# Tuning Spectral Properties of Individual and Multiple Quantum Emitters in
Noisy Environments

Herbert F Fotso  Department of Physics, University at Albany SUNY, Albany, New
York 12222, USA  Department of Physics, University at Buffalo SUNY, Buffalo,
New York 14260, USA

###### Abstract

A quantum emitter in a dynamic environment may have its energy levels drift
uncontrollably in time with the fluctuating bath. This can result in an
emission/absorption spectrum that is spread over a broad range of frequencies
and presents a challenging hurdle for various applications. We consider a
quantum emitter in an environment that alters the energy levels so that the
emission frequency is represented by a Gaussian random distribution around a
given mean value with given standard deviation and correlation time. We study
the emission spectrum of this system when it is placed under the influence of
a periodic sequence of finite width $\pi$ pulses. We show that this external
field protocol can effectively overcome spectral diffusion in this system by
refocusing the bulk of the emission spectrum onto the pulse carrier frequency.
We further consider two such emitters in different noisy environments and find
that the two-photon interference operation can be made efficient by the
sequence of finite width pulses applied on both systems. Finally, we show that
an ensemble of nominally similar emitters, each with its different
environment, and thus randomly shifted emission frequency, can have its
overall emission spectrum that would otherwise be inhomogeneously broadened
according to the random distribution, refocused onto a lineshape with a well-
defined central peak that has the linewidth of an individual isolated non-
noisy emitter. These results demonstrate for this specific model of noisy
environments, the protection of spectral properties by an external control
protocol here represented by a periodic sequence of finite width pulses.

##  I Introduction

The ability to control and protect from environmental variations the spectral
properties of a quantum emitter in a dynamic environment is of significant
importance for numerous applications extending from spectroscopy to a variety
of fundamental operations of quantum information processing (QIP). Indeed,
spectral diffusion, the random drift of the emission frequency of a quantum
emitter with time AmbroseMoerner_spectralDiffusion_Nature1991 ;
KMFuBeausoleil_PRL2009 ; PfaffHanson_Science2014 ; LyasotaKapon_SciRep2019 ;
GaoHolmes_APL2019 ; VuralMichler_PRB2020 ; VuralMichler_PRB2020 , reduces the
efficiency of essential QIP procedures such as two-photon interference,
entanglement generation between distant quantum nodes, and coupling to
cavitiesKambsBecher_NJP2018 ; AwschalomHansonZhou_opticsQIP ;
AtatureEnglund_Wrachtrup_NatRevMat_2018 ; HansonAwschalom_QIP_ss_08 ;
NV_Review_PhysRep2013 ; CarterGammonQDcavity ; JeffKimble_qtmInternet ;
NielsenChuangBook . These operations typically require well behaved spectral
signatures of the involved quantum emitters. For this reason, spectral
features dominated by random fluctuations, are significant obstacles to the
scalability of photon-mediated operations or QIP interfaces.
Gao_Imamoglu_NatComm2013 ; Kuhlmann_Warburton_QDOT_NatPhys2013 ;
SantoriVuckovicYamamoto_QDOT_Nat2002 ; Humphreys_Hanson_NV_entanglement ;
Hanson_loopholeFree_Nature2015

One avenue to help overcome these fluctuations is the design of increasingly
pristine systems which is an onerous task. Although this approach is indeed
important to reduce unwanted defects and randomness, it remains limited in its
effectiveness given that some minimal fluctuations will likely persist in most
realizations of solid state systems for instance. For this reason solutions
based on external control field can play a unique role in addressing this
problem Schroeder_Englund_SPE_NatComm2017 ; QDOTs_HOM_Atature ;
Aharonovich_Englund_Toth_SS_SPE ; KMFuBeausoleil_PRL2009 ;
Acosta_Beausoleil_PRL2012 ; Dreau_Jacques ; Pfaff_Hanson_Science2014 ;
Basset_Awschalom_PRL2011 ; FaraonEtalNatPhot ; Hansom_Atature_APL2014 ;
Crooker_Bayer_PRL2010 ; Calajo_Passante_PRA2017 ; JSLee_Khitrin_JPhysB2008 ;
IDS_2017 ; LukinVuckovic_npj2020 ; LiuKumano_OptExpress2017 .

In earlier studies, we examined the emission and absorption spectrum of two-
level systems when they are driven by a variety of pulse sequences including a
periodic sequence of $\pi_{x}$ pulses FotsoEtal_PRL2016 ;
FotsoDobrovitski_Absorption ; Fotso_JPhysB2019 . We showed that for an emitter
with emission frequency $\omega$, the emission spectrum could be made mostly
independent of the constant detuning $\Delta$ with respect to the pulse
carrier frequency $\omega_{0}$. We also showed that the Hong-Ou-Mandel (HOM)
two-photon interference (TPI) HOM  could have its efficiency enhanced for two
distant emitters with different respective emission frequencies when they are
both driven by the same periodic sequence of instantaneous $\pi_{x}$ pulses
Fotso_TPI_PRB_2019 .

In the present paper, we examine the situation of emitters in explicitly noisy
environments. In particular, we study the emission spectrum of a TLS with
detuning $\Delta(t)$ with respect to a reference frequency $\omega_{0}$ such
that $\Delta(t)$ follows a random Gaussian distribution with standard
deviation $\sigma_{\Delta}$, mean value $\Delta_{0}$ and correlation time
$\tau_{c}$. We show that for a periodic sequence of finite width $\pi$-pulses
well away from the ideal pulse limit, the emission spectrum of the noisy
quantum emitter can be made minimally dependent on the noisy environment with
the bulk of the emission spectrum occurring at the pulse carrier frequency and
satellite peaks at $\pm\pi/\tau$ similar to the static detuning case. Next, we
demonstrate enhancement of the HOM-type two-photon interference experiment for
two explicitly noisy emitters in different environments such that their
detunings are $\Delta_{1}(t)$ and $\Delta_{2}(t)$ with mean values
$\Delta_{01}$ and $\Delta_{02}$, standard deviations $\sigma_{\Delta 1}$ and
$\sigma_{\Delta 2}$, correlation times $\tau_{c1}$ and $\tau_{c2}$. Finally,
we show that for a dilute ensemble of emitters with randomly distributed
emission frequencies among individual emitters, the emission spectrum that
would otherwise be inhomgeneously broadened, can be refocused by the periodic
sequence of finite width pulses.

The rest of the paper is organized as follows. In section II, we discuss the
model for the emitter in a dynamic environment, describing the Hamiltonian and
the master equation for the density matrix operator of the two-level system in
the radiation bath under the influence of the control field. In section III,
we describe the methods that are used to obtain the emission spectrum for an
individual pulse-driven noisy emitter subject to spectral diffusion, to
characterize the two-photon interference operation between two such emitters
and, finally, to obtain the emission spectrum of a dilute inhomogeneous
ensemble of two-level systems under the influence of the control protocol. In
section IV, we present the results for the different situations discussed
above before finishing with our conclusions in section V.

##  II Model, Emitter in dynamic environment

We consider a quantum emitter represented by a two-level system (TLS). Its
ground state $|g\rangle$ and its excited state $|e\rangle$ are separated by an
energy $E_{e}-E_{g}=\hbar\omega_{1}=\hbar(\omega_{0}+\Delta)$. $\Delta$ is the
detuning with respect to a target frequency $\omega_{0}$. In what follows, we
set $\hbar=1$. Because of the fluctuations in the environment, this detuning
can vary randomly in time. Here, we will specifically consider the situation
in which the fluctuations lead to a time-dependent detuning that follows a
random Gaussian distribution centered around an average value $\Delta_{0}$,
that has a standard deviation $\sigma_{\Delta}$ and a correlation time
$\tau_{c}$. The TLS is coupled to a bosonic bath representing the normal modes
of the radiation field. The protocol of interest in the present studies is
represented by pulses at the target frequency $\omega_{0}$ with Rabbi
frequency $\Omega_{x}(t)$ that is timed so as to impart on the emitter an
appropriate $\pi$ rotation over a finite time $t_{\pi}$ before being switched
off, allowing the system to evolve freely for a subsequent time $\tau-
t_{\pi}$. The process is repeated periodically so that the entire sequence has
period $\tau$. FIG.1 illustrates schematically the random drift in time of the
emission frequency of a two level system: spectral diffusion. As a function of
time, the detuning takes different random values $\Delta_{1}$, $\Delta_{2}$,
$\Delta_{3}$, $\Delta_{4}$… As a results, the emission spectrum can be broadly
ill-defined. FIG.2 shows a sampling of the detuning as a function of time for
a random Gaussian distribution with average value $\Delta_{0}=4.0$, standard
deviation $\sigma_{\Delta}=4.0$ and correlation time $\tau_{c}=0.03$ (a); and
an illustration of the sequence of finite width pulses (b).

Figure 1: Schematic representation of spectral diffusion for a quantum emitter
in a dynamic environment. The energy levels for the ground and excited states,
$|g\rangle$ and $|e\rangle$ respectively, drift randomly in time around a
target frequency $\omega_{0}$. Thus, at different points in time, the detuning
of the emission with respect to $\omega_{0}$ takes random values $\Delta_{1}$,
$\Delta_{2}$, $\Delta_{3}$, $\Delta_{4}$… As a result, the emission spectrum
of the system over time adds up to a broad ill-defined lineshape. Figure 2:
(a): We study the emission spectrum of a TLS for which the emission spectrum
with respect to a target frequency $\omega_{0}$ has a detuning $\Delta(t)$
that fluctuates in time following a random Gaussian distribution pictured here
with $\Delta_{0}=4.0$ (dashed blue line), $\sigma_{\Delta}=4.0$ and
correlation time $\tau_{c}=0.03$. (b): We will examine the spectral properties
of the system under the influence of a periodic sequence of finite width
$\pi$-pulses with inter-pulse delay $\tau$ and Rabbi frequency $\Omega$. The
driving field is applied for a time $t_{\pi}$ that amounts to a $\pi_{x}$
rotation.

In the rotating wave approximation (RWA) and in the rotating frame, so that
all energies are measured with respect to the target frequency $\omega_{0}$,
the Hamiltonian describing this system can be written as:

$\displaystyle H=\sum_{k}\omega_{k}a^{\dagger}_{k}a_{k}$ $\displaystyle+$
$\displaystyle\frac{\Delta(t)}{2}\sigma_{z}-i\sum_{k}g_{k}\left(a^{\dagger}_{k}\sigma_{-}-a_{k}\sigma_{+}\right)$
(1) $\displaystyle+$
$\displaystyle\frac{\Omega_{x}(t)}{2}(\sigma_{+}+\sigma_{-}).$

The operators $\sigma_{z}=|e\rangle\langle e|-|g\rangle\langle g|$,
$\sigma_{+}=|e\rangle\langle g|$, and $\sigma_{-}=|g\rangle\langle
e|=(\sigma_{+})^{\dagger}$ are respectively, the $z$-axis Pauli matrix, the
raising, and the lowering operators for the two-level system. $a_{k}$
($a^{\dagger}_{k}$) is the annihilation (creation) operator of the $k$-th
photon mode, $g_{k}$ is its coupling strength to the emitter, and $\omega_{k}$
is the detuning from $\omega_{0}$ of mode $k$. We consider pulses such that
$\Omega_{x}(t)=\Omega_{x}=\Omega$ during the time $t_{\pi}$ of the
$\pi$-pulses and zero otherwise. $\Delta(t)=\omega_{1}(t)-\omega_{0}$ is the
time-dependent detuning of the TLS’s transition frequency from the pulse
carrier frequency.

We assume the system to be initially prepared in the excited state. In the
absence of all control ($\Omega_{x}(t)=0$ for all times), spontaneous decay
will occur, and for a static detuning, the corresponding emission rate is
$\Gamma=2\pi\int g_{k}^{2}\;\delta(\omega_{k}-\Delta)\;dk$; We normalize our
energy and time units so that this relaxation rate is $\Gamma=2$, and the
corresponding spontaneous emission line has a simple Lorentzian shape
$1/(\omega^{2}+1)$, with half-width equal to $1$. By this process, all
frequencies are measured in units of $\Gamma/2$. Previous studies have only
studied the problem with static detuning and idealized instantaneous pulses
FotsoEtal_PRL2016 ; Fotso_JPhysB2019 ; LukinVuckovic_npj2020 . In the present
treatment of the noisy quantum emitter, we assume the same relaxation rate
$\Gamma$ as in the static detuning problem and focus on the time-dependence of
$\Delta(t)$.

To study the dynamics of the TLS and evaluate its spectral properties, we
analyze the time evolution of the emitter’s density matrix operator:

$\displaystyle\rho(t)$ $\displaystyle=$
$\displaystyle\rho_{ee}(t)|e\rangle\langle e|+\rho_{eg}(t)|e\rangle\langle g|$
(2) $\displaystyle+$ $\displaystyle\rho_{ge}(t)|g\rangle\langle
e|+\rho_{gg}(t)|g\rangle\langle g|\;,$

with the identities $\rho_{ge}^{*}=\rho_{eg}$, and $\rho_{ee}+\rho_{gg}=1$.
Using the approximation of independent rates of variation, the master equation
governing the time-evolution of the density matrix is obtained by
independently adding up, in the time-evolution of the matrix elements of
$\rho$, terms due to the radiation bath, to the incident field, and the
damping terms responsible for spontaneous emission.Cohen_Tannoudji_Book1992

For the model described above, the master equations characterizing the
dynamics of the density matrix operator or optical Bloch equations in the
rotating wave approximation can then be written as:

$\begin{bmatrix}\dot{\rho}_{ee}\\\ \dot{\rho}_{gg}\\\ \dot{\rho}_{ge}\\\
\dot{\rho}_{eg}\end{bmatrix}=\begin{bmatrix}-\Gamma&0&-i\frac{\Omega_{x}(t)}{2}&i\frac{\Omega_{x}(t)}{2}\\\
\Gamma&0&i\frac{\Omega_{x}(t)}{2}&-i\frac{\Omega_{x}(t)}{2}\\\
-i\frac{\Omega_{x}(t)}{2}&i\frac{\Omega_{x}(t)}{2}&i\Delta-\frac{\Gamma}{2}&0\\\
i\frac{\Omega_{x}(t)}{2}&-i\frac{\Omega_{x}(t)}{2}&0&-i\Delta-\frac{\Gamma}{2}\end{bmatrix}\begin{bmatrix}\rho_{ee}\\\
\rho_{gg}\\\ \rho_{ge}\\\ \rho_{eg}\end{bmatrix}$ (3)

##  III Methods

###  III.1 Emission Spectrum

We will calculate the emission spectrum of the TLS that corresponds to the
excitation probability of the detector in the narrow-band detector approach
where the detector is modeled by a two-level absorber with a very sharp
transition frequency.Scully_Zubairy_book1997  At a long time $T$, the emission
spectrum can be expressed as:

$\displaystyle P(\omega)$ $\displaystyle=$ $\displaystyle 2A^{2}$
$\displaystyle\times$
$\displaystyle\mathrm{Re}\left\\{\int_{0}^{T}dt\int_{0}^{T-t}d\theta\langle\sigma_{+}(t+\theta)\sigma_{-}(t)\rangle
exp\left[-i\omega\theta\right]\right\\}.$

Here, $A$ is a constant independent of the driving field parameters that does
not affect the spectral shape but only affects the absolute scale of the
spectrum. $\sigma_{-}(t)$ and $\sigma_{+}(t+\theta)$ are the time-dependent
operators of the TLS in the Heisenberg representation, and the angled brackets
represent the expectation values that are taken with respect to the initial
state.
To evaluate the two-time correlation function
$\langle\sigma_{+}(t+\theta)\sigma_{-}(t)\rangle$, it is typically rewritten
as a single-time expectation value RF_Mollow_PhysRev1969 ;
Scully_Zubairy_book1997 ; Loudon_book1983 :

$\langle\sigma_{+}(t+\theta)\sigma_{-}(t)\rangle=\mathrm{Tr}\left[\rho^{\prime}(t+\theta)\sigma_{+}\right].$
(5)

Here, $\sigma_{+}$ and $\sigma_{-}$ are the time-independent operators in the
Schrödinger picture and $\rho^{\prime}(t+\theta)$ is obtained from the
original density matrix operator by $\rho^{\prime}(t)=\sigma_{-}\rho(t)$ at
time $t$ and then evolved under the same master equations (3) from time $t$ up
to time $t+\theta$.

Taking advantage of expression (5), the emission spectrum is calculated
numerically using the following recipe. The time axis is discretized into
equal time slices of width $\Delta t=\tau/N_{t}$. Where $N_{t}$ is the number
of time slices in a pulse interval of width $\tau$. Starting at time $t=0$
where the initial conditions are known
($\rho_{ee}=1,\;\rho_{gg}=0,\;\rho_{eg}=0,\;\rho_{ge}=0$), we integrate the
master equation to obtain the matrix elements
$\rho_{ee},\;\rho_{eg},\;\rho_{ge},\;\rho_{gg}$ from $t$ to $t+\Delta t$,
first for a freely evolving TLS with Rabbi frequency $\Omega_{x}(t)=0$ until
time $\tau-t_{\pi}$, then from time $\tau-t_{\pi}$ to time $\tau$, in the
presence of the driving field ($\Omega_{x}(t)=\Omega$). The matrix elements of
$\rho$ at the end of this pulse interval are then used as initial values for
the next pulse interval and the process is repeated for a number $N_{p}$ of
consecutive pulse intervals, resulting in the knowledge of $\rho(t)$ and thus
$\rho^{\prime}(t)$ for $t\in[0,T]$ with $T=N_{p}\tau$. This is then followed
of by the integration of the master equation starting from each time
$t\in[0,T]$ to produce $\rho^{\prime}(t+\theta)$ for $\theta\in[0,T-t]$. From
this, we obtain the the correlation function
$\langle\sigma_{+}(t+\theta)\sigma_{-}(t)\rangle$. Finally, we perform the
Fourier transform with respect to $\theta$ and the integration over $t$ to
obtain $P(\omega)$ which is our emission spectrum. Throughout this integration
process, we use a time-dependent detuning that is obtained by generating a
random Gaussian distribution with mean value $\Delta_{0}$, standard deviation
$\sigma_{\Delta}$ and a correlation time $\tau_{c}$.

###  III.2 Two-Photon Interference

After studying the emission spectrum, we next consider a HOM-type two-photon
interference operation between two distant quantum emitters, each in its own
noisy environment resulting in different inhomogeneously broadened spectral
signatures. Photons from emitter $E_{1}$ with mean detuning $\Delta_{01}$ and
standard deviation $\sigma_{\Delta 1}$ and from emitter $E_{2}$ with mean
detuning $\Delta_{02}$ and standard deviation $\sigma_{\Delta 2}$, at
spacetime locations $1$ and $2$ respectively, are sent to the input ports of a
$50:50$ beam splitter and then measured at detectors $D_{1}$ and $D_{2}$ at
spacetime locations $3$ and $4$ beyond the output ports of the beam splitter.
The emitters can each be independently modeled by the Hamiltonian (1). We want
to evaluate the second order coherence equivalent to the intensity correlation
at the detectors $D_{1}$ and $D_{2}$ in the presence of the control protocols
made of identical finite width pulses driving the respective emitters $E_{1}$
and $E_{2}$. This intensity correlation is:

$G^{(2)}_{34}(t,\theta)=\langle
a_{3}^{\dagger}(t)a_{4}^{\dagger}(t+\theta)a_{4}(t+\theta)a_{3}(t)\rangle.$
(6)

$a_{i}^{\dagger}(t)$ ($a_{i}(t)$) is the creation (destruction) operator of a
photon at detector $i$. From (6), we extract the intensity correlation
corresponding to the measured cross-correlation in the Hanburry Brown and
Twiss setupHanburryBrownTwiss_1956 ; KirazAtature_PRA_2004 :

$g^{(2)}_{34}(\theta)=\lim_{T\to\infty}\int_{0}^{T}G_{34}^{(2)}(t,\theta)\;dt.$
(7)

This correlation function is rewritten in terms of the two-time correlation
functions at the emitters denoted by the $i$ index,
$g_{i}(t,\theta)=\langle\sigma_{+i}(t)\sigma_{-i}(t+\theta)\rangle$, that can
then be expressed as single-time expectation values similarly to (5)
RF_Mollow_PhysRev1969 ; Scully_Zubairy_book1997 ; Loudon_book1983 ;
Scully_Zubairy_book1997 ; Fotso_JPhysB2019 ; Fotso_TPI_PRB_2019 . To evaluate
$g_{i}(t,\theta)$, we can then use the same procedure employed in
Ref.Fotso_TPI_PRB_2019  where the master equation (3) is integrated on the
discretized time axis following steps similar to those highlighted above for
the emission spectrumFotsoDobrovitski_Absorption ; Fotso_JPhysB2019 .

###  III.3 Ensemble of Quantum Emitters

Finally, we consider the case of an ensemble of two-level systems with each
emitter sitting in its own specific environment that sets its emission
frequency to be independent of other emitters’ in the ensemble. Here, we
consider the case where the ensemble has emitters with static detunings
distributed randomly according to a Gaussian distribution. The ensemble is
assumed to be dilute so that we can neglect dipole-dipole interactions between
member emitters. In this situation, often encountered in spectroscopy
experiments, the emission frequency of the ensemble can be obtained by adding
up contributions from individual emitters. For a typical ensemble, this
emission spectrum may become broadly spread out without clearly identifiable
spectral featuresRF_Heitler_Book1960 . This situation can be illustrated in a
manner similar to that of FIG.1 where instead of a time axis to track the
individual emitter, with snapshots at different times, one observes at a given
moment multiple quantum emitters in the ensemble. We consider this system when
it is placed under the influence of our finite-width periodic pulse sequence
and we aim to assess the effect on the emission spectrum of the ensemble. Each
emitter in the ensemble can be described by the Hamiltonian (1) with its
detuning $\Delta$ with respect to the pulse carrier frequency. The
contributions to the emission spectrum of the individual emitters are obtained
following the procedure highlighted in section (III.1).

Figure 3: Emission spectrum for the fluctuating quantum emitter following a
Gaussian random distribution of $\Delta(t)$ with $\Delta_{0}=3.0$ for
$\tau=0.1$ (green), $\tau=0.2$ (blue), $\tau=0.3$ (magenta) after a total time
$t=2.4$ in all cases with Rabbi frequency $\Omega=35$. The dashed brown line
shows the emission spectrum of the system in the absence of the control pulse
sequence. Figure 4:  Emission spectrum of the noisy TLS under a periodic pulse
sequence with inter-pulse delay $\tau=0.3$ and Rabbi frequency $\Omega=35.$
(a) For fixed variance ($\sigma_{\Delta}=4.0$ ) with average detuning values
$\Delta_{0}=1.0$ (black), $\Delta_{0}=2.0$ (red), $\Delta_{0}=3.0$ (green),
$\Delta_{0}=4.0$ (blue). (b) For fixed average detuning value $\Delta_{0}=3.0$
with standard deviation $\sigma_{\Delta}=1.0$ (green), $\sigma_{\Delta}=2.0$
(blue), $\sigma_{\Delta}=3.0$ (red), $\sigma_{\Delta}=4.0$ (orange),
$\sigma_{\Delta}=6.0$ (brown).

##  IV Results

###  IV.1 Emission Spectrum

We consider random Gaussian distributions with a correlation time that is of
the order of the inter-pulse delay or less. We find that our emission spectra
have little dependence on this parameter and so in what follows our results
are presented for $\tau_{c}\sim 0.03$. FIG.3 presents the emission spectrum
for a TLS with $\Delta(t)$ such that the average value is $\Delta_{0}=3.0$ and
the standard deviation is $\sigma_{\Delta}=4.0$. The solid green line
corresponds to inter-pulse time delay of $\tau=0.1$, the blue line to
$\tau=0.2$, and the pink line to $\tau=0.3$. The Rabbi frequency is
$\Omega=35$ for all pulse sequences.

The dashed brown line shows the emission spectrum of this system measured over
the same duration when the system is not subject to any control field.
Clearly, the pulse sequence produces on this noisy system, a spectrum similar
to that reported in the case of a static detuning. The protocol maintains
nearly $50\%$ of the spectral weight at the pulse carrier frequency. The
controlled spectrum also features satellite peaks at integer multiples of
$\pm\pi/\tau$ with spectral weights suppressed away from the central peak. The
bulk of the emission spectrum is refocused to the pulse carrier frequency even
for fairly broad pulses.

Next, we examine this emission spectrum as a function of random distributions
of detunings (as a function of the mean value and the standard deviation of
the distribution or the typical width of the distribution). FIG.4-(a) shows
the emission spectrum under the same pulse sequence of period $\tau=0.3$ and
Rabbi frequency $\Omega=35$ for the same standard deviation
$\sigma_{\Delta}=4.0$ with average detuning value $\Delta_{0}=1.0$ (black
line), $\Delta_{0}=2.0$ (red line), $\Delta_{0}=3.0$ (green line),
$\Delta_{0}=4.0$ (blue line). FIG.4-(b) shows the emission spectrum under the
same pulse sequence (inter-pulse delay $\tau=0.3$ and Rabbi frequency
$\Omega=35$), for fixed average detuning value $\Delta_{0}=3.0$ with standard
deviation $\sigma_{\Delta}=1.0$ (green line), $\sigma_{\Delta}=2.0$ (blue
line), $\sigma_{\Delta}=3.0$ (red line), $\sigma_{\Delta}=4.0$ (orange line),
$\sigma_{\Delta}=6.0$ (brown line). The lineshape is overall preserved for a
broad range of parameters. For the narrower distribution and for the smaller
average detuning, the central peak contains more of the spectral weight. The
refocusing of the spectral peak is deteriorated for broader distributions and
for large average detuning values. However, we observe that overall, the
protocol remains effective as long as the distribution of detunings is such
that $\Delta\times\tau\lesssim 1$ for most of the detuning values.

Figure 5: (a) Schematic representation of the two-photon interference
operation between two distant emitters at spacetime locations $1$ and $2$ with
fluctuating emission frequencies characterized by $E_{1}(t)$ and $E_{2}(t)$.
Photons from the emitters are sent to a 50:50 beam splitter and then measured
at detectors $D_{1}$ and $D_{2}$ at locations $3$ and $4$. Cross-correlation
function at the detectors as a function of the delay time for two emitters
with Gaussian random detunings with average values $\Delta_{01}=4.0$ and
$\Delta_{02}=-3.0$ (blue solid line), $\Delta_{01}=4.0$ and $\Delta_{02}=-4.0$
(red dotted line), $\Delta_{01}=5.0$ and $\Delta_{02}=-4.0$ (green dashed
line) in the absence of any control protocol (b) and under a periodic pulse
sequence with inter-pulse delay $\tau=0.3$ (c). The standard deviation is
$\sigma_{\Delta 1}=\sigma_{\Delta 2}=6.0$ in all cases.

###  IV.2 Two-Photon Interference

After studying the control of the emission spectrum of the isolated quantum
emitter in a noisy environment, we consider the two-photon interference
operation between two such emitters as depicted schematically in FIG.5-(a).
The cross-correlation function at the detectors $g^{(2)}_{34}(\theta)$ as a
function of delay time $\theta$ is shown in FIG.5-(b) and FIG.5-(c)
respectively without control protocols and when the two emitters are driven by
a pulse sequence of period $\tau=0.3$ with Rabbi frequency $\Omega=35$. The
figures show $g^{(2)}_{34}(\theta)$ as a function of $\theta$ for pairs of
emitters with average detuning values of $\Delta_{01}=4.0$ and
$\Delta_{02}=3.0$ (blue solid line), $\Delta_{01}=4.0$ and $\Delta_{02}=-4.0$
(red dotted line), $\Delta_{01}=5.0$ and $\Delta_{02}=-4.0$ (green dashed
line). Note that $g^{(2)}_{34}(\theta)$ vanishes in both cases at $\theta=0$.
However, in the absence of the control protocol it also vanishes periodically
at times that are integer multiples of $\sim\pi/(\Delta_{01}-\Delta_{02})$ and
overall decreases in magnitude with increasing delay times, while it stays
finite for all other times in the presence of the control protocol including
for emitters in significantly different environments
$\Delta_{01}-\Delta_{02}\sim 10\Gamma$. Where $\Gamma$ is the free emission
lifetime of an individual emitter.

Figure 6: Emission spectrum of an ensemble of quantum emitters with the
emission frequencies of individual emitters spread across a random Gaussian
distribution of detunings with standard deviation $\sigma_{\Delta}=15.0$ and
average value $\Delta_{ave}=0$. The dashed red line shows the emission
spectrum of the system in the absence of any control protocol while the blue
solid line shows the emission spectrum when the system is under the influence
of a periodic sequence of $\pi$ pulses with inter-pulse delay $\tau=0.2$ and
with Rabbi frequency $\Omega=50$ after 8 pulses. The spectrum with no control
has been rescaled to have the same maximum as the spectrum under the pulse
sequence.

###  IV.3 Ensemble of Two-Level Systems

We now consider a dilute ensemble of quantum emitters described by two-level
systems. An emitter $i$ in the ensemble has its own particular environment and
its detuning is set to a value $\Delta_{i}$. The $\Delta_{i}$’s follow a
random Gaussian distribution with average value $\Delta_{ave}=0$ and a
standard deviation $\sigma_{\Delta}=15.0$. FIG.6 shows the emission spectrum
of this ensemble when it is placed under the influence of a pulse sequence of
period $\tau=0.2$ with Rabbi frequency $\Omega=50$. The spectrum is calculated
for 8 pulses but the general lineshape is established after 2 to 4 pulses and
further time mostly results in larger peak amplitudes. Note that the amplitude
of the spectrum in the absence of the control protocol is rescaled to match
that of the spectrum in the presence of the control fields. Overall, the
emission spectrum that has a broad Gaussian lineshape in the absence of the
control protocol is refocused by the control protocol to result in a lineshape
with a central peak at the pulse carrier frequency ($\omega=0$ in the rotating
frame) that has the linewidth $\Gamma$ of an individual isolated emitter
flanked by satellite peaks at integer multiples of $\pm\pi/\tau$.

##  V conclusion

We have examined spectral properties of quantum emitters in noisy environments
manifested by a random Gaussian distribution of detunings, as a function of
time for individual quantum emitters and by a random distribution of detunings
across an ensemble of two-level systems. For individual emitters, we
characterized the emission spectrum under the effect of a periodic sequence of
finite-width pulses. Our results indicate that for a broad range of
parameters, the emission spectrum of a noisy quantum emitter can be well
controlled by the pulse sequence. When two different such noisy individual
two-level systems are used in a HOM-type two-photon interference, we find that
the periodic sequence of finite width pulses effectively restores two-photon
indistinguishability between two the two spectrally different systems.
Finally, for an ensemble of quantum emitters, with individual emitters that
have randomly distributed emission frequencies so that the ensemble would
produce an inhomogeneously broadened emission spectrum, we show that the
control protocol can refocus the emission spectrum to a lineshape with a
central peak that has the linewidth of an individual quantum emitters. These
results demonstrate for a model of spectrally noisy two-level system the
ability to optimize spectral properties with an external control field in the
form of a pulse sequence.

## Acknowledgments

This work is supported by the National Science Foundation under Grant No.
PHY-2014023. We thank Tim Schröder for useful conversations.

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## Figure Legends

Figure 1: A schematic representation of the flow of our method. The boxes
colored blue are items provided by the investigator (GWAS SNP effect sizes,
the frequency of the GWAS SNPs across populations, and a environmental
variable). The boxes colored red make use of random SNPs sampled to match the
GWAS set as described in “Choosing null SNPs” in the methods section. For each
box featuring a calculated quantity a set of equation numbers are provided for
the relevant calculation. The Z score uses the untransformed genetic values,
rather than the transformed genetic values, but this relationship is not
depicted in the figure for the sake of readability.  Figure 2: Power of our
statistics as compared to alternative approaches (A) across a range of
selection gradients ($\delta$) of latitude, and when we hold $\delta$ constant
at 0.14 and (B) decrease $\phi$, the genetic correlation between the trait of
interest and the selected trait, (C) vary the number of loci, and (D) vary the
number of loci while holding the fraction of variance explained constant.
Bottom panels show power of the Z-test and $Q_{X}$approaches to detect
selection affecting (E) a single population, and (F) multiple populations in a
given region. See main text for simulation details. Figure 3: Histogram of the
empirical null distribution of $Q_{X}$ for each trait, obtained from genome-
wide resampling of well matched SNPs. The mean of each distribution is marked
with a vertical black bar and the observed value is marked by a red arrow. The
expected $\chi^{2}_{M-1}$ density is shown as a black curve. Figure 4: The two
components of $Q_{X}$ for the height dataset, as described by the left and
right terms in (14). The null distribution of each statistic is shown as a
histogram. The mean value is shown as a black bar, and the observed value as a
red arrow. Figure 5: Visual representation of outlier analysis at the regional
and individual population level for (A) Height, (B) Skin Pigmentation, and (C)
Body Mass Index. For each geographic region we plot the expectation of the
regional average, given the observed values in the rest of the dataset as a
grey dashed line. The true regional average is plotted as a solid bar, with
darkness and thickness proportional to the regional Z score. For each
population we plot the observed value as a colored circle, with circle size
proportional to the population specific Z score. For example, in (A), one can
see that estimated genetic height is systematically lower than expected across
Africa. Similarly, estimated genetic height is significantly higher (lower) in
the French (Sardinian) population than expected, given the values observed for
all other populations in the dataset. Figure 6: Visual representation of
outlier analysis at the regional and individual population level for (A) Type
2 Diabetes, (B) Crohn’s Disease, and (C) Ulcerative Colitis. See Figure 5 for
explanation.  Figure 7: Estimated genetic height (A) and skin pigmentation
score (B) plotted against winter PC2 and absolute latitude respectively. Both
correlations are significant against the genome wide background after
controlling for population structure (Table 2).  Figure 8: Estimated genetic
risk score for Crohn’s Disease (A) and Ulcerative Colitis (B) risk plotted
against summer PC2. Both correlations are significant against the genome wide
background after controlling for population structure (Table 2). Since a large
proportion of SNPs underlying these traits are shared, we note that these
results are not independent.

## Tables

Table 1: The contribution of each geo-climatic variable to each of our four principal components, scaled such that the absolute value of the entries in each column sum to one (up to rounding error). We also show for each principal component the percent of the total variance across all eight variables that is explained by the PC. Geo-Climatic Variable | SUMPC1 | SUMPC2 | WINPC1 | WINPC2
---|---|---|---|---
Latitude | -0.16 | -0.10 | -0.17 | -0.01
Longitude | 0.02 | 0.12 | 0.04 | 0.05
Maximum Temp | 0.24 | -0.08 | 0.17 | -0.03
Minimum Temp | 0.24 | 0.07 | 0.17 | 0.08
Mean Temp | 0.25 | -0.03 | 0.17 | 0.03
Precipitation Rate | -0.01 | 0.16 | 0.07 | 0.32
Relative Humidity | -0.06 | 0.21 | -0.06 | 0.34
Short Wave Radiation Flux | -0.03 | -0.22 | 0.15 | -0.13
Percent of Variance Explained | 38% | 35% | 58% | 20%
Table 2: Climate Correlations and $Q_{X}$ statistics for all six phenotypes in
the global analysis. We report $sign(\beta)r^{2}$, for the correlation
statistics, such that they have an interpretation as the fraction of variance
explained by the environmental variable, after removing that which is
explained by the relatedness structure, with sign indicating the direction of
the correlation. P-values are two–tailed for $r^{2}$ and upper tail for
$Q_{X}$. Values for $\beta$ and $\rho$ are reported in Tables S15 and S16.

Phenotype | SUMPC1 | SUMPC2 | WINPC1 | WINPC2 | Latitude | $Q_{X}$
---|---|---|---|---|---|---
Height | $-0.03\ (0.21)$ |  $10^{-5}$ (0.99) | $-0.008\ (0.52)$ | $\mathbf{0.086\ (0.035)}$ | 0.009 (0.50) | $\mathbf{86.9\ (0.002)}$
Skin Pigmentation | 0.061 (0.073) | 0.003 (0.69) | 0.048 (0.13) | $-0.008\ (0.51)$ | $\mathbf{-0.085\ (0.038)}$ | $\mathbf{79.1\ (0.006)}$
Body Mass Index | $-0.034\ (0.19)$ | 0.001 (0.82) | $-0.022\ (0.31)$ | 0.044 (0.14) | 0.031 (0.22) | $67.2\ (0.087)$
Type 2 Diabetes | 0.014 (0.40) | 0.012 (0.45) | 0.025 (0.27) | $-0.006\ (0.573)$ | $-0.05\ (0.11)$ | 39.3 (0.902)
Crohn’s Disease | 0.07 (0.062) | $\mathbf{-0.099\ (0.022)}$ | 0.0001 (0.94) | $\mathbf{-0.09\ (0.039)}$ | 0.01 (0.55) | 47.1 (0.68)
Ulcerative Colitis | 0.03 (0.21) | $\mathbf{-0.087\ (0.034)}$ | 0.004 (0.67) | $-0.049\ (0.12)$ | 0.01 (0.43) | 48.58 (0.61)

Table 3: $Q_{X}$ statistics and their empirical p-values for each of our six
traits in each of the seven geographic regions delimited by [61]. The
theoretical expected value of the statistic under neutrality for each region
is equal to $M-1$, where $M$ is the number of populations in the region. We
report the value of $M-1$ next to each region for reference.

| Europe (7) | Middle East (3) | Central Asia (8) | East Asia (16) | Americas (4) | Oceania (1) | Africa (6)
---|---|---|---|---|---|---|---
Height | $\mathbf{32.6\ (<10^{-4})}$ | 7.3 (0.07) | $\mathbf{15.5\ (0.05)}$ | 18.2 (0.33) | 4.2 (0.43) | 0.007 (0.94) | 5.4 (0.53)
Skin Pigmentation | 9.7 (0.22) | $\mathbf{9.6\ (0.026)}$ | $\mathbf{23.4\ (0.002)}$ | 13.8 (0.62) | 1.3 (0.89) | 0.38 (0.57) | $\mathbf{16.2\ (0.011)}$
Body Mass Index | 9.1 (0.24) | 1.6 (0.66) | 9.3 (0.32) | $\mathbf{28.4\ (0.03)}$ | $\mathbf{13.1\ (0.016)}$ | 1.2 (0.31) | 1.9 (0.94)
Type 2 Diabetes | 2.0 (0.96) | 0.90 (0.83) | 8.1 (0.43) | 7.5 (0.96) | 8.0 (0.13) | 2.5 (0.15) | 2.5 (0.88)
Crohn’s Disease | 6.6 (0.47) | 0.87 (0.84) | 7.56 (0.48) | 15.5 (0.52) | 1.3 (0.88) | 2.5 (0.13) | 2.6 (0.82)
Ulcerative Colitis | 8.4 (0.30) | 2.6 (0.48) | 10.9 (0.21) | 9.2 (0.907) | 0.43 (0.986) | 2.6 (0.12) | 3.5 (0.77)

## Supplementary Figure Legends

Figure S1: Power of tests described in the main text to detect a signal of
selection on the mapped genetic basis of skin pigmentation [66] as an
increasing function of the strength of selection (A), and a decreasing
function of the genetic correlation between skin pigmentation and the selected
trait with the effect of selection held constant at $\delta=0.13$ (B). Figure
S2: Power of tests described in the main text to detect a signal of selection
on the mapped genetic basis of BMI [73] as an increasing function of the
strength of selection (A), and a decreasing function of the genetic
correlation between BMI and the selected trait with the effect of selection
held constant at $\delta=0.07$ (B). Figure S3: Power of tests described in the
main text to detect a signal of selection on the mapped genetic basis of T2D
[74] as an increasing function of the strength of selection (A), and a
decreasing function of the genetic correlation between height and the selected
trait with the effect of selection held constant at $\delta=0.08$ (B). Figure
S4: Power of tests described in the main text to detect a signal of selection
on the mapped genetic basis of CD [25] as an increasing function of the
strength of selection (A), and a decreasing function of the genetic
correlation between CD and the selected trait with the effect of selection
held constant at $\delta=0.05$ (B). Figure S5: Power of tests described in the
main text to detect a signal of selection on the mapped genetic basis of UC
[25] as an increasing function of the strength of selection (A), and a
decreasing function of the genetic correlation between UC and the selected
trait with the effect of selection held constant at $\delta=0.05$ (B). Figure
S6: The two components of $Q_{X}$ for the skin pigmentation dataset, as
described by the left and right terms in (14). The null distribution of each
component is shows as a histogram. The expected value is shown as a black bar,
and the observed value as a red arrow. Figure S7: The two components of
$Q_{X}$ for the BMI dataset, as described by the left and right terms in (14).
The null distribution of each component is shows as a histogram. The expected
value is shown as a black bar, and the observed value as a red arrow. Figure
S8: The two components of $Q_{X}$ for the T2D dataset, as described by the
left and right terms in (14). The null distribution of each component is shows
as a histogram. The expected value is shown as a black bar, and the observed
value as a red arrow. Figure S9: The two components of $Q_{X}$ for the CD
dataset, as described by the left and right terms in (14). The null
distribution of each component is shows as a histogram. The expected value is
shown as a black bar, and the observed value as a red arrow. Figure S10: The
two components of $Q_{X}$ for the UC dataset, as described by the left and
right terms in (14). The null distribution of each component is shows as a
histogram. The expected value is shown as a black bar, and the observed value
as a red arrow. Figure S11: The genetic values for height in each HGDP
population plotted against the measured sex averaged height taken from [126].
Only the subset of populations with an appropriately close match in the named
population in [126]’s Appendix I are shown, values used are given in
Supplementary table S1 Figure S12: The genetic skin pigmentation score for a
each HGDP population plotted against the HGDP populations values on the skin
pigmentation index map of Biasutti 1959. Data obtained from Supplementary
table of [68]. Note that Biasutti map is interpolated, and so values are known
to be imperfect. Values used are given in Supplementary table S2 Figure S13:
The genetic skin pigmentation score for a each HGDP population plotted against
the HGDP populations values from the [67] mean skin reflectance (685nm) data
(their Table 6). Only the subset of populations with an appropriately close
match were used as in the Supplementary table of [68]. Values and populations
used are given in Table S2 Figure S14: The distribution of genetic height
score across all 52 HGDP populations. Grey bars represent the $95\%$
confidence interval for the genetic height score of an individual randomly
chosen from that population under Hardy-Weinberg assumptions Figure S15: The
distribution of genetic skin pigmentation score across all 52 HGDP
populations. Grey bars represent the $95\%$ confidence interval for the
genetic skin pigmentation score of an individual randomly chosen from that
population under Hardy-Weinberg assumptions Figure S16: The distribution of
genetic BMI score across all 52 HGDP populations. Grey bars represent the
$95\%$ confidence interval for the genetic BMI score of an individual randomly
chosen from that population under Hardy-Weinberg assumptions Figure S17: The
distribution of genetic T2D risk score across all 52 HGDP populations. Grey
bars represent the $95\%$ confidence interval for the genetic T2D risk score
of an individual randomly chosen from that population under Hardy-Weinberg
assumptions Figure S18: The distribution of genetic CD risk score across all
52 HGDP populations. Grey bars represent the $95\%$ confidence interval for
the genetic CD risk score of an individual randomly chosen from that
population under Hardy-Weinberg assumptions

Figure S19: The distribution of genetic UC risk score across all 52 HGDP
populations. Grey bars represent the $95\%$ confidence interval for the
genetic UC risk score of an individual randomly chosen from that population
under Hardy-Weinberg assumptions

## Supplementary Tables

Table S1: Genetic height scores as compared to true heights for populations
with a suitably close match in the dataset of [126]. See Figure S11 for a plot
of genetic height score against sex averaged height. Table S2: Genetic skin
pigmentation score as compared to values from Biasutti [127, 68] and [67]. We
also calculate a genetic skin pigmentation score including previously reported
associations at KITLG and OCA2 for comparisson. See also Figures S12 and S13.
Table S3: Conditional analysis at the regional level for the height dataset
Table S4: Conditional analysis at the individual population level for the
height dataset Table S5: Conditional analysis at the regional level for the
skin pigmentation dataset Table S6: Conditional analysis at the individual
population level for the skin pigmentation dataset Table S7: Condtional
analysis at the regional level for the BMI dataset Table S8: Conditional
analysis at the individual population level for the BMI dataset Table S9:
Conditional analysis at the regional level for the T2D dataset. Table S10:
Conditional analysis at the individual population level for the T2D dataset.
Table S11: Conditional analysis at the regional level for the CD dataset.
Table S12: Conditional analysis at the individual population level for the CD
dataset. Table S13: Conditional analysis at the regional level for the UC
dataset. Table S14: Conditional analysis at the individual population level
for the UC dataset Table S15: Corresponding $\beta$ statistics for all
analyses presented in Table 2. Table S16: Corresponding $\rho$ statistics for
all analyses presented in Table 2.