Enhanced signal-to-noise ratio in quantum plasmonic image sensing including loss and varying photon number

The signal-to-noise ratio is an important quantity in signal and image analysis that gives information about the quality of the signal and/or image of interest. When plasmonic biosensors are used to study how molecules interact in intermolecular binding reactions, the output signal and/or image must be of the highest quality to get the best value from our biosensors. Images of interest in this work are images of the binding region at the metal surface of the plasmonic biosensor. Improving the signal-to-noise ratio of these signals and/or images is a key area of research that can help scientists learn more about how different molecules interact with each other. Because these molecular entities can include a wide range of biomolecules, we can investigate different types of binding interactions. In this paper, we look at a theoretical two-mode image sensing framework that we use to model the signal-to-noise ratio in images generated by a plasmonic image-based biosensor. A Krestchmann configuration-based surface resonance sensor is used as a plasmonic biosensor. In the model, an example of how BSA and an antibody called IgG1 bind to the surface of a plasmonic biosensor are examined. Traditionally, classical states of light are used as probe states in the Krestchman configuration; in this paper, quantum states of light are considered alternative probe states. The effect of using quantum states of light, such as the Fock state, squeezed displaced states, and squeezed vacuum states, on the signal-to-noise ratio of images is investigated. This work also looks at the effect of losses in the sensing environment and changes in photon numbers in the input signal on the average signal-to-noise ratio of the output of the plasmonic biosensor. The novelty in the described work lies in the exploration of using a variety quantum states of light as probe states in a plasmonic image-based biosensor, specifically in the context of improving the signal-to-noise ratio of images captured from the binding region at the metal surface accounting for the impact of losses. It was found that some quantum states improve the signal-to-noise ratio of the plasmonic biosensor output image.


Introduction
As new viral strains cause more and more epidemics and pandemics around the world, there is a growing need for quick and accurate diagnostic tools [1].As a result of their portability, with their reproducibility, sensitivity, and specificity, surface plasmon resonance (SPR)-based biosensors have emerged as reliable diagnostic devices.Imaging tools have been widely and successfully used in disease diagnostics, so the development of such tools is crucial in the battle against pandemics and in drug development.A combination of plasmonic sensing and imaging leads to surface plasmon resonance imaging (SPRi), a powerful tool for diagnosing diseases and finding new drugs.It can be used to find diseases and analyze them [2].SPRi, also known as surface plasmon microscopy in the literature, was discovered in 1988 by Rothenhäuslar and Knoll [3].SPRi is an analytical tool that combines imaging of metal surfaces with surface plasmon resonance spectroscopy.It is label-free, has high throughput, the prism, the light couples with electrons in the metal, which results in an evanescent field that propagates on the metal surface.Excitation occurs through an evanescent field generated by total internal reflection on the rear side of the sensor's metallic surface [31].
The goal of this work is to model a sensorgram for a binding reaction of interest in the ideal case using the Krestchmann configuration and then to model the measurement noise for the binding reaction.This varies for different states of light used to probe the setup.Then, after this, the SNR will be calculated.Because the sensorgram and noise function are time-dependent functions, the SNR will also be time-dependent.The SNR will also vary depending on the state of the light used to probe the setup and will be distributed over time for each state.To compare the SNR values for the different states resulting from varying the number of photons and varying losses in the system, we took a central measure of the distribution of the SNR values over time.In this work, we find that the distribution of the SNR values over time is bimodal, so the centrality measure used to calculate the average SNR for each state is the mode value.With the mode value, we can establish the average performance of each state.The binding reaction of interest in this work is the binding reaction between the antigen BSA interacting with the antibody, rabbit anti-cow albumin IgG1 (anti-BSA), found in this article by Kausaite et al [32], in the Kausaite paper, the sensorgram is an angle-shift based sensorgram, but is transformed into a transmittance based sensorgram in the article by Mpofu et al [28].The tansmittance sensorgram is useful for introducing quantum states of light.
Here we will look at the theoretical image sensing framework shown in figure 2 and use it to show how the use of quantum light states as probe states enhances the resolution of the image obtained from an SPRi sensor over the use of classical states.We show that using quantum states, for example, the two-mode Fock (TMF) states and the two-mode squeeze-displaced (TMSD) states, enhances the signal-to-noise ratio in the obtained image.The image detection framework that we use here is a modification of the framework proposed by Nair and Yen [33].In the framework that we propose here, as in the framework proposed in [33], an image is considered to be a collection of pixels.Each image pixel corresponds to a specific binding region on the reaction surface of our biosensor.This means that each image pixel can be seen as a separate SPR experiment that generates its own sensorgram and can be studied separately.In an experimental setup, it may be ideal to capture a video of the signal.This is because a video can be thought of as a series of frames of the reaction over time, and this would help us measure dynamic properties such as the reaction kinetics.In this work, we will look at the time-dependent SNR of each individual pixel when we use different quantum states as probes for the setup and show that the individual image pixels will have enhanced SNR, and by extension, the entire image will have enhanced SNR. Figure 2 displays the general image sensing model's framework.
SPR-based biosensors are highly sensitive (and can easily be modified to increase their sensitivity) and have high throughput; however, their precision is still bound by the shot-noise limit.When it comes to imaging, these precision bounds greatly affect the resolution of images.The need to go beyond the limit of shot noise and improve the resolution of images generated by means based on surface plasmon resonance has led to an increasing interest in the development of quantum biosensors [25-30, 32, 34-37].In this paper, we theoretically Figure 1.Surface plasmon resonance imaging of a binding interaction on the metal surface of a Krestchmann-configuration-based plasmonic biosensor setup.In the image, we see an SPR-based biosensor with ligands (antibody IgG1) binding to receptors (BSA) bound to the metal surface (metal film).When the analyte is placed in the setup, it is introduced into a ligand.To establish a baseline, baseline images can be taken before the analyte is introduced, and as the analyte is introduced, the binding reaction goes on.The images track the binding at different regions of the binding surface.Intensity increases as binding interactions occur in a particular region.By taking a continuous series of pictures, we can track the intensity change in different regions.
study the SNR in signals generated in a binding reaction between a BSA and Immunoglobin G1 (IgG1) which is an antibody.We will model the same binding reaction using different quantum states, including the TMF state, two-mode squeezed vacuum states (TMSV), two-mode squeezed displaced states (TMSD), product squeezed state (PS), and two-mode coherent (TMC) states, as possible states to enhance the SNR of SPR-based imaging.
In the next section, we will discuss the different quantum states that are used to probe our two-mode Krestchmann setup.The different quantum states of interest are the TMF state, TMSV state, TMSD state, PS state, and NOON state.The classical limit of the two-mode coherent state will be used to compare the quantum states.The two-mode model as shown in figure 3 will be considered in our analysis and will be used to calculate the time-based SNR of the images extracted from the setup.The SNRs for the different states will be compared.In the later sections, we will look at the analysis of the SNR and compare performance for the different quantum states.Framework for sensing an unknown image.This is a two-mode imaging framework in which there is a signal mode that goes through the sensing mechanism before going to the detector and a reference mode that goes directly to the detector.Both modes undergo loss η where η reference is loss in the reference mode and η signal is loss in the signal mode.The dark regions of the image represent the baseline before any binding activity occurs.As binding activity begins, there is an increase in intensity; therefore, the regions where binding activity occurs will be brighter.This figure was redrawn from the figure in [33].
Figure 3. Two-mode plasmonic sensing setup with Krestchmann configuration.In this setup, the signal mode is sent toward the Krestchmann configuration setup, used to probe the setup, and then reflected towards detector A, where the signal is imaged.The reference signal is sent directly to detector B, where it is imaged.An intensity difference between the images for the corresponding pixel positions is taken to give a final output image that is clear of background noise.Different input states of light will be used in the models, and an analysis of the SNR will be conducted.This figure was redrawn from the figure in [38].

SNR and quantum states
In this section, we will look at the different quantum states used to calculate the time-dependent SNR of each pixel (SNR p ) in the sensing images of the two-mode biosensing setup shown in figure 3

ˆ( ) ˆˆˆˆ( ) † †
Here a ˆ † and a ˆare the creation and annihilation operators, respectively, and when coupled together they constitute the number operator, which counts the photon numbers in each mode.The two modes s and r are the signal and reference modes, respectively.The SNR p is thus given by equation 2, ( ˆˆ) .ΔM p , is the uncertainty (precision) of the measurement.M p ˆis a measurement of the intensity difference between the corresponding pixels of the image detected in the signal mode versus those generated in the reference mode (see figure 3).From figure 4 it is clear that by taking a series of images over time of the binding interactions at the binding surface, we can track the binding activity across the entire binding surface.Tracking each pixel in the series of images generated in the SPRi setup allows us to generate a sensorgram in each pixel, as shown in figure 4 where the intensity or transmission of the signal at time t is measured for each pixel as T p (t).The measurement of intensity difference M p ˆhas a dependence on transmittance, T p , as seen in equation (3) where N is the number of photons in the signal mode, s, or the reference mode, r, and η is the loss to the environment in the different modes.áD ñ M p ˆcan be interpreted as the degree to which the anticipated value of a measurement, represented by á ñ M p ˆ, is altered when there is a change in the transmittance for each pixel T p .for a single shot measurement á ñ M p ˆ,the precision in the estimation of T p is given by equation 4 we study how the intensity of the p th pixel changes as binding reactions of the third ligand occur.In the image taken at time, t 1 we see that the p th pixel (which corresponds to the third ligand) is dark, indicating that no binding has occurred, at t 2 the color starts to change slightly (becomes brighter), indicating some interaction between the ligand and the analyte molecule, and at t n depending on how much the ligand and the analyte molecule are bound, the color is further enhanced.We see from the arrow below that we track the intensity of the p-th pixel over time.For the n images, we will get a sensorgram that describes the binding kinetics.This can be done for any of the pixels.It is clear that the better the SNR of our images, the better the quality of any analysis we may be interested in doing.
For an experiment in which we repeat the measurements ν times, this leads to an estimate T p ¯using the mean as an estimator.The precision of the estimation of T p ¯becomes n DT p , which simply expresses that the estimation of T p ¯becomes more precise as the sample size increases.The values for ΔM Fock , ΔM TMSV , ΔM Product−Squeezed , ΔM Coherent and ΔM NOON calculated from the paper by Lee et al [26] are shown in [38].
From figure 4 it is clear that the change in transmittance can be tracked over time, T p (t).The change in refractive index as ligands and receptors bind to complexes on the gold surface causes a change in T p value over time [39].In an ideal case, when the measurement noise ≈0, on the sensorgram, T p can be written as shown in equation (5).
where T a and T c are constants [28].The value of T τ is given by = - where k a is the association constant measured in M −1 s −1 (per molarity per second), k d is the dissociation constant measured in s −1 and [L 0 ] is the initial concentration of the ligand, [28].In this work the value for T a =0.31, T c is 0.29 and T τ is 0.28.The other parameters have values k s = 0.0105 s −1 , k d = 7.771 × 10 −3 s −1 and k a = 10.029 × 10 3 M −1 s −1 , and L 0 = 274 × 10 −9 M. Quantum states of light that have inter-mode correlations can be used in a two-mode sensing configuration to lower measurement noise below the SNL.This has been demonstrated in earlier research that examined several two-mode quantum states for plasmonic sensing of a static quantity [26,30].From previous work, we chose the best-performing quantum states for this study and went beyond them by examining their capacity to increase the SNR in images produced by the setup.
Here we will look at the different quantum states used in this setup.Beginning with describing the two-mode coherent (TMC) state, which we consider to be the classical limit state whose measurement noise is the SNL.Coherent light is light with a constrained range of frequencies and waves that are all in phase with one another.Incoherent light, such as multi-mode thermal states, is referred to as having multiple frequencies and out-ofphase waves.A laser light source is typically used in the Kretschmann configuration.Typically, a laser serves as the light source in the Kretschmann configuration setup, as previously mentioned.In the quantum approach, this source of conventional light is effectively modeled as a coherent state [31].The TMC which is expressed in equation ˆ| | , respectively [28].The values of |α| 2 and |β| 2 are chosen to correspond to the mean number of photons in the signal and reference modes in order to ensure that the various quantum states may be fairly compared to the TMC state [28].We set |α| 2 = |β| 2 = N, where N is the mean number of photons in either input mode, for states that have an equal amount of photons in either or both modes (that is the balanced case).
The two-mode Fock (TMF) state is a further quantum state that we take into consideration in this article, which is expressed as The average number of photons in the TMF state is N since there are N photons in each mode, hence á ñ = N N a ând á ñ = N N b ˆin modes a and b, respectively.A Fock state, also known as a number state, is a quantum state that is a part of a Fock space and has a specific number of particles (or quanta).N-photon Fock state generation has experimentally been studied using different mechanism which include artificial quantum emitters [40], atoms in cavities [41][42][43][44], linear optics [45], and super-conducting quantum circuits [46,47].Alternative theoretical schemes have also been proposed for the generation of the N-photon Fock state, including artificial emitters [48,49], linear optics [50,51], and atoms in cavities [52].State-of-the-art systems generate Fock states of low photon number (N∼ (7-15)) [44,46].
The third quantum state we will look at in this paper is the TMSV state, which is expressed as is a squeeze operation applied to the vacuum state.The squeezing parameter c = q re i s , where r represents the amount of squeezing and θ s is a phase.The mean photon number in the modes is given by á ˆˆ, with the value of N set by the squeezing parameter r.A two-mode squeezed vacuum state is a quantum state that can be represented as the tensor product of two single-mode squeezed vacuum states.In other words, it is a state in which the quantum fluctuations of two different modes of the electromagnetic field have been squeezed below the SNL.The TMSV state, which can be produced optically utilizing a spontaneous parametric down-conversion (SPDC) setup, has been used in numerous investigations [53][54][55].The number of photons in each of the two modes is often limited to low values of N for practical reasons, despite being more empirically accessible than the Fock state for a given mean photon number.The two-mode squeezed displaced (TMSD) state is the fourth quantum state that we take into consideration, which is expressed as The mean number of photons in each mode is ˆ| | , which are dependent on the squeezing parameter r and the initial value of the mean number of photons of the coherent state |α| 2 [28].We set |α| 2 and r so that á ñ = N N a ˆ, which then gives ˆ| | .The TMSD can be generated via a four-wave mixing process [56][57][58].In quantum sensing studies, intensities of up to several tens of μW (effectively very high N) have been attained at the expense of lowering the photon-number correlation between the two modes.This allows for the generation of the TMSD state [59,60].The value = r cosh 4.5 squeezing is used in this work [61] and setting |α| 2 of the coherent state appropriately in order to satisfy á ñ = N N a ˆ.It should be noted that all of the instances we investigate in this work are subject to the energy constraint of N photons in the signal mode.As a result, the applications of the various probe states may be fairly compared.
The fifth state we consider is the product squeezed state, which can be expressed as which is expressed as * ˆ( ) ˆˆ † is a squeeze operation applied to the vacuum state.This is the same for c S b ˆ( ) applied to mode b.The sixth state we consider is the NOON state, which is an entangled photon number state.A N00N state is written as follows: The N00N state is a quantum-mechanical concept that describes a unique type of entangled state involving multiple particles.In simpler terms, it represents a situation where N particles can simultaneously exist in two different modes, labeled as 'a' and 'b'.The particles are entangled, meaning their properties are interconnected and dependent on one another, even though they might be in different modes.In the N00N state, all N particles are either found collectively in mode 'a', with none in mode 'b', or they are all in mode 'b', with none in mode 'a'.This simultaneous existence in both modes is called a 'superposition'.It is a fundamental characteristic of quantum mechanics, which allows particles to occupy multiple states at once, until a measurement is made that collapses the superposition into a single, definite state.NOON states can be generated by mixing quantum and classical light [62] and probabilistically via post-selection using SPDC [63,64].The state of the art sytems have produced N ∼ 5 photons [63,64].

Sensing model and SNR analysis
The SNR for the signal generated by the binding reaction between BSA and an antibody IgG1 considering different quantum states is shown in figure 5. Here, the time-dependent signal function, T p (t), is given by the equation below.The measurement noise functions for each state are also time-dependent.Therefore, the SNR is also a time-dependent function.The imaging process to describe a binding signal would involve taking multiple snapshots at multiple time intervals, and the finer the time intervals, the finer the changes that can be tracked.It is interesting that, when working with SPR images, we can focus on a particular binding area for which we can formulate a binding curve, or we can focus on the entire surface area and formulate a single binding curve.SPR imaging gives us the ability to zoom in on a particular area or region of interest and study more specific binding regions.In the framework, an image is described as a group of unit pixels, each of which represents a point on an SPR sensorgram.The intensity of a pixel changes over time, and by plotting each pixel's intensity change over time, we can draw a sensorgram for it.The SNR of this sensorgram at different points in time corresponds to the SNR of the pixel.The SNR of the individual pixels of the signal can be used to estimate its quality.We look at how quantum states of light could be used to improve the SNR of images (pixels) from an SPRi sensor that changes over time.The sensor response, which is the sensorgram that would be measured in an ideal experiment (i.e., with no noise), is represented theoretically by equation (5).Next, we will consider different loss modes and see how the different losses affect the SNR in the measured output for each quantum state.The different loss modes are reflective of losses that can occur in actual experiments.

Mean SNR in the standard two-mode case
The standard two-mode sensing case refers to a case where the loss in both modes is the same.To establish a baseline so that we can see the impact of losses in our system, we also consider an ideal scenario where there are no losses to the environment.The general sensing model depicted in figure 3, where the loss in each mode is the same (the standard two-mode sensing case), is taken into consideration in this situation, which is the typical two-mode sensing case.Suppose the ideal situation where neither mode suffers a loss, i.e., η signal = η reference = 1 and consider a change in photon number.In figure 6(b) we see how the mean SNR changes with increasing photon number in an ideal case where there are no losses in either mode.Increasing the photon number improves the SNR of the image pixel output, which we get from our SPRi setup.We also consider a lossy scenario where there is some loss to the environment and the loss is equal in both modes discussed in the following subsection.This can offer a better representation of a real experimental setting.

Lossy standard two-mode case
Here we consider a lossy scenario where we have the same loss in both modes, i.e., η signal = η reference = 0.8 .In figure 6(a) we show the mean SNR changes with increasing photon number for a lossy two-mode setup.We can see that in general, when we compare ideal and lossy setups, the SNR numbers in the ideal case are significantly higher than in the lossy case; hence, losses in the setups actually affect the performance of the plasmonic SPRi sensor.Losses affect the mean SNR of all states.In both the ideal and lossy standard two-mode setups, we observe that the Fock state (TMF) far surpasses the performance of all the other states and breaks the SNL set by the coherent state.The difference is much more apparent when we consider the highest photon number-based SNR measurements (10 000 photons in this case).The coherent state in this instance outperforms all the other quantum states, such as the PS, NOON, and TMSV states.We also see that the PS, NOON, and TMSV perform quite poorly even with increasing photon numbers, and in fact, the SNR is not enhanced with the change in photon numbers for these states.Arguably, these states may not be suitable for plasmonic biosensing experiments.While the TMSD state seems to perform well in this case, it does not beat the SNL set by the coherent state, and hence, in this case, it would not be a suitable state to break the SNL.

Mean SNR in the balanced two-mode case
In this case study, we adopt the approach introduced in [30] in a static plasmonic sensing experiment, where the authors set the parameter η reference equal to η signal multiplied by a factor, T. This choice is made to achieve a further reduction in the overall measurement noise for the squeezed states.It's a form of optimization specifically tailored to the sensing model, providing significant benefits to TMSD and TMSV states, especially in the static case.However, in the dynamic case which we consider in this work, setting η reference = η signal T in the reference mode requires knowing the value of, T, in each time instance, which is impractical from an experimental standpoint.To address this issue, we opt to keep T fixed at a specific value, which is the midpoint value T mid = 0.4507.In other words, we set η reference = η signal multiplied by T mid , while T is allowed to vary in the signal mode.This decision is motivated by the observation made in the that the overall SNR of the system is approximately at its maximum value when evaluated at the midpoint of the sensorgram.This figure is for the case where the photon number is 10 and there is no loss in the setup that is η r and η s both equal to 1.The red line is the TMF, black line is the TMC, the blue line is the TMSD, the green line is the TMSV, the pink line is the PS, and the brown line is the NOON state.This is a figure for the ideal case where there are no losses in the system.The TMF has the best SNR compared to the other states.This is due to the measurement noise in the setup being reduced to below the SNL imposed by the TMC state.
The balanced two-mode sensing case refers to a special case when the loss in both modes is different/ unequal.The loss in the signal mode will be equal to 1 in the ideal case and 0.8 in the lossy case.One of the factors used to calculate the loss in the reference value (T mid , which is a median value of the binding sensorgram signal) is extracted from the sensorgram of the binding reaction of interest.The loss in the reference is taken as the product of the signal loss (either 1 for the ideal no-loss case or 0.8 for the lossy case) and the median of the signal intensity (transmittance) of the binding sensorgram signal.In this second scenario, we set the loss in the reference mode η reference = η signal T mid , where is T mid = 0.4507.In figure 7(b) we show the SNR as we change the photon number N from 10 to 10 4 for the ideal case of η signal = 1 and η reference = T mid .It can be seen that as N increases, the TMF state provides a better SNR mean for the TMSD, TMF, and coherent states.The other states, that is, the TMSV, PS, and NOON states, are not enhanced by changes in the photon number.It is worth noting that the mean for the TMSD state is higher than that of the mean for the coherent state in this particular case.

Lossy balanced two-mode case
In the balanced case, we also consider the impact of loss in the signal mode.In figure 7(a) we show the SNR as the photon number N goes from 10 to 10 4 , with η signal = 0.8 and η b = 0.8T mid .We observe that with increasing photon number N the SNR improves.It is interesting to note that when the losses are unbalanced in this way, the TMSD now outperforms the coherent state and hence breaks the SNL.The enhancement of the Fock state drops, but it still performs better than the other states.There is still no enhancement observed for the TMSV, NOON, or PS states.It is worth noting in this case that the mean SNR for the lossy case is slightly higher than that for the lossless case.This could be because the closer the balance between the losses, the higher the SNR that is measured.

Mean SNR in the single-mode case
In this final case, we effectively eliminate the reference mode by setting η reference = 0, to transform the two-mode sensing model into a single-mode model.The intensity-difference measurement then turns into an intensity Figure 6.The figure depicts the statistical mean of the SNR values in a standard two-mode sensing system using the TMSV, TMSD, TMF, TMC, PS, and NOON states taken over the entire sensing time period.The mean is taken for the signals generated over a range of varying photon numbers.In the standard two-mode case, we consider the losses in both arms to be balanced.The figure also depicts how the mean value for the SNR changes as the number of photons increases.The impact of increasing photon numbers is observed by looking at the dots associated with each state going upward.The lowest dot for each state represents the mean of the SNR when the photon number is 10, then the second lowest is the SNR mode for a photon number of 100, then photon 1000, and photon number 10000.(a) shows the standard two-mode setup with loss.In this analysis, we consider a lossy case where there is loss in both modes, i.e η signal = η reference = 0.8.Here we see that the Fock state offers the best performance at the low-photon number mark and even as the photon number increases in both modes.(b) shows a standard two-mode setup without loss.In this analysis, we consider an ideal case where there is no loss in either mode, i.e η signal = η reference = 1.We see that in the ideal case, the mean values of the SNR are comparatively higher than those in the lossy case.Hence, losses in the system have an impact on the mean SNR.The means of the TMSV, PS, and NOON states here were not affected by a changing photon number in the modes, and hence their SNR mean values were not improved by the increasing photon numbers.They are not photon number-dependent.measurement of the signal mode because there will be no transmittance in that mode.Most experiments are more like this scenario.
In figure 8(b), the variation in SNR mean as the photon number N increases is illustrated.It is evident that the Fock state (TMF state with η b = 0) offers a higher SNR average compared to a coherent state having a matching average photon number in the signal mode (TMC state with η reference = 0).The corresponding enhancement is depicted in figure 8(b).All enhancements are fairly similar and align with the expected value derived from the midpoint value R M (dotted line).A similar observation can be made for figure 8(a), where the TMF state clearly surpasses the TMC state and all other states as the number of photons increases.

Lossy single-mode case
We also take into account the effects of loss in a single-mode situation.The mean SNR for increasing photon number, with η a = 0.8 and η b = 0, is shown in figure 8(a).We also demonstrate in 8(a).As can be seen, the TMF state again offers the greatest SNR, even in the presence of a substantial loss.

Summary and outlook
In this paper, we look at a two-mode image sensing model and apply it to a SPR setup to image the binding surface of the SPR sensor.The SNR of the output image from the SPR was studied when different quantum states of light were used to probe the system.The states used to probe the SPR setup and image the binding surface are the TMC, TMSD, TMSV, PS, TMC and NOON states.The SNR was also studied for a varying number of photons in the input state and losses in the setup.We see that, as the photon number increases, the SNR generally increases.Losses in the setup also affect the performance of the different states, where the TMF state consistently gives the best SNR regardless of the losses we use, making it the best state to do imaging with.The TMC state is the state that gives the classical limit of precision, and hence its SNR is bound by the SNL.We see here that for the standard two-mode case with and without loss as well as in the single-mode case, the TMC outperforms all the The figure shows the statistical mean of the SNR values for TMSV, TMSD, TMF, TMC, PS, and NOON states in an optimized two-mode sensing system.The mean is taken over the time-dependent SNR curve for different photon numbers of the curves.In the standard two-mode case, we consider the losses in both arms to be balanced.The figure also depicts how the statistical mode of the SNR values changes as the photon number increases in each mode.The impact of increasing photon numbers is observed by looking at the dots associated with each state going upward.The lowest dot for each state represents the mean of the SNR when the photon number is 10, then the second lowest is the SNR statistical mean value for a photon number of 100 then photon 1000 and photon number 10000.(a) shows the standard two-mode setup with loss.In this analysis, we consider a lossy case where there is loss in both modes, i.e η reference = η signal T mid .Here we see that the Fock state offers the best performance at the low-photon number mark and even as the photon number increases in both modes.(b) shows a standard two-mode setup without loss.In this analysis, we consider an ideal case where there is no loss in either mode, i.e η reference = T mid .We see that in the ideal case, the means of the SNR values are comparatively higher than in the lossy case.Hence, losses in the system have an impact on the SNR.In this case, changing the number of photons in the modes did not change the SNR mean values of the TMSV, PS, and NOON states.As a result, the increase in the number of photons did not improve the SNR mean values.They do not have a photon number dependency.
other states apart from the TMF; hence, the TMF state is the only state that goes beyond the SNL when we consider the impact of loss and changing photon numbers on the mean SNR.However, there is a special case in which the TMSD state outperforms the TMC in terms of the SNR; this occurs when the losses are balanced in a specific way, as shown in the optimized two-mode case, where the mode value of the sensorgram of the binding reaction is used to weight the loss in the reference arm.We see here that the TMSD state outperforms the TMC under this condition and goes beyond the limit imposed by the SNL.Although it is entirely possible to enhance the SNR of our setup by simply increasing the photon number (which corresponds to increasing the intensity of our light), we find that when working with biological samples, this will not be an option as the high intensity can damage biological samples.Using quantum states of light, such as the TMF state, improves our setup's SNR even at low photon numbers, resulting in higher-resolution images.There are a wide range of systems to produce the different quantum states defined in this work, with the current state of quantum technologies a photon number range which may be generally acceptable across all the states would be N ∼ (4-14) from the systems the researchers have studied.As such, the authors believe that the analysis in the range of N ∼ 10 conducted in the research could give a reasonable picture of the current state of technology.In this work the authors find that for the N = 10 range in the different cases, in the ideal standard two-mode case only the Fock state outperforms the coherent state and does so by an enhancement factor (measured as a ratio of the SNR of the quantum state versus that of the coherent state) of 2.5, in the lossy standard two-mode case the enhancement factor drops to 1.74, in the idea balanced case the Fock, TMSD, and TMSV state outperform the coherent state by an enhancement factor of 1.39, 1.28 and, 1.20 respectively, in the lossy balanced case the Fock, TMSD, and TMSV state outperform the coherent state by an enhancement factor of 1.34, 1.24 and, 1.12 respectively and in the ideal single mode case the Fock state outperforms the coherent state by a factor of 1.38 which drops to a factor of 1.26 in the lossy single mode case.This research could lead to better biosensor design and higher-resolution image output as quantum technologies continue to develop.In the standard two-mode case, we consider the losses in both arms to be balanced.The figure also depicts how the mean SNR value changes as the photon number increases.The impact of increasing photon numbers is observed by looking at the dots associated with each state going upward.The lowest dot for each state represents the mode of the SNR when the photon number is 10, then the second lowest is the SNR mean for a photon number of 100, then photon 1000, and photon number 10000.(a) shows the standard two-mode setup with loss.In this analysis, we consider a lossy case where there is loss in the signal mode, i.e η signal = 0.8 and η reference = 0.Here we see that the Fock state offers the best performance at the low-photon number mark and even as the photon number increases in both modes.(b) shows a standard two-mode setup without loss.In this analysis, we consider a ideal case where there is no loss in either mode, i.e η signal = η reference = 1.We see that in the ideal case, the mean values of the SNR are comparatively higher than in the lossy case.Hence, losses in the system have an impact on the SNR.The means of the TMSV, PS, and NOON states here were not affected by a changing photon number in the modes, and hence their SNR mean values were not improved by the increasing photon numbers.They do not have a photon number dependance.

Figure 2 .
Figure2.Framework for sensing an unknown image.This is a two-mode imaging framework in which there is a signal mode that goes through the sensing mechanism before going to the detector and a reference mode that goes directly to the detector.Both modes undergo loss η where η reference is loss in the reference mode and η signal is loss in the signal mode.The dark regions of the image represent the baseline before any binding activity occurs.As binding activity begins, there is an increase in intensity; therefore, the regions where binding activity occurs will be brighter.This figure was redrawn from the figure in[33].

Figure 4 .
Figure 4. Detecting the change in the intensity of the pixels as a result of the binding interaction on the sensing surface.In this figure,we study how the intensity of the p th pixel changes as binding reactions of the third ligand occur.In the image taken at time, t 1 we see that the p th pixel (which corresponds to the third ligand) is dark, indicating that no binding has occurred, at t 2 the color starts to change slightly (becomes brighter), indicating some interaction between the ligand and the analyte molecule, and at t n depending on how much the ligand and the analyte molecule are bound, the color is further enhanced.We see from the arrow below that we track the intensity of the p-th pixel over time.For the n images, we will get a sensorgram that describes the binding kinetics.This can be done for any of the pixels.It is clear that the better the SNR of our images, the better the quality of any analysis we may be interested in doing.

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called the displacement operator for the mode a, which has a displacement parameter a Î . a ˆ † and a ˆare the creation and annihilation operators, respectively, for the mode a. a ˆ † and a ˆobey the bosonic commutation relation = a a , 1 [ ˆˆ] † [31], for mode b, the displacement operator can be expressed as b = b b -The displacement operator in equation (6) produces a coherent state añ | in mode a with a mean photon number of a á ñ = á ñ = N a a a 2 ˆˆˆ| | † and one in mode b with b á ñ = N b 2

Figure 5 .
Figure5.This figure shows the SNR for different quantum states.This figure is for the case where the photon number is 10 and there is no loss in the setup that is η r and η s both equal to 1.The red line is the TMF, black line is the TMC, the blue line is the TMSD, the green line is the TMSV, the pink line is the PS, and the brown line is the NOON state.This is a figure for the ideal case where there are no losses in the system.The TMF has the best SNR compared to the other states.This is due to the measurement noise in the setup being reduced to below the SNL imposed by the TMC state.

Figure 7 .
Figure7.Balanced two-mode sensing using Fock and coherent states (loss, η signal = 0.8 and η reference = η signal T mid ).The figure shows the statistical mean of the SNR values for TMSV, TMSD, TMF, TMC, PS, and NOON states in an optimized two-mode sensing system.The mean is taken over the time-dependent SNR curve for different photon numbers of the curves.In the standard two-mode case, we consider the losses in both arms to be balanced.The figure also depicts how the statistical mode of the SNR values changes as the photon number increases in each mode.The impact of increasing photon numbers is observed by looking at the dots associated with each state going upward.The lowest dot for each state represents the mean of the SNR when the photon number is 10, then the second lowest is the SNR statistical mean value for a photon number of 100 then photon 1000 and photon number 10000.(a) shows the standard two-mode setup with loss.In this analysis, we consider a lossy case where there is loss in both modes, i.e η reference = η signal T mid .Here we see that the Fock state offers the best performance at the low-photon number mark and even as the photon number increases in both modes.(b) shows a standard two-mode setup without loss.In this analysis, we consider an ideal case where there is no loss in either mode, i.e η reference = T mid .We see that in the ideal case, the means of the SNR values are comparatively higher than in the lossy case.Hence, losses in the system have an impact on the SNR.In this case, changing the number of photons in the modes did not change the SNR mean values of the TMSV, PS, and NOON states.As a result, the increase in the number of photons did not improve the SNR mean values.They do not have a photon number dependency.

Figure 8 .
Figure 8.The figure shows the statistical mean values of the SNR in a single-mode sensing setup using TMSV, TMSD,TMF,TMC, PS, and NOON states.In the standard two-mode case, we consider the losses in both arms to be balanced.The figure also depicts how the mean SNR value changes as the photon number increases.The impact of increasing photon numbers is observed by looking at the dots associated with each state going upward.The lowest dot for each state represents the mode of the SNR when the photon number is 10, then the second lowest is the SNR mean for a photon number of 100, then photon 1000, and photon number 10000.(a) shows the standard two-mode setup with loss.In this analysis, we consider a lossy case where there is loss in the signal mode, i.e η signal = 0.8 and η reference = 0.Here we see that the Fock state offers the best performance at the low-photon number mark and even as the photon number increases in both modes.(b) shows a standard two-mode setup without loss.In this analysis, we consider a ideal case where there is no loss in either mode, i.e η signal = η reference = 1.We see that in the ideal case, the mean values of the SNR are comparatively higher than in the lossy case.Hence, losses in the system have an impact on the SNR.The means of the TMSV, PS, and NOON states here were not affected by a changing photon number in the modes, and hence their SNR mean values were not improved by the increasing photon numbers.They do not have a photon number dependance.