# Dynamics of translational and rotational thermalization
of AlF molecules via collisions with cryogenic helium

M. Karra  Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6,
14195 Berlin, Germany  M. T. Cretu  Fritz-Haber-Institut der Max-Planck-
Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany  B. Friedrich  Fritz-
Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin,
Germany  S. Truppe  Fritz-Haber-Institut der Max-Planck-Gesellschaft,
Faradayweg 4-6, 14195 Berlin, Germany  G. Meijer  Fritz-Haber-Institut der
Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany  J. Pérez-Ríos
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195
Berlin, Germany

###### Abstract

We investigated helium-mediated translational and rotational thermalization of
the aluminum monofluoride (AlF) molecule at cryogenic temperatures via a new
_ab initio_ potential energy surface (PES) and quantum multichannel scattering
theory. Our examination of the elastic and rotationally inelastic channels
revealed that helium is an efficient quencher of AlF at temperatures relevant
to buffer gas cooling experiments ($\sim 1$ mK to $10$ K). We also showed that
this conclusion is robust against possible inaccuracies of the PES.

##  I Introduction

The ability to cool gaseous ensembles of molecules to cold ($1$ mK to $10$ K)
and ultracold ($\lesssim$1 mK) temperatures has been impacting research areas
as diverse as testing the standard model of particle physics Safronova  _et
al._ (2018), quantum-logic spectroscopy Mur-Petit  _et al._ (2012); Shi  _et
al._ (2013); Wolf  _et al._ (2016), action-spectroscopy Puri  _et al._ (2014);
BrÃ¼nken  _et al._ (2017); McGuire  _et al._ (2020), chemical reaction
dynamics Doyle  _et al._ (2004a); Krems (2008); Carr  _et al._ (2009); Schnell
and Meijer (2009); Quéméner and Julienne (2012); Balakrishnan (2016); Ríos
(2020), as well as quantum computing DeMille (2002); Karra  _et al._ (2016);
Yu  _et al._ (2019) and quantum simulation Madison  _et al._ (2013); Blackmore
_et al._ (2018). In particular, ultracold _polar_ molecules afford tunability
of interactions, coveted for applications in quantum information protocols Zhu
_et al._ (2013); Sawant  _et al._ (2020) and the simulation of many-body
Hamiltonians Micheli  _et al._ (2006); Herrera and Krems (2011).

Ultracold molecules can be produced via direct or indirect cooling techniques.
The _indirect_ techniques rely on photo-association Fioretti  _et al._ (1998);
Bohn and Julienne (1999); Jones  _et al._ (2006); Pérez-Ríos  _et al._ (2015);
Blasing  _et al._ (2016) or magneto-association Köhler  _et al._ (2006); Chin
_et al._ (2010) of a pair of ultracold atoms and have proved especially
successful for generating bi-alkali ultracold molecules Ni  _et al._ (2008);
Danzl  _et al._ (2008); Ye  _et al._ (2018); De Marco  _et al._ (2019); Yang
_et al._ (2019); Voges  _et al._ (2020). The yield of the indirect techniques
is limited by the number of available ultracold atoms that serve as precursors
of the ultracold molecules. However, bi-alkali molecules are prone to
undergoing sticky collisions: these correspond to the formation of an
intermediate complex whose lifetime can be long enough to amount to a loss of
the molecules formed Mayle  _et al._ (2012); Christianen  _et al._ (2019);
Gregory  _et al._ (2019).

On the other hand, _direct_ cooling techniques are based on the dissipation of
translational – and internal – energy of pre-existing molecules via
thermalization with a cryogenic buffer gas Weinstein  _et al._ (1998), Stark
Bethlem  _et al._ (1999) or Zeeman Vanhaecke  _et al._ (2007); Narevicius  _et
al._ (2008) slowing of molecules cooled by a supersonic expansion, Sisyphus
cooling Zeppenfeld  _et al._ (2012) or, most recently, laser cooling Tarbutt
(2018); Hudson (2020). Long considered impractical if not impossible Doyle
_et al._ (2004b), laser cooling of molecules has led, for the first time, to
the creation of ultracold molecular ensembles Mitra  _et al._ (2020);
Augenbraun  _et al._ (2020a, b). As laser cooling of molecules requires near-
unity Franck-Condon factors between the molecular ground and excited
electronic states Di Rosa (2004); Shuman  _et al._ (2010), only a handful of
molecules have been laser-cooled so far Ivanov  _et al._ (2019, 2020);
Augenbraun  _et al._ (2020c); Kłos and Kotochigova (2020).

Recently, we identified the aluminum monofluoride (AlF) molecule as a
promising candidate for laser cooling. Its virtues include a large photon
scattering rate and favorable Franck-Condon factors Truppe  _et al._ (2019);
Hofsäss  _et al._ (2021); Doppelbauer  _et al._ (2020). Moreover, we
demonstrated that AlF molecules can be efficiently produced in a pulsed
buffer-gas source, with a yield of $\gtrsim 10^{12}$ molecules per steradian
per pulse Hofsäss  _et al._ (2021). A high yield of cryogenically thermalized
molecules is a key prerequisite for reaching quantum degeneracy in a cooling
step that follows upon laser cooling, namely forced evaporation.

In order to gain insight into the thermalization of AlF by cryogenic helium,
we undertook a study of the quantum dynamics of the elastic and rotationally
inelastic scattering of AlF by He at collision energies ranging from $1$ mK to
$10$ K. We made use of a new and accurate ab initio potential energy surface
(PES) and calculated the relevant scattering observables via the coupled-
channel method. The hyperfine structure of AlF has been omitted as it is
expected to play only a minor role in the collision dynamics. We also discuss
what our findings about the AlF-He system imply about the thermalization in a
buffer gas of other ${}^{1}\Sigma^{+}$ diatomic fluorides, such as MgF and
BaF. While it is known that AlF forms in highly vibrationally excited
levelsRosenwaks  _et al._ (1976), and that the vibrational quenching also
needs to be treated in detail, it is beyond the scope of the present work and
left for a future article.

We further note that, following the discovery of its presence in the envelope
of proto-planetary nebulae Highberger  _et al._ (2001), the AlF molecule and
its spectroscopic properties have been also of astronomical interest. More
recently, the $\gamma$-ray emission from ${}^{26}\textrm{AlF}$ has served as a
stepping stone to elucidating the Galactic sources of ${}^{26}\textrm{Al}$
Kamiński  _et al._ (2018), which is of consequence for understanding the
frequency of core collapse of supernovae Diehl  _et al._ (2006).

##  II Potential energy surface

The potential energy surface of the 27Al19F(X${}^{1}\Sigma^{+}$)+4He(1$S$)
system was calculated by treating AlF(X${}^{1}\Sigma^{+}$) like a
vibrationless rigid rotor. Thus, the PES only depends on two parameters: the
distance $R=|\bf{R}|$ between the center of mass of the molecule and the atom
and the angle $\theta$ between $\bf{R}$ and the internuclear axis $\bf{r}$ of
the AlF molecule, see the inset in Fig. 1. The PES was calculated at the
coupled-cluster level of theory, including singles and doubles excitations;
triples excitations were treated perturbatively [RCCSD(T)] as implemented in
Molpro Werner  _et al._ (2019) using the aug-cc-pv5Z basis set of Dunning
Schuchardt  _et al._ (2007). The Al-F equilibrium distance was fixed to that
accurately determined in a recent experiment Truppe  _et al._ (2019); Liu  _et
al._ (2020), $R_{e}=1.654369$ Å. The superposition error was corrected using
the counterpoise method. A contour plot of the PES is shown in Fig. 1. Note
that the AlF-He attraction is quite weak, as it arises from the van der Waals
interaction between a closed-shell atom and a closed-shell molecule. The PES
exhibits a global minimum of $\sim 22$ cm-1 close to the F atom, lending it a
high anisotropy characteristic of fluorine-containing heteronuclear molecules
interacting with He Tscherbul  _et al._ (2007).

Figure 1: Potential energy surface (PES) of the
AlF(X${}^{1}\Sigma^{+}$)–He(${}^{1}S$) system. The contours show fixed
electronic energies in cm-1 of the system as a function of $R$ and $\theta$,
whose meaning is defined in the inset. See also text.

The PES was calculated for 1500 geometries including 20 angles, 18 of which
corresponded to the Gauss-Legendre quadrature points and to $\theta=0$ and
$\theta=\pi$, and 75 radial points between 1.6 and 30 Å with a larger density
of points for $R$ between 3 and 5 Å. The global minimum of the PES was found
at $\theta=\pi$ and $R=3.8~{}\textrm{\AA}$, in close agreement with previously
reported calculations Gotoum  _et al._ (2011). The technical details of
calculating the PES will be published elsewhere.

For scattering calculations, it is advantageous to expand the PES in a
Legendre series

$V(R,\theta)=\sum_{\lambda=0}^{\lambda_{\text{max}}}V_{\lambda}(R)P_{\lambda}(\cos{\theta})$
(1)

with $P_{\lambda}(\cos\theta)$ Legendre polynomials of degree $\lambda$ and
expansion coefficients

$V_{\lambda}(R)=\left(\frac{2\lambda+1}{2}\right)\int_{0}^{\pi}V(R,\theta)P_{\lambda}(\cos{\theta})\sin{\theta}d\theta$
(2)

that depend solely on the radial distance $R$. Note that $V_{0}(R)$ represents
the only isotropic (spherical) contribution to the PES whereas all other
contributions are anisotropic. The integral in Eq. (2) was performed using the
Gauss-Legendre quadrature scheme employing 16 quadrature points. The
dependence of the first five $V_{\lambda}(R)$ terms on $R$ is shown in Fig. 2.
As expected, the most significant radial term is the spherical one,
$V_{0}(R)$. By including radial terms with up to $\lambda_{\text{max}}=12$ in
the Legendre series expansion of Eq. (1), we ensured a relative error of the
Legendre expansion of only 0.1 $\%$ with respect to the _ab initio_ PES.

Figure 2: The first five radial terms $V_{\lambda}(R)$ of the potential energy
surface of the AlF(X${}^{1}\Sigma^{+}$)-He(1$S$) system, see text.

For cold and ultracold collisions, the long-range tail of the PES plays a
critical role for evaluating the scattering cross sections. Therefore, we
chose a certain $R_{\textrm{mid}}$ for each $V_{\lambda}$ beyond which the
radial coefficient was replaced by the long-range dipole–induced dipole
interaction,

$V_{\lambda}(R\geq R_{\textrm{mid}})=\frac{C_{6}^{\lambda}}{R^{6}}$ (3)

The values of $C_{6}^{\lambda}$ were obtained by fitting the numerical values
from Eq. (2) between $R_{\textrm{mid}}$ and 30 Å. Additionally, a polynomial
interpolation was used to ensure the continuity of the potential and its first
derivative at $R_{\textrm{mid}}$.

##  III Collision dynamics

In evaluating the quantum dynamics of the AlF+He collisions, we make use of
the framework developed by Arthurs, Dalgarno, and Bates Arthurs  _et al._
(1960) for rigid rotor–atom collisions, omitting hyperfine structure.

###  III.1 State-to-state integral cross sections

The state-to-state integral scattering cross section for collisions leading
from an initial state $j$ of the rotor to its final state $j^{\prime}$ is
given by

$\sigma_{j\rightarrow
j^{\prime}}(E_{k})=\frac{4\pi}{k^{2}(2j+1)}\sum_{Jll^{\prime}}(2J+1)|T^{JM}_{j,l;j^{\prime}l^{\prime}}|^{2}$
(4)

where $E_{k}$ is the collision energy, $k$ the wavenumber of the relative
motion atom-molecule motion, $k^{2}=2\mu E_{k}$, with $\mu$ the reduced mass
of the atom-molecule system, and $T^{JM}_{j,l;j^{\prime}l^{\prime}}$ is the
transition matrix that encapsulates the transition probabilities between
different scattering channels characterized by the rotational quantum numbers
$j$ and $j^{\prime}$ and partial waves $l$ and $l^{\prime}$ before and after
the collision. In the absence of external fields, only channels $(j,l)$ and
$(j^{\prime},l^{\prime})$ are coupled that conserve the total angular momentum
$J$ and its projection $M$ on the quantization axis (scattering coordinate
$\bf{R}$). Note that throughout this section, we use atomic units.

In the center-of-mass frame, the Hamiltonian of the atom-molecule system is
given by

$H=-\frac{1}{2\mu}\nabla^{2}_{R}+\frac{\hat{l}^{2}}{2\mu
R^{2}}+V(R,\theta)+H_{\text{rot}}$ (5)

where the first term represents the radial kinetic energy operator along the
scattering coordinate $\bf{R}$, the second term denotes the centrifugal term
of the kinetic energy operator, $V(R,\theta)$ the atom-molecule PES, cf. Sec.
II, and $H_{\text{rot}}$ the rotational Hamiltonian of the molecule. The last
is given by

$H_{\text{rot}}Y_{j}^{m_{j}}(\hat{r})=[B_{0}j(j+1)-D_{0}(j(j+1))^{2}]Y_{j}^{m_{j}}(\hat{r})$
(6)

with $Y_{j}^{m_{j}}(\hat{r})$ the spherical harmonics, $B_{0}$ the rotational
constant of the vibrational ground state of the molecule, $D_{0}$ is the
centrifugal distortion constant, $j$ the rotational quantum number and $m_{j}$
its projection on the direction of the molecular axis,
$\hat{r}=\frac{\bf{r}}{|\bf{r}|}$.

The total energy of the rotor is comprised of its kinetic/collision energy and
the internal energy of the fragment111A fragment represents the distinct
energy levels of the system under consideration in the asymptotic region.
Therefore, a fragment corresponds to many channels in the interaction region.
it corresponds to

$E_{j}=E_{k}+B_{0}j(j+1)-D_{0}(j(j+1))^{2}$ (7)

which is an eigenvalue of the Schrödinger equation

$H\Psi^{JM}_{jl}=E_{j}\Psi^{JM}_{jl}$ (8)

In the coupled basis, preferable in the absence of external fields, the
eigenfunctions of Eq.(8) are given by

$\Psi^{JM}_{jl}=\sum_{j^{\prime}l^{\prime}}\frac{g_{jl,j^{\prime}l^{\prime}}^{JM}(R)}{R}I_{j^{\prime}l^{\prime}}^{JM}(\hat{r},\hat{R})$
(9)

with

$I_{jl}^{JM}(\hat{r},\hat{R})=\sum_{m_{j}m_{l}}C^{JM}_{jm_{j}lm_{l}}|jm_{j}lm_{l}\rangle$
(10)

the angular part of the wave function that entails the coupling of the
molecule-fixed coordinate $\hat{r}$ to the scattering coordinate
$\hat{R}=\frac{\bf{R}}{|\bf{R}|}$ via the Clebsch-Gordan coefficients
$C^{JM}_{jm_{j}lm_{l}}$.

By substituting Eq.(9) into Eq.(8) we obtain

$\displaystyle\bigg{[}-\frac{d^{2}}{dR^{2}}+k_{j^{\prime}}^{2}+\frac{l^{\prime}(l^{\prime}+1)}{R^{2}}\bigg{]}g_{jl,j^{\prime}l^{\prime}}^{JM}(R)=$
$\displaystyle
2\mu\sum_{j^{\prime\prime}l^{\prime\prime}}V^{J}_{j^{\prime\prime}l^{\prime\prime};j^{\prime}l^{\prime}}(R)g_{jl,j^{\prime\prime}l^{\prime\prime}}^{JM}(R)$
(11)

where

$\displaystyle
k_{j^{\prime}}^{2}=2\mu\bigg{[}E_{k}+B_{0}j(j+1)-D_{0}(j(j+1))^{2}-$
$\displaystyle
B_{0}j^{\prime}(j^{\prime}+1)+D_{0}(j^{\prime}(j^{\prime}+1))^{2}\bigg{]}$
(12)

is the wavenumber squared pertaining to the $(j,j^{\prime})$ channel, and

$V^{J}_{j^{\prime}l^{\prime};jl}(R)=\int\int
I^{JM*}_{j^{\prime}l^{\prime}}(\hat{r},\hat{R})V(R,\theta)I^{JM}_{jl}(\hat{r},\hat{R})d\Omega_{r}$
(13)

are the potential matrix elements. By substituting from Eq. (1), we obtain

$\displaystyle
V^{J}_{j^{\prime}l^{\prime};jl}(R)=\sum_{\lambda}^{\lambda_{\textrm{max}}}V_{\lambda}(R)(-1)^{J+l^{\prime}+l}$
$\displaystyle\sqrt{(2j+1)(2l+1)(2j^{\prime}+1)(2l^{\prime}+1)}$
$\displaystyle\begin{pmatrix}l&\lambda&l^{\prime}\\\
0&0&0\end{pmatrix}\begin{pmatrix}j&\lambda&j^{\prime}\\\
0&0&0\end{pmatrix}\begin{Bmatrix}j&\lambda&j^{\prime}\\\
l^{\prime}&J&l\end{Bmatrix}$ (14)

with $(...)$ and $\\{...\\}$ the 3j and 6j symbols, respectively. Thus the
simple form of Eq. (III.1) results by virtue of the Legendre expansion of the
PES, introduced in Sec. II.

The set of coupled differential equations (III.1) is solved numerically. In
the asymptotic region where the interaction potential in negligible compared
to the collision energy, the numerical results can be matched to the expected
analytic asymptotic behavior

$\displaystyle
g_{jl,j^{\prime}l^{\prime}}^{JM}(R\rightarrow\infty)\sim\delta_{jj^{\prime}}\delta_{ll^{\prime}}\sin{\left(k_{j^{\prime}}R-\frac{l^{\prime}\pi}{2}\right)}$
$\displaystyle+\frac{e^{\imath(k_{j^{\prime}}R-l^{\prime}\pi/2)}}{\sqrt{k_{j^{\prime}}}}T^{JM}_{jl,j^{\prime}l^{\prime}}$
(15)

leading to the numerical transition matrix and thus, via Eq.(4), to the state-
to-state cross section.

###  III.2 Diffusion and viscosity transport cross sections

The diffusion and viscosity cross sections are defined by Mott and Massey
(1965)

$\sigma_{D}(E_{k})=\int\frac{d\sigma_{\textrm{el}}(E_{k})}{d\Omega}(1-\cos{\theta})d\Omega$
(16)

and

$\sigma_{\eta}(E_{k})=\int\frac{d\sigma_{\textrm{el}}(E_{k})}{d\Omega}(1-\cos^{2}{\theta})d\Omega$
(17)

where $\frac{d\sigma_{\textrm{el}}(E_{k})}{d\Omega}$ is the elastic
differential cross section, $d\Omega=2\pi\sin\theta d\theta$ the solid angle
element, and $\theta$ the scattering angle. Quantum mechanically, these cross-
sections evaluate to Mott and Massey (1965)

$\sigma_{D}(E_{k})=\frac{4\pi}{k^{2}}\sum_{l=0}^{\infty}(l+1)\sin^{2}{[\delta_{l+1}(E_{k})-\delta_{l}(E_{k})]}$
(18)

and

$\sigma_{\eta}(E_{k})=\frac{4\pi}{k^{2}}\sum_{l=0}^{\infty}\frac{(l+1)(l+2)}{2l+3}\sin^{2}{[\delta_{l+2}(E_{k})-\delta_{l}(E_{k})]}$
(19)

respectively, wherein $\delta_{l}(E_{k})$ is the phase-shift for a given
partial wave $l$ and collision energy $E_{k}$. The quantum elastic scattering
cross-section expressed in terms of the phase shifts boils down to

$\sigma_{\textrm{el}}(E_{k})=\frac{4\pi}{k^{2}}\sum_{l=0}^{\infty}(2l+1)\sin^{2}\delta_{l}$
(20)

Classically, the transport cross-sections are given by Mason and Marrero
(1970); Hirschfelder  _et al._ (1964)

$\displaystyle\sigma^{C}_{D}(E_{k})=2\pi\int_{0}^{\infty}\left[1-\cos{(\chi(E_{k},b))}\right]b\,db$
(21)
$\displaystyle\sigma^{C}_{\eta}(E_{k})=2\pi\int_{0}^{\infty}\left[1-\cos^{2}{(\chi(E_{k},b))}\right]b\,db$
(22)

with $\chi$ the deflection angle and $b$ the impact parameter:

$\chi(E_{k},b)=\pi-2b\int_{R_{c}}^{\infty}\frac{R^{-2}dR}{\sqrt{1-V_{0}(R)/E_{k}-b^{2}/R^{2}}}$
(23)

where $R_{c}$ is the distance of closest approach in the collision, obtained
by solving

$1-V_{0}(R_{c})/E_{k}-b^{2}/R_{c}^{2}=0$ (24)

Integration of $\chi$ over all impact parameters then yields $\sigma_{D}$ and
$\sigma_{\eta}$ as given above by Eqs. (21) and (22).

The expressions for the transport cross sections can be thermally averaged,
resulting in

$\sigma^{C}_{D}(T)=\frac{1}{2(k_{B}T)^{3}}\int_{0}^{\infty}\sigma_{D}(E_{k})\exp\bigg{[}-\frac{E_{k}}{k_{B}T}\bigg{]}E_{k}^{2}dE_{k}$
(25)

and

$\sigma^{C}_{\eta}(T)=\frac{1}{6(k_{B}T)^{4}}\int\sigma_{\eta}(E_{k})\exp\bigg{[}-\frac{E_{k}}{k_{B}T}\bigg{]}E_{k}^{3}dE_{k}$
(26)

with $k_{B}$ the Boltzmann constant and $T$ the temperature. These thermally
averaged cross sections occur in the kinetic theory of gas transport
Hirschfelder  _et al._ (1964).

##  IV Computational details

###  IV.1 Classical cross sections

The classical cross-sections were calculated by solving Eqs. (21) to (24)
numerically using Python. The integration was performed using scipy’s
integrate.quad command for an initial guess of the point of closest approach.
Only the spherical component of the PES, $V_{0}(R)$, was taken into account,
cf. Eqs. (23) and (24).

$\mu$ (a.m.u.) | 3.68207364173
---|---
$B_{0}^{\textrm{AlF}}$ (cm-1) | 0.5499923150168107
$D_{0}^{\textrm{AlF}}$ (cm-1) | 0.000001040719977
$R_{\textrm{min}}(\textrm{\AA})$ | 1.6
$R_{\textrm{max}}(\textrm{\AA})$ | 120
$\textrm{DR}(\textrm{\AA})$ | 0.008
OTOL $(\textrm{\AA}^{2})$ | 0.001
DTOL $(\textrm{\AA}^{2})$ | 0.1
NLEVEL | 12
Table 1: Summary of parameters used for the quantum coupled-channel
calculations. Here DR denotes the step-size for integration, OTOL and DTOL the
off-diagonal and diagonal cross section tolerance thresholds for convergence,
and NLEVEL the number of angular momentum quantum levels used to construct the
basis set.

###  IV.2 Quantum elastic cross-sections

The quantum elastic cross sections were obtained by solving the single-channel
Schrödinger equation using Numerov’s method. We employed 105 steps between 3.6
$a_{0}$ and $R_{\text{max}}$ (300 $a_{0}$ for the lowest collision energy and
60 $a_{0}$ for the highest) including an appropriate number of partial waves
to guarantee a convergence better than 1$\%$. And, as in the case of the
classical calculations, only the spherical component $V_{0}(R)$ of the
potential was used.

###  IV.3 Quantum multi-channel calculations

Throughout this study, we made use of the spectroscopic constants summarized
in Table III of Ref. (Truppe  _et al._ , 2019) (i.e., $B_{0}=16488.3548$ MHz
and $D_{0}=0.0312$ MHz) to calculate the rotational levels of the AlF rotor in
its electronic and vibrational ground state. We used the MOLSCAT software
package(Jeremy M. Hutson and C. Ruth Le Sueur, ; Hutson and Le Sueur, 2019) to
perform quantum multi-channel scattering calculations of the rotationally
inelastic cross-sections based on the coupled-channel approach of Dalgarno et
al. described in Sec. III. The coupled-channel equations were solved using the
log-derivative method of Manolopoulos (Manolopoulos, 1986) between 1.6 Å and
120 Å with a step size of 0.008 Å. This method is an improved version of the
Johnson’s log-derivative algorithm (Johnson, 1973), which propagates the log-
derivative matrix $Y(R)=\Psi^{\prime}(R)[\Psi(R)]^{-1}$ instead of the wave
function. The maximum total angular momentum of the system required to reach
convergence is automatically decided by the code by setting the parameter
JTOTU $\geq 99999$, and is found not to exceed a maximum value of 50. In
addition, all the multi-channel calculations comprised 12 rotational states
with $j=0-11$. A summary of the parameters used for the scattering
calculations is given in Table 1.

##  V Results and discussion

The calculated quantum elastic and transport cross-sections are shown in panel
(a) of Fig. 3 as functions of collision energy. The cross-sections are seen to
exhibit three shape resonances, which occur at collision energies between 0.5
and 10 K, which happens to be the energy range relevant for buffer-gas cooling
experiments. At collision energies below 0.1 K, one can see the emergence of
the Wigner threshold behavior. This is consistent with the value of the van
der Waals energy Jones  _et al._ (2006); Ríos (2020)

$E_{\text{vdW}}=\left(\frac{2\hbar^{6}}{\mu^{3}C_{6}}\right)^{1/2}$ (27)

which, for the AlF+He system, evaluates to $E_{\text{vdW}}$=0.168 K, where we
made use of the _ab initio_ value of the $C_{6}^{0}$ pertaining to the
spherical component $V_{0}$ of the potential, $C_{6}^{0}=-24.6$ a.u.

In the ultracold regime which, for the system considered, sets in below 10 mK,
the diffusion cross section is seen to converge to the elastic one, whereas
the viscosity cross section converges to a limit of its own. This behavior is
due to the s-wave scattering that dominates the ultracold collision regime.
For s-wave scattering, the phase shift is related to the scattering length $a$
via

$\lim_{k\to 0}\delta_{0}(k)\rightarrow-ka$ (28)

By substituting this result into Eqs. (18), (19), and (20), we obtain

$\displaystyle\lim_{k\to 0}\sigma_{D}(E_{k})\rightarrow 4\pi a^{2}$
$\displaystyle\lim_{k\to 0}\sigma_{\eta}(E_{k})\rightarrow\frac{2}{3}4\pi
a^{2}$ $\displaystyle\lim_{k\to 0}\sigma_{\textrm{el}}(E_{k})\rightarrow 4\pi
a^{2}$ (29)

which confirms the numerically found convergence behavior of the diffusion and
viscosity cross-sections displayed in panel (a) of Fig. 3.

Figure 3: Transport cross-sections for AlF in cryogenic helium. Panel (a)
displays the elastic, diffusion and viscosity cross sections as functions of
the collision energy. Cf. the $0\to 0$ elastic cross section in Fig. 5 (thick
blue). Panel (b) displays the temperature-averaged quantum transport cross
sections (full curves) on temperature. The corresponding classical cross
sections are shown by dashed curves. Figure 4: Elastic cross-sections and
partial wave contributions. The s-, p-, d-, f-, g- and h-wave partial
contributions explain specific features of the elastic cross section curve for
the ground state ($j=0$) of the AlF rigid rotor. The prominent shape
resonances around $2.5$ K occur in a buffer gas cell experiment.

A comparison between the classical and quantum temperature-averaged transport
cross sections is shown in panel (b) of Fig. 3, which reveals that both
treatments predict similar trends in the temperature dependence. For higher
temperatures, the classical and quantum results are in a better agreement than
at lower temperatures, due to the washing out of contributions from a large
number of partial waves. In the temperature range relevant for buffer-gas
cooling experiments, i.e., $T\lesssim 10$ K, the discrepancy between the
classical and quantum mechanical results becomes more significant – and thus a
quantum treatment of the transport cross-sections desirable.

We have also calculated the classical thermally averaged diffusion cross
section at 20 K in order to compare it with the previously reported results
for YbF(${}^{2}\Sigma^{+}$) in He Skoff  _et al._ (2011) at that temperature.
It turns out that the diffusion cross section of AlF in He is 70.13 Å2, which
comes quite close to the value of 79.6 Å2 for YbF in He. This suggests that
the transport cross sections for the two fluoride diatomics are essentially
independent of either molecular symmetry (${}^{2}\Sigma$ versus
${}^{1}\Sigma$) or atom properties (Yb versus Al).

A molecule injected into a buffer-gas cell undergoes also relaxation of its
internal degrees of freedom, in particular of molecular rotation. This
process, termed rotational quenching, is driven by rotationally inelastic
collisions from a given initial rotational state. In what follows, we present
the results of our calculations of the rotationally inelastic AlF+ He
collisions that are based on the multi-channel coupled-channel approach
described in Sec. III.1.

Figure 5: Elastic and rotationally inelastic scattering cross-sections for the
first three initial rotational states ($j=0,1,2$) of the AlF rotor. For
collision energies greater than about $1.7$ K, elastic scattering begins to
dominate over all inelastic scattering channels.

Inelastic channels affect the elastic one via the unitarity of the scattering
matrix Ríos (2020), p. 26, as can be gleaned from Fig. 4, where we find a
larger number of resonances than in the elastic cross-section, cf. panel (a)
of Fig. 3. In addition, the resonances are often less distinct than in the
single-channel, elastic case because of their mutual proximity. These
additional resonances come about for two main reasons: (1) availability of
more channels and (2) the anisotropy of the PES. Having more channels
increases the probability of resonances, provided the underlying atom-molecule
interaction couples the channels. After performing a decomposition of the
cross sections into partial waves, we were able to conclude that, barring two
resonances at $0.09$ K and $0.33$ K, the s-wave contribution dominates at
collision energies of up to 0.67 K. The remaining partial-wave contributions
account for the first few prominent features of the elastic cross section. In
particular, for collision energies $\gtrsim 10$ K, the different partial waves
exhibit a rather oscillatory behavior that is averaged out once the partial
waves are summed up to yield the elastic cross section.

Figure 6: Rate constants for elastic (blue curves) and rotationally inelastic
(black curves) processes as functions of temperature. The elastic rates
dominate over the inelastic ones beyond collision energies in excess of $0.1$
K. For detailed legend, see Fig. 5.

In Fig. 5 we show both elastic and inelastic cross sections for the first
three rotational states of AlF ($j=0,1,2$) as functions of collision energy.
At low collision energies (the left-most regime), quantum behavior dominates
the dynamics, leading to a pronounced state-dependent effect on the elastic
and inelastic cross sections. For instance, the elastic cross section tends to
a different scattering length depending on the rotational state considered.
The positions of the resonances in both the elastic and inelastic cross
sections depend significantly on the initial and final rotational states. The
rotational-state dependence of the inelastic cross-sections follows in part
from energetic considerations (the larger the energy gap between different
initial and final states, the lower the transition probability). However,
additional features of the system in question need to be included in order to
rationalize its behavior: For instance, the inelastic cross-section
$\sigma_{2\rightarrow 0}(E_{k})$ is less than $\sigma_{1\rightarrow 0}(E_{k})$
because the two processes are driven, respectively, by the $V_{2}(R)$ and
$V_{1}(R)$ components of the potential and $V_{2}(R)<V_{1}(R)$, as revealed by
our _ab initio_ PES calculations.

System |  $T=5$ K |  $T=10$ K
---|---|---
HF-He | $5\times 10^{-12}$ | $5\times 10^{-12}$
CO-He | $3.4\times 10^{-11}$ | $3.2\times 10^{-11}$
AlF-He | $5.4\times 10^{-11}$ | $4\times 10^{-11}$
Table 2: State-to-state rate constants $k_{1\to 0}(T)$ in cm3s-1 for different
${}^{1}\Sigma^{+}$-He scattering systems. The values of the constants for the
HF-He and CO-He systems have been taken from Refs. (Yang, B.  _et al._ , 2015;
Cecchi-Pestellini  _et al._ , 2002).

The transport rate constants of a fluid Hirschfelder  _et al._ (1964) can be
obtained from the transport cross sections introduced in Section III.2. In the
absence of molecular vibration, the rotational state-to-state rate constants
are given by Montero and Pérez-Ríos (2014),

$\displaystyle k_{j\rightarrow j^{\prime}}(T)=<\sigma_{j\rightarrow
j^{\prime}}(E_{k})v>=\left(\frac{8}{\pi\mu\beta^{3}}\right)^{1/2}$
$\displaystyle\int_{0}^{\infty}\sigma_{j\rightarrow
j^{\prime}}(E_{k})\exp{(-E_{k}/\beta)}E_{k}dE_{k}$ (30)

with $\beta=k_{B}T$. The dependence of the state-to-state rate constants on
temperature is displayed in Fig. 6 for a temperature range relevant to buffer-
gas cooling. What we see is a large difference between the rates of elastic
and inelastic scattering in the cryogenic buffer gas. Similarly as in the case
of the cross sections, the rate constants for the exchange of the largest
rotational quanta are suppressed. However, the elastic rate constants are
almost independent of the rotational state, which reflects the average nature
of the rate coefficient.

Figure 7: Sensitivity to shifts in the PES up ($\uparrow$) or down
($\downarrow$) by 5%. The density of resonances essentially remains unchanged
within the collision energy regime accessible to buffer gas experiments. Inset
shows rate coefficients for rotational quenching as a function of temperature.

Fig. 6 also reveals that the state-to-state rate constants corresponding to a
relaxation of the rotational degrees of freedom of the molecule are only
weakly temperature-dependent. A similar behavior has been reported for other
${}^{1}\Sigma^{+}$-He collision systems, as summarized in Table 2. The more
efficient relaxation of AlF in He can be rationalized by invoking the
adiabaticity parameter, $\xi=\tau_{\textrm{coll}}/\tau_{\textrm{rot}}\propto
B_{0}/\sqrt{\mu}$, where $\tau_{\textrm{coll}}$ and $\tau_{\textrm{rot}}$ are,
respectively, the collision time and rotational period of the molecule, see
also Table 1. For $\xi\gg 1$, the collision time exceeds the rotational period
and hence the collision proceeds adiabatically, without changing the
rotational state of the molecule. On the other hand, for $\xi\ll 1$ the
collision process is nonadiabatic, leading to efficient rotational energy
transfer. Therefore, heavier molecules with smaller rotational constants, such
as AlF compared with CO or HF, lead to small values of $\xi$ and thus a more
nonadiabatic behavior (more facile rotational quenching). Based on the same
argument, we conclude that the MgF-He system will show relaxation rates
similar to those of the AlF-He system, but that BaF-He will exhibit a more
efficient quenching due to BaF’s smaller rotational constant Liu  _et al._
(2020).

Given that the inaccuracy of the ab initio quantum chemistry calculations
exceeds the collision energies of $\lesssim$100 K that are at play in
cryogenic helium cooling, we tested how sensitive the elastic scattering cross
sections are to changes of the PES. To this end, the PES was either raised or
lowered throughout by $5\%$ and the calculations repeated. The error thus
simulated overestimates the inaccuracies of our method in the long-range tail
of the potential but may underestimate them in the short-range region. The
simulations are displayed in Fig. 7, which attests that the positions and the
widths of the resonances are affected and how. Nevertheless, the density of
the resonances remains unchanged. The simulation also reveals that in the
thermal regime, for energies greater than 10 K, the cross sections are less
sensitive to changes in the PES, as expected. The inset of Fig. 7 shows the
effects of varying the PES on the state-to-state rate constants, which are
seen to remain qualitatively unaffected.

##  VI Conclusions

We undertook a study of the collisions between aluminum monofluoride (AlF)
molecules and helium (He) atoms at collision energies relevant to buffer-gas
cooling by cryogenic helium. Our study, based on an accurate ab initio
potential energy surface and quantum multichannel scattering theory, revealed
high thermalization rates for both translational and rotational degrees of
freedom of the molecule, on the order of $10^{10}$ and $10^{11}$ cm3s-1,
respectively.

The large anisotropy of the AlF-He PES (due to the preferred He-F attraction
over that between He and Al) and a small rotational constant of AlF translates
into a complex resonance structure in elastic and rotationally inelastic
scattering cross sections. However, at collision energies $\gtrsim 10$ K, the
quantum effects are found to average out and most of the scattering
observables become independent of the rotational state of the molecule.
Classical calculations of the transport cross sections are found to be valid
up to temperatures of $5$ K with a relative error $\lesssim$ $15\%$.

Although possible inaccuracies of the PES may affect the positions of the
resonances, their density was found to remain the same. Moreover, we showed
that the state-to-state rate constants are quite robust against inaccuracies
of the PES for temperatures relevant to buffer-gas cell experiments. These
findings confirm AlF as a serious candidate for efficient sympathetic cooling.

Our results on the AlF-He system are of a more general interest. For one, our
classical transport cross sections benchmark the accuracy of such methods for
${}^{1}\Sigma^{+}$ molecule-He collisions. Furthermore, considering the
adiabaticity parameter, we estimate that MgF will be quenched as efficiently
as AlF. Finally, owing to its smaller rotational constant, we expect BaF to be
quenched more efficiently than AlF.

##  VII Acknowledgments

MK acknowledges support by the IMPRS for Elementary Processes in Physical
Chemistry. BF thanks John Doyle and Hossein Sadeghpour for their hospitality
during his stay at the Harvard Department of Physics and at the Harvard-
Smithsonian Institute for Theoretical Atomic, Molecular, and Optical Physics
(ITAMP).

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# Fano factor, $\Delta T$-noise and cross-correlations in double quantum dots

A. Crépieux  T.Q. Duong  Aix Marseille Univ, Université de Toulon, CNRS, CPT,
Marseille, France  M. Lavagna  Univ. Grenoble Alpes, CEA, INAC, PHELIQS,
F-38000 Grenoble, France

###### Abstract

We present a theoretical study of electrical current fluctuations and finite-
frequency noise in a double quantum dot connected to two electron reservoirs
with the aim of deriving the Fano factor, the $\Delta T$-noise and the cross-
correlations. This allows one to highlight several interesting features.
Firstly the possibility of getting a significant reduction of current noise
and Fano factor either when the system is placed in a given operating regime,
or when a temperature gradient is applied between the two reservoirs,
resulting from the fact that a negative $\Delta T$-noise is generated. The
second feature is the sign change found in the cross-correlator between the
two reservoirs with increasing frequencies. This study clarifies the
understanding of the results obtained experimentally in such systems.

Introduction – The reduction of electrical noise in double quantum dots is a
major issue if one wishes to finely control the electric charge transfer and
improve the performance and the quality factor, in spin-qubits in
particular[1, 2, 3]. There are some theoretical studies devoted to the
characterization of current fluctuations and electrical noise in double
quantum dots[4], but they are most often limited to the calculation of noise
at zero frequency[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] or use
perturbative approaches[19, 20, 21] based mainly on master equation technique.
However we now have experimental works studying fluctuations in double quantum
dots, built from GaAs/AlGaAs or Si/SiGe heterostructures, that are expected to
undergo a considerable development in the coming years. Part of these works
are looking for the optimal conditions to suppress the sensitivity of the
device to electrical noise, in order to obtain long-lived and high-fidelity
spin qubits[1, 2, 3], while others are devoted to the measurement of
electrical current cross-correlations[22] or to the measurement of entropy
fluctuations[23]. All these experimental works provide the motivation for
developing further theoretical studies of electrical noise and current
fluctuations in double quantum dots.

In this letter, we present a non-perturbative approach to determine the
finite-frequency noise in a double quantum dot system using the non-
equilibrium Green function technique. After outlining the formalism used to
model the double quantum dot connected to left and right electron reservoirs,
we give the results for the general expression of the noise. We numerically
calculate the noise and the Fano factor in various geometries: double quantum
dot connected either in series or in parallel, and discuss the results. We
next focus our interest to the $\Delta T$-noise produced when the two
reservoirs are raised to different temperatures[24, 25, 26, 27, 28, 29]. We
show that the latter quantity exhibits specific characteristics in its
evolution[30] that have never been demonstrated before in double quantum dots.
To end up, we determine the cross-correlator between the left and right
reservoirs and compare the obtained results with the experimental ones [22].

Model and results – The Hamiltonian of two coupled quantum dots connected to
left (L) and right (R) reservoirs is given by

$\displaystyle\mathcal{\widehat{H}}=\sum_{\begin{subarray}{c}\alpha=L,R\\\
k\in\alpha\end{subarray}}\varepsilon_{\alpha k}^{\,}\widehat{c}_{\alpha
k}^{\;\dagger}\widehat{c}_{\alpha k}^{\,}+\sum_{\begin{subarray}{c}i=1,2\\\
n\in
i\end{subarray}}\varepsilon_{in}^{\,}\widehat{d}_{in}^{\;\dagger}\widehat{d}_{in}^{\,}$
$\displaystyle+\sum_{\begin{subarray}{c}n\in 1\\\ m\in
2\end{subarray}}\mathcal{V}_{12}^{\,}\widehat{d}_{2m}^{\;\dagger}\widehat{d}_{1n}^{\,}+\sum_{\begin{subarray}{c}\alpha=L,R\\\
k\in\alpha\end{subarray}}\sum_{\begin{subarray}{c}i=1,2\\\ n\in
i\end{subarray}}V_{i\alpha}^{\,}\widehat{c}_{\alpha
k}^{\;\dagger}\widehat{d}_{in}^{\,}+h.c.~{},$ (1)

where $\widehat{c}_{\alpha k}^{\;\dagger}$ ($\widehat{c}_{\alpha k}$) is the
creation (annihilation) operator related to the reservoir $\alpha$ with
momentum $k$ and energy $\varepsilon_{\alpha k}$, and
$\widehat{d}_{in}^{\;\dagger}$ ($\widehat{d}_{in}$) is the creation
(annihilation) operator related to the dot $i$, with $i=1,2$. Each dot $i$
contains $N$ discrete energy levels denoted $\varepsilon_{in}$, with
$n\in[1,N]$. The notation $h.c.$ corresponds to the hermitian conjugate terms
associated with the third and fourth contributions in Eq. (Fano factor,
$\Delta T$-noise and cross-correlations in double quantum dots). One assumes
that the inter-dot coupling $\mathcal{V}_{12}$ between the states $|1n\rangle$
and $|2m\rangle$ in the dots, and the hopping integral $V_{i\alpha}$ between
the states $|in\rangle$ in the dot $i$ and $|\alpha k\rangle$ in the reservoir
$\alpha$ do not depend on the indices $n$ and $m$, nor on the momentum $k$.

The finite-frequency non-symmetrized noise in the DQD is defined as the
Fourier transform of the current fluctuations,
$\mathcal{S}_{\alpha\beta}(\omega)=\int_{-\infty}^{\infty}\langle\Delta\hat{I}_{\alpha}(t)\Delta\hat{I}_{\beta}(0)\rangle
e^{-i\omega t}dt$, where
$\Delta\hat{I}_{\alpha}(t)=\hat{I}_{\alpha}(t)-\langle\hat{I}_{\alpha}\rangle$
is the deviation of the current $\hat{I}_{\alpha}(t)$ from its average value
$\langle\hat{I}_{\alpha}\rangle$. The calculations lead to the expression
$\mathcal{S}_{\alpha\beta}(\omega)=\frac{e^{2}}{h}\sum_{i=1}^{5}\int_{-\infty}^{\infty}\text{Tr}\Big{\\{}\underline{\underline{\mathcal{C}_{\alpha\beta}^{(i)}}}\Big{\\}}d\varepsilon$,
where $\text{Tr}\\{\,\\}$ corresponds to the trace, and where the matrices
$\underline{\underline{\mathcal{C}_{\alpha\beta}^{(i)}}}$ are given by

$\displaystyle\underline{\underline{\mathcal{C}_{\alpha\beta}^{(i)}}}$
$\displaystyle=$
$\displaystyle\delta_{\alpha\beta}\Big{[}\underline{\underline{\bf
G^{<}}}(\varepsilon)\underline{\underline{\mathbb{\Sigma}^{>}_{\alpha}}}(\varepsilon-\hbar\omega)$
(2) $\displaystyle+\underline{\underline{\bf
G^{>}}}(\varepsilon-\hbar\omega)\underline{\underline{\mathbb{\Sigma}^{<}_{\alpha}}}(\varepsilon)\Big{]}~{},$
$\displaystyle\underline{\underline{\mathcal{C}_{\alpha\beta}^{(2)}}}$
$\displaystyle=$ $\displaystyle-\Big{[}\underline{\underline{{\bf
G}^{r}}}(\varepsilon-\hbar\omega)\underline{\underline{\mathbb{\Sigma}^{>}_{\beta}}}(\varepsilon-\hbar\omega)$
(3) $\displaystyle+\underline{\underline{\bf
G^{>}}}(\varepsilon-\hbar\omega)\underline{\underline{\mathbb{\Sigma}^{a}_{\beta}}}(\varepsilon-\hbar\omega)\Big{]}$
$\displaystyle\times\Big{[}\underline{\underline{{\bf
G}^{r}}}(\varepsilon)\underline{\underline{\mathbb{\Sigma}^{<}_{\alpha}}}(\varepsilon)+\underline{\underline{\bf
G^{<}}}(\varepsilon)\underline{\underline{\mathbb{\Sigma}^{a}_{\alpha}}}(\varepsilon)\Big{]}~{},$
$\displaystyle\underline{\underline{\mathcal{C}_{\alpha\beta}^{(3)}}}$
$\displaystyle=$ $\displaystyle\underline{\underline{\bf
G^{>}}}(\varepsilon-\hbar\omega)\Big{[}\underline{\underline{\mathbb{\Sigma}^{<}_{\beta}}}(\varepsilon)\underline{\underline{{\bf
G}^{a}}}(\varepsilon)\underline{\underline{\mathbb{\Sigma}^{a}_{\alpha}}}(\varepsilon)$
(4)
$\displaystyle+\underline{\underline{\mathbb{\Sigma}^{r}_{\beta}}}(\varepsilon)\underline{\underline{\bf
G^{<}}}(\varepsilon)\underline{\underline{\mathbb{\Sigma}^{a}_{\alpha}}}(\varepsilon)+\underline{\underline{\mathbb{\Sigma}^{r}_{\beta}}}(\varepsilon)\underline{\underline{{\bf
G}^{r}}}(\varepsilon)\underline{\underline{\mathbb{\Sigma}^{<}_{\alpha}}}(\varepsilon)\Big{]}~{},$
$\displaystyle\underline{\underline{\mathcal{C}_{\alpha\beta}^{(4)}}}$
$\displaystyle=$ $\displaystyle\underline{\underline{\bf
G^{<}}}(\varepsilon)\Big{[}\underline{\underline{\mathbb{\Sigma}^{r}_{\alpha}}}(\varepsilon-\hbar\omega)\underline{\underline{{\bf
G}^{r}}}(\varepsilon-\hbar\omega)\underline{\underline{\mathbb{\Sigma}^{>}_{\beta}}}(\varepsilon-\hbar\omega)$
(5)
$\displaystyle+\underline{\underline{\mathbb{\Sigma}^{r}_{\alpha}}}(\varepsilon-\hbar\omega)\underline{\underline{\bf
G^{>}}}(\varepsilon-\hbar\omega)\underline{\underline{\mathbb{\Sigma}^{a}_{\beta}}}(\varepsilon-\hbar\omega)$
$\displaystyle+\underline{\underline{\mathbb{\Sigma}^{>}_{\alpha}}}(\varepsilon-\hbar\omega)\underline{\underline{{\bf
G}^{a}}}(\varepsilon-\hbar\omega)\underline{\underline{\mathbb{\Sigma}^{a}_{\beta}}}(\varepsilon-\hbar\omega)\Big{]}~{},$

and,

$\displaystyle\underline{\underline{\mathcal{C}_{\alpha\beta}^{(5)}}}$
$\displaystyle=$
$\displaystyle-\Big{[}\underline{\underline{\mathbb{\Sigma}^{r}_{\alpha}}}(\varepsilon-\hbar\omega)\underline{\underline{\bf
G^{>}}}(\varepsilon-\hbar\omega)$ (6)
$\displaystyle+\underline{\underline{\mathbb{\Sigma}^{>}_{\alpha}}}(\varepsilon-\hbar\omega)\underline{\underline{{\bf
G}^{a}}}(\varepsilon-\hbar\omega)\Big{]}$
$\displaystyle\times\Big{[}\underline{\underline{\mathbb{\Sigma}^{r}_{\beta}}}(\varepsilon)\underline{\underline{\bf
G^{<}}}(\varepsilon)+\underline{\underline{\mathbb{\Sigma}^{<}_{\beta}}}(\varepsilon)\underline{\underline{{\bf
G}^{a}}}(\varepsilon)\Big{]}~{}.$

The self-energy matrices associated with the reservoir $\alpha$ are given by
$\underline{\underline{\mathbb{\Sigma}_{\alpha}^{r,a}}}(\varepsilon)=\mp(i/2)\underline{\underline{\Gamma_{\alpha}}}$,
$\underline{\underline{\mathbb{\Sigma}_{\alpha}^{<}}}(\varepsilon)=if_{\alpha}^{e}(\varepsilon)\underline{\underline{\Gamma_{\alpha}}}$
and
$\underline{\underline{\mathbb{\Sigma}_{\alpha}^{>}}}(\varepsilon)=-if_{\alpha}^{h}(\varepsilon)\underline{\underline{\Gamma_{\alpha}}}$,
where
$f_{\alpha}^{e}(\varepsilon)=1/(1+\exp(\varepsilon-\mu_{\alpha})/k_{B}T_{\alpha})$
and $f_{\alpha}^{h}(\varepsilon)=1-f_{\alpha}^{e}(\varepsilon)$ are the Fermi-
Dirac distribution functions for electrons and holes respectively,
$T_{\alpha}$ is the temperature and $\mu_{\alpha}$ the chemical potential of
the reservoir $\alpha$ with density of states $\rho_{\alpha}$, which is energy
independent in the wide-band limit, and where the elements of the dot-
reservoir coupling matrix $\underline{\underline{\Gamma_{\alpha}}}$ are
$\Gamma_{\alpha,ij}=2\pi\rho_{\alpha}V^{*}_{i\alpha}V_{j\alpha}$. The retarded
Green function associated with the double quantum dot is a $2\times 2$ matrix
given by[31]

$\displaystyle\underline{\underline{{\bf
G}^{r}}}(\varepsilon)=\frac{1}{D^{r}(\varepsilon)}\left(\begin{array}[]{cc}\widetilde{\bf
g}_{1}^{r}(\varepsilon)&\widetilde{\bf
g}_{1}^{r}(\varepsilon)\mathbb{\widetilde{\Sigma}}_{12}^{r}(\varepsilon)\widetilde{\bf
g}_{2}^{r}(\varepsilon)\\\ \widetilde{\bf
g}_{2}^{r}(\varepsilon)\mathbb{\widetilde{\Sigma}}_{21}^{r}(\varepsilon)\widetilde{\bf
g}_{1}^{r}(\varepsilon)&\widetilde{\bf
g}_{2}^{r}(\varepsilon)\end{array}\right)~{},$ (9)

where $D^{r}(\varepsilon)=1-\widetilde{\bf
g}_{1}^{r}(\varepsilon)\mathbb{\widetilde{\Sigma}}_{12}^{r}(\varepsilon)\widetilde{\bf
g}_{2}^{r}(\varepsilon)\mathbb{\widetilde{\Sigma}}_{21}^{r}(\varepsilon)$ and
${\widetilde{\bf g}_{i}^{r}}(\varepsilon)={\bf
g}_{i}^{r}(\varepsilon)/(1-{{\mathbb{\widetilde{\Sigma}}}_{ii}^{r}}(\varepsilon){\bf
g}_{i}^{r}(\varepsilon))$. In these expressions appear the retarded Green
function of the disconnected dot $i$, defined as ${\bf
g}_{i}^{r}(\varepsilon)=\sum_{n\in i}g_{in}^{r}(\varepsilon)$ with
$g_{in}^{r}(\varepsilon)=1/(\varepsilon-\varepsilon_{in}+i0^{+})$, and the
total self-energy:
$\underline{\underline{\mathbb{\widetilde{\Sigma}}^{r}}}(\varepsilon)=\sum_{\alpha=L,R}\underline{\underline{\mathbb{\Sigma}^{r}_{\alpha}}}(\varepsilon)+\underline{\underline{\mathbb{\Sigma}^{r}_{\text{int}}}}$,
where the matrix $\underline{\underline{\mathbb{\Sigma}^{r}_{\text{int}}}}$ is
given by

$\displaystyle\underline{\underline{\mathbb{\Sigma}^{r}_{\text{int}}}}=\left(\begin{array}[]{cc}0&\mathcal{V}^{*}_{12}\\\
\mathcal{V}^{*}_{21}&0\end{array}\right)~{}.$ (13)

The advanced Green function matrix $\underline{\underline{{\bf
G}^{a}}}(\varepsilon)$ is obtained from the retarded one by replacing the
superscript $r$ by the superscript $a$ in the expression giving
$\underline{\underline{{\bf G}^{r}}}(\varepsilon)$. Moreover, one has
$\underline{\underline{{\bf
G}^{\lessgtr}}}(\varepsilon)=\sum_{\alpha=L,R}\underline{\underline{{\bf
G}^{r}}}(\varepsilon)\underline{\underline{\mathbb{\Sigma}^{\lessgtr}_{\alpha}}}(\varepsilon)\underline{\underline{{\bf
G}^{a}}}(\varepsilon)$.

We have established the following expression for the auto-correlator
$\mathcal{S}_{LL}(\omega)$ calculated in the $L$-reservoir

$\displaystyle\mathcal{S}_{LL}(\omega)=\frac{e^{2}}{h}\int_{-\infty}^{\infty}d\varepsilon\,\text{Tr}\bigg{\\{}f^{e}_{L}(\varepsilon)f^{h}_{L}(\varepsilon-\hbar\omega)$
$\displaystyle\times\left[\underline{\underline{\mathcal{T}_{LL}^{\text{eff}}}}(\varepsilon)\underline{\underline{\mathcal{T}_{LL}^{\text{eff}}}}(\varepsilon-\hbar\omega)+\left|\underline{\underline{t_{LL}}}(\varepsilon)-\underline{\underline{t_{LL}}}(\varepsilon-\hbar\omega)\right|^{2}\right]$
$\displaystyle+f^{e}_{R}(\varepsilon)f^{h}_{R}(\varepsilon-\hbar\omega)\underline{\underline{\mathcal{T}_{RL}}}(\varepsilon)\underline{\underline{\mathcal{T}_{RL}}}(\varepsilon-\hbar\omega)$
$\displaystyle+f^{e}_{L}(\varepsilon)f^{h}_{R}(\varepsilon-\hbar\omega)\left[1-\underline{\underline{\mathcal{T}_{LL}^{\text{eff}}}}(\varepsilon)\right]\underline{\underline{\mathcal{T}_{RL}}}(\varepsilon-\hbar\omega)$
$\displaystyle+f^{e}_{R}(\varepsilon)f^{h}_{L}(\varepsilon-\hbar\omega)\underline{\underline{\mathcal{T}_{RL}}}(\varepsilon)\left[1-\underline{\underline{\mathcal{T}_{LL}^{\text{eff}}}}(\varepsilon-\hbar\omega)\right]\bigg{\\}}~{},$
(14)

where we have defined the transmission amplitude matrix
$\underline{\underline{t_{\alpha\alpha}}}(\varepsilon)=i\underline{\underline{{\bf
G}^{r}}}(\varepsilon)\underline{\underline{\Gamma_{\alpha}}}$, the
transmission coefficient matrix
$\underline{\underline{\mathcal{T}_{\alpha\beta}}}(\varepsilon)=\underline{\underline{{\bf
G}^{r}}}(\varepsilon)\,\underline{\underline{\Gamma_{\alpha}}}\;\underline{\underline{{\bf
G}^{a}}}(\varepsilon)\,\underline{\underline{\Gamma_{\beta}}}$, and the
effective transmission coefficient matrix
$\underline{\underline{\mathcal{T}_{\alpha\alpha}^{\text{eff}}}}(\varepsilon)=\underline{\underline{t_{\alpha\alpha}}}(\varepsilon)+\underline{\underline{t_{\alpha\alpha}^{+}}}(\varepsilon)-\underline{\underline{\mathcal{T}_{\alpha\alpha}}}(\varepsilon)$,
where $\underline{\underline{t_{\alpha\alpha}^{+}}}(\varepsilon)$ is the
conjugate transpose of
$\underline{\underline{t_{\alpha\alpha}}}(\varepsilon)$. The expression for
the cross-correlator $\mathcal{S}_{LR}(\omega)$ is given by

$\displaystyle\mathcal{S}_{LR}(\omega)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{h}\int_{-\infty}^{\infty}d\varepsilon\,\text{Tr}\bigg{\\{}f^{e}_{L}(\varepsilon)f^{h}_{L}(\varepsilon-\hbar\omega)\left[\left[\underline{\underline{\mathcal{T}_{LL}}}(\varepsilon)-\underline{\underline{t_{LL}}}(\varepsilon)\right]\underline{\underline{\mathcal{T}_{LR}}}(\varepsilon-\hbar\omega)-\underline{\underline{\mathcal{T}_{LL}}}(\varepsilon)\underline{\underline{t_{RR}^{+}}}(\varepsilon-\hbar\omega)\right]$
(15)
$\displaystyle+f^{e}_{R}(\varepsilon)f^{h}_{R}(\varepsilon-\hbar\omega)\left[\left[\underline{\underline{\mathcal{T}_{RL}}}(\varepsilon)-\underline{\underline{t_{LL}^{+}}}(\varepsilon)\right]\underline{\underline{\mathcal{T}_{RR}}}(\varepsilon-\hbar\omega)-\underline{\underline{\mathcal{T}_{RL}}}(\varepsilon)\underline{\underline{t_{RR}}}(\varepsilon-\hbar\omega)\right]$
$\displaystyle+f^{e}_{L}(\varepsilon)f^{h}_{R}(\varepsilon-\hbar\omega)\left[\underline{\underline{\mathcal{T}_{LL}}}(\varepsilon)-\underline{\underline{t_{LL}}}(\varepsilon)\right]\left[\underline{\underline{\mathcal{T}_{RR}}}(\varepsilon-\hbar\omega)-\underline{\underline{t_{RR}}}(\varepsilon-\hbar\omega)\right]$
$\displaystyle+f^{e}_{R}(\varepsilon)f^{h}_{L}(\varepsilon-\hbar\omega)\left[\underline{\underline{\mathcal{T}_{RL}}}(\varepsilon)-\underline{\underline{t_{LL}^{+}}}(\varepsilon)\right]\left[\underline{\underline{\mathcal{T}_{LR}}}(\varepsilon-\hbar\omega)-\underline{\underline{t_{RR}^{+}}}(\varepsilon-\hbar\omega)\right]\bigg{\\}}~{}.$

The auto-correlator $\mathcal{S}_{RR}(\omega)$ and cross-correlator
$\mathcal{S}_{RL}(\omega)$ are obtained by interchanging the indices $L$ and
$R$ in Eq. (Fano factor, $\Delta T$-noise and cross-correlations in double
quantum dots) and Eq. (15), respectively. These results are a generalization
to double quantum dots of the results obtained for a single dot[32, 33]. The
main differences are the expressions themselves of the Green functions and of
the transmission amplitudes and coefficients, and the presence of matrix
products rather than scalar products. Eqs. (Fano factor, $\Delta T$-noise and
cross-correlations in double quantum dots) and (15) apply for both serial and
parallel double quantum dots since one can freely play with the values of the
dot-reservoir couplings $\Gamma_{\alpha,ij}$. In the following we consider
only symmetrical dot-reservoir couplings, meaning that
$\Gamma\equiv\Gamma_{L,11}=\Gamma_{R,22}$ and $\Gamma_{L,22}=\Gamma_{R,11}=0$
for a serial double dot, and $\Gamma\equiv\Gamma_{\alpha,ij}$,
$\forall\\{\alpha,i,j\\}$, for a parallel double dot.

Discussion – We first study the Fano factor, defined as
$\mathcal{F}_{\alpha}=\mathcal{S}_{\alpha\alpha}(0)/(eI_{\alpha})$, in order
to identify the regions where the zero-frequency noise is reduced compared to
the current, i.e. such that one has $\mathcal{F}_{\alpha}\ll 1$. Figure 1
displays the color-scale plot of the Fano factor associated to the
L-reservoir, $\mathcal{F}_{L}$, as a function of bias voltage
$V=\mu_{L}-\mu_{R}$ and detuning energy
$\varepsilon_{d}=\varepsilon_{2}-\varepsilon_{1}$ for a double quantum dot in
series with $N=3$ energy levels in each dot ($n=1,2,3$). It shows that when
the energy $\varepsilon_{1}$ of dot 1 is aligned with $\mu_{L}+\mu_{R}$, here
equal to zero since we set $\mu_{L}=V/2$ and $\mu_{R}=-V/2$, then the value of
$\mathcal{F}_{L}$ at low voltage is most of the time about 0.5 or larger,
excepted in some narrow purple stripes along the lines
$\varepsilon_{d}=n\varepsilon_{0}$ when $V>0.5$ (see left panel of Fig. 1). On
the contrary, when the energy $\varepsilon_{1}$ of dot 1 is not aligned with
$\mu_{L}+\mu_{R}$ (the most probable situation), one observes that the value
of $\mathcal{F}_{L}$ is strongly reduced at low voltage for some values of the
detuning energy (see the purple color regions in the right panel of Fig. 1),
leading to the possibility of reducing the noise, relatively to the current,
even at low voltage. In the case of a double quantum dot arranged in parallel,
the situation is opposite. Indeed, it is when the detuning energy is aligned
with $\mu_{L}+\mu_{R}$ that one observes a strong reduction of noise at low
voltage (see the purple color regions in the left panel of Fig. 2), whereas
for $\varepsilon_{d}\neq\mu_{L}+\mu_{R}$, one has $\mathcal{F}_{L}\approx 0.5$
at low voltage (see the right panel of Fig. 2), meaning that the noise is high
compared to the current. These results help to identify the regions of the
$(V,\varepsilon_{d})$-plane where one has a noise reduction.

Figure 1: Color-scale plot of the Fano factor $\mathcal{F}_{L}$ as a function
of the bias voltage $V$ and the detuning energy $\varepsilon_{d}$ for a double
quantum dot in series, with three energy levels in each dot, equal to
$\varepsilon_{in}=\varepsilon_{i}+n\varepsilon_{0}$ where $n=1,2,3$, with
(left) $\varepsilon_{1}=0$, and (right) $\varepsilon_{1}=0.3$. The other
parameters are $\varepsilon_{0}=1$, $T_{L,R}=0.01$, $\Gamma=0.1$, and
$\mathcal{V}_{12}=0.2$.

Figure 2: Same as in Fig. 1 for a double quantum dot in parallel.

Figure 3: Color-scale plot of the $\Delta T$-noise as a function of
$\varepsilon_{1}$ and $\varepsilon_{2}$ for a double quantum dot in series
(a.u.), with three energy levels in each dot, equal to
$\varepsilon_{in}=\varepsilon_{i}+n\varepsilon_{0}$ where $n=1,2,3$, with
(left) $T=0.01$, $\Gamma=0.1$, $\mathcal{V}_{12}=0.1$, and (right) $T=0.1$,
$\Gamma=0.05$, $\mathcal{V}_{12}=0.1$. The other parameters are $\delta
T=T/2$, $\varepsilon_{0}=1$, and $V=0$.

Figure 4: Same as in Fig. 3 for a double quantum dot in parallel. Figure 5:
Value of the minimum of the $\Delta T$-noise $\Delta\mathcal{S}_{LL}$,
according to $\varepsilon_{1}$ and $\varepsilon_{2}$, as a function of the
ratio $\Gamma/T$ at $V=0$, $\mathcal{V}_{12}=0.1$, $\Gamma=0.1$, and
$\varepsilon_{0}=1$. One takes $\delta T=T/2$. For both simple and double
quantum dots, three energy levels in each dot have been included. In the
parallel geometry case, a scaling factor 1/2 has been added in order to be
able to compare the different curves.

We now turn our interest to the $\Delta T$-noise defined as
$\Delta\mathcal{S}_{\alpha\alpha}=\mathcal{S}_{\alpha\alpha}^{\delta
T}(0)-\mathcal{S}_{\alpha\alpha}^{0}(0)$, where
$\mathcal{S}_{\alpha\alpha}^{\delta T}(0)$ is the zero-frequency auto-
correlator at zero voltage when the reservoir are raised to distinct
temperatures, i.e., $T_{L}=T+\delta T/2$ and $T_{R}=T-\delta T/2$, and
$\mathcal{S}_{\alpha\alpha}^{0}(0)$ is the zero-frequency auto-correlator
calculated for identical reservoir temperatures, i.e., $T_{L,R}=T$. Figure 3
displays the color-scale plots of the $\Delta T$-noise in a double quantum dot
in series for two different sets of parameters. It shows that in the regime
where $\Gamma\gtrsim T$, the $\Delta T$-noise remains positive (see left panel
of Fig. 3), whereas in the regime where $\Gamma\lesssim T$, the sign of the
$\Delta T$-noise changes (see right panel of Fig. 3), meaning that the noise
is reduced in some regions of the $(\varepsilon_{1},\varepsilon_{2})$-plane
when a temperature gradient between the two reservoirs is applied. The fact
that one has a change of behavior in the $\Delta T$-noise when reducing
$\Gamma$ is in perfect agreement with what has been obtained and generically
explained in Ref. 30, with a change of behavior at $\Gamma/T\approx 2.6$,
principally due to the fact that the energy dependencies of the transmission
amplitude $\underline{\underline{t_{\alpha\alpha}}}(\varepsilon)$ and
transmission coefficients
$\underline{\underline{\mathcal{T}_{\alpha\alpha}}}(\varepsilon)$ become
relevant when $\Gamma\lesssim T$. In the case of a double quantum dot in
parallel, such a change of behavior in the $\Delta T$-noise, between the
regime where $\Gamma\lesssim T$ and the regime where $\Gamma\gtrsim T$, is
also observed (see Fig. 4). In order to further understand this phenomenon, we
have reported in Fig. 5 the evolution of the $\Delta T$-noise minimum as a
function of the $\Gamma/T$ ratio for three different quantum dot geometries:
single dot, double dot in series, and double dot in parallel. We see that the
value of this minimum is negative at low $\Gamma/T$ ratio and converges to
zero at $\Gamma/T\approx 3$ whatever the geometry is. Again, this variation is
in agreement with previous results on $\Delta T$-noise obtained for quantum
systems caracterized by energy-dependent transmission[30]. This is an
important result because it means that there is an operating regime in which
the noise can be reduced by applying a temperature gradient between the right
and left reservoirs, in both single and double quantum dot systems.

Finally, we study the finite-frequency cross-correlator for a double quantum
dot in series in order to make a comparison with the experimental observations
presented in Ref. 22. Figure 6 shows the cross-correlator
$\mathcal{S}_{LR}(\omega)$ near the honeycomb vertex located in the central
region of the $(\varepsilon_{1},\varepsilon_{2})$-plane, at both zero-
frequency ($\omega=0$) and finite-frequency ($\omega=0.1$). At zero-frequency,
the cross-correlator is negative in sign (see left panel of Fig. 6) as
expected since one has $\mathcal{S}_{LR}(0)=-\mathcal{S}_{LL}(0)$ with
strictly positive auto-correlator, whereas at finite-frequency the sign of the
cross-correlator becomes positive in some regions of the
$(\varepsilon_{1},\varepsilon_{2})$-plane (see right panel of Fig. 6).
Remarkably, and this deserves to be emphasized, the entire evolution of the
cross-correlator that we obtained at finite-frequency is in perfect agreement
with the experimental results presented in Ref. 22 which show a sign change.
And identically to what is experimentally observed, one obtains a vanishing
finite-frequency cross-correlator at zero voltage due to the fact that at low
temperature the system can not emit noise at frequency larger than the
voltage[34, 35], so that for $\hbar\omega>eV$, one has
$\mathcal{S}_{LR}(\omega)=\mathcal{S}_{LL}(\omega)=0$.

Figure 6: Color-scale plot of the cross-correlator $\mathcal{S}_{LR}(\omega)$
as a function of $\varepsilon_{1}$ and $\varepsilon_{2}$ for a double quantum
dot in series (a.u.), with three energy levels in each dot, equal to
$\varepsilon_{in}=\varepsilon_{i}+n\varepsilon_{0}$ where $n=1,2,3$, with
(left) $\omega=0$, and (right) $\omega=0.1$. The other parameters are $V=0.2$,
$\Gamma=0.1$, $\mathcal{V}_{12}=0.1$, $T_{L,R}=0.01$, and $\varepsilon_{0}=1$.

Summary – We have derived expressions for the auto-correlators and cross-
correlators of the current fluctuations in a double quantum dot which apply to
any frequency, voltage and temperature values, whatever the values of inter-
dot and dot-reservoirs couplings are. They allow to highlight specific
features such as the reduction of the noise and of the Fano factor, as well as
the possibility of having a negative $\Delta T$-noise, meaning that the noise
can be even more reduced by applying a temperature gradient between the two
reservoirs. Moreover, it leads to behavior for the cross-correlator which is
in perfect agreement with the experimental measurements[22] with a change of
sign near a honeycomb vertex. The approach presented in this Letter can be
extended to take electron-electron and electron-photon interactions into
account, which are all the more important to be able to describe realistic
situations of double quantum dot systems, as experimentally studied. Indeed,
in the limit where two-electron processes are negligible compared to single-
electron ones, one can insert the Green functions of the double quantum dot,
calculated in the presence of interactions, into the formula we have obtained
for the finite-frequency noise in order to study their effects. This has been
done successfully in the case of a single quantum dot in the Kondo regime[33]
and has allowed to explain the main features of the experimental curves[36],
so it would be worthwhile to do it for a double quantum dot as well.

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