The Hong–Ou–Mandel (HOM) effect is a fascinating quantum phenomenon that defies classical explanation. Traditionally, remote nonlinear sources have been used to achieve coincident photons at the HOM beam splitter. Here, we suggest that the coincident emission source required for HOM interference can be created locally using superradiant near field coupled emitters positioned across the beam splitter gap. We show that sensitivity to permittivity changes in the beam splitter gap, and corresponding Fisher information can be substantially enhanced with HOM photon detection. Subsequently, we outline several strategies for integration of superradiant emitters with practical sensor systems. Taken together, these findings should pave a way for a wide array of near field HOM quantum sensors and novel quantum devices.

Detecting small changes in permittivity is critically important for bioscience, environmental, and industrial applications. In the rapidly growing field of photonic sensing, subtle optical refractive index changes can help pinpoint biological activity, level of protein production, cell structure/disease, and can even be used for early cancer detection.1–4 Furthermore, increased sensitivity is vitally important for the array of remote sensors interrogated by fiber optic networks.5,6 These types of sensors are critical for energy infrastructure, remote chemical and industrial process sensing, and the multitude of additional applications. Modern optical microscopy techniques are reaching the limits of performance imposed by classical laws of physics. New types of refractive sensors and probes are needed for ever evolving level of resolution and contrast. Quantum sensors can offer new and unexplored modalities for optical and refractive index sensing.7,8

Most commonly, the quantum techniques based on coincident photon detection have been based on nonlinear processes such as parametric downconversion.9,10 While offering a reliable source of entangled photons, these types of sources suffer from low photon flux and generally require high instantaneous power levels. To make this kind of sensor suitable for widespread use, alternative means of generating coincident photons are needed.

Among the single photon quantum phenomena, the Hong–Ou–Mandel (HOM) effect stands out as an unambiguous manifestation of the quantum nature of light.11,12 The effect is fundamentally linked to the requirement for unitarity of the beam splitter operation. The HOM effect is manifested by quantum cancelation of coincident states after passing through the optical beamsplitter. Here, in the ideal case, two single photons incident on the two input sides of the beam splitter can only result in two photon output states, contrary to the intuition that predicts nonzero probabilities of single photon and dual photon states generation. The Pauli's statement on the indivisibility of photon stands as it travels though the beamsplitter since the photon does not give up its wave properties until the measurement is made.

HOM sensing using parametric down conversion sources has been demonstrated in several mediums such as fiber and integrated circuits.13,14 In this work, we propose to forgo the use of low flux sources of coincident photons and instead utilize the efficient superradiant fluorescent light originating from discrete coupled light sources such as color centers in crystals positioned at the sensor location.

Several color center mediums have been proposed and demonstrated over the last decade. Color centers based on diamond, hexagonal boron nitride, transition metal oxides, and rear earth doped crystals have shown its uses as quantum and single photon emitters.15–18 Among these diamond crystals offer a wide variety of color centers with the narrow zero phonon line (ZPL) and energy state properties attractive for quantum applications.15,19

Shallow lying surface diamond color centers can be generated via several methods such as ion implantation and plasma assisted processes.20–22 Silicon color centers with surface depth of 10 nm have been previously demonstrated.

Superradiant emission and time domain resolved superradiant emission since its inception23 have been experimentally and theoretically characterized by multiple studies.24–26 Superradiant emission from the two independent discrete emitters has been considered in the past in the context of homogeneous medium.27 The major manifestation of this phenomena for an ensemble of N emitters is the scaling of the emission rate, which is proportional to N2 due to coupled emission effects. In this work, we consider a case superradiant emission from two discrete in inhomogeneous structured geometry.

Normally, emission from two separate discrete fluorescent emitters is random and asynchronous. However, when they are brought together, the mutual coupling and phase locking take place. The phase locking is analogous to behavior of classical coupled pendulums and metronomes.28,29 Furthermore, upon emission of first photon, the divergent near field of the dipole emitter enhances density of states for the second photon emission. Therefore, in the simplest realization, the near-field quantum HOM sensor would consist of a pair of color centers brought to close proximity separated by a thin film beam splitter. In this work, we first analyze the case when the direction of sensed incident and reflected photons on the beam splitter is orthogonal to the plane of the beam splitter interfacial layers. This is followed by analysis of improved systems suitable for practical implementation. In this case, the sensing parameter is a coincidence rate of the photon emission from both sides of beam splitter that is sensitive to small refractive index changes in thin film.

Spontaneous emission from discrete emitters on planar interfaces has been theoretically investigated in the past.30,31 We have extended this theory to the case of superradiant emission from two symmetrically positioned emitters at the single layer interface [Fig. 1(a)]. In this approach, the added electric field contribution at the emitter location is composed of contributions from reflected Im(Gr(r0,r0))μ0 and transmitted Im(Gt(r0,r1))μ1 waves, where Grr0,r0 and Gt(r0,r1) are corresponding Green's function of two separate emitters.

FIG. 1.

Scaling of emission enhancement as a function of the gap size. Analytical (red solid) and finite element modeling (black dotted) results. Inset: electric field magnitude during superradiant emission from the quantum sensor (a); negative of imaginary part (b), real part (c), and absolute value (d) of effective transmission Green's function G̃tr0,r1 for various gap distances. Results for free space emitting dipole are shown for comparison.

FIG. 1.

Scaling of emission enhancement as a function of the gap size. Analytical (red solid) and finite element modeling (black dotted) results. Inset: electric field magnitude during superradiant emission from the quantum sensor (a); negative of imaginary part (b), real part (c), and absolute value (d) of effective transmission Green's function G̃tr0,r1 for various gap distances. Results for free space emitting dipole are shown for comparison.

Close modal

Analytical approach that we employed in our calculations can be described in terms of Green's function theory.31 In this framework, the electric field produced by the point source at r0=x0,y0,z0 with electric dipole moment μ is given by

Er=ω2μ0μ1G0(r,r0)μ,
(1)

where G0(r,r0) is a dyadic Green's function, ω is the oscillating frequency, μ0 and μ1 are the permeability of free space and the relative permeability of the medium, respectively. G0 needs to correlate electric dipole moment with resulting electric field; therefore, we can work with the vector potential Ar,

2+k12Ar=μ0μ1j(r),
(2)

where jr=iωδ(rr0)μ is the current density corresponding to electric dipole.

Using the definition of scalar Green's function for Helmholtz operator G0(r,r0)=ekrr0/4πrr0, we obtain

Ar=μk12iωε0ε1eik1rr0rr0.
(3)

In order to employ angular spectral expansion, we can apply Weyl's identity to Eq. (3) along with the standard transformation E=iω1+k12·A, which produces the desired dyadic Green's function,

G0r,r0=i8π2Meiikxxx0+kyyy0+kzzz0,M=1k12kz1k12kx2kxkykxkz1kxkyk12ky2kykz1kxkz1kykz1k12kz12,
(4)

where terms in M matrix have negative sign for z>z0 and positive for <z0. Equation (4) along with Eq. (1) allows us to expand the fields from arbitrary oriented dipole in terms of evanescent and plane waves.

In order to investigate the planar interfaces in the vicinity of the dipole, we can split the dyadic Green's function into s and p polarized waves. This way we can simply multiply the individual plane waves in the Green's function expansion by generalized Fresnel transmission and reflection coefficients. The final Green's function becomes a sum of primary, reflected, and transmitted Green's functions. A further simplification of the final equation is brought by converting to the cylindrical coordinates and performing analytical integration over ϕ.

Following these steps, the spontaneous emission enhancement for the system of two emitters can be expressed as

PP0=1+340Resszrssz2rpe2ik1z0szds+340Resszts+sz2tpe2ik1z0szds,
(5)

where sandsz are normalized k vectors, s=kp/k1, where kp=kx2+ky2, rsandrp are, respectively, the s and p-polarized effective reflection coefficients for symmetric single layer interface given by30 

r(p,s)=r1,2p,s+r2,1(p,s)exp(2ik2zd)1+r1,2(p,s)r2,1(p,s)exp(2ik2zd);t(p,s)=t1,2(p,s)t2,1(p,s)exp(ik2zd)1+r1,2(p,s)r2,1(p,s)exp(2ik2zd),
(6)

where r1,2(p,s) and r2,1(p,s) are the Fresnel reflection coefficients for respective interface. Second term on the right-hand side of Eq. (5) corresponds to reflected field and the third term to transmitted (superradiant) field contributions. Note that we have combined imaginary i terms resulting in the change of effective Green's function, G̃tr0,r1, contributions from imaginary to real. Refractive indices for the layers were assumed that of diamond and air (n1,n3=2.4 and n2=1), and distance of emitters to the surface was set at d0=10 nm.

Our comparative results [Fig. 1(a)] for spontaneous emission enhancements demonstrate a good match between analytical and full finite element method (FEM) simulations. In all cases, in this study, we have used the COMSOL solver for FEM modeling. Both dipoles were assumed to be oriented parallel to the beamsplitter interface in the x̂ direction and emit synchronously with the same phase. Emission wavelength was chosen to correspond to diamond silicon vacancy ZPL, λ0=738 nm. As expected from the superradiant theory, we have observed the increase in the emission enhancement from 2P0 for infinite separation toward 4P0 for closely spaced emitters, where P0 is the emission rate for isolated single emitter.

We have also compared emission for various gap thickness beamsplitters with the emission of the free space dipole. Here, the contribution of various k vectors to free space density of states was assumed to be fully uniform and angle independent. The total dipole emission was normalized to unity.

We can observe that the overall shape changes substantially with change in gap size with rich interplay of k-modes leading to broad multipeak profile. From Figs. 1(b) and 1(c), we can also see that the imaginary k vectors (s>1) while heavily present in the real part of the effective transmission Green's function responsible for steady state emission rate enhancement are entirely absent in the imaginary part corresponding to energy shift. The perturbation that the second dipole would be exposed to after the first photon emission would be proportional to absolute value of Green's function, and for small beamsplitter thicknesses can greatly exceed that of the free space dipole [Fig. 1(d)]. We can also observe that for small values of s corresponding to normal incident beamslitter photons, the Green’s function is approaching that of the free space dipole and is largely independent of the spacing. These results suggest that a good degree of superradiant coupling and, consequently, enhanced synchronization of emission across the gap can be achieved for the small ∼200 nm gap size.

The quantum physics of beamsplitter has attracted significant attention in the past few decades.32,33 Through previous research, it has been conclusively determined that in order for the beamsplitter transform to be unitary, a π/2 phase shift must exist between the reflected and transmitted waves. In terms of reflection coefficient, the general condition is expressed as ϕ31+ϕ42ϕ32ϕ41=±π, where ports 1and2 are the inputs and 3and4 are the outputs of the beamsplitter. Additionally, energy conservation requires R342+T412=R422+T322=1. Assuming symmetric beamsplitter, the transfer matrix can be represented as32 

cd=12i·RTTi·Rab,
(7)

where R and T are magnitudes of reflection and transmission coefficients, respectively. From this transformation between input a,b and output c,d creation operators becomes aiRc+Td2 and bTciRd2. Applying transformed operators to the ground state, we obtain for the input state |1,1>,

ab|0,0>ab=12iRc+TdTciRd|0,0>cd=12(iRTc2R2cd+T2dciRTd2)|0,0>cd,

which for R2=T2=0.5 results in cancelation of coincident states. After using the operation N3N4=ccdd, where c and d are corresponding annihilation operators, we arrive to the results for coincidence probability,

P(1,1)=T4+R42R2T2,
(8)

which confirms earlier results for coincidence detection in the beamsplitter.34 

Similarly, two photon detection probability can be found from (1/2)<N3(N31)> to be

P2,0=2R2T2.
(9)

From the classical statistical theory, the fundamental lower bound on the variance of any unbiased estimator is dictated by Cramer–Rao bound,

Var(R̃2)1NF,
(10)

where N is the number of measurements and F is the Fisher information defined in our case as

FR2=iR2Pi|R22Pi|R2.
(11)

We have used R2 as a proxy for permittivity sensing in the gap, which can be found from corresponding Fresnel coefficients for the single layer structure [Eq. (6)]. Written out explicitly for each outcome, the Fisher information terms become

(d(P(1,1))/d(R2))2P(1,1)=(8R24)2T4+R42R2T2,
(12)
(d(P(2,0))/d(R2))2P(2,0)=(24R2)22R2T2.
(13)

Figure 2 demonstrates the changes in relevant parameters for the single layer superradiant case analyzed earlier. Examining Fig. 2(a) and Eq. (12), we observe that for P1,1, the Fisher information is independent of R2 and equals to 16, which implies the reduction of the standard deviation of R2 estimation by a factor of 4 with this measurement. For P2,0, reduction in probability of refection at low gap distances leads to a large increase in Fisher information; however, the total NF product would remain constant. It may be possile to gain further advantage by operating in this regime due to Poissonian nature of spontaneous emission, i.e., the reduced signal would also translate to decreased variance, i.e., Δn=n¯, where n¯ is the average number of detected photons.

FIG. 2.

Hong–Ou–Mandel sensor properties, λ0=738 nm. Fisher information terms for P(1,1) (a) and P(2,0) (d) corresponding probabilities for P(1,1) (b) and P(2,0) (e). Square of reflection coefficient (c) and total Fisher information (f) versus gap size of the beamsplitter.

FIG. 2.

Hong–Ou–Mandel sensor properties, λ0=738 nm. Fisher information terms for P(1,1) (a) and P(2,0) (d) corresponding probabilities for P(1,1) (b) and P(2,0) (e). Square of reflection coefficient (c) and total Fisher information (f) versus gap size of the beamsplitter.

Close modal

Both regimes would normally be challenging to explore with low flux nonlinear single photon excitation. With near field spontaneous emission, it should be possible to use this phenomenon for quantum enhanced sensing. Comparing to remote nonlinear single photon excitation, the path through medium taken by single photons is reduced by a factor of 2, which would be advantageous for sensing in the turbid, high optical loss environment such as biological tissue. Additionally, in random biological tissue, there would be differences in arrival time of photons on two sides of the beamsplitter using remote excitation. These differences will also be eliminated using our approach.

Normally, for discrete dipole emitters in addition to the photons reflected from the boundary, there will exist spurious photons reaching the detector emitted directly from color center bypassing the beamsplitter reflection. Additionally, there will be the two-photon states whose emission probability peak coincides with the resonant maximum sensitivity region.

There are several approaches through which the spurious signals can be isolated. The coincidence detection method using two separate detectors automatically results in cancelation of two-photon states. The main source of interference in coincident photon measurement is the dipole radiation that emits into free space and shares the same k vector as the photons reflected from the interface. Since the spatial separation between the reflected signal and dipole signal is deeply subwavelength, it would be difficult to directly spatially isolate them due to resolution imposed by diffraction limit.

One viable approach to isolating the signals is by introducing a polarizer layer as a buffer between the beamsplitter layer and the emitter or within the beamsplitter itself. Several techniques to induce polarization rotation using thin planar chiral metamaterials have been suggested recently.35,36 The detected signal would be filtered through polarizer, and a fractional signal will be fed to coincidence detectors. As most of the chiral metamaterials are lossy, the signal coincidence rate would be reduced; however, the HOM effect will hold as long as the π/2 phase relation between transmitted and reflected signals is maintained.34 

Alternatively, it may also be possible to induce polarization change by orienting the dipole at an angle to the surface and choosing the sampled k vector incident on the interface that results in superposition of s and p polarized wave at the interface. Difference in s and p reflection coefficients will induce polarization rotation of reflected signal. However, in the general form, this method may not be optimal as s and p polarized signals will have different gap thicknesses corresponding to a point of R2=0.5. Therefore, it might be challenging to directly apply this approach.

A simplified version of this technique may work as a better isolation scheme. We can collect a p polarized signal for dipole positioned at a 45° angle to the interface [Fig. 3(a)]. In this configuration, the null of dipole emission coincides with the direction of reflected photons; therefore, a very large degree of isolation can be achieved while high sensitivity is still maintained. Figure 3(b) shows the polar emission plot for the 45° dipole indicating direction of reflected and transmitted signals. The transmission Green's function equation in this case takes form of

G̃tr0,r1=120sszts+sz2tpe2ik1z0szds+0ssztpe2ik1z0szds.
(14)
FIG. 3.

Schematic of the Hong–Ou–Mandel setup excited by 45° oriented color centers. Dipole emission pattern is shown in dashed line (a); polar plot of the emission profile for 45° oriented color centers (b); far field emission pattern of the multilayer planar structure (c).

FIG. 3.

Schematic of the Hong–Ou–Mandel setup excited by 45° oriented color centers. Dipole emission pattern is shown in dashed line (a); polar plot of the emission profile for 45° oriented color centers (b); far field emission pattern of the multilayer planar structure (c).

Close modal

Figure 4 demonstrates the nearfield transmission Green's function results for the case of 45° oriented dipoles.

FIG. 4.

Negative imaginary (a) real part (b) and absolute value (c) of transmission Green's function G̃tr0,r1 for 45° superradiant dipole. Results for free space emitting dipole are shown for comparison.

FIG. 4.

Negative imaginary (a) real part (b) and absolute value (c) of transmission Green's function G̃tr0,r1 for 45° superradiant dipole. Results for free space emitting dipole are shown for comparison.

Close modal

As an additional consideration, the semi-omnidirectional emission profile of the color centers needs to be taken into account. Normally, the far field single photons emit according to the dipole emission pattern without the preferential emission direction. This pattern can be modified due to quantum coupling of the cascaded superradiant emission states,27 but, overall, a large photon flux is emitted semi-uniformly across solid angle sphere. Therefore, the probability of coincident photon emission in the z direction is reduced. We can engineer the local density of states, such that the emission takes place preferentially in the desired direction. Multiple structures based on metamaterials, plasmonics, and photonic crystals have been used in the past for tailoring/enhancing local photonic density of states.37–39 In our case, a simple multilayer nanostructure can help to achieve tailored spontaneous emission by selectively enhancing z and/or suppressing x, y density of states. Figure 3(c) shows emission profile for the silicon nanostructure with dimensions 2 μm × 150 nm, which shows high degree of directionality in the orthogonal direction indicating enhanced z density of states. In this simulation, we have used general analytical 1D solution for reflection from multilayer structures to determine the thickness of symmetric structure producing 50% reflection at the beamsplitter.

Fabrication of the sensor can be done using standard nanofabrication techniques. The non-coincident distribution of the color centers across two sides of the beamsplitter can be tuned by using a high-resolution alignment stage to position the two fluorescent color centers. Sufficiently, low concentration of dopants needs to be used to avoid superradiant emission from color centers on the same side of the beamsplitter. Doping of the crystals with the impurities can be done through the focused ion beam techniques.40 For angled emission, the specific dipole orientation can be achieved with crystals cut at prescribed crystalline planes.

The techniques presented here can be further extended to the systems with number of emitters N>2 for the generation of complex quantum states for potential applications in quantum computing such as utilizing the cluster state approach.41 Additionally, a gating mechanism via selective control of emission from individual emitters employing, for example, microwave-controlled NV centers may also be implemented. Nanoparticle designs based on multilayer disk structures presented in this work can be readily used as enhanced local permittivity sensing probes for biomedical applications. Separately, controllably launching phonons in the gap material or inducing gap modulation can also potentially provide an avenue for single phonon/single photon coupling as well as single phonon detection.

In this work, we have proposed and theoretically demonstrated a new approach for quantum sensing based on the near field Hong–Ou–Mandel effect. These findings can enable a new generation of quantum enhanced local sensors suitable for deployment in a wide variety of applications. The general method of combing superradiant emission with local environment constrained by unitarity can have applications in quantum computing and quantum sensing as well as fundamental explorations of properties of coupled quantum systems.

This work was performed under the support of the United States Department of Energy's Office of Fossil Energy Crosscutting Technology Research Program or Subsurface Science, Technology, Engineering, and R&D (SubTER) Crosscut Program. The research was executed through the NETL Research and Innovation Center's Carbon Storage Field Work Proposal. This research was also supported in part by an appointment to the NETL Research Participation Program, sponsored by the U.S. Department of Energy and administered by the Oak Ridge Institute for Science and Education. Research performed by Leidos Research Support Team staff was conducted under the RSS Contract No. 89243318CFE000003. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

The authors have no conflicts to disclose.

Roman Shugayev: Conceptualization (lead); Investigation (lead). Ping Lu: Validation (supporting). Yuhua Duan: Supervision (equal). Michael Buric: Supervision (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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