An integrated quantum photonic sensor based on Hong-Ou-Mandel interference

Photonic-crystal-based integrated optical systems have been used for a broad range of sensing applications with great success. This has been motivated by several advantages such as high sensitivity, miniaturization, remote sensing, selectivity and stability. Many photonic crystal sensors have been proposed with various fabrication designs that result in improved optical properties. In parallel, integrated optical systems are being pursued as a platform for photonic quantum information processing using linear optics and Fock states. Here we propose a novel integrated Fock state optical sensor architecture that can be used for force, refractive index and possibly local temperature detection. In this scheme, two coupled cavities behave as an"effective beam splitter". The sensor works based on fourth order interference (the Hong-Ou-Mandel effect) and requires a sequence of single photon pulses and consequently has low pulse power. Changes in the parameter to be measured induce variations in the effective beam splitter reflectivity and result in changes to the visibility of interference. We demonstrate this generic scheme in coupled L3 photonic crystal cavities as an example and find that this system, which only relies on photon coincidence detection and does not need any spectral resolution, can estimate forces as small as $10^{-7}$ Newtons and can measure one part per million change in refractive index using a very low input power of $10^{-10}$W. Thus linear optical quantum photonic architectures can achieve comparable sensor performance to semiclassical devices.

In parallel, integrated optical systems are being pursued as a platform for photonic quantum information processing using linear optics and Fock states. Here we propose a novel integrated Fock state optical sensor architecture that can be used for force, refractive index and possibly local temperature detection. In this scheme, two coupled cavities behave as an "effective beam splitter".
The sensor works based on fourth order interference (the Hong-Ou-Mandel effect) and requires a sequence of single photon pulses and consequently has low pulse power. Changes in the parameter to be measured induce variations in the effective beam splitter reflectivity and result in changes to the visibility of interference. We demonstrate this generic scheme in coupled L3 photonic crystal cavities as an example and find that this system, which only relies on photon coincidence detection and does not need any spectral resolution, can estimate forces as small as 10 −7 Newtons and can measure one part per million change in refractive index using a very low input power of 10 −10 W.
Thus linear optical quantum photonic architectures can achieve comparable sensor performance to semiclassical devices.

I. INTRODUCTION
Integrated photonics based on photonic crystal (PhC) structures provides a path to extremely small optical sensors with applications to biology [1,2], chemistry [3] and engineering [4]. PhC devices with various geometries and structures such as hollow core PhC fibers [5], 1D and 2D waveguides [6,7] and nano-cavities [8][9][10][11][12] have been fabricated and used for sensing applications. Among these devices, PhC cavity-based sensors offer important advantages over PhC waveguide sensors since they can be made much smaller, thus reducing vulnerability from impurities and losses. Moreover, exploiting high Q cavities with large mode volume are advantageous for sensors based on refractive index (RI) changes, for example in bio-pathogen detection [13], chemical sensing [14] and single particle detection [15]. All these schemes make use of second order interference of coherent states of the optical field and are thus basically classical phenomenon.
In parallel to integrated optical sensors, considerable progress has been in using integrated optical systems for single photon optical quantum computing using linear optics, so called LOQC [16]. These schemes are enabled by the uniquely quantum-optical phenomenon of Hong-Ou-Mandel (HOM) interference [17]. This opens up a new perspective for optical quantum metrology that combines ideas from photonic crystal sensors with linear optical quantum information processing using single photon states. We make a first step in this direction by presenting a scheme that uses HOM interference to make sensors from optical pulses prepared in single photon pulses. As HOM interference does not arise for coherent optical pulses, our proposal is a true quantum metrology scheme and realizes the gain in sensitivity such schemes offer. Furthermore, it opens up the prospect of using LOQC protocols to construct more sophisticated quantum metrology protocols that are compatible with integrated optical systems.
Recent demonstrations of cutting edge sensors that exploit quantum mechanics have been shown to outperform their classical counterparts in achieving higher sensitivities [18][19][20]. Many applications, e.g. biological sensing [21], require low power to preserve delicate samples destroyed by: photo-decomposition, photo-thermal effects, and photon pressure for example. This requirement is in addition to the usual requirements of high input-output gain (responsivity), low noise and high bandwidth. In that regard, weak coherent light offers a route to low power sensing. However, the use of weak coherent pulses lowers a sensor's bandwidth. Consider for example a series of weak coherent pulses with on average one photon per pulse, in this case roughly 37% of pulses have no photons at all and 26% have more than one photon per pulse. Clearly the ultimate low pulse power limit is achieved by single photon pulses with only one photon per pulse. A sensor operating with single photon states offers low power suitable for deployment in lab on a chip applications [22] and compatible with attojoule all-optical switching [23] and opto-mechanical devices for strain sensors [24] and accelerometers [25].
While single photon states are not easy to make there is a very large research effort underway driven by their potential application in quantum information processing [16]. For our purposes it suffices to note that PhC devices are compatible with a number of quantum dot single photon sources [26] and that technological advances in integrated multiplexed single photon sources in PhCs are very encouraging [27]. The fundamental quantum nature of photons is usually observed through the HOM effect which has now been demonstrated in a variety of physical systems such as evanescently coupled optical waveguides [28] and microwave devices [29]. In the HOM effect indistinguishable photons simultaneously arrive at each of the two input ports of a 50/50 beam splitter, after which the photons "bunch" together so that both photons are either in one output port or the other. Never will you observe one photon in both outputs.
The use of dual Fock states was proposed in 1993 in the context of quantum metrology to reduce the uncertainty of phase measurements [30]. In this article, we propose a novel scheme for a quantum photonic sensor based on coupled PhC cavities that exploits the HOM effect, shown in Fig. 1. The coupled PhC cavities form an "effective beam splitter" for two incident photons. The central idea is that a parameter to be estimated, call it ψ, modulates the coupling between the optical cavities, g. This can be done by changing the distance between the cavities through compressing or stretching the dielectric material (e.g. for force and strain sensing) or by changing the refractive index of the media between the two cavities (e.g. for RI, temperature and single particle sensing). The change in g modifies the reflection and transmission of the effective beam splitter which changes the visibility of HOM interference. Therefore, by measuring the change in HOM visibility, we can sense the variation in g and thus estimate ψ. This scheme is independent of transmission/reflection spectra normally used for classical cavity-based sensors [31] and neither a dispersive element nor spectral resolution for the measurement is required. First, we characterize the proposed sensor in terms of its performance metrics, the responsivity and minimum detectable value for the parameter to be estimated. This characterization in terms of the working parameters of the sensor is expressed in a general way, with no assumptions set for the values for the cavity damping rate, cavities coupling strength, PhC refractive index, etc. Then, a more specific example is provided by considering this scheme with previously reported experimental parameters for GaAs/AlGaAs PhC structures. We theoretically predict that such a system can measure one part per million change in refractive index as well as forces on the order of 10 −7 N. These results are not obtained by using experimental values specifically optimized for our scheme. However, the results obtained for refractive index and force sensing are promising for integrated on-chip sensing.

II. HONG-OU-MANDEL SENSOR
As HOM interference is a uniquely quantum mechanical phenomenon we must necessarily proceed with a full quantum description. Consider the double optical resonator scheme composed of two optical cavities with resonance frequency ω, depicted in Fig. 1. The optical fields in the two cavity modes are described by the bosonic annihilation operators a 1 and a 2 . The interaction picture Hamiltonian is given by where g is the effective interaction strength that depends on the parameter ψ. We couple this system, via the evanescent field, to the input and output channels comprising two optical wave-guides. The relation between the respective input and output fields is [32] a j, where κ j (j = 1, 2) is the damping rate of cavity j. Cavity modes a 1 (t) and a 2 (t) can be related to the input modes using the input-output stochastic differential equations [32,33] where the solution to these equations is given in appendix A. To operate this device as a sensor we then load the two input ports with single photons and perform coincidence detection at the outputs. The upper input port of the beam splitter is loaded with a single photon in the state |1 ξ = dsξ(s)a † 1,in (s)|0 having pulse shape ξ(t) = √ γe − 1 2 γt with the normalization condition dt|ξ(t)| 2 = 1. The lower input port is loaded with a single photon having exactly the same amplitude function but time shifted with respect to the top , where γ is the input photon bandwidth. This state has zero field amplitude, a in (t) = 0, so conventional (second order) interference cannot be used.
However a † in (t)a in (t) = 0 so fourth order interference will reflect the quantum coherence inherent in the pure state |1 ξ .
The probability of one and only one click occurring at both detectors D 1 and D 2 is given by the fourth order correlation function It should be noted that in this expression the time τ is not the delay between detection events but a temporal separation of the two input photons. In practice, the integration time need not and should not be infinite as it sets the time interval between successive pulses. In fact the integration time needs to be of the order τ rep ∼ max{1/κ, 1/γ}. In what follows we work in regimes where κ γ > 1, which is compatible with available experimental realizations, so we have τ rep ∼ 1/γ. Through equations (1), (2) and (3) the explicit dependence of G (2) (τ ) on g can be seen. By monitoring changes in G (2) (τ ) we can infer changes in g. In the ideal case, we would like to detect both photons.
In practice, either one or two photons could fail to be counted at the detectors due to optical losses, imperfect single photon sources or non-unit detection efficiency. The case of photon loss before detection (or detector inefficiency) can be modelled as usual [34] by inserting a beam splitter with transmissivity amplitude t k in the path of the output fields a k,out (t). The field that reaches an ideal detector is given by the transformation is a vacuum field mode annihilation operator. Substitution into the equation (4) shows that the only term that contributes to the both the numerator and denominator are multiplied by the factor T 1 .T 2 , where T k = |t k | 2 is the conditional probability for a single photon in the output field to reach an ideal detector. This is because this average is normally ordered. Thus G (2) (τ ) is unchanged by loss since we have normalised it by the intensity that actually reaches the detector from each mode: in effect G (2) (τ ) is a conditional probability conditioned on only those detection events that give two counts, one at each detector. Single counts and no counts are discarded. These cases should be considered as failed, but heralded trials in which we discard and simply run again with another two single photons. However, this lowers the sensor's bandwidth. In section III, we explicitly include an optical loss factor, ε, which is defined as the number of failed trials over the total number of trials, to take the effect of these imperfections into account. As photon loss is heralded, LOQC error correction techniques might be employed to mitigate the loss of signal on such events.
Photon loss before the device, or failure of a source to produce a photon can likewise be modelled by inserting a beam splitter into the path of a in (t). This changes the input state to a mixed state as follows. The input state is a two photon state of the form is a vacuum field mode annihilation operator. Thus the total state after the beam splitter is but the actual input state to the device is given by tracing out over the two vacuum modes. This gives the input state as a mixed state of the form This also indicates that loss at the input is detected as the conditional input state, conditioned on counting two photons in total at ideal detectors. This is simply the first term in the above sum. This is the same pure state as for the case of perfect sources. The coefficient T 1 T 2 is simply the conditional probability that two input photons in each of the inputs enter the device. Input loss or source inefficiency is also heralded in the detectors and those trials can be discarded. The fraction discarded in total including input inefficiency and detectors inefficiency is simply T 1,in T 2,in T 1,out T 2,out , where T k,in and T k,out are the conditional probabilities that a single photon enters the device in mode-k and that a single photon for output mode-k is detected.
In Fig. 1, the HOM dip for our system is depicted for particular values of κ/γ and g/γ. For τ = 0, where input photons are indistinguishable, quantum interference results in photon bunching, or photon pairs, and we see the minimum of the coincidence probability i.e. the HOM dip. As τ increases or decreases the coincidence probability increases.
We define the responsivity of the sensor to detect the changes in g as Operating at τ = 0 is optimal for most combinations of γ and κ and maximizes the responsivity. We then optimize the values of κ/γ and g/γ so that the derivative of G (2) (0) with respect to g is maximized. By maximizing the responsivity over our device parameters, g 0 the initial beam splitter coefficient and the cavity damping rate κ, we can optimize the performance of our sensor. Due to the fact that our sensor is a linear quantum system, we can analytically calculate G (2) (τ ) and its derivative for the initial state |ψ(0) = |1 a 1 ,ξ , 1 a 2 ,η , the full expression for G 2 (τ ) is given in appendix A. Figure 2 can serve as a guide for experimental implementations and device fabrication. Figure 2(a) shows G (2) (0) as a function of g/γ and κ/γ. Figure 2(b) shows the behaviour of the system response for different operating points g/γ and κ/γ. The dashed line on Fig. 2(b) demonstrates the operating points at which dR g dg = 0 where we can take advantage of maximum sensor response. In addition, at this maximum sensor response, the estimator G 2 (0) behaves linearly with small signal variations as will be described below.

III. NOISE CHARACTERISTICS
Another important measure in characterizing the sensor performance is the Linear Dynamic Range (LDR) which is related to the estimation error and the sensor linearity which we now explore. The error in estimating δg is related to the error in estimating δG (2) (0) in a finite number of samples where is the standard deviation of a Bernoulli distribution with N trials. The loss factor, ε, has If we operate the sensor on a bias g 0 where sensor response is maximum, we can take advantage of the sensor linear response, up to small variations in g. (b) Shows LDR for bias g 0 γ = 1.8 shown in (a) for different detection frequency bandwidths over γ, f γ . The red star shows the upper LDR limit that is the point up to which sensor responds linearly within 1% variation. already been introduced in section II. The minimum detectable shift in g from the bias g 0 should be larger than this error, i.e. δg min > δg noise , so that we are able to measure it. For a large number of samples (N → ∞), δg noise is negligible (up to accidental coincidences caused by dark counts or stray light). This result is useful for the estimation of a static or quasi static parameter.
We now give an order of magnitude estimate for δg min when the parameter is time varying.
If T meas = τ rep N is the time between our samples of g(t), naive arguments from the Nyquist-Shannon sampling theorem imply that we can not determine frequency components of g(t) greater than f = 1/(2T meas ), which is called the detection frequency bandwidth. For a onesigma level of confidence we should have N ≥ min{γ, κ}/(2f ) and the noise equivalent δg, given in equation (7), becomes Now we can calculate the LDR which is defined as where δg max is the point bellow which the sensor response is linear within 1% variation, i.e.
In Fig. 3(a), responsivity is plotted with respect to g for an arbitrary value of κ. We bias the initial coupling between the optical resonators (g 0 ) where the responsivity peaks.
Therefore, there is a range of δg = g − g 0 for which the sensor behaves linearly. LDR is shown in Fig. 3(b) for some arbitrary detection bandwidth in units of γ. For smaller choices of f /γ, δg noise will be decreased, so the sensor can resolve smaller shifts in g.

IV. HOM SENSOR IMPLEMENTATION
We now consider specific physical applications for our sensor, first as a force sensor and then as a refractive index sensor, employing coupled L3 PhC cavities [8][9][10][11][12] experimental data to estimate its responsivity and minimum resolvable shift in signal for each case. By examining the normal mode splitting reported in these references we infer the coupling strength g between the PhC resonators is of the order of 10 11 − 10 15 Hz. The evanescent coupling strength between the resonators and waveguides κ can be tailored, so that κ ∼ g for example. Operating as a force sensor, the measured signal is the shift in cavity separation induced by an applied force or a strain, while operating as a refractive index sensor, the signal to be measured is a change in refractive index induced by the presence of a molecule dropped on the air holes between the PhC resonators, for a constant bias cavity separation.
A shift in either cavity separation, call it x, or refractive index, call it n, modifies the coupling strength between the resonators which will be detected by measuring G 2 (0). Therefore, to give an order of magnitude estimation of the responsivity and minimum detectable signal in each case we need to investigate the dependence of g as a function of x and n. To do this we used a 1D model analysed by the transfer matrix method [35] (see Appendix B) to investigate the dependence of the cavity normal mode splitting on the change in cavity separation or refractive index.
In the case of identical resonators, ω 1 = ω 2 = ω and κ 1 = κ 2 , the splitting in frequencies of the symmetric and asymmetric normal cavity modes is ∆Ω = 2g [8,36]. Therefore, we can write g = πc∆λ/λ 2 , where c is the speed of light and λ is the cavity mode wavelength.
Since πc/λ 2 is a constant, to find the functionality of g with x and n, we need to find the functionality of ∆λ with those parameters. Numerics show that an exponential function of the form g = ae −bx fits very well on data achieved for normal mode splitting change versus different cavity separations (see Appendix B) and an exponential of the form g = ae bn 2 can describe the changes with respect to refractive index (see Appendix B). Hence, we can generally write g(x, n) = ae −bx+dn 2 . We extract the coefficients a, b and d by fitting data from figure 2 of citation [12] for a PhC made of GaAs/AlGaAs (see Appendix C). According to their data g is on the order of 10 12 − 10 13 Hz for this range of x bias that is shown in Fig.   4. We have chosen κ of the order of 10 13 Hz.

A. HOM force sensor
First we investigate the efficiency of our system operating as a force sensor. In this case n = 1, so by substituting g(x, 1) into equation (4) we can see how the probability of joint detections changes for different operating points x bias (Fig. 4(a)). The sensor response to changes in x is calculated as R Figure 4(b) shows that for an input photon bandwidth on the order of γ = 1GHz, which is experimentally feasible at the moment [37,38], sensor response to shifts in x is of the order of 10 −3 (nm) −1 . Minimum detectable x can be easily related to δg min as δx min = −1 bg δg min . Figure 4(c) shows this noise equivalent x is of the order of 10 −3 (nm/ √ Hz). Young's modulus for GaAs is E = 85.5GPa [39]. Therefore, for the given lattice with a thickness of t 1µm the stiffness of GaAs is k = E t 85.5 kN m . Minimum detectable force is shown in Fig. 4(d) and is of the order of 10 −7 N which compares rather well with the high resolution PhC force sensors [40,41] exploiting coherent light. However, these schemes use significantly larger input power while in our results not only is the pulse power (1 ph/pulse) low but also the average power (10 −10  W), which is defined by the emission rate of the current single photon sources (∼ GHz), is also low.
Importantly, fabricating the cavities with a smaller κ does not affect the sensor resolution but shifts the optimum operating points at which the sensor behaves linearly (white dashed lines in Fig. 4) towards larger x bias .

B. HOM refractive index sensor
To operate the system as a refractive index sensor we operate at a fixed x bias , so g(n) = ae −bx bias +dn 2 . System response to refractive index shift is R n (x 0 , κ) = | dG 2 (0) dn | n=1 and the minimum detectable refractive index shift is calculated as δn min = (δg min /2dgn)| n=1 . Figure   5(a) is a fabrication guide for γ = 1GHz to find the best operating points to achieve maximum responsivity together with linear response. Figure 5(b) predicts a resolution of the order of 10 −6 refractive index unit (RIU) per √ Hz for single photon bandwidth of γ = 1GHz.
Up to the best of our knowledge the best resolution achieved in schemes [42,43] is of the order of 10 −7 RIU per √ Hz, however these use more input power. The advantages of the presented scheme are as follows. This scheme can be implemented on chip and fabricated in micro-scale dimensions. Moreover, unlike sensing approaches based on transmission spectrum of a L3 cavity coupled to a waveguide, this approach does not require spectral resolution that reduces the bandwidth. Additionally, this scheme can be a multi-purpose sensor. In this article, we have discussed force (strain) and refractive index sensing. With minor modifications, it can be used for other targets such as local temperature, pressure and particle detection and analysis.
A √ 2 improvement in estimation accuracy over schemes using coherent light and a √ 2 improvement in bandwidth over schemes using serial single photons can be achieved. The improvement of √ 2 in estimation accuracy may sound underwhelming. But a different accounting philosophy shows why this factor is important (see Appendix D for further details). to the serial single photon case. Due to this the reduction in samples the bandwith of our sensor relative to both cases is increased.
The disadvantage of this single-photon-based scheme compared to those using coherent light is the difficulty in building reliable single photon sources and detectors.
To summarise, the key point we are making in the paper is that if one has already made a commitment to single photonics in order to gain access to the quantum information processing that this provides, single-photon metrology can also be added to the suite of tools. HOM interference is the key phenomena that enables scalable quantum information processing in single photonics with linear optics.

VI. APPENDIX A
The solutions to the quantum Langevin equations (3) are given by where A(t) = e −κt/2 cos(gt), B(t) = −ie −κt/2 sin(gt), C(t) = e κt/2 cos(gt) and D(t) = ie κt/2 sin(gt). By using the above solutions for the cavity mode in the input-output relation (2), we can analytically calculate the joint detection probability as where

VII. APPENDIX B
To find the functionality of coupling strength g with separation distance between the cavities and refractive index of the media in between the two cavities, we can use the analogy of the coupled cavities with a quantum double-well problem with a potential barrier in between, where we need to find the tunnelling rate g. Solving this problem shows the functionality of g with the width of the barrier x and the height of the barrier V scales as g ∝ exp{−bx − dV }. According to citations [44,45], which introduce the optical equivalence of the Scrödinger equation, V ∝ −n 2 . Therefore, we expect the coupling strength between the cavities to scale as g ∝ exp{−bx + dn 2 }. To be more precise a one dimensional optical modelling simulation has been performed to find the functionality of coupling strength g with x and n, which suggest a very good fit can be achieved with the above given functionality.
The simulations are done by the transfer matrix method described in [35]. The results are shown in Fig. 6 Refractive index of the air holes, n

VIII. APPENDIX C
We choose experimental data for a PhC lattice made of GaAs/AlGaAs given in [12] as an example to find the functionality of coupling strength g in terms of cavities separation x and the refractive index of the dielectric material n. Figure 7 shows the numbers for the best found fitted function.  [12] to find the coefficient a, b and d in functionality of g versus x and n that we found of the form g(x, n) = πc∆λ/λ 2 = πc λ 2 ae −bx+dn 2 where λ = 1000nm.

IX. APPENDIX D
Here we back up the claims made in the conclusion, further details will be available in a future publication. Here we assume a mean flux greater than two photons may damage a sample. We model our coupled cavity sensor as an effective beam splitter between the input modes such that the beam splitter reflectivity g is a function of the parameter to be estimated, x, i.e g(x). In the case of very small changes in x, this functional relationship can be linearized about an operating point x 0 so that changes in θ are linearly proportional to changes in g.
Now we ask what is the ultimate limit imposed by quantum theory on the precision of our estimate g est of g. This, of course depends on the state used to probe the beam-splitter. If the mean squared error (MSE) i.e. E[(g −g est ) 2 ] quantifies the performance of our estimation scheme then the quantum Cramer-Rao bound provides a method to answer this question.
The quantum Cramer-Rao bound states that asymptotically the precision of an unbiased estimation scheme is bounded below by 1/ N I(g) where N is the number of experimental trials and I(g) is the quantum Fisher information with respect to the input state.
If the input state is a coherent state then the quantum Fisher information is generally I α (g) = |α| 4 where |α| 2 is the mean photon number at the input. So for the cases of interest I α (g) = 1 and I α (g) = 4. If the input state is a Fock state then the quantum Fisher information is [34]: I F (g) = 4 (input photon number = 1), I F (g) = 16 (input photon number = 2). The numbers quoted in the main text can be arrived at by setting MSE = 10 −4 and solving for N .
Loss is easily included using the standard beam splitter model discussed in section 2.
Coherent states are not entangled on beam splitters so tracing out the vacuum modes leaves the signal mode in a coherent state with amplitude reduced by α → tα, where |t| 2 = T is the probability for a photon not to be lost. The Fisher information [46] is then simply rescaled to reflect the loss of amplitude. Single photon product states at the input to a beam splitter do become entangled at the output. However, we can trace out the vacuum modes to give a mixed input state for the case of n = 2 in a HOM experiment of the form The Fisher information for a mixed state is more difficult to calculate as it involves the logarithmic derivative. However, because the mixed state above is so simple it is easy to see that, conditioned on detecting two photons for correct heralded operation, the Fisher information is rescaled by T 2 1 T 2 2 .

ACKNOWLEDGMENTS
We acknowledge the support of the Australian Research Council Centre of Excellence for Engineered Quantum Systems, CE110001013. SBE and AA were funded by the University of Queensland International Scholarship. JC was supported in part by National Science