Multiparticle quantum plasmonics

2.1 Quantum properties of the electromagnetic field

In this section, we review the quantization of Maxwell’s equations. The reader can find additional discussions and details in Refs. [42], [43], [44], [45]. Our treatment assumes the absence of sources of radiation and vacuum propagation only, where Maxwell’s equations read as

Here, c is the speed of light in vacuum and E and B represent the electric and magnetic fields, respectively.

Then considering a one-dimensional cavity of length L with perfectly conducting walls, we obtain the single-mode field solution,

where we have taken the wave’s propagation along the z direction and polarization along the x direction. Moreover, V represents the effective volume of the cavity and Q(t) defines a canonical position. In addition, here k and ω are the wavenumber and frequency of the mode, respectively. As the one-dimensional boundary condition must be met, where k=jπ/L(j=1, 2, 3, …), the magnetic field in the cavity is

Then, the classical field energy of the single-mode field, i.e. the Hamiltonian H, is given by

where p=q˙; these are canonical variables for classical systems. Then, it is clear that the single-mode field is equivalent to a harmonic oscillator of unit mass. To quantize the single-mode field, one can then introduce the canonical commutation relation,

where ℏ=h/2π. Then, by convention, we introduce the annihilation a^ and creation a^† operators

The annihilation operator a^ and the creation operator a^† satisfy the commutation relation

Therefore, the Hamiltonian can be written as

This description enables us to quantize the electromagnetic field.

2.2 Quantum states of light

The annihilation and creation operators can be used to describe quantum states of light. Remarkable examples include Fock states, coherent states, squeezed states, and thermal states. The possibility of preparing light in these states has been extensively used to prepare many-body systems of photons.

2.3 Classical and quantum coherence

The advent of the laser gave an enormous impulse to the development of the theory of optical coherence [56]. Nowadays, photonic technologies depend to an important extent on our ability to manipulate the coherence properties of the electromagnetic field. In this regard, plasmonic systems have been extensively used to engineer the spatial and temporal properties of photons. In this section, we provide a brief review of the concept of spatial coherence, a property that will be used in several parts of this review. We illustrate this concept through the famous Young’s two-slit experiment (see Figure 1).

Figure 1:

Schematic of Young’s double-slit setup.

Here, r₁ and r₂ represent the distances from the source to the upper and lower slits, respectively. The distances from the upper and lower slits to the detector D are represented by d₁ and d₂, respectively.

The double-slit experiment was introduced by Thomas Young to illustrate the wave nature of light. This beautiful experiment has also been used to quantify spatial coherence of light. We show a simplified version of this experiment in Figure 1. We assume that a double-slit structure is illuminated with a quasi-monochromatic source of light. At time t, the total electric field at the detector E_d(r, t) is given by

which is the sum of the field amplitudes E(r₁, t₁) and E(r₂, t₂) produced by each of the slits. Here, t₁=t−d₁/c and t₂=t−d₂/c represent the times at which the photons leave the slits, and k₀=ω/c is the wavenumber in vacuum. The values for a₁ and a₂ are defined by the geometry of the slits. Then, the intensity I_d measured by the detector D is given by

where I₁=⟨E*(r₁, t₁)E(r₁, t₁)⟩ and I₂=⟨E*(r₂, t₂)E(r₂, t₂)⟩. The notation ⟨···⟩ represents an ensemble average. These quantities can be used to introduce the mutual coherence function

Indeed, it is possible to express Eq. (30) as

The first two terms in Eq. (32) are transmission contributions from the first and second slits, respectively. Furthermore, the last term in Eq. (32) describes interference. The intensities I₁(r₁, t₁) and I₂(r₂, t₂) provide information about self-field correlations. These can be described by the functions of first-order coherence Γ⁽¹⁾(r₁, r₁, t₁, t₁) and Γ⁽¹⁾(r₂, r₂, t₂, t₂). Similarly, the mutual-field correlations can be described by the function of first-order coherence Γ⁽¹⁾(r₁, r₂, t₁, t₂). It is worth noting that the interference fringes are formed when the length of the spatial coherence of the illuminating beam is larger than the separation between the slits. In other words, interference fringes are produced when the properties of light are similar at the spatial locations defined by the two slits. The quality of the formed fringes can be quantified through Rayleigh’s definition of fringe visibility,

where I_max and I_min are the maximum and minimum intensity values in the interference pattern, respectively. The visibility V is equal to zero for incoherent sources. Furthermore, V=1 describes a coherent source. Also, sources of light characterized by visibilities in the range 0≤V≤1 are considered partially coherent.

A quantum formulation of coherence can be constructed using similar ideas to those described above. In this regard, we introduce the general first-order correlation function [42], [44], [45],

where ρ^ is the density matrix of a quantum state, and E^(+) is the electric-field operator and E^(−)=[E^(+)]†. In addition, the normalized first-order correlation function is defined as

As discussed above, the first-order coherence function can be used to determine the spatial coherence of the electromagnetic field. However, additional information can be gained through the implementation of intensity correlations. In this regard, in 1956, Hanbury Brown and Twiss (HBT) performed a novel interference experiment through the use of measurements of intensity correlation [57]. The original HBT stellar interferometer was designed to determine diameters of stars [58]. This experiment used two detectors located at different positions on Earth that collected light produced by independent sources on the disc of a star.

A simplified schematic of the HBT experiment is shown in Figure 2. Here, two detectors D₁ and D₂ are placed at the same distance from the beam splitter. The setup measures intensity correlations as a function of the time delay between the signals generated by the two detectors. Here, the coincident count rate is given by

Figure 2:

Diagram of the HBT interferometer.

A pseudothermal beam of light is passed through a beam splitter (BS) and measured by two detectors D₁ and D₂. The time delay between the two detectors is controlled in this experiment, and the output signals produced by both detectors are correlated. This measurement is equivalent to the implementation of correlations of intensity fluctuations.

where I(t₁) and I(t₂) are the intensities measured by the two detectors D₁ and D₂. The generic function for second-order coherence is defined as

which describes a statistical average of the product of intensities associated to the fields E(t₁) and E(t₂). In general, the fields are detected at two different spatial and temporal positions. For practical purposes, we use the normalized version of the classical second-order coherence function

where τ=t₁−t₂ is the time delay between the two light beams, and it is smaller than the coherence time of the source.

Similar to the first-order quantum coherence function, we can introduce the second-order quantum coherence function [42], [44], [45],

and the normalized second-order quantum coherence function, g⁽²⁾, is given by

Notably, for a single-mode field, it is possible to reduce Eq. (40) to

We can observe that Eq. (41) does not depend on the time difference τ. The second-order quantum coherence function thus becomes a powerful tool to probe the underlying statistical properties of light.

It is worth noting the values of g⁽²⁾(0) for different light sources. For an attenuated laser, which is described by coherent states |α⟩, g⁽²⁾(0)=1. Furthermore, a single-mode thermal state ρ^th is characterized by g⁽²⁾(0)=2. Indeed, any classical electric field satisfies g⁽²⁾(0)≥1. For photon number states (Fock states) represented by |n⟩, for a situation in which n≥1, it can be shown that g⁽²⁾(0)=1−1/n. Particularly, for a single-photon state |1⟩, one expects g⁽²⁾(0)=0. By comparing the aforementioned examples, we conclude that the measurement of g⁽²⁾ can be used to characterize nonclassical properties of light [59]. Later in this review, we will discuss how recent work on quantum plasmonic relies on the measurements of first- and second-order quantum coherence.

2.4 Hong, Ou, and Mandel (HOM) interference

In 1987, Hong, Ou and Mandel, unveiled the fundamental physics behind bosonic interference through an emblematic experiment that cannot be explained using the classical theory of light [60]. As illustrated in Figure 3, in this experiment, two indistinguishable photons are injected into the input ports of a beam splitter. The pair of photons is described by the state |1, 1〉in=a^in†b^in†|0, 0〉in, where a^in† and b^in† denote the two modes of the injected photons. Following the transformation of a 50/50 beam splitter, we have

Figure 3:

Schematic representation of the HOM two-photon interference experiment.

Two single photons with a relative time delay τ are injected into the input ports of a 50/50 beam splitter (BS) and measured by two detectors D₁ and D₂. The signals produced by the two detectors are analyzed using a correlation circuit.

According to Eq. (42), the two interfering photons are expected to emerge through the same output port of the beam splitter if these were simultaneously injected. Thus, one should observe photon bunching and the absence of simultaneous photon events. It is worth noting that the bunching effect depends on the degree of indistinguishability of the interfering photons. In the famous HOM experiment, the single-photon pairs were generated through spontaneous PDC (SPDC) in a source with a spectral bandwidth Δω. In this case, the coincidence probability as a function of time delay τ is given by

At τ=0, the two single photons are perfectly indistinguishable and the HOM dip is observed. Nowadays, the HOM experiment represents one of the basic tools to control multiphoton processes. In addition, it also serves as a platform to test indistinguishability among photons.

2.5 Quantum entanglement

Entanglement constitutes one of the most remarkable consequences of quantum mechanics. This interesting property of quantum mechanical systems was used as an argument to question the validity of quantum mechanics. Indeed, its origin can be traced back to 1935 when Einstein, Podolsky, and Rosen (EPR) pointed out its nonlocal nature [61]. The arguments presented by EPR aimed to demonstrate that quantum mechanics was an incomplete theory. In their seminal paper, EPR analyzed a system of two distant particles entangled simultaneously in their position and momentum properties. They pointed out that, in a system with these properties, one could perform a measurement of either position or momentum of one of the particles and infer, with complete certainty, either the position or the momentum, respectively, of the unmeasured particle. In their emblematic “gedanken” experiment, the two distant particles do not interact; thus, the possibility of inferring information of a distant particle would imply that the position and momentum of the unmeasured particle were simultaneous realities, leading to a violation of Heisenberg’s uncertainty principle. Remarkably, over the past 25 years, a series of systematical experimental tests have demonstrated that entanglement is in fact a real property of quantum mechanical entities such as molecules, atoms, and photons [62], [63], [64].

Notably, SPDC offers the possibility of generating pairs of entangled photons optically [62], [63], [65]. As depicted in Figure 4, SPDC is a χ⁽²⁾ nonlinear process in which one pump photon is annihilated to generate entangled photon pairs [63]. In Figure 4B and C, linear momentum and energy are conserved in this nonlinear process. As shown in Figure 4A, a pair of entangled photons is generated at the same crystal position; consequently, the photons are correlated in the variable of linear position. In Figure 4A and B, the conservation of linear momentum forces the two beams to propagate with opposite spatial frequencies [67]. Thus, SPDC photons are characterized by opposite transverse wavevectors that induce anticorrelations in linear momentum. In Figure 4C, the energy conservation results in the wavelengths of the pump, signal, and idler photons satisfying 1/λ_p=1/λ_s+1/λ_i. These conditions enable the generation of photons entangled in multiple degrees of freedom, such as in energy and time, angular position, and orbital angular momentum and polarization [68]. Below, we discuss experiments that rely on these forms of entanglement.

Figure 4:

Spontaneous parametric down-conversion (SPDC) process and the conservation of momentum and energy.

(A) The process of SPDC is produced by pumping a χ⁽²⁾ nonlinear crystal. SPDC photons are generated at the same spatial position in the crystal; consequently, photon pairs are correlated in the variable of spatial position. The conservation of linear momentum shown in (B) forces a pair of SPDC photons to be anticorrelated in the variable of linear momentum; consequently, photons located in opposite positions of the down-conversion cone show nonclassical correlations. (C) The process of SPDC satisfies the conservation of energy; in this case, a blue photon is annihilated to create two red photons. Reproduced from Ref. [66].

3.1 Classical picture

As indicated earlier, surface plasmons are surface electromagnetic waves formed at the interface between a metal and a dielectric due to the interaction of surface charges and an electromagnetic field. These waves can propagate along the surface of a metal (SPPs) or can be localized to metallic nanoparticles (LSPs). For instance, consider the SPP of a single planar interface between a metal and a dielectric with permittivities, ε_m and ε_d, respectively, as shown in Figure 5A and B. Solving Maxwell’s equations with the appropriate boundary conditions yields the following expression for the SPP wavenumber [69],

Figure 5:

Electromagnetic properties of plasmonic structures.

(A) Illustration of the electric field lines and surface charge distribution associated with the SPP supported by a metal-dielectric interface. (B) Magnitude of the electric field across the interface showing the extend of electric field confinement. The penetration depth into the metal (δ_m) and dielectric (δ_d) characterize the SPP mode confinement. (C) Dispersion relations of SPPs (red line) and photons (blue line). The wavenumber of an SPP is larger than that of a photon in the dielectric medium and can acquire large values near the resonance frequency asymptote, ω_as. (D) Illustration of the electric field lines, E, associated with the LSP excited on a metallic nanosphere by an incident electric field, E₀. (E) Schematic illustration of the charge distribution in a metal disk (top) and a circular aperture (bottom) due to an incident electric field, E₀. (F) Calculated near-field distribution in the x, y-plane for a gold disk (top row) and a circular aperture (bottom row) having a diameter of 150 nm and a thickness of 30 nm. The calculations were carried out using the finite-difference time-domain method using a y-polarized plane wave excitation at a wavelength of 750 nm. The white circle indicates the position of the nanostructures.

where k₀=ω/c is the magnitude of the wavevector in free space. K is a complex quantity because ε_m is generally a complex quantity. The real and imaginary parts of K can be approximated as

where the prime and double prime symbols indicate real and imaginary parts, respectively. The approximation is valid when |ε′m| ≪ |ε″m|, which is a reasonable approximation for noble metals in the visible and near-infrared region of the spectrum (away from interband transitions). The real part, K′, is related to the phase evolution of the SPP and hence defines its dispersion relation (Figure 5C). The expression for K′ predicts a significant increase in the SPP momentum at frequencies near the resonance-frequency asymptote ω_as, which is the frequency at which ε′m=−εd, i.e. the frequency at which the denominator in Eq. (45) vanishes. Also, note that because of their surface-wave nature, SPPs possess wavenumbers that are larger than those of photons in the dielectric for a given frequency. Hence, momentum matching mechanisms, such as grating coupling or prism coupling, are required to excite them with photons. In contrast, the imaginary part K″ is associated with the SPP field attenuation, which limits the propagation range of the wave. The propagation length of the SPP is defined as the distance over which the field intensity is reduced by a factor of e⁻¹ and is given by ℓSPP=1/2K″.

The SPP electric field is p-polarized [electric field oscillation confined to (z-x) plane] and its distribution has the form

where the subindex jε{m, d} identifies the medium. Here, we have taken z as the direction normal to the metal-dielectric interface and x as the propagation direction. Note that the fields decay exponentially away from the interface and have their maximum amplitude right at the interface (Figure 5B). The degree of confinement to the metal-dielectric interface is determined by the penetration depth into the two media, δ_j=1/k_zj. Here, k_zj is the normal wavevector component in medium j and is given by

Near the resonance-frequency asymptote, the magnitude of k_zj can be very large, resulting in a subwavelength mode confinement. The degree of mode confinement is only limited by the losses of the system, which increase significantly near the resonance-frequency asymptote.

Variants of this single-interface plasmonic structure described here may include metal films or stripes bounded by dielectrics [70], [71]. Such variants support SPP modes that share similar characteristics to the single-interface SPP discussed here. Nevertheless, the additional degrees of freedom associated with those structures allow them to support various modes with particular field symmetries [72], [73], [74], [75].

For metallic nanoparticles with dimension smaller or comparable to the skin depth of the metal, the electric field can penetrate the nanoparticle and excite directly LSPs [69]. As these nanoparticles typically have dimensions much smaller than the wavelength, their electromagnetic response is well described by the first-order dipolar contribution. Furthermore, their excitation can be described under the quasi-static approximation, in which retardation effects are neglected and a constant phase over the particle is assumed. Using this approximation, the electromagnetic response of the particle takes the form of a static electric dipole with a harmonic time dependence [76]. The dipolar charge distribution (see Figure 5D) induced in the particle by the external electric field, E₀, is described by the electric dipole moment p=αε_dE₀, where α is the electric polarizability of the particle. For the case of a spherical nanoparticle, one obtains [76]

where V is the volume of the nanoparticle. In this case, the LSP resonance occurs at the frequency that satisfies the relation ε′m=−2εd. Clearly, for the case of a spherical nanoparticle, the resonance frequency only depends on the material properties. This is not the case for other particle shapes. For instance, the electric polarizability of a prolate ellipsoidal nanoparticle is given by [76]

where L_j is a function of the aspect ratio (the ratio of the long axis to the short axis) of the nanoparticle along the axis jε{x, y, z}. Hence, its resonance frequency depends on the aspect ratio through L_i. In general, the LSP resonance frequency critically depends on the shape and dimensions of the nanoparticle, enabling a broad tunability across the visible and near-infrared spectrum [77].

At the LSP resonance frequency, a large electric field develops in the nanoparticle due to the large polarizability. Similar to the SPP, the LSP fields peak at the metal surface and decay exponentially away from it. The field enhancement factors exhibited by single nanoparticles, due to this resonant behavior, can vary significantly depending on their shapes. These range from a modest enhancement in gold nanospheres to an enhancement of several orders of magnitude in particles with sharp edges.

LSPs can also exist in subwavelength apertures constructed on thin metallic films. These plasmonic modes are similar to the LSPs supported by nanoparticles in the sense that their electromagnetic response depend critically on geometrical factors. For example, size and shape of the aperture as well as the thickness of the metal film and the surrounding dielectric. The near fields of LSPs supported by planar particles and apertures can be qualitatively related by Babinet’s principle of complementarity [78], [79], which results in electric and magnetic near-fields with complementary distributions for both types of structures. For instance, Figure 5E shows schematically the charge distribution induced by a y-polarized electric field, E₀, on a metallic disk and a circular aperture. Dipole moments with opposite directions develop in the nanostructures because the field lines in the metallic region oppose the external field E₀. Figure 5F shows the magnitude of the main electric and magnetic field components, E_y and H_x, of the LSPs supported by a gold disk and a circular aperture with the same diameter (150 nm) and thickness (30 nm). These results clearly show the complementarity of the spatial distribution of electric and magnetic fields [79]. For a more detailed review on the electromagnetic properties of plasmonic nanoholes, the readers are referred to Refs. [80], [81].

3.2 Quantization of surface plasmons

Various methods to quantize surface plasmon fields have been proposed in the past. Elson and Ritchie [82] developed the first quantum theory for SPPs. The first microscopic theory to quantize the matter field and the light field was given by Huttner and Barnett [83], extending the Hopfield approach [84] to include material losses. Also, a macroscopic quantization method was recently proposed by Philbin [85] using Green’s functions. Here, we briefly describe two basic quantization methods for SPPs and LSPs [86], [87], which start from the classical macroscopic description of the fields accounting for the dispersive nature of plasmonic material systems. We consider first the case of an SPP mode supported by the metal-dielectric interface illustrated in Figure 5A. Using Coulomb’s gauge, the electric and magnetic fields can be obtained from the vector potential, A(r, t), which can be cast in the following form:

Here, K is the SPP wavevector and α_K is the corresponding complex amplitude associated with each of the K vectors. At this point, we ignore the losses of the system and assume that K is a real valued quantity. The effect of losses will be incorporated later in this procedure, rendering K a complex quantity as it is physically required. The vector, u_K, which represents the plasmonic mode, can be written as

where L(ω) is a normalization constant with dimensions of length. K^ and z^ are unit vectors along the x and z axes, respectively. Now, consider a square of sides L_x and L_y in the x-y plane whose area, S=L_x×L_y. This leads to quantized values of wavevector K_x=n_x2π/L_x and K_y=n_y2π/L_y, where n_x, n_y∈ℤ; now, Eq. (50) can be cast into a summation as

Using this expression and recasting the electric and magnetic fields in the expression for free space energy density [76], u=1/2(ε₀E(r, t)²+1/μ₀B(r, t)²), gives the total energy of the surface wave [86]

Note that, for each of the modes labeled by K, the energy of the surface wave has the same form as that of the energy of a harmonic oscillator. Hence, by mapping Ak→ℏ2εoωSa^K and A*k→ℏ2εoωSa^K† and using the Hamiltonian of a harmonic oscillator, we obtain the Hamiltonian of the system H^=∑Kℏω2[a^Ka^K†+a^K†a^K]. As in the formalism of the harmonic oscillator [43], a^K is the annihilation and a^K† is the creation operators as defined in Eq. 11 and Eq. 12, such that a^K†|nk〉=nK+1|nk+1〉 and a^K|nk〉=nK|nk−1〉. The fields can now be written as operators acting on the eigenstate |n_k⟩ as

where bK=(K+γiz^)×uK and H.c is the Hermitian conjugate.

The effect of losses due to material dispersion can be incorporated into the formalism by including interactions of the SPP with a continuum of bath modes [20]. Here, the interaction of the SPP with the bath modes upon propagation modifies the SPP wavevector, K, such that it accounts for the complex permittivity of the metal [20].

The quantization of LSPs in metallic nanospheres is carried out by considering a nanoparticle as an optical resonator and by finding its quantized optical modes. The discrete modes are identified as localized solutions of Maxwell’s equations [87]. For instance, the electric field distribution associated with the resonant modes of a nanosphere with permittivity ε_m and radius d, at position r with respect to its center, is given by

where i^={x^, y^, z^} represents the unit vector along different coordinates. The quantized electric field operator is

Here, a^i is the bosonic field operator [87] and

is the mode volume, which depends on the dimensions and material properties of the nanosphere. As for the SPP case, the losses of the system can be incorporated in the quantization through the interaction of the fields with a reservoir of bath modes. Other more rigorous approaches based on Green’s function formalism have been proposed, enabling the quantization of surface plasmons in more general three-dimensional systems, including the effect of losses, dispersion, and material inhomogeneities [88], [89].

3.3 Quantum effects in plasmonic systems

An important property of plasmonic systems is their capacity to enhance the spontaneous emission rate of quantum emitters, such as molecules or quantum dots. This enhancement, known as the Purcell effect, results from the large density of electromagnetic states existing in the close vicinity of plasmonic structures [90], [91], [92], [93], [94]. The Purcell enhancement factor, which is inversely proportional to the mode volume, can be extremely large in plasmonic structures because of their ability to confine the fields to subwavelength dimensions [95], [96], [97]. As a result, plasmonic systems have been recognized as important candidates for developing novel single-photon sources with enhanced efficiency [98], [99], [100]. Within this context, a recent analysis reported that the theoretical limit for photon-rate enhancement in plasmonic systems can be two orders of magnitude larger than those in their photonic counterparts (i.e. dielectric cavities) [101]. Indeed, these results together with the recent demonstration of room-temperature strong coupling between quantum emitters and LSPs [102] suggest an exciting road ahead for the development of efficient single-photon sources. Although the topic of plasmon-enhanced single-photon emission is quite relevant in quantum plasmonics, a review of this topic is beyond the scope of this work. For more details about this subject, we refer the reader to the following references [23], [95], [103], [104], [105], [106].

It has also been demonstrated over the past decades that surface plasmons show exotic quantum properties just like photons. Remarkable examples include wave-particle duality [107], nonlocality [108], [109], [110], [111], [112], [113], [114], entanglement [115], [116], tunneling [117], [118], [119], bunching and antibunching [120], [121], etc. However, the possibility of preserving quantum mechanical properties in scattering interactions among photons and plasmons has attracted particular attention from multiple research communities in recent years. As described below, this peculiar feature of hybrid photonic-plasmonic systems offers new mechanisms to control fundamental properties of quantum many-body systems [115], [116], [122].

The wave-particle duality is one of the fundamental properties of quantum mechanical systems. In this regard, plasmons have shown self-interference and antibunching effects that demonstrate their undulatory and corpuscular behaviors, respectively [107]. Furthermore, plasmons have also shown wave-particle duality simultaneously [120]. Last but not least, quantum tunneling has been observed in plasmonic systems, for cases in which the separation between nanoparticles is of the order of few angstroms. It has also been shown that tunneling effects can modify the optical response of plasmonic nanogaps in dramatic fashions [21]. There has been an enormous interest in exploring wave-particle duality and quantum tunneling in plasmonic systems. The reader can find additional discussions in the review articles by Zhu et al. [22] and by Xu et al. [23].

4.1 Multiphoton and multiparticle interactions

As illustrated in Figure 6A, multiparticle scattering can be controlled through electromagnetic near-field effects [32], [33], [123], [124], [125], [126]. Remarkably, the possibility of manipulating quantum systems at this fundamental level represents one of the most important motivations behind the field of quantum plasmonics and, in general, behind the field of quantum optics [31], [127]. Indeed, the performance of a significant number of quantum protocols depends on the control of complex quantum interactions in systems composed of multiple particles [128], [129], [130], [131], [132], [133]. In fact, the future of multiple quantum photonic technologies for information processing depends, to an important extent, on the level of control of quantum effects in multiphoton systems [134]. In this regard, photonic-plasmonic systems provide a flexible platform to investigate quantum multiparticle dynamics. The interest in these hybrid platforms has been triggered by recent progress in the generation of nonclassical multiphoton sources [32], [33], [115], [116], [135], [136], [137], [138], [139], [140], [141], [142], [143], [144], [145]. Although, the losses in plasmonic systems and the low photon fluxes that characterize typical quantum sources of light impose challenges to the field of quantum plasmonics. Hitherto, these problems have been partially alleviated using multiple single-photon sources. However, there have been recent breakthroughs in the development of nonclassical single-photon and multiphoton sources. Indeed, the improvement in the generation and engineering of mesoscopic states of light have enabled the investigation of quantum properties of hybrid photonic-plasmonic systems at the mesoscopic scale. For example, in 2016, Harder et al. demonstrated a source for mesoscopic quantum optics with mean photon numbers above 20. This source used SPDC processes in nonlinear waveguides [146]. Later, Magaña-Loaiza et al. demonstrated a tunable multiphoton source that enabled the engineering of quantum statistics and nonclassical correlations of entangled multiphoton states [147].

Figure 6:

Multiparticle scattering in hybrid photonic-plasmonic systems.

(A) Concept behind the control of multiphoton dynamics through the manipulation of electromagnetic near-fields. Interestingly, multiphoton interference in multiport devices can be used to investigate fundamental physics behind multiparticle scattering. (B) Summary of the importance of indistinguishability and the overall phase φ for two- and three-photon interactions. (i) Concept of indistinguishability for two photons. (ii) A simplified version of the HOM experiment. (iii and iv) Process of three-photon interference and the idea behind the collective phase φ, respectively. Reproduced from Ref. [123]. (C) Remarkably, these effects can be manipulated in plasmonic nanostructures through the engineering of metallic gratings and slits. This hybrid quantum device enables the possibility of engineering multiparticle scattering and consequently multiparticle interference. (D) Photon-plasmon scattering has been demonstrated in hybrid photonic-plasmonic tritters [(i) and (ii)]. The plasmonic tritter couples a photonic mode a^ to two plasmonic modes b^1 and b^3. Reproduced from Ref. [33].

The large variety of scattering events that can take place in many-body systems of photons offers new paths to perform quantum information processing. Interestingly, the complexity and control of underlying scattering events can be boosted through the enhancement of electromagnetic near-fields by means of surface plasmons. In general, surface plasmons offer the possibility of achieving novel light-matter interactions [127]. Below, we review recent research achievements along this research line. However, for the sake of clarity, we first describe underlying ideas behind multiparticle interactions.

In Section 2.4, we revisited the HOM experiment, where two photons injected into the input ports of a 50/50 beam splitter interfere and exhibit photon bunching. As illustrated in Figure 6B (i), the distinguishability of two photons in states |ϕ_i⟩ and |ϕ_j⟩ can be quantified through the inner scalar product of the two states as 〈ϕi|ϕj〉=rijeiφij. Here, the modulus of r_ij can be interpreted as a measure of distinguishability between the two photons. The probability of detecting one photon in each of the output ports of a beam splitter only depends on the degree of distinguishability as P11=12(1−rij2). In general, the argument eiφij does not play a relevant role in two-photon interference. Surprisingly, quantum interactions in systems with more than two photons cannot be exclusively described by the degree of indistinguishability between pairs of photons [125], [148], [149]. In 2017, Menssen et al. demonstrated that particle indistinguishability is not enough to describe multiphoton interactions involving more than two photons [123]. In this work, the team unveiled the complexity behind multiphoton interference in a quantum mechanical tritter, a multiport device with three inputs and three equally likely outputs. As shown in Figure 6B (iii), three-photon interference shows a strong dependence on the collective triad phase defined as φ=φ₁₂+φ₂₃+φ₃₁ [Figure 6B (iv)]. Generally, three-photon interference can be described as

These quantum interactions in photonic systems can be controlled through plasmonic effects. As illustrated in Figure 6C, the series of scattering events induced by photon-plasmon-photon conversions can be used to control global phases and thus multiphoton interference in quantum many-body systems. These ideas have been discussed in the context of plasmonic multiport devices in metallic slits (see Figure 6D). As discussed below, these mechanisms also offer the possibility to preserve and manipulate multiphoton coherence [121], [127], [150], [151], [152], [153], [154]. Later in this review, we will describe recent demonstrations of quantum control in hybrid photonic-plasmonic systems.

4.2 Quantum statistics of photonic-plasmonic systems

As previously discussed, plasmonic effects can be used to control the quantum interactions in quantum systems. Despite the inherent losses that characterize quantum plasmonic systems, properties in photon-plasmon scattering processes offer new mechanisms to exert control of multiparticle systems [115], [116], [136], [141]. For example, in 2009, Huck et al. demonstrated that quadrature squeezing was preserved in the photon-plasmon-photon process even in the presence of linear loss and decoherence [137]. In their experimental setup (Figure 7A), squeezed light was generated in an optical parametric oscillator and then coupled to a gold stripe. Then, the scattered photons from the plasmonic setup were collected and analyzed to certify squeezing. In Figure 7B, one can observe that, although quadrature squeezing is degraded, the squeezing was coherently transferred through the photon-plasmon-photon process. It was further confirmed that plasmonic interactions can be described by an unitary beam splitter transformation [20], [137].

Figure 7:

Experiments that demonstrate the preservation of quantum statistics in hybrid photonic-plasmonic systems.

(A) Schematic of the experimental setup used for demonstration of quadrature squeezing preservation. OPO, optical parametric oscillator; PPKTP, periodically poled potassium titanyl phosphate crystal; SV, squeezed vacuum. (v) Experimentally generated and transmitted squeezed vacuum states after a photon-plasmon-photon process. The quadrature of the electric field, absolute values of the reconstructed density matrices, Wigner functions, and noise power relative to the shot noise level of input and output are also presented. (A and B) Reproduced from Ref. [137]. (C) Schematic of the setup used for the experimental demonstration of the preservation of quantum statistics in plasmonic waveguides. (D) Conditional second-order quantum coherence functions g⁽²⁾(τ) for light with nonclassical and classical statistical properties. The dotted blue line illustrates the classical limit. Right, corresponding two-photon number distributions. (c and d) Reproduced from Ref. [138]. (E) Schematic of the experimental setup for plasmonic control of the Pancharatnam-Berry geometric phase. This is demonstrated through the interference patterns in (F). The shift of interference fringes unveils the possibility of preserving geometric phases in plasmonic systems. AP, aperture; BS, 50/50 beam splitter; L, lens; M, mirror; ND, neutral density filter; OB, microscope objective; P, linear polarizer; SP, spatial filter; Q, quarter-wave plate. (e and f) Reproduced from Ref. [155].

More recently, Martino et al. characterized the role that losses play in the quantum statistics of the electromagnetic field in plasmonic waveguides [138]. In their setup (Figure 7C), a type I SPDC source was used to generate entangled photon pairs. The team used one of the generated modes to excite an SPP mode in the thin metallic stripe waveguide. By measuring the photon number distribution of multiple output states and the second-order quantum coherence functions g⁽²⁾(τ) (Figure 7D), it was demonstrated that losses in quantum plasmonic systems behave similarly to that predicted by the classical theory of uncorrelated Markovian linear loss. Furthermore, due to the fact that photon-plasmon-photon processes do not modify the second-order coherence function g⁽²⁾(τ), it is possible to fabricate larger SPP waveguide structures for quantum control.

Undoubtedly, the phase associated to quantum states plays a fundamental role in multiple applications [156]. In the context of quantum control, the manipulation of sequential photon-plasmon scattering processes in plasmonic gratings has enabled the control of geometric phases in single-plasmon systems. The phase of quantum systems plays an important role in quantum interactions, particularly in interference effects [123]. In this regard, Daniel et al. showed that photon-plasmon-photon processes preserve the well-known Pancharatnam-Berry geometric phase [155]. This study was carried out through a Mach-Zehnder interferometer (Figure 7E). The averaged and fitted data for different polarization states are shown in Figure 7F. These figures illustrate that the shifts of the peaks are proportional to the change in polarization, which is an indication of the conservation geometric phase in photon-plasmon-photon processes.

4.3 Preservation of entanglement in plasmonic systems

Entanglement is one of the most remarkable consequences of quantum physics [68]. In addition, it constitutes a critical resource for multiple applications, including quantum imaging [66], [157], [158], [159], [160], quantum key distribution [161], [162], [163], [164], [165], quantum computation [166], [167], [168], and quantum communication [169], [170]. Given the potential of entanglement for the development of quantum technologies, it is always desirable to investigate paths to preserve it in complex processes involving light-matter interactions. The possibility of preserving quantum entanglement in photon-plasmon-photon interactions makes quantum plasmonic systems a promising platform for quantum technologies [23], [171], [172]. In this regard, Altewischer et al. [115] initiated the field of quantum plasmonics with their experimental verification of the preservation of polarization entanglement in plasmonic systems. This seminal work demonstrated the possibility of preserving polarization entanglement in photon-plasmon-photon conversion processes. As shown in Figure 8A, correlated photon pairs generated through SPDC were transmitted through an array of holes in a gold film where the photon-plasmon-photon conversion takes place. The team certified entanglement between two detected light beams. This experiment demonstrated the possibility of generating entanglement between plasmonic and photonic modes. Although entanglement is preserved, the team also showed that, when focusing one of the beams onto a single hole in the metallic structure, the degree of entanglement decreased. This was observed through a reduction in the visibility associated to biphoton interference.

Figure 8:

Experiments demonstrating entanglement preservation in plasmonic systems.

(A) First experiment that demonstrated plasmon-assisted transmission of entangled photons. This was confirmed through the measurement of two-photon quantum interference after photon-plasmon-photon conversion, which demonstrated the preservation of entanglement. The experimental setup used a β barium borate (BBO) nonlinear crystal to generate entangled photon pairs that were passed through an array of gold. The generated light beams were detected by two single-photon detectors. C, compensating crystals; HWP, half-wave plate; IF, interference filters; L, lens; P1 and P2, polarizers; TEL, confocal telescope. Reproduced from Ref. [115]. (B) Schematic of the experimental setup demonstrating energy-time entanglement in a plasmonic system. APD, avalanche photodiode; DM, dichroic mirror; NL, nonlinear crystal; PC, polarization controller; TAC, time to analogue converter. (C) Schematic of the LR-SP waveguide. BCB, benzocyclobutene. (D) Experimental interference fringes produced by the setups shown in (B) and (C). V_p represents the visibility of the interference fringes after plasmonic conversion. Furthermore, V_net describes the visibility in the absence of plasmonic conversion, and T is the transmittance of the plasmonic setup. (B–D) Reproduced from Ref. [116].

This work stimulated new fundamental research. For example, Moreno et al. [173] reported a quantum description of the underlying physics behind plasmon-assisted transmission. They theoretically showed that the degradation of entanglement arose from the polarization sensitivity of gold gratings. Furthermore, the scattering theory of plasmon-assisted entanglement transfer and distillation was introduced by van Velsen et al. [174]. Since then, the preservation of entanglement has been further explored by different groups under multiple experimental conditions [136], [140], [175]. Fakonas et al. showed path entanglement between surface plasmons [176]. Ren et al. further demonstrated that entanglement based on orbital angular momentum is preserved in photon-plasmon-photon processes [139]. Last but not least, Asano et al. showed that plasmonic metamaterials can be used for entanglement distillation [177].

Particularly, in 2005, Fasel et al. showed that energy-time entanglement was also persevered in photon-plasmon-photon scattering processes [116]. In their experiment, the preservation of energy-time entanglement was demonstrated using subwavelength metallic hole arrays and long-range surface plasmons (LR-SP) propagating in metallic waveguides. Their setup shown in Figure 8B used two different wavelengths of 810 and 1550 nm. The metallic hole arrays were carefully designed to maximize photon-plasmon-photon conversion. In addition, the setup for LR-SP waveguide shown in Figure 8C was used to confirm photon-plasmon-photon scattering. This arrangement enabled the induction of time delays that are essential to test energy-time entanglement. The interference visibility shown in Figure 8D demonstrated preservation of energy-time entanglement in photon-plasmon-photon processes.

The experiments described above exploited quantum interactions among photons in plasmonic structures with overall sizes larger than their wavelengths λ. More recently, smaller metallic structures with sizes comparable to λ have been used to perform quantum plasmonics. Li et al. experimentally demonstrated that polarization entanglement can be maintained in these metallic structures [178]. Their setup shown in Figure 9A used a type I SPDC source and a nanoscale hybrid plasmonic waveguide to preserve quantum polarization entanglement. The generated polarization entangled two-photon Bell state |Φ+〉=1/2(|HH〉 +|VV〉) was tuned to achieve an output fidelity of 0.932 after the photon-plasmon-photon conversion. Additionally, the team verified entanglement by performing a series of projective measurements. This was confirmed by the measured fourth-order quantum interference structure shown in Figure 9B. This work was further explored through a demonstration of the propagation of polarization entangled two-photon N00N states in a nanowire [179]. In their work, Chen et al. generated the polarization entangled N00N state |N00N〉=1/2(|2H, 0V〉 −|0H, 2V〉), which was coupled into a nanowire. The team verified the preservation of the N00N state using two-photon HOM interference shown in Figure 9C. The team was able to observe visibilities of 0.737±0.007 and 0.880±0.013 for single- and two-photon N00N state input, respectively. In Figure 9D, it is possible to identify the doubled oscillation period produced by particles prepared in a N00N state, where N=2.

Figure 9:

Experiments that demonstrate the preservation of entanglement in plasmonic systems.

(A) Experimental setup used for polarization entanglement in a nanoscale hybrid plasmonic waveguide. In this case, one of the correlated photon pairs is sent to a silver nanowire, whereas the other photon partner is sent to a module for structures state tomography (QST). (B) Quantum interference structures observed through correlation measurements demonstrate the preservation of entanglement. Blue and red dots represent coincidence rates for photons projected onto different polarization states. (A and B) Reproduced from Ref. [178]. (C) Experimental setup for the demonstration of quantum plasmonic N00N states in a silver nanowire. The generated N00N state is transmitted and preserved in the nanowire. The transmitted photons are then forced to interfere in an HOM setup (ii). The arrangement used to measure the de Broglie wavelength of biphotons (iii). The transmitted two-photon entangled state is characterized by quantum state tomography (ii). (D) Two-photon coincidence counts and single-photon counts of the plasmonic N00N state transmitted in (C). The red curve represents the two-photon coincidence, whereas the blue curve represents single-photon counts. (C and D) Reproduced from Ref. [179]. (E) Schematic of the experimental setup for the symmetry protection of entanglement in a single nanoaperture. (F) Quantum tomography of the state |Ψ₊⟩ and |Ψ₋⟩ with and without the nanoaperture interaction. (E and F) Reproduced from Ref. [180].

Furthermore, Büse et al. demonstrated symmetry protection of quantum entanglement through the interaction with a single nanoaperture [180]. As shown in Figure 9E, the group of scientists used Bell states that were strongly focused on a metallic nanoaperture to demonstrate the potential of dissipative near-fields to preserve quantum coherence.

Indeed, the protection of two-photon entangled states based on polarization exploited the polarization sensitivity of a metallic nanoaperture. Interestingly, this simple system acts as a lossy beam splitter that endowed this protocol with a unique sensitivity to the relative quantum phase between the entangled modes [127], [181], [182]. The experimental results that demonstrate symmetry protection are presented in Figure 9F. The quantum tomography of the transmitted state suggests that the state |Ψ₋⟩ is unaffected, whereas the state |Ψ₊⟩ is mixed with |Ψ₀⟩, where

and a^0,+/−†|0〉→|R/L〉.

5.1 Multiparticle quantum interference in photonic-plasmonic systems

Quantum interference has been extensively used as a tool to manipulate quantum mechanical systems [183]. Interestingly, the conditions under which multiparticle interference occurs can be controlled through photon-plasmon scattering. In 2013, Heeres et al. demonstrated the first quantum interference experiment in a plasmonic platform [151]. As shown in Figure 10A, the team used a free-space source of SPDC, a plasmonic directional coupler as beam splitter, and on-chip detectors to perform HOM interference. The ideal HOM dip visibility for their SPDC source is estimated to be V=0.92±0.01. The excited single plasmons interfere on the chip leading to HOM dips with visibilities of the order of V=0.43±0.02 and V=0.39±0.01 for two different plasmonic devices. Typically, the asymmetric losses in photonic-plasmonic systems will decrease the second-order quantum interference visibility. However, careful symmetric designs of plasmonic nanostructures enables one to preserve indistinguishability that leads to high visibilities. As discussed in Section 2.4, the visibility is determined by the degree of indistinguishability between two photons. In this case, reduced visibility was attributed to the excitation of long-range plasmon modes, which are loosely confined plasmonic modes [184].

Figure 10:

HOM interference with different plasmonic structures for beam splitters.

(A) On-chip HOM interference with single plasmons. The team used photon pairs generated through SPDC to excite single plasmons that were forced to interfere in a plasmonic directional coupler. The scanning electron microscope (SEM) image of the gold plasmonic directional coupler device is also shown. (B) HOM interference dip of the photon pair source and quantum interference traces of single plasmons in two devices. (A and B) Reproduced from Ref. [151]. (C) DLSPPWs patterned by polymethyl methacrylate (PMMA) used in Ref. [152]. Reproduced from Ref. [152]. (D) Optical image of the plasmonic beam splitter used in Ref. [121]. The coupling and decoupling gratings of the plasmonic beam splitter consist of 11 ridges. Reproduced from Ref. [121]. (E) SEM image of part of a typical plasmonic DC structure used in Ref. [153]. Reproduced from Ref. [153]. Reproduced from Ref. [154]. (F) Microscope image of the fabricated LRSPP device used in Ref. [154]. (G) SEM top view of the photon-to-SPP launcher used in Ref. [150]. Reproduced from Ref. [150].

Despite the fundamental relevance of this experiment, the nonclassical nature of the interference patterns remains unclear. This is due to the fact that the measured interference patterns did not exceed a visibility of 50%. Nevertheless, this pioneering experiment motivated the development of hybrid quantum networks on chip [185], [186]. Later in 2014, Fakonas et al. used a dielectric-loaded SPP waveguide (DLSPPW; Figure 10C) to achieve an HOM visibility of 0.932±0.01 [152]. In addition, Martino et al. also certified the nonclassical nature of interference effects in plasmonic systems [121]. Similar to Ref. [151], Martino et al. used an X-shaped plasmonic beam splitter to demonstrate an HOM visibility of V=0.72±0.13 (see Figure 10D). These works were followed by multiple experiments that aimed to validate quantum interference in a large variety of plasmonic platforms. For example, Cai et al. observed a visibility of 0.957±0.089 in a similar DLSPPW platform (Figure 10E) [153]. Recently, Fujii et al. observed a visibility of 0.843±0.091 in long-range SPP (LRSPP) waveguides (Figure 10F) [154]. These beautiful experiments indicate that, although surface plasmons are formed from photons (bosons) and electrons (fermions), they indeed exhibit bosonic behaviors in the limit of many-electron regimes. Moreover, quantum plasmonic systems have shown potential for quantum metrology, this due to the importance of quantum interference for metrology. In fact, quantum plasmonic technologies offer a solid platform to implement quantum metrology on chip. Interestingly, the additional quantum interference paths, provided by photon-plasmon scattering processes, have been used to demonstrate coalescence and anticoalescence of a pair of bosons. In 2016, Dheur et al. experimentally verified the wave-particle duality for a single surface plasmon [150]. This experiment made use of asymmetric 11-groove gratings as those illustrated in Figure 10G. By measuring the intensity correlation function of heralded SPPs, they observed single SPPs antibunching in the single-photon-level regime, showing the particle-like nature of SPPs. Moreover, they observed single SPP interference, which certified the wave-like properties of the SPP. This experimental work also revealed fundamental similarities between plasmons and photons.

5.2 Controlling quantum multiparticle interactions

As previously discussed, the control of quantum interference in plasmonic systems has been improved through the use of multiple degrees of freedom of photons and SPPs. Although surface plasmons are well known to suffer from losses, the underlying scattering processes among photons and plasmons provide additional quantum interference paths that are of fundamental importance for controlling decoherence of quantum systems.

Until recently, researchers started to exploit the design of nanostructures as well as their inherent losses to control scattering processes [30], [172], [187], [188], [189]. Inspired by their previous research [150], Vest et al. recently modified the reflection and transmission coefficients of a surface plasmonic beam splitter (SPBS) to demonstrate coalescence and anticoalescence of single SPPs [127]. As shown in Figure 11A, the change in the width w, metal gap g, and groove depth h led to different reflection and transmission factors r and t. Furthermore, the parameters r and t provide a unique control of the phase difference between the reflected and transmitted plasmons. Remarkably, the phase in the system leads to exotic interactions that were measured through the implementation coincidence measurements P(1, 1). Indeed, the coincidence detection probability can be modeled as

Figure 11:

Control of quantum multiparticle interactions.

(A) (i) SEM image of an integrated plasmonic beam splitter. (ii) Cross-section diagram of the integrated plasmonic beam splitter showing the photon-plasmon-photon process. The photons are converted to plasmon on the left, then launched into the plasmonic beam splitter (enclosed by black dashed line), and converted back into photons in the silica substrate. (iii, inset) Grating parameters that are used to control the transmission and reflection of photons in the plasmonic beam splitter. (B) Experimental observation of the plasmonic HOM coalescence/anticoalescence effects. (A and B) Reproduced from Ref. [127]. (C) Control of spatial coherence through surface plasmons in a metallic slit. Reproduced from Ref. [31].

where 2ℜ(t2r2)=t2r*2+t*2r2, and I is the overlap between two particles’ wave packets, similar to the modulus of r_ij described in Figure 6B (i). When two particles’ wave packets overlap (I=1), and when the SPBS is 50/50 (t=±ir and t=1 / 2), the coincidence P(1, 1)=0.

In this case, the completely vanished coincidence resembles the HOM interference effect discussed in Section 2.4, where two single photons coalesce on a lossless beam splitter. Additionally, when the two particles’ wave packets are independent (I=0), the coincidence probability resembles the classical case, where

For the case in which |t|=|r|=1/2 and t=±r, the coincidence probability P(1, 1)=2P_cl(1, 1). This suggests that, instead of getting a dip in the coincidence probability, one would instead get a peak, which indicates the anticoalescence of SPPs. As shown in Figure 11B, two samples of SPBS exhibit coalescence and anticoalescence of SPPs. Particularly, this experiment demonstrated that, although the surface plasmons are typically treated as bosons, they can exhibit fermionic properties.

Additionally, single plasmons can also modulate spatial coherence of optical fields [190], [191], [192], [193], [194], [195]. Using this feature, Li and Pacifici recently achieved continuous modulation of the degree of spatial coherence with amplitudes ranging from 0 to 80% [31]. By varying the slit’s separation, wavelength, and polarization of the incident light beam, they demonstrated the possibility of transforming incoherent optical fields to partially coherent fields and vice versa. As shown in Figure 11C, surface plasmons are generated in a metallic double-slit structure when this is illuminated by light polarized along the incidence plane. These near fields induce a change in the interference pattern formed in the far field of the double slit. These interesting effects demonstrated that, despite the losses in metallic nanostructures, plasmons can be used to enhance spatial coherence of optical fields. Interestingly, the gratings in plasmonics systems can induce phase shifts that can be used to control quantum interactions. Undoubtedly, this possibility enables the control quantum many-body systems of photons.