Photon-Assisted Tunneling: Theory & Applications

• Photon-assisted tunneling is a quantum phenomenon in which photon energy modulates electron tunneling across barriers, creating characteristic sidebands and sub-gap features.
• It is applied across diverse platforms—such as mesoscopic superconductors, semiconductor nanostructures, and cold-atom systems—to probe quantum coherence and many-body interactions.
• Advanced models like the Tien-Gordon and P(E) theories quantitatively describe PAT processes, guiding improvements in quantum metrology and device control.

Photon-assisted tunneling (PAT) is a quantum transport phenomenon in which the transfer of particles across an energy barrier is enabled, enhanced, or otherwise modified by the exchange of energy quanta (photons) between the tunnel system and an external time-dependent electromagnetic field or environment. In a broad sense, PAT encompasses both single-particle and correlated multi-particle tunneling processes driven by periodic excitations ranging from classical electromagnetic fields to engineered photon or phonon modes, as well as vacuum or non-classical quantum fluctuations. PAT has become a powerful and versatile tool for probing quantum coherence, many-body interactions, charge-spin dynamics, and environmental coupling in platforms including mesoscopic superconductors, semiconductor nanostructures, strongly correlated lattice systems, photonic crystals, and molecular electronics.

1. Theoretical Frameworks for Photon-Assisted Tunneling

Photon-assisted tunneling fundamentally arises from the modification of the tunnel matrix element by a time-dependent perturbation, typically periodic in time. For a wide class of junctions (e.g., normal metal-insulator-superconductor (NIS)), the canonical description uses the Tien-Gordon approach, in which the time-dependent phase imposed by an ac drive leads to sidebands in the tunneling current at energies shifted by integer multiples of the photon energy $n\hbar\omega$ (Peters et al., 2020, Carrad et al., 2022). The tunneling current $I_{\rm dc}(V)$ takes the form

$$I_{\rm dc}(V) = \sum_{l} J_l^2(\alpha) I^0\left( V + \frac{l\hbar\omega}{q} \right),$$

where $J_l$ is the $l$th Bessel function, $\alpha$ is proportional to the ac drive amplitude, $I^0(V)$ is the background current, $q$ is the effective charge, and the sum runs over photon sidebands.

In junctions with dissipation or environmental coupling, the rate of photon absorption and emission processes is encoded in a probability distribution $P(E)$, frequently evaluated via the $P(E)$-theory formalism. For example, in NIS junctions in a resistive electromagnetic environment of resistance $R$ and temperature $T_{\rm env}$, the $P(E)$-function can become Lorentzian, leading to an effective density of states (DOS) given by the convolution of the BCS DOS with $P(E)$ (Pekola et al., 2010),

$n_S^\sigma(E) = \int dE'~n_S(E')\, P(E-E').$

This "environmental dressing" transforms the ideal BCS gap into an effective Dynes DOS with smearing parameter $\gamma \propto R T_{\text{env}}$.

PAT is not restricted to single-particle processes. In multi-electron tunneling such as multiple Andreev reflections (MAR) in Josephson junctions, the photon energy is shared among $n$ correlated electrons, resulting in voltage spacings between PAT sidebands of $\delta V = hf/(n e)$ (Carrad et al., 2022). The modification of the Tien-Gordon approach incorporates the effective charge $n e$, and the sideband spectra serve as a direct probe of charge transfer quanta in correlated processes.

2. Photon-Assisted Tunneling in Mesoscopic Superconducting Devices

In mesoscopic NIS junctions, PAT manifests as sub-gap leakage currents, even when the underlying superconducting DOS is ideally gapped. Theoretical and experimental evidence demonstrates that a dissipative, high-temperature electromagnetic environment can "smear" the BCS DOS into the empirical Dynes form,

$$n_S^{\mathrm{Dynes}}(E) = \left| \Re \frac{(E/\Delta + i\gamma)}{\sqrt{(E/\Delta + i\gamma)^2 - 1}} \right|,$$

with the broadening parameter $\gamma$ arising not from superconductor intrinsic properties, but from the environment through photon-assisted tunneling (Pekola et al., 2010). The broadening is quantified as

$\gamma = \frac{e^2 k_B T_{\rm env} R}{\hbar^2}.$

This approach resolves long-standing questions around the origin of sub-gap leakage currents in NIS devices. Experimental validation includes engineering the electromagnetic environment by adding capacitive shunting, which reduces the effect of environmental photons and suppresses sub-gap conduction proportionally.

In SINIS turnstile devices deployed as metrological current sources, the accuracy is limited by residual sub-gap leakage linked to photon-assisted tunneling. Introducing a ground plane for additional capacitance suppresses the leakage, enabling improved current quantization. This provides a direct pathway to reducing errors in quantum current standards.

3. PAT in Strongly Correlated and Driven Lattice Systems

Beyond mesoscale tunneling, PAT is central in the manipulation of quantum states in cold atom and ion-trap systems, as well as in nanowire-based Josephson junctions. In optically driven double-well systems, the time-periodic modulation induces resonant single- and multi-particle tunneling. Notably, "fractional" photon-assisted tunneling arises as a distinct many-body effect: when interactions are strong, the energy of a single photon can drive the simultaneous coherent tunneling of, e.g., two or three particles, leading to resonant features at fractional multiples of the photon energy (Esmann et al., 2011).

In quantum gas experiments, lattice modulation at frequencies resonant with the tilt or interaction-shifted energy differences induces coherent photon-assisted tunneling. The inclusion of higher-orbital and multi-particle interaction shifts enables the tunneling to become occupation-sensitive, and further, driving across such resonances can realize quantum phase transitions in many-body systems (Ma et al., 2011). The technique enables high-bandwidth, site-resolved lattice operations, with prospects for dynamically engineered gauge fields.

In Josephson junctions with MAR, PAT results in sidebands whose spacings provide a direct measure of the effective charge transferred in the process, confirming the MAR order by the voltage spacing $\delta V = hf/(n e)$ and allowing identification of the dominant tunneling mechanism (Carrad et al., 2022).

4. Spin, Valley, and Quantum Control via PAT

Photon-assisted tunneling is heavily exploited for quantum control in double quantum dots, artificial molecules, and related low-dimensional systems. In semiconductor quantum dots, PAT is conventionally associated with electric-dipole, spin-conserving processes. However, spin–orbit coupling introduces a "spin-flip" channel: microwave-induced tunneling transitions can change both the spin and valley quantum numbers, enabling manipulation and spectroscopy of spin and valley states (Schreiber et al., 2010, Osika et al., 2017, Braakman et al., 2014). The effective spin-flip tunneling amplitude is frequently orders of magnitude weaker than the spin-conserving term, but becomes the dominant mechanism in specific field and geometry regimes.

Landau–Zener–Stückelberg interference and microwave-driven avoided crossings enrich the PAT landscape, producing oscillatory patterns in the transition probabilities as a function of drive parameters and revealing coherent quantum dynamics even in non-equilibrium regimes (Shang et al., 2013). These techniques underpin proposed schemes for quantum gate operations, spin qubit readout, and all-electric control in quantum information applications.

In designer photonic systems, the bosonic nature of photons enables both photon-aided and photon-inhibited tunneling, manifesting as a competition between bosonic enhancement and quantum statistical inhibition. The presence of squeezed light or nonclassical states in the environment further modifies the PAT rates, introducing negative quasi-probabilities for energy exchange with the tunneling electrons and leading to characteristic conductance anomalies (Souquet et al., 2014, Liu et al., 2013).

5. Role of Environment and Advanced Theoretical Extensions

The electromagnetic environment in which PAT occurs critically shapes the tunneling dynamics. The $P(E)$ theory provides a comprehensive approach, where the environmental spectral density, temperature, and impedance enter the probability distribution for energy exchange with tunneling electrons. This formalism precisely predicts the width and structure of observed sidebands, sub-gap leakage, and their dependence on device architecture (including capacitively-shunted or dynamically filtered setups).

Recent theoretical works extend the density matrix and Keldysh approaches to treat environments ranging from classical coherent fields to quantum cavity photon states. For strongly confined fields, as in nanoantenna-enhanced tunnel diodes, PAT not only dominates the nonlinear current response but defines a nonlinear conductance hierarchy that parallels higher-order susceptibilities in nonlinear optics (Davids et al., 2019, Davids et al., 2017). In these devices, harmonic generation, optical rectification, and THz signal emission emerge directly from multiphoton-assisted tunneling.

For subgap states in superconductors—such as Yu–Shiba–Rusinov or Majorana bound states—PAT under microwave drive acts as a sensitive spectroscopic tool. Sideband spacings tied to the charge transferred in the elementary event, as well as "V- and Y-shaped" patterns resolving electron–hole asymmetry, allow unambiguous differentiation between conventional and topological subgap states (Peters et al., 2020, Gonzalez et al., 2021, Zanten et al., 2019). Systematic Keldysh calculations capture deviations from standard Tien–Gordon predictions when resonance conditions and environmental effects dominate.

6. Applications and Broader Implications

Photon-assisted tunneling enables a rich spectrum of applications:

• Quantum metrology: Suppression of residual leakage through environmental engineering, improved quantization in electron turnstiles, and direct assignment of MAR processes for current standards (Pekola et al., 2010, Carrad et al., 2022).
• Quantum simulation: Implementation of driven many-body Hamiltonians, density-dependent tunneling, tunable gauge fields, and quantum phase transitions in cold atoms and ion traps (Bermudez et al., 2015, Bermudez et al., 2012, Ma et al., 2011).
• Quantum control: Spin manipulation, valley-state resolution, and quantum non-demolition readout in quantum dots and artificial molecules (Schreiber et al., 2010, Osika et al., 2017, Braakman et al., 2014).
• Nanoscale optoelectronics: Enhanced control of tunneling in molecular devices with graphene electrodes, using polarization and surface-plasmon excitations to boost and direct photocurrent (Fainberg, 2013).
• Nonlinear nanophotonics: Rectification, harmonic generation, and frequency mixing in metasurface-enhanced tunnel systems, leveraging the direct mapping from field-induced currents to nonlinear conductances (Davids et al., 2019).

7. Limitations, Challenges, and Future Perspectives

While PAT has evolved into a diagnostic and control tool of broad scope, several open challenges remain:

• Theoretical limitations: Many theoretical frameworks assume weak environmental coupling and linear perturbations; strong environmental dissipation, non-Markovian effects, or high-order processes often require nonperturbative approaches and improved models.
• Environmental engineering: Direct measurement or deterministic control of environment parameters (e.g., $T_{\rm env}$, $R$) can be challenging, necessitating inference from fits or the design of explicit filters and shunts.
• Intrinsic vs. extrinsic broadening: Disentangling photon-assisted leakage from intrinsic DoS broadening or structural imperfections remains complex, particularly in complex multi-junction circuits or devices harnessing topological states.
• Extending to hybrid platforms: Integrating PAT schemes with cavity QED, topological nanowires, or quantum simulation of dynamical gauge fields requires precise engineering of interactions and environmental couplings.

Overall, photon-assisted tunneling stands as both a window into nonequilibrium, many-body, and quantum dynamics in nanostructures and a platform for developing applications in quantum metrology, information processing, and nonlinear quantum devices. The continued convergence of experimental advances and theoretical generalizations will expand the frontier of PAT-enabled technologies.