Single-Electron Double Quantum Dots (DQD)

• Single-electron DQDs are quantum systems consisting of two spatially separated dots sharing one electron, enabling clear control over charge and spin states.
• They offer tunable interdot coupling and detuning parameters that facilitate coherent qubit operations through precise gate-defined control.
• Applications include advanced charge detection, spin qubit manipulation, and integration with cavity QED architectures for scalable quantum information processing.

A single-electron double quantum dot (DQD) consists of two spatially separated potential minima, each capable of confining a single electron, and coupled via an interdot tunnel barrier. In the single-electron regime, precisely one conduction electron is shared between the two dots, yielding a minimal Hilbert space and sharply defined charge and spin states. Such systems have become foundational architectures for the paper of coherent charge and spin manipulation, the engineering of charge/spin qubits, and the implementation of quantum information protocols in solid-state nanostructures.

1. Theoretical Formalism and Hamiltonian Structure

A rigorous treatment of the single-electron DQD begins with the effective-mass many-body Hamiltonian (for two electrons, but easily specialized to the one-electron case):

$$\mathcal{H}_\mathrm{MB} = \sum_i \mathcal{H}_i + \sum_{i<j} \frac{e^2}{\kappa | \vec{r}_i - \vec{r}_j |}$$

$$\mathcal{H}_i = \frac{(\vec{p} - e \vec{A})^2}{2 m^*} + V(\vec{r}) + \frac{e}{m^*} \vec{S} \cdot \vec{B},$$

where $m^*$ is the effective mass, $\vec{A}$ is the vector potential for external magnetic fields ($\vec{B} = \vec{\nabla} \times \vec{A}$), and $V(\vec{r})$ is the confining DQD potential (Nielsen et al., 2010). For single-electron occupation, the electron is described by a Hamiltonian with a DQD potential that, for model precision, typically takes the form:

$$V(x, y, z) = \frac{1}{2} \begin{bmatrix} m_x^* \omega_x^2 \min \left((x - L)^2 + \epsilon, (x + L)^2\right) \ + m_y^* \omega_y^2 y^2 + m_z^* \omega_z^2 z^2 \end{bmatrix},$$

where $L$ is half the dot separation and $\epsilon$ is the detuning (bias) parameter.

The basis for a single electron in a DQD is most naturally described by the symmetric (“bonding,” $|s\rangle$) and antisymmetric (“antibonding,” $|a\rangle$) superpositions of localized states. The relevant low-energy subspace is spanned by:

$$|s\rangle \propto |L\rangle + |R\rangle, \quad |a\rangle \propto |L\rangle - |R\rangle,$$

where $|L\rangle$, $|R\rangle$ are states localized in the left or right dot. The energy splitting between $|s\rangle$ and $|a\rangle$ is determined by the interdot tunnel coupling $t$, with the two-level effective Hamiltonian:

$$\mathcal{H} = \frac{\epsilon}{2} \sigma_z + t \sigma_x,$$

where $\epsilon$ is the detuning and $\sigma_{x,z}$ are Pauli operators.

2. Charge, Spin, and Valley States: Regimes and Manipulation

Charge States and Readout

Charge occupation in a single-electron DQD is quantized: the system toggles between $(1, 0)$ and $(0, 1)$ configurations, or occupies a coherent superposition determined by $t$ and $\epsilon$. Charge detection is performed through sensitive electrometry, typically using a nearby single-electron transistor (SET) (Rossi et al., 2010) or via radiofrequency reflectometry (Jung et al., 2012). A classical capacitance model describes the influence of occupancy on the detector; steps in the SET current or shifts in Coulomb blockade peaks directly map onto single-electron transitions.

Spin and Valley Degrees of Freedom

The electron spin in a single-electron DQD forms the basis for spin qubit implementations. The valley degree of freedom is particularly relevant for silicon and carbon-based DQDs. The splitting of valley states (valley splitting $\Delta_v$) is critical for defining the effective qubit Hilbert space and can be characterized by magnetospectroscopy, with level crossings and "kinks" in energy vs. magnetic field diagnostics (Zajac et al., 2015).

Spin control is enabled via tunneling (exchange interaction), magnetic field gradients (Zeeman splitting), and spin-orbit coupling. Kondo effect signatures, singlet-triplet spin blockade, and valley-orbit mixing are direct probes of the underlying quantum degree of freedom (Mittag et al., 2020).

3. Control Methodologies and Gate Architectures

Gate-defined, tunable DQDs are routinely realized in III–V (e.g., GaAs, InAs) and group IV (e.g., Si/SiGe, graphene) heterostructures. Devices employ lithographically patterned gate electrodes to form electrostatic potentials, define single- or double-dot regions, and control tunnel barriers. Reconfigurable overlapping gate geometries (multiple layers of gates with independently tunable voltages) permit transformations between single-dot and double-dot regimes, and allow for the precise adjustment of charging energies, interdot coupling, and dot occupancy (Zajac et al., 2015).

The transition into the single-electron regime is achieved by decreasing the electrochemical potential well below the Fermi level until only the last electron remains. Automated tuning algorithms have been developed to perform this process with minimal user supervision, relying on pinch-off characterization, image analysis, and template matching to identify the few-electron regime (Baart et al., 2016).

4. Quantum Coherence: Decoherence Mechanisms and Optimal Qubit States

Decoherence in single-electron DQDs is primarily induced by charge noise, phonon coupling, nuclear spin fluctuations (hyperfine interaction), and, in some materials, poorly split valley states or fluctuating spin-orbit interactions. Electron temperature in the DQD is a critical parameter for quantum operations; isolated (lead-less) DQDs in isotopically purified Si have demonstrated suppression of electron temperature below that of the leads, reducing thermal occupation of excited states and associated dephasing (Rossi et al., 2011).

The optimal charge qubit basis is constructed from the symmetric and antisymmetric superpositions of the zero-bias bonding and antibonding states, as these maximize overlap with the true localized states whenever detuning is varied (Mosakowski et al., 2016). Initialization into these optimal qubit states requires a precise pulse protocol, carefully tailored to the system dynamics including realistic electronic rise times; arbitrary rotations on the Bloch sphere are realized with composite, spin-echo-type gates.

5. Interdot Coupling, Exchange, and Qubit Operation

The interdot tunnel coupling $t$ sets the bonding-antibonding splitting and, for two-electron configurations, mediates the exchange interaction $J$, fundamental for singlet-triplet qubit operation:

$J = E_{T_0} - E_S,$

where $E_{T_0}$ and $E_S$ are the energies of the unpolarized triplet and singlet states, respectively (Nielsen et al., 2010). Configuration interaction (CI) methods employing Gaussian basis functions accurately describe $J$ as a function of detuning, dot size (confinement energy $E_0$), and separation $L$, capturing both the (1,1) and the (0,2) charge sectors—critical for robust gate operation and noise insensitivity.

For the single-electron DQD, $t$ determines the energy gap $E_q = \sqrt{\epsilon^2 + 4t^2}$, which can be matched to external drives for resonant photon absorption in cavity quantum electrodynamics (cQED) architectures. Qubit manipulations (e.g., rotations) are achieved via voltage pulses on plunger or detuning gates, with arbitrary single-qubit operations attainable through carefully designed pulse sequences.

6. Measurement, Readout, and Quantum Circuit Integration

Readout in single-electron DQDs is typically dispersive, employing charge sensors or high-fidelity radiofrequency detection. Embedded in a resonant LC or microwave cavity, the DQD modifies the resonator’s impedance through its quantum capacitance and tunneling capacitance, producing measurable frequency shifts and damping (Esterli et al., 2018). The small-signal equivalent circuit captures both the dissipative (Sisyphus resistance) and dispersive (quantum and tunneling capacitance) elements, enabling parametric amplification and non-resonant state readout.

In cQED architectures, the DQD is capacitively coupled to a microwave cavity, with the charge-cavity interaction strength $g_c$ dictating the rate of dispersive shifts in cavity transmission. Recent advances have achieved charge-cavity coupling rates $g_c/2\pi \sim 23$–$40$ MHz, transition frequencies finely tunable by electrostatic gates, and photon-to-electron conversion efficiencies up to 25%, with single-photon sensitivity on-chip (Basset et al., 2013, Mi et al., 2016, Haldar et al., 5 Jun 2024).

7. Material Platforms and Outlook

Single-electron DQD architectures have been demonstrated across a variety of material systems:

Material Platform	Features for Single-Electron DQDs	Notable Outcomes
Si/SiGe	Large valley splittings (Δ_v ~ 35–70 μeV), low decoherence	High-fidelity qubits, cQED integration
InAs/InSb nanowire	Strong spin-orbit coupling, flexible gate patterns	Fast spin manipulation, high fidelity tunneling control
Graphene (BLG)	Ambipolarity, strong field-induced band gaps	Tunable occupancy, interdot coupling up to GHz
Phosphorus-doped silicon	Suppressed hyperfine interaction (with ²⁸Si isotope)	Long coherence in electrically isolated DQDs
InAs 2DEG	High g-factor (~16), long spin-orbit length	Reduced hyperfine, Kondo/spin blockade physics

Single-electron DQDs in these systems can be engineered for qubit operations that are robust to charge and spin noise, and scalable to large quantum register arrays (Vyurkov et al., 11 Jul 2025). Techniques such as ensemble averaging over arrays of nanowires provide environmental noise immunity, while spatially resolved control over symmetric and antisymmetric states permits quantum information encoding without net charge transfer.

Advances in automated tuning (Baart et al., 2016), precise charge and spin measurement (Rossi et al., 2010, Jung et al., 2012), and circuit-level equivalent modeling (Esterli et al., 2018), combined with integration into cavity QED platforms (Basset et al., 2013, Mi et al., 2016, Haldar et al., 5 Jun 2024), position single-electron DQDs as a central technology for scalable, coherent quantum information processing and fundamental studies of condensed matter quantum dynamics.