathrm{eV}$)$ |  $\gamma=$28\text{\,}\mathrm{GHz}\text{\,}{\mathrm{T}}^{-1}$$ |  Quantum computation
Group IV-Vacancy |  |  |  Cryogenic temperature. Becker and Neu (2020)
centers (Diamond) |  |  |  Quantum networks Chen, Zheludev, and Gao (2020)
SiV- |  $($737\text{\,}\mathrm{nm}$)$ |  $D=$50\text{\,}\mathrm{GHz}$$ |  Pingault  _et al._ (2017)
SiV0 |  $($946\text{\,}\mathrm{nm}$)$ |  $D=$944\text{\,}\mathrm{GHz}$$ |  Zhang  _et al._ (2020)
GeV |  $($612\text{\,}\mathrm{nm}$)$ |  $D=$170\text{\,}\mathrm{GHz}$$ |  Siyushev  _et al._ (2017)
SnV |  $($260\text{\,}\mathrm{nm}$)$ |  |  Iwasaki  _et al._ (2017)
V1 center |  $\left($861\text{\,}\mathrm{nm}$\right)$ |  $D=$2\text{\,}\mathrm{MHz}$$ |  Spin quartet Nagy  _et al._ (2018)
PL8 center |  [$920\text{\,}\mathrm{nm}$] |  $D=$1.39\text{\,}\mathrm{GHz}$$ |  Room temperature Yan  _et al._ (2020)
(4H-Silicon carbide) |  $\left(1007\textrm{ to }$1024\text{\,}\mathrm{nm}$\right)$ |  $E=$4\text{\,}\mathrm{MHz}$$ |  Castelletto and Boretti (2020)
Unknown defects |  [$532\text{\,}\mathrm{nm}$] |  $D=0.1\textrm{ to }$2.4\text{\,}\mathrm{GHz}$$ |  Stern  _et al._ (2022)
(Boron Nitride) |  |  |
Rare earth ions |  |  |  Quantum memories Zhong and Goldner (2019); Grimm  _et al._ (2021)
$Ce^{3+}$ (YAG) |  $\left(460\textrm{ and }$486\text{\,}\mathrm{nm}$\right)$ |  $<2$2.2\text{\,}\mathrm{GHz}$$ |  Xia  _et al._ (2015)
$Yb^{3+}$ ($Y_{2}SiO_{5}$) |  $\left($980\text{\,}\mathrm{nm}$\right)$ |  $<$2.62\text{\,}\mathrm{GHz}$$ |  Zhou  _et al._ (2020)
$Eu^{3+}$ ($Y_{2}SiO_{5}$) |  $\left($580\text{\,}\mathrm{nm}$\right)$ |  $34.5$ and $46.2\text{\,}\mathrm{MHz}$ |  Ortu  _et al._ (2022)
Quantum dots |  |  |
(In,Al)As/(AlAs) |  [$360\textrm{ and }$404\text{\,}\mathrm{nm}$]$ |  |  Ivanov  _et al._ (2018)
CdSe/(Cd,Mn)S |  [$405\text{\,}\mathrm{nm}$] |  $60\text{\,}\mathrm{GHz}$ |  O. Tolmachev  _et al._ (2020)
Table 1: Summary of some popular ODMR systems and their main parameters.

###  I.4 Main ODMR techniques

####  I.4.1 Continuous wave mode

ODMR can be performed under continuous wave (CW) optical and MW excitation. A
CW-ODMR spectrum is obtained by recording the optical emission or absorption
as function of the MW frequency that is being swept. When spin resonances are
reached, dips or peaks appear in the recorded optical intensity (see Fig. 2),
allowing to measure physical parameters affecting the resonance frequencies or
to determine the gyromagnetic ratio of the system itself (EPR spectroscopy).
For a given system, several external parameters affect the precision of the
spectrum obtained by CW-ODMR through the change in signal-to-noise ratio and
spin resonance linewidth. These parameters include MW power (via power
broadening Dréau  _et al._ (2011) and sample heating), laser power (via change
in excitation rate, governing spin polarization, optical emission and photo-
ionization rates), MW sweeping method and pace, etc. Optimization of CW-ODMR
signal acquisition is discussed in Sec. V.2.

Figure 2: Example of a continuous-wave ODMR spectrum from an NV-center
ensemble under d.c. magnetic field, displaying eight resonances (hyperfine
splitting not resolved) with contrasts on the order of 1%.

####  I.4.2 Pulse sequences

Pulsed ODMR has the first advantage of mitigating power broadening by letting
the system evolve freely after optical and MW excitation. It also allows to
tune the system for being specifically sensitive to certain quantities and
insensitive to unwanted perturbations, taking advantage of decades of
development in the field of magnetic resonance spectroscopy Vandersypen and
Chuang (2005). Good practices for optimizing a pulsed ODMR signal are
discussed in Sec. V.3. Below, we first introduce the most elementary pulse
sequences (in the case of a spin triplet).

Common to most sequences, a first laser pulse initializes the system (e.g. in
state $|m_{S}=0\rangle$). After a period of spin manipulation with MW pulses
and/or free evolution, a second laser pulse interrogates the spin state
projection along the $\left(|m_{S}=0\rangle,|m_{S}=+1\rangle\right)$ basis
through the optical emission or absorption intensity. It also re-initializes
the system for the subsequent measurement. In the Bloch sphere representation,
a MW $\pi$-pulse performs a $\pi$ rotation along a certain axis; for instance,
it can swap the $|m_{S}=0\rangle$ and $|m_{S}=+1\rangle$ (or
$|m_{S}=-1\rangle$). A $\pi/2$ pulse only performs a quarter turn, placing an
initial $|m_{S}=0\rangle$ state into an equal superposition
$\frac{1}{\sqrt{2}}\left(|m_{S}=0\rangle+e^{i\phi}|m_{S}=+1\rangle\right)$ (or
$|-1\rangle$).

##### Relaxometry – $T_{1}$ measurement

— This sequence, reported in Fig. 3(a), requires no MW fields. The two
initialization and readout optical pulses are separated by a varying dark time
$\tau$. It allows for probing the relaxation dynamics from an initialized pure
state to a thermal equilibrium. Interesting for applications, the relaxation
time, or $T_{1}$, can sense the magnetic noise of the local environment
Tetienne  _et al._ (2013); Rollo  _et al._ (2021) such as the one created by
free radicals species in living cells Sigaeva  _et al._ (2022a); Morita  _et
al._ (2023) or by chemical reactions Barton  _et al._ (2020).

##### Rabi sequence – $\pi$-pulse calibration

— In this sequence, reported in Fig. 3(b), the delay between the optical
pulses is fixed and a MW pulse is applied for a varying duration $\tau$
(alternatively, its power can be varied). The measured optical signal as a
function of $\tau$ shows Rabi oscillations. From this measurement the duration
of $\pi$ and $\frac{\pi}{2}$-pulses can be inferred as half and quarter
periods of the oscillations, respectively.

##### Ramsey sequence – $T_{2}^{*}$ measurement

— This sequence, reported in Fig. 3(c), consists in a pump-probe experiment
where two $\pi/2$ pulses are inserted. The first $\pi/2$, immediately after
the initialization laser pulse, induces spin precession. After the free
evolution time $\tau$ the precession is halted by a second $\pi/2$ pulse
before the read-out laser pulse is applied. The optical signal as function of
$\tau$ presents oscillations at the difference-frequency between the applied
MW field and the spin transition being probed. Therefore, this sequence can be
used for d.c. sensing and offers better sensitivity compared to the CW-ODMR
approach Dréau  _et al._ (2011) by eluding the detrimental effect of power
broadening. Ramsey fringes are damped with a time constant reflecting the spin
dephasing time or heterogeneous decoherence time called $T_{2}^{*}$.

##### Hahn echo sequence – $T_{2}$ measurement

— This sequence, reported in Fig. 3(d), uses a $\pi$-pulse in the middle of
the Ramsey sequence. This approach mitigates the effect of slowly fluctuating
fields (such as from the nuclear spin bath) by canceling in the second half of
the evolution the accumulated phase difference from the first half and
restoring the original coherence. The characteristic decay time measured with
this protocol is usually called $T_{2}$, and can typically be one or two
orders of magnitude longer than $T_{2}^{*}$. More and more complex and faster
dynamical decoupling sequences can be employed to extend $T_{2}$; yet, it
remains bounded by relaxation time through $T_{2}<2\cdot T_{1}$. Such echo
sequence is particularly suitable of a.c. field sensing as it can filter a
narrow frequency window Degen, Reinhard, and Cappellaro (2017).

Many efforts have been devoted to develop and optimize pulse sequences for
increasing sensitivities to specific quantities and frequencies while
canceling other noises. A comprehensive review dedicated to NV centers can be
found in Barry  _et al._ (2020).

Figure 3: Most common pulse sequences for ODMR: a) Relaxometry b) Rabi
sequence c) Ramsey sequence d) Hahn echo sequence. In each panel, laser and MW
pulses are represented schematically in the top part. In the bottom part, NV
centers’ experimental data obtained in our set-up from are shown.

##  II Hardware for continuous and pulsed ODMR

We review here the instrumental requirements for ODMR measurements in terms of
optical setup, MW generation and biased magnetic field. Both CW and pulsed
ODMR measurements are discussed. We take the diamond NV center as an
accessible example (green pumping, red photoluminescence (PL)). However, the
properties of other defects summarized in Table 1 can be consulted in order to
adapt the equipment for their study, with a special care given on the optical
wavelengths, microwave frequencies and possible need for cryogenic
environment.

###  II.1 Photoluminescence microscopy

As mentioned in Sec. I, ODMR relies on monitoring an optical signal such as
the PL. It is obtained by directing a pumping beam (a laser or LED) onto the
system of interest and routing the luminescence signal onto a photodectector
(such as an amplified photodiode, a single photon counting module or a
camera).

The simple optical setup reported in Fig. 4 analogous to an epifluorescence
microscope can represent the minimal required optical setup to perform ODMR.
An excitation laser beam is reflected on a dichroic mirror and focused on the
sample via a microscope objective or a simple lens. The sample’s luminescence
is collected by the same lens and sent to a photodetector. Alternatively, the
sample can be pumped from another side or by total internal reflection Chipaux
_et al._ (2015b); Wolf  _et al._ (2015); Clevenson  _et al._ (2015) to relax
the need for a dichroic mirror or optimize pump power utilization and spatial
homogeneity of excitation. An inverted configuration of the microscope (e.g.
where the objective is below the sample) is preferred when using liquid
immersion objectives, especially for biological applications Webb  _et al._
(2021).

Figure 4: Schematic of a simple OMDR setup.

####  II.1.1 Light detector and imaging mode

The choice of the light detector is probably the most impacting point in the
development of a new setup. It is governed by two main aspects:

##### Single-pixel vs. wide-field imaging

— In the former case, a photodetector is sufficient, with appropriate filters
placed in front of it to discriminate the signal of interest from the
scattered laser light and the possible background luminescence from ODMR-
inactive centers Jonkman  _et al._ (2020). At the cost of small loss of
transmitted power, spatially filtering the luminescence signal through a
confocal pinhole or an optical fiber reduces the out-of-focus background and
collection area (see Fig. 5). If imaging capability is desired it can be
obtained by confocal scanning (see II.1.5 and Fig. 5).

For wide-field imaging, a pixel array (camera) is required in combination with
a large and homogeneous illumination area. Comprehensive reviews about wide-
field imaging in the particular case of NV centers can be found in Levine  _et
al._ (2019); Nomura (2021); Scholten  _et al._ (2021). Besides, accordingly to
the specific experiments, different cameras (e.g. CCD or a CMOS) can be
preferred Wojciechowski  _et al._ (2018).

##### Strength of optical signal

— It depends on the number of luminescent centers or molecules being
addressed, their individual quantum yields and the collection + detection
efficiency. For the study of single or few emitters, a digital photon counter
(d-PC) is necessary, such as a reverse-biased avalanche photodiode operated in
Geiger mode or a superconducting nanowire detector. d-PCs generate electrical
pulses (e.g. TTL) upon photon absorption with a quantum efficiency that can
exceed 80%, allowing for the study of photon fluxes down to sub-kHz level,
limited by the dark count rate of the detector.

For measurements on large ensembles, d-PCs typically saturate at count rates
of few tens of MHz (i.e. few picowatts at visible wavelengths). Above that
analog photodectors (a-PDs) can be used, such as amplified or avalanche
photodiodes. In a wide-field configuration with a weakly emitting system,
particularly sensitive cameras are required such as Electron Multiplying CCD
(low noise) or phosphorus intensified camera (gated).

To enhance the sensitivity in under significant background signal, one can
modulate and demodulate the ODMR signal (e.g. through the applied MW
frequency) with the help of a lock-in amplifier Kuwahata  _et al._ (2020).
Such methods can be of great help in the context of biosensing where
background emission is strong Miller  _et al._ (2020). For wide-field imaging,
the recently developed arrays for lock-in detector Wojciechowski  _et al._
(2018) may find numerous applications in the upcoming years.

Figure 5: Schematic of a typical home-build confocal microscope used for low
light ODMR experiments. The collimated laser beam is spectrally filtered (to
remove broadband spontaneous emission) before being reflected on the dichroic
mirror and focused on the sample using a high-NA objective lens.
Photoluminescence is collected by the same objective, passes through the
dichroic mirror and an interference filter (band-pass or long-pass) to reject
the excitation light, and is optionally spatially filtered using a confocal
pinhole or an optical fiber before its intensity is measured by the detector.
In order to scan the sample both a scanning mirror or a piezo stage can be
used. To study single emitters, a Hanbury-Brown and Twiss interferometer can
be added.

####  II.1.2 Optical components and alignment

When addressing individual emitters, one of the most important parameters is
the objective’s numerical aperture ($NA$) defined as $NA=n\,\sin{\theta}$,
with $n$ the refractive index of the immersion medium and ${\theta}$ the half-
cone angle of collection. For isotropic emission pattern, it limits the
fraction of collected to emitted light to:

$\eta<\frac{1}{2}\left(1-\cos{\theta}\right)$ (1)

with $\cos{\theta}={\sqrt{1-{(NA/n)}^{2}}}$.

Through diffraction, it also limits the achievable resolution $r_{min}$
defined as the minimum distance at which two objects can be distinguished. As
commonly determined by the Rayleigh criterion:

$r_{min}=\frac{r_{Airy}}{2}=0.61\cdot\frac{\lambda}{NA}$ (2)

where $\lambda$ is the wavelength of the optical source and $r_{Airy}$ the
radius of its diffraction pattern on the sample. Similarly, the axial (depth)
resolution is set by:

$z_{min}=1.4\frac{\lambda n}{{NA}^{2}}$ (3)

A larger $NA$ is usually obtained at the cost of a smaller working distance.
As a good compromise, an air objective with $NA$ up to 0.9 can offer a working
distance of 1 mm. In contrast, in a confocal setting, the magnification
$M_{obj}$ of the microscope objective bears no direct relation to spatial
resolution and collection efficiency. It matters however for the design of the
excitation and collection optics as well as in the case of wide-field imaging
on a pixel array.

To achieve a spatial resolution limited by Abbe's diffraction and avoid
unnecessary light loss, care should be taken to the following aspects:

  * •

Firstly, the excitation beam diameter $\Phi$ should match or exceed the back
aperture diameter of the microscope objective given by:

$\Phi=2f^{\prime}_{obj}\tan{\theta}$ (4)

with $\tan{\theta}=\frac{NA/n}{\sqrt{1-{(NA/n)}^{2}}}$,
$f^{\prime}_{obj}=\frac{l_{tube}}{M_{obj}}$ the effective focal lens of the
microscope objective, $l_{tube}$ being the tube lens as defined by the
microscope supplier (e.g. $180\text{\,}\mathrm{m}\mathrm{m}$ for Olympus,
$200\text{\,}\mathrm{m}\mathrm{m}$ for Nikon and Mitutoyo and
$165\text{\,}\mathrm{m}\mathrm{m}$ for Zeiss).

  * •

Secondly, the confocal pinhole of radius $r_{hole}$ (or optical fiber of
numerical aperture $NA_{fiber}$) has to be chosen to match the diffraction
pattern of the photoluminescence beam on the pinhole (or fiber). When the
image is sent to infinity after the objective the total magnification up to
the pinhole (or fiber) equals:

$M_{tot}=M_{obj}*M_{lens}=M_{obj}\frac{f^{\prime}_{lens}}{l_{tube}}$ (5)

where $f^{\prime}_{lens}$ is the focal length of the lens used to focus on the
pinhole. So to provide good confocal resolution while avoiding significant
optical loss, $r_{hole}$ can be chosen to match the radius of the Airy
diffraction pattern:

$r_{hole}=1.22\times M_{tot}\frac{\lambda_{PL}}{NA}$ (6)

Here, $\lambda_{PL}$ refers to the wavelength of the photoluminescent signal.
The pinhole can also be chosen smaller. This would enhance the resolution
(e.g. reduce $r_{min}$) by a factor down to $2/3$, at the cost of more optical
loss.

  * •

If an optical fiber is used instead, its numerical aperture can be taken as:

$NA_{fiber}=\frac{NA}{M_{tot}}$ (7)

In those cases, either the focusing lens, the pinhole or optical fiber can be
changed to obtain the desired spatial filtering. Further details about the
basics of confocal microscopy can be found in Jerome and Price (2018).

Many efforts have been devoted to surpass the diffraction limit in ODMR using
super-resolution techniques such as stochastic optical reconstruction
microscopy (STORM) and stimulated emission depletion (STED) microscopy Hsiao
_et al._ (2016); Jin  _et al._ (2018).

####  II.1.3 Light source

For the most studied ODMR systems (Sec. I.3), the optical pumping can be
performed off resonance. Consequently, there is no stringent requirement on
temporal coherence and linewidth of the light source, so that even a white-
light LED can be employed, as long as suitable filter sets are used to prevent
leakage of excitation light into the detector. However, fluctuations in laser
power affects the contrast of the ODMR, as well as the stability and coherence
of the NV centers. Therefore, it is important to carefully control the laser
power fluctuation against temperature.

Transverse spatial coherence (single mode emission) is required in a confocal
configuration in order to obtain a diffraction-limited spot size at the focus.
As shown in Fig. 5, a spatial filter (pinhole or mono-mode fiber) can be
introduced in the excitation path to improve transverse coherence. In this
case, and using large NA, optical powers of milliwatts or less are often
sufficient to saturate the emitter. Therefore, commercially available solid
state lasers are a great and affordable option. Optionally, neutral density
filters allow to adjust the power incident on the sample while maintaining the
laser at its optimal operating power. In the wide-field configuration, the
laser power is diluted over the entire area of interest, requiring high power
lasers up to watts level. In all cases, temporal fluctuation in laser power
represents the main source of technical noise, which can be mitigated by
adding a feedback control loop or monitoring laser power in real time through
a second detector. Other forms of technical noises related to pulsed
measurements are discussed in Sec. II.3.

####  II.1.4 PL digitization and visualization

Different approaches are used to measure the PL as a function of the type of
signal coming from the photodetector.

Analog photodetectors (a-PDs) deliver an electric signal proportional to light
intensity (temporally averaged over a given bandwidth) that can be visualized
directly on an oscilloscope for both continuous and pulsed measurements.
Alternatively, a computer interfaced acquisition card that embeds an Analog to
Digital Converter (ADC) can be used. In this context, the most relevant
features of these data acquisition cards (DAQ) are the number of input and
output channels and the maximum sampling rate. The sampling rate sets an upper
limit to the fastest dynamics that can be measured under analog acquisition.
While the oscilloscope sampling rate can be chosen up to the GHz range, the
DAQ analog inputs are often limited to a few MHz. Modern oscilloscopes can be
interfaced on a computer for direct collection of the data temporally stored
in the oscilloscope (see Sec. IV.1). In case of pulsed acquisition (see Sec.
II.3) averaging over several repetition is needed to obtain sufficient signal
to noise ratio and/or to reduce the communication bandwidth requirement with
the computer. Consequently, it demands a sync pulse for each repetition and
limits the averaging strategy to the $NP$ method described in Sec. III.4.

Digital photocounters (d-PCs) deliver nanosecond pulses (e.g. TTL) with sharp
rising edge. In such case, the use of a computer becomes almost unavoidable.
Conventionally, a time tagging instrument or counter (e.g. from Swabian
instruments or PicoQuant) provides time resolution down to picosecond range,
below the typical timing jitter of the detector. Such resolution can be useful
to resolve the PL lifetime (see Sec. V), or to acquire second order
autocorrelation function $(g^{(2)})$ (e.g. to evidence single photon
operationsAharonovich  _et al._ (2011); Tchernij  _et al._ (2017); Monticone
_et al._ (2014)). Otherwise, ODMR rarely involves that fast processes. Such a
time precision can then become a disadvantage as the instrument buffer becomes
more rapidly saturated by the recorded time tags. A field-programmable gate
array (FPGA) can be used as a cost effective alternative to time-tagging
instrument, but it requires dedicated programming skills. DAQ cards also
include general purpose timers/counters. As presented in Sec. III.1 it is
possible to configure them for photon counting with sampling rate up to twice
their internal clock, which is two orders of magnitude faster than their
analog inputs.

Finally, in the case of a wide-field configuration, the camera conventionally
embeds its own ADC and directly delivers a digital image to the computer. In
such cases, however, acquisition rates are at best up to hundreds of hertz,
often too slow to resolve the pulse dynamics. When pulses cannot be isolated
from each others, the averaging possibility is limited to the $NP$ method
described in Sec. III.4.

####  II.1.5 Scanning in a confocal microscope

In a confocal microscopy configuration, locating individual emitters or
mapping an area requires scanning the focal spot over the sample. Two
possibilities arise:

  * •

The sample or the microscope objective is mounted on an $xy$ (or $xyz$) piezo-
based nanopositioning stage so as to scan their respective position. In such
case, the range is often limited by the stage itself from a few tens to a
couple of hundreds of micrometers. If moving the objective, the field of view
can also be limited by the shift of its back aperture with respect to the
excitation and collection beams.

  * •

The optical beam is angle-scanned by means of a 2-axis oscillating mirror and
a $4f$ relay lens system. In such case, the field of view can be chosen by the
$4f$ lenses and is ultimately limited by the field number of the microscope
objectives.

While the first method is significantly less bulky, the second option is
typically more affordable and offers a faster scanning rate. It can also offer
the flexibility of changing the objective to adjust the field of view and
resolution. Note that stepper motors and slip-stick piezo stages have bad
repeatably: for accurate mapping and positioning, closed-loop positioners
should be privileged. We also warn that magnetic materials should be avoided,
in particular in the sample stage. In both cases, controlling the scanning
device requires a computer program and a Digital to Analog Converter
interface. Such functions are often provided via the aforementioned DAQ cards.

Component |  Key specs |  Ex. of Brand (and model) |  Price (€) |  Sec.
---|---|---|---|---
Light source |  wavelength |  any diode laser |  $\sim$ 300-7k |  II.1.3
intensity stability |  Thorlabs (CPS 532) |  |
transverse coherence |  Coherent (OBIS 532 nm) |  |
power (for wide-field) |  Labs-electronic (DLnSec 520nm) |  |
intensity modulation option |  _Cobolt (06-MLD 515 nm)_ |  |
|  diffraction efficiency |  Gooch & housego (AOMO 3350-199) |  $\sim$ 4k |  II.3
AOM + RF driver |  switching time (rise/fall) |  _AA Opto Electronic (MT350-A0.12-xx)_ |  |
|  numerical aperture |  Nikon, Leica, Zeiss, etc. |  $\sim$ 500-5k |  II.1.2
Microscope Objective |  working distance |  _Mitutoyo (0.6NA 1.3mm WD)_ |  |
|  Immersion medium (air, oil…) |  _Olympus (LMPLFLN 50X 0.5NA)_ |  |
Dichroic mirror / fluorescence filter |  cut-on wavelength |  depending on the ODMR system |  $\sim$ 100-500 |  II.1.2
absorption losses |  Semroc, Thorlabs, Chroma. etc. |  |
extinction outside transm. band |  |  |
Light detector |  wavelength range |  |  |  II.1.1
Analog (a-PD) |  bandwidth |  Newport |  $\sim$ 1k-2k |
noise equivalent power |  _Thorlabs (APD410A/M)_ |  |
Digital (d-PC) |  quantum efficiency |  IDquantique (ID100), (ID120) |  $\sim$ 2k-8k |
dark count rate |  _Excelitas (SPCM-AQRH)_ |  |
|  timing jitter |  |  |
Camera |  pixel size/number |  Thorlabs, UEye |  $\sim$ 500-1500 |
repetition rate, Gating |  Andor, Princeton instruments |  $\sim$ 2k-20k |
|  quantum Efficiency, readout noise |  |  |
Piezo stage or scanning mirror |  scanning speed |  Thorlabs (GVS212/M) |  |  II.1.5
resolution, repeatability |  _MadCityLab (Nano-3D200)_ |  |
travel range |  _Newport (FMS-300)_ |  |
|  closed vs. open loop |  |  |
MW source |  frequency range |  Wainvam (Wainvam-e1) |  $\sim$ 500- |  II.2
phase noise |  DS Instruments (SG4400L,SG6000L) |  600 |
sweep rate |  Anritsu (MG3691c), Agilent (n9310a) |  50k |
power |  Keysight (M9384B) |  |
|  pulse modulation |  _Rohde &Schwarz (SMB100B)_ |  |
|  frequency range |  |  |  II.2
MW amplifier |  gain and maximal power |  _MiniCircuit (ZHL-16W-43-S+)_ |  $\sim$ 5k |
|  field intensity |  $\Omega$-shaped antenna Opaluch  _et al._ (2021) |  $<10$ |  II.2
MW antenna |  bandwidth |  wire loop, straight wire |  |
|  spatial homogeneity |  _Sasaki et al. Sasaki  _et al._ (2016)_ |  |
|  insertion loss |  Minicircuit ZFSW-2-4, |  |  II.2
MW switch |  switching time (rise/fall) |  Minicircuit ZASWA-2-50DRA+, |  |
Acquisition card |  maximum ADC sampling rate |  Keysight-U2300A |  $\sim$ 3k-7k |  II.1.4
(DAQ) |  number of i/o channels |  _National Instrument (PCIe-6363)_ |  |  II.1.5
|  minimum pulse width |  Spincore (PulseBlasterESR-PRO) |  $\sim$ 4k |  II.3
|  number of channels |  Zurich Instrument |  |
Pulse generator |  max. number of pulses |  _Swabian Instrument (Pulse streamer 8/2)_ |  |
|  minimum bin width |  PicoQuant (PicoHarp300), |  $\sim$ 3k-10k |  II.1.4
Time-tagger |  dead time |  IDquantique (ID900) |  |
|  timing jitter |  |  |
|  bulk or nanodiamonds (ND) |  _Element6_ , _Appsilon B.V._ (bulk) |  $\sim$ 100-3k |
Diamond sample |  NV density |  Van Moppes, Pureon, |  |
|  NV coherence/relaxation time |  Adamas nanotechnologies (ND) |  |
Table 2: Necessary equipment for a confocal ODMR instrument, with remarks and
examples of suitable providers. The cells in gray color highlight specific
equipment for pulsed experiments. In italic is what we used in our own NV
setups. We underline the minimum equipment required to perform a ODMR confocal
microscopy experiment.

Apart from the essential components described in the previous paragraphs, the
setup can be improved to allow additional characterization features. For
example, sending the luminescence to a spectrometer allows to verify its
origin and to quantify the amount of background emission (for example, for NV
centers, emission from $\mathrm{NV}^{0}$ is detrimental to the contrast).
Moreover, to confirm that a single emitter contributes to the signal, a
Hanbury-Brown and Twiss (HBT) interferometer can be used to measure the
second-order autocorrelation function $(g^{(2)})$.

###  II.2 Microwave instrumentation

The manipulation of electron spins is most readily achieved with near-
resonant, coherent MW radiation. According to the targeted spin resonance(s)
the MW source should have an appropriate frequency range (See Table 1). Also,
the phase (or frequency) noise of the source can be the limiting factor when
studying highly coherent spin transitions. While a variable crystal oscillator
(VCO) can be used as a very cost effective solution, tabletop MW generators
such as from Rohde & Shwartz or Anritsu are often preferred in metrology and
high-resolution spectroscopy for their spectral purity and power stability. A
key specification for CW-ODMR is the dwell time in frequency sweep, which
affects the acquisition time. MW powers above 30 dBm that can be achieved with
standard MW amplifiers. An insulator should be placed after the MW source
and/or the amplifier to prevent damage from back reflections.

MW pulses can be carved out of a CW MW source using external switches. Some MW
generators may also directly embed an internal switch, which typically offers
better performance to measure coherence times less than
$100\text{\,}\mathrm{ns}$ due to lower phase noise between pulses.
Furthermore, the MW source must be capable of frequency modulation (FM) for
lock-in measurements. Compatibility between the various MW components must be
checked.

The MW radiation is delivered to the sample by MW antennas. Working with
ensembles may require spatially homogeneous MW amplitude over several hundreds
of $\mu m^{2}$. Several designs focusing on optimizing their field intensity
and spatial homogeneity can be found in Abe and Sasaki (2018); Sasaki  _et
al._ (2016); Opaluch  _et al._ (2021); Herrmann  _et al._ (2016); Bayat  _et
al._ (2014). Particularly, a trade-off needs to be found between the power
conversion required to reach a fast enough Rabi oscillation frequency, the
area of field homogeneity, and the bandwidth of the antenna (important when
bias magnetic fields are applied to split the resonances). When spatial
homogeneity is not essential, e.g. when working with a single spin, a small
wire loop or straight wire can serve as MW antenna Zheng  _et al._ (2019).

###  II.3 Pulse generation

For time domain measurements (such the ones described in Sec. I.4) a pulse
generator must be included in the setup. While a minimum of two output
channels is required to control both the MW and the laser, more channels can
be useful for synchronization or for more versatile pulse schemes (such as
rotating the spin around different axes on the Bloch sphere, or driving with
multiple MW frequencies at once). The pulse generator must provide a minimum
pulse length significantly shorter than the minimum coherence time to be
measured. To allow multi-pulse protocols (e.g for Ramsey fringes), it must be
possible to fire more than one pulse in each channel after a single trigger.

Two different options are available to produce optical pulses, starting with
acousto-optic modulators (AOM). For a wide-field illumination, special care
should be taken about the maximum optical admissible beam intensity to prevent
any damage of the device. This is to be considered together with the desired
rise and fall times, since all those parameters depend on the beam diameter on
the AOM. As a handy alternative, laser diodes may embed pulsing capabilities.
Unfortunately, overshoot can be present at the beginning of the pulse and
depend on the preceding off state duration (through thermal relaxation). It
may cause measurement artifacts, in particular when the repetition rate of the
laser is varied, like in $T_{1}$ measurement (see Sec. I.4). While post
treatment mitigation methods are presented in Sec. V.3, the use of an AOM
reduces such artifacts.

###  II.4 Bias magnetic field

Several applications require a bias d.c. magnetic field and precise controls
of its strength and orientation, which allows to tailor the system eigenstates
so that they are most susceptible to the perturbation of interest Dolde  _et
al._ (2011); Michl  _et al._ (2019). For example, when NV centers are used for
magnetometry and temperature sensing, a bias axial magnetic field often allows
to improve the sensitivity Barry  _et al._ (2020); Moreva  _et al._ (2020);
Kraus  _et al._ (2014). In addition, applying a field of the proper magnitude
parallel to the quantization axis allows to work in specific level
configurations such as at the ground state or exited state-level anti-crossing
(GSLAC or ESLAC) Ivády  _et al._ (2021). Cross-relaxation conditions between
different spin systems can be matched by properly tuning the magnitude of an
axial magnetic field Broadway  _et al._ (2016); Lazda  _et al._ (2021). These
configurations can be particularly relevant in the context of all optical
sensing Wickenbrock  _et al._ (2016); Broadway  _et al._ (2016) or spin
polarization Jacques  _et al._ (2009). Finally, a transverse magnetic field
can be relevant in specific situations, e.g. for electrometry Dolde  _et al._
(2011).

Different options are available and reported in literature to achieve this:

  * •

Place a permanent magnet on a micrometer stage Bauch  _et al._ (2018); Hart
_et al._ (2021); Moreva  _et al._ (2020). Controlling the direction,
uniformity and intensity of the field can be challenging. Using a pair of
permanent magnets mounted on a rotation stage can be helpful. Fine calibration
is required for precise magnetic field alignment and repeatability.

  * •

Design a system of three Helmholtz coils, each addressing one axis Choi
(2019); Zhang  _et al._ (2018b). This approach requires more initial efforts,
but leads to higher precision and control. It allows to generate a bias field
in any possible direction and to swiftly switch between different bias fields.
Ref. Crosser  _et al._ (2010) presents a useful numerical model to simulate
the magnetic field in the center of the coils for varying parameters. If the
bias field needs to be modified during the measurement (as for example in real
time electrometry Yang  _et al._ (2020); Zhang  _et al._ (2021)), this
approach is particularly convenient.

  * •

Hybrid solutions, matching the specific application requirements Wickenbrock
_et al._ (2016); Webb  _et al._ (2020); Yang  _et al._ (2020); Stürner  _et
al._ (2021). It is often not necessary to generate a bias field in any
possible directions. Simpler systems compound of one or a pair of Helmholtz
coils and permanent magnets are often sufficient. Placing the sample on a
rotation stage can offer more convenience and versatility.

To conclude this Section, we refer to Table 2 that summarizes hardware
requirement for a typical room-temperature confocal ODMR setup.

##  III Connection and synchronization

We previously mentioned that the visualization of ODMR signals can be either
performed on an oscilloscope or through an acquisition card on a computer and
we described when the second option is necessary. Here we explain how to
connect and synchronize all devices in either case, focusing on four typical
ODMR measurements.

###  III.1 Photon detection

As mentioned in sec. II.1.4, two main types of detectors (analog photodectors,
a-PD vs. digital photon counters, d-PC) can be employed depending mainly on
the signal intensity. The required connections for each case are illustrated
in Fig. 6. On the one hand, the analog signal coming from an a-PD can be
measured either directly with the help of an oscilloscope or on a computer
after digitization via DAQ. In the latter case, the a-PD is plugged to an
analog input. The analog acquisition is limited to the clock frequency of the
ADC, $f_{ADC}$, such that the temporal resolution cannot be better than
$T_{Sampling}=1/f_{ADC}$. For example, in the case of the NI-6364 card and for
single channel, $T_{Sampling}=$0.5\text{\,}\mathrm{\SIUnitSymbolMicro s}$$,
increased to $n_{c}\times T_{Sampling}$ when $n_{c}$ channels are used.

On the other hand, the output of a d-PC is a train of pulses (e.g. TTL), each
corresponding to one (or more) detected photon. A DAQ can be used to measure
this signal. Commonly, one of its internal counters is configured to count the
number of received pulses through one of its PFI (Programmable Function
Input). The count is then periodically stored in the DAQ buffer and reset. As
a result, the number of counts per interval is obtained. In such a case, the
sampling rate can be similar to analog signals.

In order to increase the temporal resolution, which is most desired in pulsed
measurements, three counters of DAQ can be used for time tagging. Two counters
are set at a high data rate as inputs to count the arriving signal from the
d-PC and sync signal that triggers the acquisition. The third counter is set
as output and it generates a periodic TTL signal to internally reset the other
two. Their count are then saved in the buffer after receiving the TTL reset
signal. The reset signal determines the temporal time-tagging resolution.
Knowing the relative time difference between sync and signal, the averaged
temporal signal can be reconstructed.

Alternatively, the role of the clock and the d-PC pulses can be inverted.The
pulse count is then increased by one at each period of the clock, while each
pulse, when detected, triggers its storage to the buffer. This results in a
list of time-stamped events. At the cost of more information saved, which can
slow down the computer, this architecture can achieve a better time
resolution. With the same NI-6363 card considered above, one can achieve the
temporal resolution of
$T_{Sampling}=1/(2*f_{counter_{clock}})=$5\text{\,}\mathrm{ns}$$; the factor 2
accounts for both rising and falling edges being counted.

Figure 6: Hardware connections for monitoring a photoluminescence signal. The
solid and dashed/dotted lines indicate the three hardware alternatives. The
green color refers to the pumping beam and the red to the sample
photoluminescence (PL). DAQ card internal connections are shown in gray. The
analog, digital and oscilloscope-based acquisition blocks introduced here, are
also used in Fig.s 7, 8, and 9 along with the same conventions.

###  III.2 Confocal scanning

In confocal microscopy, a scanning device (such as a piezo stage or a scanning
mirror) is controlled while the sample PL is monitored. Synchronizing the two
allows to attribute to each pixels its corresponding PL intensity, creating an
image. As illustrated in Fig. 7, these controls and PL monitoring are commonly
performed with a DAQ card in which an analog output is used for each spatial
axes $(x),(y)$ and $(z)$. The measurement is performed by generating a sync
signal of frequency $f_{sync}$ in the acquisition card and three step-ramp
functions ranging from the lowest to the highest voltage are sent to the
control ports of the scanning device. Each sync period therefore corresponds
to a specific position of the scanning device for which the PL is collected.
The sync frequency and order of the generated step ramp functions are
eventually used to build the PL image.

Figure 7: Hardware connections and synchronization links for confocal
scanning. The analog and digital data acquisition blocks are introduced in
Fig. 6.

###  III.3 CW ODMR

Continuous wave ODMR spectra are measured in a similar way, with the
difference that the sweep is applied on the MW frequency instead of the beam
(or sample) position. Each optical intensity must be attributed to the
corresponding MW frequency. As described in Sec. I.4, the plot of the PL
versus MW frequency mirrors the magnetic resonance spectrum. As reported in
Fig. 8 the DAQ sends a TTL sync signal to the MW source and triggers frequency
change while the corresponding PL signal is recorded simultaneously.

In case of using an oscilloscope for acquisition, the MW source, set in
frequency modulation mode, becomes the synchronization master. It sends either
a sweep or a TTL signal with the same period to a second input of the
oscilloscope to trigger at the beginning of each sweep. The horizontal
temporal scale of the oscilloscope can then be adjusted to display the PL as a
function of the frequency. In the case of using a VCO, the frequency has to be
set by an external voltage source, such as generated by a DAQ or waveform
generator. The visualization on an oscilloscope can be performed in the same
way.

Typical MW sources offer two modes to change the MW frequency: sweep and list.
In the sweep mode, the minimum and maximum frequency and the step size are
defined. As the MW generator receives a TTL signal, it changes the frequency
by the defined step. In the list mode, the list of frequencies are previously
stored in the generator, then as a TTL signal is received, the frequency is
changed to the next one in the list. The list mode allows to perform random
sweeping of the MW frequency mitigating problems such as hysteresis due to
frequency-dependent MW heating through the antenna resonance. In practice, the
acquisition speed of a CW-ODMR spectrum is limited by how fast the MW
generation hardware can switch the output frequency. This values is for
example around 1 ms minimum per step for R&S SMF100A.

Figure 8: Hardware connections and synchronization links for CW-ODMR, MW
connections are shown in orange. Other conventions are the same as in Fig. 6.

###  III.4 Pulsed measurements

As depicted in Fig. 9, pulsed measurements require synchronizing the laser and
MW pulses with the optical signal acquisition. A pulse generator sends a TTL
signal to command the laser and MW output switch. An additional TTL signal is
also generated within the pulse generator to trigger the acquisition and set
the time reference for each measurement. This signal is sent to the DAQ or
oscilloscope. For each data point of a pulsed experiment, the optical signal
is analyzed against the change of one variable (e.g., duration of the MW pulse
or wait time between MW and laser pulses, see Sec.I.4).

##### Averaging strategies

— We discuss two different options, where $i\in\\{1;P\\}$ labels the value of
a parameter (e.g. waiting time) to be swept among $P$ values and $N$ is the
number of repetition over which the signal is to be averaged.

  * •

A first averaging method is denoted as $"NP"$, where each $i^{th}$ pulse
sequence is individually repeated $N$ times, before moving to the next
parameter value from the $1^{st}$ to the $P^{th}$.

  * •

In the second one, denoted as $"PN"$, a full series of $P$ sequences
(consisting of $P+1$ optical pulses since the first only serves as initial
polarization) is sent at once while the PL is recorded. This sequence is then
repeated $N$ times to reach the desired signal to noise ratio. In this method,
a slow varying noise similarly affects signals corresponding to all parameter
values and is therefore averaged out. It also allows to monitor how the signal
to noise ratio improves over the successive repetitions. However, it demands
sufficient time resolution to isolate signals from each pulse sequence making
it hardly compatible with the use of a camera. It also requires digitizing and
storing the full sequence at once, which also limits the use of an
oscilloscope (see Sec. II.1.4).

##### Synchronization strategies

— In either cases, two distinct synchronization methods are possible:

  * •

‘Sync method 1’: Only two TTL sync pulses are generated, at the start and at
end of each series of $N$ (or $P$) sequences.The incoming optical signal is
continuously recorded between the initial and final pulses according to the
sampling rate. The position of each pulse is then calculated by software (see
Sec. V.3). This cannot be done when using an oscilloscope.

  * •

‘Sync method 2’: A TTL pulse is sent at the beginning of each pulse to be
either timestamped by the counter with a DAQ card or used to trigger the
oscilloscope.

Fig. 9b depicts two out of the four possible combinations of averaging and
synchronization strategies in the case of a Rabi oscillation measurements.

##### Implementation with a DAQ card

— The implementation of aforementioned methods with an NI card can be
elaborated based on the type of the acquired signal.

  * •

In the analog case, the a-PD signal is sent to an analog input channel of the
DAQ card. As discussed in section III.1, the ADC captures and saves the analog
signal into the buffer until it is read out. With ‘Sync method 1’, the
generated sync pulses are transmitted to another analog channel of the DAQ
card and the acquisition starts as soon as it receives a sync pulse. As a
consequence, the optical signal is recorded even when the laser is off.
Identification of each pulse then requires post-processing as described in
Sec. V. With Â´Sync method 2’, the generated sync pulses are sent to a digital
channel of the DAQ card. The channel is adjusted to acquire the pulses during
the laser pulse windows using start edge trigger and pause trigger protocols
of the DAQ card.

  * •

Acquiring data from a d-PC, however, demands time-tagging as described in
section III.1. Similarly, the ‘sync method 1’ records all the photon received.
the pulses are found and reconstructed by post-processing. With ‘sync method
2’ the trigger signals are time-tagged and mark the beginning of each pulse.

One chooses the synchronization method depending on hardware limitations and
desired averaging strategy discussed above.

Figure 9: (a) Hardware connections and synchronization links for pulsed
experiments. Solid and dashed lines indicate different alternatives depending
on available hardware. (For example, if the laser permits pulsed modulation,
the AOM is not required.) (b) Averaging and synchronization strategies in the
case of a Rabi measurement. Top: $PN$ strategy with ‘Sync method 1’, Bottom:
$NP$ method with ‘Sync method 2’

##  IV Qudi, an open-source interface for ODMR experiments

As discussed in Secs. II and III, the use of a dedicated software can be
necessary to control a confocal microscope or deal with photon counting. It
can also automatize the control of ODMR instruments to speedup the
measurements and ease their reproducibility, either for fundamental studies or
application developments. Ideally, the software interface should deal with
both single photon counters in low light conditions and analog photodetectors,
process confocal and wide-field images, and freely combine different
measurement parameters and methods.

Several options exist for coding such interface program. On the one hand, mid-
level languages such as C++ or FORTRAN, down to the lower-level ones such as
assembly, offer the best possibilities to optimize computer resources. They
are often the best option for demanding computational physics and simulations,
artificial intelligence related research, or for processing of complex or
voluminous data Mirams  _et al._ (2013). On the other hand, higher level
programming languages, such as Python or Java, greatly facilitate design,
development and maintenance of codes. They also ensure better portability
between different hardware configurations. They are therefore particularly
appropriate for interfacing laboratory experiments Cao  _et al._ (2013);
Chapman and Chang (2000). A comparison between programming languages including
C, C++, Java and Python can be found in Prechelt (2000).

Among others, Labview (for Laboratory Virtual Instrument Engineering
Workbench), a proprietary programming environment developed by National
Instruments Bitter, Mohiuddin, and Nawrocki (2006), offers a graphical diagram
coding style making it more intuitive to newcomers in programming. Moreover,
it permits to easily interface with laboratory instruments and offers
libraries for signal generation, data acquisition and processing that allow to
rapidly setup basic experiments. However, when the experimental complexity
increases, this graphical programming style can become difficult to handle. It
also remains incompatible with versioning and collaborative tools such as
Github for sharing codes and subroutine within a community.

In this context, Python, a high-level programming language, offers a great
alternative Cahn and Russo (2007). Coding environments and interpreters being
open source, it is particularly popular and used in various domains, including
experiment control. It supports versatile and efficient community-developed
libraries. Moreover, it offers sufficient level of abstraction to allow
portability between computers of different configurations and making it
upgradeable to new functionalities, embeds efficient multi-threading and
signaling capabilities allowing for parallel operations (data acquisition,
processing and display).

Here, we introduce Qudi, an open-source Python code dedicated to ODMR
measurements. First presented in Binder  _et al._ (2017), it is an open-source
collaborative project shared on Github and it benefits from a well programmed
Python-based architecture. We first present Qudi’s original features and the
ones we added to improve its performance and extend its applicability. We then
detail Qudi’s architecture and presents how to configure and use it.The
official Qudi’s version is available on GithubQud (a) and a general
documentation based on developers "in code" comments along with a brief
general tutorial can be found here Qud (b). For those not already merged
within the official Github, the added features can be found on our Github page
forked from it Our (a).

###  IV.1 Qudi’s features

Qudi Binder  _et al._ (2017); Qud (a); Our (a), initially released in 2017,
integrates modules for performing ODMR experiments, instrument control, and
real-time data acquisition and processing. It enables fast imaging by scanning
the beam position while collecting luminescence. Qudi can locate ODMR emitters
and track them during acquisition, compensating for sample drift. It is
particularly useful for studying nanoparticles in liquid suspensions
McGuinness  _et al._ (2011); Sigaeva  _et al._ (2022b). It provides useful
tools for interfacing a scanning confocal microscope by controlling piezo-
based nanopositioning stages or scanning mirrors and collecting the optical
emission signal. Possibilities include in-plane scanning $(xy)$, $z$-stack
acquisitions (to obtain 3D images), out of plane cross-sections $(xz)$ or
$(yz)$ as well as any other direction cuts depending on the wiring and
instrument. Associated to confocal microscope hardware (see Sec. II.1), Qudi
can locate and track ODMR emitters, compensating for sample drift. This
feature is particularly interesting when dealing with nanoparticles in a
liquid suspension such as in biomedical applications McGuinness  _et al._
(2011); Sigaeva  _et al._ (2022b).

Qudi efficiently controls the instruments required for acquiring CW ODMR
spectra and determining spin resonance frequencies. Real-time fitting of
analytical curves on ODMR data allows for adjustment of acquisition parameters
and quick sharing of results with collaborators. In addition to the CW ODMR
signal, Qudi is designed for the study of spin dynamics with pulse sequences
such as for measuring longitudinal spin relaxation curves ($T_{1}$), Rabi
oscilations, Ramsey ($T^{*}_{2}$) or Hahn Echo ($T_{2}$) sequences (see Sec.
I.4). It can control the pulse generator for triggering laser and MW pulses
(see Sec. II.3) in a synchronous manner with the collection of the
photoluminescence (see Sec. III.4). It also enables the design of complex
pulse sequences, including dynamical decoupling sequences. Predefined methods
like XY8 pulses are available via the graphical interface, and new pulse
sequences can be easily added to the program Qud (c).

Qudi offers tools for on-the-go pulse extraction, analysis, and automated
experimental parameter settings. This simplifies long measurement sequences
across different materials. Using Qudi’s Jupiter notebooks Qud (d), one can
design and automate independent experiments to vary multiple control
variables, enabling efficient exploration of parameter space.

Module |  Original features |  Added features
---|---|---
|  Analog acquisition |  Acquisition with oscilloscope
Counting |  Digital acquisition |  Faster analog and digital DAQ acquisition
|  Analog acquisition |  Acquisition with oscilloscope
Confocal |  Digital acquisition |  Control with Pulse Streamer
|  |  Arbitrary/random MW sweep
|  |  Acquisition with oscilloscope
|  |  Acquisition with Spectrometer
ODMR |  Analog acquisition Digital acquisition |  Lock-in detection*
|  |  Analog acquisition with DAQ
|  |  Analog acquisition with oscilloscope
|  |  Digital acquisition with DAQ
|  |  Pulse extraction improved
Pulsed |  Digital acquisition with time tagging device |  New pulse sequences
Table 3: Summary of Qudi main features in the original release Qud (a) and in
our upgraded version Our (a). *Under development.

The initial version of Qudi was focused for experiments on single quantum
emitters, thus requiring digital single photon counting modules as well as
time tagging electronics in the pulsed measurements. In order to broaden its
contextual applicability, we added several features.

At first, a direct oscilloscope interface was implemented for data acquisition
(see Sec. II.1.4) while, conversly, NI card (or similar DAQ device)’s digital
inputs can now also be used as a fast event counter and the minimum temporal
resolution has been reduced to half the card internal clock period, making it
sufficient for ODMR experiments without need of dedicated time tagging
instruments.

Two strategies for photon counting were implemented (see Sec. III.1) and a new
logic for pulsed measurements was coded to enhance acquisition speed and
eliminate the need for post-treatment and pulse extraction described in Sec.
III.4. Pulsed measurements using analog signals were also added, making the
system compatible with analog photodiode signals for studies of ensemble (see
Sec. II.1.1). Analog acquisition in CW-ODMR was also improved by increasing
the sample rate per frequency sweep, optimizing signal-to-noise ratio (see
Sec. III.1).

Finally, an arbitrary/random MW frequency sweep was added to minimize thermal
effects of the resonator on the ODMR signal and an optical spectrometer module
was incorporated for ODMR signal acquisition using a user-specified spectral
window. The summary of the added features to the Qudi platform is shown in
Table 3.

###  IV.2 Main architecture

Qudi is made of different modules that are loaded and connected together by a
manager component. The science modules responsible for experiment control are
divided into three categories: GUI (graphical user interface), logic, and
hardware. As detailed in the following, this division is based on a clear
separation of tasks among the categories and offers a reliable and flexible
software architecture and simultaneous (multi-threading) acquisition, data
treatment and visualization. Importantly, the setting of those modules and the
links between them is to be defined in a configuration file (See Sec.
IV.3).The important notion of hardware interface (higher abstraction level of
hardware module) is also introduced.

####  IV.2.1 Modules

##### GUI modules —

They create an interactive graphical interface that allows the user to control
the experiment and to visualize the acquired data. In particular, the GUI
allows to start and stop the acquisition, to adjust the experimental
parameters (e.g. MW power, frequency sweep range, acquisition time…), to save
the data and to fit them with pre-defined functions, simply pushing buttons or
inserting values into the interactive windows. Changes made in the graphic
interface, such as a pushed button or a modified parameter, trigger a
broadcast signal sent to the related logic modules. Although GUI modules offer
an intuitive and efficient way to interact with the logic modules, Qudi can
also be fully functional without them via the integrated Ipython console or
via a Jupyter notebook.

##### Logic modules —

They form the basis of every experiment designed in Qudi. Their main function
is to coordinate the different tasks necessary to perform a complete
experiment. They serve as a bridge between the two other kinds of modules. On
one side they receive the emitted signals from the GUI modules and they send
others to the hardware modules to configure the instruments sig . Conversely,
they gather and analyze data coming from the hardware modules and pass them to
the GUI modules for display.

Each experiment depends on a variety of generic tasks that are common to
different types of measurements, such as fitting appropriate curves, saving
data, etc. Therefore, rather than having a single logic module for each
experiment performing all these tasks, they are divided in multiple logic
modules used in a versatile manner. Besides communication with hardware and
GUI modules, each logic module can receive inputs from and send outputs to
other related logic modules. Note that logic modules are the only ones allowed
to talk to modules of the same category.

##### Hardware modules —

The main task of these modules is to allow an effective communication between
the logic modules and the specific hardware. Receiving orders from the logic
modules, they run the equipment drivers (often provided by the supplier) to
act onto the experiment. They also send the output signals of the instruments
to the logic modules, to be further analyzed, displayed and eventually saved.
Moreover, hardware modules can also work as virtual dummy or mock hardware,
emulating the functionality of a real device. This possibility can be very
helpful for configuring a new setup. As an example, modules and links between
them that are required for the four typical experiments described in Sec. III
are presented in Figure 10.

Figure 10: Block diagrams for the software connections of counting, confocal
scan, CW ODMR, and pulsed measurement which needs to be included in the
configuration file.

####  IV.2.2 Hardware interfaces

Even if a set of instruments have common features for performing similar
tasks, there is, between different models and suppliers, a wide difference in
command structure, grammar and connection methods. In order to handle this
complexity, a specific interface is defined for each category of instruments.
In the context of object oriented programming, hardware modules are classes
that inherit from hardware interfaces. The running of a laboratory equipment
consists in defining and operating instances (objects) from the instruments-
related classes. The interface therefore consists of a set of functions that
each hardware module within the same category must implement. Therefore, each
instrument used in an experiment should have its own class compatible with the
related interface class in Qudi. This class needs to control the instrument
while respecting constraints on its operation as specified in its data sheet.
To accomplish this task, the class may use drivers provided by the
instrument’s manufacturer. Various instruments have already been introduced in
Qudi in different categories (such as spectrometers, cameras, MW generators,
pulse generators, etc.). However, due to the wide variety of systems,
applications, suppliers and models available, it may be necessary to add a new
device or modify the class for an instrument whose model is slightly different
from the one defined in the code.

###  IV.3 Utilization

Installing Qudi is rather straightforward from Qudi Documentation's
instructions doc . It can be cloned from the original Qud (a) or our Our (a)
Github.

Performing experiments with Qudi, also requires to tune it specifically for
the user’s hardware environment and specific needs by writing a configuration
file, which, in particular, defines the connections between the modules that
are used such as represented in Figure 10. Besides, installing an instrument
that is not yet supported requires to write the corresponding hardware module
class (see above section IV.2.1). Finally, for important coding project such
as Qudi, it can be useful to configure the Python environment to ensure the
good functioning of the different packages and libraries that are used in
different places of the code. This can be performed through a conda
environment installer such as the one called Anaconda available in the Github
subfolder qudi\tools.

The utilization of the module we added in our Github Our (a) (see Sec. IV.1)
is detailed in Babashah (2023), Our (b).

In the following section, we will focus on the characterization of an NV
center ensemble to propose some practical advice and discuss good practices
when performing typical ODMR experiments with a home-build confocal microscope
(see part. II) controlled using Qudi.

##  V Good practices for ODMR measurements: NV Center characterization

As described in Sec. I.4 the main figures of merit characterizing an ODMR
system are the longitudinal spin relaxation time $T_{1}$, dephasing time
$T_{2}^{*}$ and coherence time $T_{2}$ (the latter being dependent on the type
of echo sequence used to measure it). In this section, we describe the
procedure and methods to reliably perform these measurements such as presented
in Fig. 3. As an illustration, we characterize a small ensemble of NV centers
in bulk diamond using a single pixel detector in a confocal geometry (see Sec.
II.1.1). While an extensive review on optimizing sensitivity with NV centers
can be found in Barry  _et al._ (2020), we aim here to provide practical
advice on how to easily reach good measurement quality in terms of acquisition
speed, contrast and repeatability.

The measurements presented in this section are performed on a bulk diamond
sample enriched in NV centers and provided by Appsilon B.V. Other
possibilities for commercial diamond providers are summarized in Table 2. It
should be noted that the figures of merit discussed here can vary
significantly depending on the material quality. For practical purposes, we
provide throughout the section typical values for these figures of merit in an
NV ensemble. They are given for a sample at room temperature, in the absence
of external perturbation (magnetic field, electric field, pressure, etc.).

###  V.1 Procedure

The sample is first placed in the optical setup and its position adjusted to
match the object focal plane of the objective. A confocal scan allows to image
the sample and identify the region of interest. At this step, further
alignment of the setup to ensure that maximum signal is emitted by and
collected from the NV centers is possible. A single NV center can provide
photon count rates above $100\text{\,}\mathrm{kHz}$ when using a numerical
aperture exceeding 0.9 and a laser excitation power in the mW range. The total
PL depends linearly on the number of NVs in the ensemble. According to the
level of brightness of the sample and collection efficiency of the setup, a
proper detector (e.g. photon counter or analog photodiode) and possibly a
neutral density filter are selected. Then, we use Qudi ODMR interface to
perform a CW-ODMR scan. The goal of this step is to adjust the bias magnetic
field (by monitoring the different resonances in the spectrum) and to identify
the resonance frequency that will be used for the following pulsed
measurements.

One specificity with NVs is that each center can take one of the four $(111)$
crystallographic orientations of the diamond lattice. Considering the two
allowed MW transitions from $\left|m_{s}=0\right>$ to $\left|m_{s}=-1\right>$
and $\left|m_{s}=+1\right>$, this can lead to eight distinct resonances (that
are further split by hyperfine interaction) whose frequencies correspond to
the projection of the magnetic field along each orientation. It allows to
perform vectorial magnetometry Maertz  _et al._ (2010); Chipaux  _et al._
(2015b), even without biased applied fieldsLenz  _et al._ (2021); Wang  _et
al._ (2022). Certain applications require a specific magnetic field alignment
(see Sec. II.4). In this example, we target a bias field such that the Zeeman
splitting for at least one NV orientation is different from all other
orientations, i.e at least one pair of transitions in the spectrum is non-
degenerate.

We select a MW frequency resonant with one of these two transitions
(corresponding to either $\left|m_{s}=-1\right>$ or $\left|m_{s}=+1\right>$)
and perform pulsed measurements to characterize $T_{1}$, $T_{2}$ and
$T_{2}^{*}$. To this end, we move to Qudi interface for pulsed Measurement.
Rabi oscillations are first measured to determine the duration of a
$\pi$-pulse which is half the oscillation period. We note in passing that the
decay of Rabi oscillations is not simply related to any of the time constants
mentioned above because increasing the MW pulse duration reduces is spectral
bandwidth and thus progressively filters a smaller set of NV centers among the
inhomogeneously broadened ensemble. Since a good estimate of $T_{2}^{*}$ can
be obtained from the CW-ODMR linewidth (in the limit of low MW powers) it is
recommended to use sufficient MW power in the pulse sequences so that the
$\pi$-pulse is shorter than $T_{2}^{*}$ in order to efficiently drive all NV
centers in the ensemble. The last step consists in applying each of the
dedicated pulse sequences to extract $T_{1}$, $T_{2}$ and $T_{2}^{*}$,
previously presented in Sec. I.4.

###  V.2 CW ODMR

Figure 11: (a, c) Contrast and (b, d) FWHM of a single resonance in the CW
ODMR spectrum, versus (a, b) MW Rabi frequency and (c, d) laser power.

In order to perform CW ODMR with highest possible contrast and narrow
linewidth, a compromise must be reached in terms of both MW and laser powers,
as illustrated in Fig. 11. While the optical excitation rate must be faster
than the $1/T_{1}$ relaxation rate to ensure that all spins are well
polarized, a too high rate can decrease the ODMR contrast by competing with
the MW excitation and preventing a large population difference between states
$\left|m_{s}=0\right>$ and $\left|m_{s}=1\right>$ Jensen  _et al._ (2013). On
the other hand, while increasing the MW power first enhances the contrast, it
then broadens the resonances through power broadeningDréau  _et al._ (2011)
and eventually reaches saturation of the spin transition.

As practical guidelines (for NV ensembles) the optimal laser power is the one
maximizing contrast and the optimal MW power is the highest power for which
the linewidth remain close to the $T_{2}^{*}$ limit. For further optimization,
the optimum condition for the laser and MW power can be derived analytically
using a rate equation approach Dréau  _et al._ (2011); Jensen  _et al._
(2013). Typically, a single NV center exhibit an ODMR contrast above $10\%$.
For an ensemble with 4 possible NV orientations, this results in best contrast
of around $2\%$ for each resonance in the ODMR spectrum, close to the values
reported in Fig. 11. The presence of other photoluminsecent defects, such as
neutral NV0, causes a reduction of the ODMR contrast.

###  V.3 Pulsed experiments

To improve the quality of a pulsed experiment several factors should be taken
into account. Some are general to all pulsed protocols, while others are more
specific. Common requirements for all pulsed experiments are reliable
initialization and readout. Laser pulse duration needs to be long enough to
ensure maximum polarization in state $\left|m_{s}=0\right>$. The
characteristic time for spin polarization depends on the optical power density
and typically ranges from $100\text{\,}\mathrm{ns}$ to
$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$. For NV centers, the initialization
pulse must be followed by a waiting time, typically $1~{}{\mathrm{\mu s}}$, to
allow relaxation of population trapped in the singlet state towards the ground
state $\left|m_{s}=0\right>$ Robledo  _et al._ (2011). The characteristic
polarization time also impacts the readout phase. Depending on the time
resolution of the hardware (see Sec. II.1.4 and III.1), it can be convenient
to reduce the laser power, to polarize the spin slower and be able to resolve
it dynamics as in Fig. 12. The duration of the optical pulse needs to be
adjusted accordingly.

In the case of ‘Synchronization method 1’ (see Sec. III.4) in which only the
beginning and the end of a sequence is triggered, it is necessary to locate
each pulse within the entire recorded optical signal. In this case, two
standard approaches are implemented in Qudi for pulse recognition. The first
uses a threshold value to identify rising and falling edges of a pulse. The
second relies on a convolution of the signal with a Gaussian distribution and
takes its first derivative, resulting in a pulse identification signal
$S_{p}(t)$ given by:

${S_{p}}(t)=\frac{{d}}{{dt}}\left[{S_{r}}(t)\otimes\frac{1}{{\sigma\sqrt{2\pi}}}{e^{-\frac{1}{2}{{(\frac{{t}}{\sigma})}^{2}}}}\right]$
(8)

where $t$ represents the time and $S_{r}(t)$ is the raw signal acquired during
the pulsed measurement; $\sigma$ is the standard deviation for the Gaussian
filter. Rising and falling edges of pulses correspond to local maxima and
minima of ${S_{p}}(t)$. The Gaussian filter allows to smooth the signal and
reduces false pulse detection. This second method is thus generally more
robust than the first one. Optimal extraction is achieved by adjusting the
parameter $\sigma$.

Figure 12: Normalized luminescence intensity during the readout laser pulse,
for spins initialized in state $\left|0\right>$ (blue curve) and
$\left|1\right>$ (orange, using a $\pi$-pulse between optical pumping and
readout), collected with NA 0.65 objective and 35 mW incident laser power. The
shaded gray region indicates a good choice of time window for signal
integration, yielding high SNR.

Figure 12 displays the time evolution of luminescence intensity during a laser
pulse starting at $t=0$, for spins initially prepared either in
$\left|m_{s}=0\right>$ or $\left|m_{s}=1\right>$. The different levels of PL
in the two cases and the spin polarization dynamics, are clearly resolved. To
obtain a measure of the spin projection along the quantization axis the PL
intensity is integrated over an early time window marked as gray shaded area,
and normalized to the PL intensity integrated at the end of the pulse (not
represented), when spins are polarized again. It allows to compensate possible
slow fluctuations of the laser intensity or collection efficiency.

To maximize the signal to noise ratio, the time window should start with the
laser pulse, where the PL difference between $\left|m_{s}=0\right>$ and
$\left|m_{s}=1\right>$ is maximal. If the initial time of a pulse is not
properly identified by the pulse extraction method for all pulses in the
sequence (e.g. due to slow laser rise time), the resulting timing jitter
translates into noise on the total integrated signal. In this case excluding
very early times of the readout pulse can be beneficial.

Figure 13: (a) Pulse protocol for optimized $T_{1}$ measurement, with the
addition of a $\pi$-pulse to extract $S_{1}$. (b) Bloch sphere representation
of the spin state at times of the pulse protocol indicated in (a). (c)
Resulting signal versus delay $\tau$, fitted with an exponential decay (solid
line)

Several other sources of noise can affect readout fidelity. Among them, some
are spin-independent, for example due to NV ionization and NV- recombination
caused by the laser pulse Aslam  _et al._ (2013); Giri  _et al._ (2018). A
powerful solution to suppress this kind of noise is to repeat the pulsed
protocol but inverse the spin state onto $\left|m_{s}=1\right>$ just before
readout using a $\pi$-pulse as illustrated in Fig. 13 Jarmola  _et al._
(2012). The extracted signals with and without inserted $\pi$-pulse are
labeled $S_{1}$ and $S_{0}$, respectively. The expectation value of the spin
state projection is linearly related to the difference signal $S_{0}-S_{1}$
from which spin-independent common noise is removed. The result of this
procedure for a $T_{1}$ measurement is illustrated in Fig. 13.

Another important source of artifacts arises when the dark time $\Delta T$
between consecutive laser pulses is swept. As mentioned in Sec. II.3 it can
lead to variation in laser pulse intensities over for the entire sequence. In
most cases, a simple solution is to keep $\Delta T$ constant, adjusting some
waiting times before and after the microwave pulses accordingly, as long as
$\Delta T\ll T_{1}$. It is however not possible for a $T_{1}$ measurement.
Fortunately, the protocol discussed just above allows to partially mitigate
such artifact by normalizing the signal by $S_{0}+S_{1}$. Such protocol also
allows to extract a relevant signal even when the setup temporal resolution
does not allow to resolve the pulse dynamic, e.g. when using a camera in a
widefield configuration.

A final knob for optimization of pulsed measurements is in the definition of
$\pi$, $\pi/2$ and other pulses. The duration of such pulses is in principle
extracted from the measured period of Rabi oscillations. However, in practice
small deviations can arise, e.g. due to not perfectly square MW pulses and
fitting errors. It is thus good practice to perform a calibration step. To
this end, we use a Hahn echo sequence with fixed delay and monitor the echo
amplitude for varying initial, central and final MW pulses. The result of this
procedure is illustrated in Fig. 14, where the initial and final pulses are
half the central pulse. Every second sequence, we also replace the final pulse
with a $3\pi/2$-pulse, to apply the above common noise rejection strategy. The
optimum pulse duration corresponds to the simultaneous maximum echo amplitude
when the final pulse is $\pi/2$ and minimum when it is $3\pi/2$. This ensures
maximum contrast for a $T_{2}$ measurement, using the Hahn echo sequence (Fig.
3(d)). The calibrated $\pi/2$-pulse is also used for $T_{2}^{*}$ measurement
using Ramsey interferometry (Fig. 3(c)).

As stated earlier, $T_{1}$, $T_{2}$ and $T_{2}^{*}$ depend greatly on the
diamond quality, as well as on the presence of external perturbations. For an
NV ensemble with moderate concentration (below 1ppm), $T_{1}$ is usually on
the order of several milliseconds, $T_{2}^{*}\approx 0.5-1\mathrm{\mu s}$ and
$T_{2}$ is limited to few 10s of $\mathrm{\mu s}$ with a simple spin-echo
sequence, but can be extended to several 100 $\mathrm{\mu s}$ with better
dynamical decoupling protocols. The interested reader should refer to e.g.
Ref. Barry  _et al._ (2020) for a more quantitative, in-depth discussion of
these coherence times, which is beyond the scope of the present paper.

Figure 14: (a) Hahn echo sequence used for optimization of $\pi$ and
$\pi/2$-pulses. (b) Rabi oscillations with cosine fit (solid line). (c) Echo
amplitude versus pulse duration $\tau_{p}$. The delay is
$\tau=500~{}\mathrm{ns}$. Dashed line indicates the $\pi$-pulse duration
extracted from fit of Rabi oscillations in (b) which appears not to be
optimal.

## Conclusions and perspectives

In this article, we present the settings of an experimental breadboard for
ODMR experiments with both commercially available instruments and an open-
source interface. In particular, we show that a relatively inexpensive
acquisition card equipped with analog-to-digital converter, together with our
upgrades on the Python-based software Qudi, allow anyone with minimal
experimental skills to setup ODMR instrumentation capable of advanced magnetic
resonance pulse sequences and address both individual optically-active spin-
systems and large ensembles. We believe that our work will help foster the
development of ODMR studies and their applications to sensing and metrology,
quantum technologies, and material science, while also making ODRM methods
more accessible to non-specialists. We encourage Qudi users to develop new
instrument modules and make them available on GitHub to participate to wider
openness and collaboration in research. We conclude by mentioning a few
envisioned directions for both hardware and software developments in the
coming years.

Performing ODMR at room temperature without space constraints does not present
significant hardware challenges anymore. Yet, implementing similar
measurements in confined environments, in ultra-high vacuum or at cryogenic
temperatures, can rapidly become much more complex. While optical excitation
and collection through free space or optical fiber is usually feasible in such
circumstances, it can be challenging to locally provide enough MW power
(without spurious heating) and apply strong enough and precisely oriented bias
magnetic fields. Therefore, an integrated solution in the form of a chip-scale
cryo-compatible device, with footprint below a few square millimeters, and
which embeds a d.c. supplied MW oscillator and antenna, would be a valuable
commodity for ODMR experiments. Such devices have already been developed in
the context of conventional electron and nuclear spin resonance spectroscopy
Matheoud, Sahin, and Boero (2018).

Regarding MW generation, voltage-controlled oscillators (VCOs) are inexpensive
and compact alternatives to high-end MW sources, and lowering their phase
noise would make them more useful for advanced measurements. Design of MW
antennas delivering strong and homogeneous driving fields over wide areas is
still in progress.

For single-emitter ODMR, both the digital single photon counting modules and
the time tagging electronics drive the costs of the setup higher (typically
5’000 EUR each). As a cheaper alternative to the latter, FPGAs can be
programmed for photon counting to yield much better temporal resolution than
an NI card. An open-source package based on a commonly available FPGA for a
do-it-yourself time tagger would make single emitter measurements more
accessible.

For generating complex, long and low noise pulse sequences, NI cards reach
their limits. As above, FPGA-based open-source solutions have the potential to
make such experiments accessible at lower costs and with a high degree of
customization for specific needs.

###### Acknowledgements.

This project has received funding from the Swiss National Science Foundation
(grants No. 185824, 170684, 198898), from the European Union's Horizon 2020
research and innovation programme under the Marie Skł odowska-Curie grant
agreement No. 754354, and from EPFL Interdisciplinary Seed Fund.
We acknowledge Umut Yazlar from Appsilon B.V. (Delft, Netherlands) for his
valuable contribution in providing single crystal diamond samples with desired
characteristics.

## Author declarations

### Conflict of Interest

The authors declare no conflicts of interest.

### Author Contributions

H.S and H.B. contributed equally to this work. H.B contributed significantly
to the coding and assisted with the writing. H.S played a key role in writing
the paper and valuably contributed to the coding. E.L contributed to the
writing and revisions. H.B built the characterization setup for measurements.
V.G. conducted, analyzed, and reported the measurements. M.C guided the
writing throughout the project. M.C and C.G supervised the work and
contributed to the writing and revisions.

## Data availability

The original and updated version of qudi are available on github Qud (a); Our
(a).

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