confined field (Takahara  _et al._ , 1997; Takahara, 2009). A nanowire also
has the potential to be modified into more advanced nanophotonic circuitry for
complex functionality Bozhevolnyi  _et al._ (2006); Gramotnev and Bozhevolnyi
(2010) and has already been considered for classical phase sensing Homola  _et
al._ (1999a).

####  IV.2.1 Phase sensing with discrete variable states

The theoretical work by Lee et al. Lee  _et al._ (2016) first studied the use
of a nanowire in quantum phase sensing and the model considered is shown in
Figure 35(a). Here, a source produces a two-mode quantum state of light,
$\left|{\Psi_{\rm in}}\right\rangle$, and a silver nanowire is placed in one
of the modes as a probe ($p$), while the other mode is used as a reference
($r$). A biological medium is considered to surround the nanowire and for a
given change in a dynamical quantity, such as concentration, a change in the
refractive index, $n$, occurs which changes the relative phase between the
probe and reference modes by $\phi(n)=\ell\beta(n)$, where $\ell$ is the
length of the nanowire and $\beta$ is the propagation constant in the
nanowire, which is a function of $n$. A measurement of the quantum state is
then performed in order to obtain an estimate of the phase and therefore an
estimate of the refractive index. The dynamical quantity in the biological
medium, such as concentration, can then be estimated from the refractive
index.

Figure 36:  Experimental investigation of quantum plasmonic phase sensing
using a silver nanowire. (a) Generation of polarization encoded NOON state
($N=2$) via parametric down-conversion. The two spatial modes of the state (1
and 2) are combined into a single mode (4), while maintaining the two-mode
scenario in the polarization degree of freedom. The entangled photons in the
single spatial mode are injected into an optical fiber. (b) The fiber is
tapered with a silver nanowire fixed to the end. The out-coupled light is
collected and sent to a measurement stage. (c) Scanning electron microscope
image of the tapered fiber and nanowire. (d) Measurement signal for the two-
photon NOON state (red) and classical case (blue). The y-axes on the left
(right) is for the NOON state (classical case). Adapted with permission from
ref 358. Copyright 2018 The Optical Society.

In the ideal case when there is no loss, a NOON state is well known to be an
optimal state for quantum phase sensing Walther  _et al._ (2004); Crespi  _et
al._ (2012); Afek  _et al._ (2010), as discussed in section III.4.2, allowing
one to reach the HL for the precision. It can be used as the two-mode quantum
state produced by the source in Figure 35(a), i.e., $\left|{\Psi_{\rm
in}}\right\rangle=\frac{1}{\sqrt{2}}(\left|{N,0}\right\rangle_{\text{p,r}}+\left|{0,N}\right\rangle_{\text{p,r}})$,
where $N$ denotes the number of photons in a given mode. The relative phase in
the nanowire is picked up by the first term in the state,
$\left|{N,0}\right\rangle\to e^{iN\phi(n)}\left|{N,0}\right\rangle$. The
resulting state is measured using the optimal quantum observable
$\hat{A}=|0,N\rangle\langle N,0|+|N,0\rangle\langle 0,N|$ (see section
III.4.2). This leads to a measurement signal
$\langle\hat{A}\rangle=A_{0}\cos[N\phi(n)]$ and a corresponding precision in
the estimation of the refractive index of $\Delta
n=\frac{1}{N}\left|\frac{\text{d}\phi}{\text{d}n}\right|^{-1}$, which is the
HL for a single-shot measurement, i.e., $\nu=1$, as discussed in section
III.4.2. In contrast, the best classical strategy uses the two-mode classical
coherent state $\left|{\Psi_{\rm
in}}\right\rangle=\left|{\alpha/\sqrt{2}}\right\rangle_{p}\left|{\alpha/\sqrt{2}}\right\rangle_{r}$
produced by the source, with $|\alpha|^{2}=N$, which picks up the relative
phase in the first term:
$\left|{\alpha/\sqrt{2}}\right\rangle_{p}\to\left|{e^{i\phi(n)}\alpha/\sqrt{2}}\right\rangle_{p}$.
The optimal measurement consists of a BS and measurement of the observable
$\hat{M}=\hat{I}_{p}-\hat{I}_{r}$, which is the intensity-difference, as
exploited in section III.3.3. This leads to a measurement signal
$\langle\hat{M}\rangle=M_{0}\cos[\phi(n)]$ and a corresponding precision
$\Delta n=\frac{1}{\sqrt{N}}\left|\frac{\text{d}\phi}{\text{d}n}\right|^{-1}$,
which is the SNL or SQL. The NOON state therefore provides a $\sqrt{N}$
improvement in the precision compared to the classical case for a given $\nu$.
The signals $\langle\hat{A}\rangle$ and $\langle\hat{M}\rangle$, and their
associated precisions are shown in Figures 35(b) and (c) for $N=4$, where they
are compared with the case in which a conventional dielectic nanowire is used.
The results clearly show the benefit of using a plasmonic nanowire with a
quantum source and measurement for phase sensing.

The improvement in the phase sensing precision comes from two distinct factors
that are revealed by writing the precision (in this case the LOD) as $\Delta
n=\Delta\phi\left|\frac{\text{d}\phi}{\text{d}n}\right|^{-1}$, where
$\Delta\phi=\Delta
O\left|\frac{\text{d}\langle\hat{O}\rangle}{\text{d}\phi}\right|^{-1}$ is the
precision of the phase estimation and $\hat{O}$ is the observable in the
measurement ($\hat{O}=\hat{A}$ or $\hat{M}$). $\Delta\phi$ obviously depends
on the observable $\hat{O}$ and state $\left|{\Psi_{\rm in}}\right\rangle$,
but interestingly not on the nanowire properties, and is therefore quantum in
origin. On the other hand, $\left|\frac{\text{d}\phi}{\text{d}n}\right|$
represents the sensitivity, $S_{\phi}$, which depends on the length and
propagation constant [$\phi(n)=\ell\beta(n)$], i.e., the nanowire properties,
and is therefore classical in origin. Both classical and quantum factors are
needed to obtain a good precision, however, it is the quantum factor that
improves the precision beyond the SQL, as discussed in section II.2.3.

In a more realistic scenario, losses in the nanowire must be considered. In
this case, even for moderate losses, the NOON state is no longer the optimal
quantum state. A more general analysis using the QFI, $H$, must then be
undertaken in order to find the optimal state to use. In this case, a more
general state $\left|{\Psi_{\rm
in}}\right\rangle=\sum_{n=0}^{N}c_{n}\left|{n,N-n}\right\rangle_{p,r}$ can be
considered Dorner  _et al._ (2009), with the coefficients $c_{n}$ optimized in
order to maximize $H$ and thus minimize the QCR bound $\Delta\phi_{\rm
min}=(\max_{\\{c_{n}\\}}H)^{-1/2}$, which in turn minimizes the estimation
uncertainty $\Delta n$. In Lee et al. Lee  _et al._ (2016) it was found that
plasmonic sensors using these optimized states provide a precision beyond the
SIL, which is the SNL of standard interferometers when loss is present Dorner
_et al._ (2009) and discussed in section III.2.2, even for a moderate amount
of loss in a nanowire. Results were obtained for arbitrary $N$, where the
precision of the optimized states was shown to always be an improvement over
the SIL precision.

While the use of the QFI enables an optimization of the quantum state used by
the source, it is not clear what the optimal measurement is that allows one to
reach the lower bound in the estimation uncertainty of $\Delta\phi_{\rm min}$,
i.e., the QCR bound introduced in section III.1. Further work in this
direction will be needed to uncover practical measurement schemes for discrete
variable quantum phase sensing in plasmonics. Furthermore, the use of
different types of plasmonic material Tassin  _et al._ (2012) for the nanowire
could be considered in order to reduce loss while maintaining the high field
confinement, including the use of metamaterials Cai and Shalaev (2010);
Soukoulis and Wegener (2011); Chen  _et al._ (2012); Meinzer  _et al._ (2014),
graphene Grigorenko  _et al._ (2012) and hybrid material systems Ozbay (2006);
Sorger  _et al._ (2012); Fang and Sun (2015).

The use of a plasmonic nanowire for phase sensing has recently been
experimentally investigated for the simple case of a two-photon NOON state
($N=2$) generated by parametric down-conversion Chen  _et al._ (2018), as
shown in Figure 36(a). The two-mode state used takes the form
$\left|{\Psi_{\rm
in}}\right\rangle=\frac{1}{\sqrt{2}}(\left|{2_{H},0}\right\rangle_{1,2}+\left|{0,2_{V}}\right\rangle_{1,2})$,
where $\left|{2_{H}}\right\rangle$ ($\left|{2_{V}}\right\rangle$) corresponds
to two horizontal (vertical) polarized photons in a given spatial mode. The
polarization dependence of the photons enables the two spatial modes to be
combined into a single spatial mode in an optical fiber, while maintaining an
effective ‘two-mode’ setting – both polarization ‘modes’ co-propagate in the
same spatial mode. The vertical (horizontal) polarization represents the probe
(reference) mode. The optical fiber is then tapered and a silver nanowire
attached to the end, which enables the efficient excitation of SPPs and is
shown in Figure 36(b) and (c). The output of the signal is collected by a
microscope objective and sent to a measurement stage with optical elements and
single-photon detectors for the measurement of the phase.

In the experiment, the phase was modified using a liquid crystal phase
retarder outside the nanowire in order to gain an understanding of the
performance of the nanowire for phase sensing in a controlled manner. While
not a direct demonstration, the approach is equivalent to the case that the
phase is picked up directly in the nanowire, as phase shift operations commute
with the loss accumulated in the nanowire Demkowicz-Dobrzański  _et al._
(2015). A more direct demonstration is yet to be performed, however, this is
more challenging due to the requirement of a ligand coating along one of the
axes of the nanowire in order for only one polarization mode to interact with
the biochemical substance being sensed. Fortunately, this can be achieved in a
number of ways, for example using nanografting Salaita  _et al._ (2007);
Leggett (2012). By modifying the phase of the probe mode, then converting the
probe and reference modes back into spatial modes and measuring the
coincidences, a measurement equivalent to that of the observable $\hat{A}$ can
be performed. The signal of this measurement is shown in Figure 36(d) as the
red curve, along with the corresponding classical signal as the blue curve,
equivalent to the observable $\hat{M}$. The NOON state signal clearly shows an
oscillation that occurs over a phase twice as small as the classical case, as
expected – a phenomenon known as ‘super resolution’ Resch  _et al._ (2007).

The coincidence counts in Figure 36(d) can be normalized to give a probability
of a coincidence Crespi  _et al._ (2012), according to eq 69 in section
III.4.2, i.e., $p_{\text{coin}}(\phi)=f_{2}(1+{\cal V}_{2}\cos{2\phi})/2$,
where $f_{2}$ represents the total proportion of photons that lead to a two-
photon coincidence and ${\cal V}_{2}$ is the two-photon visibility. This
coincidence probability can then be used to infer the uncertainty in the
measurement of the phase, $\Delta\phi_{\text{coin}}$. In this realistic and
lossy setting, the SIL for the precision in estimation of the phase is given
by Demkowicz-Dobrzański  _et al._ (2009); Thomas-Peter  _et al._ (2011)
$\Delta\phi_{\rm SIL}=1/\sqrt{N\eta}$, which is simply the SNL of standard
interferometers in the case of no loss multiplied by the factor
$1/\sqrt{\eta}$, where $\eta$ is the total loss factor. In order to show an
improvement, the uncertainty in the experiment must satisfy
$\Delta\phi_{\text{coin}}<\Delta\phi_{\rm SIL}$, after which it can be called
‘super sensitive’ Thomas-Peter  _et al._ (2011). Using the normalized
probability function $p_{\text{coin}}(\phi)$, the bound can be reformulated as
Chen  _et al._ (2018) $1<2f_{2}^{2}{\cal V}_{2}^{2}/\eta$, which can be
inverted to give a threshold visibility of ${\cal
V}_{2}^{\text{(th)}}=\sqrt{\eta/2f_{2}^{2}}$. A visibility above this value
shows an improvement in the precision beyond the SIL. In the absence of loss,
$f_{2}=1$ and $\eta=1$, leading to ${\cal V}_{2}^{\text{(th)}}=2^{-1/2}\approx
0.707$. In the experiment, the visibility from the data shown in Figure 36(d)
was calculated to be ${\cal V}_{2}\approx 0.880$, clearly above the threshold.
However, once loss is included, ${\cal V}_{2}^{\text{(th)}}$ rises. It was
concluded that even if ${\cal V}_{2}=1$ in the experiment, the threshold could
not be surpassed without a reduction in loss in the setup. Fortunately, it was
shown that in principle the loss could be reduced by improving collection
efficiencies at the various components in the setup, while still accommodating
for loss in the plasmonic nanowire. Thus, the precision could be improved
beyond the SIL with further experimental improvements.

It is clear from the above discussion that various technical challenges must
be overcome for a complete experimental demonstration of quantum phase sensing
using a plasmonic system and discrete variable quantum states. Reducing losses
in the components of the setup, implementing the phase change at the nanowire
itself, as well as the potential use of more robust-to-loss states other than
the NOON state are all avenues for further study to address the challenges.
These would enable the realization of practical nanoscale quantum plasmonic
interferometric sensors.

In light of these challenges, it is interesting to note that the generation of
plasmonic NOON states has also been achieved in several other recent
experiments. By building on previous demonstrations of two-plasmon quantum
interference Heeres  _et al._ (2013); Martino  _et al._ (2014); Fakonas  _et
al._ (2014); Cai  _et al._ (2014); Fujii  _et al._ (2014), Fakonas et al.
Fakonas  _et al._ (2015) generated NOON states with $N=1$ and $N=2$, and
studied the impact of decoherence on these states in low-loss dielectrically
loaded SPP waveguides. They found that for the type of waveguide studied,
decoherence in the form of phase damping had only a minimal impact on the
visibility of the interference fringes. Taken together with the low-loss
nature of the waveguides, this demonstrates the ability for plasmonic
waveguides to generate and maintain coherence well, highlighting their
potential use for practical quantum phase sensing. Other types of plasmonic
waveguides have also been used to generate NOON states with $N=2$, including
planar Vest  _et al._ (2018) and stripe Dieleman  _et al._ (2017) waveguides.

By considering larger photon number NOON states ($N>2$) it may then be
possible to demonstrate a Heisenberg scaling in the precision and obtain close
to a $\sqrt{N}$ improvement compared to the classical case. Several photonics
experiments have already shown how to generate NOON states with $N=3$ photons
Mitchell  _et al._ (2004), $N=4$ photons Walther  _et al._ (2004); Nagata  _et
al._ (2007) and $N=5$ photons Afek  _et al._ (2010); Israel  _et al._ (2012).
Most recently, the generation of NOON-like states up to $N=8$ photons has been
demonstrated Thekkadath  _et al._ (2020). Many theoretical schemes have also
been proposed for going beyond these experiments to generate NOON states with
a higher photon number Kok  _et al._ (2002); Lee  _et al._ (2002); Dowling
(2008); Afek  _et al._ (2010) and other related states Huver  _et al._ (2008);
Lee  _et al._ (2009); Dorner  _et al._ (2009). In addition to the source of
quantum states, e.g., NOON states, and the plasmonic system being used, e.g.,
plasmonic nanowire, the experimental resources required include single-photon
and/or photon number resolving detectors, a coincidence counting interface and
quantum statistical measurement acquisition software Chen  _et al._ (2018).

####  IV.2.2 Phase sensing with continuous variable states

Despite the substantial developments in classical interferometric plasmonic
sensing and the progress in interferometric quantum plasmonic sensing with
discrete variable quantum states described above, very little progress has
been made in the development of squeezed interferometric plasmonic sensors.
One reason for this is the increased loss in interferometric sensors.
Intensity-based plasmonic sensors are most sensitive at the inflection point
of the plasmon absorption spectrum, so the detrimental effect of the plasmon
absorption can be mitigated somewhat. Phase-based plasmonic sensors are most
sensitive at the maximum of the plasmon absorption spectrum Kabashin  _et al._
(2009a), so as described in section II.3, little squeezing would remain in a
squeezed interferometric plasmonic sensor after accounting for plasmonic
absorption. Nevertheless, some progress has been made in the characterization
of plasmons exhibiting squeezing in the phase-sum quadrature, notably in a
report that characterized the two-mode squeezing in intensity-difference and
phase-sum quadratures for LSPs supported in triangular nanoapertures
Holtfrerich  _et al._ (2016). Further, substantial progress has been made in
the development of squeezed interferometric sensors for other applications
Lawrie  _et al._ (2019). Moreover, as described in section III.4.5, nonlinear
interferometers offer favorable scaling with loss compared with conventional
squeezed light readouts, and could plausibly be integrated with plasmonic
sensors in order to obtain a quantum advantage. Truncated nonlinear
interferometers in particular offer favorable loss scaling and the ability to
use a high power local oscillator in order to improve shot-noise limited
precision without concern for photochemical or photothermal effects Pooser
_et al._ (2020); Anderson  _et al._ (2017a); Gupta  _et al._ (2018). Many
opportunities still exist to develop loss-resilient squeezed interferometric
plasmonic sensors based on lower loss plasmonic sensor designs and new
approaches to SU(2) and SU(1,1) interferometry.

###  IV.3 Quantum plasmonic sensing based on emitter-plasmon coupling

Outside the field of plasmonic sensing, much attention has been paid to the
plasmonic control of emitters Hümmer  _et al._ (2013); Kristensen and Hughes
(2014); Lodahl  _et al._ (2015); Pelton (2015); Lawrie  _et al._ (2012);
Gonçalves  _et al._ (2020b), including semiconductor quantum dots Buckley  _et
al._ (2012) and color centers Aharonovich  _et al._ (2011) in the weak and
strong coupling regimes Tame  _et al._ (2013); Hümmer  _et al._ (2013). It is
increasingly evident that these effects can be leveraged for alternative
approaches to quantum plasmonic sensing, building on the PEF and SERS sensors
described in section II.4.4. The optical field at metallic nanostructures can
be significantly enhanced via the excitation of LSPs, enabling strong coupling
with matter in the vicinity of metallic structures Tame  _et al._ (2013). The
effects that originate from strong coupling are inherently nonclassical Hümmer
_et al._ (2013), and therefore the quantitative analysis and understanding of
the various phenomena measured in experiments requires a full-quantum
mechanical description, whereby the plasmonic nanostructures supporting LSP
excitations can be treated as a lossy and small-mode-volume resonator in
cavity quantum electrodynamics Hümmer  _et al._ (2013); Törmä and Barnes
(2014). A result of this is that strong coupling is very sensitive to various
structural and material parameters of an analyte being placed in the proximity
of a metallic structure, which has led to several types of quantum plasmonic
sensors.

In one of the first examples of a quantum sensor based on emitter-plasmon
coupling, a coupled system of a quantum dot and metallic nanorod was studied
as a means to sense a small change in the local refractive index around the
system in the near-infrared regime Hatef  _et al._ (2013), as shown in Figure
37(a). The quantum phase-dependent changes in the coherent exciton-plasmon
coupling provide the capability of detecting and distinguishing adsorption or
detachment of target molecules. A similar structure has also been considered
for optical detection and recognition of single biological molecules Sadeghi
_et al._ (2013a). Adsorption of a specific molecule to the nanorod results in
the ultrafast upheaval of coherent dynamics of the system, that turns off the
blockage of energy transfer between the quantum dot and the nanorod. The
emission of the system is thus strongly modified depending on the adsorption
event. Measuring the emission pattern from a quantum dot and metallic
nanoshell system would also enable the remote detection of the spatial
coordinates and movement of biological molecules or nanostructures Sadeghi
_et al._ (2013b), as shown in Figure 37(b). Refractive index sensing is also
possible by analysing the Rabi-splitting spectrum of the exciton-plasmon
strong coupling in the gap between a plasmonic nanorod and plasmonic nanowire
Qian  _et al._ (2020). In another study, a full quantum mechanical theory was
developed to model how a small amount of absorbing Trinitrotoluene (TNT)
molecules influences the spectrum of a graphene spaser based on a graphene
flake with quantum dot emitters Nechepurenko  _et al._ (2018). All these
studies of the emission spectrum from an emitter-plasmon quantum dynamical
system for sensing purposes open up a promising direction for future work,
although a careful analysis is required in order to determine whether the
quantum effects being exploited provide a quantum advantage in terms of sub-
shot-noise sensing, an improvement in the sensitivity, or both.

Figure 37:  Quantum plasmonic sensors based on emitter-plasmon coupling. (a) A
single quantum nanosensor in a to-the-end configuration, where a gold nanorod
and quantum dot are functionalized. Reproduced with permission from ref Hatef
_et al._ (2013). Copyright 2013 IOP Publishing. (b) Illustration of quantum
detection and ranging. The antenna includes a quantum dot and a metallic
nanoshell. $F(r,t)$ and $G(r^{\prime},t^{\prime})$ refer to two different
time-dependent emission intensity patterns of an optically active nanoscale
system at two different locations. Reproduced with permission from ref Sadeghi
_et al._ (2013b). Copyright 2013 AIP Publishing. (c) An illustration of a
strong-coupling immunoassay setup. A gold hemisphere nanodimer cavity captures
an immunoassay complex in the proximity of the plasmonic hotspot. Reproduced
with permission from ref Kongsuwan  _et al._ (2019). Copyright 2019 American
Chemical Society. (d) A nanofiber with dark-field heterodyne illumination.
Nanoparticles in a droplet of ultrapure water are detected when entering the
probe beam waist next to the nanofiber. Reproduced from ref Mauranyapin  _et
al._ (2017). Copyright 2017 Springer Nature.

A related recent theory proposal has suggested the use of quantum dots as
quantum labels bonded to individual antibody-antigen-antibody complexes being
placed inside or close to a nanoplasmonic dimer Kongsuwan  _et al._ (2019), as
shown in Figure 37(c). As the surface density of the analyte-emitter complexes
changes, the extinction cross section spectrum is modified, which not only
causes spectrum shifting, but also brings multiple resonance peaks due to the
strong coupling between the emitters and LSP at the nanoplasmonic dimer.
Through a statistical study of multiple analyte-emitter complexes for the
theoretical simulation of realistic conditions, the proposed splitting-type
sensing approach using quantum emitter labels has been shown to achieve a
nearly 15-fold sensitivity enhancement in comparison with conventional
shifting-type label-free plasmonic sensors. Such a study demonstrates the
potential of quantum plasmonic sensors to detect a single analyte, which is
usually challenging since the size of an analyte (typically $<100$ nm) is far
smaller than the optical wavelength of the light exciting the system.

Another theoretical work focusing on the strong coupling regime has considered
a three-level quantum system, e.g., atom or quantum dot, that by interacting
with a plasmonic cavity mode supports a so-called ‘embedded superstate’
Nefedkin  _et al._ (2020). The resulting system relies on quantum interference
between multiple atomic transition pathways and strong coupling, and exhibits
unboundedly narrow emission lines that enable a significant reduction in noise
in the scattering spectra. The result thereby provides a means to improve the
sensitivity significantly.

On the experimental front, metallic nanorods have been used in a recent study,
where an optical nanofiber is immersed in a droplet of water containing
nanoparticles, such as silica nanospheres or gold nanorods, and biomolecules,
as shown in Figure 37(d). Heterodyne interferometry was used together with a
dark-field illumination approach Mauranyapin  _et al._ (2017). The experiment
demonstrated a quantum-noise limited precision of evanescent single-molecule
biosensing, allowing a reduction of four orders of magnitude in the optical
intensity that is required to maintain state-of-the-art sensitivity. In a
related theoretical study, the fraction of the total spontaneous emission
energy from an emitter coupled to SPPs, called the $\beta$-factor Chen  _et
al._ (2010b), has been investigated for two nitrogen vacancy centers (NVCs) in
diamond placed in a plasmonic waveguide Jahromi (2018). It was shown that a
maximally entangled state of NVCs and a product NVC state provide the optimal
estimation of the $\beta$-factor at initial times and at long times,
respectively.

Finally, the use of color center emitters for quantum sensing of magnetic
Balasubramanian  _et al._ (2008); Schirhagl  _et al._ (2014) and electric
Dolde  _et al._ (2011) fields has experienced significant attention in the
past few years. The use of a plasmonic system as a mediator of the sensing
signal is an interesting development. A recent experiment has demonstrated the
use of a plasmonic groove waveguide interacting with multiple NVCs, which are
positioned at the end of the groove waveguide milled in a thick gold film
Shalaginov  _et al._ (2020). The gold film carries the microwave control
signal for the NVCs, while the groove waveguide acts as a fluorescence
collector. The fluorescence signal enables the readout of the NV spin
population (or specifically a spin sub-level population) as a function of an
applied external magnetic field. This is known as optically detected magnetic
resonance (ODMR). It is particularly appealing due to its simplicity and room-
temperature operation, as well as scope for also measuring strain or
temperature fields.

A related theoretical work has proposed the use of a plasmonic metasurface as
a means to achieve an infrared absorption-based readout of the magnetic field
resonance for an ensemble of NVCs Kim  _et al._ (2020). This is important as
standard fluorescence-based ODMR techniques are limited by low-photon
collection efficiency and modulation contrast. The work exploits the
enhancement of an infrared probe field via plasmonic excitation, which
interacts with an NVC infrared singlet state transition, and whose absorption
then carries information about spin state populations that change as a
magnetic field is applied. The result is a new kind of microscopic ODMR sensor
with infrared readout providing enhanced sensitivity.

##  V Perspective and outlook

In this review we covered the background and latest developments in the
emerging field of quantum plasmonic sensing. This is a research field that
sits at the interface between plasmonic sensing and quantum metrology – the
former provides researchers working in the field with decades of extensive
knowledge in classical optical sensing across a wide array of plasmonic
systems, their commercial development and applications, while the latter
provides new concepts and methods intensively developed in recent years that
enable the performance of the sensors to be improved by exploiting quantum
mechanical features in various quantum systems.

The review started with a discussion of conventional plasmonic sensors and
their basic working principles. The excitation of SPPs was shown to enable
enhanced sensing compared to standard optical sensors due to their high
electromagnetic field confinement. Two main types of plasmonic sensing were
discussed – intensity and phase sensing – with the benefits of each described
as depending on the specific application. This was followed by a discussion of
estimation theory and the limits of sensing in plasmonic systems using
classical light, i.e., the SNL. We then showed how the use of quantum states
of light and quantum measurements enables sensing beyond this SNL, where we
elaborated on single- and two-mode sensing schemes while reviewing recent
works on multimode or multiple parameter sensing. Recent works combining these
quantum sensing techniques with plasmonic sensing were then reviewed. We
covered the basic theory behind the work and highlighted its motivation in
relation to applications, including biosensing, monitoring chemical reactions
and ultrasound sensing. It was shown that despite the presence of loss in
plasmonic systems, one can use techniques from quantum sensing to obtain
improvements in sensing performance.

The field of quantum plasmonic sensing has grown steadily over the last five
years, with researchers studying many ways in which to combine plasmonic and
quantum sensing. However, much work still needs to be done to address the
current challenges and bring quantum plasmonic sensors to the same level of
maturity as their classical counterparts, including successful
commercialization. In the short term (next five years), it is likely that
plasmonic sensors based on SPP sensing will provide the most direct and
accessible route for incorporating the quantum intensity and phase sensing
techniques discussed in section IV. SPP sensing using the Kretschmann
configuration is still the main approach used by researchers and in industry.
Advancements made in the experimental generation of quantum states of light,
such as NOON states and TMSD states, will help bring quantum SPP sensing to
maturity. New results from the field of quantum sensing for dealing with loss
Oh  _et al._ (2017); Nair (2018); Degen  _et al._ (2017); Braun  _et al._
(2018); Pirandola  _et al._ (2018), including error correction Arrad  _et al._
(2014); Kessler  _et al._ (2014); Cohen  _et al._ (2016); Layden  _et al._
(2019); Zhuang  _et al._ (2020); Shettell  _et al._ (2021) and novel resource
states Demkowicz-Dobrzański  _et al._ (2015); Wang  _et al._ (2019c); Shettell
and Markham (2020); Ouyang  _et al._ (2019), may lead to a widening in the
range of plasmonic systems that can be used for quantum sensing and their
related applications.

In the mid to long term (five to ten years) the next steps would be the
miniaturization and integration of the sensors. Research currently being done
in the field of classical plasmonic sensors Hoa  _et al._ (2007); Caucheteur
_et al._ (2015); Niu  _et al._ (2015); Li  _et al._ (2020a), including the use
of fibers, waveguides and nanoparticles, would then enable compact quantum
plasmonic sensors to be realized for deployment in commercial applications.
Furthermore, non-standard material systems, including metamaterials Cai and
Shalaev (2010); Soukoulis and Wegener (2011); Chen  _et al._ (2012); Meinzer
_et al._ (2014); Wang  _et al._ (2019a); Consales  _et al._ (2020b); Hassan
_et al._ (2020), graphene Grigorenko  _et al._ (2012); Rodrigo  _et al._
(2015); Heydari and Samiei (2020) and more exotic two-dimensional materials
Das  _et al._ (2015); Zeng  _et al._ (2018) may help reduce loss, further
improve the level of precision in applications and enable a range of
specificity to be achieved, such as in monitoring chemical and biological
reactions Homola (2008), food safety Mello and Kubota (2002), pathogen
detection Leonard  _et al._ (2003); Pejcic  _et al._ (2006), and environmental
monitoring Andreescu and Sadik (2004). Hybrid systems that exploit electro-
optic Ozbay (2006); Sorger  _et al._ (2012); Fang and Sun (2015), nonlinear
Kauranen and Zayats (2012), nonlocal Raza  _et al._ (2015), quantum size
effects Halperin (1986); Fitzgerald  _et al._ (2016) and electron tunelling
Zhu  _et al._ (2016) could also offer additional quantum functionality.
Another interesting direction is the incorporation of quantum emitters into
plasmonic systems, such as quantum dots Lodahl  _et al._ (2015); Pelton (2015)
and color centers Aharonovich  _et al._ (2011), which may bring further
applications, such as magnetic Balasubramanian  _et al._ (2008); Schirhagl
_et al._ (2014) and electric field sensing Dolde  _et al._ (2011), and
molecular spectroscopy Zhan  _et al._ (2018). There is also the natural
connection between sensing and imaging Rotenberg and Kuipers (2014), and it
remains to be seen how the techniques developed for quantum plasmonic sensing
can be translated to plasmonic imaging Zhang and Liu (2008); Kawata  _et al._
(2009); Willets  _et al._ (2017) in order to improve aspects such as image
resolution, feature extraction and pattern recognition for applications in the
life sciences and medicine.

The field of quantum plasmonic sensing is likely to develop into a rich
subfield of optical science and it is an exciting time to enter this emerging
research field. Multidisciplinary collaboration from researchers working in
plasmonics, quantum information science, material physics, chemistry, biology
and medicine will advance this field significantly, bringing with them new
opportunities for sensing in science and industry. For these prospective
studies, this review will provide a helpful guidance and inspire novel avenues
of research.

## Appendix

##  Appendix A Multiparameter QFIM

The QFI matrix, $\boldsymbol{H}$, in eq 38 is defined by

$\displaystyle[\boldsymbol{H}]_{jk}=\text{Tr}[\hat{\rho}_{\boldsymbol{x}}(\hat{\cal
L}_{j}\hat{\cal L}_{k}+\hat{\cal L}_{k}\hat{\cal L}_{j})/2],$ (A1)

with $\hat{\cal L}_{j}$ being a symmetric logarithmic derivative operator
associated with $j$th parameter $x_{j}$ Braunstein and Caves (1994). Here,
$\boldsymbol{F}^{-1}$ and $\boldsymbol{H}^{-1}$ are understood as the inverse
on their support if the matrices are singular, i.e., not invertible Ge  _et
al._ (2018).

For single parameter estimation, the optimal measurement setting reaching the
QCR bound given by eq 28 always exists and can be constructed with projectors
onto the eigenvectors of the SLD operator Paris (2009); Braunstein and Caves
(1994); Demkowicz-Dobrzański  _et al._ (2015). For multiparameter estimation,
on the other hand, the QCR bound has been known to be attained only if the SLD
operators commute, i.e., $[\hat{\cal L}_{j},\hat{\cal L}_{k}]=0$ for all
$j,k$, which is valid even when the generators do not commute, i.e.,
$[\hat{G}_{j}(x_{j}),\hat{G}_{k}(x_{k})]\neq 0$. This is the sufficient
condition for the saturability of the multiparameter QCR bound, in which case
the optimal set of POVMs can be constructed over the common eigenbasis of the
commuting SLD operators Baumgratz and Datta (2016). A weaker, but necessary
and sufficient condition for the saturability can be found for pure states
$\hat{\rho}_{\boldsymbol{x}}=\left|{\Psi(\boldsymbol{x})}\right\rangle\left\langle{\Psi(\boldsymbol{x})}\right|$,
written as Matsumoto (2002); Szczykulska  _et al._ (2016)

$\displaystyle\left\langle{\Psi(\boldsymbol{x})}\right|(\hat{\cal
L}_{j}\hat{\cal L}_{k}-\hat{\cal L}_{k}\hat{\cal
L}_{j})\left|{\Psi(\boldsymbol{x})}\right\rangle=0,$ (A2)

for all $j,k$. The condition of eq A2 implies the commutation relation among
the SLD operators only on average with respect to the state
$\left|{\Psi(\boldsymbol{x})}\right\rangle$. For commuting generators
$\hat{G}_{j}(x_{j})$ associated with the evolution of a pure probe state,
i.e., $[\hat{G}_{j}(x_{j}),\hat{G}_{k}(x_{k})]=0$ for all $j,k$, eq A2 is
satisfied, so that the multiparameter QCR bound can be saturated Matsumoto
(2002).

##  Appendix B Calculation of multiparameter QFIM

Generally for $M$ parameters
$\boldsymbol{\phi}=\\{\phi_{1},\cdots,\phi_{M}\\}$, the relation is given by
Liu  _et al._ (2020)

$\displaystyle\sum_{j,k}H_{jk}\text{d}\phi_{j}\text{d}\phi_{k}=4{\cal
D}_{\text{B}}^{2}(\hat{\rho}_{\boldsymbol{\phi}},\hat{\rho}_{\boldsymbol{\phi}+\text{d}\boldsymbol{\phi}}),$
(B1)

where the Bures distance can be written in terms of the quantum fidelity as

$\displaystyle{\cal
D}_{\text{B}}^{2}(\hat{\rho}_{\boldsymbol{\phi}},\hat{\rho}_{\boldsymbol{\phi}+\text{d}\boldsymbol{\phi}})=2\left[1-\sqrt{{\cal
F}(\hat{\rho}_{\boldsymbol{\phi}},\hat{\rho}_{\boldsymbol{\phi}+\text{d}\boldsymbol{\phi}})}\right]$
(B2)

and the quantum fidelity ${\cal F}$ is defined as Uhlmann (1976); Jozsa (1994)

$\displaystyle{\cal
F}(\hat{\rho}_{\boldsymbol{\phi}},\hat{\rho}_{\boldsymbol{\phi}+\text{d}\boldsymbol{\phi}})=\left(\text{Tr}\sqrt{\sqrt{\hat{\rho}_{\boldsymbol{\phi}}}\hat{\rho}_{\boldsymbol{\phi}+\text{d}\boldsymbol{\phi}}\sqrt{\hat{\rho}_{\boldsymbol{\phi}}}}\right)^{2}.$
(B3)

Thus, the calculation of the quantum fidelity leads to the calculation of QFIM
for any pair $(\phi_{j},\phi_{k})$.

In particular, the input state
$|\Psi\rangle_{\text{in}}=|\alpha\rangle_{\text{a}}|\xi\rangle_{\text{a}}$
considered in the MZI of Figure 10(b) is called a Gaussian state as its
characteristic Wigner function can be described by a Gaussian distribution.
Because the BS operation $\hat{B}$, the phase operation $\hat{U}(\phi)$, and
linear photonic loss channel, including inefficient detectors, are a Gaussian
map, the output state $\hat{\rho}_{\text{out}}$ is kept in the form of a
Gaussian state. This enables us to describe the output state
$\hat{\rho}_{\text{out}}$ in terms of the first-order moment vector
$\boldsymbol{d}$, and the second-order moment matrix $\boldsymbol{V}$
Braunstein and van Loock (2005); Weedbrook  _et al._ (2012). The displacement
vector $\boldsymbol{d}$ is defined as
$d_{j}=\text{Tr}[\hat{\rho}_{\boldsymbol{\phi}}\hat{Q}_{j}]$, whereas the
covariance matrix $\boldsymbol{V}$ is defined by
$V_{jk}=\text{Tr}[\hat{\rho}_{\boldsymbol{\phi}}\\{\hat{Q}_{j}-d_{j},\hat{Q}_{k}-d_{k}\\}/2]$,
where $\\{\hat{A},\hat{B}\\}\equiv\hat{A}\hat{B}+\hat{B}\hat{A}$. Here,
$\hat{\boldsymbol{Q}}$ denotes a quadrature operator vector for a two-mode
continuous variable quantum system and is written as
$\hat{\boldsymbol{Q}}=(\hat{x}_{1},\hat{p}_{1},\hat{x}_{2},\hat{p}_{2})^{\text{T}}$,
satisfying the canonical commutation relation,
$[\hat{Q}_{j},\hat{Q}_{k}]=i\Omega_{jk}$, where

$\displaystyle\boldsymbol{\Omega}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\times\mathbb{I}_{2}$ (B4)

and $\mathbb{I}_{n}$ is the $n\times n$ identity matrix.

Due to the input state remaining a Gaussian state throughout the evolution,
the analytical form of the quantum fidelity can be found for the displacement
vectors and covariance matrices Marian and Marian (2012); Banchi  _et al._
(2015), so one can readily calculate the quantum fidelity. For two Gaussian
states $\hat{\rho}_{j=1,2}$ with $\boldsymbol{d}_{j=1,2}$ and
$\boldsymbol{V}_{j=1,2}$, the quantum fidelity can be calculated via Marian
and Marian (2012); Banchi  _et al._ (2015)

$\displaystyle{\cal
F}(\hat{\rho}_{1},\hat{\rho}_{2})=\frac{\exp\left[-\frac{1}{2}\delta\boldsymbol{d}^{\text{T}}(\boldsymbol{V}_{1}+\boldsymbol{V}_{2})^{-1}\delta\boldsymbol{d}\right]}{(\sqrt{\Gamma}+\sqrt{\Lambda})-\sqrt{(\sqrt{\Gamma}+\sqrt{\Lambda})^{2}-\Delta}},$
(B5)

where $\Delta=\text{det}(\boldsymbol{V}_{1}+\boldsymbol{V}_{2})$,
$\Gamma=16~{}\text{det}(\boldsymbol{\Omega}\boldsymbol{V}_{1}\boldsymbol{\Omega}\boldsymbol{V}_{2}-\mathbb{I}_{4}/4)$,
$\Lambda=16~{}\text{det}(\boldsymbol{V}_{1}+i\boldsymbol{\Omega}/2)\text{det}(\boldsymbol{V}_{2}+i\boldsymbol{\Omega}/2)$,
and $\delta\boldsymbol{d}=\boldsymbol{d}_{2}-\boldsymbol{d}_{1}$.

Using eqs B1, B2 and B5, one can calculate the QFIM $\boldsymbol{H}$ generally
for multiple parameters $\boldsymbol{\phi}=\\{\phi_{1},\cdots,\phi_{M}\\}$. In
the case when only a single parameter $\phi$ encoded by $\hat{U}(\phi)$ is of
interest, the QFI is calculated by $H=8\left[1-\sqrt{{\cal
F}(\hat{\rho}_{\phi},\hat{\rho}_{\phi+\text{d}\phi})}\right]/\text{d}\phi^{2}$.

##  Appendix C CR bound in lossy MZIs

Homodyne detection is particularly useful, as the measurement bases are
represented by Gaussian states, i.e., the POVM element is written as
$\hat{\Pi}_{x}=|x_{\phi_{\text{HD}}}\rangle\langle x_{\phi_{\text{HD}}}|$,
where $|x_{\phi_{\text{HD}}}\rangle$ denotes a quadrature variable state.
Projection of the reduced output state
$\hat{\rho}_{\text{out,a}}=\text{Tr}_{\text{b}}[\hat{\rho}_{\text{out}}]$ into
the measurement basis $|x_{\phi_{\text{HD}}}\rangle$ leads to the probability
density function of the quadrature variable outcomes $\\{x\\}$, which is
written as

$\displaystyle p(x|\phi)=\langle
x_{\phi_{\text{HD}}}|\hat{\rho}_{\text{out,a}}|x_{\phi_{\text{HD}}}\rangle.$
(C1)

It can be shown that the probability density function of eq C1 follows a
Gaussian distribution for the measurement outcomes, leading to homodyne
detection often being called a Gaussian measurement Weedbrook  _et al._
(2012); Giedke and Cirac (2002). Hence, the probability density function of eq
C1 can be described by the first-order moment vector $\tilde{\boldsymbol{d}}$
and the second-order moment covariance matrix $\tilde{\boldsymbol{V}}$, for
which the FI can be calculated via Kay (1993); Porat and Friedlander (1986)

$\displaystyle F(\phi)$
$\displaystyle=\frac{\partial\tilde{\boldsymbol{d}}^{\text{T}}}{\partial\phi}\tilde{\boldsymbol{V}}^{-1}\frac{\partial\tilde{\boldsymbol{d}}}{\partial\phi}+\frac{1}{2}\text{Tr}\left[\tilde{\boldsymbol{V}}^{-1}\frac{\partial\tilde{\boldsymbol{V}}}{\partial\phi}\tilde{\boldsymbol{V}}^{-1}\frac{\partial\tilde{\boldsymbol{V}}}{\partial\phi}\right].$
(C2)

For an optimally chosen homodyne angle $\phi_{\text{HD}}$ with respect to the
phase $\phi$ being estimated, one can show that the CR bound reads Gard  _et
al._ (2017)

$\displaystyle\Delta\phi_{\text{CR}}=\frac{1}{\sqrt{\nu}}\sqrt{\frac{1}{|\alpha|^{2}e^{2r}}+\frac{1-\eta}{\eta|\alpha|^{2}}}.$
(C3)

## Acknowledgments

At KIT, this work was partially supported by the Volkswagen Foundation and by
the VIRTMAT project. At KIAS, CL was supported by a KIAS Individual Grant
(QP081101) via the Quantum Universe Center. At ORNL, BL was supported by the
U. S. Department of Energy, Office of Science, Basic Energy Sciences,
Materials Sciences and Engineering Division. At HU, KGL was supported by the
Basic Science Research Program through the National Research Foundation (NRF)
of Korea and funded by the Ministry of Science and ICT (Grants No.
2020R1A2C1010014) and Institute of Information & Communications Technology
Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.
2019-0-00296). At SU, MST was supported by the South African National Research
Foundation, the Council for Scientific and Industrial Research National Laser
Centre and the South African Research Chair Initiative of the Department of
Science and Innovation and National Research Foundation.

## Glossary

  * •

Bias - The difference between an estimate $x_{\text{est}}$ and the true value
$x$ of the parameter being estimated on average, i.e.,
$\bar{x}_{\text{est}}-x$. An unbiased estimator has a bias of zero.

  * •

Cramér-Rao bound - The lower bound of the Cramér-Rao inequality.

  * •

Cramér-Rao inequality - The inequality that the standard deviation of an
unbiased estimator of a parameter should obey for a given measurement setting.
The inequality reads $\Delta x\geq 1/\sqrt{\nu F(x)}$, where $\Delta x$ is the
standard deviation, $\nu$ is the number of measurements in a sample and $F(x)$
is the Fisher information. The equality holds when the optimal unbiased
estimator is chosen. It can also be asymptotically saturated in the limit of
large $\nu$ when maximum-likelihood-method is employed as an estimator
although it is not optimal.

  * •

Estimate - The value $x_{\text{est}}$ obtained from the estimator
$\hat{x}(y_{1},...y_{\nu})$ for a given sample (single set of data observed).
It is also known as a point estimate.

  * •

Estimation accuracy - The interpretation of the bias $\bar{x}_{\text{est}}-x$.
The accuracy becomes better as the bias decreases.

  * •

Estimation precision - The interpretation of the standard deviation, $\Delta
x$, of the estimate $x_{\text{est}}$, each taken from a sample made up of a
finite set of $\nu$ measurements of the parameter $x$. The smaller $\Delta x$
is, the better the precision is. It is commonly known as the fluctuation or
uncertainty in estimation, or can sometimes be interpreted as the resolution.
It can also be understood as the reproducibility.

  * •

Estimation uncertainty - The quantity directly given by the standard deviation
$\Delta x$. The smaller $\Delta x$ is, the smaller the uncertainty is.

  * •

Estimator - A rule, $\hat{x}$, that yields an estimate $x_{\text{est}}$ of an
unknown parameter $x$ from an underlying probability distribution based on the
observed data $\boldsymbol{y}$ from a sample of $\nu$ measurements,
$\boldsymbol{y}\equiv(y_{1},...y_{\nu})$. For example, $\hat{x}$ can be
expressed by some function $f$, i.e., $\hat{x}=f(\boldsymbol{y})$. The most
commonly used estimator is the sample mean.

  * •

Extraordinary optical transmission - the transmission of light through a
structured medium, where the transmission is due to the excitation of surface
plasmons. A similar structured medium that does not support surface plasmons
has a reduced transmission.

  * •

Fisher information - A quantification of the amount of information about a
parameter $x$ contained in the measurement results for a given measurement
setting. We express it by $F(x)$.

  * •

Heisenberg limit - The ultimate quantum limit achievable using the optimal
quantum state and measurement, or the precision scaled with $N^{-1}$ that is
often the case of interferometeric sensing. A scaling of $N^{-1}$ is called
Heisenberg scaling, where $N$ is the average number of particles in the
resource.

  * •

Limit of detection - An overall figure of merit for sensing quality that takes
into account both the sensitivity ${\cal S}_{y}$ and the minimum detectable
range $\Delta y_{\text{min}}$, or equivalently the value of the noise level.
This is often interpreted as a resolution.

  * •

Mean-squared-error - The average square distance between the estimator
$\hat{x}(y_{1},...y_{\nu})$ and true value of $x$ as the data
$y_{1},...y_{\nu}$ varies according to the underlying probability
distribution. It is a measure of average closeness of an estimator $\hat{x}$
to the true value $x$, and depends on both the standard deviation $\Delta x$
and the bias $\bar{x}_{\text{est}}-x$. If the estimator is unbiased, then the
mean-square error is equivalent to $\Delta x^{2}$, allowing to refer to the
standard deviation $\Delta x$ as the ‘estimation error’, ‘estimation
precision’, or simply ‘precision’.

  * •

Multi-parameter estimation - This is categorized into three kinds:

    1. 1.

Individual estimation - Multiple parameters $(x_{1},x_{2},\cdots,x_{n})$ are
estimated individually. This scheme is also called a ‘local strategy’.

    2. 2.

Simultaneous estimation - Multiple parameters $(x_{1},x_{2},\cdots,x_{n})$ are
estimated simultaneously. This scheme is also called a ‘global strategy’.

    3. 3.

Distributed estimation - A global parameter $X$, which is a function of
multiple parameters, i.e., $X=f(x_{1},x_{2},\cdots,x_{n})$, is estimated.

  * •

Noise reduction factor - In an intensity difference measurement, it is the
ratio of the variance of the photon number difference between the signal and
reference mode for a given quantum state to that of coherent states with
matching average photon number.

  * •

$N$-mode - A sensing scheme where $N$ spatial/temporal/angular/spectral modes
– the eigenvectors of the wave equation – are used. Here, $N$ can be “single”,
“two”, or “multi”.

  * •

Quantum Cramér-Rao bound - The lower bound of the quantum Cramér-Rao
inequality.

  * •

Quantum Cramér-Rao inequality - The same as Cramér-Rao inequality but with the
Fisher information replaced by quantum Fisher information, i.e., $\Delta x\geq
1/\sqrt{\nu H}$.

  * •

Quantum Fisher information - The maximized Fisher information over all
possible physical measurement settings $\\{\hat{\Pi}\\}$, i.e.,
$H=\max_{\\{\hat{\Pi}\\}}F(x)$.

  * •

Sensitivity - The derivative of a quantity $y$ being monitored, which is used
to obtain an estimation of some parameter $x$ that changes with $y$.
Mathematically, the sensitivity is $S_{y}=|{\rm d}y/{\rm d}x|$. By increasing
the sensitivity a sensor becomes more sensitive to changes in $x$.

  * •

Shot noise - Pure noise originating from the underlying statistical properties
of a resource to be analyzed or a measurement involving the random fluctuation
of electric signals. It has nothing to do with fundamental noise, which in an
ideal theory model can be assumed to be completely absent. The term originally
comes from electronics, where a current consists of a stream of discrete
charges, i.e., electrons. The charges are randomly distributed in space and
time, thereby following a Poisson distribution. The same random feature is
present in a coherent state of light that consists of discretized particles,
i.e., photons, that follow a Poisson distribution in photon number. This leads
to the shot noise when the coherent state’s intensity, i.e., photon number, is
measured.

  * •

Shot-noise limit - The best possible precision obtained using the classical
resource of a coherent state. It originates from statistical noise in the
resource that is mapped into noise in the quantity being measured, e.g. for an
intensity measurement the shot-noise limit comes from Poissonian noise that is
intrinsic to the coherent state’s photon number, or energy.

  * •

Single-to-noise ratio - The ratio of the measurement signal to the noise in
the measurement signal.

  * •

Standard quantum limit - The fundamental limit to precision achievable in
standard interferometers made of standard devices employing only classical
resources. The term is now generally used to denote the ultimate precision
limit achievable in arbitrary interferometric sensing when only classical
resources are used.

  * •

Unbiased estimator - The estimator has a bias of zero. When applied to many
samples from the underlying probability distribution it equals the true value
of the parameter being estimated on average. Equivalently, the sampling
distribution of the estimator fluctuates around the true value of the
parameter.

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