Quantum Plasmonics

• Quantum plasmonics is the study of quantized surface plasmon excitations at metal–dielectric interfaces, emphasizing nonclassical phenomena and deep subwavelength confinement.
• It employs advanced quantization techniques and models to assess key metrics such as the Purcell factor, vacuum Rabi coupling, and decoherence in nanoscale regimes.
• Applications include integrated quantum photonic circuits, high-fidelity state transfer, and quantum-enhanced sensing validated through experimental protocols like quantum process tomography.

Quantum plasmonics is the study of surface-plasmon modes—collective oscillations of conduction electrons at metal–dielectric or semiconductor–dielectric interfaces and in nanostructures—within a fully quantum framework. This field addresses the fundamental and applied aspects of how quantum optical fields (photons, quantum emitters) interact with quantized plasmonic excitations, and how quantum correlations, coherence, and nonclassical phenomena survive or are engineered in the deeply subwavelength regime. Quantum plasmonics spans fundamental quantum theory, device-level quantum information processing, and the essential quantum corrections to classical plasmonic models. The domain encompasses quantization procedures, quantum state transfer, decoherence physics, nonlocal and nonclassical corrections, platforms for on-chip quantum photonics, and advanced experimental validation.

1. Quantization Frameworks and Fundamental Models

Quantum plasmonics assigns quantum operators both to electromagnetic (photon) modes and to collective charge-density excitations (plasmons). For metal–dielectric interfaces, surface-plasmon polariton (SPP) modes are described by a dispersion relation

$$k_{\rm SPP}(\omega) = \frac{\omega}{c}\sqrt{\frac{\varepsilon_m(\omega)\varepsilon_d}{\varepsilon_m(\omega)+\varepsilon_d}}$$

where $\varepsilon_m(\omega)$ and %%%%1%%%% are the complex permittivities of metal and dielectric, respectively (Tame et al., 2013).

Canonical quantization proceeds via mode expansions (plane waves, confined nanostructure modes, or hybrid dielectric–metal guided modes), promoting the expansion coefficients to bosonic operators. The plasmon field Hamiltonian takes the form

$$H_{\rm pl} = \sum_p \hbar\omega_p\, a^\dagger_p a_p$$

with commutation $[a_p, a^\dagger_{p'}]=\delta_{pp'}$, and field operators constructed from normalized mode functions (Tame et al., 2013, Bozhevolnyi et al., 2016).

The interaction of quantum emitters (QEs) with plasmons is described by an interaction Hamiltonian

$$H_{\rm int} = \hbar \sum_{j,p} g_{jp} (\sigma^+_j a_p + a^\dagger_p \sigma^-_j)$$

where $g_{jp}$ is the single-photon QE–plasmon coupling constant, determined by the emitter dipole and the local plasmonic field (Bozhevolnyi et al., 2016, Tame et al., 2013).

Mode quantization is also fully developed for plasmonic nanostructures of arbitrary shape and loss, using either macroscopic QED (Green’s-tensor quantization, including loss and dispersion (Li et al., 2021, Shahbazyan, 2020)) or eigenvalue formulations based on self-consistent hydrodynamic or density-functional models (Ding et al., 2017).

2. Key Figures of Merit in Quantum Plasmonics

Quantum plasmonics exploits the extremely small effective mode volume $V_{\rm eff}$ of plasmonic modes (often orders of magnitude below $(\lambda/n)^3$), at the expense of moderate to low quality factors $Q=\omega_p/(2\gamma)$ due to strong Ohmic losses (Tame et al., 2013, Bozhevolnyi et al., 2016). This trade-off underlies the central parameters:

Vacuum Rabi coupling $g$: $$g = \frac{\mu\cdot f_p(\mathbf r)}{\hbar}\sqrt{\frac{\hbar\omega_p}{2\varepsilon_0}}$$ for an emitter with dipole $\mu$ at position $\mathbf r$; $f_p$ is the (normalized) local mode function (Tame et al., 2013).

Purcell factor $F$: $$F = \frac{3}{4\pi^2} \left(\frac{\lambda}{n}\right)^3 \frac{Q}{V_{\rm eff}}$$ describing spontaneous emission enhancement into the plasmon mode relative to free space (Tame et al., 2013, Bozhevolnyi et al., 2016).

Plasmon lifetime $\gamma^{-1}$ and propagation length $L_{\rm SPP}$:

Derived from the imaginary part of $\omega_p$ or $k_{\rm SPP}$. Typical values for $Q \sim 10$–$100$ and $L_{\rm SPP} \sim 1$–$20$\,$\mu$m at optical frequencies; can be lower in extremely confined geometries due to Fermi-surface and nonlocal losses (Tame et al., 2013, Ding et al., 2017).

Single-excitation and multiparticle quantum observables:

Second-order coherence $g^{(2)}(0)$, process fidelities, and quantum-state transfer fidelities are used to quantify the preservation of nonclassical statistics and coherence (Mollet et al., 2012, Tang et al., 2019, Hong, 28 Mar 2025).

3. Quantum Corrections: Nonlocality, Tunneling, and Quantum Hydrodynamic Models

As structural features reach nanometer and sub-nanometer scales, classical (local-response) Maxwell theory becomes insufficient. Quantum plasmonics provides a suite of approaches to capture quantum corrections:

Nonlocal hydrodynamic models:

These add Fermi pressure to the electron response, leading to a wavevector-dependent dielectric function

$$\varepsilon_L(\omega,k) = 1 - \frac{\omega_p^2}{\omega(\omega+i\gamma) - \beta^2 k^2}$$

where $\beta$ is proportional to $v_F$ (Yan et al., 2015, Teperik et al., 2013). These models predict blueshifts, saturation of field enhancement at high confinement, and set a maximum mode confinement $k_P = \omega/\beta$.

Spill-out and surface quantum corrections:

Self-consistent hydrodynamic models (SC-HDM) and ab-initio time-dependent density-functional theory (TDDFT) predict induced-charge centroid displacement and electron density spill-out, captured efficiently in the Projected-Dipole Model (PDM). The Feibelman $d$-parameter encodes the centroid shift, and the PDM incorporates this as a surface dipole boundary condition throughout arbitrary geometries, quantitatively matching ab-initio spectra in plasmonic dimers and gaps (Yan et al., 2015).

Tunneling and charge-transfer modes:

TDDFT calculations reveal that, below gaps of 1 nm, quantum tunneling leads to a smooth transition from capacitive plasmonic coupling to conductive (charge-transfer) modes and quenched field enhancement—a regime inaccessible to classical or hydrodynamic models but captured by the Quantum Corrected Model (QCM) and by direct TDDFT (Teperik et al., 2013).

Quantum energy partition and the nonclassical impact parameter (NCI):

A quantitative measure of the fraction of plasmon energy in nonclassical (quantum) degrees of freedom is given by the NCI,

$\mathrm{NCI} = 1 + L_{\rm PR}/Q_{\rm PR}$

where $L_{\rm PR}$ is the loss function and $Q_{\rm PR}$ the quality factor; $\mathrm{NCI} \to 1$ signals dominance of quantum pressure effects (Yan et al., 2015).

Madelung hydrodynamics for 2D systems:

The Madelung formulation, with explicit Bohm (quantum) pressure and Fermi pressure corrections, provides analytic predictions for quantum nonlocality and nonlinearities in 2D electron gases, including magnetoplasmon spectra and nonlinear SHG enhancement (Cardoso et al., 2024).

4. Experimental Protocols and Quantum Process Tomography

Quantum plasmonics is distinguished by rigorous experimental protocols that directly probe the quantum state evolution through plasmonic devices.

Full quantum-process tomography:

The photon-plasmon-photon conversion process in an extraordinary optical transmission (EOT) hole array has been mapped via continuous-variable quantum process tomography. The process tensor $\chi_{k\ell,mn}$, reconstructed by maximum-likelihood estimation from homodyne measurements of nine probe coherent states, shows that the EOT device acts as a single-mode beamsplitter with transmissivity $T=0.62$ and photon-independent phase shift $\phi=0.92$\,rad, with process fidelity $>99\%$ across coherent-state inputs (Tang et al., 2019). The process is indistinguishable from a pure loss + phase channel for any quantum state.

Preservation of quantum statistics and entanglement:

Direct measurements of second-order coherence, $g^{(2)}(\tau)$, and entanglement visibility confirm that quantum antibunching and entanglement survive with negligible loss of visibility or $g^{(2)}(0)$ degradation in plasmon propagation, including in highly dispersive and strongly confined regimes (Mollet et al., 2012, Tokpanov et al., 2018).

Scanned deterministic injection:

Nanodiamond-based scanning near-field tips enable deterministic, high-fidelity single plasmon injection at arbitrary positions, with preservation of quantum statistics, leakage-radiation microscopy readout, and nanometric spatial control (Mollet et al., 2013, Cuche et al., 2010).

5. Quantum Plasmonic Devices: Architectures and Transfer Protocols

Quantum plasmonics enables new paradigms for compact quantum photonics, plasmonic quantum information, and sensing.

Fully integrated quantum plasmonic circuits:

On-chip platforms have demonstrated single-plasmon generation from self-assembled quantum dots, propagation and routing in hybrid or pure metal waveguides, and photonic recovery via optical antennas, with confirmed single-plasmon statistics and photonic–plasmonic–photonic cycling rates as high as $3\times10^6\,{\rm s}^{-1}$ (Wu et al., 2016).

Nanoparticle array channels:

Linear arrays of nanospheres serve as quantum channels for state transfer and Hong-Ou-Mandel–type plasmon interference at the nanoscale. Transmission fidelity in such chains is limited by cumulative Ohmic loss, but fidelities $>0.8$ are demonstrated for $n=3$–$7$ spheres (Lee et al., 2011).

Quantum state evolution and decoherence engineering:

The steady-state evolution of QEs coupled to plasmonic nanoparticles is governed by the formation of bound states in the energy spectrum, giving rise to regimes of complete decay, population trapping, and persistent quantum oscillation, determined by emitter–plasmon coupling strengths and system geometry. The controllable number of bound states enables decoherence-free subspaces and long-lived quantum correlations (Yang et al., 2019).

Semiconductor quantum plasmonics:

Doped quantum wells permit control of plasmon resonance energies and oscillator strengths via doping density and quantum-well width, introducing quantum size effects in the confined slab and enabling tailored epsilon-near-zero and hyperbolic responses (Vasanelli et al., 2020).

6. Multiparticle Quantum Plasmonics and Advanced Sensing Strategies

Quantum plasmonics is entering a multiparticle regime in which conditional measurements, photon-number resolution, and engineered quantum correlations enable advanced sensing, imaging, and statistical control.

Multiphoton process control:

Multiparticle scattering in plasmonic systems can up-convert classical-light inputs into output fields with nontrivial quantum statistical properties, such as sub-thermal or Poissonian statistics, tunable via geometry, polarization, or conditional photon detection (Hong, 28 Mar 2025).

Projective and conditional measurement strategies:

Photon-number resolving detectors allow projective measurement onto specific photon-plasmon number states, revealing bosonic vs. fermionic correlations in near-field subsystems (Hong, 28 Mar 2025).

Quantum-enhanced sensing and imaging:

Conditional detection (photon/plasmon-subtraction, heralded events) in plasmonic interferometers increases signal-to-noise and phase-resolution in quantum plasmonic sensors. Quantum imaging with thermal light, using post-selection or subtraction, enables contrast enhancement under strong noise by isolating high-order multiparticle correlations (Hong, 28 Mar 2025).

7. Challenges, Outlook, and Future Directions

Quantum plasmonics faces fundamental and practical challenges as it targets the ultimate limits of light–matter interaction.

• Loss and decoherence: Ohmic damping, Landau damping, and quantum tunneling set limits on quantum coherence. Strategies to mitigate loss include hybrid photonic–plasmonic architectures, material optimization (single-crystalline metals, doped semiconductors, graphene), and operation at cryogenic temperatures (Bozhevolnyi et al., 2016, Tame et al., 2013).
• Nonlocal and nonclassical corrections: Reliable modeling requires quantum-corrected theories beyond local response, validated by rigorous experiment and benchmarking against ab-initio calculations (Yan et al., 2015, Ding et al., 2017, Yan et al., 2015, Teperik et al., 2013).
• Integration and scalability: Site-controlled single-photon sources, on-chip photodetection, nonlinear elements, and active phase/electro-optic control are critical for quantum circuitry (Wu et al., 2016).
• Multiphoton and many-body phenomena: Methods for preparing and detecting entangled, squeezed, or high-number Fock-like plasmon states remain an area of active research, with implications for quantum metrology and information processing (Hong, 28 Mar 2025).
• Open questions: Resolving the ultimate physical limits for Purcell enhancement, the scaling of $Q/V$ in the presence of nonlocality, and the role of quantum fluctuations and strong correlations as structures approach the atomic scale remain at the forefront.

Quantum plasmonics, as established by theory and experiment, provides a rigorous and predictive framework for investigating, engineering, and exploiting quantum coherence, entanglement, and nonlinearities in the most confined and strongly interacting regimes of light–matter physics (Tame et al., 2013, Bozhevolnyi et al., 2016, Tang et al., 2019, Yan et al., 2015, Hong, 28 Mar 2025).