# Adiabaticity of spin dynamics in diamond nitrogen vacancy centers in time-
dependent magnetic fields

Y. B. Band1 and Y. Japha2 1Department of Chemistry, Department of Physics,
Department of Electro-Optics, and the Ilse Katz Center for Nano-Science,
Ben-Gurion University, Beer-Sheva 84105, Israel
2Department of Physics, and the Ilse Katz Center for Nano-Science,
Ben-Gurion University, Beer-Sheva 84105, Israel

###### Abstract

We study the spin dynamics of diamond nitrogen vacancy (NV) centers in an
oscillating magnetic field. The coupling between the otherwise degenerate
Zeeman levels due to strain and electric fields is responsible for a Landau-
Zener process near the pseudo-crossing of the adiabatic energy levels when the
axial component of the oscillating magnetic field changes sign. If the
oscillation is slow the spin follows the direction of the axial component of
the field adiabatically, whereas Landau-Zener transitions take place if the
oscillation is fast. The role of the nuclear spin hyperfine interactions on
the dynamics is shown to reduce to independent pairs of levels with the same
pattern of evolution except for a different magnetic field offset due to the
nuclear spin. If the NV center is coupled to the environment, which is modeled
by a spin bath with Gaussian white noise spectrum as in NVs near the diamond
surface, then the spin dynamics is accompanied by relaxation of the Zeeman
level populations and decoherence with a non-monotonic decrease of the purity
of the system. Finally, we show that appropriately chosen transverse magnetic
fields can fully or partially compensate for the effect of strain and electric
fields, and one can tune the adiabaticity of spin dynamics.

###### pacs:

05.40.Ca, 05.40.-a, 07.50.Hp, 74.40.De

##  I Introduction

Negatively charged nitrogen-vacancy (NV) color centers in diamond Doherty_13
have been proposed as candidates for sensors of various physical events or
environmental changes, including biological fluorescent markers Sage_13 ,
qubits Gruber_97 ; Oort_88 ; Jiang_09 , magnetic field and electric field
sensors Balasubramanian_09 ; Dolde_11 , stress sensors Kehayias , temperature
sensors Neumann_13 ; Kucsko_13  and inertial sensors Ledbetter_12 .

The peculiar properties of a NV center in a diamond stem from the fact that it
is a hybrid creature that combines a solid object (the diamond crystal) with a
system consisting of a single spin with discrete energy levels like an atom or
a molecule. Unlike atoms, NV centers do not suffer from the broadening effects
of atomic systems such as Doppler broadening due to translational motion or
power broadening due to collisions. Unlike molecules, they do not have the
broad spectrum of rotations or vibrations, so that at room temperature NV
centers show a very precise spectrum of discrete levels. This is why NV
centers are ideal for studying fundamental quantum processes, and for ultra-
precise sensing. In addition, a levitated, cooled and trapped nano-diamond
with a single or a few spins is a good candidate for implementing a fully
quantum object with a large mass and coupling between its internal spin and
its external motion for studying the transition between quantum and classical
mechanics Hsu_2016 ; Delord_2020 . In particular, manipulation of such an
object with external electric, magnetic or optical fields requires a thorough
study of the response of a NV center to such external fields.

Here we focus on the response of the spin of the NV center to a time-dependent
magnetic field. A spin of an atom in a magnetic field would usually follow the
direction of the magnetic field adiabatically if the field varies slowly but
would show non-adiabatic dynamical evolution involving precession if the field
varies quickly. In a molecule or a non-spherical object with spin the dynamics
of the spin is coupled to the dynamics of the rotational degrees of freedom of
the whole object. In a massive diamond crystal the spin degrees of freedom are
coupled to the whole crystal whose rotational degrees of freedom are
practically fixed due to its large mass. In this case the spin cannot follow
the direction of the magnetic field and its projection along the axis of the
NV center can change sign but not size. In terms of energy eigenstates, an
avoided crossing is formed. The dynamics of the spin across such a pseudo-
crossing is the subject of this work, and in particular the question of
whether the spin projection along the axis follows the projection of the
magnetic field (adiabatic dynamics) or stays aligned along the original
direction while the magnetic field flips its projection, in a process of
Landau-Zener transition (non-adiabatic dynamics).

When considering the dynamics of a system based upon the time-dependent
Schrödinger equation, $i\hbar\,d\psi/dt=H(t)\psi$, with a time-dependent
Hamiltonian $H(t)$ that varies “slowly”, the adiabatic theorem Band_Avishai
tells us that If the system is initially in an eigenstate $u_{n}(t_{0})$ of
the initial Hamiltonian, $H(t_{0})$, it remains in the eigenstate $u_{n}(t)$
of the instantaneous Hamiltonian $H(t)$ ($t>t_{0}$) that at each instant of
time $t$ satisfies

$H(t)u_{n}(t)=E_{n}(t)u_{n}(t)~{},$ (1)

and is therefore called “an adiabatic state”. If the initial state is a
superposition of initial eigenstates,
$\psi_{\mathrm{in}}=\psi(t_{0})=\sum_{n}c_{n}(t_{0})u_{n}(t_{0})$, it evolves
to a superposition of the instantaneous eigenstates
$\psi(t)=\sum_{n}c_{n}(t_{0})u_{n}(t)\exp[\frac{-i}{\hbar}\int_{t_{0}}^{t}dt^{\prime}\,E_{n}(t^{\prime})]$
with the same probabilities (adiabatic evolution) if the Hamiltonian varies
sufficiently slowly, in the sense that Band_Avishai_ibid

$\int_{t_{0}}^{t}dt^{\prime}\,\left|\frac{\langle
u_{i}(t^{\prime})|\frac{dH(t^{\prime})}{dt^{\prime}}|u_{n}(t^{\prime})\rangle}{[E_{n}(t^{\prime})-E_{i}(t^{\prime})]}\right|\ll
1\text{ for }n\neq i.$ (2)

The 2-level Landau-Zener (LZ) problem Landau ; Zener ; Stueckelberg ; Majorana
in a system with avoided level crossing was first suggested in 1932 to
theoretically model molecular pre-dissociation. Since then LZ dynamics was
observed in many systems, including solid-state systems, such as
superconductor two-level systems Shevchenko_2010  and systems having
paramagnetic defects in semiconductors Awschalom_14 . In particular, LZ
dynamics was also observed in diamond NV centers subjected to microwave or
radio-frequency magnetic fields Dimitriev_2019 .

The simplest model for LZ dynamics is a 2-level system with Hamiltonian
$H(t)=\left(\\!\\!\begin{array}[]{cc}\varepsilon_{1}&V\\\
V^{*}&\varepsilon_{2}+\alpha t\end{array}\\!\\!\right)$. The higher energy
eigenstate at $t\to-\infty$,
$|a(-\infty)\rangle=(1,0)^{{\dagger}}\equiv|1\rangle$ with
$E_{a}(t\to-\infty)=\varepsilon_{1}$ evolves, in an infinitely small rate of
change $\alpha\to 0$, continuously into the higher energy eigenstate
$|a(t\to+\infty)\rangle=(0,1)^{{\dagger}}\equiv|2\rangle$ with energy
$E_{a}(t\to+\infty)=\varepsilon_{2}+\alpha t$. The Landau-Zener transition
probability $P_{LZ}$ of finding the system in the adiabatic state $|b\rangle$
(orthogonal to the adiabatic state $|a\rangle$, which coincides with
$|1\rangle$ at $t\to\infty$, is then negligible, as the adiabatic theorem
ensures that the system stays on the initial adiabatic state. For finite
$\alpha$, the transition probability $P_{LZ}$ at the final time depends
exponentially on the inverse rate of change of the energy difference:

$P_{LZ}=1-\exp\left(\\!-2\pi\frac{V^{2}}{\hbar\alpha}\\!\right)~{}.$ (3)

The ground electronic state of the NV is a 3-level system, i.e., a triplet
($S=1$) state, which becomes a 9-level system when the 14N nuclear spin
($I=1$) and the hyperfine interaction is included. The electronic ground state
has a large splitting [${\cal D}=h\times 2.87$ GHz, see Eq. (4)] between the
level with projection of the angular momentum along the axis parallel to the
nitrogen-vacancy vector, $M_{S}=0$, and the $M_{S}=\pm 1$ levels. Here we
consider this system subjected to a time-dependent magnetic field ${\bf B}(t)$
oscillating with a radio-frequency that is much smaller than the splitting
${\cal D}$. The electronic Hamiltonian can be written as Doherty_13

$\displaystyle H(t)$ $\displaystyle=$ $\displaystyle{\cal
D}\left(S_{z}^{2}-\frac{2}{3}{\bf 1}\right)+\mu\,{\bf S}\cdot{\bf B}(t)+H_{\rm
strain},$ (4)

where $\mu=h\times 2.8$ MHz/G is the magnetic moment of the NV, and $H_{\rm
strain}$ represents the interaction of the NV center with strain and electric
fields that couples the $M_{S}=\pm 1$ levels Doherty_13  and removes their
degeneracy. In general it has the form

$H_{\rm
strain}=\epsilon_{1}(S_{x}^{2}-S_{y}^{2})+\epsilon_{2}(S_{x}S_{y}+S_{y}S_{x}),$
(5)

where the values of $\epsilon_{1},\epsilon_{2}$ relative to each other depend
on the direction of the strain and the electric field at the site of the NV.
The spin angular momentum components $S_{i}$ ($i=x,y$ and $z$) are represented
by spin-1 3$\times$3 matrices Band_Avishai_1 . The Hamiltonian is then
represented by a 3$\times$3 matrix

$H=\left(\\!\\!\begin{array}[]{ccc}\mu
B_{z}&-\mu\frac{B_{x}+iB_{y}}{\sqrt{2}}&\epsilon\\\
-\mu\frac{B_{x}-iB_{y}}{\sqrt{2}}&-{\cal
D}&-\mu\frac{B_{x}+iB_{y}}{\sqrt{2}}\\\
\epsilon^{*}&-\mu\frac{B_{x}-iB_{y}}{\sqrt{2}}&-\mu
B_{z}\end{array}\\!\\!\right)+\frac{1}{3}{\cal D}\,{\bf 1},$ (6)

where $\epsilon\equiv\epsilon_{1}+i\epsilon_{2}$. The strain coefficient
$|\epsilon|$ is in the range of several MHz (it can vary from sample to
sample). The Hamiltonian in Eq. (6) will be time dependent, $H\to H(t)$, if
the $z$ component of magnetic field is time dependent, $B_{z}(t)$, as in the
studies discussed below.

Quite generally, any 3-level time-dependent Hamiltonian can be written in the
form

$H(t)=\sum_{j=0}^{8}\Delta_{j}(t)\lambda_{j},$ (7)

where $\lambda_{j}$ are the 3$\times$3 Gell-Mann matrices Gellmann  and the
$\Delta_{j}$ are real coefficients that can be time-dependent. Such
Hamiltonian matrices belong to su(3), i.e., the algebra of the group SU(3)
Iachello ; su2_su3 . Analytic solutions to the time-dependent Schrödinger
equation with the Hamiltonian (7) are in general not available, except for
Hamiltonians of special form belonging to an su(2) algebra, i.e., the algebra
of the group SU(2) Band_3_level_19 . In this paper we usually use the large
splitting ${\cal D}$ between the $M_{S}=0$ and $M_{S}=\pm 1$ levels to reduce
the problem effectively into a two level problem.

In this paper we start with a simple situation where the magnetic field is
parallel to the axis of the NV center ${\bf B}=B_{z}\hat{z}$. In this case the
$M_{S}=0$ level is not coupled to the $M_{S}=\pm 1$ levels and the 3$\times$3
Hamiltonian matrix for the levels of interest $M_{S}=\pm 1$ reduces to a
2$\times$2 matrix with two adiabatic eigenstates whose minimum energy
difference is given by the zero field splitting $|\epsilon|$. This implies
that the dynamics upon sweeping the magnetic field through ${\bf B}=0$ is
adiabatic and Landau-Zener transitions are suppressed unless the sweeping rate
is high enough. We shall see that by applying the transverse magnetic field
and tuning it to a certain direction and strength , one can modify the total
off-diagonal coupling and hence control the adiabaticity of the transitions.
Aside from the investigation of adiabaticity in the effective 2-level system
of $M_{s}=\pm 1$, a major aim of this paper is also to investigate the effect
of the hyperfine interactions on the dynamics and the effect of certain kinds
of decoherence processes.

The outline of the paper is as follows. Section II considers the dynamics in
the presence of a time-dependent magnetic field in the axial direction without
considering the nuclear spin (i.e., hyperfine) degrees of freedom. Section III
presents the dynamics of the system including the nuclear degrees of freedom.
Section IV considers the dynamics including decoherence due to isotropic white
noise using a formalism developed in reference Band_3_level_19  which
considered the 3-level LZ problem for open system cases where interaction with
an environment is present. When the environment can be modeled as white
Gaussian noise, the system can be treated using a master equation of the form
$i\,\partial\rho/\partial t=[H(t),\rho(t)]-\Gamma\rho(t)$, where $\rho$ is the
‘density operator’ and $\Gamma$ is the Lindblad decay operator. Section V
shows that a transverse magnetic field can be used to tune (and even zero) the
effective non-adiabaticity of the LZ dynamics by making the off-diagonal
coupling between states vanish. Such tuning by adjusting an external field can
be taken advantage of for practical applications. Finally Sec. VI presents a
summary and conclusions.

##  II Dynamics of the electronic spin

Figure 1: Adiabatic energy eigenvalues [Eq. (10)] for the case of a magnetic
field ${\bf B}(t)=\hat{\bf z}B_{0}\cos\omega t$ for the 3-level system reduced
to a 2-level Hamiltonian [Eq. (8)] ignoring hyperfine structure. For the
3-level system [Eq. (6)], the eigenvalue associated with the $M_{S}=0$
electronic state that is near -1.913 GHz lies outside the plot range. The
parameters are $\epsilon=h\times 2.87$ MHz, $\mu B_{0}=h\times 28$ MHz
(corresponding to a magnetic field of 10 Gauss) and angular frequency
$\omega=2\pi\times 0.25$ MHz. The adiabaticity parameter is
$\beta=\hbar\omega\mu B_{0}/\pi\epsilon^{2}=0.26\ll 1$. The energy eigenvalues
are shown only in the range close to the pseudo-crossing at
$t=\pi/2\omega=1000$ nm.

In this section we discuss the unitary dynamics for the electronic system
while neglecting the hyperfine interaction. As will be shown in the next
section, the nuclear degrees of freedom do not change the nature of the
dynamics but modify it in a way that allows its description on the basis of
the dynamics of the electronic system that we discuss here. We first consider
the case where the magnetic field ${\bf B}(t)$ is in the $\hat{z}$ direction,
i.e., along the axis of the NV. As we see in the Hamiltonian (6), the energies
of the levels $M_{S}=\pm 1$ depend linearly on the magnetic field component
$B_{z}$ whie the transverse components $B_{x},B_{y}$ couple between these
sublevels and the level $M_{S}=0$, whose energy is far by ${\cal D}$. As we
see below the relevant energy scale for LZ dynamics is governed by $\epsilon$,
which is 3 orders of magnitude smaller than the splitting ${\cal D}$. The
effect of transverse fields of the same order as $\epsilon$ on the dynamics is
then smaller by a factor of $\epsilon/{\cal D}\sim 10^{-3}$ than the effect of
other energy scales and therefore at this stage we consider only a magnetic
field in the $\hat{z}$ direction, while the effects of transverse fields is
discussed in Sec. V.

If we consider only the axial magnetic field $B_{z}$ then the dynamics of the
levels $M_{S}=\pm 1$ is determined by a 2$\times$2 Hamiltonian

$H(t)=\left(\begin{array}[]{cc}\mu B_{z}(t)&\epsilon\\\ \epsilon^{*}&-\mu
B_{z}(t)\end{array}\right)+\frac{1}{3}{\cal D}{\bf 1}.$ (8)

For this Hamiltonian, with trigonometric magnetic field time dependence
$B_{z}(t)=B_{0}\cos(\omega t)$, the off-diagonal coupling potential of the LZ
problem [Eq. (3)] is given by $V=\epsilon$ and the rate of energy change is
$\alpha=2\mu B_{0}\omega$ for times close to where the diabatic energies
cross, so by Eq. (3), the probability for transition between the adiabatic
levels is $P_{LZ}=1-\exp\left(\\!-\frac{\pi|\epsilon|^{2}}{4\mu
B_{0}\hbar\omega}\\!\right)$. The dimensionless quantity

$\beta\equiv\frac{\mu B_{0}\hbar\omega}{\pi|\epsilon|^{2}}$ (9)

is a measure of how adiabatic the transition is. The pseudo-crossings of the
energy eigenvalues

$E_{\rm ad}(t)=\pm\sqrt{|\epsilon|^{2}+\mu^{2}B_{z}(t)^{2}}+\frac{1}{3}{\cal
D}$ (10)

occur at times $t=j\pi/2\omega$, $j=1,3,5,\ldots$, as demonstrated in Fig. 1
for $0\leq t\leq\pi/\omega$. Here we only follow the dynamics through the
first pseudo-crossing. If we were to follow the dynamics over longer time
periods, multiple pseudo-crossings would occur and Stückelberg oscillations
would play a role Stuckelberg , but here we wish to focus on the adiabaticity
in a single pseudo-crossing rather than coherent evolution between pseudo-
crossings.

Figure 2: Nearly adiabatic dynamics of the 2-level system with the population
initially in the $M_{S}=1$ level. The population is adiabatically transferred
to the $M_{S}=-1$ level, and the $M_{S}=0$ remains zero throughout the
dynamics because with a magnetic field only along the $z$-axis, this component
is completely decoupled if the evolution is unitary. The parameters used in
this calculation are as in Fig. 1 (adiabaticity parameter $\beta=0.26\ll 1$).
Figure 3: Two-level non-adiabatic dynamics obtained with an angular frequency
20 times larger than that used in Figs. 2, corresponding to an adiabaticity
parameter $\beta=5.41\gg 1$. All the other parameters are identical. (To
compare with Fig. 2, scale the abscissa by a factor of 20). Figure 4: Two-
level non-adiabatic dynamics obtained with the same parameters used in Fig. 3,
but the magnetic field strength is reduced by a factor of 5, i.e., $\mu
B_{z}=2\pi\times 4.8$ MHz (corresponding to a magnetic field of 2 Gauss). The
adiabaticity parameter is $\beta=1.08\sim 1$, the slope of the eigenenergies
versus time is slower than in Fig. 3 and the dynamics here are less non-
adiabatic.

In the adiabatic limit where the sweeping frequency is small such that
$\beta\ll 1$ the instantaneous state of the system is given by the adiabatic
state

$|\psi(t)\rangle_{\rm ad}=\frac{\epsilon|-1\rangle+[E_{\rm ad}-\mu
B_{z}(t)]|+1\rangle}{\sqrt{(E_{\rm ad}-\mu B_{z}(t))^{2}+|\epsilon|^{2}}},$
(11)

which is dominated by the state $|+1\rangle$ at one side of the transition and
by $|-1\rangle$ at the other side of the transition. This is demonstrated in
Fig. 2, where initially ($t=0$) the population is in the state $|+1\rangle$,
which almost completely coincides with one adiabatic state and ends up in the
final time with the population almost completely in the $|-1\rangle$ state,
which coincides with the same adiabatic eigenstate [the upper branch in Fig. 1
and Eq. (10)], corresponding to the magnetic moment aligned along the
instantaneous axial magnetic field component. In the other limit where
$\beta\gg 1$, the population of the $|M_{S}=\pm 1\rangle$ states does not
change during the sweeping, so that the final magnetic moment is aligned
opposite to the axial magnetic field components. In the intermediate case of
$\beta\sim 1$ a partial transition between the two states occurs, as
demonstrated in Figs. 3 and 4.

##  III Dynamics including hyperfine interaction

If the hyperfine interaction with the 14N nuclear spin ($I=1$) is included,
the Hamiltonian can be written as a 9$\times$9 matrix,

$\displaystyle H_{\mathrm{hf}}(t)$ $\displaystyle=$ $\displaystyle
H(t)\otimes{\bf 1}^{\mathrm{nucl}}_{3\times 3}+{\bf 1}_{3\times 3}\otimes
H^{\mathrm{nucl}}(t)+H^{\mathrm{int}},$ (12)

where

$\displaystyle H^{\mathrm{nucl}}(t)={\cal D}_{n}(I_{z}^{2}-\frac{2}{3}{\bf
1})+\mu_{n}\,{\bf I}\cdot{\bf B}(t),$ (13)

and

$\displaystyle H^{\mathrm{int}}={\cal A}_{\parallel}S_{z}\otimes I_{z}+{\cal
A}_{\perp}(S_{x}\otimes I_{x}+S_{y}\otimes I_{y}).$ (14)

where the zero-field splitting of the nuclear spin is ${\cal D}_{n}=h\times
5.0$ MHz, the nitrogen magnetic moment is $\mu_{n}=(0.403/1837)\mu_{B}=0.612$
kHz/G, and the hyperfine splitting factors are ${\cal A}_{\parallel}=h\times
2.2$ MHz and ${\cal A}_{\perp}=h\times 2.1$ MHz.

In the case where the magnetic field is directed only in the $\hat{z}$
direction, the Hamiltonian can be decoupled into three blocks of 2$\times$2
matrices for $M_{S}=\pm 1$ levels and three blocks of 1$\times$1 for $M_{S}=0$
levels. The 2$\times$2 matrices can be written as

$\displaystyle H_{\pm 1,M_{I}}$ $\displaystyle=\left[\frac{1}{3}{\cal D}+{\cal
D}_{n}(M_{I}^{2}-\frac{2}{3})+\mu_{n}M_{I}B_{z}\right]{\bf 1}_{2\times 2}+$
(17) $\displaystyle+\left(\\!\\!\begin{array}[]{cc}{\cal
A}_{\parallel}M_{I}+\mu B_{z}&\epsilon\\\ \epsilon^{*}&-{\cal
A}_{\parallel}M_{I}-\mu B_{z}\end{array}\\!\\!\right).$

The adiabatic eigenstates and eigenvalues of this Hamiltonian are then exactly
the same as those of the 2-level Hamiltonian [Eq. (8), except that the
position of the pseudo-crossing as a function of $B_{z}$ is shifted by $\Delta
B_{z}(M_{I})=-{\cal A}_{\parallel}/\mu$ and the energy eigenvalues of each
$M_{I}$ pair of levels is shifted by ${\cal D}_{n}(M_{I}^{2}-\frac{2}{3})$.
This implies that when the magnetic field is swept through the pseudo-crossing
the transitions between the $M_{s}=\pm 1$ states occur at different times for
each $M_{I}$ state. Transverse field components $B_{x},B_{y}$ couple between
different $M_{I}$ blocks, but their effect is expected to be negligible when
$|B_{x}|,|B_{y}|\ll{\cal D}_{n}/\mu_{n}$, i.e., when these components are less
than hundreds of Gauss.

Figure 5 shows the adiabatic eigenstates of the 9-level Hamiltonian as
calculated for the same parameters and on the same scale used in Fig. 1. This
description allows us to understand the adiabatic or non-adiabatic behavior of
the dynamics in the presence of the hyperfine interaction.

Figure 5: Adiabatic energy eigenvalues for the case of a magnetic field ${\bf
B}(t)=\hat{\bf z}B_{0}\cos\omega t$ with all parameters as used in Figs. 1 and
2, for the 9-level (practically, 6-level) Hamiltonian in Eq. (12). The three
eigenvalues associated with the $M_{S}=0$ electronic state that are near
-1.913 GHz lie outside the plot range. The 6 energy eigenvalues consist of
three pairs of eigenstates of the Hamiltonians $H_{\pm 1,M_{I}}$ [Eq. (17)]
for $M_{I}=0,\pm 1$. The two energy eigenvalues of each pair are similar to
those of the pair of electronic levels shown in Fig. 2, except for an energy
shift ${\cal D}_{n}(M_{I}^{2}-\frac{2}{3})$ and a shift along the $t$ axis
$\cos\omega t\to\cos\omega t-{\cal A}_{\parallel}M_{I}/\mu B_{0}$. The
Hamiltonian does not involve transitions between pairs of levels with
different $M_{I}$ and the pseudo-crossing within adiabatic levels of the same
pair is similar to the one described by the electronic levels only (Sec. II).

Now we examine the dynamics of the system with the same parameters used in the
previous section, but now we include the nuclear degrees of freedom, i.e., we
treat the system with hyperfine interactions included. Figure 6 shows the
9-level population dynamics corresponding to the parameters used to obtain the
2-level dynamics in Fig. 2. The curves in Fig. 6(a) show the populations
versus time summed over the hyperfine levels. The curves are very similar to
those in Fig. 2, except that the oscillations are more ragged and the slopes
of the curves at the transition point are a bit smaller because the
populations in the individual hyperfine pairs of levels [see Figs. 6(b) and
(c)] evolve with slight temporal offsets from one another, hence the features
of the curve with a sum over hyperfine levels are somewhat smeared.

Figure 6: Almost fully adiabatic dynamics for the same parameters used in Fig.
2 but including the nuclear degrees of freedom, with the population initially
distributed equally in the hyperfine sublevels of the $M_{S}=1$ level. (a) sum
over the hyperfine levels. Here $\rho_{j\,j}$, $j=+1,0,-1$, is a sum over all
three $M_{I}$ sublevels of the $M_{S}=j$ manifold. (b) $\rho_{k,k}(t)$ for
$k=1,2,3$ – the probabilities for the hyperfine levels with $M_{S}=1$. Here
the subscripts $k$ refer to the 9 levels $k=5-3M_{S}-M_{I}$ (($k=1,\ldots,9$).
These curves are labeled $\rho h_{k,k}(t)$ in the figure labels. (c) Same as
(b) except here, only the probabilities $\rho_{k,k}(t)$ for $k=7,8,9$
associated with the $M_{S}=-1$ level are shown. Probabilities associated with
the $M_{S}=0$ level are zero throughout the dynamics and are not shown. For
the sum probabilities in (a) the transition from $M_{S}=1$ to $M_{S}=-1$ is a
bit less sharp than in Fig. 2 and the oscillations are smeared due to the
offset in the time of transition among the three hyperfine levels, as shown in
(b) and (c) and as explained in Fg. 5.

Figure 7 shows the 9-level population dynamics corresponding to the parameters
used in Fig. 3. As in Fig. 6, Fig. 7(a) shows the sum populations
corresponding to the electronic states $M_{S}=\pm 1$ and Figs. 7(b) and (c)
show the populations of the different hyperfine levels. Comparison to Fig. 3
reveal the same features discussed above with respect to Fig. 6.

Figure 7: Non-adiabatic dynamics including the effect of nuclear spin for the
same parameters used in Fig. 3. The initial state and hyperfine parameters are
as in Fig. 5. (a) Sum over the hyperfine levels. (b) $\rho_{k,k}(t)$ for
$k=1,2,3$ associated with the $M_{S}=+1$ manifold. (c) $\rho_{k,k}(t)$ for
$k=7,8,9$ associated with the $M_{S}=-1$ manifold.

Figure 8: Non-adiabatic dynamics including nuclear spin obtained with the same
parameters used in Fig. 4, except that the time of the dynamical calculation
has been extended by a factor of four (the dynamics are followed for four
periods of the radio-frequency field). (a) The sum over the hyperfine levels,
with the population initially distributed equally among the hyperfine $M_{I}$
sub-levels of the $M_{S}=1$ level. (b) $\rho_{k,k}(t)$ for $k=1,2,3$
associated with the $M_{S}=+1$ manifold. (c) $\rho_{k,k}(t)$ for $k=7,8,9$
associated with the $M_{S}=-1$ manifold. Probabilities associated with the
$M_{S}=0$ level are almost zero throughout the dynamics.

Figure 8 shows the 9-level population dynamics corresponding to the parameters
used in Fig. 4. Note that we have extended the range of time in the dynamical
calculation to a final time of $t=400$ ns (recall that the first pseudo-
crossing occurs at 50 ns for this set of parameters). Here too, the curves of
the different hyperfine levels appear at a time offset from each other and
hence the sum curves show the same smearing features shown in Figs. 6 and 7.
Except for these features, the sum population behaves quite similarly to that
calculated for the model that neglected the hyperfine interaction.

Finally, let us note that the separation of the 9-level LZ problem into
2-level problems is facilitated by the conservation of the nuclear spin during
the evolution, i.e., the non-adiabaticity of the nuclear spin dynamics. This
property of the nuclear spin follows from the lack of direct coupling between
the $M_{I}=\pm 1$ levels in the Hamiltonian (13), so that no avoided crossing
appears at zero magnetic field. Moreover, even if such a coupling term between
the nuclear spin levels $M_{I}=\pm 1$ appeared (for example, due to strong
transverse magnetic fields as in Sec. V), the magnitude $\epsilon_{n}$ of such
a coupling is expected to be about 3 orders of magnitude smaller than the
equivalent coupling $\epsilon$ between the electronic spin states $M_{S}=\pm
1$. The adiabaticity parameter $\beta_{n}$ for the nuclear spin dynamics at
the pseudo-crossing that then appears is expected to be about 3 orders of
magnitude larger than the equivalent adiabaticity parameter $\beta$ of the
electronic spin: $\beta_{n}\propto\mu_{n}/|\epsilon_{n}|^{2}\sim
10^{3}\mu/|\epsilon|^{2}$. Thus, the nuclear spin dynamics would be completely
non-adiabatic. Hence the nuclear spin does not flip direction when the
magnetic field $B_{z}$ changes sign.

##  IV Dynamics including decoherence

Open systems, i.e., systems that interact with their environment, undergo
dephasing, decoherence and relaxaton. For systems that are coupled to Gaussian
white noise, the stochastic dynamics can be described using the
Schrödinger–Langevin equation vanKampenBook . If one averages over the
stochasticity, one can obtain a Markovian quantum master equation for the
density matrix $\rho(t)$ with Lindblad operators ${\cal V}_{j}$ master_eq ;
vanKampenBook :

$\displaystyle{\dot{\rho}}$ $\displaystyle=$ $\displaystyle-i[H(t),\rho(t)]$
$\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{j}\xi_{0,j}^{2}\left(2{\cal
V}_{j}\rho(t){\cal V}^{{\dagger}}_{j}-\rho(t){\cal V}^{{\dagger}}_{j}{\cal
V}_{j}-{\cal V}^{{\dagger}}_{j}{\cal V}_{j}\rho(t)\right).$

The Lindblad coefficients (volatilities) $\xi_{0,j}$ specify the strength of
the white noise. We may assume that the interaction of the 3-level system of
the electronic ground state of the NV center with the environment is
equivalent to the interaction of the NV spin with magnetic noise arising from
a spin bath or several spin baths. The Lindblad operators for this case can be
taken to be the three spin-1 operators, ${\cal V}_{j}=S_{j}$
($S_{x},S_{y},S_{z}$ for $j=x,y,z$). A NV near the diamond surface experiences
a bath due to noise originating from the diamond surface that has fast
correlation times, perhaps even as fast as $\tau_{c}\approx 10^{-11}$ s
Rosskopf14 . Therefore, shallow NVs in diamond have an environment correlation
time comparable to, or shorter than, the energy splitting parameter ${\cal
D}^{-1}$, and one approaches the white noise limit. Hence, shallow NVs can be
modeled by the Lindblad master equation (IV) Ajisaka_16 . Note that
experimental NV coherence times reported in Ref. Rosskopf14  are on the order
of tens to hundreds of $\mu$s.

For simplicity we take the interaction of the NV with the environment to be
isotropic, such that the volatilities $\xi_{0,j}$ are equal for $j=x,y,z$. In
this case the Lindblad part of the master equations is given by the following
decoherence and relaxation terms,

$\displaystyle\dot{\rho}_{11}^{\gamma}$ $\displaystyle=$
$\displaystyle\gamma(\rho_{00}-\rho_{11})$
$\displaystyle\dot{\rho}_{00}^{\gamma}$ $\displaystyle=$
$\displaystyle\gamma(\rho_{11}+\rho_{-1\,-1}-2\rho_{00})$
$\displaystyle\dot{\rho}_{-1\,-1}^{\gamma}$ $\displaystyle=$
$\displaystyle\gamma(\rho_{00}-\rho_{-1\,-1})$
$\displaystyle\dot{\rho}_{0\,\pm 1}$ $\displaystyle=$
$\displaystyle\gamma(\rho_{\mp 1\,0}-2\rho_{0\,\pm 1})$
$\displaystyle\dot{\rho}_{1\,-1}^{\gamma}$ $\displaystyle=$
$\displaystyle-3\gamma\rho_{1\,-1}$ (19)

where $\gamma=2\xi_{0,j}^{2}$. These equations (without the LZ dynamics) can
be solved analytically. The dynamics of the diagonal elements of the density
matrix (the occupation probabilities of the levels) leads to their relaxation
into a state of equal population, $\rho_{ii}=1/3$. This evolution of the
populations involves terms that decay exponentially like $e^{-\gamma
t},e^{-2\gamma t}$ and $e^{-3\gamma t}$. The decoherence of the off-diagonal
elements is uncoupled from the relaxation of the diagonal elements. The LZ
dynamics may create superpositions of the $M_{S}=\pm 1$ levels such that
$\rho_{1\,-1}\neq 0$. This coherence decays at a rate of $3\gamma$, such that
the coherence time of the superposition is shorter than the relaxation time.

To demonstrate the LZ dynamics in the presence of relaxation and decoherence
we take $\gamma=2\times 10^{5}$ s-1, corresponding to a $\xi_{0,j}^{2}=1\times
10^{5}$ s-1 for $j=x,y,z$ and a relaxation time of $5\times 10^{-6}$ s. Hence
relaxation and decoherence have a significant effect on the time scale of a
few periods of oscillation of the magnetic field. In Fig. 9 we show the
occupation probabilities $\rho_{11}(t)$, $\rho_{00}(t)$ and $\rho_{-1\,-1}(t)$
versus time. Relaxation is apparent in each of the probabilities and can be
compared with the results in Fig. 2 obtained for the same system parameters
but without the presence of noise. At very long time, the population is
equally distributed among all three levels (the environment is not a thermal
bath). Of course the total probability $\sum_{i}\rho_{ii}(t)$ remains unity
(see red curve in Fig. 9) throughout the course of the dynamics. The purity
$p(t)\equiv{\mathrm{Tr}}[\rho^{2}(t)]$ decreases with time, but not
monotonically. Asymptotically at large time, the purity reaches a value of
1/3, and this despite the fact that the $M_{S}=0$ level would not participate
in the dynamics at all and would remain with zero population where it not for
the decoherence. Figure 9 shows that after a time comparable to the coherence
time of $t=10\,\mu$s each of the populations of the three levels is already
close to 1/3 and hence the effect of LZ transitions on the populations is
damped.

Figure 9: LZ dynamics in the presence of decoherence. Probabilities
$\rho_{11}$ (pale blue curve on top left), $\rho_{00}$ (orange curve on bottom
left) and $\rho_{-1\,-1}$ (green curve on bottom right) versus time [same
parameters as in the unitary case (without decoherence) shown in Fig. 2. We
use isotropic noise with volatilities $\xi_{0,j}^{2}=1\times 10^{5}$ s-1,
$j=x,y,z$, which corresponds to $\gamma=2\times 10^{5}$ s-1. The red curve
shows the sum of the probabilities versus time, ${\mathrm{Tr}}[\rho(t)]$,
which remains unity, and the blue curve shows the purity,
$p(t)\equiv{\mathrm{Tr}}[\rho^{2}(t)]$.

Quite generally, for Lindblad master equations describing 3-level systems, the
decoherence cannot be described using a single exponential decay rate
$\gamma$. A more complicated temporal dependence results because of the
presence of multiple decay timescales. The decoherence behavior can be
understood as follows. For a time-independent Hamiltonian, each of the matrix
elements of the density matrix can be expressed as

$\rho_{\alpha\beta}(t)=a_{0,\alpha\beta}+\sum_{i=1}^{8}a_{i,\alpha\beta}\exp(-\gamma_{i}t),\
\ (\alpha,\beta=1,2,3)$

where the $\gamma_{i}$ ($i=0,1,\ldots,8$) are the 9 eigenvalues of the
9$\times$9 Liouvillian operator, and $\gamma_{0}$ is zero. The real parts of
$\gamma_{i}$ determine decay rates and the imaginary parts determine energy
eigenvalue differences, $a_{i}$ ($i=1,\ldots,8$) are the amplitude
coefficients, and the coefficient $a_{0}$ corresponds to the amplitude of the
steady state whose existence is guaranteed by trace preservation. Hence, for a
time-independent 3-level system, the maximum number of possible timescales
that determine the population decay and the coherence dynamics is 8 (the
number of non-zero eigenvalues), but there may be a lower the number due to
symmetry. For a time-dependent Hamiltonian in the adiabatic regime, the
eigenvalues $\gamma_{i}$ and the amplitudes $a_{i}$ are time-dependent, but an
adiabatic expansion can still be carried through Band_92 . In any case, it is
clear from this analysis that more than one decay rate is in general required
to describe the decoherence of a 3-level system. Our use of equal decay rates
for three Lindblad operators while setting the remaining decay rates to zero
is therefore an approximation.

##  V Transverse Magnetic Field Tuning of the Adiabaticity

This section shows that it is possible to tune adiabaticity (or, more
precisely, non-adiabaticity) with a transverse magnetic field. The quantity
which is responsible for the pseudo-crossing of the adiabatic eigenvalues is
the strain coefficient $\epsilon$. Here we show that it is possible to
counterbalance the strain (and electric field) effects with a transverse
magnetic field, and even cause the effective interaction between the
$M_{S}=\pm 1$ levels to vanish, by tuning the strength and direction of the
transverse magnetic field. In addition to the longitudinal field $B_{z}$ we
now include also a transverse magnetic field
$(B_{x},B_{y})=B_{\perp}(\cos\phi,\sin\phi)$. In this case the 3$\times$3
Hamiltonia (6) cannot be decoupled into two blocks as in the case where
$B_{\perp}=0$. However, we can eliminate the $M_{S}=0$ level due to its high
energy separation from the other enenergy scales involved. Using the Feshbach
formalism Feshbach , we apply the projection operators $P=|1\rangle\langle
1|+|-1\rangle\langle-1|$ and $Q=1-P=|0\rangle\langle 0|$ to write the
Hamiltonian as

$H=H_{M_{S}=0}+H_{M_{S}=\pm 1}+H_{0,\pm 1}+H_{\pm 1,0},$ (20)

where $H_{M_{S}=0}=QHQ=-{\cal D}$, $H_{M_{S}=\pm 1}=PHP$, $H_{0,\pm 1}=QHP$
and $H_{\pm 1,0}=PHQ$. Since ${\cal D}$ is much larger than any of the other
terms, we can eliminate the $M_{S}=0$ level and obtain a reduced 2$\times$2
effective Hamiltonian Feshbach

$H_{\rm eff}=H_{Ms=\pm 1}-\frac{H_{\pm 1,0}H_{0,\pm 1}}{\cal D}.$ (21)

For a time-independent problem this is equivalent to obtaining an approximate
solution for the Schrödinger equation $H\psi=E\psi$ for
$\psi=(\psi_{1},\psi_{-1},\psi_{0})$ by eliminating the weakly coupled
component $\psi_{0}$. The effective Schrödinger equation then takes the form

$H_{\rm eff}\left(\begin{array}[]{c}\psi_{1}\\\ \psi_{-1}\\\
\end{array}\right)=E\left(\begin{array}[]{c}\psi_{1}\\\ \psi_{-1}\\\
\end{array}\right),$

where $(H_{\rm eff})_{ij}=H_{ij}+H_{i0}G_{0}(E)H_{0j}$ and
$G_{0}(E)=(E-H_{00})^{-1}$. Equation (21) represents an approximation for
$H_{\rm eff}$ that holds for a time dependent Hamiltonian $H(t)$ as well,
given that the energies involved in the 2$\times$2 sub-space evolution are
much smaller than the magnitude of the energy splitting $H_{00}=-{\cal D}$.

In our case this effective Hamiltonian has the explicit form

$H_{\rm eff}=\left(\\!\\!\begin{array}[]{cc}\mu B_{z}+\eta
B_{\perp}^{2}&\epsilon+\eta B_{\perp}^{2}e^{2i\phi}\\\ \epsilon^{*}+\eta
B_{\perp}^{2}e^{-2i\phi}&-\mu B_{z}+\eta
B_{\perp}^{2}\end{array}\\!\\!\right),$ (22)

and $\eta=\mu^{2}/2{\cal D}$. The eigenvalues of this effective Hamiltonian
are

$E_{\pm}({\bf B})=\eta B_{\perp}^{2}\pm\sqrt{\mu^{2}B_{z}^{2}+|\epsilon+\eta
B_{\perp}^{2}e^{2i\phi}|^{2}}.$ (23)

The splitting between the two states at $B_{z}=0$ vanishes when $\eta
B_{\perp}^{2}=|\epsilon|$ and the direction of the transverse field is such
that the phase $\phi$ in the $x$-$y$ plane of the transverse field direction
is equal to $\pi$ plus half the phase of $\epsilon$ in the complex plane. In
the example below we use real $\epsilon$ and $B_{\perp}=B_{y}$, with
$\epsilon=h\times 2.87$ MHz, hence $\eta=h\times
2.8^{2}\,($MHz/G$)^{2}/(2\cdot 2.87$ GHz$)=h\times 1.35$ kHz/G2. It follows
that the critical magnetic field at which the levels become degenerate is
$B_{c}\approx 46$ G. Figure 10 shows the adiabatic eigenvalues for a magnetic
field with components $B_{z}(t)=B_{z0}\cos(\omega t)$ as in Fig. 2 and
$B_{y}=B_{c}$. Clearly, the eigenvalues cross, rather than pseudo-cross. We
numerically solved the Schrödinger equation with the effective Hamiltonian
(22), taking the initial state to be $M_{S}=1$ and all the other parameters,
including the time-dependence of the longitudinal magnetic field $B_{z}(t)$ as
in Fig. 2. The results are shown in Fig. 11. Clearly, coupling is turned off
and non-adiabaticity ensues (e.g., for $B_{y}=0.9995\,B_{c}$, the final
probability to stay in the adiabatic level is $\approx 3.6\times 10^{-6}$). A
full calculation with 9-levels including hyperfine are interaction is shown in
Fig. 12 and confirms this behavior. The probability of staying in the
adiabatic state (i.e., to be in the $M_{S}=-1$ manifold at the end) is still
very small even when the hyperfine interactions are taken into account. Note
the small probability for building up population in the $M_{S}=0$ states
during intermediate times, which do not fully return to the $M_{S}=1$ states
at large times. However, over the time scale of microseconds, effects of
decoherence, which were not taken into account in this calculation, may be
more significant than the small changes observed in this calculation (see Sec.
IV above).

Figure 10: Adiabatic eigenvalues of the 3-level Hamiltonian in Eq. (6) with
real $\epsilon$ and $B_{y}=B_{c}$ (other parameters as in Fig. 1). The plot
shows that the adiabatic eigenvalues cross, rather than pseudo-cross, hence
the dynamics is expected to be non-adiabatic. The third adiabatic eigenvalue
of the 3-level Hamiltonian with $M_{S}=0$ is a straight horizontal line at
energy $-2.87$ GHz$-2\eta B_{y}^{2}/h$ (where the second term is $\approx
5.74$ MHz, and is due to the second-order interaction with the transverse
field), and lies well outside of the plot range. Figure 11: Non-adiabatic
dynamics with the population initially in the $M_{S}=1$ level and
$B_{y}=0,0.9487\,B_{c},0.995\,B_{c},0.9995\,B_{c}$ corresponding to $1-\eta
B_{y}^{2}/\epsilon=1,\,0.1,\,0.01$ and $0.001$. The other parameters are
identical to those used to obtain Fig. 2. The figure plots the probability
$P_{M_{S}=1}(t)$, on a log scale, versus $\omega t$.

Figure 12: 9-level non-adiabatic dynamics obtained with the same parameters
used in Fig. 11 but a transverse magnetic field exactly tuned to the critical
field $B_{c}$. (a) $\rho_{k,k}(t)$ for $k=1,2,3$ associated with the
$M_{S}=+1$ manifold. (b) $\rho_{k,k}(t)$ for $k=7,8,9$ associated with the
$M_{S}=-1$ manifold. (c) $\rho_{k,k}(t)$ for $k=4,5,6$ associated with the
$M_{S}=0$ manifold. Transitions to the $M_{S}=-1$ manifold are of the order of
1%, showing that adiabaticity cancellation by a transverse field is effective
even if hyperfine interactions are taken into account.

The transverse magnetic field $(B_{x},B_{y})$ needed to eliminate the
adiabaticity in the general case where $\epsilon=\epsilon_{1}+i\epsilon_{2}$
is complex must satisfy the equations $B_{y}^{2}-B_{x}^{2}=\epsilon_{1}/\eta$
and $2B_{x}B_{y}=-\epsilon_{2}/\eta$. For example, if $\epsilon=i\epsilon_{2}$
is purely complex the adiabaticity is eliminated by
$B_{x}=-B_{y}=\pm\epsilon_{2}/2\eta$.

##  VI Summary and Conclusions

We studied the dynamics of negatively charged NV color centers in diamond
under the influence of a sinusoidally varying magnetic field which induces
energy eigenvalue pseudo-crossings. The avoided crossing is due to and zero-
field splitting between spin states that are otherwise degenerate in the
absence of magnetic field is caused by strain and electric field and the
oscillating magnetic fields are used for scanning the pseudo-crossing and
inducing Landau-Zener dynamics. The electronic ground state of the NV center
is a 3-level (spin $S=1$) system and a 9-level system together with the
nitrogen 14N nuclear spin ($I=1$), which interacts with the electronic spin to
form a hyperfine structure. However, we find that in relatively small magnetic
fields varying with a radio frequency the system can be described adequately
as a two-level system because the energy of the spin state with $M_{S}=0$ is
far from the scale of energies involved in the dynamics of the spin of the two
levels with $M_{S}=1$ and $M_{S}=-1$, so that the population stays in these
two levels. Under these conditions only the magnetic field component along the
axis of the NV center plays a role in the dynamics and induces Landau-Zener
dynamics at the pseudo-crossing between these levels. In contrast, transverse
components of the magnetic field with comprble strength couple the two
$M_{S}=\pm 1$ levels to the far detuned$M_{S}=0$ level and do not induce
significant transitions or energy shifts that would influence the dynamics.

We focused on the adiabaticity of the NV dynamics and the influence of the
hyperfine interaction, coupling to the environment and transverse magnetic
fields that do not oscillate together with the axial component. We found that
the Hamiltonian for these 6 levels corresponding to the $M_{S}=\pm 1$ states
can be decoupled into three pairs of levels with the same nuclear spin
projection $M_{I}$, as the nuclear spin does not change its state when the
magnetic field is swept through zero. Each of these pairs has exactly the same
structure as the pair of levels studied without the nuclear spin, except for a
magnetic field offset induced by the hyperfine interaction. It follows that
for each of the pairs transitions between the $M_{S}=\pm 1$ states, or
equivalently Landau-Zener transitions between the adiabatic levels, occur at
slightly different times for different nuclear spin states when the external
oscillating field is swept through the pseudo-crossings of the pairs of
levels. When all nuclear spin states are occupied it therefore appears that
the transitions are smeared.

We studied the effects of coupling of shallow NVs near diamond surfaces to an
environment modeled by a spin-bath with short fluctuation times (Gaussian
white noise). Such a coupling leads to decoherence and relaxation of the spin
population into a mixed state with equal occupation of all $M_{S}$ states. We
showed that the combination of the relaxation and decoherence with the effect
of the oscillating field gives rise to a non-monotonic decrease of the purity
of the state.

Finally, we showed that a transverse static magnetic field tunes (and can even
zero) the effective adiabaticity of the LZ dynamics by making the effective
coupling between states vanish. This effect can be used to probe this coupling
and thereby extract information about the direction and strength of the strain
and external or internal electric fields that cause the zero-field splitting
between the otherwise degenerate states $M_{S}=\pm 1$.

###### Acknowledgements.

We thank Yosef Rosenzweig, Yechezkel Schlussel and Ron Folman for initiating
the motivation for this paper and for useful conversations. This work was
supported in part by grants from the DFG through the DIP program (FO703/2-1).

## References

  * (1) M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. L. Hollenberg, Physics Reports 528, 1-45 (2013), and references therein.
  * (2) D. L. Sage, et al., Nature 496, 486 (2013).
  * (3) A. Gruber, et al., Science 276, 5321 (1997).
  * (4) E. van Oort, N. B. Manson, M. Glasbeekm, J. Phys. C: Solid State Physics 21, 4385 (1988).
  * (5) L. Jiang, et al., Science 326, 267 (2009).
  * (6) G. Balasubramanian, et al., Nature Materials 8, 383 (2009).
  * (7) F. Dolde, et al., Nature Phys. 7, 459 (2011).
  * (8) P. Kehayias, M. J. Turner, R. Trubko, J. M. Schloss, C. A. Hart, M. Wesson, D. R. Glenn, and R. L. Walsworth, Phys. Rev. B 100, 174103 (2019).
  * (9) P. Neumann, et al., Nano Lett. 13(6), 2738 (2013).
  * (10) G. Kucsko, et al., Nature 500, 54 (2013).
  * (11) M. P. Ledbetter, K. Jensen, R. Fischer, A. Jarmola, D. Budker, Phys. Rev. A 86, 052116 (2012).
  * (12) J.-F. Hsu, P. Ji, C. W. Lewandowski, and B. D’Urso, Sci. Rep. 6, 30125 (2016).
  * (13) T. Delord, P. Huillery, L. Nicolas and G. Hetet, Nature 580, 56 (2020).
  * (14) A. K. Dmitriev, H. Y. Chen, G. D. Fuchs, and A. K. Vershovskii, Phys. Rev. A 100, 011801(R) (2019)
  * (15) L. D. Landau, Phys. Z. Sowjetunion 2, 46 (1932).
  * (16) C. Zener, Proc. R. Soc. (London) A 137, 696 (1932).
  * (17) E. C. G. Stückelberg, Helv. Phys. Acta 5, 369 (1932).
  * (18) E. Majorana, Nuovo Cimento 9, 45 (1932).
  * (19) S. N. Shevchenko, S. Ashhab and F. Nori, Phys. Rep. 492, 1 (2010).
  * (20) P. V. Klimov, A. L. Falk, B. B. Buckley, and D. D. Awschalom, Phys. Rev. Lett. 112, 087601 (2014).
  * (21) Y. B. Band and Y. Avishai, Y. B. Band and Y. Avishai, Quantum Mechanics with Applications to Nanotechnology and Quantum Information Science, Sec. 7.8.
  * (22) Y. B. Band and Y. Avishai, Phys. Rev. A99, 032112 (2019).
  * (23) Y. B. Band and Y. Avishai, Quantum Mechanics, with Applications to Nanotechnology and Quantum Information Science, (Elsevier, 2013), p. 288.
  * (24) F. Iachello, Lie Algebras and Applications, (Springer, Heidelberg, 2015).
  * (25) 3-level Hamiltonians can be classified according to whether they can be expanded in terms of su(2) generators (the 3$\times$3 spin-1 matrices $S_{x}$, $S_{x}$ and $S_{z}$) or in terms of the more general su(3) generators given by the 3$\times$3 Gellman matrices $\lambda_{j}$, $j=1,\ldots 8$.
  * (26) Y. B. Band and Y. Avishai, ibid., Sec. 7.9.4.
  * (27) Y. B. Band and Y. Avishai, ibid., p. 335.
  * (28) U, Weiss, Quantum Dissipative Systems, (World Scientific, Singapore, 1999); H.-P. Breuer and F. Petruccione, Theory of Open Quantum Systems, (Oxford University Press, Oxford, 2002); M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition, (Springer, Berlin, 2007).
  * (29) C. E. Carroll and F. T. Hioe, J. Phys. A: Math. Gen. 19, 2061-2073 (1986).
  * (30) A. V. Paralo and M. N. Kiselev, arXiv:1807.11604.
  * (31) S. N. Shevchenko, S. Ashhabb and F. Nori, Physics Reports 492, 1-30 (2010).
  * (32) Handbook of Mathematical Functions, edited by M. Abramowitz and I.A. Stegun (National Bureau of Standards, Washington, DC, 1964).
  * (33) Y. Avishai and Y. B. Band, Phys. Rev. A 90, 032116 (2014).
  * (34) E. C. G. Stückelberg, Helv. Phys. Acta 5, 369 (1932).
  * (35) N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, (Elsevier, Amsterdam, 1997).
  * (36) K. Molmer, Y. Castin and J. Dalibard, J. Opt. Soc. Am B10, 524 (1993).
  * (37) We did not pause to calculate the general stochastic Schrödinger–Langevin equation. This would have allowed us to also determine the standard deviation of the probabilities as a function of time [with average equal to the results obtained using the Markovian quantum master equation (IV)].
  * (38) T. Rosskopf, et al., Phys. Rev. Lett. 112, 147602 (2014).
  * (39) S. Ajisaka and Y. B. Band, Phys. Rev. B 94 (2016).
  * (40) Y. B. Band, Phys. Rev. A45, 6643 (1992).
  * (41) H. Feshbach, Ann. of Phys. 19, 287 (1962).


# Ultrafast Collective Excited State Dynamics of a Virus-supported Fluorophore
Antenna

Joseph Holmes  [  Arathi Anil Sushma  [  Irina B. Tsvetkova  [
<EMAIL_ADDRESS>William L. Schaich  [  Richard D. Schaller  [  Bogdan
Dragnea  [<EMAIL_ADDRESS>
###### Abstract

Radiation brightening was recently observed in a multi-fluorophore-conjugated
brome mosaic virus (BMV) particle, at room temperature under pulsed
excitation. Based on its nonlinear dependence on the number of fluorophores,
the origins of the phenomenon were attributed to a collective relaxation.
However, the mechanism remains unknown. We present ultrafast transient
absorption and fluorescence spectroscopic studies which shed new light on the
collective nature of the relaxation dynamics in such radiation-brightened,
multi-fluorophore particles. Our findings indicate that the emission dynamics
is consistent with a superradiance mechanism. The ratio between the rates of
competing radiative and non-radiative relaxation pathways depends on the
number of fluorophores per virus. We also discuss the evidence of coherent
oscillations in the transient absorption trace from multi-fluorophore
conjugated which last for $\sim 100$s of picoseconds, at room temperature. The
findings suggest that small icosahedral virus shells provide a unique
biological scaffold for developing non-classical, deep subwavelength light
sources, and may open new realms for the development of photonic probes for
medical imaging applications.

###### keywords:

biophotonics, nanolaser, sub-wavelength, superradiance, quantum coherence

Department of Chemistry] Department of Chemistry, Indiana University,
Bloomington, IN 47405, U.S. Contributed equally to this work Department of
Chemistry] Department of Chemistry, Indiana University, Bloomington, IN 47405,
U.S. Contributed equally to this work Department of Chemistry] Department of
Chemistry, Indiana University, Bloomington, IN 47405, U.S. Physics Department]
Physics Department, Indiana University, Bloomington, IN 47405, U.S. CNM] The
Center for Nanoscale Materials at Argonne National Laboratory, Lemont, IL
60439, U.S. Department of Chemistry] Department of Chemistry, Indiana
University, Bloomington, IN 47405, U.S.

High-contrast luminescent nanoprobes enable a myriad applications including
biological detection1, therapeutics2, 3, sensing4, 5, optogenetics6, 7, and
anti-counterfeiting8, 9. For the vast majority of current probes, radiation is
the result of random, spontaneous relaxation. Consequently, emission dynamics
obeys the classical exponential decay 10. Since background emission has
similar dynamics, time-domain background removal to improve contrast is seldom
a viable option. Increasing the number of emitters per nanoprobe to augment
brightness generally results in self-quenching due to inter-emitter distances
becoming short enough ($\lesssim 5$ nm) for efficient resonant energy transfer
to occur. Both challenges could be addressed by constructing a multi-emitter
nanoprobe with correlated, non-classical emission. In this letter we present
experimental evidence for this behavior, which occurs after pulsed excitation
of a dense array of hundreds of fluorescent dyes, deterministically-arranged
on a 28 nm diameter icosahedral virus template.

Instead of the several ns exponential decay expected from individual
fluorophores in solution, emission from a multi-fluorophore virus particle, at
saturation coverage, occurs as a short burst of $\approx 20$ ps. Peak
intensity is attained at $\approx 25$ ps after the ultrafast excitation pulse.
Instead of being nearly quenched like under cw excitation, the estimated
quantum yield in burst-mode is comparable to that of free, individual
fluorophores. These unusual characteristics occur at room temperature, in a
biocompatible setting. Therefore, such viromimetic probes are promising to
overcome some of the current limitations of classical biophotonic probes and
open new venues in fluorescence imaging.

Radiation brightening from dye-conjugated fluorescent virus-like particles
(fVLPs) was first reported by Tsvetkova _et al_ 11. It was found that when the
fluorophores are conjugated with reactive residues of the brome mosaic virus
(BMV) capsid interfaces, emission by the complex is strongly accelerated with
respect to that of the free dye 11. The brightening effect was found to be a
nonlinear function of $N$, the average number of fluorophores per particle.
Thus, the origins of the phenomenon were attributed to a collective effect 11,
12. However, emission dynamics, which carries potential clues about the
mechanism, remained unknown. To obtain additional information about the
mechanism of radiation brightening in fVLPs, we performed measurements of the
emission and the excited state relaxation dynamics.

Figure 1: A) Schematic of accessible labelling sites on BMV capsid: red –
external lysines K64,K105, K165 and green – internal lysines K41, K44 and K86.
B) and C) transient absorption surface plots of free OG and BMV-OG278
following an initial excitation at 7.1 $\text{mJ}/\text{cm}^{2}$. The color
scale corresponds to the difference between the absorbance of the samples in
excited and ground-state (units of m$\Delta A$)

In this work, a fluorescein-derived dye, Oregon GreenTM 488, was covalently
bound to the BMV capsid _via_ NHS ester labelling of exposed lysines11.
Maximum labeling density was $\sim 300$ dyes/virus11(see Figure SI1). Figure
1A shows a molecular model of BMV, with the dye-accessible external and
internal lysines colored in red and green respectively12. Here ”internal”
means located between the lumenal or outer capsid surfaces. The minimum
nearest neighbour distance between lysines was estimated to be $1.9~{}$nm11.

To follow the excited state dynamics of coupled fluorophores in fVLPs, we
performed pump-probe femtosecond, transient absorption (TA) spectroscopy.
Differential absorption spectra of isolated fluorophores and fVLP samples with
an average number of 278 dyes per particle (hereby called BMV-OG278) are
presented as a function of the wavelength and time-delay in Figure 1B and
Figure 1C, respectively. The ultrafast excitation pulse at 488 nm was provided
by an optical parametric amplifier pumped by a regenerative amplifier ($\sim
120$ fs pulse width).

In both the control and BMV-OG278 sample, two intense negative signals are
evident at very early times in the transient spectra. One appears at $570-580$
nm and it can be attributed to Raman scattering of water13. Since this event
is simultaneous with the excitation pulse, its appearance defines the zero
delay between pump and probe pulses. The second, early negative signal is due
to the coherent interaction between pump and probe pulses, which leads to
stimulated Raman amplification appearing at 650-750 nm14. Both Raman signals
vanish within $0.4$ ps from the pump pulse.

A spectral region of particular interest is $500-530$ nm, where ground state
bleach (GSB) is expected. Indeed, both sample and control exhibit a prominent
feature in this spectral region. However, there are also some stark
differences between the two, Figure 1B and Figure 1C: In particular, the
excited state decay is much faster for the BMV-OG278 sample than for the free
dye solution.

Minima in $\Delta A$ spectra were at $498$ nm for free dye and $504$ nm for
VLP-bound fluorophores, and are consistent with the peak wavelength in steady-
state absorption spectra (Figure SI1). The TA spectral evolution of the free
dye solution and BMV-OG278 samples depends on pump energy, Figures SI3-7. The
$5-6$ nm shift between free and bound fluorophores is attributable to the
difference in dielectric constant between protein and water.

For quantitative time-domain analysis and comparison, TA spectra were
integrated from 520 to 530 nm and fitted with an exponential model from $400$
fs to $2.6$ ns (Figure SI8-12). For analysis, the wavelength and time-delay
range were chosen past the ground state bleach minimum to avoid artifacts from
pump laser scattering as well to remove coherent artifacts which arise around
the zero delay time where the pump and probe beams are temporally
overlapping15, 16. Exponential fit parameters (amplitude and decay time) were
obtained by means of nonlinear least squares, and the estimates of the errors
from the model were computed from the sample covariance matrix. Best fit
kinetic models for the transients were found to be predominantly mono-
exponential for the free fluorophore and bi-exponential for the virus-bound
fluorophore. Normalized amplitudes ($A_{i}$) and decay times ($\tau_{i}$)
obtained from the fit procedure are presented as a function of $\braket{N}$ in
the plots in Figure 2, with $\braket{N}=0$ corresponding to free fluorophores
in solution.

Figure 2: Fit components for the excited state decay dynamics plotted against
varying $\braket{N}$ _i.e._ number of fluorophores/virus; Top panel:
amplitudes $A_{1}$ and $A_{2}$; Bottom panel: Lifetimes ($\tau_{1}$) and
($\tau_{2}$)

At low power, the excited state lifetime ($\tau_{1}$) of free dyes measured by
TA was in average $\approx 3.3$ ns, which matches the spontaneous emission
lifetime of OG in solution. Since the quantum yield is high for this dye,
fluorescence emission is the dominant pathway in the relaxation dynamics of
the free fluorophore excited at $488~{}$nm. The long lifetime component
($\tau_{1}$) exhibits a monotonic shortening as the number of bound dyes
increases (Figure 2C, red curve). At high labeling density, the $\tau_{1}$ is
reduced by a factor of $6$ when compared to the free fluorophore. In previous
fluorescence lifetime experiments at similar excitation fluence and
$\braket{N}$ the fluorescence lifetime was short, and the photon counts were
quenched 11. A possible explanation is that at low pump power, only a few
fluorophores are excited while the majority are in the ground state and there
is a high probability for homo- resonance energy transfer (RET) to occur, with
the result of an increase in fluorescence quenching. In any event, we note a
correlation between the evolution of the TA $\tau_{1}$ evolution and that of
the fluorescence decay measured previously.

As the pump fluence increases, $\tau_{1}$ decreases for the free dye, possibly
due to photobleaching or to an increase in inter-system crossing and triplet
state formation 17. Indeed, some photobleaching was observed in amount of
$10-20\%$ between the first and second run for each sample. However, at high
labeling density ($\braket{N}\gtrsim 200$) $\tau_{1}$ increases from $\approx
600$ ps at lowest pump fluence up to $\approx 2000$ ps for highest pump
fluence. Thus, it appears that the nonradiative rate, which was deemed
responsible for the initial shortening of the lifetime at low fluences and
high density 11, slows down to a value that is observed at lower labeling
densities.

The second fit component (parameters, $A_{2}$ and $\tau_{2}$) dominates BMV-OG
dynamics at short times suggesting a new relaxation channel that only operates
in the multi-fluorophore VLP. The associated amplitude $A_{2}$ is
significantly greater than $A_{1}$, Figure 2B. Its time constant is
$\tau_{2}\approx 25-35$ ps for low and medium fluence and $\approx 20-25$ ps
for the highest pump fluence, Figure 2D. This component appears to decrease
with the number of dyes per virus although the trends are noisy (Figure 2D).

To summarize up to this point, two factors appear to lead to changes in the
excited state dynamics: i) pump fluence at fixed $\braket{N}$, and ii)
$\braket{N}$ at all fluences. Specifically, relaxation dynamics is accelerated
when $\braket{N}$ increases.

While fluorophore coupling clearly affects relaxation lifetimes, the result of
the competition between nonradiative and radiative relaxation channels cannot
be solely gauged based on TA. To obtain further spectral and temporal
information on fluorescence emission, we have performed time-resolved
fluorescence spectroscopy with a streak-camera detector. Figures 3A and B
present spectrally and time-resolved fluorescence emission for free
fluorophore and BMV-OG204, respectively. The pump pulse arrives at $t_{0}\sim
110$ ps. Raman scattering from water, at $\approx 560-580$ nm, is simultaneous
with the pump pulse. Also, a faint spot at $t_{0}$ can be observed at $\approx
500-510$ nm in free OG and BMV-OG samples. This is due to elastic scattering
of the laser line being incompletely suppressed by laser rejection filters.
The scattering peak is very faint in the case of free OG because this sample
does not scatter as strongly as the virus particles. The inelastic and the
elastic scattering peaks are convenient because they facilitate the timing of
fluorescence emission with respect to the pump pulse.

For free dye samples emission is spread in time and decays uniformly along the
entire measured time interval, Figure 3 A,C. In stark contrast, the
fluorescence from the BMV-OG samples is emitted as a burst of $\approx 20$ ps
duration. The burst is delayed with respect to the pump pulse, with a maximum
occurring at $t\approx 25$ ps after excitation, Figure 3B,C. At this point, we
note that the value of $\tau_{2}$ from TA measurements is close to the
characteristic time of the fluorescence burst observed in the streak camera
experiment. This observation is consistent with the short relaxation pathway
observed in TA corresponding to radiative relaxation. The wavelength-
integrated peak amplitude of the burst shows a $2\times$ increase when
$\braket{N}$ is increased from $\approx 140$ to $\approx 200$ dyes per virus,
Figure 3C.

Figure 3: Dynamics of fluorescence emission using a streak-camera detector.The
total dye concentration for all samples was kept at 200 nM, and the laser
energy density was 16.2 mJ/cm2. A) Free OG dye. B) BMV-OG204. ; C) Wavelength-
integrated (510-540 nm), time-resolved fluorescence decay traces for BMV-OG
and controls.

What phenomena could potentially be responsible for delayed, non-exponential,
accelerated radiation in VLPs? We consider superfluorescence (SF),
superradiance (SR), and amplified spontaneous emission (ASE)18, 19. Although
we are not completely excluding the possibility of lasing, i.e. light
amplification by stimulated emission, we note that it is less likely, at least
in the conventional sense, because of the absence of a resonator and the
extremely small VLP volume and thus, small Purcell factor20.

There are qualitative differences between ASE and SF in both time and
frequency domain. In the frequency domain, SF(SR) emission occurs in a broader
window21 while in ASE, spectral narrowing is expected as pump intensity
increases22. We have not observed spectral narrowing in our experiments11.

In the time domain, the SF pulse is sharp and develops after a time delay 23.
In ASE, the time delay is vanishingly small, the output pulse is longer than
in SF and noisy, and the pulse duration is insensitive to dephasing factors
(e.g. local viscosity and temperature)24, 25, 26. From previous work 11, 12 we
know that the radiation brightening from multi-fluorophore VLPs is very
sensitive to the local environment, unlike ASE. Therefore, the significant
shortening of excited state lifetime observed in TA and streak camera
experiments, the delayed intense pulse in fluorescence emission, the broad
fluorescence spectrum, the sensitivity to the fluorophore local environment,
all point to a superradiance-like mechanism responsible for radiation
brightening. We discuss a simple superradiance model in the SI, which
illustrates the burst emission.

An intriguing feature here is the fact that the phenomenon occurs at room
temperature. SR emission dynamics was experimentally studied first in atomic
gases at very low temperatures 27. Later, it was detected in low-temperature
solids28, 29, 30, 31, 32. Excitonic superradiance was studied extensively in
molecular aggregates, at cryogenic temperatures as well. 33, 34.

Only recently SR was shown to occur at room temperature in photosynthetic
complexes 35, 36. The structured environment within the photosynthetic complex
is believed to promote collective fluorophore behavior since rigidity and
proper orientation can alleviate thermal dephasing 37. In support of this
idea, alterations in the relative position and orientation of fluorophores in
the protein pockets of an artificial photosynthetic system were found to
strongly affect the excited state dynamics of protein-bound fluorophores 38,
39. However, it is worth emphasizing that interfluorophore interactions in
light harvesting complexes and in most molecular J-aggregates are
characterized by strong coupling 40, 41, 42. By comparison, the nearest-
neighbor fluorophore distances on BMV-OG are much longer than that of
molecular aggregates, which precludes electron tunnelling.

In our case, collective emission of radiation arise from weak coupling at room
temperature. Until now, this coupling range has not been given much attention.
Dipole-dipole coupling is considered detrimental to the formation of coherent
multi-fluorophore states 27. Could the virus template play a role in coupling?
The original model of Dicke superradiance (see SI) does not take into account
the spatial organization or orientation of the fluorophores. However, from
previous work it was clear that the nature of the template is important for
the radiation brightening effect 11, 12. Below we provide further evidence for
it from a study of relaxation dynamics.

We performed additional TA experiments with a set-up in which pump and probe
beams were counter-propagating, at a small angle. This set-up change allowed
us to reduce laser scattering artifacts and measure the excited state dynamics
at high laser powers with the probe wavelength closer to ground state bleach
peak position $\lambda=500-520$ nm.

The higher signal-to-noise ratio afforded by this setup change has allowed us
to make an interesting observation. Figure 4 presents normalized absorbance
difference curves obtained for free dye and BMV-OG with $\braket{N}$ ranging
from 200 to 300 dyes/virus. The data were tail-fitted with double exponential
decay and the residuals from the fitting are presented on the top panels of
the graph. At high dye loads, oscillations in TA can be observed, even in the
raw data but most prominently in the residuals.

Oscillating TA signals previously observed in photosynthetic complexes at room
temperature were attributed to vibrations of the protein matrix encapsulating
the fluorophores43, 44. The oscillations observed on high-load VLPs in Figure
4 persist for a longer time, _i.e._ up to 100 ps, and have lower frequency.
They cannot be assigned to quantum beats from closely spaced levels since
those are not expected to be observed at room temperature. We hypothesise that
the oscillations have to do with the template. Two mechanisms could be
envisioned by which the virus template may affect the collective emission of
the multi-fluorophore antenna. First, we note that the frequency of the
observed oscillations falls within the range of computed low-frequency modes
(vibrational/breathing) of the virus shells 45, 46, 47. Since coupling in
near-field is very sensitive to inter-fluorophore distance and orientation,
such low-frequency vibrations could modulate the TA trace. Second, the
template may modulate the radiative rate of the antenna via oscillations in
polarizability due to virus shell vibrations 48. Could such virus-shell modes
be involved in maintaining fluorophore correlations? At this point such
questions concerning the role of spatial symmetry and the nature of coupling
remain open, and warrant future studies.

Figure 4: Oscillating signals derived from TA decay fitting. The normalized TA
amplitude and fitting curves are on bottom panel of the graph and the box-
filter smoothed (10 data points) residuals are on the top panels of the graph.

In conclusion, the symmetric BMV capsids provides a biological scaffold for a
multi-fluorophore array that acts an antenna with accelerated emission rate at
room temperature. We have employed ultrafast, time-resolved transient
absorption and fluorescence spectroscopy to provide further evidence for the
mechanism of radiation brightening. Taken together with results of previous
work, the new data strongly suggest that the virus-supported multi-fluorophore
array exhibits a form of superradiant emission. Moreover, coherent, low-
frequency virus shell vibrations could be observed as they modulate the
relaxation rate for nearly 100 ps at room temperature. This finding highlights
the involvement of the virus scaffold in the excited state relaxation
dynamics. Enhanced light intensity, non-classical emission dynamics, and
biocompatibility make these viromimetic superfluorescent particles interesting
for biological imaging at high background conditions, for instance as deep
tissue optical probes.

##  1 Experimental

###  1.1 BMV-OG sample preparation

BMV-OG samples were prepared according to protocols previously described 11.
Briefly, the lysines of wtBMV were modified by covalent conjugation with
Oregon Green™ 488 carboxylic acid, succinimidyl ester49. wtBMV stock in SAMA
buffer (50 mM NaOAc, 8 mM Mg(OAc)2(pH 4.6)) was mixed with sodium bicarbonate
buffer (100 mM, pH 8.2) in 1:1 ratio to a final concentration of $10^{13}$
particles/ml. Dye stock was freshly prepared in DMSO and was added to virus
solution at varied ratios ranging from 500 to 10000 dyes/capsid. The DMSO
concentration in reaction mixture was kept constant at $\sim 5$ w/v%. The
reaction was left for 1-2 hours at room temperature, followed by removal of
free dye via filter wash or dialysis with SAMA buffer. UV-Vis spectroscopy was
performed to estimate the number of dyes per virus, and fluorescence
spectroscopy was used to confirm absence of free dyes in the purified samples.

###  1.2 Transmission Electron Microscopy

Electron-transparent samples were prepared by placing the dilute sample (10
$\mu$L) onto a carbon-coated copper grid. After 10 mins, excess solution on
the grid was removed with filter paper. The grid was stained with uranyl
acetate (10 $\mu$L of 2% solution) for 10 minutes and the excess solution was
removed by blotting with filter paper. The sample was then left to dry for
several minutes. Images were acquired at an accelerating voltage of 80 kV on a
JEOL JEM 1010 Transmission electron microscope and analyzed with the ImageJ
Processing Toolkit (see SI2).

###  1.3 Time-resolved Pulsed Excitation Fluorescence Spectroscopy with Streak
camera

The sample was excited by a pulse generated through an optical parametric
amplifier using a Ti:sapphire femtosecond laser (Spectra-Physics Spitfire
Pro). The pulses have a wavelength of $\lambda=480$ nm, a duration of 35 fs,
and a repetition rate of 2000 Hz. The focused spot size was measured to be 160
$\mu$m and the average laser power varied from 1 to 6.5 mW (or 0.5 to 3.25
$\mu$J/pulse). The excitation beam was directed at the cuvette at a small
angle to prevent cavity effect. The sample fluorescence was collected with a
lens, directed through long pass filter (LP508) and a 150 mm spectrograph,
then was detected as a function of wavelength and time after excitation using
a photon counting streak camera (Hamamatsu C5680). Data was collected and pre-
processed by HPDTA software and further analysis was done with Igor Pro
software.

###  1.4 Transient absorption spectroscopy

Ultrafast transient absorption measurements were carried out using an output
of regeneratively amplified Ti:sapphire laser (800 nm, 120 fs, 2 kHz
repetition rate) which was split into two beams. The first beam, containing
10% power, was focused into a sapphire crystal to generate a white light
continuum (440-750 nm), which serves as the probe laser. The other beam,
containing 90% of the power, was sent into an optical parametric amplifier to
generate the 480 nm pump beam. After passing through a depolarizer, the pump
beam is focused and overlapped with the probe beam at the sample (focal
diameter being 300 $\mu$m). The pump pulse fluence was varied from 0.71
mJ/$\text{cm}^{2}$ to 14.5 mJ/$\text{cm}^{2}$, while the probe pulse fluence
was significantly smaller and kept constant in all experiments. The upper and
lower limits over the range of excitation energies correspond to photon
densities that were close to but greater than the number of fluorophores
attached in VLP antennas. Prior to any analysis, spectra were corrected for
chirp and group dispersion velocity using Surface Explorer (Ultrafast
Systems).

###  1.5 Transient absorption set-up

The tunable wavelength output of optical parametric amplifier pumped by a
Ti:sapphire regenerative amplifier at 250 kHz repetition rate(180 fs
pulsewidth), seeded by a mode-locked oscillator at 76 MHz (Coherent Inc.) was
set at 480 nm and used as the pump pulse for transient absorption
measurements. The part of white light output from OPA passed through a 490/20
optical filter and used as probe pulse. The fluence of the pump was varied
from from 0.1 to 3.5 mJ/cm2 and the probe pulse was set at $\sim
5~{}\mu$J/mm2. For measurements of relaxation decay, the probe pulse arrived
at a delayed time after the pump that was continuously adjusted using a
variable length delay line (600 ps max.). A mechanical chopper was used to
modulate the pump beam at 1000 Hz. The lock-in amplifier was used to measure
probe signal at the modulation frequency. The transient absorption data
processing included a background subtraction step.

The work was supported by the Army Research Office, under awards W911NF2010072
and W911NF2010071, and by the National Science Foundation, under award
CBET-1803440. This work was performed, in part, at the Center for Nanoscale
Materials, a U.S. Department of Energy Office of Science User Facility, and
supported by the U.S. Department of Energy, Office of Science, under Contract
No. DE-AC02-06CH11357. We are grateful to the Center for Bioanalytical
Metrology (CBM), an NSF Industry-University Cooperative Research Center, for
providing funding for this project under grant NSF IIP 1916645, and to members
of the industry advisory board of the CBM for valuable discussions and
feedback.

The Supporting Information is available free of charge on the ACS Publications
website at DOI: 10.1021/acs.jpclett.0000000. Experimental methods, sample
characterization, TA fitting results and model of superradiant emission from N
fluorophores are presented in the supporting information file.

## References

  * Wolfbeis 2015 Wolfbeis, O. S. An overview of nanoparticles commonly used in fluorescent bioimaging. _Chem. Soc. Rev._ 2015, _44_ , 4743–4768
  * Yi et al. 2014 Yi, X.; Wang, F.; Qin, W.; Yang, X.; Yuan, J. Near-infrared fluorescent probes in cancer imaging and therapy: an emerging field. _International Journal of nanomedicine_ 2014, _9_ , 1347
  * Vats et al. 2017 Vats, M.; Mishra, S. K.; Baghini, M. S.; Chauhan, D. S.; Srivastava, R.; De, A. Near infrared fluorescence imaging in nano-therapeutics and photo-thermal evaluation. _International journal of molecular sciences_ 2017, _18_ , 924
  * Ueno and Nagano 2011 Ueno, T.; Nagano, T. Fluorescent probes for sensing and imaging. _Nature methods_ 2011, _8_ , 642–645
  * Gao et al. 2021 Gao, L.; Wang, W.; Wang, X.; Yang, F.; Xie, L.; Shen, J.; Brimble, M. A.; Xiao, Q.; Yao, S. Q. Fluorescent probes for bioimaging of potential biomarkers in Parkinson’s disease. _Chemical Society Reviews_ 2021, _50_ , 1219–1250
  * Leopold et al. 2019 Leopold, A. V.; Shcherbakova, D. M.; Verkhusha, V. V. Fluorescent biosensors for neurotransmission and neuromodulation: engineering and applications. _Frontiers in cellular neuroscience_ 2019, _13_ , 474
  * Oheim et al. 2014 Oheim, M.; van’t Hoff, M.; Feltz, A.; Zamaleeva, A.; Mallet, J.-M.; Collot, M. New red-fluorescent calcium indicators for optogenetics, photoactivation and multi-color imaging. _Biochimica et Biophysica Acta (BBA)-Molecular Cell Research_ 2014, _1843_ , 2284–2306
  * Abdollahi et al. 2020 Abdollahi, A.; Roghani-Mamaqani, H.; Razavi, B.; Salami-Kalajahi, M. Photoluminescent and chromic nanomaterials for anticounterfeiting technologies: recent advances and future challenges. _ACS nano_ 2020, _14_ , 14417–14492
  * Li et al. 2018 Li, M.; Feng, Y.; Tian, Q.; Yao, W.; Liu, L.; Li, X.; Wang, H.; Wu, W. Tunable and ultra-stable UV light-switchable fluorescent composites for information hiding and storage. _Dalton Transactions_ 2018, _47_ , 11264–11271
  * Valeur and Berberan-Santos 2013 Valeur, B.; Berberan-Santos, M. r. N. _John Wiley Sons_ , 2nd ed.; Molecular Fluorescence: Principles and Applications; John Wiley & Sons: New York, 2013
  * Tsvetkova et al. 2019 Tsvetkova, I. B.; Anil Sushma, A.; Wang, J. C.-Y.; Schaich, W. L.; Dragnea, B. Radiation Brightening from Virus-like Particles. _ACS Nano_ 2019, _13_ , 11401–11408
  * Anil Sushma et al. 2021 Anil Sushma, A.; Zhao, B.; Tsvetkova, I. B.; Pérez-Segura, C.; Hadden-Perilla, J. A.; Reilly, J. P.; Dragnea, B. Subset of Fluorophores Is Responsible for Radiation Brightening in Viromimetic Particles. _J. Phys. Chem. B_ 2021, _125_ , 10494–10505
  * Parker 1959 Parker, C. Raman spectra in spectrofluorimetry. _Analyst_ 1959, _84_ , 446–453
  * Lorenc et al. 2002 Lorenc, M.; Ziolek, M.; Naskrecki, R.; Karolczak, J.; Kubicki, J.; Maciejewski, A. Artifacts in femtosecond transient absorption spectroscopy. _Applied Physics B_ 2002, _74_ , 19–27
  * Vardeny and Tauc 1981 Vardeny, Z.; Tauc, J. Picosecond coherence coupling in the pump and probe technique. _Optics Communications_ 1981, _39_ , 396–400
  * Hui et al. 2011 Hui, L.; Hang, Z.; Jin-Hai, S.; Li-He, Y.; Feng, C.; Xun, H. Elimination of the coherent artifact in a pump-probe experiment by directly detecting the background-free diffraction signal. _Chinese Physics Letters_ 2011, _28_ , 086602
  * Demchenko 2020 Demchenko, A. P. Photobleaching of organic fluorophores: quantitative characterization, mechanisms, protection. _Methods and Applications in Fluorescence_ 2020, _8_ , 022001
  * Cong et al. 2016 Cong, K.; Zhang, Q.; Wang, Y.; Noe, G. T.; Belyanin, A.; Kono, J. Dicke Superradiance in Solids [Invited]. _J. Opt. Soc. Am. B_ 2016, _33_ , C80
  * Dai 2011 Dai, D. Brief comment: Dicke superradiance and superfluorescence find application for remote sensing in air. _arXiv preprint arXiv:1108.5360_ 2011,
  * Samuel et al. 2009 Samuel, I. D. W.; Namdas, E. B.; Turnbull, G. A. How to Recognize Lasing. _Nat. Photonics_ 2009, _3_ , 546–549
  * Dicke 1954 Dicke, R. H. Coherence in Spontaneous Radiation Processes. _Phys. Rev._ 1954, _93_ , 99–110
  * Allen and Peters 1973 Allen, L.; Peters, G. Amplified spontaneous emission and external signal amplification in an inverted medium. _Physical review A_ 1973, _8_ , 2031
  * Malcuit et al. 1987 Malcuit, M. S.; Maki, J. J.; Simkin, D. J.; Boyd, R. W. Transition from Superfluorescence to Amplified Spontaneous Emission. _Phys. Rev. Lett._ 1987, _59_ , 1189–1192
  * Wilk et al. 1983 Wilk, S. R.; Boyd, R. W.; Teegarden, K. J. Laser characteristics of KCL: O-2. _Optics communications_ 1983, _47_ , 404–406
  * Rai and Bowden 1992 Rai, J.; Bowden, C. M. Quantum-statistical analysis of superfluorescence and amplified spontaneous emission in dense media. _Physical Review A_ 1992, _46_ , 1522
  * Maki et al. 1989 Maki, J. J.; Malcuit, M. S.; Raymer, M. G.; Boyd, R. W.; Drummond, P. D. Influence of collisional dephasing processes on superfluorescence. _Physical Review A_ 1989, _40_ , 5135
  * Gross and Haroche 1982 Gross, M.; Haroche, S. Superradiance: An Essay on the Theory of Collective Spontaneous Emission. _Phys. Rep._ 1982, _93_ , 301–396
  * Miyajima et al. 2017 Miyajima, K.; Kumagai, Y.; Ishikawa, A. Ultrashort radiation of biexcitonic superfluorescence from high-density assembly of semiconductor quantum dots. _The Journal of Physical Chemistry C_ 2017, _121_ , 27751–27757
  * Tighineanu et al. 2016 Tighineanu, P.; Daveau, R. S.; Lehmann, T. B.; Beere, H. E.; Ritchie, D. A.; Lodahl, P.; Stobbe, S. Single-photon superradiance from a quantum dot. _Physical review letters_ 2016, _116_ , 163604
  * Rainò et al. 2018 Rainò, G.; Becker, M. A.; Bodnarchuk, M. I.; Mahrt, R. F.; Kovalenko, M. V.; Stöferle, T. Superfluorescence from lead halide perovskite quantum dot superlattices. _Nature_ 2018, _563_ , 671–675
  * Krieg et al. 2020 Krieg, F.; Sercel, P. C.; Burian, M.; Andrusiv, H.; Bodnarchuk, M. I.; St0̈ferle, T.; Mahrt, R. F.; Naumenko, D.; Amenitsch, H.; Rainò, G., et al. Monodisperse long-chain sulfobetaine-capped CsPbBr3 nanocrystals and their superfluorescent assemblies. _ACS central science_ 2020, _7_ , 135–144
  * Haider et al. 2021 Haider, G.; Sampathkumar, K.; Verhagen, T.; Nádvorník, L.; Sonia, F. J.; Valeš, V.; Sỳkora, J.; Kapusta, P.; Němec, P.; Hof, M., et al. Superradiant Emission from Coherent Excitons in van Der Waals Heterostructures. _Advanced Functional Materials_ 2021, 2102196
  * Palacios et al. 2002 Palacios, M. A.; de Weerd, F. L.; Ihalainen, J. A.; van Grondelle, R.; van Amerongen, H. Superradiance and Exciton (De)localization in Light-Harvesting Complex II from Green Plants? _The Journal of Physical Chemistry B_ 2002, _106_ , 5782–5787
  * Engel et al. 2007 Engel, G. S.; Calhoun, T. R.; Read, E. L.; Ahn, T.-K.; Mančal, T.; Cheng, Y.-C.; Blankenship, R. E.; Fleming, G. R. Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. _Nature_ 2007, _446_ , 782–786
  * Monshouwer et al. 1997 Monshouwer, R.; Abrahamsson, M.; van Mourik, F.; van Grondelle, R. Superradiance and Exciton Delocalization in Bacterial Photosynthetic Light-Harvesting Systems. _J. Phys. Chem. B_ 1997, _101_ , 7241–7248
  * Malina et al. 2021 Malina, T.; Koehorst, R.; Bína, D.; Pšenčík, J.; van Amerongen, H. Superradiance of bacteriochlorophyll c aggregates in chlorosomes of green photosynthetic bacteria. _Scientific Reports_ 2021, _11_ , 8354
  * Rolczynski et al. 2018 Rolczynski, B. S.; Zheng, H.; Singh, V. P.; Navotnaya, P.; Ginzburg, A. R.; Caram, J. R.; Ashraf, K.; Gardiner, A. T.; Yeh, S.-H.; Kais, S., et al. Correlated protein environments drive quantum coherence lifetimes in photosynthetic pigment-protein complexes. _Chem_ 2018, _4_ , 138–149
  * Noriega et al. 2015 Noriega, R.; Finley, D. T.; Haberstroh, J.; Geissler, P. L.; Francis, M. B.; Ginsberg, N. S. Manipulating excited-state dynamics of individual light-harvesting chromophores through restricted motions in a hydrated nanoscale protein cavity. _J. Phys. Chem. B_ 2015, _119_ , 6963–6973
  * Delor et al. 2018 Delor, M.; Dai, J.; Roberts, T. D.; Rogers, J. R.; Hamed, S. M.; Neaton, J. B.; Geissler, P. L.; Francis, M. B.; Ginsberg, N. S. Exploiting chromophore–protein interactions through linker engineering to tune photoinduced dynamics in a biomimetic light-harvesting platform. _Journal of the American Chemical Society_ 2018, _140_ , 6278–6287
  * Spano and Mukamel 1989 Spano, F. C.; Mukamel, S. Superradiance in Molecular Aggregates. _J. Chem. Phys._ 1989, _91_ , 683–700
  * Meinardi et al. 2003 Meinardi, F.; Cerminara, M.; Sassella, A.; Bonifacio, R.; Tubino, R. Superradiance in molecular H aggregates. _Physical review letters_ 2003, _91_ , 247401
  * Ratner 2003 Ratner, M. Superradiance of J-aggregates: correspondence between an infinite disordered chain and a regular finite chain. _Low Temperature Physics_ 2003, _29_ , 602–605
  * Panitchayangkoon et al. 2010 Panitchayangkoon, G.; Hayes, D.; Fransted, K. A.; Caram, J. R.; Harel, E.; Wen, J.; Blankenship, R. E.; Engel, G. S. Long-lived quantum coherence in photosynthetic complexes at physiological temperature. _Proceedings of the National Academy of Sciences_ 2010, _107_ , 12766–12770
  * Engel 2011 Engel, G. S. Quantum coherence in photosynthesis. _Procedia Chemistry_ 2011, _3_ , 222–231
  * van Vlijmen and Karplus 2005 van Vlijmen, H. W.; Karplus, M. Normal mode calculations of icosahedral viruses with full dihedral flexibility by use of molecular symmetry. _J Mol Biol_ 2005, _350_ , 528–42
  * Dykeman and Sankey 2008 Dykeman, E. C.; Sankey, O. F. Low Frequency Mechanical Modes of Viral Capsids: An Atomistic Approach. _Phys. Rev. Lett._ 2008, _100_ , 028101
  * Hadden et al. 2018 Hadden, J. A.; Perilla, J. R.; Schlicksup, C. J.; Venkatakrishnan, B.; Zlotnick, A.; Schulten, K. All-atom molecular dynamics of the HBV capsid reveals insights into biological function and cryo-EM resolution limits. _Elife_ 2018, _7_
  * Young et al. 2018 Young, G. et al. Quantitative mass imaging of single biological macromolecules. _Science_ 2018, _360_ , 423–427
  * Johnson and Spence 2010 Johnson, I., Spence, M., Eds. _The Molecular Probes Handbook : a guide to fluorescent probes and labeling technologies._ , 11th ed.; Life Technologies, 2010