# Quantum metric on the Brillouin Zone in correlated electron systems and its
relation to topology for Chern insulators

Takahiro Kashihara  Department of Physics, Kyoto University, Kyoto 606-8502,
Japan  Yoshihiro Michishita  RIKEN Center for Emergent Matter Science (CEMS),
Wako, Saitama 351-0198, Japan  Robert Peters<EMAIL_ADDRESS>Department of Physics, Kyoto University, Kyoto 606-8502, Japan

(August 28, 2024)

###### Abstract

Geometric aspects of physics play a crucial role in modern condensed matter
physics. The quantum metric is one of these geometric quantities which defines
the distance on a parameter space and contributes to various physical
phenomena, such as superconductivity and nonlinear conductivity. Despite its
importance, the quantum metric in interacting systems is poorly understood. In
this paper, we introduce a generalized quantum metric(GQM) on the Brillouin
zone for correlated electron systems. This quantum metric is based on the
optical conductivity that is written by single-particle Green’s functions. We
analytically prove that this definition is equivalent to the existing
definition of the quantum metric in noninteracting systems and that it is
positive semi-definite as necessary for a metric. Furthermore, we point out
the relationship between the GQM and the Chern number in interacting systems.
We then numerically confirm these properties of the GQM in the Qi-Wu-Zhang
model with and without interaction. We believe that the GQM will be a step
toward generalizing the quantum metric to the interacting regime.

##  I Introduction

In various areas of physics, geometric aspects play an essential role in
understanding the fundamental structure, such as in the gauge theory or the
topological classification of defects[1]. In condensed matter physics, the
geometry of eigenstates in the momentum space has attracted much attention in
recent years [2, 3, 4, 5, 6]. One of the most important geometric quantities
in momentum space is the Berry curvature. The integral of the Berry curvature
over the momentum space is quantized, and the value is proportional to the
Chern number, which contributes to the quantum Hall effect [7].

The counterpart to the Berry curvature is the quantum metric. While the
imaginary part of the quantum geometric tensor is proportional to the Berry
curvature, the real part of the tensor is the quantum metric [8]. The quantum
metric is a metric that defines the distance on the momentum space. In recent
studies, it has been revealed that the quantum metric is equipped with rich
mathematical properties [9, 10, 11] and plays a crucial role in various fields
of condensed matter physics such as superconductivity [12, 13, 14, 15],
stability of the fractional quantum Hall states [16, 17], electrical
conductivity [18, 19, 20, 21, 22, 23, 24, 25, 26, 27], orbital magnetism [28],
nonadiabatic quantum evolution[29], energy shift in excitons [30], maximally
localized Wannier function [31], and non-Hermitian physics [32, 33].

The quantum metric has also been used as an important tool to characterize
quantum phase transitions since the pioneering works Ref. [34] and Ref. [35].
Although the integral of the quantum metric is not quantized, the relationship
between the quantum metric and topological invariants is also actively
investigated in recent studies. For example, inequalities between integrals of
the quantum metric (concretely, the trace of the quantum metric and the
quantum volume defined by the quantum metric) and the Chern number have been
noted [12, 36, 37, 38]. It was numerically confirmed that the Chern number is
inferred by the quantum volume [36]. Furthermore, the relation to fragile
topology [39, 13] and the winding number of the SSH chain (see the
supplemental material of Ref. [39]) have also been pointed out, and the
general relationship between the quantum volume and the winding number of a
chiral symmetric system, the Chern number, including the second Chern number,
are investigated in detail in Ref. [40].

The topology of systems has been intensively studied even in correlated
systems [41, 42]. One of the strategies employed is to describe the
topological invariant by the single-particle Green’s function[43, 44, 45, 46].
Since Green’s functions can be defined even in the presence of correlations,
topological invariants such as the TKNN invariant can be defined in correlated
systems. It has been shown that these topological invariants are also
quantized at zero temperature [43, 47]. The single-particle Green’s function
describes the quasi-particle states in the system, and the Chern number
defined by the Green’s function can be experimentally detected via the integer
quantum Hall conductivity in correlated materials.

On the other hand, the quantum metric in interacting systems is not well
understood. A very recent proposal of a definition of the quantum metric on
the momentum space in interacting systems is given in Ref. [48]. The authors
propose a generalization of the quantum metric using the relationship between
the quantum metric and the charge polarization susceptibility. In Ref. [48],
the quantum metric is called ”the dressed quantum metric.” They define the
dressed quantum metric using the Bloch wave function and the two-particles
Green’s function. Although they analyze the dressed quantum metric with
impurity scattering by approximating the two-particles Green’s function, the
behavior of the dressed quantum metric in a general interacting system and its
expression by the single-particle state are not investigated. Furthermore,
they do not mention the positive semi-definiteness of the dressed quantum
metric in interacting systems, which must be satisfied as a metric. Thus, the
quantum metric on the momentum space in interacting systems is still veiled.

Note that the quantum metric on the twist angle space can be defined even in
interacting systems by using the ground state wave function [22, 23, 24].
However, it is essentially different from the conventional definition of the
quantum metric on the momentum space[31] because it cannot be described in
single-particle states.

In this paper, we define a generalized quantum metric(GQM) on the momentum
space in interacting systems based on the single-particle Green’s function
without using Bloch wave functions. The GQM can be expressed by the
eigenvalues and the eigenvectors of the single-particle spectral function.
Thus, the GQM reflects the properties of the single-particle states in general
systems. Generalizing the quantum metric on the momentum space using the
single-particle spectral function will be a helpful tool in analyzing material
properties because the quantum metric should affect physical properties even
in interacting systems in the same way as in noninteracting systems [18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. We prove that this quantum metric
agrees with the usual definition of the quantum metric in noninteracting
systems and is positive semi-definite even in an interacting system, as
necessary for a metric. Furthermore, we show that the GQM is related to the
Chern number at $T=0$. We then use this definition to calculate the quantum
metric of an interacting system going through a topological phase transition
and show that the GQM is sensitive to topological phase transitions. While
previous works characterized quantum phase transitions by the quantum metric
calculated from the quantum states[34, 35, 36, 40], we utilize the Green’s
function to detect the transition.

The rest of the paper is organized as follows: In section II, we define the
generalized quantum metric in interacting systems and demonstrate its
correspondence to the usual quantum metric in noninteracting systems. We
furthermore prove that the GQM is positive semi-definite and relates to the
Chern number at $T=0$. In section III, we use this definition to study the
topological phase transition in interacting systems. Finally, we summarize and
conclude this paper in section IV.

##  II Quantum metric in correlated systems

###  II.1 Definition

In previous studies [22, 23, 24], the following relationship between the
quantum metric and the optical conductivity at $T=0$ has been pointed out in
noninteracting systems:

$\displaystyle\mathrm{Re}\int_{0}^{\infty}d\omega_{1}\;\frac{\sigma_{\alpha\beta}(\omega_{1})+\sigma_{\beta{\alpha}}(\omega_{1})}{2\pi\omega_{1}}$
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{d}}\int
d\bm{k}\;g_{\alpha\beta}(\bm{k}),$

where $\sigma_{\alpha\beta}(\omega)$ is the optical conductivity and $d$ is
the dimension of the system. Although the original work [23] deals with three-
dimensional systems, it is easily seen that the relation holds for any
dimension. Throughout this paper, we set the Planck constant and the lattice
constant to unity, $\hbar=a=1$. We also set the electron charge and the
Boltzmann constant to unity, $e=k_{B}=1$. Eq. (LABEL:SWM_k) holds for
noninteracting systems, where $g_{\alpha\beta}(\bm{k})$ is the existing
definition of the quantum metric on the Brillouin Zone (BZ).

Taking Eq. (LABEL:SWM_k) as a starting point, we propose a generalization of
$g_{\alpha\beta}$ on the BZ. In this paper, the generalized quantum metric is
called GQM. When defining the GQM, we impose the following conditions: (i) The
GQM is identical to the existing definition in noninteracting systems. (ii)
The GQM is positive semi-definite even in correlated systems. (iii) The GQM
can be expressed by the single-particle spectral function. (iv) The GQM is
related to the Chern number. We impose condition (i) so that we can regard the
GQM as a generalization of the conventional definition. The GQM should satisfy
(ii) because the GQM should serve as a metric. Condition (iii) is imposed
because the GQM should be based on single-particle states as in noninteracting
systems. In addition, condition (iv) is imposed because the relationship
between the conventional quantum metric and the Chern number has been pointed
out in noninteracting systems [12, 36, 37, 38, 40]. Although the existing
quantum metric has a rich mathematical structure[9, 10, 11], we do not
restrict it when defining the GQM. This paper aims to provide a step forward
in generalizing the quantum metric to interacting systems.

To satisfy the above conditions (i)-(iv), we define the GQM by the following
equation:

$\displaystyle\tilde{g}_{\alpha\beta}(\bm{k})$ $\displaystyle=$
$\displaystyle\mathrm{Re}\int_{0}^{\infty}d\omega_{1}\frac{\sigma_{\alpha\beta}(\omega_{1},\bm{k})+\sigma_{\beta{\alpha}}(\omega_{1},\bm{k})}{2\pi\omega_{1}}$
(2) $\displaystyle\sigma_{\alpha\beta}(\omega_{1},\bm{k})$ $\displaystyle=$
$\displaystyle-\frac{1}{\omega_{1}}\int\frac{d\omega}{2\pi}f(\omega)$ (3)
$\displaystyle\times\mathrm{Tr}\Bigl{[}J_{\alpha\beta}G^{R-A}(\omega)$
$\displaystyle\ \ \ \ \ \
+J_{\alpha}G^{R}(\omega+\omega_{1})J_{\beta}G^{R-A}(\omega)$ $\displaystyle\ \
\ \ \ \ +J_{\alpha}G^{R-A}(\omega)J_{\beta}G^{A}(\omega-\omega_{1})\Bigr{]},$

where $f(\omega)=1/(\exp(\beta\omega)+1)$ is the Fermi distribution function,
$J_{\alpha}=\frac{\partial H}{\partial k_{\alpha}}$, $J_{\beta}=\frac{\partial
H}{\partial k_{\beta}}$ are the current operators. $G^{R}$ and $G^{A}$ are the
retarded and advanced single-particle Green’s functions, respectively. We
define $G^{R-A}(\omega)=G^{R}(\omega)-G^{A}(\omega)$. We omit explicitly
writing the momentum dependence of $J$ and $G^{R/A}$. These single-particle
Green’s functions describe the single-particle properties of a system even in
the presence of correlations. They can be measured by experiments, such as
angle-resolved photoemission spectroscopy and tunneling experiments. Because
single-particle Green’s functions can include correlation effects via the
self-energy, this equation does describe not only noninteracting systems but
also correlated systems. The optical conductivity is expressed as the integral
of $\sigma_{\alpha\beta}(\omega_{1},\bm{k})$ [49]:

$\displaystyle\sigma_{\alpha\beta}(\omega_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{d}}\int
d\bm{k}\sigma_{\alpha\beta}(\omega_{1},\bm{k})$ (4)

We note, however, that vertex corrections are ignored when expressing the
optical conductivity by single-particle Green’s functions in Eq. (2) and Eq.
(4). We will show next that Eq. (2) corresponds to the usual definition of the
quantum metric in noninteracting systems and is positive semi-definite as
necessary for a metric. Furthermore, we will point out the relationship
between the GQM and Chern number at $T=0$.

###  II.2 Correspondence to noninteracting systems

First, we will show that the GQM coincides with the conventional quantum
metric in noninteracting systems.

Because the first term in Eq. (3) is generally purely imaginary, it can be
ignored. In a noninteracting system, the Hamiltonian can be diagonalized by
single-particle wave functions $|n\rangle$ yielding
$H|n\rangle=E_{n}|n\rangle$, and the Green’s function can also be diagonalized
by the same eigenvector as $G^{R/A}\ket{n}=(\omega-E_{n}\pm
i\eta)^{-1}\ket{n}$. Because $\eta\rightarrow 0$ holds in noninteracting
systems, $G^{R-A}\ket{n}=-2\pi i\delta(\omega-E_{n})\ket{n}$. Using these
eigenfunctions and equations, we can write the second term in Eq. (3) as

$\displaystyle\mathrm{Tr}\Bigl{[}{J_{\alpha}G^{R}(\omega+\omega_{1})J_{\beta}G^{R-A}(\omega)}\Bigr{]}$
$\displaystyle=-2\pi
i\sum_{n,m}\frac{(J_{\alpha})_{nm}(J_{\beta})_{mn}}{\omega+\omega_{1}-E_{m}+i\eta}\delta(\omega-
E_{n}).$ (5)

Similarly,

$\displaystyle\mathrm{Tr}\Bigl{[}{J_{\alpha}G^{R-A}(\omega)J_{\beta}G^{A}(\omega-\omega_{1})}\Bigr{]}$
$\displaystyle=2\pi
i\sum_{n,m}\frac{(J_{\alpha})_{nm}(J_{\beta})_{mn}}{\omega-\omega_{1}-E_{n}-i\eta}\delta(\omega-
E_{m}).$ (6)

Inserting Eq. (5) and (6) into Eq. (3), we obtain

$\displaystyle\mathrm{Re}\Bigl{[}{\sigma_{\alpha\beta}(\omega_{1},\bm{k})}\Bigr{]}$
$\displaystyle=\frac{1}{\omega_{1}}\sum_{nm}\Bigl{(}f(E_{n})-f(E_{m})\Bigr{)}\mathrm{Re}\Bigl{[}\frac{i(J_{\alpha})_{nm}(J_{\beta})_{mn}}{\omega_{1}-(E_{m}-E_{n})+i\eta}\Bigr{]}.$
(7)

Symmetrizing this formula, we find

$\displaystyle\mathrm{Re}\Bigl{[}{\frac{\sigma_{\alpha\beta}(\omega_{1},\bm{k})+\sigma_{\beta\alpha}(\omega_{1},\bm{k})}{2}}\Bigr{]}$
$\displaystyle=\frac{1}{2\omega_{1}}\sum_{n\neq
m}\Bigl{(}f(E_{n})-f(E_{m})\Bigr{)}$ $\displaystyle\ \ \ \
\times\mathrm{Re}\Bigl{[}\frac{i((J_{\alpha})_{nm}(J_{\beta})_{mn}+(J_{\beta})_{nm}(J_{\alpha})_{mn})}{\omega_{1}-(E_{m}-E_{n})+i\eta}\Bigr{]}$
(8)

By using
$(J_{\alpha})_{nm}(J_{\beta})_{mn}+(J_{\beta})_{nm}(J_{\alpha})_{mn}=2\mathrm{Re}[(J_{\alpha})_{nm}(J_{\beta})_{mn}]$,

$\displaystyle\mathrm{Re}\Bigl{[}{\frac{\sigma_{\alpha\beta}(\omega_{1},\bm{k})+\sigma_{\beta\alpha}(\omega_{1},\bm{k})}{2}}\Bigr{]}$
$\displaystyle=\frac{\pi}{\omega_{1}}\sum_{n\neq
m}\delta(\omega_{1}-(E_{m}-E_{n}))\Bigl{(}f(E_{n})-f(E_{m})\Bigr{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
\times\mathrm{Re}\Bigl{[}{(J_{\alpha})_{nm}(J_{\beta})_{mn}}\Bigr{]}$ (9)

For $T=0$, this can be written as

$\displaystyle\frac{\pi}{2\omega_{1}}\displaystyle\sum_{n:E_{n}<0}\displaystyle\sum_{m:E_{m}>0}((J_{\alpha})_{nm}(J_{\beta})_{mn}+(J_{\beta})_{nm}(J_{\alpha})_{mn})$
$\displaystyle\times[\delta(\omega_{1}-(E_{m}-E_{n}))-\delta(\omega_{1}-(E_{n}-E_{m}))].$
(10)

Finally, we can write the GQM as

$\displaystyle\tilde{g}_{\alpha\beta}(\bm{k})$ $\displaystyle=$
$\displaystyle\mathrm{Re}\;\int_{0}^{\infty}d\omega_{1}\frac{\sigma_{\alpha\beta}(\omega_{1},\bm{k})+\sigma_{\beta{\alpha}}(\omega_{1},\bm{k})}{2\pi\omega_{1}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\displaystyle\sum_{n:E_{n}<0}\displaystyle\sum_{m:E_{m}>0}\Bigl{[}\frac{\langle
n|\partial_{\alpha}H|m\rangle\langle
m|\partial_{\beta}H|n\rangle}{(E_{n}-E_{m})^{2}}$ $\displaystyle\ \ \ \ \ \
+(\alpha\leftrightarrow\beta)\Bigr{]}$ $\displaystyle=$
$\displaystyle\mathrm{Re}\displaystyle\sum_{E_{n}<0}\displaystyle\sum_{E_{m}>0}\frac{\langle
n|\partial_{\alpha}H|m\rangle\langle
m|\partial_{\beta}H|n\rangle}{(E_{n}-E_{m})^{2}},$ (12)

which agrees with the definition of the quantum metric in noninteracting
systems [31].

###  II.3 Proof of positive semi-definiteness

As a metric, the GQM must be positive semi-definite. This must hold in
interacting systems as well. The strategy employed here is to use the
eigenstates of $G^{R-A}(\omega)$ (not $G^{R}$ or only $G^{A}$). This can be
done by symmetrizing with respect to $\alpha$ and $\beta$. For example, after
symmetrizing Eq. (6),

$\displaystyle\mathrm{Re}\Bigl{[}{\mathrm{Tr}\Bigl{[}{J_{\alpha}G^{R}(\omega+\omega_{1})J_{\beta}G^{R-A}(\omega)+(\alpha\leftrightarrow\beta)}\Bigr{]}}\Bigr{]}$
$\displaystyle=\mathrm{Re}\Bigl{[}{\mathrm{Tr}\Bigl{[}{J_{\alpha}G^{R}(\omega+\omega_{1})J_{\beta}G^{R-A}(\omega)}}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
+\\{J_{\beta}G^{R}(\omega+\omega_{1})J_{\alpha}G^{R-A}(\omega)\\}^{\dagger}\Bigr{]}\Bigr{]}$
(13)
$\displaystyle=\mathrm{Re}\Bigl{[}{\mathrm{Tr}\Bigl{[}{J_{\alpha}G^{R-A}(\omega+\omega_{1})J_{\beta}G^{R-A}(\omega)}\Bigr{]}}\Bigr{]}$
(14)

Since $G^{A}=(G^{R})^{\dagger}$, $i(G^{R}-G^{A})$ is a Hermitian matrix. Thus,
$i(G^{R}-G^{A})$ is diagonalizable by a unitary matrix, and the eigenvalues
are real.

We denote the eigenvectors as $\ket{n(\omega)}$, and the corresponding
eigenvalues as $A_{n}(\omega)(\in\mathbb{R})$. Note that $A_{n}(\omega)\geq 0$
because $G^{R-A}(\omega)$ corresponds to the spectral function. Then, Eq. (14)
is equal to

$\displaystyle\displaystyle\sum_{n,m}-A_{n}(\omega)A_{m}(\omega+\omega_{1})\times\Bigl{(}\langle
n(\omega)|J_{\alpha}|m(\omega+\omega_{1})\rangle$ $\displaystyle\times\langle
m(\omega+\omega_{1})|J_{\beta}|n(\omega)\rangle+(\alpha\leftrightarrow\beta)\Bigr{)}/2.$
(15)

For the third term in Eq. (3), we utilize the same method, substituting

$\displaystyle\omega-\omega_{1}\rightarrow\omega,\ \ \ f(\omega)\rightarrow
f(\omega+\omega_{1})$ (16)

Finally, we obtain all terms which contribute to the GQM. The argument for the
frequency integration can be written as

$\displaystyle\displaystyle\sum_{n,m}A_{n}(\omega)A_{m}(\omega+\omega_{1})$
$\displaystyle\times\Bigl{(}\langle
n(\omega)|J_{\alpha}|m(\omega+\omega_{1})\rangle\langle
m(\omega+\omega_{1})|J_{\beta}|n(\omega)\rangle$
$\displaystyle+(\alpha\leftrightarrow\beta)\Bigr{)}\times\frac{f(\omega)-f(\omega+\omega_{1})}{\omega_{1}^{2}}\times\frac{1}{8\pi^{2}}.$
(17)

We define

$\displaystyle
h(n,m,\omega,\omega_{1})=A_{n}(\omega)A_{m}(\omega+\omega_{1})\frac{f(\omega)-f(\omega+\omega_{1})}{8\pi^{2}\omega_{1}^{2}}$
(18)

It is important to notice that $h(n,m,\omega,\omega_{1})$ is non-negative
because of the positive semi-definiteness of the spectral function and the
monotonicity of the Fermi distribution function. Then, for arbitrary real
numbers $c_{\alpha}$,

$\displaystyle\sum_{\alpha,\beta}c_{\alpha}\tilde{g}_{\alpha\beta}c_{\beta}$
$\displaystyle=\displaystyle
2\sum_{n,m,\alpha,\beta}\int_{0}^{\infty}d\omega_{1}\int_{-\infty}^{\infty}d\omega\;h(n,m,\omega,\omega_{1})\;$
$\displaystyle\ \ \ \times\langle
n(\omega)|c_{\alpha}J_{\alpha}|m(\omega+\omega_{1})\rangle\langle
m(\omega+\omega_{1})|c_{\beta}J_{\beta}|n(\omega)\rangle.$ (19)

Denoting

$\displaystyle|\phi_{n}(\omega)\rangle=\displaystyle\sum_{\alpha}c_{\alpha}J_{\alpha}|n(\omega)\rangle,$
(20)

we complete the proof of the positive semi-definiteness;

$\displaystyle\displaystyle\sum_{\alpha,\beta}c_{\alpha}\tilde{g}_{\alpha\beta}c_{\beta}$
$\displaystyle=$ $\displaystyle\displaystyle
2\sum_{n,m}\int_{0}^{\infty}d\omega_{1}\int_{-\infty}^{\infty}d\omega
h(n,m,\omega,\omega_{1})$ (21) $\displaystyle\
\times\langle\phi_{n}(\omega)|m(\omega+\omega_{1})\rangle\langle
m(\omega+\omega_{1})|\phi_{n}(\omega)\rangle$ $\displaystyle\geq$
$\displaystyle 0$

Thus, the GQM serves as a metric even if correlation effects are present. As
we can see from this proof, the GQM is expressed as

$\displaystyle\tilde{g}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{n,m}\int_{0}^{\infty}d\omega_{1}\int_{-\infty}^{\infty}d\omega
A_{n}(\omega)A_{m}(\omega+\omega_{1})$ (22)
$\displaystyle\times\Bigl{(}\langle
n(\omega)|J_{\alpha}|m(\omega+\omega_{1})\rangle\langle
m(\omega+\omega_{1})|J_{\beta}|n(\omega)\rangle$
$\displaystyle+(\alpha\leftrightarrow\beta)\Bigr{)}\times\frac{f(\omega)-f(\omega+\omega_{1})}{\omega_{1}^{2}}\times\frac{1}{8\pi^{2}}.$

The GQM is described by the single-particle spectral function and its
eigenvectors and does not include the Bloch wave function. This point is the
major difference between the GQM and the dressed quantum metric in Ref. [48].
We utilize the eigenvectors of the single-particle spectral function. They
coincide with the Bloch wave functions in noninteracting systems but are
generally different in interacting systems. The GQM can capture the feature of
the single-particle states even in correlated systems.

###  II.4 Relation to the Chern number

In two-dimensional systems, the following equation and inequality for the
quantum metric on the twist-angle space are satisfied at $T=0$:

$\displaystyle\mathrm{Re}\int^{\infty}_{0}d\omega_{1}\frac{\sigma_{xx}(\omega_{1})+\sigma_{yy}(\omega_{1})}{\pi\omega_{1}}$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int
d\bm{\kappa}\mathrm{Tr}g_{\mathrm{TBC}}(\bm{\kappa})$ (23)
$\displaystyle\geq\frac{1}{(2\pi)^{2}}\int
d\bm{\kappa}|\Omega_{\mathrm{TBC}}(\bm{\kappa})|$ (24)
$\displaystyle\geq\frac{1}{2\pi}|Ch|,$ (25)

where $g_{\mathrm{TBC}}(\kappa)$ and $\Omega_{\mathrm{TBC}}(\bm{\kappa})$ are
the quantum metric and the Berry Curvature introduced by imposing twisted
boundary conditions, respectively [22, 23, 24]. Eq. (23) is proportional to
the localization tensor, which measures the degree of the localization of the
wave function [23]. $Ch$ is the Chern number determined by
$\Omega_{\mathrm{TBC}}(\bm{\kappa})$. This Chern number is the so-called Niu-
Thouless-Wu invariant, which contributes to the quantized Hall conductivity in
interacting systems [50]. Eq. (23) has been pointed out in Ref. [23] assuming
the ground state is non-degenerate and gapped for any twist angle
$\bm{\kappa}$. When deriving the inequality Eq. (24), the general inequality
$\mathrm{Tr}g_{\mathrm{TBC}}(\bm{\kappa})\geq|\Omega_{\mathrm{TBC}}(\bm{\kappa})|$
is used [12]. Thus, the GQM is also expected to be large in Chern insulators
at $T=0$ due to the following inequality:

$\displaystyle\frac{1}{(2\pi)^{2}}\int d\bm{k}\mathrm{Tr}\tilde{g}$
$\displaystyle\approx$
$\displaystyle\mathrm{Re}\int^{\infty}_{0}d\omega_{1}\frac{\sigma_{xx}(\omega_{1})+\sigma_{yy}(\omega_{1})}{\pi\omega_{1}}$
(26) $\displaystyle\geq$ $\displaystyle\frac{1}{2\pi}|Ch|$ (27)

We note that this inequality is not rigorous because vertex corrections are
ignored in Eq. (26). In the next section, we numerically verify the
relationship between the GQM and the Chern number and check the above
inequality.

##  III Analysis of correlation effects on the GQM

In this section, we numerically confirm the properties of the GQM pointed out
in the previous section and investigate the relationship between the GQM and
the Chern number. For this purpose, we calculate the GQM for the following
Hamiltonian:

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{\mathrm{QWZ}}+H_{\mathrm{int}}$ (28) $\displaystyle H_{\mathrm{QWZ}}$
$\displaystyle=$
$\displaystyle\displaystyle\sum_{\bm{k}}(c^{\dagger}_{\bm{k},A}\;c^{\dagger}_{\bm{k},B})\;H(\bm{k})\;(c_{\bm{k},A}\;c_{\bm{k},B})^{T}$
(29) $\displaystyle H(\bm{k})$ $\displaystyle=$ $\displaystyle\sin
k_{x}\sigma_{x}+\sin k_{y}\sigma_{y}$ (30) $\displaystyle\ \ \ +(M+\cos
k_{x}+\cos k_{y})\sigma_{z}$ $\displaystyle H_{\mathrm{int}}$ $\displaystyle=$
$\displaystyle U\displaystyle\sum_{i}n_{i,A}n_{i,B},$ (31)

where $A$ and $B$ denote pseudo spin indices, $i$ labels the sites on the
lattice, and $\sigma_{i}$ are the Pauli matrices. $M$ plays the role of a
staggered potential [51](or magnetic field). $H_{\mathrm{QWZ}}$ is known as
the QWZ model, which is a toy model of a Chern insulator on a square lattice
introduced by X.-L. Qi, Y.-S. Wu, and S.-C. Zhang [52, 51]. The Chern number
of this model is $Ch=1$ when $0<M<2$ and $Ch=-1$ when $-2<M<0$. This model is
topologically trivial if $|M|>2$. $M=\pm 2$ and $M=0$ are quantum critical
points, for which the gap at the Fermi energy closes [51]. In our
calculations, we set the chemical potential as $\mu=-U/2$, corresponding to a
half-filled system.

###  III.1 $U=0$ cases

Before investigating the $U$-dependence of the GQM, we confirm the
correspondence to the existing definition of the quantum metric in the
noninteracting system in Eq. (12). When evaluating the GQM, all calculations
are performed at $T=0$. In our numerical calculations, we use $\eta=0.02$ in
the noninteracting Green’s function, $G^{R/A}=(\omega-H(\bm{k})\pm
i\eta)^{-1}$. $\eta$ can be regarded as an inverse lifetime of the
quasiparticles due to scattering in the model. Thus, the spectral function at
the Fermi energy becomes finite even in the insulating states due to the
finiteness of $\eta$. This results in a finite DC conductivity, leading to a
divergence of the GQM. To avoid such an artificial divergence, the frequency
integral in Eq. (2) needs to be cut at a small frequency of the order of
$\eta$. In this paper, we have used $\omega_{0}=0.01$.

In Fig. 1(top), we compare the momentum-dependence of $\mathrm{Tr}(g(\bm{k}))$
and $\sqrt{\mathrm{det}(g(\bm{k}))}$ calculated by the existing definition and
the GQM. We see that they coincide very well, but the GQM is a little larger,
particularly near the $\mathrm{M}$ point, where the gap is smallest. This
enhancement can be traced back to $\eta$ in the noninteracting Green’s
function. (A more detailed analysis of the $\eta$ dependence is given in
Appendix A.)

This broadening also results in a change of the behavior of
$\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}$, which is called quantum volume in
Ref. [36], in the noninteracting system, shown in Fig. 1(bottom). Near the
topological phase transition, $M=2$, a small peak is observed for
$\int\sqrt{\mathrm{det}(g)}_{\mathrm{GQM}}dk_{x}dk_{y}$ while the conventional
definition does not show such a peak.

Furthermore, peaks in $\int\mathrm{Tr}(g)dk_{x}dk_{y}$ of the GQM are observed
at the critical points ($M=0,\;2$). The reason for these peaks is a finite
longitudinal optical conductivity at small $\omega$. We note that we omitted
the point at $M=2$, where the existing definition of
$\int\mathrm{Tr}(g)dk_{x}dk_{y}$ diverges. Overall, we see that the existing
definition of the quantum metric and the GQM agree very well for small $\eta$.

In addition, Fig. (1)(bottom) shows that the inequality
$\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}\geq\pi|Ch|=\pi$ pointed out in Ref.
[36] is satisfied even for the GQM in the topological phase. This inequality
can be used to distinguish the topologically trivial phase from the
topologically nontrivial phase in the QWZ model,[36] and we will later show
this detection is possible even in the interacting QWZ model by using the GQM.
Furthermore, we see that $\int\mathrm{Tr}(g)dk_{x}dk_{y}$ and
$\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}$ tend to be larger in the topological
phase than in the trivial phase. (The same behavior for
$\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}$ has been confirmed numerically in
various models of Chern insulators [36].)

Figure 1: top: The momentum-dependence of the quantum metric calculated by the
existing formula (squares) compared to the GQM (Eq. (2)) (circles) at $M=1.7$
and $U=0$. bottom: $M$-dependence of the integral of the quantum metric
calculated by the existing formula (squares) compared to the GQM (Eq. (2))
(circles) at $U=0$. We note that we do not explicitly show $M=2$ as the
existing definition diverges at this point.

###  III.2 $U\neq 0$ cases

Next, we focus on the interacting system and show the results for $U>0$. We
analyze the effect of interaction by using the dynamical mean-field theory
(DMFT). DMFT self-consistently maps a lattice model to an impurity model [53].
DMFT neglects the momentum dependence of the self-energy but incorporates the
energy dependence of it [53]. It has been frequently utilized to analyze the
topology of correlated electron systems [54, 55, 56, 57, 58, 59, 60, 61, 62].
To solve the impurity model, we utilize the numerical renormalization group
(NRG) [63, 64].

Besides using the GQM, we use Wilson Loops to analyze the topology of the
system[51, 65, 66] focusing on the effective Hamiltonian [44, 45]

$\displaystyle H_{\mathrm{eff}}(\bm{k})=H(\bm{k})+\Sigma^{R}(\omega=0),$ (32)

where $\Sigma^{R}(\omega)$ is the self-energy of the retarded Green’s function
calculated by DMFT. Because $\omega=0$ lies in the gap in our setup, this
effective Hamiltonian can detect the topology of the system. In the numerical
calculations, we employ the discretized formula of the Wilson Loop,

$\displaystyle W(k_{x})=\langle
u(k_{x},2\pi)|\displaystyle\prod_{n=1}^{N}P(k_{x},2\pi
n/N)|u(k_{x},2\pi)\rangle,$ (33)

where $|u(k_{x},k_{y})\rangle$ is the occupied eigenstate of
$H_{\mathrm{eff}}(\bm{k})$ and $P(\bm{k})=|u(k_{x},k_{y})\rangle\langle
u(k_{x},k_{y})|$ is the projection operator. The product is arranged so that
$k_{y}$ increases when going from the left to the right. Furthermore, $N$ must
be large enough in a numerical calculation. A finite winding number is a
signal of a nontrivial topology of the filled bands [51, 65, 66]. If the
winding number is zero, the topology of the bands can be judged to be trivial.
Because the imaginary part of the self-energy at the Fermi energy is almost
zero (a typical value is $10^{-4}$) in the parameter region analyzed in our
numerical calculations, we can neglect the imaginary part. Thus, the effective
Hamiltonian is Hermitian, and we can utilize the eigenvectors corresponding to
the smallest eigenvalues when calculating Wilson loops.

It is expected that the system will undergo a topological phase transition as
$U$ increases. This phase transition can be understood as follows: The self-
energy affects the band structure around the Fermi energy by changing the
staggered potential to
$M_{\mathrm{eff}}=M+(\mathrm{Re}\Sigma^{R}(\omega=0)_{AA}-\mathrm{Re}\Sigma^{R}(\omega=0)_{BB})/2$,
where $\mathrm{Re}\Sigma^{R}(\omega=0)_{AA}$ and
$\mathrm{Re}\Sigma^{R}(\omega=0)_{BB}$ are the real parts of the diagonal
elements of the self-energies. Thus, we expect topological phase transitions
in the interacting system when $M_{\mathrm{eff}}=\\{-2,0,2\\}$.

In fact, for $U=3.5$, the system undergoes a topological phase transition from
the topological phase to the trivial phase as $U$ increases. This is confirmed
by the Wilson Loop, shown in Fig. 2(top). The winding number of the Wilson
Loop is one when $U<U_{c}\approx 3.5$, and it is zero when $U>U_{c}$.

Figure 2: top: The $k_{x}$-dependence of the phase of $W(k_{x})$ at $M=1.5$
for two different interaction strengths. The Wilson loop is winding for
$U=3.4$ (blue dots). The Wilson loop is not winding for $U=3.5$ (orange dots).
bottom: Interaction dependence of $\int\mathrm{Tr}(g)dk_{x}dk_{y}$(blue line)
and $\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}$(red line) calculated by Eq. (2).
At the transition point, the effective staggered potential (green line)
$M_{\mathrm{eff}}=M+(\mathrm{Re}\Sigma^{R}(\omega=0)_{AA}-\mathrm{Re}\Sigma^{R}(\omega=0)_{BB})/2=2$,
which is the same value as in the noninteracting system. The parameters are
$M=1.5,\mu=-U/2,\eta=0.02$.

In Fig. 2(bottom), we show the corresponding GQM. We see a small peak in
$\int\mathrm{Tr}(g)dk_{x}dk_{y}$ and $\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}$
at $U=3.5$ indicating the topological phase transition. This behavior is
similar to that in the noninteracting system. A small peak in
$\int\mathrm{Tr}(g)dk_{x}dk_{y}$ also appears in a correlated system at the
topological phase transition. The reason for this small peak is the finite DC
conductivity due to the gap closing. Remarkably, the inequality
$\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}\geq\pi$ still holds in the topological
phase of the correlated system. Due to the general inequality for the positive
semi-definite tensor $\mathrm{Tr}(g(\bm{k}))\geq
2\sqrt{\mathrm{det}(g(\bm{k}))}$, we see that
$\int\mathrm{Tr}(g)dk_{x}dk_{y}\geq 2\pi$ holds for the topological phase.
This corresponds to the inequality Eq. (27). Although vertex corrections are
ignored when deriving Eq. (27), the results show that the inequality holds
even in the interacting systems in this model. In addition,
$\int\mathrm{Tr}(g)dk_{x}dk_{y}$ and $\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}$
are larger in the topological phase compared to the trivial phase. Thus,
similar to the noninteracting system,
$\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}\geq\pi$ can be used as a probe to
judge the topology of the system.

We believe that the small peak in $\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}$ at
$U=3.5$ is caused by the small $\eta$. We note that the behavior of
$\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}$ in the correlated system near the
transition point is similar to that in the noninteracting system. The effects
of $\eta$ on the GQM are also investigated in Appendix A.

Next, we demonstrate that the GQM is positive semi-definite in the correlated
system. To demonstrate this, we show the momentum dependence of
$\mathrm{Tr}g(\bm{k})$ and $\mathrm{det}g(\bm{k})$ in Fig. 3(top). We note
that in a two-dimensional system, the positive semi-definiteness is equivalent
to $\mathrm{Tr}g(\bm{k})\geq 0$ and $\mathrm{det}g(\bm{k})\geq 0$ for any
$\bm{k}$.

Figure 3: top: Momentum-dependence of $\mathrm{Tr}g(\bm{k})$ and
$\mathrm{det}g(\bm{k})$ demonstrating the positive semi-definiteness of the
GQM. bottom: Spectral function
$A(\omega,\bm{k})=-\frac{1}{\pi}\text{Tr}\;\text{Im}G^{R}(\omega,\bm{k})$. The
gap becomes small near the $\mathrm{M}$ point. In both figures, the parameters
are $M=1.5,U=2.2,\mu=-U/2,\eta=0.02$.

In Fig. 3(top), we see that the GQM becomes large near the $M$ point
($k_{x}=k_{y}=\pi$). This is understood from the behavior of the spectral
function shown in Fig. 3(bottom). The energy gap is small in this region.
Thus, $\sigma_{\alpha\beta}(\omega_{1},\bm{k})$ becomes finite at small
$\omega$ and the GQM becomes large due to the factor $1/\omega_{1}$ in Eq.
(2).

Figure 4:  top: The $k_{x}$-dependence of the phase of $W(k_{x})$ at $U=2.2$.
It is winding when $M=1.67$(blue dots) and is not winding when $M=1.69$(orange
dots). bottom: The $M$-dependence of the $\int\mathrm{Tr}(g)dk_{x}dk_{y}$(blue
line) and $\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}$(red line) calculated by Eq.
(2). We also include $M_{\mathrm{eff}}$ as a green line, indicating that the
phase transition occurs at $M_{\mathrm{eff}}=2$. The parameters are
$U=2.2,\mu=-U/2,\eta=0.02$.

Finally, we repeat our analysis for $U=2.2$ varying $M$. The topology of the
system changes as $M$ is increased. From the Wilson loop, shown in Fig.
4(top), we see that a topological phase transition from the topological phase
to the trivial phase occurs at $M\approx 1.69$. Because of the presence of
interactions, the value at which the topological phase transition occurs is
reduced in the correlated system.

The corresponding $M$ dependence of the GQM is shown in Fig. 4(bottom). We see
that the GQM shows a small peak again exactly at the topological phase
transition. Furthermore, from the comparison between the values of the GQM on
both sides of the topological phase transition, we infer that the phase for
$M<1.69$ is topological nontrivial, in agreement with the Wilson loop
analysis. We see that the inequalities
$\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}\geq\pi$ and
$\int\mathrm{Tr}(g)dk_{x}dk_{y}\geq 2\pi$ hold again in the whole parameter
region of the topological phase.

##  IV Conclusions

In summary, we have defined a generalized quantum metric(GQM) on the BZ using
the single-particle Green’s function in interacting systems. The GQM is based
on the optical conductivity of the system. The major difference from the
previous work is that the GQM is written by the eigenvectors of the spectral
function, not the Bloch wave functions. Correlation effects are included via
the self-energy in the single-particle Green’s function. We have shown
analytically that the GQM agrees with the existing definition in a
noninteracting system at zero temperature and is positive semi-definite as
necessary for a metric. Furthermore, we have pointed out the relationship
between the GQM and the Chern number ignoring vertex corrections.

To numerically verify our results, we have analyzed the QWZ model, a toy model
of a Chern insulator, with additional repulsive interaction. We have used the
DMFT and NRG to obtain the self-energy of the system and calculate the GQM. To
further diagnose the topology of the system, we have computed the Wilson Loop
of the effective Hamiltonian at the Fermi energy.

With the help of the Wilson Loop analysis, we have found that the GQM tends to
be larger in the topologically nontrivial phase than in the trivial phase. In
addition, the inequality between the quantum volume and the Chern number still
holds in the correlated system. Thus, the GQM inherits the properties of the
conventional quantum metric even in interacting systems. We expect that the
GQM affects the physical quantity as in free systems, and this point is left
as future work.

## Acknowledgements

T. K deeply appreciates Ken Mochizuki, Tomoki Ozawa, Taisei Kitamura, and
Shuntaro Sumita for their fruitful discussions. Y. M. is supported by RIKEN
Special Postdoctoral Researcher Program and was partially supported by the
WISE program. R.P. is supported by JSPS, KAKENHI Grant No. JP18K03511. The
computation in this work has been done using the facilities of the
Supercomputer Center, the Institute for Solid State Physics, and the
University of Tokyo.

##  Appendix A Effect of dissipation on the GQM

Although $\eta$ should be infinitesimally small, it is finite in numerical
calculations. In a noninteracting system, if $\eta$ is infinitesimally small,
the integrand in Eq. 2 becomes finite only when $\omega=E_{n}$ and
$\omega+\omega_{1}=E_{m}$ due to the factor
$A_{n}(\omega)A_{m}(\omega+\omega_{1})$ in Eq. 22. It follows that the real
part of the symmetrized optical conductivity becomes finite only when the
input frequency $\omega_{1}$ is equal to $E_{m}-E_{n}$. However, when $\eta$
is finite, the delta peaks in $A_{n}(\omega)$ and $A_{m}(\omega+\omega_{1})$
obtain a finite width. Due to this broadening, the gap effectively becomes
smaller than it would be for an infinitesimally small $\eta$, resulting in an
increase of the GQM because of the factor $1/\omega_{1}$ in Eq. (2). To
confirm the validity of the above discussion, we investigated the $\eta$
dependence of the GQM in Fig. 5. This figure shows that the general behavior
of the GQM follows the behavior of the existing definition in the
noninteracting system for any $\eta$. Figure 5 shows that the GQM becomes
large when $\eta$ is increased. For any $\eta$, the difference between the
existing quantum metric and the GQM becomes maximal at the $\mathrm{M}$ point
of the BZ, where the gap is the smallest. Because $\eta$ can be regarded as a
scattering rate due to defects or impurities([19, 67, 20]), our result implies
that the GQM will increase in the presence of the defects or impurities.

Figure 5: Momentum resolved GQM in the noninteracting system ($M=1.7$ in the
Hamiltonian) for different $\eta$. In these figures, the difference between
the GQM and the existing definition is shown. The left (right) panel shows
$\mathrm{Tr}\Bigl{[}{g(k)}\Bigr{]}$ ($\sqrt{\text{det}(g(k)}$).

Finally, in Fig. 6, the $\eta$ dependence of the integral of the GQM is shown
for different interaction strengths. We see that the integral increases with
increasing $\eta$ and $\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}$ at $U=3.4$,
nearest to the topological phase transition, is the most sensitive to $\eta$.
These results suggest that a small peak of
$\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}$ in Fig. 2(bottom) and Fig. 4(bottom)
are numerical artifacts created by the finite $\eta$.

Figure 6: Dependence of $\int\mathrm{Tr}(g)dk_{x}dk_{y}$ and
$\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}$ on $\eta$ for different interaction
strengths. The points in the figure represent the value of the integrals
divided by $\int\mathrm{Tr}(g)dk_{x}dk_{y}$ at $\eta=0.02$ and
$\int\sqrt{\mathrm{det}(g)}dk_{x}dk_{y}$ at $\eta=0.02$

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# Event-Based Imaging of Levitated Microparticles

Yugang Ren  Joint first authors  Department of Physics, King’s College London,
Strand, London, WC2R 2LS, UK.  Enrique Benedetto  Joint first authors
Department of Physics, King’s College London, Strand, London, WC2R 2LS, UK.
Harry Borrill  Department of Physics, King’s College London, Strand, London,
WC2R 2LS, UK.  Yelizaveta Savchuk  Department of Physics, King’s College
London, Strand, London, WC2R 2LS, UK.  Molly Message  Department of Physics,
King’s College London, Strand, London, WC2R 2LS, UK.  Katie O’Flynn
Department of Physics, King’s College London, Strand, London, WC2R 2LS, UK.
Muddassar Rashid  Department of Physics, King’s College London, Strand,
London, WC2R 2LS, UK.  James Millen<EMAIL_ADDRESS>Department of
Physics, King’s College London, Strand, London, WC2R 2LS, UK.  London Centre
for Nanotechnology, Department of Physics, King’s College London, Strand,
London, WC2R 2LS, UK.

###### Abstract

ABSTRACT
Event-based imaging is a neuromorphic detection technique whereby an array of
pixels detects a positive or negative change in light intensity at each pixel,
and is hence particularly well suited to detecting motion. As compared to
standard camera technology, an event-based camera reduces redundancy by not
detecting regions of the image where there is no motion, allowing increased
frame-rates without compromising on field-of-view. Here, we apply event-based
imaging to detect the motion of a microparticle levitated under vacuum
conditions, which greatly facilitates the study of nanothermodynamics and
enables the independent detection and control of arrays of many particles.

††preprint: AIP/123-QED

When unravelling the underlying physics of particles interacting with external
forces, or of interacting multi-particle systems, object tracking is key. One
must consider a range of detection metrics such as field-of-view, resolution,
latency, sensitivity, bandwidth, signal-to-noise ratio (SNR) and the ability
to detect multiple objects.

In this work, we consider tracking the motion of microparticles levitated
under vacuum conditions by optical, electrical or magnetic fields Millen  _et
al._ (2020); Gonzalez-Ballestero  _et al._ (2021). Such systems are of
interest for studies of fundamental quantum science Millen and Stickler
(2020), nano-thermodynamics Millen and Gieseler (2018) and advanced sensing
Rademacher, Millen, and Li (2020); Moore and Geraci (2021).

When working with particles optically trapped in liquid, it is sufficient to
use standard CMOS or CCD cameras to track their motion, since the viscous
damping provided by the liquid reduces dynamical timescales to a level
suitable for camera frame-rates. However, once objects are levitated in
vacuum, their motion is underdamped and faster tracking is required Li  _et
al._ (2010). Although this is possible at the 100 kHz level with high-speed
CMOS cameras Svak  _et al._ (2021), this requires significant reduction in the
sensor resolution. Hence, particle tracking is usually performed using
photodiodes, balanced photodetectors or quadrant photodetectors. Whilst these
devices are fast, even tracking at GHz rates Ahn  _et al._ (2018); Reimann
_et al._ (2018), they have limited field-of-view, restricting tracking to
scales not much larger than the optical wavelength. This can limit studies of
nanothermodynamics or nonlinear motion where particles may explore large
regions of space Millen and Stickler (2020).

Figure 1: (a) Left image is taken with a conventional CMOS camera, right image
is taken by a camera using event-based imaging (EBI). The black and blue
pixels in the image correspond to negative / positive changes in light
intensity, respectively. (b) Information pipeline from the point of data
capture in an event-based camera (EBC).

Additionally, a key technique when working with levitated objects is the
application of real-time feedback onto their motion. This has enabled cooling
to the quantum ground state of a harmonic potential Magrini  _et al._ (2021),
but more generally is essential for stabilization under vacuum conditions.
Balanced photodetectors are the standard tool to realise real-time feedback
Gieseler  _et al._ (2012), although recent work employing powerful graphics
cards with a limited number of pixels Bullier, Pontin, and Barker (2019) or
on-board microprocessors Minowa  _et al._ (2022) has enabled feedback control
of sub-500 Hz oscillators via CMOS camera detection. We also note that cameras
have excellent SNR and allow super-resolution detection Bullier, Pontin, and
Barker (2019), allowing one to minimize the amount of light required to detect
levitated particles avoiding absorption Millen  _et al._ (2014) and photon
recoil heating Jain  _et al._ (2016).

Finally, the prospect of levitating systems of multiple interacting particles
has emerged Gonzalez-Ballestero  _et al._ (2021), for distributed sensing
Afek, Carney, and Moore (2022) or generation of entanglement Rudolph  _et al._
(2022). Single photodetectors can only track single particles, whereas cameras
are well suited to multi-particle detection.

Conventional, CMOS/CCD based cameras work using a specified region of interest
(ROI) or the whole pixel array to capture light from a scene. An alternative
approach is that of event-based imaging (EBI), where pixels work independently
of each other, triggering only when the change in light intensity is above a
preset threshold Gallego  _et al._ (2022). This enables a dynamic ROI, thus
enabling decreased informational load compared to conventional cameras.

In this paper, we apply EBI for detecting the motion of microparticles
levitated in vacuum. This imaging technique offers the potential for tracking
single and multiple objects with high bandwidth, whilst an integrated tracking
algorithm provides the real-time position of each object for use in feedback
for state control.

Conventional cameras, such as CMOS/CCD, capture continuous movement as a
sequence of still images (frames) formed from every pixel of the sensor. As a
result, stationary elements are unnecessarily replicated, while moving
elements are under-sampled Pro .

EBI provides an approach to image acquisition by only capturing changes in
images through the detection of modifications in light intensity on each pixel
Gallego  _et al._ (2022). Pixels in these neuromorphic sensors (sensors that
try to mimic the neural structure of the brain) are completely independent.
Each one of them contains a contrast detector (CD) that continuously tracks
photocurrents. When the variation of a photocurrent crosses a threshold, the
CD triggers a contrast detection event, which represents a relative increase
(positive) or decrease (negative) in light intensity. It then initiates the
measurement of a new value, as outlined in fig. 1(b). Pixels which do not
observe changes in light intensities that exceed the threshold do not produce
output.

This can be seen in fig. 1(a) where two images obtained with different cameras
are shown. The event-based image consists of only three colours: white pixels
indicate no change in light intensity across the threshold; blue and black
pixels represent positive and negative changes in light intensity across the
threshold, respectively. Only the blue and black pixels are transmitted as
data. By comparing both sets of images in fig. 1(a), it is clear that the
amount of data transferred is suppressed with EBI.

In conventional cameras the bandwidth of the communication link is usually a
constraint whenever higher acquisition rates are needed to track rapid
movement. This leads users to reduce the ROI of the sensor to decrease the
amount of data per frame Bullier, Pontin, and Barker (2019). Due to the
suppression of data redundancy, and the fact that EBI sensors have an
effective pixel depth of 1-bit (an event is either detected or not), the data
volume transmitted is considerably reduced, allowing acquisition rates over 1
GHz Gallego  _et al._ (2022).

Figure 1(b) shows the detailed process of object tracking based on EBI. In an
event-based camera (EBC), the EBI sensor is packaged with hardware which
performs object tracking, such that the output of the EBC lists the position
of each detected object as a function of time. All of the input information
related to the camera sensor is read, and in the setup period camera sensor
parameters are tuned. An event producer, which is contained in the event
buffer, is used to generate a stream of events. For each stream of events, a
noise filter is applied to pick up events in the neighboring 8 pixels during a
certain time. A trail filter then accepts an event if the last event is
detected at the same position within an accumulation time. All of the data
collected by the sensor pixels generates a frame and a proprietary generic
tracking algorithm (GTA) analyzes these frames to extract detected objects and
associate data to previous frames. The deployed tracking algorithm limits the
effective frame-rate of the EBC as compared to the read-out rate of the EBI
sensor. When the detected object is recognized, there is a trigger in callback
and the motional information of detected particle is obtained from output.

Figure 2: (a) Time domain motion of one degree-of-freedom of a levitated
microparticle, obtained via the GTA of an EBC (blue) and via a QPD (red). (b)
Corresponding PSDs, illustrating the varying noise characteristics of the two
detection methods.

Detection based on object tracking (rather than, for example, measuring the
intensity of light) has been shown to allow sub-pixel resolution and low noise
Bullier, Pontin, and Barker (2019). The dynamic range of the EBI sensor used
in this study 111Prophesee PPS3MVCD, $640\times 480$ pixels is 120 dB, which
is high compared to standard CMOS / CCD ($\sim 70\,$dB) or EMCCD ($\sim
100\,$dB) sensors. It is hard to quantify the effect of shot noise or dark
counts on the GTA, and hence make a direct comparison to photodiode-based
tracking.

Figure 3: (a) Schematic overview of experimental setup. Silica microspheres
are levitated in a Paul trap, illuminated by a laser, and the scattered light
is imaged onto an EBC or QPD. (b) EBC SNR at the motional frequency of the
levitated microparticle, as a function of the maximal scattered laser power
reaching the detector. (c) Average position of a levitated microparticle in
response to a potential difference across the endcap electrodes. It is evident
that the response of the EBC (blue) is linear across the full range of motion,
whereas the QPD (red) responds non-linearly at large displacements. (d)
Position probability histogram for the motion of a levitated microparticle,
recorded by the EBC (blue) and QPD (red), where it is evident that the QPD
cannot pick up large displacements.

We use an EBC222Prophesee Evaluation Kit Gen3M VGA CD 1.1 to track the motion
of levitated microspheres. Charged silica microspheres of diameter 5 $\mu$m
and charge $Q$ are levitated in partial vacuum using a Paul trap, made with
four 3mm-diameter rods and two 1mm-diameter endcap rods (not shown, aligned
axially along the centre of the structure), as illustrated in fig. 3(a). The
microsphere is trapped using an oscillating electric field with frequency
$\Omega_{RF}=2\pi\times 800$ rad s-1 and amplitude $V_{RF}=750$ V. The two
endcap electrodes are held at -4V DC. The particles are loaded into the Paul
trap using laser-induced acoustic desorption (LIAD) Bykov  _et al._ (2019);
Nikkhou  _et al._ (2021) at a pressure of $2\times 10^{-2}$ mbar.

An illumination laser (532 nm) is focused onto the trapped particle, with a
beam waist of $\sim 80\,\mu$m. The scattered light is collected by a
microscope ($16\times$ magnification) and directed onto the EBC for motion
detection. In fig. 3(b) we present the SNR at the motional frequency of the
particle as recorded by the EBC, as we vary the power of the illumination
laser (see Supplementary Information IV). We estimate the maximal scattered
power reaching the EBC sensor, illustrating the excellent sensitivity of the
detector.

Figure 2(a) shows the output from the EBC compared to the output of a quadrant
photodiode (QPD) 333New Focus 2901 Quadrant Cell Photoreceiver, when tracking
the oscillatory motion of a microparticle using identical imaging systems. The
GTA acts like an effective filter, removing high-frequency noise. Figure 2(b)
shows the corresponding derived power spectral densities (PSDs) of the
particle motion. Like all balanced detection methods, the QPD minimizes $1/f$
noise at low frequencies, whereas the EBC has lower-noise performance at
higher frequencies.

Regardless of the detection method used, the detector has to be calibrated.
The motion of the particle is calibrated by applying a known potential
difference $\Delta U$ across the endcap electrodes, causing the particle to
move in the $z$-direction. The particle oscillates at the centre of the trap
when $\Delta U=0$ V. If the voltage of one of the endcaps is modified from
$U_{0}\to U_{1}$, the resulting voltage difference $\Delta U=U_{1}-U_{0}$
exerts a force on the particle $\vec{F}=F_{z}\hat{z}$, where $\hat{z}$ denotes
a unit vector along the $z$-axis, determined by:

$F_{z}=\frac{Q\left.\Delta U\right.}{d}\>,$ (1)

where $d$ is the distance between the endcap electrodes. We confirm this
analytic model for our specific trap geometry in Supplementary Information II.
Since the Paul trap provides a harmonic pseudo-potential, the particle also
experiences a linear restoring force:

$F_{z}=-k\left<z\right>\>,$ (2)

where $k$ is the trap stiffness and $\left<z\right>$ is the average position
of the particle (assuming $\left<z\right>=0$ when $\Delta U=0$).

Noting that the measured $\langle z_{m}\rangle$ is in volts for the QPD, and
pixels for the EBC, and related to the true value of $\langle z\rangle$
through a conversion factor $\gamma$, with units V/m and pixels/m,
respectively, we can equate these two equations. Considering further that
$k=m\omega^{2}_{z}$ for a harmonic oscillator of mass $m$ and oscillation
frequency $\omega_{z}$, then:

$\left<z_{m}\right>=-\gamma\frac{Q}{m}\frac{\left.\Delta
U\right.}{d\omega^{2}_{z}}\>.$ (3)

By measuring the mean displacement of the particle as $\Delta U$ is varied, we
can determine the conversion factor $\gamma$ for each detector. The
oscillation frequency $\omega_{z}$ can be obtained from the power spectral
density (fig. 2(b)) and $Q/m$ obtained by solving the Mathieu equations (see
Supplementary Information I). In our system we typically trap particles of
charge $Q=2\times 10^{4}\,e$.

The calibrated particle position as a response to a potential difference can
be seen in fig. 3(c). It is apparent that as $\left|\Delta U\right|$ becomes
large, the response of the QPD becomes nonlinear, whereas eqn. (3) predicts a
linear response. This can be somewhat mitigated through a nonlinear
calibration (see Supplementary Information II), but limits the field-of-view
of the QPD. On the other hand, the EBC has a linear response to the particle
displacement across the full range. Figure 3(d) shows a histogram of the
equilibrium motion of the trapped particle, again showing that large
displacements are missing when using the QPD. Hence, we can track both small
(i.e. the oscillatory motion) and large displacements using EBI.

Levitated particles provide an ideal system for probing stochastic
thermodynamics Millen and Gieseler (2018). This is in part due to the
characteristic energy of the system being comparable to that of thermal
fluctuations of the bath, which enable levitated systems to be highly
sensitive to surrounding fluctuations. The coupling to the bath is
characterized by the ratio of its oscillation frequency $\omega_{z}$ to its
momentum damping rate $\Gamma$, yielding overdamped ($\Gamma\gg\omega_{z}$)
and underdamped ($\Gamma\ll\omega_{z}$) regimes. In the overdamped regime,
trapped particles have been used for studying heat engines Martinez  _et al._
(2017), non-thermal baths Wei and Ou-Yang (2010), and for testing Landauer’s
principle Martinez  _et al._ (2015). In the underdamped regime, the studies in
stochastic thermodynamics have been extended to observing Kramer’s turnover
Rondin  _et al._ (2017), studying non-equilibrium Landauer’s principle Jun,
Gavrilov, and Bechhoefer (2014) and testing fluctuation-dissipation theorems
Hoang  _et al._ (2018).

Often when such systems exhibit non-equilibrium dynamics, they go beyond the
linear detection regime of photodiode-based detection systems, and CMOS
cameras would need a large ROI to capture the dynamics, at the expense of
bandwidth.

EBI employs a dynamic ROI, based on only triggered pixels. To characterise the
capability of EBI, and the GTA of the EBC, we cause random jumps in the
particles’ position with varying time intervals, $\tau$ (see Supplementary
Information III). These random jumps, as seen in fig. 4(a), are driven by
voltage changes applied to the endcap electrodes which follow telegraph noise
statistics (see Supplementary Information III) distributed about a mean
waiting time $\bar{\tau}$. For fast switching times relative to the gas
damping rate $\Gamma_{0}$, the position probability distribution of the
particle is Gaussian, and for slow switching times it is bimodal, as shown in
fig. 4(b). Comparing to fig. 3(c), the particle’s motion would go deep into
the nonlinear range of the QPD. However, the EBC has no such limitation, and
position shifts greater than $100\,\mu$m are tracked, without compromising on
position sensitivity, which is approximately $30\,$nm Hz-1/2 in this work for
both devices, as can be extracted from fig. 2(b). For the QPD to track such a
range of motion, one would have to use a lower magnification imaging system,
with a corresponding reduction in position sensitivity.

Figure 4: (a) Particle motion, as tracked by an EBC, when the particle is
driven to make large jumps in position. Different coloured lines indicate
different jump time-constants $\tau$. (b) Position probability distributions
for different jump time-constants. As predicted, when $\tau=1\times 10^{-4}$s,
the distribution is Gaussian, and for slower jump time-constants the
distribution is bimodal.

The factor limiting the bandwidth of our EBC is the GTA. For shifts in
position that are in quick succession and large in displacement, the GTA lags
behind or misses consecutive shifts. This is illustrated in fig. 4(a), where
the GTA of the EBC faithfully tracks the particle position when
$\bar{\tau}>1\,$s. For shorter mean waiting times, $\bar{\tau}<500\,$ms the
GTA struggles to track the jumps, as evidenced by apparent spikes in the time
domain signal. The GTA doesn’t represent a true bandwidth limit to EBI and
advantageously, as shown in fig. 4(b), our EBC can simultaneously track large
displacements and the smaller oscillations of the particle about its
equilibrium position.

We extend our study to multi-particle tracking using EBI. The ability to track
arrays of particles would enable the study of quantum correlations Kotler  _et
al._ (2021); Brandão, Tandeitnik, and Guerreiro (2021); Mercier de Lépinay
_et al._ (2021); Roque and Roversi (2012), non-hermitian systems Qi  _et al._
(2020), and the detection of dark matter Carney  _et al._ (2021); Moore and
Geraci (2021), vacuum friction Zhao  _et al._ (2012) and differential force-
sensing Rudolph  _et al._ (2022). To date, few have experimentally explored
multi-particle physics, with only two particles trapped to demonstrate state-
swapping and sympathetic cooling Penny, Pontin, and Barker (2021), dipole-
dipole Rieser  _et al._ (2022); Arita, Wright, and Dholakia (2018) and
coulomb-coulomb Slezak and D’Urso (2019) interactions, and cold damping and
state control Vijayan  _et al._ (2022).

Motional detection of individual particles in a multiple particle trap is a
non-trivial problem. A single focused light beam can carry motional
information of two particles Praveen  _et al._ (2016), or an additional beam
can be added for detection of the second particle Rieser  _et al._ (2022).
Going beyond two particles in this way would require a complex and non-
scalable optical setup. Within the context of optical tweezers, there are
numerous approaches most suited to tracking multiple particles, using high-
speed cameras Gibson  _et al._ (2008) and multiple-beams in conjunction with a
QPD Ott  _et al._ (2014), but these detection methods face the same
limitations outlined earlier in this paper.

Figure 5: (a) EBC image of two microspheres of $5\,\mu$m diameter trapped in a
Paul trap. The image shows the bounding boxes which the GTA uses to track the
two-dimensional position of the particles, labelled "986" and "987". (b) EBC
image illustrating tracking of five particles. (c) PSDs reconstructed from the
output of the GTA for the particles in (a).

Figure 5(a) shows an image of two microspheres of $5\,\mu$m diameter trapped
in a Paul trap, captured on our EBC. The image shows the bounding boxes which
the GTA uses to track the two-dimensional position of the particles, labelled
"986" and "987".

The levitated microspheres are tracked and their motional information
separately reconstructed in fig. 5(c). We observe oscillation frequencies of
both particles independently. For particle ID 986, the motional frequencies
are $\\{\omega_{x}^{986}=2\pi\times 68\,$rad s-1, $\omega_{y}^{986}=2\pi\times
114\,$rad s-1, $\omega_{z}^{986}=2\pi\times 108\,$rad s${}^{-1}\\}$, whilst
particle ID 987 has frequencies $\\{\omega_{x}^{987}=2\pi\times 20\,$rad s-1,
$\omega_{y}^{987}=2\pi\times 122\,$rad s-1, $\omega_{z}^{987}=2\pi\times
184\,$rad s${}^{-1}\\}$. We observe all three centre-of-mass degrees-of-
freedom due to imperfect alignment between the coordinate axis of the Paul
trap and our imaging system. The additional frequency components in the
spectrum for ID 987 could be other degrees-of-freedom (e.g. librational) or
evidence of multi-particle collective modes, but that is beyond the scope of
this study.

The EBC is not limited to tracking two particles, and in fig. 5(b) we show
that the GTA has identified 5 separate particles. Due to varying $Q/m$, each
particle has different resonant motional frequencies, and hence it is possible
to individually excite them. We believe this opens the door to the study of a
wide range of non-equilibrium phenomena.

In conclusion, we have shown that EBI is an interesting alternative to
conventional detection schemes used for tracking levitated particles. The key
enabling feature of EBI is the low data transfer, which enables EBC to track
multiple objects at higher speeds than conventional cameras, and when combined
with natural low pixel latencies Gallego  _et al._ (2022) will allow the
experimenter to implement real-time feedback for state control. In this study,
the bandwidth of tracking is limited to 1 kHz by the very general and
proprietary GTA employed. The underlying dynamics of our system are well
known, therefore a more precise filter, like an asynchronous Kalman filter
Wang  _et al._ (2021); Afshar  _et al._ (2020) in which we can input the
expected equation of motion, will enable faster and more accurate tracking.

As compared to photodiode-based detection schemes, EBCs feature a dynamic ROI,
enabling tracking over a wide field-of-view, with particular application in
the study of non-equilibrium physics. We have demonstrated tracking over 100
micrometres whist retaining 30 nm Hz-1/2 resolution. Finally, we have
introduced an imaging technique suitable for fast tracking of a large number
of particles, reaching MHz rates with the application of tailored particle
tracking algorithms.

## Supplementary Information

Supplementary Material is provided for detailed information about how to solve
charge-mass ratio, conduct nonlinear calibration of QPD, generate random jumps
and obtain signal-to-noise ratio.

JM would like to thank Dr. John Dale for the inspiration to start this
project. This research has been supported by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (Grant Agreement Nos. 803277 & 957463) and by EPSRC New Investigator
Award EP/S004777/1.

##  I Calibration: Solving for $Q/m$

Considering that our system follows a Mathieu equation of motion then the
frequency of the different degrees-of-freedom can be written as,

$\omega_{i}\cong\frac{1}{2}\Omega_{RF}\sqrt{a_{i}+\frac{1}{2}q_{i}^{2}},$ (4)

where $\omega_{i}$ the harmonic oscillation frequency, $\Omega_{RF}$ is the
driving RF frequency, and $a_{i}$, $q_{i}$ with $i=\\{x,y,z\\}$ are known as
the stability parameters Drewsen and Brøner (2000).

Using the stability parameters we can write a general statement for all three
axis:

$\omega^{2}_{i}=q_{m}^{2}\left(\frac{V_{RF}^{2}}{2\Omega_{RF}^{2}r^{4}}\alpha_{i}^{2}\right)+q_{m}\left(\frac{2U}{d^{2}}\right)\beta_{i},$
(5)

where we $q_{m}=Q/M$ is the charge-mass ratio, $V_{RF}$ is the RF voltage
amplitude, $U$ is the DC voltage at the endcaps, $r$ is the distance between
RF electrodes from the centre of the trap and $d$ is the distance between the
endcap electrodes. The variables $\alpha_{i}$ and $\beta_{i}$ are geometric
factors that are obtained through SIMION numerical simulation of the Paul trap
used in the experiments (see Table S1) for each axis. The $Q/m$ ratio can then
be obtained by solving for $q_{m}$.

Table S1: Geometric factors obtained from SIMION numerical simulation of our Paul Trap. | $x$ | $y$ | $z$
---|---|---|---
$\alpha$ | 2.32 | 3.74 | 6.37
$\beta$ | -0.090 | -0.090 | -0.198

##  II Calibration: Nonlinear Calibration with the QPD

To calibrate the system and obtain a conversion function we apply an electric
force, $F_{z}$ to the particle via a potential difference $\Delta U$ across
the endcap electrodes:

$F_{z}=\frac{Q\Delta U}{d},$ (6)

where $d$ is the spacing between the endcap electrodes, which is equal to
spring force experienced in the trap: $m\omega^{2}_{z}\langle z\rangle$. When
equated with the above equation we get a theoretical value for positional
shift in metres for an applied voltage difference $\Delta U$:

$\langle z\rangle=\frac{Q}{m}\frac{\Delta U}{\omega_{z}^{2}d},$ (7)

FIG. S6:  (a) Paul trap geometry in SIMION simulation environment which is set
with the same size of experimental setup. (b) The mean position variations of
the trapped particle with different voltage differences of two endcap rods,
where a clear linear relationship can be seen.

To further verify eqn. S7, simulations based on SIMION are conducted. Figure
S6(a) shows a trap geometry in the simulation environment which is with the
same size as our experimental system. Figure S6(b) shows that when the voltage
of one endcap is tuned by -0.5V to 0.5V, related mean positions of the trapped
particle linearly changes, verifiying the analytical expression used in the
manuscript

The quadrant photodetector (QPD) has a nonlinear response to the movement of
an image if the image deviates too far from the centre, i.e. if the particle
moves too far from the centre of the trap. We can fit an error function to the
mean measured position of the particle (recorded in Volts, $\langle
x_{V}\rangle$), with the expected actual position $\langle x_{m}\rangle$ based
on the amount of applied potential difference, eqn. S7.

$\langle x_{V}\rangle=a+b*\rm erf(\it\frac{\langle x_{m}\rangle_{m}-c}{d}),$
(8)

where $(a,b,c,g)$ are fitting constants obtained by fitting the above equation
to the data as shown in fig. S7. The inverse of this error function then
enables us to convert the positional information in volts $\langle
x_{V}\rangle$ to metres $\langle x_{m}\rangle$.

$\langle x_{m}\rangle=g\,\rm{erf}^{-1}{\it\left(\frac{\langle
x_{V}\rangle-a}{b}\right)+c}.$ (9)

This enables us to extend the detection range of the QPD by a few microns, but
not by the tens of microns offered by the event-based camera.

FIG. S7:  Error function fitted to the measured mean position in volts,
$\langle x_{V}\rangle$ of the particle when a potential difference $\Delta U$
is applied across the endcaps. The measured position $\langle x_{V}\rangle$ is
converted to the actual position $\langle
x_{m}\rangle=\frac{qV}{m\omega^{2}d}$. The inverse of the error function then
converts Volts into metres

##  III Generating Random Jumps

Random jumps in position of the particle are implemented by applying telegraph
noise statistics to the particle. To achieve this, we generate random numbers
normally distributed, which then are multiplied by a mean waiting time
$\bar{\tau}$. The waiting time is used to delay a voltage applied to the
particle $\pm V_{\rm{tel}}$ via the endcap electrodes. This applied voltage is
experienced by the trapped particle as an electric force,
$\pm\eta_{\rm{tel}}=\frac{Q}{m}V_{\rm{tel}}/(d)$. In this work,
$V_{\rm{tel}}=\pm 1\,$V.

This applied force has noise statistics governed by:

$\langle\eta_{t}\eta_{t}\rangle=\eta_{\rm{tel}}^{2}e^{(-2|t-t^{\prime}|/\tau)}.$
(10)

##  IV Detection Signal-to-noise ratio (SNR)

We measure the SNR of the EBC at the resonant frequency of the particle motion
$f_{z}$. To do this, we fit the following model to the noise floor of the
detector, by analysing its power spectral density (PSD):

${\rm Noise}(f)=\log_{10}(\frac{a_{0}}{f}+\frac{b_{0}}{f^{2}}),$ (11)

where $a_{0},b_{0}$ are fitting constants. We found that both terms were
required to get a good fit. We fit the PSD of the particle motion with a
standard modelMillen and Gieseler (2018), with the above noise model added:

${\rm
Signal}(f)=\log_{10}(\frac{ac}{(b^{2}-f^{2})^{2}+(cf)^{2}}+\frac{a_{0}}{f}+\frac{b_{0}}{f^{2}}),$
(12)

where $a,b,c$ are fitting constants.Therefore, the SNR at the motional
frequency is obtained via ${\rm SNR}(f_{z})={\rm Signal}(f_{z})-{\rm
Noise}(f_{z})$.

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