Photon-Assisted Tunneling in Double Quantum Dot: Application of Scattering Theory

Technical Summary: Photon-Assisted Tunneling in Double Quantum Dot: Application of Scattering Theory

Problem Statement
This paper addresses the theoretical understanding of photon-assisted tunneling (PAT) in quantum dot (QD) systems subjected to an alternating current (AC) field. Specifically, it investigates transport through a single QD and a double quantum dot (DQD) where one dot possesses an oscillating energy level, $\varepsilon(t) = \varepsilon_0 + eV_{AC} \cos \omega t$. The study focuses on two distinct regimes defined by the relationship between the tunnel coupling strength to external leads ($\Gamma$) and the photon energy ($\hbar\omega$): the PAT regime ($\Gamma \ll \hbar\omega$) and the adiabatic transport regime ($\Gamma \gg \hbar\omega$). Furthermore, the paper examines the DQD in the presence of a magnetic flux penetrating the area between the dots to elucidate the Aharonov–Bohm (AB) effect on transport through polariton states and to explore phase measurements in a three-terminal geometry.

Methodology
The authors employ a scattering theory approach to solve the time-dependent Schrödinger equation, avoiding the use of Keldysh nonequilibrium Green's functions or Floquet theory for the primary derivation, though they note the relationship between their Green's functions and Floquet theory.

• Model: The system is modeled using a Hamiltonian comprising the dot energy levels, the leads, and the tunnel coupling. For the DQD, the model includes a magnetic flux $\Phi$ introducing an AB phase $\phi$ in the tunnel coupling between the dots and the leads.
• Perturbation and Scattering: The electric current is derived by performing a perturbative expansion with respect to the tunnel Hamiltonian ($H_T$) to infinite order. This yields a scattering matrix and a T-matrix that accounts for the emission and absorption of $N$ photons.
• Approximations: The electron-electron interaction ($U$) is neglected, as the authors argue it does not qualitatively alter the current peaks studied. The calculation for the DQD is carried out to the second order in the dimensionless AC amplitude $\tilde{V}_{AC}/2 = eV_{AC}/(2\hbar\omega)$.
• Geometries: The theory is applied to two configurations: a two-terminal setup (for current and AB oscillation analysis) and a three-terminal setup (for phase shift measurement).

Key Results

1. Single Quantum Dot (QD):

   • $\Gamma \ll \hbar\omega$ (PAT): The conductance exhibits a main peak at the Fermi level matching $\varepsilon_0$ and subpeaks at $\varepsilon_0 + N\hbar\omega$ ($N = \pm 1, \pm 2, \dots$). These peaks correspond to resonant tunneling through polariton states formed by the coupling of the electron to the AC field. The weight of the subpeaks is determined by the square of the Bessel function $[J_N(\tilde{V}_{AC})]^2$.
   • $\Gamma \gg \hbar\omega$ (Adiabatic): Electrons do not perceive the oscillation during transport. Resonant tunneling occurs through the instantaneous energy level $\varepsilon(t)$. The time-averaged conductance shows peaks corresponding to the extreme values of the oscillating level.
   • Unification: The scattering theory formulation provides a unified, intuitive physical picture for both regimes, recovering known results derived via other methods (e.g., Keldysh Green's functions) while offering a direct derivation of the transition rates.

2. Double Quantum Dot (DQD) in Two-Terminal Geometry:

   • AB Effect on Polariton States: The authors observe the AB effect not only in the main peak ($N=0$) but also in the subpeaks ($N \neq 0$). This indicates coherent transport through the polariton states.
   • Suppression and Enhancement: The AC field suppresses the amplitude of the AB oscillation at the main peak while enhancing it at the subpeaks.
   • Mechanism: The interference at subpeaks is attributed to intermediate processes in the T-matrix where an electron traverses both QDs. For the main peak, the interference is partially suppressed because certain intermediate processes involve only the irradiated QD.

3. Double Quantum Dot in Three-Terminal Geometry:

   • Phase Measurement: Using the non-irradiated QD as a reference arm (analogous to a double-slit experiment), the phase shift through the irradiated QD is measured.
   • Continuous Phase Shift: The measured phase shift ($\phi_{max}$) changes continuously from $0$ to $\pi$ around both the main peak and the subpeaks.
   • Absence of Phase Lapse: Unlike previous studies of QDs without AC fields where a "phase lapse" (abrupt $\pi$ jump) occurs between successive conductance peaks, this study finds no phase lapse between the main peak and subpeaks. The phase shift behavior follows the Breit-Wigner resonance form, $\tan \theta_{QD} \propto \Gamma/(\varepsilon_0 - \varepsilon_{center})$, even in the presence of the AC field.

Significance and Claims
The paper claims that its primary contribution is the reformulation of PAT and adiabatic transport in QDs using scattering theory, which offers an intuitive physical picture of time-dependent transport phenomena.

• Theoretical Clarity: The method successfully bridges the gap between the sequential tunneling limit and the resonant tunneling limit, providing exact expressions for the current that are valid across temperature regimes.
• Coherence in PAT: The observation of the AB effect in subpeaks confirms that the polariton states maintain quantum coherence, allowing for interference effects even when photon emission or absorption occurs.
• Phase Behavior: The demonstration of continuous phase shifts without phase lapses in the presence of an AC field provides new insight into the nature of resonant tunneling in driven systems, contrasting with the behavior of static multi-level systems.
• Applicability: The theory is presented as applicable to both two-terminal and three-terminal geometries, offering a framework for interpreting phase measurements in future experiments involving driven quantum dots, such as those utilizing terahertz light or cavity QED setups.

The authors remain modest regarding the scope, noting that while their scattering theory approach is intuitive, it is less suited for treating strong electron-electron interactions beyond the mean-field level compared to Green's function methods. They do not propose specific new experimental setups but rather provide the theoretical foundation for understanding existing and future PAT experiments in DQDs.