# Characterization of the photon emission statistics in nitrogen-vacancy
centers

I. Panadero  Arquimea Research Center, Camino las Mantecas s/n, 38320 San
Cristobal de La Laguna, Spain  Department of Physics, Universidad Carlos III
de Madrid, Avda. de la Universidad 30, 28911 Leganés, Spain  Department of
Physical Chemistry, University of the Basque Country UPV/EHU, 48080 Bilbao,
Spain  H. Espinós  Department of Physics, Universidad Carlos III de Madrid,
Avda. de la Universidad 30, 28911 Leganés, Spain  L. Tsunaki  Helmholtz-
Zentrum Berlin für Materialien und Energie GmbH, Hahn-Meitner-Platz 1, 14109
Berlin, Germany  K. Volkova  Helmholtz-Zentrum Berlin für Materialien und
Energie GmbH, Hahn-Meitner-Platz 1, 14109 Berlin, Germany  A. Tobalina
Arquimea Research Center, Camino las Mantecas s/n, 38320 San Cristobal de La
Laguna, Spain  J. Casanova  Department of Physical Chemistry, University of
the Basque Country UPV/EHU, 48080 Bilbao, Spain  EHU Quantum Center,
University of the Basque Country, 48940 Leioa, Spain  IKERBASQUE, Basque
Foundation for Science, Plaza Euskadi 5, 48009 Bilbao, Spain  P. Acedo
Department of Electronic Technology, Universidad Carlos III de Madrid, Avda.
de la Universidad 30, 28911 Leganés, Spain  B. Naydenov  Helmholtz-Zentrum
Berlin für Materialien und Energie GmbH, Hahn-Meitner-Platz 1, 14109 Berlin,
Germany  R. Puebla  Department of Physics, Universidad Carlos III de Madrid,
Avda. de la Universidad 30, 28911 Leganés, Spain  E. Torrontegui
<EMAIL_ADDRESS>Department of Physics, Universidad Carlos III de
Madrid, Avda. de la Universidad 30, 28911 Leganés, Spain
<EMAIL_ADDRESS>
###### Abstract

We model and experimentally demonstrate the full time-dependent counting
statistics of photons emitted by a single nitrogen-vacancy (NV) center in
diamond under non-resonant laser excitation and resonant microwave control. A
generalization of the quantum jump formalism for the seven electronic states
involved in the fast intrinsic dynamics of an NV center provides a self-
contained model that allows for the characterization of its emission and
clarifies the relation between the quantum system internal states and the
measurable detected photon counts. The model allows the elaboration of
detection protocols to optimize the energy and time resources while maximizing
the system sensitivity to magnetic-field measurements.

Keywords: Nitrogen vacancy centers, Full photon statistics.

##  I Introduction

The nitrogen-vacancy (NV) center in diamond has emerged as a prominent
platform in the field of quantum technologies [1, 2] due to its formidable
optical transitions that enable initialization and spin-state discrimination,
in addition to exceptional stability, extended coherence time of up to several
milliseconds, even at room temperature, and effective control using both
microwave (MW) and optical methods [3, 4]. Its level structure and coherence
time are sensitive to different physical fields, making it a versatile sensor
with nano-scale spatial resolution and high sensitivity. NV centers have
demonstrated great potential in applications of magnetic field [5, 6],
temperature [7], electric field [8], and strain [9] sensing, as well as in
nuclear magnetic resonance [10]. Additionally, they can be used as quantum
emitters [11, 12]: when optically excited, they can absorb a photon and
transition to an excited state, and then relax back to the ground state by the
emission of a single photon. The fluorescence emission properties depend on
the NV spin state, allowing for a spin-dependent readout [13, 14].
Understanding the spin-dependent photon emission is crucial for characterizing
the dynamics of the NV center and optimizing its performance in quantum
information processing, sensing and communication tasks [12, 15, 16].

Fluorescence properties of the NV center have been modeled through rate
equations for specific energy levels [4, 13, 17]. This method provides the
fluorescence properties based on the populations of the NV electronic states.
Different energy models and sets of transition rates have been explored for
the NV center in various contexts, including ionization dynamics [18],
hyperfine couplings [19, 20], level anti-crossing [21] and optically detected
magnetic resonance (ODMR) experiments [4]. However, these methods typically
either rely on a steady-state approximation of the NV dynamics or simply focus
on state populations via rate equations. In either case, these approaches do
not directly allow for a precise characterization of the key experimental
quantity, namely, the emitted stream of photons [22, 23].

Given the limitations of current approaches, we make use of a formalism based
on quantum jumps [24]. This technique has been previously used to model the
emission of different quantum systems such as single molecules [25], qubit
readout of transient charge states [26], quantum dots [27], electron transport
[28], and NV ionization [29]. As we experimentally demonstrate, the quantum
jump formalism can be adapted to account for the full photon emission
statistics (PES) of the NV center within a modelization of the seven
electronic states involved in the fast intrinsic dynamics [30]. The PES are
defined as the probabilities $P(n,t)$ of emitting a certain number $n$ of
photons at specific time $t$, and serve as a phenomenological framework for
establishing the connection between the quantum state of an NV center and the
stream of detected photons. The proposed model fully characterizes the
emission properties of a single NV, providing physical insight of the NV
fluorescence, giving access to different relevant measurable quantities, and
allowing for the design of efficient magnetic-field sensing protocols.

This work is structured as follows. The PES model is described in Sect. II,
introducing first in Sect. II.1 the relevant energy levels, and solving their
dynamics in Sect. II.2 using generalized multi-level Bloch equations, also
referred to as a quantum jump formalism. Section III.1 presents the
experimental setup for benchmark and continues with a set of applications.
Section III.2 discusses how this statistics allows for optimized state
readout, and in Sect. III.3 we analyze the properties of the emitter by
computing the autocorrelation function from the photon statistics.
Additionally, we simulate characterization measurements such as the saturation
curve of the NV in Sect. III.4, and optimize the sensitivity in continuous
wave ODMRs, which combine both microwave and laser excitation, in Sect. III.5.
The main conclusions of this work are summarized in Sect. IV.

##  II NV photon counting statistics

###  II.1 Energy levels

As we are interested in the polarization and readout of a negatively charged
NV- (in the following NV), we consider the fast intrinsic dynamics [30]
responsible for the radiative and non-radiative transitions between the
electronic states. The NV relevant states are distributed in two different
spin $S=1$ manifolds, the ${}^{3}A_{2}$ electronic ground state with the
$\\{\ket{0},\ket{1},\ket{-1}\\}$ states used to encode a qubit and the
${}^{3}E$ excited manifold containing the $\\{\ket{e0},\ket{e1},\ket{-e1}\\}$
states. Laser light is used to couple the ground states to the spin-triplet
excited states. When excited, the NV center relaxes either directly through a
spin-conserving transition emitting red fluorescence of $\sim 637-800$ nm
wavelength, or across a non spin-conserving channel through a meta-stable
spin-singlet state $\ket{s}$ responsible for the initialization and readout of
the NV. The energy levels together with the allowed transitions between them
are depicted in Fig. 1.

The coherent dynamics of the electronic ground state under MW resonant control
and a static magnetic field $B_{z}$ aligned with the NV axis is described by
the Hamiltonian

$\hat{H}=D\,\hat{S}_{z}^{2}-\gamma_{e}\,B_{z}\,\hat{S}_{z}\
+\Omega\hat{S}_{x}\cos(\omega t),$ (1)

where we have neglected perpendicular terms and hyper-fine interactions. The
NV operators $\hat{S}_{x,y,z}$ correspond to spin-1 matrices. The first term
is the intrinsic zero-field splitting with $D=(2\pi)2.87\,$ GHz. The second is
the Zeeman splitting of the $\ket{m_{s}}=\ket{\pm 1}$ states due to the static
magnetic field $B_{z}$, proportional to the electron gyromagnetic ratio
$\gamma_{e}=(2\pi)28.024$ $\frac{\text{MHz}}{\text{mT}}$. The last term
represents the MW control with a Rabi frequency $\Omega$ proportional to the
square root of the microwave power $\Omega\propto\sqrt{P_{MW}}$. A proper
selection of the driving frequency $\omega$ enables the qubit encoding either
in the $\ket{0}\leftrightarrow\ket{1}$ or $\ket{0}\leftrightarrow\ket{-1}$
states with transition frequency $\omega_{0}=D\pm\gamma_{e}B_{z}$. This is
done by setting the frequency of the MW source close to the transition
$\omega=\omega_{0}+\Delta$, with $\Delta$ a small detuning. Under this
condition and moving to an interaction picture with respect to
$\hat{H}_{0}=D\hat{S}_{z}^{2}-\gamma_{e}B_{z}\hat{S}_{z}$, we can neglect
transitions to the remaining spin level, and consider our system as a
spin-$\frac{1}{2}$ in which the Hamiltonian takes the form

$\hat{H}_{2}=\Delta\hat{\sigma}_{z}+\Omega\hat{\sigma}_{x},$ (2)

with $\hat{\sigma}_{i}$ the Pauli matrices. Without loss of generality, let us
assume that we select the resonance $\omega=D+\gamma_{e}B_{z}$ such the
$\ket{-1}$ is left out of the spin dynamics.

Figure 1: Representation of the levels of an NV center showing the system
dynamics. The electronic ground states $\ket{0}$ and $\ket{1}$ encode the
qubit and are coupled with a MW field of Rabi frequency $\Omega$ and detuning
$\Delta$. Laser light excites the states $\ket{0}$ and $\ket{1}$ with equal
rate $\Gamma_{P}$ to the excited states $\ket{e0}$ and $\ket{e1}$. The excited
states decay to the ground state with rate $\Gamma_{0}$ emitting a photon. The
state $\ket{e1}$ also decays to the singlet state $\ket{s}$ with rate
$\Gamma_{f1}\gg\Gamma_{f0}$. The state $\ket{s}$ can decay to either ground
state $\ket{0}$ or $\ket{1}$ with rates $\Gamma_{s0}$ and $\Gamma_{s1}$,
respectively.

On the other hand, the non-unitary dynamics of the system, described by the
density matrix $\varrho(t)$, follows the Lindblad master equation

$\dot{\varrho}=-i[\hat{H}_{2},\varrho]+L[\varrho]=\mathcal{L}[\varrho],$ (3)

with

$L[\varrho]=\sum_{k}\Gamma_{k}(\hat{A}_{k}\varrho\hat{A}_{k}^{\dagger}-\frac{1}{2}\\{\hat{A}_{k}^{\dagger}\hat{A}_{k},\varrho\\}),$
(4)

where $\hat{A}_{k}$ is a jump operator that accounts for the dissipative part
of the dynamics with rate $\Gamma_{k}$ [19].

Using this master equation, it is possible to extend beyond a basic rate
equation model and comprehensively consider the impact of transverse fields
and coherent spin polarization exchange. Laser light excites the electronic
ground states to the excited states with a rate
$\Gamma_{0}^{eo}=\Gamma_{1}^{e1}=\Gamma_{-1}^{-e1}=\Gamma_{P}$ through a spin
conserving transition, where their associated jump operators $\hat{A}_{k}$ are
of the form $\ket{e0}\bra{0}$, $\ket{e1}\bra{1}$ and $\ket{-e1}\bra{-1}$,
respectively. The excitation rate $\Gamma_{P}$ is modeled assuming a linear
dependency with the laser power $P_{L}$, $\Gamma_{P}=c_{L}P_{L}$ [4]. The
parameter $c_{L}\simeq 0.1$ MHz/$\mu$W, is estimated for our current setup by
fitting the PES model to the experimental data through the saturation curve
III.4. The electronic excited states can spontaneously decay with rates
$\Gamma_{e0}^{0}=\Gamma_{e1}^{1}=\Gamma_{-e1}^{-1}=\Gamma_{0}$ to the ground
states.

The operators $\hat{A}_{k}$ do not only represent incoherent jumps between
different states, but can also account for other non unitary processes such as
those producing longitudinal and transverse relaxation of the ground state,
with rates $\gamma_{1}=1/T_{1}$ and $\gamma_{2}=1/T_{2}$. Without external
excitation, the populations of the ground states $\ket{0}$ and $\ket{\pm 1}$
are assumed to decay towards equal populations at a rate $\gamma_{1}$ through
the operators $\hat{A}_{+}=\hat{S}^{+}$ and $\hat{A}_{-}=\hat{S}^{-}$ with
rate $\Gamma_{\pm}=\sqrt{\gamma_{1}/2}=\sqrt{1/(2T_{1})}$. The transverse
relaxation enters via a jump operator $\hat{A}_{z}=\hat{S}_{z}$, whose
associated rate is given by $\Gamma_{z}=\sqrt{\gamma_{2}}=\sqrt{1/T_{2}}$.

Optical readout is possible because the $\ket{e1}$ and $\ket{-e1}$ states may
also decay to the metastable singlet state $\ket{s}$ with rates
$\Gamma_{e1}^{s}=\Gamma_{-e1}^{s}=\Gamma_{f1}$ much faster than the decay from
the $\ket{e0}$ to the metastable state $\Gamma_{e0}^{s}=\Gamma_{f0}$. Thus,
the $\ket{0}$ state is brighter than the $\ket{\pm 1}$ states when excited
with the green source, as more population relaxes back to the ground state
through radiative transitions in that case. Furthermore, optical polarization
occurs because the state $\ket{s}$ decays with different rates
$\Gamma_{s}^{0}=\Gamma_{s0}$ and $\Gamma_{s}^{1}=\Gamma_{s}^{-1}=\Gamma_{s1}$
to the ground states $\ket{0}$ and $\ket{\pm 1}$. Typical values of the decay
rates are $\Gamma_{0}\simeq 63$ MHz, $\Gamma_{f0}\simeq 12$ MHz,
$\Gamma_{f1}\simeq 80$ MHz, $\Gamma_{s1}\simeq 2.4$ MHz and $\Gamma_{s0}\simeq
3.3$ MHz [19]. Similar values can be found in [4, 13, 18, 31].

###  II.2 Quantum jumps

The master equation given by Eq. (3) provides information about the time
evolution of the system density matrix, but it does not directly give us
information about the number of photons that are emitted during the evolution.
To address this, the density matrix $\varrho(t)$ is generalized to include
information about the number of spontaneous photon emission events (or jumps)
that have occurred prior to time $t$ [24]. This generalization allows us to
resolve the full density matrix into individual components $\rho(n,t)$,
representing a quantum trajectory with $n$ photons being spontaneously emitted
by the NV center during the time interval $[0,t]$,

$\varrho(t)=\sum_{n}\rho(n,t).$ (5)

The equations of motion for these generalized matrix elements follow from Eq.
(3) via an exact operator expansion for $\rho$ in Liouville space [27],

$\dot{\rho}(n,t)=\mathcal{L}_{0}[\rho(n,t)]+\mathcal{L}_{1}[\rho(n-1,t)],$ (6)

where we split the Liouville operator from Eq. (3)
$\mathcal{L}[\varrho]=\mathcal{L}_{0}[\varrho]+\mathcal{L}_{1}[\varrho]$ in
such a way that we have singled out the jump operator associated to the
radiative spontaneous decays $\mathcal{L}_{1}$. This necessarily advances the
cumulative photon count by one $\rho(n-1,t)\rightarrow\rho(n,t)$, with rate
$\Gamma_{0}$, of the excited states $\\{\ket{e0},\ket{e1},\ket{-e1}\\}$ into
the ground states $\\{\ket{0},\ket{1},\ket{-1}\\}$

$\mathcal{L}_{1}[\varrho]=\Gamma_{0}\sum_{i\in\\{0,1,-1\\}}(\outerproduct{i}{ei}\varrho\outerproduct{ei}{i}-\frac{1}{2}\\{\outerproduct{ei}{ei},\varrho\\}),$
(7)

and $\mathcal{L}_{0}[\varrho]=\mathcal{L}[\varrho]-\mathcal{L}_{1}[\varrho]$.

We numerically solve the set of coupled differential equations (see Appendix
A) that dictate the dynamics described by Eq. (6) under time-dependent laser
intensity, MW power and detuning by iterating up to a specific cutoff of
emitted photons. Recall that for smaller and simpler systems the problem can
be solved analytically by introducing generating functions [24, 25, 26, 27],
this is no longer the case for general time-dependent laser and/or MW
controls.

The PES of the system is obtained by tracing $\rho(n,t)$,

$P(n,t)=\Tr{\rho(n,t)}.$ (8)

This formalism captures the probability distributions of an NV regardless of
its initial configuration, going beyond the steady-state configuration. Two
examples of $P(n,t)$ are represented in Fig.2a as a function of time for
different $n$ and in Fig.2b as a function of $n$ at a fixed time for two
different initializations of the NV center.

Figure 2: (a) Emission probability of different number of photons $n$ as a
function of time when exciting the NV center initially in a thermal state with
a laser source of $\Gamma_{P}=10$ MHZ. (b) After 700 ns of laser irradiation,
we plot the probability distributions of measuring $n$ photons for the initial
state $\ket{0}$ in red ($P_{0}(n)$) and $\ket{1}$ in blue ($P_{1}(n)$). The
error $\epsilon=(\epsilon_{0}+\epsilon_{1})/2$ is represented by the area of
the overlap between both functions. We plot in green the log-likelihood ratio
$\lambda$ given by Eq. (11), which can be understood as the value of how
confidently we can assign a photon number $n$ to each state.

From these distributions we compute the expectation value of the number of
emitted photons as a function of time, as well as higher moments of the
distribution,

$\expectationvalue{n^{m}(t)}=\sum_{n}P(n,t)n^{m},$ (9)

and also derive the counts per seconds, i.e. the photon intensity
$\expectationvalue{I}$, that the NV center emits through the time derivative
of the average number of photons,

$\expectationvalue{I(t)}=\expectationvalue{\dot{n}(t)}.$ (10)

Employing these tools, we performed numerical simulations to compute the PES
and used them to analyze different physical scenarios of interest. Overall,
the model provides a comprehensive framework for understanding the photon
emission dynamics of NV centers, and can be used to make predictions under a
range of experimental conditions.

##  III Applications

###  III.1 Experimental methods

The experimental measurements to benchmark the model presented in Sect. II
were performed on an NV center from a diamond layer grown by chemical vapour
deposition on a high pressure high temperature substrate. Optical
characterization of NV centers was performed using a custom confocal
microscope with continuous wave excitation for optically detected magnetic
resonance (ODMR) experiments. The confocal setup used to interrogate single NV
centers is similar to the one used in [32]. For NV optical excitation we use a
diode laser with a wavelength of 515 nm. The NV center fluorescence was
collected using a $100\times$ oil objective with a $1.4$ numerical aperture.
Emitted photons were detected by a SPAD and counted with a FAST ComTec MCS8.
The Qudi software [33] suite was used to control the experiments and for data
collection. The static magnetic field was generated with a permanent magnet.
The microwave source used for ODMR experiments was TTi TGR600 with the
amplifier Mini-Circuits ZHL-16W-43-S+. The AWG used to create the pulses for
the Rabi experiment was Tektronix AWG7122C. Both MW signals were delivered to
the sample through a cable antenna. All the experiments were performed
employing the same selected NV center except for the Rabi oscillations where a
second NV with a a 15N $20-30$ keV implantation was used.

###  III.2 State readout

NV-based sensing applications rely on standard control sequences that involve
spin initialization, manipulation, and readout. The effectiveness of the spin
readout process is crucial to the overall performance of the sensor, which can
be evaluated within the PES model [34]. PES provides a useful framework for
proposing new measurement strategies, improving sensitivity based on Bayesian
estimation [35], performing efficient single shot readout measurements [36] or
speeding them up through a real-time adaptive decision rule [29], which can
substantially decrease the average measurement time without significantly
affecting readout fidelity. In the context of NV centers, a readout process
usually consist on the discrimination between the $\ket{0}$ and $\ket{1}$
states of the NV center that encode the qubit. These states can be
distinguished through the number of emitted photons having different
statistical properties depending on the internal levels involved in the
transition [14, 37].

The single-repetition error rate $\epsilon$ is a commonly used figure of merit
for the readout performance in a single repetition [34], defined as the
average probability in a binary assignment of the incorrect state to the
observed outcome. For an ideal measurement with $\epsilon=0$ the distributions
associated to each state do not overlap [38], so we can discriminate between
the $\ket{0}$ and $\ket{1}$ states in a single readout. However, a readout is
subject to uncertainties in the discrimination leading to non-ideal
measurements and overlap between the probability distributions $P_{0}(n,t)$
and $P_{1}(n,t)$ that hinder the $\\{\ket{0},\ket{1}\\}$ state discrimination,
see Fig. 2b. The assignment rule that minimizes $\epsilon$ is obtained by
calculating the log-likelihood ratio

$\lambda(n,t)=\ln{\frac{P_{0}(n,t)}{P_{1}(n,t)}}.$ (11)

When $\lambda(n,t)$ is larger (smaller) than 0, the state $\ket{1}$
($\ket{0}$) is assigned, see Fig. 2b. The log-likelihood ratio can be
interpreted as the level of confidence of the observer in the state assignment
given the collected number of photons $n$. The average single-repetition error
rate is $\bar{\epsilon}=\frac{1}{2}(\epsilon_{0}+\epsilon_{1})$, where

$\epsilon_{0}(t)=\sum_{n\leq n_{c}}P_{0}(n,t),\hskip
28.45274pt\epsilon_{1}(t)=\sum_{n>n_{c}}P_{1}(n,t),$ (12)

are the error rates conditioned on preparation of $\ket{0}$ and $\ket{1}$ with
$n_{c}$ defined such as $\lambda(n_{c},t)=0$. These error rates $\epsilon_{0}$
and $\epsilon_{1}$ are represented by color bars in Fig. 2b.

Although a single readout cannot determine with certainty whether the NV
center is in the $\ket{0}$ or $\ket{1}$ state due to measurement uncertainty,
repeated readouts “average out” the noise, mitigating readout errors. By
performing $N$ repetitions of the readout process provides a set of photon
counts outcomes $\\{n_{1},n_{2},...n_{N}\\}$ with an associated cumulative
error rate $e_{N}$, that decreases asymptotically as the number of
measurements $N$ grows [34]

$\ln e_{N}(t)\rightarrow-C(t)N\text{ as }N\rightarrow\infty.$ (13)

with

$C(t)=-\min_{0\leq s\leq 1}\ln{[\sum_{n}P_{0}(n,t)^{s}P_{1}(n,t)^{1-s}]}.$
(14)

The quantity $C$ is known as the Chernoff information [39], a symmetric
distance measure between the distributions $P_{0}(n,t)$ and $P_{1}(n,t)$ that
can be interpreted as a rate of information gain per repetition. Readout
outcome distributions $P_{0}(n,t)$ and $P_{1}(n,t)$ with the same single
repetition error rates $\epsilon$ do not necessarily have the same Chernoff
information.

Figure 3: (a) Calculated Chernoff information for different laser powers as a
function of time. The read circle and black square correspond to the pulse
power and length at which we simulate two different Rabi oscillations in Fig.
4b. (b) The maximum value achieved by the Chernoff information
$max\\{C(t)\\}$, and the time at which this value is achieved $t_{max}$, as a
function of the laser intensity. This allows to investigate how fast and well
we are able to discriminate the NV center state and the required power. The
maximum value and time saturates as the power increases.

The calculated Chernoff information is a useful tool for analyzing the
statistical properties of the fluorescence. It can be used to estimate the
minimum number of photons that need to be measured in order to discriminate
between two different quantum states of a light source with a certain level of
confidence [34, 40], thereby providing us with the sensitivity limits of a
quantum system. Thus, the Chernoff information provides useful insight into
the optimal experimental parameters for maximizing state discrimination.

Figure 4: (a) Measured Rabi oscillations for initialization and readout pulses
of $2$ $\mu$s and a MW field with $\Omega=270$ MHz, $\Delta=0$ MHz and
$\Gamma_{P}=20$ MHz. The dashed line represents the mean photon number
$\langle n\rangle$ computed by Eq. (9) from the theoretical $P(n,t)$, symbols
correspond to sampling $N$ photon counts, while the black-solid line is the
measured Rabi oscillations using the procedure and setup described in Sect.
III.1. (b) Simulated Rabi oscillations for different Chernoff informations
with $\Omega=\Delta=10$ MHz and readout excitation rate $\Gamma_{P}=12.5$ MHz
for two different time pulses. In red $t=300$ ns corresponding to the Chernoff
maximum $C=4\cdot 10^{-3}$ and in black $t=2$ $\mu s$ for $C=1\cdot 10^{-3}$.
We plot the normalized counts deduced from the same emission probability
$P(n,t)$ considering $N=3000$ measurements (points) and its average (lines).
Figure 5: Measured auto-correlation functions (black solid line) and the
photon statistics model predictions (red dashed line) according to (18) for
different excitation rates: (a) $\Gamma_{P}=5$ MHz, (b) $\Gamma_{P}=15$ MHz
and (c) $\Gamma_{P}=20$ MHz

In Fig. 3a, the Chernoff information is plotted for different laser
intensities as a function of time taking a constant laser irradiation. For a
specific laser intensity, the figure shows a maximum, indicating the optimal
duration of the laser pulse for reading out the ground state of the NV center.
In Fig. 3b, the maximum value of the peak and the time $t_{max}$ at which it
occurs are plotted for different values of $\Gamma_{P}$. As the laser
intensity is increased, the peak becomes higher and occurs faster, resulting
in better and faster state discrimination. However, it also increases the
background noise introduced in the measurement. This will be discussed in
Sect. III.4. Both the Chernoff peak and $t_{max}$ saturate for high laser
intensities, meaning that increasing the laser power beyond this point will
not result in better measurement outcomes.

To illustrate the relevance of the Chernoff information we have used the PES
to simulate the readout of Rabi oscillations. A MW pulse takes the NV to a
superposition state, that after a readout laser pulse emits fluorescence with
a $P(n,t)$ distribution for a given irradiation time $t$. A photon measurement
$n$ can be simulated by sampling a point from this distribution. Thus, $N$
measurements consist on sampling a set of readouts outcomes
$\\{n_{1},n_{2},...,n_{N}\\}$. On the one hand, as the number of repetitions
increases (see symbols of Fig. 4a) the distribution of the sampled photons
converges to the average of the distributions (red-dashed line), since the
error of repeated measurements averages out the uncertainties. Note, how the
model distribution perfectly cast the experimental data (black-solid line). On
the other hand, working conditions for a larger Chernoff parameter (see Fig.
3a) favors the discrimination of the $\ket{0}$ and $\ket{1}$ states, enhancing
the Rabi oscillations amplitude, cf. Fig. 4b. The increased amplitude of the
Rabi oscillations resulting from larger Chernoff value leads to improved
discrimination of the state of the NV center, higher readout fidelity, and
ultimately, better experimental outcomes. As seen, the photon statistic model
allows to predict and improve measurement sequences by optimizing relevant
experimental parameters.

###  III.3 Autocorrelation measurements

Autocorrelation measurements are a valuable technique for studying the
temporal dynamics of fluorescence signals and the statistical behavior of
photons in a light source [41, 42]. By measuring the correlation between the
intensity of the signal at different delay times, it is possible to extract
information about the decay rates of the states of an NV center [43],
categorize an emitter as a single photon source [44], investigate NV centers
embedded in different physical structures [32, 45, 46, 47] and characterize
the interactions with other quantum systems [11, 48, 49, 50]. The second-order
temporal coherence $g^{(2)}(\tau)$ measures the photon autocorrelation, that,
in terms of the intensity $I(t)$, reads

$g^{(2)}(\tau)=\frac{\expectationvalue{I(t)I(t+\tau)}}{\expectationvalue{I(t)}^{2}}.$
(15)

Alternatively, it can be also computed using the delay function $D_{1}(t)$
describing the distribution of delay times before the next emission occurs.
This function is calculated from the survival probability $P(0,t)$ as [51]

$D_{1}(t)=-\dot{P}(0,t).$ (16)

The conditional probability of an NV center emitting any photon between time
$\tau$ and $\tau+d\tau$, after having emitted a photon at time $\tau_{0}$ = 0
is $D(\tau)d\tau$. The photon emitted at time $\tau$ can be the first to be
emitted after that at time $\tau_{0}=0$ or the next after any other
$\tau^{\prime}$ ($0<\tau^{\prime}<\tau)$, so that

$D(\tau)=D_{1}(\tau)+\int_{0}^{\tau}d\tau^{\prime}D(\tau^{\prime})D_{1}(\tau-\tau^{\prime}).$
(17)

The function $g^{(2)}(\tau)$ is the normalized correlation function for a
photon detection at $\tau_{0}=0$ followed by the detection of any photon (not
necessarily the next) at time $\tau$. It follows that

$g^{(2)}(\tau)=\frac{D(\tau)}{\lim_{\tau\to\infty}D(\tau)}.$ (18)

In Fig. 5 we plot the background noise corrected $g^{2}(\tau)$ [52] from a
single NV center in a Hanbury-Brown-Twiss correlation experiment [50]. The
results predicted from the photon statistics model (18) perfectly fit the
experimental data.

Alternatively, the Mandel $Q$ parameter [25]

$Q(t)=\frac{\expectationvalue{n^{2}(t)}-\expectationvalue{n(t)}^{2}}{\expectationvalue{n(t)}}-1,$
(19)

is another measurement for the correlation of emitted photons that relies in
the variance of the photon number in the detected fluorescence. When $Q=0$,
the counting statistics $P(n,t)$ follow Poisson distributions with no
correlation between emitted photons. On the other hand, when $Q>0$, the
statistics is super-Poissonian, indicating photon bunching, where photons tend
to be emitted together. Conversely, sub-Poissonian statistics ($Q<0$) indicate
photon anti-bunching, where photons tend to be emitted at well-separated
times.

Using PES, in Fig. 6a we calculate $Q(t)$ at different power excitation,
observing different bunching behaviours as a function of time for a thermal
initialization of the NV center. The photon emission begins with some degree
of anti-bunching because the excited states are not yet populated. At time
$t_{min}$, such that $Q(t_{min})=\min\\{Q(t)\\}$, the emitted photons are
maximally anticorrelated, showing a non-classical anti-bunched character.
Then, the statistics starts to bunch at time $t_{0}$, until it saturates at
long times and the emitted distribution of photons becomes thermal. As
presented in Figs. 6b and 6c, the model predicts a higher degree of anti-
bunching for increasing power, albeit the time interval during which anti-
bunching takes place is reduced.

This information is helpful to investigate the capabilities of NV centers as
photon emitters since it has been found that NV centers exhibit both bunching
and antibunching behavior depending on the specific excitation conditions [13,
53]. Bunching and antibunching phenomena have important consequences on the
performance of devices that use quantum systems as a source of photons, such
as those used in quantum communication. For example, bunching can lead to
increased noise in the emitted light [54, 55], while antibunching can improve
the use of NV centers for single photon generation [56, 57, 58].

Figure 6: (a) Mandel $Q$ parameter (19) as a function of time for a thermal
initialization of the NV center. The characteristic times $t_{min}$ and
$t_{0}$ correspond to a Mandel paramater value $Q(t_{min})=\min\\{Q(t)\\}$ and
$Q(t_{0})=0$ of the photon emission, respectively. (b) Both $t_{min}$ and
$t_{0}$, and (c) $Q(t_{min})$ as a function of the laser power.

###  III.4 Fluorescence saturation

As the laser power is increased, the NV center absorbs more photons and
becomes excited more frequently. Eventually, as the laser power continues to
increase, the populations of the excited state and the ground state become
approximately equal, resulting in a saturation of the emission rate of the NV
center. Using PES to compute the saturation curve for an NV center allows to
optimize and compare the performance of an NV-based sensor [59, 60], study its
quantum efficiency [61] and its signal-to-noise ratio [62].

Figure 7: Benchmark of the predicted saturation curve for an NV center as a
function of the laser power computed by the photon statistic model (red line)
against the detector photon counts (black points). The NV counts were computed
by subtracting the background counts (green triangles) to the total measured
counts (blue crosses).

In Fig. 7 we plot the experimental saturation curve of the selected NV center
(black dots), on top of the theoretical curve using the photon statistic model
(dashed-red line). Note that the photon statistic model matches the NV counts
computed by subtracting the background from the total measured counts. We also
have taken into account that not all the emitted photons are collected due to
the geometry of the experiment and the efficiency of the detector by
multiplying the saturation curve by a scaling factor. This scaling factor
provides insight into the collection efficiency of the setup and indicates
improvement potential. Modeling of such saturation curves may be useful in
scenarios where subtracting the background from the total counts can be
harder, for instance in experiments with NV ensembles.

###  III.5 Continuous Wave Optically Detected Magnetic Resonance (cwODMR)

The NV centers in diamond have demonstrated remarkable potential as a tool for
detecting magnetic fields. This ability is based on the NV electronic spin
level structure and the Zeeman effect, which results in a lifting of
degeneracy between the states $\ket{\pm 1}$ when a magnetic field is applied,
as discussed in Sect. II. The energy splitting between these states is
proportional to the applied magnetic field and can be measured via optically
detected magnetic resonance (ODMR) experiments.

In an ODMR experiment, a MW field is applied to the NV center while a green
laser initializes the NV center to the $\ket{0}$ state. When the MW field is
resonant with the transition between the $\ket{0}$ and one of the $\ket{\pm
1}$ states, fluorescence from the NV center drops, as population is
transferred from the bright $\ket{0}$ state to the darker states $\ket{\pm 1}$
[4, 43]. This phenomenon enables the detection of magnetic fields by measuring
the resonant transition frequency between the $\ket{0}$ and $\ket{\pm 1}$
states.

We perform Continuous Wave ODMR (cwODMR) experiments to benchmark the photon
statistic model. These experiments involve the continuous application of both
the laser and the MW field to the NV center. While Pulsed ODMR (pODMR)
experiments provide greater sensitivity, they require the generation and
synchronization of fast MW and laser pulses, which can be demanding in terms
of experimental setup. The advantages of cwODMR experiments include their
simplicity, ability to measure a wide range of magnetic fields, and capability
to detect weak magnetic fields. As a result, the cwODMR method is a valuable
technique for a broad range of applications, such as sensing magnetic fields
from biological systems [1], imaging magnetic materials [63], and detecting
single-spin dynamics [64].

Figure 8: (a) The black points correspond to an ODMR experiment with a MW
power of $\Omega=8$ MHz and an excitation rate of $\Gamma_{P}=20$ MHz. We
obtained a contrast $c=15\%$. The red line is the result from the photon
statistics simulation. (b) Contrast as a function of the laser power at a
fixed MW power of $\Omega=8$ MHz. The black points correspond to the values
from Lorentzian fits to the experimental data and the red line the photon
statistics model.

The accurate estimation of cwODMR center resonance frequency and precise
computation of the corresponding magnetic field requires a high contrast $c$
[4], quantity that depend on the MW and laser powers, as well as other
physical parameters. The photon statistics model predicts the behavior of $c$
allowing the optimization of MW and laser powers that leads to the maximal
sensitivity of the NV. Note that the model was successfully used to simulate
cwODMR, as demonstrated in Fig. 8a. In Figure 8b, the contrast is plotted for
different laser powers, with a fixed MW power of $\Omega=8$ MHz. The contrast
increases at lower powers and reaches a maximum around $200$ $\mu$W before
decreasing. This maximum depends on the applied MW field and occurs due to the
NV center being optically excited at a faster rate than the spin transitions.

##  IV Conclusions

The work provides physical insight to the photon emission of an NV allowing
its full characterization. The experimental benchmark validates the presented
PES model for the fast intrinsic dynamics of the NV center. Within this
framework, we were able to relate the qubit state of an NV with the detected
stream of photons, and investigate the NV properties in a variety of
scenarios, including the state readout optimization through the Chernoff
information and exemplified with the Rabi oscillations, characterizing
experimental autocorrelation measurements, and examine the properties of the
emitted photons by the bunching and antibunching regimes. Finally, the model
allows us to determine the optimal regimes for laser and MW intensities in
magnetic sensing experiments. Originally proposed to account for the PES of a
two-level atom, the method introduced in [24] can be extended to include the
relevant seven-electronic levels of an NV center. This technique can be
adapted to benchmark other color centers and quantum systems, as well as to
include other phenomena such as ionization [18] or two photon absorption [65].
Additionally, our work can also help to refine current measurements on the NV
center energy levels, leading to improved accuracy and precision in future
experiments. Furthermore, PES are useful for proposing and evaluating new
readout techniques [66] to improve the NV center sensitivity. Overall, our
findings contributes to the understanding of NV centers and have potential
implications for quantum sensing and related applications.

##  V Acknowledgements

We thank J. J. García-Ripoll for fruitful discussions and T. Pregnolato and T.
Schröder from Ferdinand-Braun-Institute for the 15N implantation on the
diamond sample. This work is supported by Arquimea Research Center and by
Horizon Europe, Teaming for Excellence, under grant agreement No 101059999,
project QCircle. We acknowledge financial support form the Spanish Government
via the projects PID2021-126694NA-C22 and PID2021-126694NB-C21
(MCIU/AEI/FEDER, EU), the German Research Foundation DFG (project numbers
410866378 and 410866565) and the German Federal Ministry of Education and
Research BMBF (Project DIQTOK, number 16KISQ034K), the ELKARTEK project
Dispositivos en Tecnologías Cuánticas (KK-2022/00062), the Basque Government
grant IT1470-22, by Comunidad de Madrid-EPUC3M14. H.E. acknowledges the
Spanish Ministry of Science, Innovation and Universities for funding through
the FPU program (FPU20/03409). J. C and E.T. acknowledge the Ramón y Cajal
(RYC2018-025197-I and RYC2020-030060-I) research fellowship.

##  Appendix A Generalized Bloch equations

The Lindblad equation (6) can be explicitly expanded into the density matrix
projectors $\rho_{ij}(n,t)\equiv\rho_{ij}^{n}$ leading to a set of generalized
Bloch equations. Defining the populations $\rho_{ii}(n,t)=\rho_{i}^{n}$ these
equations read

$\displaystyle\dot{\rho}_{0}^{n}$
$\displaystyle=-i\frac{\Omega}{2}(\rho_{01}^{n}-\rho_{10}^{n})-\frac{\gamma_{1}}{2}(\rho_{0}^{n}-\rho_{1}^{n})-\frac{\gamma_{1}}{2}(\rho_{0}^{n}-\rho_{-1}^{n})$
$\displaystyle-\Gamma_{P}\rho_{0}^{n}+\Gamma_{0}\rho_{e0}^{n-1}+\Gamma_{s0}\rho_{s}^{n},$
$\displaystyle\dot{\rho}_{1}^{n}$
$\displaystyle=i\frac{\Omega}{2}(\rho_{01}^{n}-\rho_{10}^{n})+\frac{\gamma_{1}}{2}(\rho_{0}^{n}-\rho_{1}^{n})$
$\displaystyle-\Gamma_{P}\rho_{1}^{n}+\Gamma_{0}\rho_{e1}^{n-1}+\Gamma_{s1}\rho_{s}^{n},$
$\displaystyle\dot{\rho}_{-1}^{n}$
$\displaystyle=\frac{\gamma_{1}}{2}(\rho_{0}^{n}-\rho_{-1}^{n})-\Gamma_{P}\rho_{1}^{n}+\Gamma_{0}\rho_{-e1}^{n-1}+\Gamma_{s1}\rho_{s}^{n},$
$\displaystyle\dot{\rho}_{01}^{n}$
$\displaystyle=-(\frac{\gamma_{1}}{2}+\frac{\gamma_{2}}{2}+\Gamma_{P}-i\Delta)\rho_{01}^{n}+i\frac{\Omega}{2}(\rho_{1}^{n}-\rho_{0}^{n}),$
$\displaystyle\dot{\rho}_{10}^{n}$
$\displaystyle=-(\frac{\gamma_{1}}{2}+\frac{\gamma_{2}}{2}+\Gamma_{P}+i\Delta)\rho_{10}^{n}-i\frac{\Omega}{2}(\rho_{1}^{n}-\rho_{0}^{n}),$
$\displaystyle\dot{\rho}_{e0}^{n}$
$\displaystyle=\Gamma_{P}\rho_{0}^{n}-\Gamma_{0}\rho_{e0}^{n}-\Gamma_{f0}\rho_{e0}^{n},$
$\displaystyle\dot{\rho}_{e1}^{n}$
$\displaystyle=\Gamma_{P}\rho_{1}^{n}-\Gamma_{0}\rho_{e1}^{n}-\Gamma_{f1}\rho_{e1}^{n},$
$\displaystyle\dot{\rho}_{-e1}^{n}$
$\displaystyle=\Gamma_{P}\rho_{-1}^{n}-\Gamma_{0}\rho_{-e1}^{n}-\Gamma_{f1}\rho_{-e1}^{n},$
$\displaystyle\dot{\rho}_{s}^{n}$
$\displaystyle=\Gamma_{f1}\rho_{e1}^{n}+\Gamma_{f1}\rho_{-e1}^{n}+\Gamma_{f0}\rho_{e0}^{n}-(\Gamma_{s0}+\Gamma_{s1})\rho_{s}^{n}.$
(20)

From the three first equations, note that the emission of $n$ photons is
affected by the $n-1$ photon process, where the state decays from the
electronic ${}^{3}E$ manifold to the corresponding ${}^{3}A_{2}$ electronic
ground state.

## References

  * Schirhagl  _et al._ [2014] R. Schirhagl, K. Chang, M. Loretz, and C. L. Degen, Nitrogen-vacancy centers in diamond: nanoscale sensors for physics and biology, Annu. Rev. Phys. Chem 65, 83 (2014).
  * Rondin  _et al._ [2014] L. Rondin, J.-P. Tetienne, T. Hingant, J.-F. Roch, P. Maletinsky, and V. Jacques, Magnetometry with nitrogen-vacancy defects in diamond, Reports on progress in physics 77, 056503 (2014).
  * Jelezko and Wrachtrup [2006] F. Jelezko and J. Wrachtrup, Single defect centres in diamond: A review, physica status solidi (a) 203, 3207 (2006).
  * Jensen  _et al._ [2013] K. Jensen, V. M. Acosta, A. Jarmola, and D. Budker, Light narrowing of magnetic resonances in ensembles of nitrogen-vacancy centers in diamond, Physical Review B 87, 10.1103/physrevb.87.014115 (2013).
  * Hong  _et al._ [2013] S. Hong, M. S. Grinolds, L. M. Pham, D. Le Sage, L. Luan, R. L. Walsworth, and A. Yacoby, Nanoscale magnetometry with nv centers in diamond, MRS bulletin 38, 155 (2013).
  * Bao  _et al._ [2023] B. Bao, Y. Hua, R. Wang, and D. Li, Quantum-based magnetic field sensors for biosensing, Advanced Quantum Technologies , 2200146 (2023).
  * Hsiao  _et al._ [2016] W. W.-W. Hsiao, Y. Y. Hui, P.-C. Tsai, and H.-C. Chang, Fluorescent nanodiamond: a versatile tool for long-term cell tracking, super-resolution imaging, and nanoscale temperature sensing, Accounts of chemical research 49, 400 (2016).
  * Dolde  _et al._ [2011] F. Dolde, H. Fedder, M. W. Doherty, T. Nöbauer, F. Rempp, G. Balasubramanian, T. Wolf, F. Reinhard, L. C. Hollenberg, F. Jelezko, _et al._ , Electric-field sensing using single diamond spins, Nature Physics 7, 459 (2011).
  * Ho  _et al._ [2021] K. O. Ho, K. C. Wong, M. Y. Leung, Y. Y. Pang, W. K. Leung, K. Y. Yip, W. Zhang, J. Xie, S. K. Goh, and S. Yang, Recent developments of quantum sensing under pressurized environment using the nitrogen vacancy (nv) center in diamond, Journal of Applied Physics 129, 241101 (2021).
  * Mamin  _et al._ [2013] H. Mamin, M. Kim, M. Sherwood, C. Rettner, K. Ohno, D. Awschalom, and D. Rugar, Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor, Science 339, 557 (2013).
  * Sipahigil  _et al._ [2012] A. Sipahigil, M. L. Goldman, E. Togan, Y. Chu, M. Markham, D. J. Twitchen, A. S. Zibrov, A. Kubanek, and M. D. Lukin, Quantum interference of single photons from remote nitrogen-vacancy centers in diamond, Physical review letters 108, 143601 (2012).
  * Leifgen  _et al._ [2014] M. Leifgen, T. Schröder, F. Gädeke, R. Riemann, V. Métillon, E. Neu, C. Hepp, C. Arend, C. Becher, K. Lauritsen, _et al._ , Evaluation of nitrogen-and silicon-vacancy defect centres as single photon sources in quantum key distribution, New journal of physics 16, 023021 (2014).
  * Nizovtsev  _et al._ [2001] A. Nizovtsev, S. Y. Kilin, C. Tietz, F. Jelezko, and J. Wrachtrup, Modeling fluorescence of single nitrogen–vacancy defect centers in diamond, Physica B: Condensed Matter 308, 608 (2001).
  * Hopper  _et al._ [2018] D. A. Hopper, H. J. Shulevitz, and L. C. Bassett, Spin readout techniques of the nitrogen-vacancy center in diamond, Micromachines 9, 437 (2018).
  * Su  _et al._ [2008] C.-H. Su, A. D. Greentree, and L. C. Hollenberg, Towards a picosecond transform-limited nitrogen-vacancy based single photon source, Optics Express 16, 6240 (2008).
  * Rodiek  _et al._ [2017] B. Rodiek, M. Lopez, H. Hofer, G. Porrovecchio, M. Smid, X.-L. Chu, S. Gotzinger, V. Sandoghdar, S. Lindner, C. Becher, _et al._ , Experimental realization of an absolute single-photon source based on a single nitrogen vacancy center in a nanodiamond, Optica 4, 71 (2017).
  * Robledo  _et al._ [2011a] L. Robledo, H. Bernien, T. Van Der Sar, and R. Hanson, Spin dynamics in the optical cycle of single nitrogen-vacancy centres in diamond, New Journal of Physics 13, 025013 (2011a).
  * Wirtitsch  _et al._ [2023] D. Wirtitsch, G. Wachter, S. Reisenbauer, M. Gulka, V. Ivády, F. Jelezko, A. Gali, M. Nesladek, and M. Trupke, Exploiting ionization dynamics in the nitrogen vacancy center for rapid, high-contrast spin, and charge state initialization, Physical Review Research 5, 013014 (2023).
  * Poggiali  _et al._ [2017] F. Poggiali, P. Cappellaro, and N. Fabbri, Measurement of the excited-state transverse hyperfine coupling in nv centers via dynamic nuclear polarization, Phys. Rev. B 95, 195308 (2017).
  * Auzinsh  _et al._ [2019] M. Auzinsh, A. Berzins, D. Budker, L. Busaite, R. Ferber, F. Gahbauer, R. Lazda, A. Wickenbrock, and H. Zheng, Hyperfine level structure in nitrogen-vacancy centers near the ground-state level anticrossing, Physical Review B 100, 075204 (2019).
  * Ivády  _et al._ [2021] V. Ivády, H. Zheng, A. Wickenbrock, L. Bougas, G. Chatzidrosos, K. Nakamura, H. Sumiya, T. Ohshima, J. Isoya, D. Budker, _et al._ , Photoluminescence at the ground-state level anticrossing of the nitrogen-vacancy center in diamond: A comprehensive study, Physical Review B 103, 035307 (2021).
  * Schmunk  _et al._ [2012] W. Schmunk, M. Gramegna, G. Brida, I. P. Degiovanni, M. Genovese, H. Hofer, S. Kück, L. Lolli, M. Paris, S. Peters, _et al._ , Photon number statistics of nv centre emission, Metrologia 49, S156 (2012).
  * Temnov and Woggon [2009] V. V. Temnov and U. Woggon, Photon statistics in the cooperative spontaneous emission, Optics express 17, 5774 (2009).
  * Cook [1981] R. J. Cook, Photon number statistics in resonance fluorescence, Phys. Rev. A 23, 1243 (1981).
  * Zheng and Brown [2003] Y. Zheng and F. L. H. Brown, Single-molecule photon counting statistics via generalized optical bloch equations, Phys. Rev. Lett. 90, 238305 (2003).
  * D’Anjou and Coish [2017] B. D’Anjou and W. A. Coish, Enhancing qubit readout through dissipative sub-poissonian dynamics, Physical Review A 96, 052321 (2017).
  * Xu and Vavilov [2013] C. Xu and M. G. Vavilov, Full counting statistics of photons emitted by a double quantum dot, Phys. Rev. B 88, 195307 (2013).
  * Rudge and Kosov [2019] S. L. Rudge and D. S. Kosov, Counting quantum jumps: A summary and comparison of fixed-time and fluctuating-time statistics in electron transport, The Journal of Chemical Physics 151, 034107 (2019).
  * D’Anjou  _et al._ [2016] B. D’Anjou, L. Kuret, L. Childress, and W. A. Coish, Maximal adaptive-decision speedups in quantum-state readout, Physical Review X 6, 011017 (2016).
  * Manson  _et al._ [2006] N. B. Manson, J. P. Harrison, and M. J. Sellars, Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics, Phys. Rev. B 74, 104303 (2006).
  * Robledo  _et al._ [2011b] L. Robledo, H. Bernien, T. van der Sar, and R. Hanson, Spin dynamics in the optical cycle of single nitrogen-vacancy centres in diamond, New Journal of Physics 13, 025013 (2011b).
  * Volkova  _et al._ [2022] K. Volkova, J. Heupel, S. Trofimov, F. Betz, R. Colom, R. MacQueen, S. Akhundzada, M. Reginka, A. Ehresmann, J. Reithmaier, S. Burger, C. Popov, and B. Naydenov, Optical and spin properties of nv center ensembles in diamond nano-pillars, Nanomaterials 12, 1516 (2022).
  * Binder  _et al._ [2017] J. M. Binder, A. Stark, N. Tomek, J. Scheuer, F. Frank, K. D. Jahnke, C. Müller, S. Schmitt, M. H. Metsch, T. Unden, _et al._ , Qudi: A modular python suite for experiment control and data processing, SoftwareX 6, 85 (2017).
  * D'Anjou [2021] B. D'Anjou, Generalized figure of merit for qubit readout, Physical Review A 103, 10.1103/physreva.103.042404 (2021).
  * Zhang  _et al._ [2023] J. Zhang, T. Liu, S. Xia, G. Bian, P. Fan, M. Li, S. Wang, X. Li, C. Zhang, S. Zhang, _et al._ , Efficient real-time spin readout of nitrogen-vacancy centers based on bayesian estimation, arXiv preprint arXiv:2302.06310 (2023).
  * Neumann  _et al._ [2010] P. Neumann, J. Beck, M. Steiner, F. Rempp, H. Fedder, P. R. Hemmer, J. Wrachtrup, and F. Jelezko, Single-shot readout of a single nuclear spin, science 329, 542 (2010).
  * Steiner  _et al._ [2010] M. Steiner, P. Neumann, J. Beck, F. Jelezko, and J. Wrachtrup, Universal enhancement of the optical readout fidelity of single electron spins at nitrogen-vacancy centers in diamond, Physical Review B 81, 035205 (2010).
  * Degen  _et al._ [2017] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Reviews of modern physics 89, 035002 (2017).
  * Chernoff [1952] H. Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, The Annals of Mathematical Statistics , 493 (1952).
  * Audenaert  _et al._ [2007] K. M. Audenaert, J. Calsamiglia, R. Munoz-Tapia, E. Bagan, L. Masanes, A. Acin, and F. Verstraete, Discriminating states: The quantum chernoff bound, Physical review letters 98, 160501 (2007).
  * Fishman  _et al._ [2021] R. E. Fishman, R. N. Patel, D. A. Hopper, T.-Y. Huang, and L. C. Bassett, Photon emission correlation spectroscopy as an analytical tool for quantum defects, arXiv preprint arXiv:2111.01252 (2021).
  * Buller and Collins [2009] G. Buller and R. Collins, Single-photon generation and detection, Measurement Science and Technology 21, 012002 (2009).
  * Doherty  _et al._ [2013] M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, The nitrogen-vacancy colour centre in diamond, Physics Reports 528, 1 (2013).
  * Morrow and Ma [2022] D. J. Morrow and X. Ma, Noisy detectors with unmatched detection efficiency thwart identification of single photon emitters via photon coincidence correlation, arXiv preprint arXiv:2204.07654 (2022).
  * Andersen  _et al._ [2017] S. K. Andersen, S. Kumar, and S. I. Bozhevolnyi, Ultrabright linearly polarized photon generation from a nitrogen vacancy center in a nanocube dimer antenna, Nano letters 17, 3889 (2017).
  * Berthel  _et al._ [2015] M. Berthel, O. Mollet, G. Dantelle, T. Gacoin, S. Huant, and A. Drezet, Photophysics of single nitrogen-vacancy centers in diamond nanocrystals, Physical Review B 91, 035308 (2015).
  * Albrecht  _et al._ [2014] R. Albrecht, A. Bommer, C. Pauly, F. Mücklich, A. W. Schell, P. Engel, T. Schröder, O. Benson, J. Reichel, and C. Becher, Narrow-band single photon emission at room temperature based on a single nitrogen-vacancy center coupled to an all-fiber-cavity, Applied Physics Letters 105, 073113 (2014).
  * Englund  _et al._ [2010] D. Englund, B. Shields, K. Rivoire, F. Hatami, J. Vuckovic, H. Park, and M. D. Lukin, Deterministic coupling of a single nitrogen vacancy center to a photonic crystal cavity, Nano letters 10, 3922 (2010).
  * Siampour  _et al._ [2017] H. Siampour, S. Kumar, and S. I. Bozhevolnyi, Chip-integrated plasmonic cavity-enhanced single nitrogen-vacancy center emission, Nanoscale 9, 17902 (2017).
  * Faraon  _et al._ [2012] A. Faraon, C. Santori, Z. Huang, V. M. Acosta, and R. G. Beausoleil, Coupling of nitrogen-vacancy centers to photonic crystal cavities in monocrystalline diamond, Physical review letters 109, 033604 (2012).
  * Plenio and Knight [1998] M. B. Plenio and P. L. Knight, The quantum-jump approach to dissipative dynamics in quantum optics, Reviews of Modern Physics 70, 101 (1998).
  * Vorobyov  _et al._ [2017] V. V. Vorobyov, A. Y. Kazakov, V. V. Soshenko, A. A. Korneev, M. Y. Shalaginov, S. V. Bolshedvorskii, V. N. Sorokin, A. V. Divochiy, Y. B. Vakhtomin, K. V. Smirnov, _et al._ , Superconducting detector for visible and near-infrared quantum emitters, Optical Materials Express 7, 513 (2017).
  * Beveratos  _et al._ [2000] A. Beveratos, R. Brouri, J. P. Poizat, and P. Grangier, Bunching and antibunching from single nv color centers in diamond (2000).
  * Burkard  _et al._ [2000] G. Burkard, D. Loss, and E. V. Sukhorukov, Noise of entangled electrons: Bunching and antibunching, Physical Review B 61, R16303 (2000).
  * Lachs [1968] G. Lachs, Effects of photon bunching on shot noise in photoelectric detection, Journal of Applied Physics 39, 4193 (1968).
  * Beveratos  _et al._ [2002] A. Beveratos, S. Kühn, R. Brouri, T. Gacoin, J.-P. Poizat, and P. Grangier, Room temperature stable single-photon source, The European Physical Journal D-Atomic, Molecular, Optical and Plasma Physics 18, 191 (2002).
  * Aharonovich  _et al._ [2009] I. Aharonovich, S. Castelletto, D. Simpson, A. Greentree, and S. Prawer, Photophysics of novel diamond based single photon emitters, arXiv preprint arXiv:0909.1873 (2009).
  * Schröder  _et al._ [2011] T. Schröder, F. Gädeke, M. J. Banholzer, and O. Benson, Ultrabright and efficient single-photon generation based on nitrogen-vacancy centres in nanodiamonds on a solid immersion lens, New Journal of Physics 13, 055017 (2011).
  * Li  _et al._ [2017] S. Li, C.-H. Li, B.-W. Zhao, Y. Dong, C.-C. Li, X.-D. Chen, Y.-S. Ge, and F.-W. Sun, A bright single-photon source from nitrogen-vacancy centers in diamond nanowires, Chinese Physics Letters 34, 096101 (2017).
  * Obydennov  _et al._ [2018] D. V. Obydennov, D. A. Shilkin, E. V. Lyubin, M. R. Shcherbakov, E. F. Ekimov, O. S. Kudryavtsev, I. I. Vlasov, and A. A. Fedyanin, Saturation of fluorescence from nv centers in mie-resonant diamond particles,  _Journal of Physics: Conference Series_ , J. Phys.: Conf. Ser. 1092, 012102 (2018).
  * Radko  _et al._ [2016] I. P. Radko, M. Boll, N. M. Israelsen, N. Raatz, J. Meijer, F. Jelezko, U. L. Andersen, and A. Huck, Determining the internal quantum efficiency of shallow-implanted nitrogen-vacancy defects in bulk diamond, Optics Express 24, 27715 (2016).
  * Gupta  _et al._ [2016] A. Gupta, L. Hacquebard, and L. Childress, Efficient signal processing for time-resolved fluorescence detection of nitrogen-vacancy spins in diamond, Journal of the Optical Society of America B 33, B28 (2016).
  * Maletinsky  _et al._ [2011] P. Maletinsky, S. Hong, M. Grinolds, B. Hausmann, M. Lukin, R.-L. Walsworth, M. Loncar, and A. Yacoby, A robust, scanning quantum system for nanoscale sensing and imaging, arXiv preprint arXiv:1108.4437 (2011).
  * Jacques  _et al._ [2008] V. Jacques, P. Neumann, J. Beck, M. Markham, D. Twitchen, J. Meijer, F. Kaiser, G. Balasubramanian, F. Jelezko, and J. Wrachtrup, Dynamic polarization of single nuclear spins by optical pumping of nv color centers in diamond at room temperature, arXiv preprint arXiv:0808.0154 (2008).
  * Wee  _et al._ [2007] T.-L. Wee, Y.-K. Tzeng, C.-C. Han, H.-C. Chang, W. Fann, J.-H. Hsu, K.-M. Chen, and Y.-C. Yu, Two-photon excited fluorescence of nitrogen-vacancy centers in proton-irradiated type ib diamond, The Journal of Physical Chemistry A 111, 9379 (2007).
  * Tian  _et al._ [2023] J. Tian, R. S. Said, F. Jelezko, J. Cai, and L. Xiao, Bayesian-based hybrid method for rapid optimization of nv center sensors, Sensors 23, 3244 (2023).


# Schrödinger Operators with
Purely Discrete Spectrum

Barry Simon  Dedicated to A. Ya. Povzner

(Date: October 13, 2008)

###### Abstract.

We prove $-\Delta+V$ has purely discrete spectrum if $V\geq 0$ and, for all
$M$, $\lvert\\{x\mid V(x)<M\\}\rvert<\infty$ and various extensions.

###### Key words and phrases:

compact resolvent, Schrödinger operators

###### 2000 Mathematics Subject Classification:

47B07, 35Q40, 47N50

Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125.
E-mail<EMAIL_ADDRESS>Supported in part by NSF grant DMS-0652919 and by
Grant No. 2006483 from the United States-Israel Binational Science Foundation
(BSF), Jerusalem, Israel

To appear in the journal, Methods of Functional Analysis and Topology, volume
in memory of A. Ya. Povzner

##  1\. Introduction

Our main goal in this note is to explore one aspect of the study of
Schrödinger operators

$H=-\Delta+V$ (1.1)

which we’ll suppose have $V$’s which are nonnegative and in
$L_{\text{\rm{loc}}}^{1}({\mathbb{R}}^{\nu})$, in which case (see, e.g., Simon
[16]) $H$ can be defined as a form sum. We’re interested here in criteria
under which $H$ has purely discrete spectrum, that is,
$\sigma_{\text{\rm{ess}}}(H)$ is empty. This is well known to be equivalent to
proving $(H+1)^{-1}$ or $e^{-sH}$ for any (and so all) $s>0$ is compact (see
[10, Thm. XIII.16]). One of the most celebrated elementary results on
Schrödinger operators is that this is true if

$\lim_{\lvert x\rvert\to\infty}V(x)=\infty$ (1.2)

But (1.2) is not necessary. Simple examples where (1.2) fails but $H$ still
has compact resolvent were noted first by Rellich [11]—one of the most
celebrated examples is in $\nu=2$, $x=(x_{1},x_{2})$, and

$V(x_{1},x_{2})=x_{1}^{2}x_{2}^{2}$ (1.3)

where (1.2) fails in a neighborhood of the axes. For proof of this and
discussions of eigenvalue asymptotics, see [12, 17, 18, 21, 22].

There are known necessary and sufficient conditions on $V$ for discrete
spectrum in terms of capacities of certain sets (see, e.g., Maz’ya [7]), but
the criteria are not always so easy to check. Thus, I was struck by the
following simple and elegant theorem:

######  Theorem 1.

Define

$\Omega_{M}(V)=\\{x\mid 0\leq V(x)<M\\}$ (1.4)

If (with $\lvert\,\cdot\,\rvert$ Lebesgue measure)

$\lvert\Omega_{M}(V)\rvert<\infty$ (1.5)

for all $M$, then $H$ has purely discrete spectrum.

I learned of this result from Wang–Wu [26], but there is much related work. I
found an elementary proof of Theorem 1 and decided to write it up as a
suitable tribute and appreciation of A. Ya. Povzner, whose work on continuum
eigenfunction expansions for Schrödinger operators in scattering situation [8]
was seminal and inspired me as a graduate student forty years ago!

The proof has a natural abstraction:

######  Theorem 2.

Let $\mu$ be a measure on a locally compact space, $X$ with $L^{2}(X,d\mu)$
separable. Let $L_{0}$ be a selfadjoint operator on $L^{2}(X,d\mu)$ so that
its semigroup is ultracontractive ([2]): For some $s>0$, $e^{-sL_{0}}$ maps
$L^{2}$ to $L^{\infty}(X,d\mu)$. Suppose $V$ is a nonnegative multiplication
operator so that

$\mu(\\{x\mid 0\leq V(x)<M\\})<\infty$ (1.6)

for all $M$. Then $L=L_{0}+V$ has purely discrete spectrum.

######  Remark.

By $L_{0}+V$, we mean the operator obtained by applying the monotone
convergence theorem for forms (see, e.g., [14, 15]) to $L_{0}+\min(V(x),k)$ as
$k\to\infty$.

The reader may have noticed that (1.3) does not obey Theorem 1 (but, e.g.,
$V(x_{1},x_{2})=x_{1}^{2}x_{2}^{4}+x_{1}^{4}x_{2}^{2}$ does). But out proof
can be modified to a result that does include (1.3). Given a set $\Omega$ in
${\mathbb{R}}^{\nu}$, define for any $x$ and any $\ell>0$,

$\omega_{x}^{\ell}(\Omega)=\lvert\Omega\cap\\{y\mid\lvert
y-x\rvert\leq\ell\\}\rvert$ (1.7)

For example, for (1.3), for $x\in\Omega_{M}$,

$\omega_{x}^{\ell}(\Omega_{M})\leq\frac{C_{\ell}}{\lvert x\rvert+1}$ (1.8)

We will say a set $\Omega$ is $r$-polynomially thin if

$\int_{x\in\Omega}\omega_{x}^{\ell}(\Omega)^{r}\,d^{\nu}x<\infty$ (1.9)

for all $\ell$. For the example in (1.3), $\Omega_{M}$ is $r$-polynomially
thin for any $M$ and any $r>0$. We’ll prove

######  Theorem 3.

Let $V$ be a nonnegative potential so that for any $M$, there is an $r>0$ so
that $\Omega_{M}$ is $r$-polynomially thin. Then $H$ has purely discrete
spectrum.

As mentioned, this covers the example in (1.3). It is not hard to see that if
$P(x)$ is any polynomial in $x_{1},\dots,x_{\nu}$ so that for no
$v\in{\mathbb{R}}^{\nu}$ is $\vec{v}\cdot\vec{\nabla}P\equiv 0$ (i.e., $P$
isn’t a function of fewer than $\nu$ linear variables), then $V(x)=P(x)^{2}$
obeys the hypotheses of Theorem 3.

In Section 2, we’ll present a simple compactness criterion on which all
theorems rely. In Section 3, we’ll prove Theorems 1 and 2. In Section 4, we’ll
prove Theorem 3.

It is a pleasure to thank Peter Stollmann for useful correspondence and Ehud
de Shalit for the hospitality of Hebrew University where some of the work
presented here was done.

##  2\. Segal’s Lemma

Segal [13] proved the following result, sometimes called Segal’s lemma:

######  Proposition 2.1.

For $A,B$ positive selfadjoint operators,

$\lVert e^{-(A+B)}\rVert\leq\lVert e^{-A}e^{-B}\rVert$ (2.1)

######  Remarks.

1\. $A+B$ can always be defined as a closed quadratic form on $Q(A)\cap Q(B)$.
That defines $e^{-(A+B)}$ on $\overline{Q(A)\cap Q(B)}$ and we set it to $0$
on the orthogonal complement. Since the Trotter product formula is known in
this generality (see Kato [6]), (2.1) holds in that generality.

2\. Since $\lVert C^{*}C\rVert=\lVert C\rVert^{2}$, $\lVert
e^{-A/2}e^{-B/2}\rVert^{2}=\lVert e^{-B/2}e^{-A}e^{-B/2}\rVert$, and since
$\lVert e^{-(A+B)/2}\rVert^{2}=\lVert e^{-(A+B)}\rVert$, (2.1) is equivalent
to

$\lVert e^{-A+B}\rVert\leq\lVert e^{-B/2}e^{-A}e^{-B/2}\rVert$ (2.2)

which is the way Segal [13] stated it.

3\. Somewhat earlier, Golden [5] and Thompson [23] proved

$\text{\rm{Tr}}(e^{-(A+B)})\leq\text{\rm{Tr}}(e^{-A}e^{-B})$ (2.3)

and Thompson [24] later extended this to any symmetrically normed operator
ideal.

###### Proof.

There are many; see, for example, Simon [19, 20]. Here is the simplest, due to
Deift [3, 4]: If $\sigma$ is the spectrum of an operator

$\sigma(C\\!D)\setminus\\{0\\}=\sigma(DC)\setminus\\{0\\}$ (2.4)

so with $\sigma_{r}$ the spectral radius,

$\sigma_{r}(C\\!D)=\sigma_{r}(DC)\leq\lVert DC\rVert$ (2.5)

If $C\\!D$ is selfadjoint, $\sigma_{r}(C\\!D)=\lVert C\\!D\rVert$, so

$C\\!D\text{ selfadjoint}\Rightarrow\lVert C\\!D\rVert\leq\lVert DC\rVert$
(2.6)

Thus,

$\lVert e^{-A/2}e^{-B/2}\rVert^{2}=\lVert
e^{-B/2}e^{-A}e^{-B/2}\rVert\leq\lVert e^{-A}e^{-B}\rVert$ (2.7)

By induction,

$\lVert(e^{-A/2^{n}}e^{-B/2^{n}})^{2^{n}}\rVert\leq\lVert
e^{-A/2^{n}}e^{-B/2^{n}}\rVert^{2n}\leq\lVert e^{-A}e^{-B}\rVert$ (2.8)

Take $n\to\infty$ and use the Trotter product formula to get (2.1). ∎

In [19], I noted that this implies for any symmetrically normed trace ideal,
${\mathcal{I}}_{\Phi}$, that

$e^{-A/2}e^{-B}e^{-A/2}\in{\mathcal{I}}_{\Phi}\Rightarrow
e^{-(A+B)}\in{\mathcal{I}}_{\Phi}$ (2.9)

I explicitly excluded the case ${\mathcal{I}}_{\Phi}={\mathcal{I}}_{\infty}$
(the compact operators) because the argument there doesn’t show that, but it
is true—and the key to this paper!

Since $C\in{\mathcal{I}}_{\infty}\Leftrightarrow
C^{*}C\in{\mathcal{I}}_{\infty}$ and $e^{-(A+B)}\in{\mathcal{I}}_{\infty}$ if
and only if $e^{-\frac{1}{2}(A+B)}\in{\mathcal{I}}_{\infty}$, it doesn’t
matter if we use the symmetric form (2.2) or the following asymmetric form
which is more convenient in applications.

######  Theorem 2.2.

Let ${\mathcal{I}}_{\infty}$ be the ideal of compact operators on some Hilbert
space, ${\mathcal{H}}$. Let $A,B$ be nonnegative selfadjoint operators. Then

$e^{-A}e^{-B}\in{\mathcal{I}}_{\infty}\Rightarrow
e^{-(A+B)}\in{\mathcal{I}}_{\infty}$ (2.10)

###### Proof.

For any bounded operator, $C$, define $\mu_{n}(C)$ by

$\mu_{n}(C)=\min_{\psi_{1}\dots\psi_{n-1}}\,\sup_{\begin{subarray}{c}\lVert\varphi\rVert=1\\\
\varphi\perp\psi_{1},\dots,\psi_{n-1}\end{subarray}}\lVert C\varphi\rVert$
(2.11)

By the min-max principle (see [10, Sect. XIII.1]),

$\lim_{n\to\infty}\,\mu_{n}(C)=\sup(\sigma_{\text{\rm{ess}}}(\lvert C\rvert))$
(2.12)

and $\mu_{n}(C)$ are the singular values if $C\in{\mathcal{I}}_{\infty}$. In
particular,

$C\in{\mathcal{I}}_{\infty}\Leftrightarrow\lim_{n\to\infty}\,\mu_{n}(C)=0$
(2.13)

Let $\wedge^{\ell}({\mathcal{H}})$ be the antisymmetric tensor product (see
[9, Sects. II.4, VIII.10], [10, Sect. XIII.17], and [19, Sect. 1.5]). As usual
(see [19, eqn. (1.14)]),

$\lVert\wedge^{m}(C)\rVert=\prod_{j=1}^{m}\mu_{j}(C)$ (2.14)

Since $\mu_{1}\geq\mu_{2}\geq\cdots\geq 0$, we have

$\lim_{n\to\infty}\,\mu_{n}(C)=\lim_{n\to\infty}\,(\mu_{1}(C)\dots\mu_{n}(C))^{1/n}$
(2.15)

(2.13)–(2.15) imply

$C\in{\mathcal{I}}_{\infty}\Leftrightarrow\lim_{n\to\infty}\,\lVert\wedge^{n}(C)\rVert^{1/n}=0$
(2.16)

As usual, there is a selfadjoint operator, $d\wedge^{n}(A)$ on
$\wedge^{n}({\mathcal{H}})$ so

$\wedge^{n}(e^{-tA})=e^{-t\,d\wedge^{n}(A)}$ (2.17)

so Segal’s lemma implies that

$\displaystyle\lVert\wedge^{n}(e^{-(A+B)})\rVert$
$\displaystyle\leq\lVert\wedge^{n}(e^{-A})\wedge^{n}(e^{-B})\rVert$
$\displaystyle=\lVert\wedge^{n}(e^{-A}e^{-B})\rVert$ (2.18)

Thus,

$\lim_{n\to\infty}\,\lVert\wedge^{n}(e^{-(A+B)})\rVert^{1/n}\leq\lim_{n\to\infty}\,\lVert\wedge^{n}(e^{-A}e^{-B})\rVert^{1/n}$
(2.19)

By (2.16), we obtain (2.10). ∎

##  3\. Proofs of Theorems 1 and 2

###### Proof of Theorem 1.

By Theorem 2.2, we need only show $C=e^{\Delta}e^{-V}$ is compact. Write

$C=C_{m}+D_{m}$ (3.1)

where

$C_{m}=C\chi_{\Omega_{m}}\qquad D_{m}=C\chi_{\Omega_{m}^{c}}$ (3.2)

with $\chi_{S}$ the operator of multiplication by the characteristic function
of a set $S\subset{\mathbb{R}}^{\nu}$.

$\lVert e^{-V}\chi_{\Omega_{m}^{c}}\rVert_{\infty}\leq e^{-m}$

and $\lVert e^{\Delta}\rVert=1$, so

$\lVert D_{m}\rVert\leq e^{-m}$ (3.3)

and thus,

$\lim_{m\to\infty}\,\lVert C-C_{m}\rVert=0$ (3.4)

If we show each $C_{m}$ is compact, we are done. We know $e^{\Delta}$ has
integral kernel $f(x-y)$ with $f$ a Gaussian, so in $L^{2}$. Clearly, since
$V$ is positive, $C_{m}$ has an integral kernel $C_{m}(x,y)$ dominated by

$\lvert C_{m}(x,y)\rvert\leq f(x-y)\chi_{\Omega_{m}}(y)$ (3.5)

Thus,

$\int\lvert C_{m}(x,y)\rvert^{2}\,d^{\nu}xd^{\nu}y\leq\lVert
f\rVert^{2}_{L^{2}({\mathbb{R}}^{\nu})}\lVert\chi_{\Omega_{m}}\rVert_{L^{2}({\mathbb{R}}_{\nu})}<\infty$

since $\lvert\Omega_{m}\rvert<\infty$. Thus, $C_{m}$ is Hilbert–Schmidt, so
compact. ∎

###### Proof of Theorem 2.

We can follow the proof of Theorem 1. It suffices to prove that
$e^{-sL_{0}}e^{-sV}$ is compact, and so, that $e^{-sL_{0}}\chi_{\Omega_{m}}$
is Hilbert–Schmidt.

That $e^{-sL_{0}}$ maps $L^{2}$ to $L^{\infty}$ implies, by the Dunford–Pettis
theorem (see [25, Thm. 46.1]), that there is, for each $x\in X$, a function
$f_{x}(\,\cdot\,)\in L^{2}(X,d\mu)$ with

$(e^{-sL_{0}}g)(x)=\langle f_{x},g\rangle$ (3.6)

and

$\sup_{x}\,\lVert f_{x}\rVert_{L^{2}}=\lVert e^{-sL_{0}}\rVert_{L^{2}\to
L^{\infty}}\equiv C<\infty$ (3.7)

Thus, $e^{-sL_{0}}$ has an integral kernel $K(x,y)$ with

$\sup_{x}\int\lvert K(x,y)\rvert^{2}\,d\mu(y)=C<\infty$ (3.8)

(for $K(x,y)=f_{x}(y)$). But $e^{-sL_{0}}$ is selfadjoint, so its kernel is
complex symmetric, so

$\sup_{y}\int\lvert K(x,y)\rvert^{2}\,d\mu(x)=C<\infty$ (3.9)

Thus,

$\int\lvert K(x,y)\chi_{\Omega_{m}}(y)\rvert^{2}\,d\mu(x)d\mu(y)\leq
C\mu(\Omega_{m})<\infty$ (3.10)

and $e^{-sL_{0}}\chi_{\Omega_{m}}$ is Hilbert–Schmidt. ∎

##  4\. Proof of Theorem 3

As with the proof of Theorem 1, it suffices to prove that for each $M$,
$e^{\Delta}\chi_{\Omega_{M}}$ is compact. $e^{\Delta}$ is convolution with an
$L^{1}$ function, $f$. Let $Q_{R}$ be the characteristic function of
$\\{x\mid\lvert x\rvert<R\\}$. Let $F_{R}$ be convolution with $fQ_{R}$. Then

$\lVert e^{\Delta}-F_{R}\rVert\leq\lVert f(1-Q_{R})\rVert_{1}\to 0$ (4.1)

as $R\to\infty$, so

$\lVert e^{\Delta}\chi_{\Omega_{M}}-F_{R}\chi_{\Omega_{M}}\rVert\to 0$ (4.2)

and it suffices to prove for each $R,M$,

$C_{M,R}=F_{R}\chi_{\Omega_{M}}$ (4.3)

is compact. Clearly, this works if we show for some $k$,
$(C_{M,R}^{*}C_{M,R})^{k}$ is Hilbert–Schmidt.

Let $D$ be the operator with integral kernel

$D(x,y)=\chi_{\Omega_{M}}(x)Q_{2R}(x-y)\chi_{\Omega_{M}}(y)$ (4.4)

Since $f$ is bounded, it is easy to see that

$(C_{M,R}^{*}C_{M,R})(x,y)\leq cD(x,y)$ (4.5)

for some constant $c$, so it suffices to show $D^{k}$ is Hilbert–Schmidt.

$D^{k}$ has integral kernel

$D^{k}(x,y)=\int D(x,x_{1})D(x_{1},x_{2})\dots
D(x_{k-1},y)\,dx_{1}\dots,dx_{k-1}$ (4.6)

Fix $y$. This integral is zero unless $\lvert x-x_{1}\rvert<2R,\dots\lvert
x_{k-1}-y\rvert<2R$, so, in particular, unless $\lvert x-y\rvert\leq 2kR$.
Moreover, the integrand can certainly be restricted to the regions $\lvert
x_{j}-y\rvert\leq 2kR$. Thus,

$\displaystyle D^{k}(x,y)$ $\displaystyle\leq
Q_{2kR}(x-y)\biggl{(}\int_{\lvert x_{j}-y\rvert\leq
2kR}\,\prod_{j=1}^{k-1}\chi_{\Omega_{M}}(x_{j})\,dx_{1}\dots
dx_{k-1}\biggr{)}\chi_{\Omega_{m}}(y)$ (4.7)
$\displaystyle=Q_{2kR}(x-y)(\omega_{y}^{2kR}(\Omega_{M})^{k-1})\chi_{\Omega_{M}}(y)$
(4.8)

by the definition of $\omega_{x}^{\ell}$ in (1.7).

Thus,

$\int\lvert D^{k}(x,y)\rvert^{2}\,d^{\nu}xd^{\nu}y\leq
C(kR)^{\nu}\int_{x\in\Omega}[\omega_{x}^{2kR}(\Omega_{M})]^{2k-2}\,d^{\nu}x$

so if $2k-2>r$ and (1.9) holds, $D^{k}$ is Hilbert–Schmidt. ∎

## References

  * [1]
  * [2] E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal. 59 (1984), 335–395.
  * [3] P. A. Deift, Classical Scattering Theory With a Trace Condition, Ph.D. dissertation, Princeton University, 1976.
  * [4] P. A. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), 267–310.
  * [5] S. Golden, Lower bounds for the Helmholtz function, Phys. Rev. (2) 137 (1965), B1127–B1128.
  * [6] T. Kato, Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups, in “Topics in Functional Analysis,” pp. 185–195, Adv. in Math. Suppl. Stud., 3, Academic Press, New York-London, 1978.
  * [7] V. Maz’ya, Analytic criteria in the qualitative spectral analysis of the Schrödinger operator, in “Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th birthday,” pp. 257–288, Proc. Sympos. Pure Math., 76.1, American Mathematical Society, Providence, RI, 2007\.
  * [8] A. Ya. Povzner, On expansions in functions which are solutions of a scattering problem, Dokl. Akad. Nauk SSSR (N.S.) 104 (1955), 360–363. [Russian]
  * [9] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972.
  * [10] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, New York, 1978.
  * [11] F. Rellich, Das Eigenwertproblem von $\Delta u+\lambda u=0$ in Halbröhren, in “Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948,” pp. 329–344, Interscience Publishers, New York, 1948.
  * [12] D. Robert, Comportement asymptotique des valeurs propres d’opérateurs du type Schrödinger à potentiel “dégénéré [Asymptotic behavior of the eigenvalues of Schrd̈inger operators with “degenerate” potential], J. Math. Pures Appl. (9) 61 (1982), 275–300 (1983).
  * [13] I. Segal, Notes towards the construction of nonlinear relativistic quantum fields. III. Properties of the $C^{*}$-dynamics for a certain class of interactions, Bull. Amer. Math. Soc. 75 (1969), 1390–1395.
  * [14] B. Simon, Lower semicontinuity of positive quadratic forms, Proc. Roy. Soc. Edinburgh 29 (1977), 267–273.
  * [15] B. Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Funct. Anal. 28 (1978), 377–385.
  * [16] B. Simon, Maximal and minimal Schrödinger forms, J. Oper. Theory 1 (1979), 37–47.
  * [17] B. Simon, Nonclassical eigenvalue asymptotics, J. Funct. Anal. 53 (1983), 84–98.
  * [18] B. Simon, Some quantum operators with discrete spectrum but classically continuous spectrum, Ann. Phys. 146 (1983), 209–220.
  * [19] B. Simon, Trace Ideals and Their Applications, second edition, Mathematical Surveys and Monographs, 120, American Mathematical Society, Providence, RI, 2005.
  * [20] B. Simon, Ed Nelson’s work in quantum theory, in “Diffusion, Quantum Theory, and Radically Elementary Mathematics,” pp. 75–93, Mathematical Notes, 47, Princeton University Press, Princeton, NJ, 2006.
  * [21] M. Z. Solomyak Asymptotic behavior of the spectrum of a Schrödinger operator with nonregular homogeneous potential, Soviet Math. Dokl. 30 (1984), 379–383; Russian original in Dokl. Akad. Nauk SSSR 278 (1984), 291–295.
  * [22] H. Tamura, The asymptotic distribution of eigenvalues of the Laplace operator in an unbounded domain, Nagoya Math. J. 60 (1976), 7–33.
  * [23] C. J. Thompson, Inequality with applications in statistical mechanics, J. Math. Phys. 6 (1965), 1812–1813.
  * [24] C. J. Thompson, Inequalities and partial orders on matrix spaces, Indiana Univ. Math. J. 21 (1971/72), 469–480.
  * [25] F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York-London, 1967.
  * [26] F.-Y. Wang and J.-L. Wu, Compactness of Schrödinger semigroups with unbounded below potentials, to appear in Bull. Sci. Math.