precise measurement of an image at individual pixels of an amplitude mask
corresponds to the problem of the precise estimation of multiple loss
parameters. Increasing the resolution of thermal electromagnetic sources can
also be formulated as loss parameter estimation Tsang  _et al._ (2016); Nair
and Tsang (2016); Lupo and Pirandola (2016); Helstrom (1973); Lu  _et al._
(2018). Other examples of importance are to probe the change of intensity
parameters over different frequencies Whittaker  _et al._ (2017) or temporal
modes Taylor and Bowen (2016).

Figure 13: Multiple intensity parameter sensing. (a) General ancilla-assisted
strategy: Each signal mode (red) is sent to a fictitious BS with unknown
transmittivity $T_{j}$ in the $j$th mode, while an ancilla (yellow) is kept
lossless. The composite output state of the signal modes and the ancilla mode
is measured to estimate the vector $\boldsymbol{T}$ consisting of all
transmittivities. Virtual environment modes (green) are also considered and
assumed to be in the vacuum state before entering the BSs. Adapted with
permission from ref 269. Copyright 2018 the American Physical Society. (b)
Left: Circular dichroism (CD) is experimentally observed by measuring the
intensity-difference of the transmitted left-circularly polarized (LCP) light
and right-circularly polarized (RCP) light upon propagation through a chiral
medium. Right: General ancilla-assisted CD sensing scheme, similar to (a), but
with additional losses $\eta_{\text{L}}$ and $\eta_{\text{R}}$ for the
respective LCP and RCP modes. The transmittivities $T_{\text{L}}$ and
$T_{\text{R}}$ represent the transmittivities of the LCP and RCP modes through
a chiral medium, respectively. Reproduced from ref 270.

A recent study has derived the ultimate quantum bound to the estimation
uncertainty of multiple intensity parameters and identified the optimal
schemes reaching the ultimate quantum bound Nair (2018). Consider the problem
of the estimation of $K$ transmittivities $\\{T_{k}\\}_{k=1}^{K}$ of $K$
transmissive channels using what is called an ‘ancilla-assisted entangled
parallel’ strategy, as illustrated in Figure 13(a). The probe states are sent
through signal modes (S) representing the $K$ transmissive channels modeled by
$K$ BSs, whereas ancillary modes (A) are used to transmit states entangled
with the probe states and are kept lossless. To make the scenario more
general, multimode features at individual channels are taken into account, so
that $M_{k}$ modes are assumed to be used for probing the $k$th transmittance.
The overall probe state can thus be written as a joint pure state
$\left|{\Psi}\right\rangle_{\text{SA}}$, with energy constraints on the
individual channels, i.e.,
$\left\langle{\Psi}\right|\hat{n}_{k}\left|{\Psi}\right\rangle_{\text{SA}}=N_{k}$
for $k=1,\cdots,K$, where
$\hat{n}_{k}=\sum_{m=1}^{M_{k}}\hat{a}^{\dagger}_{k,m}\hat{a}_{k,m}$. It was
shown that the maximum QFIM $\boldsymbol{H}$ can be written as Ioannou  _et
al._ (2020)

$\displaystyle\boldsymbol{H}=\text{diag}\left(\frac{\eta_{1}N_{1}}{T_{1}(1-\eta_{1}T_{1})},\cdots,\frac{N_{K}}{\eta_{K}T_{K}(1-\eta_{K}T_{K})}\right),$
(49)

where $\eta_{k}$ is an additional but unwanted loss rate for the $k$th
channel. Such an ultimate quantum bound exhibits a quantum enhancement in
comparison with the QFIM for a product coherent (PC) state probe written as

$\displaystyle\boldsymbol{H}=\text{diag}\left(\frac{\eta_{1}N_{1}}{T_{1}},\cdots,\frac{\eta_{K}N_{K}}{T_{K}}\right).$
(50)

It has been shown that the above maximum QFIM of eq 49 can be achieved by the
product probe
$\left|{\Psi}\right\rangle=\bigotimes_{k=1}^{K}\left|{\text{NDS}}\right\rangle_{k}~{}$
Nair (2018), where $\left|{\text{NDS}}\right\rangle_{k}$ is the so-called
number-diagonal-signal (NDS) state with the energy constraint of $N_{k}$ on
the signal mode Nair (2011); Nair and Yen (2011), defined as

$\displaystyle\left|{\text{NDS}}\right\rangle_{k}=\sum_{\boldsymbol{n}_{k}\geq\boldsymbol{0}}\sqrt{p(\boldsymbol{n}_{k})}\left|{\boldsymbol{n}_{k}}\right\rangle_{\text{S}}\left|{\phi_{\boldsymbol{n}_{k}}}\right\rangle_{\text{A}},$
(51)

where
$\left|{\boldsymbol{n}_{k}}\right\rangle_{\text{S}}=\left|{n_{1}}\right\rangle_{k}\cdots\left|{n_{M_{k}}}\right\rangle_{k}$
is an $M_{k}$-mode number state basis at the $k$th signal channel,
$\\{\left|{\phi_{\boldsymbol{n}_{k}}}\right\rangle_{\text{A}}\\}$ is an
orthonormal basis at the $k$th ancillary channel, and $p(\boldsymbol{n}_{k})$
is the probability distribution of $\boldsymbol{n}_{k}$. The signal energy
constraint on $k$th signal mode then reads as
$N_{k}=\sum_{m_{k}=0}^{\infty}m_{k}p(m_{k})$ with
$p(m_{k})=\sum_{n_{1}+\cdots+n_{M_{k}}=m_{k}}p(\boldsymbol{n}_{k})$.

According to the formulation introduced in section III.1, one can find the
optimal measurement setting that reaches the ultimate multiparameter quantum
bound. It has been shown that the optimal setting is the joint measurement
with the Schmidt bases, i.e., the basis
$\left|{\phi_{\boldsymbol{n}_{k}}}\right\rangle$ on the ancillary modes and
the number basis $\left|{\boldsymbol{n}_{k}}\right\rangle_{\text{S}}$ on the
signal modes Nair (2018). This leads the FIM to be equal to the QFIM of eq 49
for any $\\{T_{k}\\}$. Note that the multiparameter QCR bounds given by eq 49
can be simultaneously achieved since the associated SLDs commute Helstrom
(1976).

The above formulation has recently been applied to one notable example, called
circular dichrosim (CD) sensing, which is shown in Figure 13(b). Typical CD
sensing schemes aim to estimate the difference of the transmittivities between
left- and right-circular polarization through a chiral medium composed of
either chiral molecules Greenfield (2006); Micsonai  _et al._ (2015) or chiral
nanophotonic structures Gansel  _et al._ (2009); Passaseo  _et al._ (2017);
Collins  _et al._ (2017). By modeling CD sensing quantum mechanically [see
Figure 13(b)] and using eq 39, Ioannou et al. identified the ultimate quantum
limit on the estimation uncertainty of an estimate
$(T_{\text{L}}-T_{\text{R}})$ and investigated the optimality in terms of
various quantum state inputs and quantum measurements Ioannou  _et al._
(2020).

####  III.3.3 Quantum noise reduction in intensity measurements

The most widely exploited scheme for measuring intensity changes in both the
classical and quantum regime is reference-assisted transmission or absorption
spectroscopy. Here, the signal mode encodes a single intensity parameter of an
object and a reference mode is kept unchanged. Examples include quantum
imaging Genovese (2016), quantum illumination Lloyd (2008); Lopaeva  _et al._
(2013) and quantum sensing Meda  _et al._ (2017). In particular, an intensity-
difference measurement between the signal and reference output modes has often
been performed in order to remove common excess noise between the two modes
Bachor and Ralph (2004). This is a technique that has recently been used in
classical plasmonic sensing to reach the SNL Piliarik and Homola (2009); Wang
_et al._ (2011); Wu  _et al._ (2004).

The ultimate aim of using quantum probe states is to reduce the noise below
the SNL. The noise reduction of the intensity-difference measurement of two
modes $a$ and $b$ can be quantified by the so-called noise reduction factor
(NRF) Jedrkiewicz  _et al._ (2004); Bondani  _et al._ (2007); Blanchet  _et
al._ (2008); Peřina  _et al._ (2012), defined by the ratio of the variance of
the photon number difference between the signal and reference mode to that of
coherent states with matching average photon number, written as

$\displaystyle\sigma=\frac{\langle[\Delta(\hat{n}_{\text{b}}-\hat{n}_{\text{a}})]^{2}\rangle}{\langle\hat{n}_{\text{a}}\rangle+\langle\hat{n}_{\text{b}}\rangle}.$
(52)

It can be shown that $\sigma\geq 1$ for all classical light, and thus the
light is non-classical when $\sigma<1$ is measured. The NRF is minimized by
light whose uncertainty in the photon number difference is minimal, i.e., the
optimal probe state generally written in the form of the photon-number-
correlated state
$\left|{\Psi}\right\rangle_{\text{ab}}=\sum_{n}w_{n}\left|{n,n}\right\rangle$
with any weights $\\{w_{n}\\}$, for which $\sigma=0$. An example is the twin
Fock (TF) state $\left|{\text{TF}}\right\rangle=\left|{N,N}\right\rangle$ with
$\sigma^{\text{(TF)}}=0$. For a PC state $\left|{\alpha,\beta}\right\rangle$,
on the other hand, $\sigma^{\text{(PC)}}=1$, which sets the SNL.

Another useful photon-number-correlated state leading to $\sigma=0$ is the
two-mode squeezed vacuum (TMSV) state, often referred to as a twin beam. It
has been widely used in experiments over the last few decades Meda  _et al._
(2017). The TMSV state can be generated from a spontaneous parametric down
conversion (SPDC) process Burnham and Weinberg (1970); Heidmann  _et al._
(1987); Schumaker and Caves (1985). It is formally written as the outcome of a
two-mode squeezing operation applied to a two-mode vacuum state and is written
in the photon number basis as

$\displaystyle\left|{\text{TMSV}}\right\rangle=\hat{S}_{\text{2}}(\xi)\left|{0,0}\right\rangle=\sum_{n=0}^{\infty}c_{n}\left|{n,n}\right\rangle,$
(53)

where
$\hat{S}_{\text{2}}(\xi)=\exp[\xi^{*}\hat{a}\hat{b}-\xi\hat{a}^{\dagger}\hat{b}^{\dagger}]$
denotes the two-mode squeezing operator with $\xi\in\mathbb{C}$ and
$c_{n}=(-e^{i\theta_{\text{s}}}\tanh r)^{n}/\cosh r$ for
$\xi=re^{i\theta_{\text{s}}}$. It is clear from eq 53 that the TMSV state
exhibits a strong quantum correlation in the photon number between the two
modes, leading to $\sigma^{\text{(TMSV)}}=0$, whereas the individual modes
follow thermal statistics, i.e.,
$\langle(\Delta\hat{n})^{2}\rangle/\langle\hat{n}\rangle=N+1$ with
$N=\sinh^{2}r$. This strong photon number correlation has been used in various
applications such as quantum ellipsometry Abouraddy  _et al._ (2001);
Toussaint  _et al._ (2004), absorption/transmission measurements Tapster  _et
al._ (1991); Jakeman and Rarity (1986); Hayat  _et al._ (1999); Moreau  _et
al._ (2017); Losero  _et al._ (2018), quantum sensing Zhang  _et al._ (2015),
quantum illumination Lopaeva  _et al._ (2013), quantum radiometry Zwinkels
_et al._ (2010); Polyakov and Migdall (2007), and quantum gravity tests
Berchera  _et al._ (2013). Figure 14(a) presents an experimental demonstration
of a quantum imaging technique using the nonclassical spatial correlation of
TMSV states for a weakly absorbing object, in direct comparison with the
experimental images obtained using classical light Brida  _et al._ (2010). The
TMSV state is normally regarded as a weak-field probe since the average photon
number $N$ is small, i.e., small $r$, due to the weak nonlinearity of the
material used for realizing the two-mode squeezing operation Ruo-Berchera
(2009).

Figure 14:  (a) (Left) Twin beam [i.e., the TMSV state of eq 53] is generated
via a SPDC process. One mode of the twin beam passes through a weakly
absorbing object and arrives at a CCD camera array, while the other mode is
directly sent to another area of the CCD array. A subtraction is performed
between the two noisy images, realizing the intensity-difference measurement
scheme, whereby an image of the object is constructed. (Right) For two example
sets (upper and lower row) of the objects, three kinds of schemes are used:
(left) differential-intensity measurement using a TMSV state, (middle)
differential-intensity measurement using a PC state, and (right) direct-
intensity measurement using single-mode coherent state. Adapted with
permission from ref 263. Copyright 2010 Springer Nature. (b) (Bottom) The
parametric process in the double-lambda scheme converts two pump (P) photons
into one probe (Pr) photon and one conjugate (C) photon, generating a bright
twin beam, i.e., the TMSD state of eq 54. In the experiment, a bright pump
beam couples with the probe and conjugate fields through a hot 85Rb vapor cell
involving the double-lambda scheme. The spatially correlated bright twin beams
are generated with cylindrical symmetry over a range of angles $\Delta\theta$
around the axis of the pump beam. When the spatial field profile of the probe
seed is modified by an amplitude mask, an entangled image can be generated
between the probe and conjugate outgoing field. (Left) Image of the probe seed
entering a hot 85Rb vapor cell. (Middle) Image of the outgoing probe beam.
(Right) Image of the outgoing conjugate beam, clearly showing anti-spatial
correlation with the image of the probe beam. Reproduced with permission from
ref 302. Copyright 2008 AAAS.

When sensors operating in the high-intensity regime are more preferable or
favorable, one can employ a brighter entangled state at the expense of
decreasing the quantum correlation. For example, the state produced by
performing the two-mode squeezing operation on a vacuum state in one mode and
coherent state in another (whose intensity can be high). This state is
formally called a two-mode squeezed displaced (TMSD) state and has been
investigated from various perspectives Marino  _et al._ (2009); Cai  _et al._
(2015); Fang and Jing (2015). The TMSD state can be written as

$\displaystyle\left|{\text{TMSD}}\right\rangle=\hat{S}_{2}(\xi)\left|{\alpha}\right\rangle\left|{0}\right\rangle,$
(54)

for which $\langle\hat{n}_{\text{a}}\rangle=\sinh^{2}r+|\alpha|^{2}\cosh^{2}r$
and $\langle\hat{n}_{\text{b}}\rangle=\sinh^{2}r+|\alpha|^{2}\sinh^{2}r$ and

$\displaystyle\sigma^{\text{(TMSD)}}=\frac{|\alpha|^{2}}{|\alpha|^{2}+2(1+|\alpha|^{2})\sinh^{2}r}.$
(55)

It is clear that for a given small $r$, the intensity of the individual modes
can be increased by cranking up the pump laser power (i.e., $|\alpha|^{2}$)
incident to the nonlinear medium that produces the TMSD state. A large
$|\alpha|^{2}$ increases the NRF, but the NRF is always less than unity,
implying that the quantum correlation of the TMSD state is always stronger
than that of the classical case and can thus be exploited in quantum imaging
or sensing. In an experiment, the TMSD state can be generated via a four-wave
mixing (FWM) process in a double-lambda configuration provided by a
${}^{85}\text{Rb}$ vapor cell McCormick  _et al._ (2008); Boyer  _et al._
(2008b); Turnbull  _et al._ (2013), where a $\text{NRF}\approx 87\%$, i.e.,
about $-9~{}\text{dB}~{}[\approx 10\log_{10}(1-0.87)]$, has been measured.
Such an intense quantum correlation with tens of $\mu$W of optical power has
been exploited in quantum sensing Pooser and Lawrie (2015); Clark  _et al._
(2012), imaging Boyer  _et al._ (2008a); Lawrie and Pooser (2013), and quantum
plasmonics Lawrie  _et al._ (2013). The application to quantum plasmonic
sensing, in particular, will be described in more detail in section IV.1.2.
Figure 14(b) presents a demonstration of entangling spatial information of the
probe beam with the conjugate beam by a FWM scheme that generates a TMSD state
Boyer  _et al._ (2008a).

An interesting comparison can be made among the aforementioned quantum states
of light in quantum imaging or intensity parameter sensing scenarios. Consider
a setup, where the object with transmittance $T$ is placed in the signal mode
$a$, and the signal and reference modes undergo inefficient transmission
channels with factors $\eta_{\text{a}}$ and $\eta_{\text{b}}$, respectively,
which include the detection efficiencies of the two detectors Mandel and Wolf
(1995). The intensity-difference measurement operator
$\hat{n}_{-}=\hat{b}_{\text{out}}^{\dagger}\hat{b}_{\text{out}}-\hat{a}_{\text{out}}^{\dagger}\hat{a}_{\text{out}}$
can be written in terms of input operators using the Heisenberg picture Loudon
(2000). Here, states are static and operators contain the relevant dynamics.
In this case, we can model the change of the transmittance and losses in the
modes as fictitious BSs. This enables the output NRF to be calculated for the
respective probe states Loudon (2000). For example, the input-output relation
given by

$\displaystyle\hat{a}_{\text{out}}$
$\displaystyle=\sqrt{G\eta_{\text{a}}T}\hat{a}_{\text{in}}+\sqrt{(G-1)\eta_{\text{a}}T}\hat{b}^{\dagger}_{\text{in}}+\sqrt{(1-\eta_{\text{a}})T}\hat{a}_{\text{bath}}$
$\displaystyle\quad+\sqrt{(1-T)}\hat{a}_{\text{obj}},$ (56)
$\displaystyle\hat{b}_{\text{out}}$
$\displaystyle=\sqrt{G\eta_{\text{b}}}\hat{b}_{\text{in}}+\sqrt{(G-1)\eta_{\text{b}}}\hat{a}^{\dagger}_{\text{in}}+\sqrt{(1-\eta_{\text{b}})}\hat{b}_{\text{bath}}$
(57)

can be applied to $\hat{n}_{-}$ and the expectation value taken with respect
to the initial vacuum state $\left|{0}\right\rangle\left|{0}\right\rangle$ for
the TMSV state probe, or the initial displaced state
$\left|{\alpha}\right\rangle\left|{0}\right\rangle$ for the TMSD state probe.
Here, $\hat{a}_{\text{bath}}~{}(\hat{b}_{\text{bath}})$ is the input operator
of a fictitious BS describing a lossy channel of the signal (reference) mode
Loudon (2000), $\hat{a}_{\text{obj}}$ denotes the virtual input operator of
the object, and $G=\cosh^{2}r$ with $r$ being the squeezing parameter and
$\theta_{\text{s}}=\pi$ assumed. The bath and object modes are taken to be in
the vacuum initially.

The NRFs of the respective output states can thus be written as

$\displaystyle\sigma_{\text{out}}^{\text{(TF)}}$
$\displaystyle=1-\frac{T^{2}\eta_{\text{a}}^{2}+\eta_{\text{b}}^{2}}{T\eta_{\text{a}}+\eta_{\text{b}}},$
(58) $\displaystyle\sigma_{\text{out}}^{\text{(TMSV)}}$
$\displaystyle=1+\frac{G|\alpha|^{2}(T\eta_{\text{a}}-\eta_{\text{b}})^{2}-(T^{2}\eta_{\text{a}}^{2}+\eta_{\text{b}}^{2})}{T\eta_{\text{a}}+\eta_{\text{b}}},$
(59) $\displaystyle\sigma_{\text{out}}^{\text{(TMSD)}}$ $\displaystyle\approx
1+\frac{2(G-1)\left[G(T\eta_{\text{a}}-\eta_{\text{b}})^{2}-\eta_{\text{b}}^{2}\right]}{GT\eta_{\text{a}}+(G-1)\eta_{\text{b}}},$
(60)

where the limit $|\alpha|^{2}\gg 1$ has been taken into account for the TMSD
state probe Jasperse  _et al._ (2011). An example of them is shown in Figure
15.

Figure 15: Comparison of the noise reduction for different quantum states: TF,
TMSV and TMSD. The parameters chosen are $G=4.5$, $|\alpha|^{2}=10$ and
$\eta_{\rm a}=\eta_{\rm b}=0.8$. The starred versions (TF∗, TMSV∗ and TMSD∗)
correspond to the case when $T\eta_{\rm a}=\eta_{\rm b}$. Inset: General two-
mode scheme using an input state that consists of the signal and reference
mode to estimate the intensity parameter $T$ in the presence of losses
$\eta_{\text{a,b}}$ for each mode. Here, the transmissive object with $T$ and
linear losses are described by fictitious BSs.

The quantum enhancement in noise reduction, i.e., a sub-shot-noise
measurement, can be identified by the condition $\sigma_{\text{out}}<1$. The
latter holds for $\sigma_{\text{out}}^{\text{(TF)}}$ at any value of $T$,
implying that the use of the TF state always achieves the sub-shot-noise
measurement. The other two states, on the other hand, reach the noise level
below the SNL when $G(T\eta_{\text{a}}-\eta_{\text{b}})^{2}$ is less than
$T^{2}\eta_{\text{a}}^{2}+\eta_{\text{b}}^{2}$ and $\eta_{\text{b}}^{2}$,
respectively. These conditions can be simply satisfied when $T\eta_{\text{a}}$
is close to $\eta_{\text{b}}$, motivating a consideration of the ideal case
that $T\eta_{\text{a}}=\eta_{\text{b}}=\eta$, for which the NRFs of eqs 58 to
60 become

$\displaystyle\sigma_{\text{out}}^{\text{(TF)}}$ $\displaystyle=1-\eta,$ (61)
$\displaystyle\sigma_{\text{out}}^{\text{(TMSV)}}$ $\displaystyle=1-\eta,$
(62) $\displaystyle\sigma_{\text{out}}^{\text{(TMSD)}}$ $\displaystyle\approx
1-\frac{2\eta(G-1)}{2G-1}.$ (63)

These NRFs are shown in Figure 15 as the starred versions. As expected, a sub-
shot-noise measurement can be achieved with all the considered states at any
value of $T$. This has been experimentally demonstrated in various studies
Boyer  _et al._ (2008b); Pooser and Lawrie (2016); McCormick  _et al._ (2007),
including the imaging of a weakly absorbing object, for which
$\eta_{\text{b}}=\eta_{\text{a}}$ and $T\approx 1$ are of interest Brambilla
_et al._ (2008); Brida  _et al._ (2010). The behaviors of the TF and TMSV
state exhibit the maximal noise reduction in intensity sensing, as predicted
from the previous discussion where the NSD state has been shown to lead to the
optimal results Nair (2018); Tame and Lee (2019).

Another useful and practical figure of merit that has often been used in the
two-mode scheme is the SNR with respect to an intensity-difference measurement
Brida  _et al._ (2010), defined by

$\displaystyle\text{SNR}=\frac{|\langle\hat{n}_{\text{b}}\rangle-\langle\hat{n}_{\text{a}}\rangle|}{\langle[\Delta(\hat{n}_{\text{b}}-\hat{n}_{\text{a}})]^{2}\rangle^{1/2}}.$
(64)

When symmetric probes are used (such as the TF, and TMSV states), i.e.,
$\langle\hat{n}_{\text{a}}\rangle_{\rm
in}=\langle\hat{n}_{\text{b}}\rangle_{\rm in}=N$ and
$\langle(\Delta\hat{n}_{\text{a}})^{2}\rangle_{\rm
in}=\langle(\Delta\hat{n}_{\text{b}})^{2}\rangle_{\rm in}$, the SNR can be
rewritten as

$\displaystyle\text{SNR}=\frac{(\eta_{\text{b}}-\eta_{\text{a}}T)N}{\sqrt{N\left[(T\eta_{\text{a}}-\eta_{\text{b}})^{2}Q_{\text{M}}+2T\eta_{\text{a}}\eta_{\text{b}}(\sigma_{\text{in}}-1)+(T\eta_{\text{a}}+\eta_{\text{b}})\right]}},$
(65)

where
$Q_{\text{M}}=\langle(\Delta\hat{n})^{2}\rangle_{\text{in}}/\langle\hat{n}\rangle_{\text{in}}-1$
is the Mandel Q-parameter Mandel (1979) and $\sigma_{\text{in}}$ is the NRF
for the input state Jedrkiewicz  _et al._ (2004); Bondani  _et al._ (2007);
Blanchet  _et al._ (2008); Peřina  _et al._ (2012). The Mandel Q-parameter is
equal to or greater than zero for all classical light, so non-classical light
can be identified by $Q_{\text{M}}<0$. As mentioned previously, the NRF
$\sigma_{\text{in}}\geq 1$ for all classical light, and thus light is non-
classical when $\sigma_{\text{in}}<1$ is observed. It is clear to see that the
SNR of eq 65 increases with decreasing $Q_{\text{M}}$ and
$\sigma_{\text{in}}$. One finds that the minimum values of $Q_{\text{M}}$ and
$\sigma_{\text{in}}$ are $-1$ and $0$, respectively, which are obtained for
the TF state
$\left|{\Psi_{\text{in}}}\right\rangle=\left|{N}\right\rangle_{\text{a}}\left|{N}\right\rangle_{\text{b}}$.

The quantum enhancement in the SNR can be quantified by the ratio of the SNR
for the particular quantum state probe to that for the balanced PC probe
state, written by

$\displaystyle R_{\text{SNR}}$
$\displaystyle=\frac{\text{SNR}_{\text{q}}}{\text{SNR}_{\text{c}}}$
$\displaystyle=\sqrt{\frac{T\eta_{\text{a}}+\eta_{\text{b}}}{(T\eta_{\text{a}}-\eta_{\text{b}})^{2}Q_{\text{M}}+2T\eta_{\text{a}}\eta_{\text{b}}(\sigma_{\text{in}}-1)+(T\eta_{\text{a}}+\eta_{\text{b}})}}.$
(66)

Note that quantum enhancement with $R_{\text{SNR}}>1$ is observed when
$(T\eta_{\text{a}}-\eta_{\text{b}})^{2}Q_{\text{M}}+2T\eta_{\text{a}}\eta_{\text{b}}(\sigma_{\text{in}}-1)<0$,
so a quantum probe with $-1\leq Q_{\text{M}}<0$ or $0\leq\sigma_{\text{in}}<1$
can be shown to be advantageous. The optimal state that maximizes the quantum
enhancement is the TF state, for which $Q_{\text{M}}=-1$ and
$\sigma_{\text{in}}=0$, which are the minimum bound for the respective
parameters. In this optimal case, $R_{\text{SNR}}>1$ regardless of $T$ and
$\eta_{\text{a,b}}$.

The use of the TMSV state of eq 53, for which $Q_{\text{M}}=N$ and
$\sigma_{\text{in}}=0$, provides a quantum enhancement only when
$(T\eta_{\text{a}}-\eta_{\text{b}})^{2}N<2T\eta_{\text{a}}\eta_{\text{b}}$
with respect to the balanced PC state with $Q_{\text{M}}=0$ and
$\sigma_{\text{in}}=1$. This indicates that probing with a TMSV state is more
advantageous as $N$ decreases. When $\eta_{\text{a}}\approx\eta_{\text{b}}$
and $T\approx 1$, or $\eta_{\text{a}}T\approx\eta_{\text{b}}$, the use of a
TMSV state becomes more useful even with high $N$. Such an enhancement has
been observed in quantum imaging experiments Brida  _et al._ (2010); Meda  _et
al._ (2017). One can also find other useful states with $\sigma_{\text{in}}=0$
and $Q_{\text{M}}<0$, for example, pair coherent states Agarwal (1986) or
finite-dimensional photon-number entangled states Lee  _et al._ (2012b). These
states have all been considered in quantum plasmonic sensing, as we will
discuss in detail in section IV.1.

###  III.4 Sub-shot-noise phase sensing

####  III.4.1 Phase sensing in Mach-Zehnder interferometers

The shot-noise limited interferometric phase sensor described in section III.2
can be improved by making use of quantum resources. The original idea proposed
by Caves, who largely contributed to the advent of the field of quantum
metrology, is to inject a squeezed vacuum state $|\xi\rangle$ into the input
mode $b$ of the first BS in the MZI shown in Figure 10(b), where previously
the input state in mode $b$ was assumed to be the vacuum in the shot-noise
limited classical sensor Caves (1981). The squeezed vacuum state is defined as
$|\xi\rangle=\hat{S}_{1}(\xi)|0\rangle$, where the single-mode squeezing
operator is represented by
$\hat{S}_{1}(\xi)=\exp(\frac{1}{2}\xi^{*}\hat{a}^{2}-\frac{1}{2}\xi\hat{a}^{\dagger
2})$ and $\xi=re^{i\theta_{\text{s}}}$, with $r\geq 0$ manifesting a squeezing
magnitude in the variance of the corresponding quadrature variable,
$x_{\theta_{\text{s}}/2}=\langle\hat{x}_{\theta_{\text{s}}/2}\rangle$, with
$\hat{x}_{\theta_{\text{s}}/2}=(e^{-i\theta_{\text{s}}/2}\hat{a}+e^{i\theta_{\text{s}}/2}\hat{a}^{\dagger})/\sqrt{2}$.
Here, $\theta_{\text{s}}/2$ represents the angle of the axis along which
squeezing takes place Scully and Zubairy (1997). The total input state is then
$\left|{\Psi}\right\rangle_{\text{in}}=\left|{\alpha}\right\rangle_{\text{a}}\left|{\xi}\right\rangle_{\text{b}}$,
with the total average photon number of the two modes being
$N=|\alpha|^{2}+\sinh^{2}r$. In the absence of loss
($\eta_{\text{a}}=\eta_{\text{b}}=1$), this input state is transformed by the
first BS into a non-separable state
$\left|{\Psi}\right\rangle_{\text{prob}}=\hat{B}(\pi/4,\pi/2)\left|{\Psi}\right\rangle_{\text{in}}$
that probes the phase information, resulting in
$\left|{\Psi(\phi)}\right\rangle=\hat{U}(\phi)\left|{\Psi}\right\rangle_{\text{prob}}$
before the measurement. For such a lossless case, Pezzé and Smerzi showed that
the QCR bound to the estimation uncertainty of the relative phase $\phi$ is
written by Pezzé and Smerzi (2008)

$\displaystyle\Delta\phi_{\text{QCR}}=\frac{1}{\sqrt{\nu}\sqrt{|\alpha|^{2}e^{2r}+\sinh^{2}r}},$
(67)

where the optimal phase condition between the displacement and squeezing
parameters is set, i.e., $\cos(2\theta_{\text{c}}-\theta_{\text{s}})=1$. Note
that the QCR bound $\Delta\phi_{\text{QCR}}$ is the same for any true value of
$\phi$, and it can reach Heisenberg scaling $\Delta\phi_{\text{QCR}}\propto
N^{-1}$ for ${N\gg 1}$, when $|\alpha|^{2}\simeq\sinh^{2}r\simeq N/2$. It can
be shown that $\Delta\phi_{\text{QCR}}\approx e^{-r}/\sqrt{\nu N}$ when
$|\alpha|^{2}\gg\sinh^{2}r$ Caves (1981) and $\Delta\phi_{\text{QCR}}\approx
e^{-r}/\sqrt{\nu N(4|\alpha|^{2}+1)}$ when $|\alpha|^{2}\ll\sinh^{2}r$ Pezzé
and Smerzi (2008). These cases show that too small or too large amounts of
squeezing cannot achieve Heisenberg scaling, but they do provide sub-shot-
noise sensing due to the additional factor that multiplies $N^{-1/2}$.

Interestingly, the squeezed vacuum state considered above has been shown to be
the optimal state that can be inserted into mode $b$ of the lossless MZI when
the coherent state is inserted into mode $a$, as it minimizes the estimation
uncertainty of the relative phase $\phi$ Lang and Caves (2013). This indicates
that the initial idea proposed by Caves turns out to be a good choice as a
modification for better phase sensing in a MZI setup. The original goal here
was to improve classical gravitational wave detectors, where a strong
classical field is used in one mode of the interferometric setup Collaboration
(2011). This has now led to the latest improvements to the LIGO detector M.
Tse et al. (2019) and the Virgo detector F. Acernese et al. (2019) (Virgo
Collaboration). Furthermore, the QCR bound of eq 67 is not only theoretical,
but practically attainable by photon-number-counting detectors at the two
output ports of the MZI Pezzé and Smerzi (2008), or a parity detection scheme
at the output port $a$ of the MZI Seshadreesan  _et al._ (2013).

When a coherent state does not need to be used in mode $a$ of the MZI, the
optimal input state for phase estimation using the MZI setup is two single-
mode squeezed vacuum states with anti-phases, i.e.,
$\left|{\Psi}\right\rangle_{\text{opt}}=\hat{S}_{\text{a}}(-r)\hat{S}_{\text{b}}(r)\left|{0,0}\right\rangle$
with $r\in\mathbb{R}$ Lang and Caves (2014), for which the QCR bound reads
$\Delta\phi_{\text{QCR}}=1/\sqrt{\nu N(N+1)}$ with $N=2\sinh^{2}r$ being the
total average photon number.

The MZI phase sensing setup has sometimes been considered with a single phase
shift in mode $a$, i.e., $\hat{U}(\phi)=e^{i\phi\hat{a}^{\dagger}\hat{a}}$
Sparaciari  _et al._ (2015, 2016). However, in this setting particular care
must be made in quantifying the extent to which the estimation uncertainty is
reduced, since an optical phase can only be defined in a relative sense with
respect to a reference phase Jarzyna and Demkowicz-Dobrzański (2012). Hence, a
sensing analysis with a single phase shift is valid only when a reference beam
with a certain phase is assumed, whose resource also needs to be counted when
determining the dependence of the uncertainty on $N$.

In the case of loss in both arms of the MZI, the calculation of the QCR bound
becomes complicated due to the probabilistic nature of photon loss that needs
to be taken into account. In this case, it is more convenient to use the
relation between the QFIM $\boldsymbol{H}$ and the Bures distance ${\cal
D}_{\text{B}}^{2}$ for the infinitesimally close states
$\hat{\rho}_{\boldsymbol{\phi}}$ and
$\hat{\rho}_{\boldsymbol{\phi}+\text{d}\boldsymbol{\phi}}$ Bures (1969);
Braunstein and Caves (1994); Facchi  _et al._ (2010). Using the relevant
formulation introduced in Appendix B, one can thus derive the QCR bound for
the case $\eta_{\text{a}}=\eta_{\text{b}}=\eta$, written as Gard  _et al._
(2017); Ono and Hofmann (2010)

$\displaystyle\Delta\phi_{\text{QCR}}=\frac{1}{\sqrt{\nu}\sqrt{|\alpha|^{2}\frac{\eta}{(1-\eta)+e^{-2r}\eta}+\eta\sinh^{2}r}}.$
(68)

It is clear that $\Delta\phi_{\text{QCR}}$ increases with loss, i.e., as
$\eta$ decreases, whereas the lossless QCR bound of eq 67 is recovered when
$\eta=1$. It can also be shown that the Heisenberg scaling
$\Delta\phi_{\text{QCR}}\propto N^{-1}$ promised for the lossless cases starts
to deteriorate as $\eta$ decreases.

The QCR bound $\Delta\phi_{\text{QCR}}$ of eq 68 has been shown to be
achievable by a linear optical setup using a weak local oscillator field and
photon counting Ono and Hofmann (2010). Other types of measurement schemes
have been investigated, for example, homodyne detection with a measurement
operator $\hat{M}=\hat{x}_{\phi_{\text{HD}}}$, a single-mode intensity
measurement with $\hat{M}=\hat{a}^{\dagger}\hat{a}$, an intensity-difference
measurement with $\hat{M}=\hat{b}^{\dagger}\hat{b}-\hat{a}^{\dagger}\hat{a}$,
and a parity measurement with $\hat{M}=(-1)^{\hat{a}^{\dagger}\hat{a}}$ Gard
_et al._ (2017); Oh  _et al._ (2017). In the case of homodyne detection (see
Appendix C), it was shown that this offers a nearly optimal measurement scheme
and approaches the QCR bound of eq 68 in the large power regime.
Alternatively, the robustness of the TMSV state against photon loss enables
one to achieve sub-shot-noise limited phase sensing for a wide range of the
phase parameter space without pre- nor post-selection You  _et al._ (2020b).

Figure 16:  (a) (left) Linear optical network to demonstrate optimal phase
estimation in the presence of loss. Initial BSs prepare the optimal probe
state depending on the amount of loss measured via a loss monitor in a proof-
of-principle demonstration. (right) Phase estimation uncertainties measured
using two-photon optimal states (blue circles), NOON (red squares) states, and
weak-coherent states to set the SIL regime (gray diamonds), for five phase
values $\phi=0,\pm 0.2,\pm 0.4$ rad under several lossy conditions, given by
different transmittivities $\eta$. Horizontal lines represent the theoretical
CR bounds for individual input states in the presence of loss. Reproduced with
permission from ref 339. Copyright 2010 Springer Nature. (b) (left) Two-photon
NOON state undergoes a phase rotation in a collinear interferometer. The
signal and idler photons are separated via a PBS and measured by
superconducting nanowire detectors. The high efficiency and visibility in the
experiment meant that post-selection was not needed, leading to the
demonstration of unconditional quantum enhancement. (right) Experimentally
measured standard deviation of the estimated phases with error bars determined
via the standard bootstrapping technique. The theoretical SNL (purple line)
and CR bound for phase sensing with NOON states (orange line) are shown with
95% confidence regions. Reproduced with permission from ref 340. Copyright
2017 Springer Nature.

####  III.4.2 Phase sensing with NOON states

In addition to exploiting continuous variable states such as the squeezed
vacuum state in phase sensing, discrete variable states or definite-photon-
number states have also been considered as probe states for phase sensing
Polino  _et al._ (2020). The most famous state is the so-called NOON state,
defined as
$\left|{\text{NOON}}\right\rangle=\frac{1}{\sqrt{2}}(\left|{N,0}\right\rangle+\left|{0,N}\right\rangle)$,
where $N$ photons are found in either mode $a$ or $b$ Bollinger  _et al._
(1996); Lee  _et al._ (2002); Dowling (2008), exhibiting $N$-particle maximal
entanglement Mermin (1990). The generation of NOON states is very difficult
with current technology, and they have been limited to $N=5$ in photonic
systems Afek  _et al._ (2010); Israel  _et al._ (2012). The NOON state
possesses a very useful feature that the variance
$\langle(\Delta\hat{n}_{-})^{2}\rangle~{}$ of the photon number difference
between the two modes is maximal among the states with $N$ photons in total,
consequently maximizing the QFI written by
$H=4\langle(\Delta\hat{G})^{2}\rangle$, where $\hat{G}=\hat{n}_{-}/2$ is the
generator of the relative phase shift. When the NOON state is employed as a
probe state, the QCR bound is found to be
$\Delta\phi_{\text{QCR}}=1/\sqrt{\nu}N$, clearly manifesting a Heisenberg
scaling enabled by the maximal variance of the photon number difference of the
probe state. It is interesting to note that injecting a coherent state and a
squeezed vacuum state into the two input ports of the first BS in a MZI
generates an effective NOON state, as the output state has a large overlap
with the NOON state, i.e.,
$\hat{B}(\pi/4,\pi/2)\left|{\alpha,\xi}\right\rangle\approx\left|{\text{NOON}}\right\rangle$
for $\sinh^{2}r=|\alpha|^{2}=N/2$ and $N\gg 1$ Pezzé and Smerzi (2008); Afek
_et al._ (2010). In the MZI setup, the coherent state and squeezed vacuum
state thus exploit a large variance $\langle(\Delta\hat{n}_{-})^{2}\rangle$,
similar to the NOON state probe, in order to achieve sub-shot-noise sensing.

The QCR bound for the NOON state has been shown to be achievable by the
measurement of the observable
$\hat{A}_{N}=\left|{0,N}\right\rangle\left\langle{N,0}\right|+\left|{N,0}\right\rangle\left\langle{0,N}\right|$
Mitchell  _et al._ (2004), or parity detection Bollinger  _et al._ (1996). One
major obstacle in using the NOON state for phase sensing from a practical
perspective is that it is extremely sensitive to photon loss, since the loss
of a single or a few photons drastically changes the state’s photon number
distribution and its variance $\langle(\Delta\hat{n}_{-})^{2}\rangle$ quickly
decreases. The estimation uncertainty associated with the NOON state thus
becomes large even with a moderate amount of photon loss Sarovar and Milburn
(2006b); Shaji and Caves (2007); Gilbert  _et al._ (2008); Huver  _et al._
(2008). Such an extreme sensitivity to photon loss can be alleviated by
exploiting partial entanglement at the expense of a degraded performance,
while still achieving sub-shot-noise limited behavior Huelga  _et al._ (1997).
Hence, adding other photon-number components (e.g., $|k,N-k\rangle$ for $k<N$)
into the NOON state helps phase sensing with NOON states become more robust to
loss and provide sub-shot-noise sensing even in the presence of loss. In
particular, in phase sensing with the definite-photon-number state
$\left|{\Psi}\right\rangle_{\text{prob}}=\sum_{k=0}^{N}c_{k}\left|{k,N-k}\right\rangle$
with $N$ particles distributed over the two modes, one can find the optimal
distribution $\\{c_{k}\\}$ that maximizes the QFI and consequently enables one
to beat the SQL or SNL at any loss level Dorner  _et al._ (2009); Demkowicz-
Dobrzański  _et al._ (2009). The latter has been experimentally demonstrated
with optimally engineered definite-photon-number states Kacprowicz  _et al._
(2010) [see Figure 16(a)], and has been considered theoretically for quantum
plasmonic sensing, as will be discussed in more detail in section IV.2.1. The
unconditional quantum enhancement in phase estimation precision with NOON
state has also been experimentally demonstrated in a nearly lossless scenario
with significantly increased efficiency and visibility Slussarenko  _et al._
(2017) [see Figure 16(b)].

Figure 17: Single-mode phase sensing. (a) A coherent state or a squeezed
vacuum state are used to measure the phase $\phi$. (b) The phase space
representation for the states. The coherent state has an estimation precision
of $\Delta\phi_{\rm coh}$ for the phase $\phi$, whereas the squeezed vacuum
state has a smaller estimation precision of $\Delta\phi_{\rm sq}$. The initial
phase of the squeezed state, $\left|{\xi}\right\rangle$, is $\theta_{\rm
s}=\pi$ and the phase affected squeezed state $\left|{\xi
e^{2i\phi}}\right\rangle$ rotates in phase space counter-clockwise by the
angle $\phi$.

The most commonly used detection scheme for phase sensing with NOON states is
the $N$-fold coincidence detection scheme that counts the number of events
when $N$ photons are detected simultaneously Wolfgramm  _et al._ (2013). This
allows one to investigate the $N$-fold detection modulation with the phase
$\phi$, leading to the probability of a coincidence detection given by

$\displaystyle p_{\text{coin}}(\phi)=\frac{f_{N}\left[1+{\cal
V}_{N}\cos(N\phi)\right]}{2},$ (69)

where $f_{N}$ is the proportion of the input state that causes an $N$-fold
coincidence detection event, and ${\cal V}_{N}$ is the $N$-photon visibility.
The CR bound for this scheme can be calculated with the underlying
probabilities $p_{\text{coin}}(\phi)$ for the detection event and
$1-p_{\text{coin}}(\phi)$ for the no detection event, and is written as

$\displaystyle\Delta\phi_{\text{coin}}=\frac{2\sqrt{p_{\text{coin}}(\phi)\left[1-p_{\text{coin}}(\phi)\right]}}{f_{N}{\cal
V}_{N}N|\sin(N\phi)|}\leq\frac{1}{f_{N}{\cal V}_{N}N|\sin(N\phi)|},$ (70)

where the upper bound takes into account the worst case of the probability
$p_{\text{coin}}(\phi)=1/2$. For comparison with the classical benchmark, the
CR bound $\Delta\phi_{\text{coin}}$ in eq 70 can be compared with
$\Delta\phi_{\text{SIL}}$ of eq 45, leading to an inequality
$\Delta\phi_{\text{coin}}<\Delta\phi_{\text{SIL}}$ for a sensor to be so-
called super-sensitive Resch  _et al._ (2007); Thomas-Peter  _et al._ (2011).
The inequality can be specified, when $|\sin(N\phi)|=1$, as

$\displaystyle 1<\frac{f_{N}^{2}{\cal V}^{2}_{N}N}{\tilde{\eta}},$ (71)

where
$\tilde{\eta}=\left[2\sqrt{\eta_{\text{a}}\eta_{\text{b}}}/(\sqrt{\eta_{\text{a}}}+\sqrt{\eta_{\text{b}}})\right]^{2}$.
For the threshold visibility defined as ${\cal
V}_{N}^{\text{(th)}}=\sqrt{\tilde{\eta}/f_{N}^{2}N}$, the above inequality can
be written as ${\cal V}_{N}>{\cal V}_{N}^{\text{(th)}}$, i.e., only a measured
visibility higher than the threshold visibility demonstrates ‘super-
sensitivity’ Resch  _et al._ (2007). The latter can be considered as a genuine
criterion for the quantum enhancement of the precision in sensing experiments
with NOON states instead of ‘super-resolution’, which has sometimes been
misinterpreted, since it can also be produced by only classical light and
projective measurements Resch  _et al._ (2007). The above theory will be used
in section IV.2 when describing quantum plasmonic sensors that have considered
the use of NOON states.

A simple example of NOON state phase sensing uses the two-photon NOON state
$\frac{1}{\sqrt{2}}(\left|{2,0}\right\rangle+\left|{0,2}\right\rangle)$. It
can be readily created by exploiting Hong-Ou-Mandel (HOM) interference Hong
_et al._ (1987). The HOM interference occurs when two single photons are
injected into a lossless 50/50 BS, i.e., the output state is written as
$\hat{B}(\pi/4,\pi/2)\left|{1,1}\right\rangle=(\left|{2,0}\right\rangle+\left|{0,2}\right\rangle)/\sqrt{2}$
up to a global phase, where the component $\left|{1,1}\right\rangle$ is absent
due to destructive interference caused by the indistinguishable paths leading
to the same output state $\left|{1,1}\right\rangle$. When the two-photon NOON
state undergoes a relative phase shift described by the operator
$\hat{U}(\phi)$, the outgoing state can be written up to a global phase as
$(\left|{2,0}\right\rangle+e^{-2i\phi}\left|{0,2}\right\rangle)/\sqrt{2}$
before the measurement. The super-resolution with $\phi$-modulation is
observed due to the multiplied constant factor of $N=2$ in the exponential
factor Wolfgramm  _et al._ (2013). For two-photon NOON states, one way to
realize the required two-fold modulation measurement is by sending the two
modes of the NOON state (now with the phase encoded) into a lossless 50/50 BS
and detecting the coincidence of single photons at the output using single-
photon detectors. The visibility of the two-photon measurement signal (as
$\phi$ is modulated) is then ${\cal
V}_{2}=2\eta_{\text{a}}\eta_{\text{b}}/(\eta_{\text{a}}^{2}+\eta_{\text{b}}^{2})$,
which enables the super-sensitivity inequality to be reduced to approximately
$(\eta_{\text{a}}+\eta_{\text{b}})/2>0.8$, i.e., super-sensitive phase sensing
with the two-photon NOON state can be achieved when the total loss is less
than $20\%$ on average Chen  _et al._ (2018). This is an important point when
using a plasmonic sensor with NOON states, as will be discussed in more detail
in section IV.2.

Interferometric phase sensing has been studied with various quantum states and
measurements that beat the SQL and achieve the HL, including the twin-Fock
state $\left|{N}\right\rangle\left|{N}\right\rangle$ incident into a MZI
Holland and Burnett (1993), the photon-number correlated state Yurke (1986);
Yurke  _et al._ (1986); Shaprio (1993); Dowling (1998); Pezzé and Smerzi
(2006); Huver  _et al._ (2008); Lee  _et al._ (2009), the TMSV state with
parity detection Anisimov  _et al._ (2010), Schrödinger’s cat states Joo  _et
al._ (2011), and multi-headed coherent states Lee  _et al._ (2015, 2020). More
details of these states and their performance can be found in ref 341.

####  III.4.3 Single-mode phase sensing

Instead of two-mode phase sensing, single-mode phase sensing has also been
investigated for the sake of simplicity. Here, the implicit assumption is that
the phase of a reference mode is set. The latter is enabled by employing
phase-sensitive detection schemes, e.g., homodyne detection Yuen and Chan
(1983); Schumaker (1984) or general-dyne measurement Genoni  _et al._ (2014).
Thus, the phase information in single-mode phase sensing cannot be extracted
from a probe state by phase-insensitive detection schemes, such as photon-
number counting, i.e., $\partial_{\phi}p(n|\phi)=0$.

Consider that a single-mode state of light $|\Psi\rangle_{\text{in}}$
undergoes a phase shift described by the operator
$\hat{U}(\phi)=e^{i\phi\hat{a}^{\dagger}\hat{a}}$, as shown in Figure 17(a).
As in the intensity sensing case, one may consider states having a small
uncertainty in their phase for single-mode phase sensing. One typical quantum
state whose phase uncertainty can be reduced below the SNL is the squeezed
vacuum state, $|\xi\rangle$. The phase shift operator $\hat{U}(\phi)$ rotates
a state in phase space about the origin and the output state is
$|\Psi\rangle_{\text{out}}=\hat{U}(\phi)|\xi\rangle=|\xi e^{2i\phi}\rangle$,
for which the QCR bound is obtained as Monras (2006); Pinel  _et al._ (2013)

$\displaystyle\Delta\phi_{\text{QCR}}=\frac{1}{\sqrt{\nu}\sqrt{8(N+N^{2})}},$
(72)

where $N=\sinh^{2}r$ represents the average photon number of the initial
squeezed vacuum state. It is clear that the QCR bound of eq 72 scales with
$N^{-1}$, not only reaching Heisenberg scaling Monras (2006); Pinel  _et al._
(2013) but also beating the SQL of $\Delta\phi=1/\sqrt{\nu 4N}$ that is
obtained for a single-mode coherent state input. Such a quantum enhancement in
phase sensing is enabled by the smaller phase uncertainty of the squeezed
vacuum state as compared to the coherent state with the same energy, as shown
in Figure 17(b). One also finds that the photon number state that is the
optimal state for single-mode intensity sensing cannot encode any phase
information since the photon number state exhibits a full phase uncertainty
Loudon (2000).

Generally speaking, to measure the change of a particular physical quantity
being induced by a parameter $x$ with better precision or lower estimation
uncertainty, the probe state showing the least uncertainty in that particular
parameter is the most useful. A more analytical understanding can be made
through the relation between the QFI and the fidelity ${\cal F}$ between two
infinitesimally close states $\hat{\rho}(x)$ and $\hat{\rho}(x+\text{d}x)$ for
a given parameter $x$ Oh  _et al._ (2019a); Banchi  _et al._ (2015), i.e.,
$H(x)=\lim_{\text{d}x\rightarrow 0}8\\{1-{\cal
F}[\hat{\rho}(x),\hat{\rho}(x+\text{d}x)]\\}/(\text{d}x)^{2}$ (see also
Appendix B). That is, the estimation capability is related to the ability to
distinguish two infinitesimally close states $\hat{\rho}(x)$ and
$\hat{\rho}(x+\text{d}x)$ and a better distinction can be made with the least
uncertainty in $x$ of the probe state.

One can show that the QCR bound of eq 72 is reached by homodyne detection,
which measures the quadrature variable of
$\hat{x}_{\theta_{\text{HD}}}=(e^{-i\theta_{\text{HD}}}\hat{a}+e^{i\theta_{\text{HD}}}\hat{a}^{\dagger})/\sqrt{2}$
Monras (2006); Olivares and Paris (2009); Oh  _et al._ (2019b) by setting the
optimal homodyne angle $\theta_{\text{HD}}$ depending on the value of
$r,\theta_{\text{s}}$ and $\phi$, Oh  _et al._ (2019b). The optimality of
homodyne detection holds in the absence of loss, i.e., probing with a pure
squeezed vacuum state, whereas a realistic squeezed state of light involves an
inevitable thermal photon contribution Vahlbruch  _et al._ (2016), for which
the QCR bound can only be obtained by performing an exotic measurement with
projectors over the eigenstates of the SLD operator Oh  _et al._ (2019a, b).

####  III.4.4 Multiple-phase sensing

Single-phase estimation can be extended to estimating multiple phases,
$\boldsymbol{\theta}=(\theta_{1},\theta_{2},\cdots,\theta_{d})^{\text{T}}$.
This is relevant to applications such as phase imaging Preza  _et al._ (1999),
which measure phase contrast and interference or gravitational wave detectors
Freise  _et al._ (2009), which measure multiple parameters. It is also
relevant to quantum plasmonic imaging, which will be discussed in more detail
in section IV.1.2. The estimation uncertainty of the total of all phases is
governed by the covariance matrix $\text{Cov}(\boldsymbol{\theta})$ and lower
bounded by the QCR bound, as given in eq 38. The uncertainty of the total of
all phases estimated can be quantified by the sum of the individual
uncertainties:
$|\Delta\boldsymbol{\theta}|^{2}=\sum_{j}(\Delta\theta_{j})^{2}=\text{Tr}[\text{Cov}(\boldsymbol{\theta})]$.
This quantity is often compared using three typical cases: (i) a scheme using
optimal classical state inputs, setting the classical benchmark or SQL, (ii) a
scheme that estimates the phases individually by using optimal separable
quantum state inputs, which is called ‘individual estimation’ or a ‘local
strategy’, and (iii) a scheme that estimates all the phases simultaneously by
using optimal entangled state inputs and a collective measurement (when
necessary), which is called ‘simultaneous estimation’ or a ‘global strategy’.
The comparison of these three cases aims to address the following questions:
Can schemes using quantum resources beat the SQL in multi-parameter
estimation? Is a simultaneous estimation approach beneficial as compared to an
optimal (maybe quantum) individual estimation approach? Relevant studies
attempting to answer these questions have recently been started from various
points of view, arising from the fact that multiparameter estimation is non-
trivial, and depends on the kind of parameters estimated, as well as type of
sensing scenario Liu  _et al._ (2020).

Figure 18: (a) (left) An experimental set-up for distributed phase sensing
with $M=4$. A displaced squeezed state input is split into four identical and
entangled probes through a BS network (BSN). The multiple-phase-encoded probe
state is measured with homodyne detection (HD) set-ups, from which the average
phase of multiple phases is estimated. (right) The precision of the estimated
average phase for different average numbers $N$ of photons per sample are
compared between the entangled scheme ($\sigma_{\text{e}}$) and the separable
scheme ($\sigma_{\text{s}}$). Their theoretical precision (solid lines) as
well as the SQL (dashed line) are also presented. Reproduced with permission
from Guo  _et al._ (2020). Copyright 2020 Springer Nature. (b) (left) A phase-
squeezed state injected into two variable BSs configures the continuous-
variable multipartite entangled probe to estimate the average displacement. An
individual displacement on the squeezed phase quadrature variable is induced
by an electro-optic modulator (EOM) driven by radio-frequency (RF) fields. The
output state is measured by homodyne detection, leading to the estimate of the
average displacement. (right) The variance of the estimated values as a
function of the transmittivity of the second BS (VBS2) is compared among
different cases: entangled sensors (circles and black curves for experiment
and theory), classical separable sensors (triangles and green curves for
experiment and theory) and the SQL (black orizontal dotted line). Reproduced
and adapted with permission from ref 380. Copyright 2020 the American Physical
Society.

In one of the earliest works on this topic, Humphreys et al. showed that the
quantum enhancement in the precision of simultaneous estimation of multiple
phases can be obtained by a coherent superposition of $N$ photons among $d$
modes, in which individual phases $\theta_{j=1,\cdots,d}$ are encoded
Humphreys  _et al._ (2013). It has been shown that the considered entangled
state is optimal among the class of definite-photon-number states, with $N$
being the total number of photons, and the advantage over individual
estimation scaling with ${\cal O}(d)$. The particular optimal measurements
identified in ref 381 have recently been generalized in ref 237, which derived
the necessary and sufficient conditions for projective measurements to
saturate the multiparameter QCR bound in the case of pure probe states.

Among $N$-particle photonic states, Holland-Burnett states Holland and Burnett
(1993) and NOON states have been shown to be the optimal states for
simultaneous multiple-phase estimation when $d=2$ in the absence of loss and
decoherence Liu and Cable (2017). Such non-Gaussian states achieve a factor of
$d$ improvement over individual estimation, as shown in ref 381, but the use
of Gaussian states offers no more than a factor of 2 improvement Gagatsos  _et
al._ (2016). Furthermore, when both definite-photon-number states and
indefinite-photon-number states are all considered together, the precision
given by the QCR bound for simultaneous estimation can be obtained or even
further enhanced through an individual estimation strategy using proper mode-
separable input states Knott  _et al._ (2016). It was found that a large
particle-number variance within each mode plays a crucial role in improving
the QCR bound of multiple-phase estimation. This comparable or even better
sensing performance provided by individual schemes suggests experimentally
more favorable settings for multiple-phase estimation. Moreover, the ultimate
quantum limit for simultaneous multiple-phase estimation is only achievable by
an optimal state that manipulates entanglement among both particles and modes
Gessner  _et al._ (2018). A more conclusive mathematical proof has been
provided by Proctor et al., showing that entanglement in simultaneous
estimation leads to no fundamental precision enhancement over individual
estimation when the generators of the multiple parameters commute and yields
no more than a factor of 2 enhancement even when the generators do not commute
Proctor  _et al._ (2018).

Another interesting but more challenging scenario is to estimate multiple
phases that are governed by non-commuting generators, e.g., the three
components of a magnetic field in terms of the spatial coordinates Jones  _et
al._ (2009). For this scenario, a three-fold improvement over individual
estimation strategies has been shown to be achievable by simultaneous
estimation schemes using permutationally invariant quantum states Baumgratz
and Datta (2016), including the superposition of three Greenberger-Horne-
Zeilinger-type states, each of which is known to attain the QCR bound of
estimating a magnetic field aligned along one of the specific axes Giovannetti
_et al._ (2006). The work in ref 387 also showed that too much entanglement is
detrimental for achieving a Heisenberg scaling in terms of the total number of
particles $N$. In general, for non-commuting generators, there exits a
tradeoff for the precisions among the individual estimators, i.e., more
precise estimation of one parameter leads to less precise estimation of the
others Vidrighin  _et al._ (2014); Crowley  _et al._ (2014); Yuan (2016). Hou
et al. have recently derived the minimal tradeoff for the precision of
simultaneous estimation of a three-dimensional magnetic field, finally leading
to the identification of the ultimate quantum limit Hou  _et al._ (2020).

Entanglement appears to lie at the heart of multiparameter estimation and has
been studied from several perspectives, but here again, in the estimation of
multiple phases, entanglement does not always guarantee a quantum enhancement
Knott  _et al._ (2016); Gessner  _et al._ (2018). However, entanglement
becomes a more significant factor in estimating a global parameter that is
composed of multiple parameters to be encoded across multiple modes or
locations, e.g., distributed sensing or networked sensing. The global phase
parameter that is often considered is a linear combination of multiple phases,
written as $\tilde{\theta}=\sum_{j}n_{j}\theta_{j}$, with positive $n_{j}$ and
normalization $\sum_{j}n_{j}=1$. For this, eq 39 needs to be used. The effect
of quantum entanglement in the uncertainty $\Delta\tilde{\theta}$ of
distributed sensing has sparked some interest. Recent work suggests useful
schemes using quantum entanglement to gain an enhancement over schemes that
measure the $\\{\theta_{j}\\}$ individually and then compute a global
parameter $\tilde{\theta}$ via classical communication.

Along the lines mentioned above, Proctor et al. have shown that entanglement
can significantly enhance the precision only when global parameters are of
interest in estimation Proctor  _et al._ (2018). Ge et al. considered a linear
optical network and separable quantum inputs to show that injecting separable
quantum states into only a few modes out of all the modes achieves a quantum
enhancement Ge  _et al._ (2018), resulting in the Heisenberg scaling $1/Nd$ in
terms of both the constrained photon number $N$ and the number of parameters
$d$. Particularly, TF states have been shown to be useful quantum states
achieving the Heisenberg scaling $\propto 1/Nd$, showing a quantum improvement
over an individual estimation approach with the precision scaled as
$1/N\sqrt{d}$. Zhuang et al. have proposed theoretical schemes that use a
squeezed vacuum input being injected into a BS array to estimate a linear
combination of displacement parameters, or that of multiple phases Zhuang  _et
al._ (2018). The proposed schemes were shown to achieve Heisenberg scaling in
the absence of loss, and have recently been experimentally demonstrated for
phase parameters Guo  _et al._ (2020) [see Figure 18(a)] and for displacement
parameters Xia  _et al._ (2020) [see Figure 18(b)]. These proposed schemes can
be enhanced by continuous-variable error correction to reinstate the
Heisenberg scaling, at least up to moderate values of $d$, even in the
presence of loss or decoherence Zhuang  _et al._ (2020). The schemes have all
used homodyne detectors, but single-photon detectors can also be used with an
anti-squeezing operation that transforms the initial squeezed vacuum state
into the vacuum, which achieves Heisenberg scaling in distributed phase
sensing Gatto  _et al._ (2019).

In the special case that Gaussian states are employed as an input into an
array of BSs to estimate the average of independent phase shifts, Oh et al.
have identified the optimal scheme that exploits partially entangled Gaussian
probe states, as maximally entangled probe states are rather detrimental Oh
_et al._ (2020).

As is evident from the above-mentioned recent work, many studies appear to
show that entanglement may not be the only quantity that determines the
characteristic behavior of the multiple-phase quantum sensors under
investigation. In such a sense, a new operational concept called
multiparameter squeezing has recently been suggested to identify
metrologically useful states and optimal estimation strategies Gessner  _et
al._ (2020). This can be seen as a counterpart to the NRF that characterizes
quantum enhancement in quantum noise reduction in intensity measurements. This
is highly relevant to quantum plasmonic imaging.

####  III.4.5 Quantum sensing with SU(1,1) interferometers

Conventional MZIs like those illustrated in Figure 10(b) consist of two inputs
and two outputs, and can be described in a group-theoretical framework as
SU(2) Yurke  _et al._ (1986), the special unitary group of degree 2. The SU(2)
group, which is a simple Lie group, is equivalent to the rotation group in
three dimensions, which has the nice feature that it allows one to visualize
the operations of BSs and phase shifters as rotations in three-dimensional
space. This is true for both classical and quantum optical readout fields.
When the BSs in linear MZIs, or other interferometers like Fabry-Perot
interferometers, are replaced by nonlinear amplifiers like four-wave mixers or
optical parametric amplifiers, as shown in Figs. 19(a) and (b), the resulting
nonlinear interferometer can be described by the SU(1,1) group Yurke  _et al._
(1986), another type of simple Lie-group where the operations can be
visualized as rotations and Lorentz boosts. Similarly, truncated nonlinear
interferometers composed of one nonlinear amplifier followed by dual homodyne
detectors, as shown in Figs. 19(c) and (f), can also be described by the
SU(1,1) group Anderson  _et al._ (2017a); Gupta  _et al._ (2018); Pooser  _et
al._ (2020). Describing interferometric sensors with such a group-theoretical
approach allows for a straightforward analysis of the quantum states that
optimize an interferometericic estimation precision. Both SU(2) and SU(1,1)
interferometers can achieve a phase precision with a scaling approaching $1/N$
for $N$ photons, but SU(2) interferometers require a squeezed vacuum state to
be injected into the second channel of the first BS in order to achieve the HL
when the first channel is fed by a coherent state of light, as discussed in
section III.4.1. In contrast, SU(1,1) interferometers can be operated with
vacuum, coherent state, or combined coherent state and squeezed vacuum state
inputs, and they offer an enhancement in precision that scales with the gain
of the nonlinear amplifier Li  _et al._ (2014); Yurke  _et al._ (1986); Plick
_et al._ (2010). However, because at least one of the nonlinear amplifiers in
a conventional SU(1,1) nonlinear interferometer relies on phase-sensitive
amplification, it is essential to maintain a near-zero phase difference within
the interferometer to maintain the precision advantages Marino  _et al._
(2012).

Figure 19:  (a-c): SU(1,1) nonlinear interferometers with direct intensity
readout, with dual homodyne readout, and with a truncated layout including
dual homodyne readout after a single nonlinear amplifier. A pump laser is
illustrated in green, probe and conjugate fields are represented in red and
blue, and BSs are illustrated to represent loss in the interferometer, with
transmission $\eta$. A transducer is shown inside the interferometer that
imparts the sensors phase $\phi$ onto the probe. Reproduced with permission
from ref 403. Copyright 2017 the American Physical Society. (d): Precision
(normalized here by the mean photon number $|\alpha|^{2}$, and presented as
the variance rather than the standard deviation of $\phi$) as a function of
the gain of the second nonlinear amplifier in a nonlinear interferometer with
direct intensity readout. The gain of the first nonlinear amplifier is set to
G=2, and the internal transmissions are set to $\eta_{\rm ai}=\eta_{\rm
bi}=\eta_{\text{i}}=1$. The blue, yellow, green, and orange lines represent
$\eta_{\text{ae}}=\eta_{\text{be}}=\eta_{\text{e}}$=0.5, 0.75, 0.9, and 0.99
respectively. Reproduced and adapted with permission from ref 403. Copyright
2017 the American Physical Society. (e): Optimal phase precision normalized to
the SNL as a function of $\eta_{\text{e}}$ for a nonlinear interferometer with
asymmetric gain between the two nonlinear amplifiers. Increased gain $r_{2}$
in the second nonlinear amplifier reduces the sensitivity to external losses
$1-\eta_{\text{e}}$. Reproduced with permission from ref 404. Copyright 2017
the American Physical Society. (f): Example of a sensor based on a truncated
nonlinear interferometer. The Rb vapor cell serves as a nonlinear amplifier
and the transducer imparts the sensor’s phase onto the probe or the probe’s
local oscillator. Reproduced with permission from ref 399. Copyright 2020 the
American Physical Society.

As discussed in section III.4.1, the main obstacle to practical quantum phase
sensing with squeezed states of light is protecting the vulnerable quantum
advantage against losses, as described in eq 68. Indeed, in the presence of
losses, SU(2) interferometers with a squeezed vacuum input asymptotically
approach the sensitivity of classical SU(2) interferometers with the same mean
photon number Aspachs  _et al._ (2009). As a result, the development of
detectors with near-unity detection efficiency has been a critical research
topic for years. While some progress has been made with Bayesian parameter
estimation as a tool for mitigating the effect of losses on squeezed SU(2)
interferometers Pezzé and Smerzi (2008), SU(1,1) interferometers offer an
alternative approach that is more robust against losses in some cases. Their
dependence on loss can be broken down into dependencies on internal and
external losses, represented here by $(1-\eta_{\text{i}})$ and
$(1-\eta_{\text{e}})$, respectively. Here, $\eta_{\text{i}}$ and
$\eta_{\text{e}}$ are assumed to be balanced for both arms of the
interferometer, i.e. $\eta_{\text{ai}}=\eta_{\text{bi}}=\eta_{\text{i}}$ and
$\eta_{\text{ae}}=\eta_{\text{be}}=\eta_{\text{e}}$, as shown in Figures
19(a)-(c). For a SU(1,1) interferometer with direct intensity readout, as
illustrated in Figure 19(a), external losses, or losses after the second
nonlinear amplifier largely represent losses due to filtering before detection
and losses due to imperfect detector efficiency. Internal losses include
losses within the nonlinear amplifiers as well as losses from optical
interactions with a sensor in the interferometer. In the limit of
$\eta_{\text{i}}=1$, and when the interferometer is operated in a balanced
mode where the two nonlinear amplifiers have identical gain and loss, with a
coherent state used in each input port, the external loss modifies the
precision $\Delta\phi$ of the SU(1,1) interferometer as Marino  _et al._
(2012)

$\displaystyle\Delta\phi_{\text{e}}=\eta_{\text{e}}^{-1/2}\Delta\phi,$ (73)

In other words, unlike SU(2) interferometers, external loss in a balanced
SU(1,1) interferometer does not change the functional description of the
precision except for the addition of a prefactor because of the second
nonlinear amplifier Kim  _et al._ (2005). In fact, because the second
nonlinear amplifier is a phase-sensitive amplifier that can exhibit noiseless
amplification, increasing its gain can compensate for external loss. The
effect of external loss in an unbalanced SU(1,1) interferometer is shown in
Figure 19(d) Anderson  _et al._ (2017b). Here, the gain of the first nonlinear
amplifier is set to 2, internal losses are neglected, and the gain of the
second nonlinear amplifier is varied. In the case of no external loss, the
precision is almost independent of gain. On the other hand, in the limit of
arbitrarily large gain, the interferometeric precision becomes insensitive to
external loss Anderson  _et al._ (2017b). Figure 19(e) expresses this
insensitivity to external loss more directly by plotting the interferometeric
precision, normalized by the SNL, as a function of $\eta_{\rm e}$ for three
different gain ratios, $r_{2}/r_{1}$. As the gain of the second amplifier,
$r_{2}$, is increased relatively to that of the first, $r_{1}$, the precision
becomes increasingly insensitive to external loss.

However, for the purposes of plasmonic sensing, internal losses that occur on
a plasmonic sensing element within one arm of the interferometer must be
considered. When $\eta_{\text{e}}$ is set equal to unity and the internal
transmission $\eta_{\text{i}}$ is varied (setting $\eta_{\rm bi}=\eta_{\rm
ai}=\eta_{\text{i}}$ such that any additional loss from the sensor is
matched), for bright coherent inputs $|\alpha\rangle$ and $|\beta\rangle$
seeding the first nonlinear amplifier and $N_{\text{i}}$ photons inside the
balanced interferometer, the precision $\Delta\phi$ of the SU(1,1)
interferometer becomes Marino  _et al._ (2012)

$\displaystyle\Delta\phi_{\text{i}}=\left[1+\frac{1-\eta_{\text{i}}}{\eta_{\text{i}}}\frac{N_{\text{i}}}{\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}}\right]^{1/2}\Delta\phi,$
(74)

This suggests that balanced SU(1,1) interferometers are robust against small
levels of loss, but as with conventional squeezed interferometers, increasing
loss will push the precision to the SNL.

Truncated nonlinear interferometers that include only a single nonlinear
amplifier followed by dual homodyne detection, as illustrated in Figs. 19(c)
and (f), provide the same phase precision as conventional nonlinear
interferometers, but they offer some substantial advantages. First, they offer
a simpler experimental design by removing the phase-sensitive nonlinear
amplifier from the nonlinear interferometer. Second, they typically exhibit
reduced loss (and thus a greater quantum enhancement) because the nonlinear
amplifier itself exhibits a tradeoff between gain and loss that always
introduces some loss into the measurement. Finally, truncated nonlinear
interferometers can use arbitrarily high power local oscillators. This is
critical because the SNL for such a measurement is defined by the sum of the
powers of the two-mode squeezed state output from the interferometer and the
local oscillators. Thus, it is possible to arbitrarily increase the power of
the local oscillators, thereby increasing the SNR of the measurement, without
introducing excess power to the sensor, and while taking advantage of the
quantum noise reduction in a two-mode squeezed state. No truncated nonlinear
interferometric plasmonic sensor has been demonstrated in the literature to
date, but a growing literature has described the precision of this approach
for photosensitive sensing applications Anderson  _et al._ (2017a); Gupta  _et
al._ (2018); Pooser  _et al._ (2020).

###  III.5 Quantum sensing beyond the Cramér-Rao bound

For phase estimation using a pure probe state in a single-mode setup with
$\hat{U}(\phi)=e^{i\phi\hat{a}^{\dagger}\hat{a}}$ or in a two-mode setup with
$\hat{U}(\phi)=e^{i\frac{\phi}{2}(\hat{a}^{\dagger}\hat{a}-\hat{b}^{\dagger}\hat{b})}$,
the QFIs of eq 35 are given as
$H=4\langle(\Delta\hat{a}^{\dagger}\hat{a})^{2}\rangle$ and
$H=\langle[\Delta(\hat{a}^{\dagger}\hat{a}-\hat{b}^{\dagger}\hat{b})]^{2}\rangle$,
respectively, where $\langle..\rangle$ denotes the average over the probe
state. This indicates that phase estimation becomes more precise by increasing
the photon number variance of the probe state in a single-mode setup and the
variance of the photon number difference of the two-mode probe state in a two-
mode setup, respectively. A natural question is then, how large can these
variances be increased by?

Rivas and Luis Rivas and Luis (2012) have suggested an estimation strategy in
a single-mode setup using a superposition of a vacuum and a squeezed vacuum
state. The considered probe state has been shown to achieve an arbitrary
scaling in precision, i.e., $\Delta\phi\propto N^{-k}$, where $N$ is the mean
photon number and $k$ is an arbitrary value with $k>1$. Such an alluring
analysis sparked an intensive debate in the literature because it seemed to
beat the Heisenberg scaling ($\Delta\phi\propto N^{-1}$) that is regarded as
the fundamental ultimate limit that can never be beaten by any other physical
strategy Giovannetti  _et al._ (2012). The state studied by Rivas and Luis is
not the only example and states with various photon number distributions have
been considered as candidates for increasing the QFI, for example, the SSW
state Shapiro  _et al._ (1989), the SS state Shapiro and Shepard (1991),
Dowling’s model Dowling (1991), and the small-peak model Braunstein (1994).
When the maximal photon number of a probe state is upper bounded, the so-
called ON state—a superposition of vacuum and a Fock state—is known to lead to
the maximal photon number variance Lee  _et al._ (2019). One can show that the
use of the ON state in single-mode phase estimation can indefinitely increase
the QFI, while keeping the average photon number fixed. The ON state is also
known to be useful in quantum computation Sabapathy and Weedbrook (2018), and
a version with 18 excitations has been realized in the harmonic motion of a
single trapped ion McCormick  _et al._ (2019).

On the other hand, when the maximal photon number of a probe state is
unbounded Johnson  _et al._ (2005), more bizarre photon number statistics can
be found, as discussed in ref 413. For example, phase estimation using a probe
state with a Borel photon number distribution Borel (1942); Tanner (1961) can
lead to sub-Heisenberg scaling $\Delta\phi\propto N^{-3/2}$, calculated by the
QCR bound Lee  _et al._ (2019). Some heavy-tailed and sub-exponential
distributions exhibit a diverging or even an infinite variance Foss  _et al._
(2013). Particular interest has been paid to the Riemann-Zeta distribution as
an example showing an infinite QFI, leading to completely precise phase
estimation without uncertainty in a two-mode scheme Zhang  _et al._ (2013).
This last example is rather mysterious. Together with the other sub-Heisenberg
strategies mentioned above, these examples need to be justified in order to
certify if their precisions are achievable in practice, putting aside the
question of how one might generate those photon number distributions in the
first place.

Apart from exploiting exotic photon number statistics, sub-Heisenberg-limited
precision can also be achieved by using nonlinear effects in many-body systems
Boixo  _et al._ (2007, 2008); Choi and Sundaram (2008); Roy and Braunstein
(2008); Woolley  _et al._ (2008); Napolitano and Mitchell (2010); Rams  _et
al._ (2018). The nonlinear effects in atomic ensemble systems have
experimentally demonstrated sub-Heisenberg-scaling Napolitano  _et al._
(2011). However, a careful examination of the total resources is needed, as
this determines how the precision scales.

Over the last decade, the debate has been devoted to address one simple
question: Can the HL or scaling be beaten? Bollinger  _et al._ (1996); Yurke
_et al._ (1986); Zwierz  _et al._ (2010); Luis and Rodil (2013); Luis (2013);
Anisimov  _et al._ (2010) Fortunately, a conclusive answer has finally been
proved Zwierz  _et al._ (2010); Tsang (2012); Giovannetti and Maccone (2012);
Giovannetti  _et al._ (2012); Berry  _et al._ (2012); Hall  _et al._ (2012);
Hall and Wiseman (2012a, b); Jarzyna and Demkowicz-Dobrzański (2015), and
shows that the overall scaling should properly include the amount of resources
required to obtain a priori probability distribution of the unknown parameter
and the number of measurements repeated to achieve the asymptotic QCR bound.
With such an accounting of the total resources, one can show that the
aforementioned scenarios all turn out to be Heisenberg scaling-limited.

In particular, when the likelihood function being used for the QCR bounds is
highly non-Gaussian and the sample size is small, Ziv-Zakai (ZZ) bounds are
known to be more appropriate Ziv and Zakai (1969); Seidman (1970); Chazan  _et
al._ (1975); Bellini and Tartara (1974); Weinstein (1988). The quantum version
of the ZZ bound has been derived, showing that the MSE would be higher and
thus tighter than a corresponding QCR bound Tsang (2012). Using ZZ bounds,
Giovannetti and Maccone have proved that sub-Heisenberg strategies are
ineffective Giovannetti and Maccone (2012). When a small amount of prior
information is given, no sub-Heisenberg scaling is achievable, i.e., sub-
Heisenberg scaling requires a large amount of prior information. When a large
amount of prior information is given, however, one can just guess a random
value based on the prior distribution without performing any measurement. A
random guess with a large amount of prior information has been proven to
achieve a comparable precision to the corresponding ZZ bound. Considering a
finite amount of prior information, Górecki et al. have recently showed that
the HL needs to be corrected by an additional constant factor of $\pi$ Górecki
_et al._ (2020).

To investigate the practical achievable precision with a finite amount of
prior information and a limited number of measurements, which often nullifies
QCR bounds in practice, Bayesian approaches have been considered in various
situations Jarzyna and Demkowicz-Dobrzański (2015); Friis  _et al._ (2017);
Rubio  _et al._ (2018); Rubio and Dunningham (2019, 2020). In this respect, it
has been shown that quantum sensors can be enhanced by machine learning Lumio
_et al._ (2018) or calibrated by neural networks Cimini  _et al._ (2019). One
can also find a review article that discusses quantum multi-parameter
estimation in terms of the Holevo CR bound, the quantum local asymptotic
normality approach and Bayesian methods Demkowicz-Dobrzański  _et al._ (2020).

Figure 20: The four different types of photonic sensors, classified depending
on the type of physical system used for the probe state, measurement scheme,
and transducer encoding the parameter to be sensed. The default type is the
classical sensor (bottom left), which can be upgraded by moving in either the
horizontal or vertical direction to become a plasmonic sensor (bottom right)
or a quantum sensor (top left). Such an upgrade involves the individual
benefits of either type, however, the benefits of both types can be achieved
by an ultimate type of photonic sensor, called a quantum plasmonic sensor (top
right).

##  IV Quantum-enhanced plasmonic sensors

As seen in section II, plasmonic structures provide sub-diffraction sensing
with high sensitivity Homola  _et al._ (1999a, b); Homola (2006, 2008); Lal
_et al._ (2007); Stewart  _et al._ (2008); Shalabney and Abdulhalim (2011); Li
_et al._ (2015), whereas from section III it is clear that quantum resources
provide the ability to reduce the noise and estimation uncertainty below the
SNL or SQL Giovannetti  _et al._ (2004); Paris (2009); Giovannetti  _et al._
(2011); Tóth and Apellaniz (2014); Demkowicz-Dobrzański  _et al._ (2015);
Taylor and Bowen (2016); Degen  _et al._ (2017); Pirandola  _et al._ (2018);
Braun  _et al._ (2018); Tan and Jeong (2019). Individual sensing techniques
have been developed for plasmonic sensing and quantum sensing independently
from each other over the last few decades, which has led to the establishment
of separate scientific fields in academia and industry. Research is now being
devoted to combining the techniques of plasmonic sensing and quantum sensing
with the aim of providing a new breed of plasmonic sensors with high
sensitivity and high precision at scales below the diffraction limit. These
new types of sensors are called ‘quantum plasmonic sensors’ and they provide a
sensing performance that cannot be obtained solely by either classical
plasmonic sensors or conventional quantum optical sensors. In this section, we
review recent studies that have exploited quantum resources to improve the
sensing performance of plasmonic sensors.

The decomposition of a general sensing procedure discussed in section III.1
(see also Figure 9) allows us to classify four different kinds of sensor Lee
_et al._ (2016): First, a sensor is called a ‘classical sensor’ if a classical
state input, an ordinary optical transducer, and a classical measurement
scheme are used. Second, when the classical input and measurement are replaced
by a quantum state input and a quantum measurement, this introduces quantum
observables, and the sensor is called a ‘quantum sensor’, enabling the
potential for sub-shot-noise sensing. Third, when the ordinary optical
transducer in a classical sensor is replaced by a plasmonic transducer, but
the state and measurement remain classical, the sensor is called a ‘plasmonic
sensor’, enabling sub-diffraction sensing with high sensitivity, but
ultimately shot-noise limited. Finally, a ‘quantum plasmonic sensor’ is formed
when a plasmonic structure is employed for the transducer, and a quantum state
input and a quantum measurement are used. In this case, the plasmonic
structure provides a high sensitivity via the sub-diffraction confinement of
light Takahara  _et al._ (1997); Takahara (2009); Gramotnev and Bozhevolnyi
(2010) , while the quantum input state and measurement provide the potential
for sub-shot-noise sensing. The four kinds of sensors are illustrated in
Figure 20, with the hope that quantum plasmonic sensors are expected to
achieve combined benefits that have only been obtained individually by
plasmonic sensors or quantum sensors.

More generally, the quantum properties of a plasmonic system can also play a
role in sensing, e.g., electron tunneling Zuloaga  _et al._ (2009); Savage
_et al._ (2012), quantum size effects Halperin (1986); Halas  _et al._ (2011);
Scholl  _et al._ (2012) or quantum surface response Gonçalves  _et al._
(2020a). In this case, the quantum plasmonic sensors do not necessarily use
external quantum resources, such as input states or measurements, but exploit
inherent quantum effects stemming from the plasmonic systems themselves. A
careful analysis of whether the quantum effects give rise to sub-shot-noise
sensing or simply improve the sensitivity must be performed. While the methods
presented in this review are invaluable for such analyses, in the following
sections we first focus on recent studies that fit into the above introduced
classification of quantum plasmonic sensors, where the plasmonic structure
provides a high sensitivity and quantum input states and measurements provide
the capability for sub-shot-noise sensing, thus improving the LOD. We then
briefly review other approaches to quantum plasmonic sensors at the end of the
section that do not fit into this classification.

Figure 21: Schematic of the prism setup probing the concentration of BSA with
heralded single photons. The photons of a photon pair generated by a PPKTP
nonlinear crystal are sent to the ATR prism setup and to a detector of the
idler channel, respectively. The detection of the idler photon heralds that
the signal photon has been injected into the ATR prism setup. The number of
transmitted signal photons are then counted for a given analyte and an
incident angle that can vary. Reproduced with permission from ref 257.
Copyright 2018 The Optical Society.

###  IV.1 Quantum plasmonic intensity sensing

The first method we describe is the use of a plasmonic transducer that encodes
parameter information into light where the intensity of the light is dependent
on the parameter to be estimated. Below, we review ‘intensity-sensitive’
plasmonic sensors that use quantum states of light and quantum measurements.

####  IV.1.1 Intensity sensing with discrete variable states

As introduced in section II.2, the ATR prism setup can be used in such a way
that the intensity of the reflected light from the prism is modulated
depending on the refractive index of an optical sample under characterization.
To use the theory of intensity parameter sensing discussed in section III.2.1,
we describe the reflection from the prism as the transmission through the
prism setup, i.e., $|r_{\text{spp}}|^{2}=T$, where $r_{\text{spp}}$ is given
by eq 13.

A practical and useful intensity-parameter SPR sensing scheme is the two-mode
scheme shown in the inset of Figure 15, but where the BS with transmittance
$T$ is replaced by a prism setup. A relevant quantum theory of the ATR prism
setup Tame  _et al._ (2008); Ballester  _et al._ (2009) can be used. Within
the ATR prism setup, the two-mode scheme can be considered in terms of a
signal mode that passes through the prism setup ($a$) and an idler mode that
is kept as a reference ($b$). In this regard, Lee et al. theoretically studied
the two-mode scheme with a particular type of two-mode state Lee  _et al._
(2017) called a twin-mode (TM) state
$\left|{\Psi_{\text{twin}}}\right\rangle=\sum
c_{n,m}\left|{n}\right\rangle_{a}\left|{m}\right\rangle_{b}$ with
$|{c_{n,m}|}=|{c_{m,n}|}$. Such a state has the following symmetric
properties:
$\langle\hat{a}^{\dagger}\hat{a}\rangle=\langle\hat{b}^{\dagger}\hat{b}\rangle~{}$
and
$\langle(\Delta\hat{a}^{\dagger}\hat{a})^{2}\rangle=\langle(\Delta\hat{b}^{\dagger}\hat{b})^{2}\rangle$,
which covers path-symmetric states with $c_{n,m}=c_{m,n}^{*}e^{-2i\gamma}$
Hofmann (2009); Seshadreesan  _et al._ (2013). In this two-mode plasmonic
sensing scheme, an intensity-difference measurement of the observable
$\hat{n}_{-}=\hat{b}^{\dagger}_{\text{out}}\hat{b}_{\text{out}}-\hat{a}^{\dagger}_{\text{out}}\hat{a}_{\text{out}}$
is considered, i.e., the intensity of the transmitted light of the signal mode
(reflected from the prism) is compared with the intensity of the reference
mode. The measured intensity-difference can be inserted into an appropriate
estimator together with eq 13 in order to estimate the refractive index of an
analyte.

As a classical benchmark, the balanced PC state input
$|\alpha\rangle_{\text{a}}|\alpha\rangle_{\text{b}}$ can be considered. When
the average photon number of the individual modes is restricted by $N$ for
both the TM state input and the PC state input, i.e.,
$\langle\hat{a}^{\dagger}_{\text{in}}\hat{a}_{\text{in}}\rangle=\langle\hat{b}^{\dagger}_{\text{in}}\hat{b}_{\text{in}}\rangle=N$,
the intensity-difference signal for both states is the same, i.e.,
$\langle\hat{n}_{-}\rangle=N(\eta_{\text{b}}-\eta_{\text{a}}T)$ with non-ideal
channel transmittance $\eta_{\text{a,b}}$, which includes the detection
efficiency, as modeled in Figure 15. However, the associated measurement noise
$\langle(\Delta\hat{n}_{-})^{2}\rangle$ is different, leading to different
SNRs. By comparing the SNR of the TM state with that of the PC state, one
finds that their SNR ratio
$R_{\text{SNR}}=\text{SNR}_{\text{TM}}/\text{SNR}_{\text{PC}}$ can be written
as eq 66, where the Mandel-Q parameter $Q_{\text{M}}$ and the NRF $\sigma$ of
eq 52 play important roles in intensity-sensitive SPR sensing. As discussed in
section III.3.3, a quantum probe with $-1\leq Q_{\text{M}}<0$ or
$0\leq\sigma_{\text{in}}<1$ leads to a quantum enhancement with
$R_{\text{SNR}}>1$. The optimal state that maximizes the ratio
$R_{\text{SNR}}$ is the TF state, which exhibits an unconditional quantum
enhancement regardless of the values of $N$, $T$ and $\eta_{\text{a,b}}$. The
TMSV state of eq 53, on the other hand, provides a conditional quantum
enhancement that depends on the values of $N$, $T$ and $\eta_{\text{a,b}}$. In
particular, when $\eta_{\text{a}}\approx\eta_{\text{b}}$ and $T\approx 1$, or
$\eta_{\text{a}}T\approx\eta_{\text{b}}$, the ratio $R_{\text{SNR}}$ for the
TMSV state case becomes almost the same as the case of using a TF state input.

Figure 22: (a) The effective transmittance through the ATR prism setup is
measured as the incidence angle $\theta_{\text{in}}$ is varied, and normalized
by the data measured with air. Two samples are used: deionized water and BSA
with $C=2$ %. (b) The standard deviation of the total transmittance histograms
[the same data as (a)] is presented as a function of the total transmittance,
in comparison with the theoretical expectation of classical sensing (upper
solid curve) and quantum sensing (lower solid curve). (c) At a particular
incident angle $\theta_{\text{in}}=67.5^{\circ}$, where the change of
intensity is expected to be maximal (the inflection point), the refractive
index of the BSA solution is estimated for samples with concentration varying
from $0~{}\%$ to $2~{}\%$ in $0.25\%$ steps. (d) The estimation uncertainty of
the refractive index, given as the error bar at each point in (c),
corresponding to the theoretical expectation for the classical and quantum
scenarios, demonstrating a $10\sim 20$% quantum enhancement. Reproduced with
permission from ref 257. Copyright 2018 The Optical Society.

As mentioned in section III, the Fock state input is known to be the optimal
state reaching the ultimate quantum limit in the estimation uncertainty in
both single parameter (section III.3.1) and multiparameter (sections III.3.2
and III.3.3) estimation scenarios that monitor the change of the intensity of
the transmitted light through an optical sample Adesso  _et al._ (2009); Nair
(2018); Ioannou  _et al._ (2020). However, Fock states with high photon number
$N\gg 1$ cannot be readily realized in experiments with current technology
Varcoe  _et al._ (2000); Bertet  _et al._ (2002); Varcoe  _et al._ (2004);
Waks  _et al._ (2006). A few novel schemes have been suggested for the
generation of large Fock states, but the fidelity is limited Clausen  _et al._
(2001); Sanaka (2005); Ou (2006); Deléglise  _et al._ (2008); Wang  _et al._
(2008); Hofheinz  _et al._ (2008); Zhou  _et al._ (2012); Uria  _et al._
(2020); Cosacchi  _et al._ (2020); Groiseau  _et al._ (2020); Zapletal  _et
al._ (2020). Alternatively, one can use $N$ single photons to achieve the same
limit that would be obtained by the Fock state with $|N\rangle$. This can be
seen by the fact that the individual single photons in the Fock state
$\left|{N}\right\rangle$ undergo an independent Bernoulli sampling. Such an
equivalence allows the use of single photons for various quantum sensors Rezus
_et al._ (2012); Whittaker  _et al._ (2017); Allen  _et al._ (2019); Yoon  _et
al._ (2020), including a plasmonic sensor Lee  _et al._ (2018) which we now
introduce.

The experimental quantum plasmonic sensing setup considered in ref 257 is
shown in Figure 21. Single-photon pairs are generated via SPDC in a nonlinear
periodically poled potassium-titanyl-phosphate (PPKTP) crystal, where the
detection of the idler photon heralds that the signal photon has been injected
into the prism setup. The detection of $N$ idler photons thus corresponds to
the use of $N$ signal photons in the signal channel, i.e., equivalent to the
use of the Fock state $|N\rangle$ in the sense of estimation uncertainty. The
experimental estimation uncertainty is measured by the repetition of an
identical experimental observation a thousand times, which yields a
distribution of the estimated total transmittance through the whole setup,
$T_{\text{total}}=N_{\text{t}}/N$, where $N_{\text{t}}$ is the number of
transmitted and detected signal photons out of $N$ injected heralded single
photons for each repetition. The effective transmittance through the prism
setup can be obtained by normalizing the total transmittance by a
normalization factor ${\cal N}$, i.e.,
$T_{\text{prism}}=T_{\text{total}}/{\cal N}$. In the experiment, the
normalization factor is defined as the average transmittance measured with an
air analyte at an individual incident angle $\theta_{\text{in}}$, i.e., ${\cal
N}(\theta_{\text{in}})=\langle T_{\text{air}}(\theta_{\text{in}})\rangle$
which leads to
$T_{\text{prism}}(\theta_{\text{in}})=T_{\text{total}}(\theta_{\text{in}})/\langle
T_{\text{air}}(\theta_{\text{in}})\rangle$. The use of air is because the
light entering the prism is off-resonant from the plasmonic excitation across
the entire range of incident angles considered. This approach enables the
elimination of the effect of an incident angle-dependent misalignment when
scanning through a wide range of incident angles.

Figure 23: (a) Schematic of the tapered hetero-core fiber quantum sensor. (b)
Experimentally obtained variance in the total transmittance of the system
shown together with the theoretical values for classical (red) and quantum
(blue) cases. Reproduced with permission from ref 478. Copyright 2020
Elsevier.

The above quantum plasmonic sensor aims to estimate the refractive index of
the blood protein of bovine serum albumin (BSA) in aqueous solution whose
concentration $C$ is controlled in this proof-of-principle demonstration. From
the measurement of the average transmittance $\langle
T_{\text{prism}}\rangle$, the refractive index of the analyte is estimated
according to the theory model of the reflectance [see eq 13] by setting
$\langle T_{\text{prism}}\rangle=R_{\text{spp}}$. The analyte is placed on the
opposite side of the gold film (see the inset in Figure 21) and its refractive
index affects the resonant condition of SPPs, changing the resonance angle
satisfying eq 12 or equivalently the intensity of the transmitted light for a
fixed incident angle, i.e., $\langle T_{\text{prism}}\rangle$.

The overall behavior of the measurements are presented in Figs. 22(a) and (b).
The average transmittance $\langle
T_{\text{prism}}(\theta_{\text{in}})\rangle$ measured in the experiment when
varying the incident angle $\theta_{\text{in}}$ is shown in Figure 22(a) for
deionized water, i.e., $C=0\%$, and BSA with $C=2\%$. At each incident angle,
the measurement is repeated a number of times, which yields a histogram [see
the inset of Figure 22(a)] whose standard deviation quantifies the error bar
in Figure 22(a). The measured errors depend on the incident angle, which
modulates the transmittance of the setup and thereby the output photon number
statistics. Note that the size of the error bars represents the estimation
uncertainty of the effective transmittance for $N$ single-photon inputs. The
corresponding classical benchmark is defined by the case of using a coherent
state with the same input power as $N$ single photons. The experimentally
measured quantum noise is thus compared with the classical benchmark, as shown
in Figure 22(b). Here, the measured uncertainty is interpreted in terms of the
total transmittance $\langle T_{\text{total}}\rangle$ that consists of the
effective transmittance through the prism setup and loss including the
detection efficiency. Such a comparison clearly exhibits a quantum enhancement
in the experiment and is in good agreement with the ultimate quantum limit
introduced in section III.3.1.

For a specific estimation of the refractive index, the incidence angle is set
to $\theta_{\text{in}}=67.5^{\circ}$ as an example and the measurement is
repeated for each concentration in a range from $0\%$ to $2\%$ in $0.25\%$
steps. From the repeated measurements, the refractive indices are estimated
using eq 13 (i.e., the reflectance $R_{\text{spp}}$). The relation between the
refractive index and the concentration of BSA solution can be determined by
the slope of the linear fitting function, yielding $\text{d}\langle
n_{\text{BSA}}\rangle/\text{d}C=(1.933\pm 0.107)\times 10^{-3}$ [see Figure
22(c)], which is in good agreement with the previously measured value of
$1.82\times 10^{-3}$ Barer and Tkaczyk (1954). The quantum property of light
plays no role in this relation of the mean values, which can be thought of as
more of a calibration, but it plays an important role in reducing the
estimation uncertainty of the refractive index. The latter is demonstrated in
the experiment by repeating the estimation, which produces a distribution of
the estimated refractive indices. The estimation uncertainties with varying
$n_{\text{BSA}}$ are shown in Figure 22(d), which evidently demonstrates a
$10\sim 20\%$ quantum enhancement in comparison with the classical benchmark.
As described in section II.2.3, this enhancement directly affects the LOD of
the sensor and improves its sensing performance.

Figure 24: (a) The experimental setup utilized by Huck et al. generated
squeezed light in an optical parametric oscillator (OPO), excited plasmon
polaritons in a gold waveguide, and characterized the squeezing in the photons
emitted from the end of the gold waveguide. PPKTP, periodically poled
potassium titanyl phosphate crystal; SV, squeezed vacuum; $\lambda/2$, half-
wave plate; $\Phi$, piezo actuated mirror for phase variation; 50:50,
symmetric beam splitter; and HD, homodyne detection scheme. The measured
squeezed vacuum state generated prior to injection into the gold waveguide (b)
and the measured quantum state after excitation of SPPs in the gold waveguide
(c) demonstrate that long range SPPs are capable of maintaining squeezing.
Reproduced with permission from ref 480. Copyright 2009 the American Physical
Society.

As discussed in section III.3.1 and shown in Figure 22(b), the quantum
enhancement in the intensity-sensitive sensor depends on the total
transmittance of the whole sensing setup. Therefore, the quantum enhancement
can be further increased by improving the total transmittivity of the setup by
minimizing channel loss and maximizing the detection efficiency.

A tapered fiber based quantum SPR sensing scheme has also been demonstrated
recently for measuring BSA concentration Peng  _et al._ (2020) and salinity
Zhao  _et al._ (2020). In these works, a tapered hetero-core structure was
fabricated, which is composed of two multi-mode fibers and a single-mode
fiber, coated with a 50 nm thick gold film [see Figure 23(a)]. Heralded single
photons were used as inputs, prepared by SPDC using a 2 mm thick type-II
metaborate crystal ($\beta$-$\text{BaB}_{2}\text{O}_{4}$, BBO) and by applying
a coincident detection scheme. Based on the long evanescent field decay of the
single-mode fiber and by reducing the diameter of the fiber, the authors
showed that it was possible to reach the theoretically predicted estimation
uncertainty below the SNL, as shown in Figure 23(b).

In the given examples and in many other quantum sensing scenarios, heralded
single photons have been used to successfully beat the classical limit in the
estimation uncertainty. Here, it is important to point out a couple of
limitations of heralded single photons for sensing purposes. First, heralded
single photons require a high total detection efficiency to truly achieve a
quantum enhancement. One main goal of quantum sensing is to achieve a high
detection resolution with limited photon intensity at the sample position to
avoid any possible damage. Therefore, all the photons impinging on the sample
should be considered, not just the post-selected ones, for a fair comparison
with the classical counterpart Slussarenko  _et al._ (2017). Second, heralded
photons are not true single-photon states. While the statistics of photon
pairs produced by SPDC follows a thermal distribution, the post-selected
photons can reveal highly anti-bunched nature at a low flux regime, however,
this degrades further at a higher flux Razavi  _et al._ (2009); Bashkansky
_et al._ (2014).

In another approach, many researchers have been pursuing the development of a
true, or so-called ‘on-demand’ single-photon source, which is an essential
ingredient in many quantum optical applications, including quantum plasmonic
sensing. These ‘deterministic’ single photons are known to be one of the
optimal states for low-noise intensity measurements Lee  _et al._ (2017). An
ideal single-photon source requires several important conditions: a high
purity (strong anti-bunching), a high level of indistinguishability depending
on the purpose, a high flux, and a high fidelity Lounis and Orrit (2005);
Aharonovich  _et al._ (2016). Current state-of-art nanophotonic methodologies
utilizing many different types of single emitters have demonstrated improved
quality of single-photon sources Berchera and Degiovanni (2019); Chu  _et al._
(2017); Grosso  _et al._ (2017); Wang  _et al._ (2019b), however, no single
one has yet satisfied all of the requirements at the same time. In particular,
the demonstrated photon fluxes are still too low (<$10^{8}$ cps) for practical
use.

Figure 25: FWM in rubidium vapor, simplified schematic for quantum noise
reduction. A pump laser (P) is used to derive a probe (Pr) field by frequency-
shifting the pump in an acousto-optic modulator (AOM). The double-pass
configuration ensures the probe is not displaced relative to the pump as the
AOM frequency is selected. The probe is offset from the pump by 3 GHz, or
approximately equal to the hyperfine ground state splitting at the D1 line
(795 nm). The pump and probe are jointly detuned from resonance by
approximately 0.8 GHz. They overlap at a small angle (0.3∘) inside the Rb
vapor cell. The resulting probe and conjugate fields are separated from the
pump by a polarizing beam splitter (PBS) and produces a TMSD state. When
incident on a balanced detector, the intensity-difference between the probe
and conjugate (C) shows quantum noise reduction, visible on a spectrum
analyzer (SA). The inset shows the energy levels in Rb associated with a
double $\Lambda$ system. The excited state hyperfine splittings are much
smaller than the ground state, and are blurred with respect to each other in
the Doppler-broadened vapor due to heating to approximately 120∘ C. As two
pump photons are absorbed, coherence between the hyperfine levels ensures that
when the probe field stimulates emission, a third field must be emitted from
the virtual level illustrated by the upper dashed line in order to maintain
energy conservation.

####  IV.1.2 Intensity sensing with continuous variable states

Sensors that exploit quantum noise reduction, or squeezed light Caves (1981),
have seen renewed interest in recent years with applications ranging from
gravitational wave detection to ultra-trace plasmonic sensing at the
nanoscale. The implementations of these sensors are increasingly limited by
their ultimate limits of detection as defined by the Heisenberg uncertainty
principle. At this limit, the noise is dominated by back action and the
quantum statistics of light, leading to the HL, or the SNL when a coherent
state of light is used. Simultaneously, many devices, including plasmonic
sensors, have reached tolerance thresholds in which power in the readout field
can no longer be increased without increasing the noise due to back action or
thermal effects. Beyond these limits, squeezed light can be used to further
improve the precision in these platforms. In recent years a growing number of
sensors based on quantum noise reduction have been demonstrated Schnabel  _et
al._ (2010); Grote  _et al._ (2013); Taylor  _et al._ (2013); Hoff  _et al._
(2013); Treps  _et al._ (2003, 2002); Brida  _et al._ (2010); Wolfgramm  _et
al._ (2010); Otterstrom  _et al._ (2014); Pooser and Lawrie (2015); Lawrie and
Pooser (2013); Lawrie  _et al._ (2013); Fan  _et al._ (2015); Pooser and
Lawrie (2016).

As discussed in section III.3.3, intensity-difference noise reduction allows
one to replace shot-noise limited sensors that operate in a differential
configuration in order to reject common-mode noise and go below the SNL by
exploiting quantum correlations present in two-mode squeezed states. The first
observation of squeezing in plasmonic media characterized the generation,
propagation, and subsequent re-emission of squeezed long-range SPPs in a gold
waveguide Huck  _et al._ (2009), as shown in Figure 24. This effort generated
a squeezed vacuum state in an optical parametric oscillator, excited long-
range SPPs in a gold strip waveguide with that squeezed vacuum state, and then
characterized the quantum state of the photons emitted from the end of the
waveguide. The authors concluded that squeezing can be efficiently and
coherently transduced from an optical field to a SPP, and back to an optical
field with no loss of quantum information except for that due to the
introduction of vacuum noise. This result led to a flurry of subsequent
demonstrations that (a) squeezing is maintained in localized surface plasmons
also Lawrie  _et al._ (2013); Holtfrerich  _et al._ (2016), and (b) this
coherent interface between squeezed states of light and plasmonic media can be
used to enable ultratrace plasmonic sensors with noise floors below the SNL
Fan  _et al._ (2015); Pooser and Lawrie (2016); Dowran  _et al._ (2018). The
results of work on this type of quantum plasmonic sensor are outlined here.

This new type of quantum plasmonic sensor generates intensity-difference
squeezing via FWM McCormick  _et al._ (2008); Boyer  _et al._ (2008b);
Turnbull  _et al._ (2013), which is based on a third-order optical
nonlinearity. An example setup is shown in Figure 25. The noise in this
amplifier can be derived in the interaction frame of the Heisenberg picture.
In this frame, the interaction Hamiltonian for the case with degenerate
pumping fields is

${\cal
H}=i\hbar\chi^{(3)}\hat{a}_{\text{pr}}\hat{a}_{\text{c}}\hat{a}^{\dagger}_{\text{p}}\hat{a}^{\dagger}_{\text{p}}+\text{H.C.},$
(75)

where $\chi^{(3)}$ is the nonlinear coefficient, $\hat{a}_{\text{p}}$ is the
pump field operator, $\hat{a}_{\text{pr}}$ is the input probe field’s
operator, and $\hat{a}_{\text{c}}$ is a third ‘conjugate’ field which is
parametrically amplified from the vacuum. The energy level diagram describing
this system is shown in Figure 25.

In many experiments the pump field is powerful relative to the probe and is
undepleted. In this simplified scenario the field operators take the form of
eqs 56 and 57 in section III.3.3. This leads to a TMSD state and intensity-
difference noise reduction as given in eq 63. The important point about eq 63
is that the noise is less than one shot-noise unit for $\eta>0$ and $G>1$.
Thus, using two-mode squeezing in lieu of a traditional reference subtraction
on a balanced detector will yield a SNR increased by $2\eta(G-1)/(2G-1)$ over
the shot-noise limited measurement McCormick  _et al._ (2007, 2008); Boyer
_et al._ (2008a). The NRF, or total ‘noise power’ contained within a power
spectral measurement, as a function of transmission on both the probe mode (a)
and conjugate mode (b), with $\eta_{\rm a}=\eta_{\rm b}=\eta$ and $G=4$ is
shown in Figure 26. While squeezing is typically known for its fragility in
the presence of loss, and for decades this has limited its usage in sensors,
Figure 26 shows that if one starts with a pure TMSD state and if losses are
symmetric and moderate, the noise reduction over classical detection still
remains substantial. Plasmonic sensors can be configured to fulfill precisely
these conditions. Despite losses, they can operate with large enhancements to
their precision over their classical counterparts, and they represent state-
of-the-art of quantum sensing with two-mode squeezed states. Here we outline
several sensors that exploit quantum noise reduction and enable practical
quantum sensing below the SNL and that beat state-of-the-art classical sensor
in some cases.

Figure 26: Quantum noise reduction as a function of transmission $\eta$ on
both the probe and conjugate fields ($\eta_{\rm a}=\eta_{\rm b}=\eta$) for
nonlinear gain $G=4$. This accounts for imperfect detection efficiency of the
two fields on identical detectors. Here, the noise reduction, or ‘noise power’
is given by $10\log_{10}(1-P/P_{0})$ dB, where $P_{0}$ is the shot-noise, $P$
is the noise from the TMSD state, and $P/P_{0}$ is the NRF $\sigma$.

The first transmission of multi-mode squeezed states through an extraordinary
optical transmission (EOT) medium was demonstrated in ref 312. Using FWM to
provide the squeezed light, the transmission of multiple optical modes through
the plasmonic medium in the form of quantum images was also demonstrated by
exploiting properties of LSPs.

Several works have shown transmission of classical light by LSPs using
triangles Rodrigo  _et al._ (2010) and crosses Lin and Roberts (2011), among
others. The inset of Figure 27(a) shows an SEM image of isosceles triangular
hole nanostructures. The transmission spectrum of this plasmonic medium has a
broad resonance centered at approximately 900 nm, with ample transmission at
the wavelength of common squeezed light sources, 795 nm.

The experimental setup for EOT of squeezed light is shown in Figure 27(a). A
digital micromirror device is used as a spatial light modulator in order to
imprint an image onto the probe field. The image is amplified in the FWM
medium due to its multi-spatial-mode nature Boyer  _et al._ (2008a), resulting
in two twin images in the probe and conjugate fields that exhibit quantum
noise reduction. The plasmonic medium transmits the probe image while the
conjugate image is attenuated by a neutral density filter. The LSPs confined
to the edges of the triangular holes transmit all of the spatial information
contained in the image, unlike the case for SPPs. Figure 27(b) shows the
incident probe image imprinted by the DMD (top) and the transmitted probe
image after the EOT (bottom).

Figure 27: (a) Optical setup showing the FWM experiment with abbreviations
DMD: digital micromirror device, PBS: polarizing beam splitter, BD: beam dump,
SA: spectrum analyzer, Rb: rubidium, $\lambda/2$: half wave plate and ND:
neutral density. One beam passes through the EOT medium while the other is
attenuated with the neutral density filter in order to balance losses and
maximize the noise reduction, or squeezing. Inset: an SEM image of a zoomed in
subarray of the nanostructures. (b) Upper right: the image of the probe beam
imprinted by the DMD before the plasmonic medium. Lower right: the transmitted
image after EOT. Only the intensity has been attenuated while the number of
spatial modes present remains the same.

The transmission is strongly dependent on polarization due to the shape of the
nanostructures, such that when the polarization overlaps with an edge of the
triangles, a LSP is excited, resulting in transmission of multiple spatial
modes. Figure 28(a) and (b) show the transmission and noise power as a
function of polarization, respectively. As the polarization becomes incident
with the base edge of the triangles, the noise power is well below the SNL,
resulting in a quantum noise reduction. Figure 28(c) shows the quantum noise
reduction (squeezing) as a function of transmission. The effects of losses in
SPP waveguides Ballester  _et al._ (2009) and scattering in metal nanoparticle
arrays Lee  _et al._ (2012b) have been previously treated as effective BSs. A
BS model also matches the experimental data, as the theoretical noise
reduction as a function of BS transmission in Figure 28(c) shows. This
experimentally demonstrates the capability for LSP-mediated EOT to transmit
quantum images while conserving macroscopic quantum information, such as
quantum noise reduction.

Figure 28:  Extraordinary optical transmission of squeezed light. (a) Single
color transmission for the hole array for a polarization of $0-175^{\circ}$,
and (b) The relative noise intensity and SNL for the rebalanced probe and
conjugate after the probe has passed through the EOT medium. (c) The measured
squeezing as a function of transmission through the hole array and through a
variable neutral density filter. The noise based on the model in eq (63) is
shown in black. The error bars in (a) and some error bars in (b) are smaller
than the symbols. Reproduced from with permission from ref 312. Copyright 2013
the American Physical Society.

Exploiting the plasmons’ propensity to conserve quantum information means that
they can serve not only as good nanoscale quantum information platforms, but
also as quantum sensors which make use of squeezing. SPPs have been used
extensively to detect trace biochemical compounds in the Kretschmann
configuration Kretschmann and Raether (1968). State-of-the-art classical SPR
sensors utilize differential detection with a reference field that does not
interact with the SPP in order to eliminate noise present in the probe laser
Blanchard-Dionne  _et al._ (2011); Wang  _et al._ (2011); Wu  _et al._ (2004).
Many of these sensors are now only limited by the SNLPiliarik and Homola
(2009); Wang  _et al._ (2011); Wu  _et al._ (2004), and quantum sensors are
required for further improvements in precision.

A plasmonic SPP sensor in the Kretschmann configuration can use quantum noise
reduction to supercede the precision of state-of-the-art classical device.
Figure 29 shows an experimental setup using FWM to produce a TMSD state and a
Kretschmann sensor to obtain a precision below the SNL.

Figure 29: Experimental setup for plasmonic sensing in the Kretschmann
configuration using FWM to provide the signal probe and reference beams, with
abbreviations: SA, spectrum analyzer; PBS, polarizing beam splitter; ND,
neutral density filter; P, pump beam; Pr, probe beam; C, conjugate beam; AOM,
acousto-optic modulator. The probe passes through a prism coated with a
plasmonic thin film on the hypoteneuse, while the conjugate beam serves as a
reference after balancing its intensity with that of the transmitted probe.
Differential detection (A-B) reveals a signal with transmission-dependent NRF.
Reproduced with permission from ref 315. Copyright 2016 American Chemical
Society.

In the experiment, a 43.5 nm thick gold film was deposited on a prism, and
index matching oils were deposited on the film in order to measure a shift in
the plasmonic resonance as a function of the refractive index. The intensity-
difference is measured at the two output channels A and B, with a ND filter
placed on the conjugate mode (A) to equal the transmittivity of mode B and
reduce the noise in the measurement signal to a minimum, as discussed in
section III.3.3. An acousto-optic modulator (AOM) was used to place a
modulation on the probe mode (B). This leads to a peak in the measurement
signal at the driving frequency 1.5 MHz when viewed on a spectrum analyzer, as
shown in Figure 30(c) and (d) for two example cases. The height of the peak is
proportional to the magnitude of the amplitude modulation on the probe field
and due to the conjugate field being unmodulated the peak is also proportional
to the magnitude of the intensity-difference signal. On the other hand, the
sidebands of the signal (either side of the peak) correspond to the noise
floor (the variance) of the measurement signal, as there is minimal amplitude
modulation of the probe (and thus of the intensity-difference measurement
signal) at frequencies away from the AOM resonance. Using a spectrum analyzer
to give a power spectrum of the measurement signal in this way allows the
intensity-difference (proportional to the peak at resonance) and the
intensity-difference noise (the side-bands) to be obtained from the same plot.
Thus, both the signal amplitude and noise can be read off simultaneously,
allowing an immediate characterization of the SNR of the system. An additional
feature is that technical noise at low frequencies (such as laser or vibration
fluctuations) can effectively be eliminated in a manner similar to lock-in
detection, allowing the sensor’s precision to be brought close to the
theoretical limit.

Figure 30: Resonance curves and noise reduction for plasmonic sensing in the
Kretschmann configuration with squeezed light. (a) Plasmonic resonance as a
function of relative angle of incidence ($0^{\circ}$ corresponds to on-
resonance for $n=1.3$) and index of refraction (green: n=1.305; blue: n=1.301;
red: n=1.300). The numerical curve fits are intended as a guide to the eye.
(b) Quantum noise reduction (squeezing) as a function of angle of incidence
and index of refraction (green: n=1.305; blue: n=1.301; red: n=1.300). (c) &
(d) The raw data for two points along the curve. The red line indicates the
SNL, while the peak is associated with the transmitted probe power. The
sidebands that indicate the minimum noise (or noise floor) associated with
each data point fall below the SNL - squeezing (noise reduction) is equal to
the difference between the black and red sidebands. Squeezing decreases as
transmission decreases, but virtually all data points are squeezed. A maximum
of 4.6 dB below the SNL is observed in the left shoulder of the peaks in (a).
A minimum of 0.3 dB is observed in the n=1.300 and n=1.301 datasets, while the
n=1.305 dataset minimum is at the SNL. Reproduced with permission from ref
315. Copyright 2016 American Chemical Society. Figure 31: Ultrasonic pressure
wave sensing. A FWM configuration is used to generate entangled probe and
conjugate beams in a TMSD state. The probe beam serves as a probe for the
plasmonic sensor while the conjugate beam serves as a quantum-correlated
reference. SA: spectrum analyzer; BD: beam dump; FG: function generator.
Adapted with permission from ref 500. Copyright 2018 The Optical Society.
Figure 32: Measured signal while linearly ramping the driving voltage of the
ultrasonic transducer, thus increasing the change of refractive index of air
($\Delta n$), when probing with coherent states, trace (i), and with twin
beams (as a TMSD state), trace (ii). The lighter-weight lines represent the
shot-noise (i) and squeezed noise floors (ii). Adapted with permission from
ref 500. Copyright 2018 The Optical Society.

Figure 30(a) shows the resulting resonance curve of the signal as a function
of the angle of incidence and the refractive index. Every data point
corresponds to a quantum noise reduction, as shown in Figure 30(b). These
points are obtained by comparing the sidebands of the signal on the spectrum
analyzer, as shown in Figure 30(c) and (d) for two example cases, where the
peak occurs at 1.5 MHz due to the frequency set by the AOM. A maximum noise
reduction of 4.6 dB is obtained on the left-hand shoulder of Figure 30(b). It
is notable that a large amount of squeezing (noise reduction) is observed near
the inflection points in Figure 30(b), meaning that one can choose to operate
this SPR sensor at a single positionWang  _et al._ (2011), where a large
quantum effect is still observed. At all points on a transmission vs index
curve, a higher SNR would be achieved than is possible with the classical
version of this sensor. Comparing the quantum sensor with a classical sensor
in which the incident power may be turned up indefinitely, the quantum sensor
still compares favorably. The quantum light source used here can be operated
at equivalent optical powers, and theoretically the amount of quantum noise
reduction does not depend on the incident optical power [see eq (63) in
section III.3.3]. On the other hand, the classical sensor cannot be used at
powers beyond the point of thermal modulationKaya  _et al._ (2013) of the
plasmon or the damage threshold of photo-sensitive ligandsRobinson  _et al._
(2014); Bartczak  _et al._ (2011). These thresholds are easily within the
power capabilities of the squeezed light source used here and many others.
Notably, it is also possible to use this configuration without amplitude
modulation to obtain a quantum-modulated signal at DC Fan  _et al._ (2015).

Similar sensing configurations have been used to detect ultrasonic pressure
waves Dowran  _et al._ (2018) and establish long range quantum plasmonic
networks Holtfrerich  _et al._ (2016). In the case of ultrasonic measurements,
the index of refraction shift is caused by local variations in air pressure
above the plasmonic nanosurface at a frequency of 199 kHz. Using LSPs and EOT,
a change in transmission serves as a transduction of index of refraction
shifts onto optical intensity modulation. The magnitude of the intensity
modulation (representing the magnitude of the refractive index modulation
$\Delta n$) can then be detected with noise below the SNL using a TMSD probe
state. Figure 31 shows the experimental setup. A chamber with an ultrasonic
transducer is used to provide pressure waves at the surface of a plasmonic
thin film, which consists of triangular hole arrays. One of the two twin beams
passes through the EOT sensor while the other continues in free space and acts
as a reference. The resulting difference measurement contains the modulation
signal, provided by the function generator to the ultrasonic transducer, on
top of a reduced noise background.

Figure 32 shows the measured signal (and noise floor) as the magnitude of the
change in refractive index $\Delta n$ increases. The classical signal
(coherent state probing) is shown in blue (i) and the TMSD state signal is
shown in red (ii). The signals are normalized by the average shot-noise of the
classical case, shown as a light blue line. Clearly the TMSD state has reduced
noise compared to the classical case: the minimum noise levels show a
difference of 4 dB. The SNR for the TMSD state is higher than the classical
case as $\Delta n$ decreases. For very small index shifts, the TMSD version of
the sensor is therefore able to discern index shifts that are smaller than the
classical sensor, enabling a detectable shift $\Delta n$ of $O(10^{-9})$ RIU.
The minimum resolution of the refractive index modulation can be determined at
a 99% confidence level in the SNR. In the quantum case this gives $\Delta
n_{\rm min}=5.5\times 10^{-9}$ RIU with the TMSD state and $\Delta n_{\rm
min}=9.6\times 10^{-9}$ RIU with classical coherent states. This demonstrates
a quantum enhancement of 56% in the precision of the sensor.

Figure 33:  (a) Schematic of quantum spectroscopy with a twin beam (i.e., the
TMSV state of eq 53) generated from a SPDC source (BBO crystals), where
coincidence photon counting is carried out by two APDs (avalanche
photodiodes). A monochromator placed in the signal channel consists of a
mirror (M), a diffraction grating (GR), a pinhole (PH) and an APD, and selects
a particular single mode of interest under investigation, realizing
spectroscopy. (b) Schematic of conventional spectroscopy with classical light
generated from a Halogen lamp, where a single APD is used together with the
same monochromator from the quantum spectroscopy scheme. (c) The MNP array
under investigation: Dark-field microscope image (top) and scanning electron
microscope image (bottom). Adapted from ref 507 (CC BY 3.0 License).

As described in section II.5, SPR imaging enables highly parallelized, label-
free, real-time, high precision biochemical sensingRothenhäusler and Knoll
(1988); Piliarik  _et al._ (2007); Smith  _et al._ (2003); Steiner (2004);
Wark  _et al._ (2005); Wong and Olivo (2014); Jordan and Corn (1997). However,
all of the quantum plasmonic sensors described in this review thus far are
single pixel sensors. Quantum plasmonic sensors that utilize multi-spatial-
mode quantum correlations in entangled optical readout fields are increasingly
a plausible idea. Early research that focused on the transduction of quantum
images by EOT showed that SPP-mediated EOT processes do not maintain spatial
information, thus limiting the potential for EOT-based spatially resolved
quantum sensorsAltewischer  _et al._ (2002). However, LSP-mediated EOT
processes do maintain spatial information, as already discussed in this
section and shown in Figure 27 Lawrie  _et al._ (2013). Kretschmann-style SPR
sensors also maintain spatial information, as discussed in section II.5.

Figure 34:  Transmission spectra measured by conventional classical (left) and
quantum (right) spectroscopy under different noise conditions
[$N_{\text{s}}=10^{2}$ (top), $2\times 10^{4}$ (middle), and $7\times
10^{4}~{}\text{c/s}$ (bottom)]. The measurement is performed for water-
glycerin solutions deposited on an array of MNPs with different concentrations
[$40\%$ (red dots) and $50\%$ (black dots)]. The photon flux of $10^{3}$ c/s
is kept in the monochromator for all the experiments and the error bars denote
the standard deviations of the measured signals. Adapted from ref 507 (CC BY
3.0 License).

Furthermore, as discussed in section III.3.2 and highlighted in recent reviews
Magaña-Loaiza and Boyd (2019); Moreau  _et al._ (2019); Genovese (2016),
substantial work has gone into the development of quantum imaging platforms
for other applications. For SPR imaging sensors that are robust to higher
power optical excitation, discrete variable entanglement is less beneficial,
but higher power multi-spatial mode squeezing could be leveraged to enable
quantum-enhanced SPR imaging. As mentioned above, and as shown in Figure 27,
plasmonic images exhibiting intensity-difference squeezing can be efficiently
transduced by LSP-mediated EOT processes. The same effect has been shown more
recently for a combination of phase-sum and intensity-difference entanglement
Holtfrerich  _et al._ (2016). Despite the demonstration of the plasmonic
transduction of multi-spatial-mode quantum states of light, there have been no
reports of quantum-enhanced SPR imaging. A key ingredient for such an advance
is the multi-pixel readout of an array of plasmonic sensors that can benefit
from quantum correlations in the readout field. A growing number of research
efforts address exactly this need. For instance, researchers are now utilizing
electron-multiplying CCD cameras for spatio-temporally resolved readout of
multi-spatial-mode squeezed states of light Kumar  _et al._ (2017); Kumar and
Marino (2019); Li  _et al._ (2020b), and preliminary research has shown that
compressive imaging techniques can be used for squeezed light imaging with
single pixel detectors Lawrie and Pooser (2013). Together, multi-spatial-mode
quantum light sources, multi-pixel plasmonic sensors, and readout schemes for
characterizing multi-spatial-mode quantum signals provide the necessary
building blocks for quantum-enhanced SPR imaging.

####  IV.1.3 Intensity sensing robust to thermal noise

The strong photon number correlation possessed by the TMSV state of eq 53,
with a NRF of $\sigma=0$, has also been used to probe an array of gold
nanoparticles and measure the refractive index change of a glycerin-water
solution Kalashnikov  _et al._ (2014). The two-mode scheme shown in Figure 15
was used to carry out quantum spectroscopy with the input state
$|\Psi\rangle_{\text{in}}=|\text{TMSV}\rangle$, where the measurement is a
coincidence detection [see Figure 33(a)]. The glycerin-water solution
surrounds a plasmonic transducer that consists of a hexagonal array of
spherical MNPs with a diameter of 130 nm and lattice period of 1.1 $\mu$m, as
shown in Figure 33(c). The idler mode of the TMSV state is sent through the
sample, while the signal mode is sent to a diffraction grating to investigate
the spectral response using the coincidence counting that follows. The sensing
performance of such a quantum scheme is compared with conventional
spectroscopy that uses a classical probe state generated from a lamp, as shown
in Figure 33(b).

The authors directly compared quantum and classical spectroscopy methods in
terms of their robustness to thermal noise influencing the detection. The
thermal noise is artificially realized by a lamp inside the monochromater part
[see Figs. 33(a) and (b)], which induces additional photon counts
$N_{\text{s}}$ in the measurement of the signal photons $S_{\text{s}}$. The
noise level is controlled by changing the brightness of the lamp, i.e.,
$N_{\text{s}}=10^{2}$, $2\times 10^{4}$, and $7\times 10^{4}~{}\text{c/s}$ are
considered in the experiment. The effect of thermal noise is examined with
respect to the distinguishability of the two resonance spectra measured for
two concentrations $C$ of glycerin-water solution on top of the array of MNPs.

The results obtained by quantum and classical spectroscopy for the two
concentrations under the three noise conditions are presented in Figure 34.
The red dots represent the spectra for $C=40\%$ and black dots for $C=50\%$.
Generally, the resonance curves in both cases become blurred due to thermal
fluctuations that increase with the thermal noise. However, the extent to
which the curves become blurred is different for the quantum and classical
spectroscopy schemes; the curves for the quantum case are more distinguishable
than those for the classical case. This is clearly seen for the case when the
noise photocount $N_{\text{s}}$ is $70$ times larger than the signal
photocount $S_{\text{s}}$ [Figs. 34(e) for classical and (f) for quantum].
Such robustness against the thermal noise can be attributed to the strong
photon number correlation of the TMSV state, which is absent in the classical
scheme.

Figure 35:  Quantum plasmonic phase sensing using a nanowire. (a) Two-mode
scheme with a quantum source, plasmonic nanowire and measurement. (b) Measured
signal for the optimal observable used ($\hat{O}=\hat{M}$ for classical (C) or
$\hat{A}$ for quantum (Q)) for a dielectric and plasmonic nanowire. (c) The
precision of the refractive index $\Delta n$ for classical (C) and quantum (Q)
states with dielectric and plasmonic nanowires. The photon number on average
is $N=4$, the nanowire radius is $50$ nm, the propagation length of the
nanowire is $4~{}\mu\text{m}$ and the free-space wavelength is
$\lambda_{0}=810$ nm. Reproduced with permission from ref 452. Copyright 2016
American Chemical Society.

###  IV.2 Quantum plasmonic phase sensing

In addition to measuring changes in the refractive index of an optical sample
using intensity sensing, it is also possible to use phase sensing, as outlined
in section II.3. In the quantum regime this provides a complementary setting
for quantum plasmonic sensing. In order to carry out quantum phase sensing
using plasmonics, the physical geometry of the system needs to be modified so
that changes in the refractive index are picked up in the phase of the optical
signal, rather than the transmission amplitude. In this case, a metal nanowire
is one example that can be considered and it provides a compact setting in
which to perform sensing below the diffraction limit due to the highly