# Quantum nonlinear spectroscopy via correlations of weak Faraday-rotation
measurements

Brian Chung Hang Cheung  Department of Physics, The Chinese University of Hong
Kong, Shatin, New Terrotories, Hong Kong, China  Ren-Bao Liu
<EMAIL_ADDRESS>Department of Physics, The Chinese University of Hong Kong,
Shatin, New Terrotories, Hong Kong, China  Centre for Quantum Coherence, The
Chinese University of Hong Kong, Shatin, New Terrotories, Hong Kong, China
The Hong Kong Institute of Quantum Information Science and Technology, The
Chinese University of Hong Kong, Shatin, New Terrotories, Hong Kong, China
New Cornerstone Science Laboratory, The Chinese University of Hong Kong,
Shatin, New Terrotories, Hong Kong, China

###### Abstract

The correlations of fluctuations are key to studying fundamental quantum
physics and quantum many-body dynamics. They are also useful information for
understanding and combating decoherence in quantum technology. Nonlinear
spectroscopy and noise spectroscopy are powerful tools to characterize
fluctuations, but they can access only very few among the many types of
higher-order correlations. A systematic quantum sensing approach, called
quantum nonlinear spectroscopy (QNS), is recently proposed for extracting
arbitrary types and orders of time-ordered correlations, using sequential weak
measurement via a spin quantum sensor. However, the requirement of a central
spin as the quantum sensor limits the versatility of the QNS since usually a
central spin interacts only with a small number of particles in proximity and
the measurement of single spins needs stringent conditions. Here we propose to
employ the polarization (a pseudo-spin) of a coherent light beam as a quantum
sensor for QNS. After interacting with a target system (such as a transparent
magnetic material), the small Faraday rotation of the linearly polarized light
can be measured, which constitutes a weak measurement of the magnetization in
the target system. Using a Mach-Zehnder interferometer with a designed phase
shift, one can post-select the effects of the light-material interaction to be
either a quantum evolution or a quantum measurement of the material
magnetization. This way, the correlated difference photon counts of a certain
numbers of measurement shots, each with a designated interference phase, can
be made proportional to a certain type and order of correlations of the
magnetic fluctuations in the material. The analysis of the signal-to-noise
ratios shows that the second-order correlations are detectable in general
under realistic conditions and higher-order correlations are significant when
the correlation lengths of the fluctuations are comparable to the laser spot
size (such as in systems near the critical points). Since the photon sensor
can interact simultaneously with many particles and the interferometry is a
standard technique, this protocol of QNS is advantageous for studying quantum
many-body systems.

††preprint: APS/123-QED

##  I Introduction

The correlations of fluctuations are key to studying the foundation of quantum
mechanics (such as via the Bell inequalities [1, 2] and the Leggett-Garg
inequalities [3]) and the dynamics and thermodynamics of quantum many-body
systems [4, 5]. They are also useful information for characterizing the noises
and environments in quantum technology [6, 7, 8, 9]. The second-order
correlations have been widely studied, such as in linear optics, transport,
thermal response, magnetic response, and neutron scattering. Higher-order
correlations are usually ignored by the assumption that they are weaker than
second-order correlations, or they can be factorized into second-order
correlations by Wick’s theorem when the fluctuations are Gaussian (which is
often the case in macroscopic systems far away from the critical points) [9].
However, there are important scenarios where the irreducible higher-order
correlations are significant. One such case is the critical fluctuations near
the phase transitions [10, 11]. Recent studies also show the importance of
high-order correlations in quantum sensing [12, 13, 14], in mesoscopic quantum
systems [15, 16, 17], and in spin systems under external noises [18]. The
characterization of higher-order correlations thus is desirable [19, 20, 21].

There is a rich structure in higher-order correlations of quantum fluctuations
[22]. Corresponding to different orderings of quantum operators, there are in
general $K!$ independent correlations among $K$ different quantities
$\hat{B}_{1},\hat{B}_{2},\ldots,\hat{B}_{K}$, given by, e.g.,
$\left\langle\hat{B}_{\sigma(1)}\hat{B}_{\sigma(2)}\cdots\hat{B}_{\sigma(K)}\right\rangle$
(with $\sigma$ denoting an element of the permutation group $S_{K}$ and
$\langle\cdots\rangle$ denoting the average over the system density operator).
For $K$ quantities at different times (specifically,
$\hat{B}_{1},\hat{B}_{2},\ldots,\hat{B}_{K}$ at $t_{1},t_{2},\ldots,t_{K}$
correspondingly with $t_{1}\leq t_{2}\cdots\leq t_{K}$), most of these
correlations are the so-called out-of-time-order correlators (OTOCs), which do
not directly affect the dynamics of the system. There are $2^{K-1}$
independent time-ordered correlations, corresponding to different arrangements
of the operators $B_{k}$ on the two branches of the contour evolution (i.e.,
on the left or right side of the system density operator $\hat{\rho}$). The
case for $K=2$ has been well studied in the form of Keldysh-Kadanoff-Baym
Green’s functions [23, 24]. The time-ordered higher-order correlations can be
formulated in an intriguing basis using the commutators or anti-commutators
between the operators. We introduce the superoperators $\mathbb{B}^{\pm}_{k}$
at time $k$ with $\mathbb{B}^{+}_{k}$ corresponding to an anticommutator,
defined as
$\mathbb{B}^{+}_{k}\hat{\rho}\equiv\left(\hat{B}_{k}\hat{\rho}+\hat{\rho}\hat{B}_{k}\right)/2$,
and $\mathbb{B}^{-}_{k}$ corresponding to a commutator, defined as
$\mathbb{B}^{-}_{k}\hat{\rho}\equiv\left(\hat{B}_{k}\hat{\rho}-\hat{\rho}\hat{B}_{k}\right)/i$.
The $2^{K-1}$ types of time-ordered $K$th-order correlations are
$C^{\eta_{K}...\eta_{1}}=\Tr{\mathbb{B}^{\eta_{K}}_{K}...\mathbb{B}^{\eta_{2}}_{2}\mathbb{B}^{\eta_{1}}_{1}\hat{\rho}}$,
where $\hat{\rho}$ is the initial state of the system and
$\eta_{k}\in\\{+,-\\}$. Note that if $\eta_{K}=-$, the correlation always
vanishes since the trace of a commutator is zero. For $\eta_{K}=+$, the
correlations can also be written as

$\displaystyle C^{+\eta_{K-1}\cdots\eta_{1}}$ $\displaystyle={\rm
Tr}\left[\hat{\rho}\left(\mathbb{B}^{\eta_{1}}_{1}\cdots\mathbb{B}^{\eta_{K-1}}_{K-1}\hat{B}_{K}\right)\right]$
$\displaystyle\equiv\eta_{1}\eta_{2}\cdots\eta_{K-1}\left\langle\mathbb{B}^{\eta_{1}}_{1}\cdots\mathbb{B}^{\eta_{K-1}}_{K-1}\hat{B}_{K}\right\rangle.$

This rich structure of correlations is important in the dynamic of quantum
system [20] but however have been largely unexplored due to the limitation of
conventional spectroscopy [25, 26, 12, 27, 18, 28, 5]. Conventional nonlinear
spectroscopy [25], using classical probes, measures only one of the $2^{K-1}$
type of correlations, namely,
$C^{+--...-}=i^{K-1}\left\langle\left[\hat{B_{1}},\left[\hat{B}_{2},\ldots\left[\hat{B}_{K-1},\hat{B}_{K}\right]\right]\right]\right\rangle$,
corresponding to the $(K-1)$-th order nonlinear susceptibility, while noise
spectroscopy usually measures only $C^{++...+}$ [26, 12, 27, 18, 28, 5].

It is recently proposed that arbitrary types and orders of time-ordered
correlations can be extracted by a systematic quantum sensing approach, coined
quantum nonlinear spectroscopy (QNS), using the correlations of sequential
weak measurement [29] via a spin quantum sensor [19]. This scheme has been
adopted in experiments to single out quantum signals from classical noises
[14, 30]. The quantum sensor interacts with the target system in a sequence of
short periods, and the projective measurement of the sensor constitutes a weak
measurement of the target. In the interaction picture, the coupled sensor-
target system evolves by the quantum Liouville equation
$\partial_{t}\hat{\rho}=-i\commutator{\hat{V}}{\hat{\rho}}$ (hereafter the
Planck constant $\hbar$ is set as unity), where
$\hat{V}=\hat{S}\otimes\hat{B}$ with $\hat{S}$ being a sensor operator and
$\hat{B}$ a target operator. For a small period of interaction, starting from
an initial state $\hat{\rho}=\hat{\rho}_{s}\otimes\hat{\rho}_{B}$ (with
$\rho_{s/B}$ denoting the sensor/target state), the coupled system evolves to
$\hat{\rho}(t+\tau)=\hat{\rho}-i\tau\commutator{\hat{V}}{\hat{\rho}}+O\left(\tau^{2}\right)$.
The commutator can be decomposed into

$\displaystyle
i^{-1}\left[{\hat{S}\otimes\hat{B}},{\hat{\rho}_{s}\otimes\hat{\rho}_{B}}\right]=\mathbb{S}^{+}\hat{\rho}_{s}\otimes\mathbb{B}^{-}\hat{\rho}_{B}+\mathbb{S}^{-}\hat{\rho}_{s}\otimes\mathbb{B}^{+}\hat{\rho}_{B}.$
(1)

The physical meanings of the two terms in the r.h.s. of Eq. (1) are clear:

  1. 1.

Considering $\Tr\mathbb{S}^{+}\hat{\rho}_{s}=\langle\hat{S}\rangle$, the first
term corresponds to the target evolving (indicated by the commutator
$\mathbb{B}^{-}_{j}\hat{\rho}_{B}$) under a quantum force $\hat{S}$ from the
sensor.

  2. 2.

Similarly, the second term is the evolution of the sensor under the quantum
force $\hat{B}$ from the target, that is, the back-action.

After the interaction, the measurement of a physical quantity $\Lambda_{j}$ of
the quantum sensor at time $t_{j}$ leads to a reduced density matrix of the
target in the form of

$\displaystyle{\mathbb{M}}_{j}\hat{\rho}_{B}(t_{j})\equiv$
$\displaystyle\Tr_{\rm
s}\left[{\hat{\Lambda}_{j}\hat{\rho}}(t_{j}+\tau)\right]$ $\displaystyle=$
$\displaystyle\Tr_{\rm s}\left(\hat{\Lambda}_{j}\hat{\rho}_{\rm
s}\right)\hat{\rho}_{B}+\tau\Tr_{\rm
s}\left(\hat{\Lambda}_{j}\mathbb{S}^{-}\hat{\rho}_{s}\right)\mathbb{B}^{+}_{j}\hat{\rho}_{B}$
$\displaystyle+\tau\Tr_{\rm
s}\left(\hat{\Lambda}_{j}\mathbb{S}^{+}\hat{\rho}_{s}\right)\mathbb{B}^{-}_{j}\hat{\rho}_{B}+O\left(\tau^{2}\right).$
(2)

The correlation of the physical quantities
$\\{\Lambda_{1},\Lambda_{2},...,\Lambda_{K}\\}$ measured at time
$t_{1},t_{2},\ldots,t_{K}$ can be written as

$G^{(K)}=\Tr_{B}{(\mathbb{M}_{K}...\mathbb{M}_{2}\mathbb{M}_{1}\hat{\rho}_{B}(t_{0})}).$
(3)

The initial state and measurement basis of the sensor can be chosen for the
pre- and post-selection of the effect of the sensor-target interaction in each
measurement period. For example, by choosing the initial state of the sensor
$\hat{\rho}_{s,j}$ and the sensor quantity to be measured $\Lambda_{j}$, one
can make ${\rm Tr}_{\rm s}\left[\hat{\Lambda}_{j}\hat{\rho}_{s,j}\right]=0$
and ${\rm Tr}_{\rm
s}\left[\hat{\Lambda}_{j}\mathbb{S}^{-}\hat{\rho}_{s,j}\right]=0$, so that the
weak measurement yields an evolution of the target, that is,
$\mathbb{B}^{-}\hat{\rho}_{B}(t_{j})$. This way, the QNS realized by
sequential weak measurement via a quantum sensor can extract arbitrary types
and orders of the time-ordered correlations in a quantum system.

However, the requirement of a central spin as the quantum sensor limits the
versatility of the QNS since usually a central spin interacts only with a
small number of particles in proximity and the measurement of single spins
demands stringent conditions. The polarization (a pseudo-spin) of a coherent
light beam can be an alternative for a central spin as a quantum sensor [12,
31]. The light beam as a quantum sensor has the advantage of large coherent
length (so that it can interact with a quantum many-body system) and has the
ease for preparation, control and detection. The pseudo-spin of light has been
routinely used in noise spectroscopy of spin systems [26, 26, 32].

In this paper, we present a scheme of QNS using polarization of a coherent
light beam as a quantum sensor. The interaction with the target system (such
as spins in a transparent material) results in small Faraday rotation of the
linearly polarized light. The measurement of the Faraday rotation constitutes
a weak measurement of the spins. The Faraday rotation can be measured with a
Mach-Zehnder interferometer with a phaseshift chosen such that the effects of
the light-material interaction is selected to be either a quantum evolution or
a quantum measurement of the magnetization of the spins (corresponding to a
commutator or anti-commutator between the magnetizaiton and the density
operator of the spins). As a result, the correlated photon counts of a
sequence of measurement shots, each with a designated phaseshift, can be made
proportional to a designated type and order of correlation of the spin
fluctuations. Analysis of the signal-to-noise ratio for the measurement of
correlations shows that the scheme is feasible under realistic conditions.

##  II Scheme

###  II.1 Basic idea

We employ the Mach-Zehnder interferometer in which the polarization of a light
beam can be measured in a designated basis by choosing a certain relative
phaseshift between the two arms (see Fig. 1). A sequence of linearly polarized
laser pulses, described by the coherent state
$\outerproduct{\alpha,H}{\alpha,H}$ (with $H/V$ denoting horizontal/vertical
polarization), are applied to a target system. The light polarization acts as
the quantum sensor, coupled to a spin system (the target). The interaction
with the spins leads to a Faraday rotation of the light polarization. After
the interaction, the light beam is separated into two arms by a polarizing
beam splitter (PBS), such that the component with horizontal polarization is
reflected and the component with vertical polarization is transmitted. A
phaseshift $\phi$ is then applied in one arm (e.g., the reflected beam) and
then the two arms are mixed through a 1:1 beam splitter (BS). The difference
between the photon counts in the two output directions is recorded to measure
the light polarization, in a basis that is selected by choosing the phase
shift $\phi$. As the light polarization is entangled with the target spins,
the measurement of the light polarization constitutes an indirect measurement
of the target spins. When the interaction between the light polarization and
the spins is weak, i.e., the Faraday rotation is small, the indirect
measurement on the spins is a weak one. The correlation among a sequence of
difference photon counts corresponds to a certain correlation in the target
spin system.

Figure 1: Schematic of a Mach-Zehnder interferometer for detecting arbitrary
correlations of a quantum spin system using a polarized light beam sensor. The
difference photon count $\Lambda$ is detected to measure the polarization of
the light after the interaction with the target system. A designated phase
shift $\phi$ is induced by a variable optical delay to select the basis of
polarization measurement.

###  II.2 Light polarization as a pseudo-spin-1/2 sensor

We can define a pseudo-spin-1/2 using Jones vectors. In the formalism of Jones
vectors, we write the horizontal (H) and vertical (V) linear polarizations as

$|H\rangle\equiv\left(\begin{array}[]{c}1\\\ 0\end{array}\right),\ \
|V\rangle\equiv\left(\begin{array}[]{c}0\\\ 1\end{array}\right),$

the diagonal (D) and anti-diagonal (A) linear polarizations as

$|D\rangle\equiv\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
1\end{array}\right),\
|A\rangle\equiv\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}+1\\\
-1\end{array}\right),$

and the two circular polarizations as

$|R\rangle\equiv\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}+1\\\
-i\end{array}\right),\
|L\rangle\equiv\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}+1\\\
+i\end{array}\right).$

In analogue to a spin-1/2, we define three photon operators

$\displaystyle\hat{S}_{1}$
$\displaystyle\equiv\frac{\hat{n}_{H}-\hat{n}_{V}}{2}=\frac{1}{2}\left(\hat{a}_{H}^{{\dagger}},\hat{a}_{V}^{{\dagger}}\right)\left(\begin{array}[]{cc}+1&0\\\
0&-1\end{array}\right)\left(\begin{array}[]{c}\hat{a}_{H}\\\
\hat{a}_{V}\end{array}\right),$ (4e) $\displaystyle\hat{S}_{2}$
$\displaystyle\equiv\frac{\hat{n}_{D}-\hat{n}_{A}}{2}=\frac{1}{2}\left(\hat{a}_{H}^{{\dagger}},\hat{a}_{V}^{{\dagger}}\right)\left(\begin{array}[]{cc}0&+1\\\
+1&0\end{array}\right)\left(\begin{array}[]{c}\hat{a}_{H}\\\
\hat{a}_{V}\end{array}\right),$ (4j) $\displaystyle\hat{S}_{3}$
$\displaystyle\equiv\frac{\hat{n}_{R}-\hat{n}_{L}}{2}=\frac{1}{2}\left(\hat{a}_{H}^{{\dagger}},\hat{a}_{V}^{{\dagger}}\right)\left(\begin{array}[]{cc}0&-i\\\
+i&0\end{array}\right)\left(\begin{array}[]{c}\hat{a}_{H}\\\
\hat{a}_{V}\end{array}\right),$ (4o)

where $\hat{a}_{\alpha}$ is the photon annihilation operator for the
polarization $\alpha$ and
$\hat{n}_{\alpha}\equiv\hat{a}_{\alpha}^{{\dagger}}\hat{a}_{\alpha}$ is the
photon number operator. The pseudo-spin operators satisfy the same commutators
as the spin operators, namely,

$\displaystyle\left[\hat{S}_{a},\hat{S}_{b}\right]=i\epsilon_{abc}\hat{S}_{c}.$
(5)

But they do not have the same anti-commutators as the spin-1/2. For example,

$\displaystyle\left\\{\hat{S}_{2},\hat{S}_{3}\right\\}$
$\displaystyle=\frac{i}{2}\hat{a}_{V}^{{\dagger}}\hat{a}_{V}^{{\dagger}}\hat{a}_{H}\hat{a}_{H}+{\rm
H.c.}\neq 0,$ (6a) $\displaystyle\left\\{\hat{S}_{3},\hat{S}_{3}\right\\}$
$\displaystyle=\hat{n}_{H}\hat{n}_{V}+\frac{\hat{n}_{H}+\hat{n}_{V}}{2}-\frac{\left(\hat{a}^{{\dagger}}_{H}\hat{a}_{V}\right)^{2}+{\rm
H.c.}}{2}\neq\frac{1}{2}.$ (6b)

###  II.3  Single Shot of measurement

Let us first consider a single shot of measurement. In the interaction
picture, the evolution of the coupled system of photons and spins is
determined by

$\partial_{t}\hat{\rho}(t)=-i\commutator{\hat{V}(t)}{\hat{\rho}},$

with the spin-photon interaction

$\displaystyle\hat{V}(t)=\hat{S}_{3}\otimes\hat{B}(t)=-\frac{i}{2}(\hat{a}^{\dagger}_{H}\hat{a}_{V}-\hat{a}^{\dagger}_{V}\hat{a}_{H})\otimes\hat{B}(t),$
(7)

where the target operator $\hat{B}$ is an effective field proportional to the
magnetization of the spins along the light propagation direction. The
magnetization induces opposite frequency shifts for the left and right
circularly polarized photons and hence opposite phase shifts, which causes the
Faraday rotation of the linear polarization.

We assume the pulse duration $\tau$ is small, that is, the Faraday rotation
angle $||\hat{B}\tau||\ll 1$ and $\hat{B}(t)\approx\hat{B}(t+\tau)$. After the
interaction for a short period of time $\tau$, starting from the initial state
$\hat{\rho}(t)=\hat{\rho}_{\rm s}\otimes\hat{\rho}_{B}$, the coupled system
evolves to the state

$\displaystyle\hat{\rho}(t+\tau)\approx\hat{\rho}(t)$
$\displaystyle+\tau\mathbb{S}_{3}^{-}\hat{\rho}_{s}\otimes\mathbb{B}^{+}\hat{\rho}_{B}+\tau\mathbb{S}_{3}^{+}\hat{\rho}_{s}\otimes\mathbb{B}^{-}\hat{\rho}_{B}.$
(8)

The physical meaning of the term
$\tau\mathbb{S}_{3}^{-}\hat{\rho}_{s}\otimes\mathbb{B}^{+}\hat{\rho}_{B}$ in
Eq. (8) is the pseudo-spin rotating about the 3rd axis under the field
$\hat{B}$ from the target. To select this effect, we can prepare the pseudo-
spin initially in an eigenstate of $\hat{S}_{1}$ and then measure the photon
polarization in the basis of $\hat{S}_{2}$, that it, in the basis of the
linear polarizations $D$ and $A$. Indeed, if the sensor initial state is
chosen as

$\hat{\rho}_{\rm s}(t_{j})=|\alpha,H\rangle\langle\alpha,H|,$

(with
$|\alpha,H\rangle\equiv\exp\left(\alpha\hat{a}_{H}^{{\dagger}}-\alpha^{*}\hat{a}_{H}\right)|{\rm
vacuum}\rangle$) and the observable to be measured as
$\hat{\Lambda}_{j}=\hat{S}_{2}$, we have $\Tr_{\rm
s}\left[\hat{S}_{2}\hat{\rho}_{\rm s}(t_{j})\right]=0$, $\Tr_{\rm
s}\left[\hat{S}_{2}\mathbb{S}_{3}^{+}\hat{\rho}_{s}(t_{j})\right]=\frac{1}{2}\left\langle\left\\{\hat{S}_{2},\hat{S}_{3}\right\\}\right\rangle=0$,
and $\Tr_{\rm
s}\left[\hat{S}_{2}\mathbb{S}_{3}^{-}\hat{\rho}_{s}(t_{j})\right]=\frac{1}{i}\left\langle\left[\hat{S}_{2},\hat{S}_{3}\right]\right\rangle=\frac{1}{2}\left|\alpha\right|^{2}$.
Therefore, the measurement induces the superoperator on the bath as

$\displaystyle{\mathbb{M}}_{j}\hat{\rho}_{B}(t_{j})=\frac{1}{2}\tau\left|\alpha\right|^{2}{\mathbb{B}}_{j}^{+}\hat{\rho}_{B}(t_{j}),\
{\rm for}\ \hat{\Lambda}_{j}=\hat{S}_{2}.$ (9)

The physical meaning of the other term
$\tau\mathbb{S}_{3}^{+}\hat{\rho}_{s}\otimes\mathbb{B}^{-}\hat{\rho}_{B}$ in
Eq. (8) is the quantum force $\hat{S}_{3}$ from the sensor driving the
evolution of the target. To select this effect, we should measure the pseudo-
spin component $\hat{S}_{3}$. Using the same initial state $\hat{\rho}_{\rm
s}(t_{j})=|\alpha,H\rangle\langle\alpha,H|$ and similar algebra for the
derivation of Eq. (9), we get

$\displaystyle{\mathbb{M}}_{j}\hat{\rho}_{B}(t_{j})=\frac{1}{2}\tau\left|\alpha\right|^{2}{\mathbb{B}}_{j}^{-}\hat{\rho}_{B}(t_{j}),\
{\rm for}\ \hat{\Lambda}_{j}=\hat{S}_{3}.$ (10)

Using the scheme illustrated in Fig. 1, we can measure either $\hat{S}_{2}$ or
$\hat{S}_{3}$ by designing the phase shift in the reflected path. Interference
occurs between transmitted path and reflected path. The 1:1 BS transforms the
input photon operators $\hat{a}$ and $\hat{b}$ into output ones $\hat{c}$ and
$\hat{d}$ by

$\begin{pmatrix}\hat{c}\\\
\hat{d}\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&i\\\
i&1\end{pmatrix}\begin{pmatrix}\hat{a}\\\ \hat{b}\end{pmatrix}.$

For the measurement of $\hat{S}_{3}$, we set the phase shift such that
$e^{i\phi}=1$ and therefore the input annihilation operators are
$\hat{a}=\hat{a}_{H}$ and $\hat{b}=\hat{a}_{V}$. Then, the outputs become

$\frac{1}{\sqrt{2}}\begin{pmatrix}1&i\\\
i&1\end{pmatrix}\begin{pmatrix}\hat{a}_{H}\\\
\hat{a}_{V}\end{pmatrix}=\begin{pmatrix}\hat{a}_{L}\\\
i\hat{a}_{R}\end{pmatrix},$

and the difference photon count $\Lambda\equiv\lambda_{d}-\lambda_{c}$ as
shown in Fig. 1 becomes
$\hat{a}_{R}^{\dagger}\hat{a}_{R}-\hat{a}_{L}^{\dagger}\hat{a}_{L}\equiv\hat{S}_{3}$.

Similarly, for the measurement of $\hat{S}_{2}$, we set the phase shift such
that $e^{i\phi}=i$ and the input annihilation operators are
$\hat{a}=\hat{a}_{H}$ and $\hat{b}=i\hat{a}_{V}$. Then, the outputs become

$\frac{1}{\sqrt{2}}\begin{pmatrix}1&i\\\
i&1\end{pmatrix}\begin{pmatrix}\hat{a}_{H}\\\
i\hat{a}_{V}\end{pmatrix}=\begin{pmatrix}\hat{a}_{A}\\\
i\hat{a}_{D}\end{pmatrix},$

and the difference photon count $\Lambda\equiv\lambda_{d}-\lambda_{c}$ becomes
$\hat{a}_{D}^{\dagger}\hat{a}_{D}-\hat{a}_{A}^{\dagger}\hat{a}_{A}\equiv\hat{S}_{2}$.

###  II.4  Correlations of sequential weak measurements

The correlation of a sequence of $K$ shots of measurement is

$\displaystyle
G^{(K)}=\Tr_{B}\left[{\mathbb{M}}_{K}\cdots{\mathbb{M}}_{2}{\mathbb{M}}_{1}\hat{\rho}_{B}\right].$
(11)

Using Eqs. (9) and (10), we find that the correlated photon count differences
$G^{(K)}$ is proportional to a certain type of correlation in the target
system as

$G^{(K)}=2^{-K}\tau^{K}\absolutevalue{\alpha}^{2K}C^{\eta_{K}...\eta_{2}\eta_{1}}$
(12)

where $\eta_{j}=+$ or $-$ if the $j^{th}$ shot of measurement is chosen as
$\hat{\Lambda}_{j}=\hat{S}_{2}$ or $\hat{S}_{3}$, respectively. For example,
choosing $\hat{\Lambda}_{2}=\hat{S}_{2}$ and $\hat{\Lambda}_{1}=\hat{S}_{3}$,
we obtain

$G^{(2)}=2^{-2}\tau^{2}\absolutevalue{\alpha}^{4}C^{+-},$

and choosing $\hat{\Lambda}_{4}=\hat{S}_{2}$, $\hat{\Lambda}_{3}=\hat{S}_{3}$,
$\hat{\Lambda}_{2}=\hat{S}_{3}$ and $\hat{\Lambda}_{1}=\hat{S}_{2}$, we get

$G^{(4)}=2^{-4}\tau^{4}\absolutevalue{\alpha}^{8}C^{+--+}.$

This way we can obtain arbitrary types and orders of correlations just by
getting the correlated photon count differences from the interferometer.

##  III  Signal-to-noise ratio

We calculate the signal-to-noise (SNR) ratio. We consider the shot noises [33]
of the photon counts and neglect the technical noises such as the laser
fluctuations and the imperfections in beam splitting, phase shift, and photon
detection, since these factors would highly depend on the specific systems
employed in experiments. The SNR is calculated for $L$ repeated sequences of
$K$ shots of measurement for obtaining a $K$-th order correlation $G^{(K)}$.

###  III.1 SNR for first-order correlations

The signal $\expectationvalue{S}$ for a first-order correlation $G^{(1)}$ is
the average difference photon count between the two outputs,

$\expectationvalue{S}=G^{(1)}=\frac{1}{2}\tau\absolutevalue{\alpha}^{2}C^{+}.$

The variance of signal is
$\sigma^{2}=\expectationvalue{S^{2}}-\expectationvalue{S}^{2}$. Since
$\expectationvalue{S}^{2}$ is of $\order{\tau^{2}}$, which is much smaller
than $\expectationvalue{S^{2}}$, the variance can be simplified as

$\sigma^{2}\approx\expectationvalue{S^{2}}=\expectationvalue{\Lambda^{2}}=\Tr[\hat{S}_{2}^{2}\hat{\rho}(t+\tau)].$

Using Eq. (8) and keeping the leading-order terms, we get

$\sigma^{2}\approx\Tr\left[\hat{S}_{2/3}^{2}\hat{\rho}(t)\right]=\absolutevalue{\alpha}^{2}.$
(13)

Therefore, the SNR for first-order correlations in the shot-noise limit is

${\rm
SNR}^{(1)}=\frac{L\expectationvalue{S}}{\sqrt{L\sigma^{2}}}=\frac{\sqrt{L}}{2}\absolutevalue{\alpha}\tau
C^{+}$ (14)

###  III.2 SNR for $K$-th order correlations

Similarly, the signal of measuring a $K$-th order correlation given by Eq.
(12) is $\expectationvalue{S}=G^{(K)}$. In the leading order, the variance
$\sigma^{2}\approx\expectationvalue{S^{2}}=\expectationvalue{\Lambda_{K}^{2}\cdots\Lambda_{2}^{2}\Lambda_{1}^{2}}=\expectationvalue{\Lambda_{K}^{2}}\cdots\expectationvalue{\Lambda_{2}^{2}}\expectationvalue{\Lambda_{1}^{2}}$,
where the subscript in $\Lambda_{k}$ labels the shot of measurement. By
straightforward calculation, we get
$\expectationvalue{\Lambda_{k}^{2}}\approx\absolutevalue{\alpha}^{2}$ in the
leading order. Thus, the variance of measuring a $K$-th order correlation is

$\expectationvalue{S^{2}}\approx\absolutevalue{\alpha}^{2K}.$

Therefore, the SNR for a $K$th order correlation is

${\rm
SNR}^{(K)}=\sqrt{L}\frac{\expectationvalue{S}}{\sigma}=2^{-K}\sqrt{L}\absolutevalue{\alpha}^{K}\tau^{K}C^{\eta_{K}...\eta_{2}\eta_{1}}.$

###  III.3  Estimation of the signal strength

We consider a target system containing an ensemble of spins. We assume that
the laser pulse length is much greater than the thickness of the sample ($D$)
and the laser pulse duration $\tau$ is much shorter than the timescale of the
spin dynamics. In this case, all the spins
$\\{\hat{\mathbf{J}}_{i}|i=1,2,\ldots,N_{\rm s}\\}$ within the laser spot
interact simultaneously with the photons. The Faraday rotation
$\hat{\theta}(t)$ is related to the magnetization $\hat{J}(t)\equiv N_{\rm
s}^{-1}\sum_{i}\hat{J}^{z}_{i}$ along the light propagation direction (the
$z$-axis) by [31]

$\hat{\theta}(t)=gD\hat{J}(t)=\tau\hat{B}(t),$ (15)

where $g$ is the coupling constant. The $K$-th order correlation is related to
the correlation of the spins by

$\tau^{K}C^{\eta_{K}\cdots\eta_{2}\eta_{1}}=g^{K}D^{K}\Tr({\mathbb{J}}^{\eta_{k}}_{K}\cdots{\mathbb{J}}_{2}^{\eta_{2}}{\mathbb{J}}_{1}^{\eta_{1}}\hat{\rho}_{B}).$
(16)

When the spins are uncorrelated, each spin contributes to the correlation
independently, and therefore the $K$th order correlation of the $N_{\rm s}$
spins is estimated to be

$\tau^{K}C^{\eta_{K}\cdots\eta_{2}\eta_{1}}\sim\left(\frac{gD}{N_{\rm
s}}\right)^{K}N_{\rm s}\expectationvalue{\left(\hat{J}_{i}^{z}\right)^{K}}.$

The SNR is

${\rm SNR}^{(K)}\sim\sqrt{L}\left(\frac{gN_{\rm ph}^{1/2}}{2n_{\rm
s}A}\right)^{K}n_{\rm
s}DA\expectationvalue{\left(\hat{J}_{i}^{z}\right)^{K}},$ (17)

where $N_{\rm ph}=\absolutevalue{\alpha}^{2}$ is the average number of photons
per laser pulse, $n_{\rm s}$ is the spin density of the target system, and $A$
is the area of the laser spot on the sample. To ensure the higher-order
correlations be observable (for $K>2$), the laser should be strong enough and
the light beam should be well focused such that ${gN_{\rm
ph}^{1/2}}/\left({2n_{\rm s}A}\right)\gtrsim 1$.

To evaluate the feasibility of the measurement, we consider LiHoF4 [34] as a
specific material system. This material is a transparent Ising spin system
with $\left(\hat{J}_{i}^{z}\right)^{2}=8^{2}$ and presents interesting quantum
phase transitions of magnetization [35]. Using the data in Ref. [34], the
coupling constant for a laser with wavelength about 435 nm (near resonant with
the band gap of the material) is $g\approx 20$ rad/cm. The density of spins of
LiHoF4 is $n_{\rm s}\approx 1.39\times 10^{28}$ cm-3. Considering a laser
pulse with focus size $\sim 10^{-4}$ cm and the number of photons $n_{\rm
ph}\sim 10^{14}$ and a sample of $1$ cm thickness, we estimate the SNR as

${\rm SNR}^{(K)}\sim\sqrt{L}\left(8\times 10^{-12}\right)^{K}\times 10^{20}.$

To get SNR$>1$, the number of measurement shots should be $L\gtrsim 10^{5}$
for measuring the second order correlations, which is quite feasible. For
measuring the fourth order correlations ($K=4$), however, the number of
measurement shots should be $L>10^{49}$, which is impossible for any practical
experiments. This difficulty in measuring the higher-order correlations arises
from the fact that uncorrelated or non-interacting spin ensembles typically
exhibit negligible higher-order correlations.

An interesting scenario is that the higher-order correlations in interacting
spin ensembles become significant near the critical points where the spins
fluctuate collectively with a diverging correlation length $\xi$. The spins
within a region of size $\xi$ can be taken as a large spin ($\sim n_{\rm
s}\xi^{3}J$). Thus, considering the collective fluctuations, the SNR for
measuring the correlations is modified to be

${\rm SNR}^{(K)}\sim\sqrt{L}\left(\frac{g\xi^{3}N_{\rm
ph}^{1/2}}{2A}\right)^{K}\frac{DA}{\xi^{3}}\expectationvalue{\left(\hat{J}_{i}^{z}\right)^{K}}.$
(18)

When the system is close to the critical point such that the correlation
length is comparable to the size of the laser focus spot, the measurement of
higher order correlations becomes feasible. Thus, the scheme presented in this
paper provides an approach to studying the effects of higher-order
correlations in interacting spin systems near the critical point.

##  IV  Conclusion

In summary, we propose an approach to selectively extracting arbitrary types
and orders of quantum correlations via weak Faraday rotation measurement.
Unlike the single spin sensors used in existing schemes [19, 13, 30, 14, 36],
which interact with only a few spins in proximity, the quantum sensor employed
in this scheme, namely, the optical polarization as a pseudo-spin, can couple
uniformly to a quantum many-body system. We have analysed the signal-to-noise
ratio to show the feasibility of the scheme. While the detection of second-
order correlations of fluctuations suffices to characterize quantum many-body
systems far away from critical points, the higher-order correlations could be
important near the critical points. When the systems approach the critical
points, the collective fluctuations have diverging correlation lengths. The
optical detection of higher-order correlations is feasible when the
correlation lengths are comparable to the focus size of the laser beam. The
scheme proposed in this paper belongs to the category of quantum nonlinear
spectroscopy [14], for its capability of extracting the rich structure of
higher-order quantum correlations, which are inaccessible to conventional
nonlinear optical spectroscopy [25]. The quantum nonlinear spectroscopy is not
only a new tool for studying quantum many-body physics, but also new
approaches to quantum science and technology, such as the study of quantum
foundation (higher-order Leggett-Garg inequalities [3] and Bell inequalities
[1, 2]), optimal quantum control [6, 7, 37, 38], and quantum sensing [13, 14,
30]).

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of
China/Hong Kong Research Council Collaborative Research Scheme Project CRS-
CUHK401/22, the Hong Kong Research Grants Council Senior Research Fellow
Scheme Project SRFS2223-4S01, and the New Cornerstone Science Foundation.

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