Circular Double Quantum Dot

• Circular double quantum dots are defined by two coupled quantum confinement regions with near-circular (harmonic) potentials, leading to symmetric orbital states.
• They support diverse qubit encodings through controlled electron configurations, tunable tunnel coupling, and robust suppression of charge noise.
• Implementations range from gate-defined lateral dots and ring-based designs to photonic-integrated structures, highlighting their flexibility in quantum device architectures.

Searching arXiv for the cited topic and papers. A circular double quantum dot is a double-well quantum-dot system in which the relevant confinement is locally circular or effectively circular, so that each constituent dot is modeled by a two-dimensional isotropic or approximately harmonic potential, or, in a distinct usage, by two quantum wells defined along a one-dimensional ring. Across the literature, the term therefore spans several closely related realizations: gate-defined lateral double dots in which smooth symmetric confinement makes each dot approximately circular; explicitly modeled double dots built from two isotropic harmonic wells; ring-based circular double quantum dots with azimuthal confinement and angular-momentum-carrying states; and vertically stacked quantum-dot molecules integrated with circular photonic structures. In all of these settings, the defining ingredients are two coupled quantum confinement regions, tunable interdot coupling, and a low-energy spectrum organized by charge configuration, orbital symmetry, spin, or spin–orbit structure (Shi et al., 2011).

1. Geometric realizations and the meaning of “circular”

In gate-defined lateral devices, a double quantum dot is formed in a two-dimensional electron gas by electrostatic gates that create two potential minima separated by a tunable barrier. In the silicon hybrid-qubit architecture, the device is explicitly a gate-defined lateral double quantum dot in a Si/SiGe heterostructure, operated in a $(2,1)$ charge configuration with two electrons in one dot and one electron in the other. Although the work does not focus on “circular” dots per se, the theoretical description assumes smooth and symmetric confinement, and for an isolated dot it notes that for a symmetric confinement potential $V_p(\mathbf{r})$, certain off-diagonal Coulomb integrals vanish. The natural model is a harmonic potential,

$$V_p(\mathbf{r}) \sim \frac{1}{2} m^* \omega_0^2 r^2,$$

so each lateral dot is treated as an approximately circular dot with well-defined orbital levels and a tunable tunnel barrier between them (Shi et al., 2011).

A more explicit notion of circularity appears in the six-electron semiconductor double-quantum-dot model, where each dot is taken to be a two-dimensional isotropic harmonic well. The double quantum dot potential is defined as the minimum of two identical parabolic wells,

$$V(x,y) = \frac{1}{2} m^* \omega^2 \min\left[(x-L)^2 + \frac{\epsilon}{2},\; (x+L)^2 - \frac{\epsilon}{2}\right] + \frac{1}{2} m^* \omega^2 y^2 .$$

Here the local confinement around each minimum is rotationally symmetric, so each dot is a near-ideal circular parabolic well in its own local coordinate system, while the full double-well structure breaks global rotational symmetry (Nielsen et al., 2013).

A distinct geometry is the circular double quantum dot on a ring. In that case, the system is reduced to a one-dimensional ring of radius $R$, with two dots defined along the azimuthal coordinate $\varphi$ by an azimuthal step potential and two barriers at $\varphi=\pm \pi/2$: $V_0(\varphi) = \beta \big[\delta(\varphi - \tfrac{\pi}{2}) + \delta(\varphi + \tfrac{\pi}{2})\big] +\begin{cases} -\dfrac{\Delta}{2}, & -\pi/2 < \varphi \le \pi/2,\[4pt] +\dfrac{\Delta}{2}, & \text{otherwise}. \end{cases}$ In this usage, “circular double quantum dot” refers not to two approximately circular lateral puddles, but to two quantum wells distributed along the circumference of a ring, with angular momentum as an explicit orbital degree of freedom (Omlor et al., 18 Sep 2025).

The phrase also appears in a photonic context through vertically stacked quantum-dot molecules integrated beneath a circular Bragg grating. There the double quantum dot is a pair of self-assembled InGaAs quantum dots separated by a $7.3$ nm GaAs spacer, while the “circular” descriptor refers to the surrounding circular Bragg grating photonic structure rather than to the electronic lateral confinement. This usage is physically distinct from lateral circular dots, but it remains a coupled two-dot system with electrically controlled molecular states (Schall et al., 2021).

2. Single-particle confinement, shell structure, and effective Hamiltonians

For lateral circular or near-circular dots, the low-energy theory begins with single-particle orbitals in smooth confinement and Coulomb-coupled tunneling between the two dots. In the hybrid silicon double-dot qubit, orbital energies and tunneling matrix elements are written as

$$\epsilon_{\alpha i} = \int d\mathbf{r} \, \phi_{\alpha i}^*(\mathbf{r})\left[ \frac{\hat{\mathbf{p}}^2}{2m^*} + V_p(\mathbf{r}) \right]\phi_{\alpha i}(\mathbf{r}),$$

$V_p(\mathbf{r})$0

with $V_p(\mathbf{r})$1 a model confinement potential. This formulation makes circularity relevant through symmetry of the orbital wavefunctions, Coulomb matrix elements, and the suppression of electric dipole moments in the harmonic limit (Shi et al., 2011).

When only the lowest orbital in each dot is retained, the double quantum dot is often reduced to a two-site Hubbard-like model. In the GaAs sensing study, the effective Hamiltonian is

$V_p(\mathbf{r})$2

with $V_p(\mathbf{r})$3 controlled by gate voltages, $V_p(\mathbf{r})$4 the intra-dot interaction, $V_p(\mathbf{r})$5 the inter-dot capacitive coupling, and $V_p(\mathbf{r})$6 the tunnel coupling. The paper states that this form is unchanged by adopting circular confinement; only the quantitative values of level spacings and wavefunctions differ (Barthel et al., 2010).

For explicitly circular harmonic dots, the single-particle Hamiltonian may be written in effective-mass form with magnetic field and Zeeman coupling: $V_p(\mathbf{r})$7

$V_p(\mathbf{r})$8

In the six-electron work, the use of symmetric gauge and Fock–Darwin basis functions makes the shell structure of circular dots explicit. Closed-shell “core” electrons occupy the lowest orbital, while “valence” electrons populate higher shells and define the low-energy singlet–triplet qubit manifold (Nielsen et al., 2013).

The ring-based circular double quantum dot has a different orbital Hamiltonian. After reducing the three-dimensional confinement to the ring, the orbital term becomes

$V_p(\mathbf{r})$9

where $$V_p(\mathbf{r}) \sim \frac{1}{2} m^* \omega_0^2 r^2,$$0 is the angular-momentum operator and $$V_p(\mathbf{r}) \sim \frac{1}{2} m^* \omega_0^2 r^2,$$1 is the magnetic flux. The presence of $$V_p(\mathbf{r}) \sim \frac{1}{2} m^* \omega_0^2 r^2,$$2 and $$V_p(\mathbf{r}) \sim \frac{1}{2} m^* \omega_0^2 r^2,$$3 means that orbital angular momentum and Aharonov–Bohm physics are intrinsic to the circular double-dot description rather than emergent approximations (Omlor et al., 18 Sep 2025).

3. Charge sectors, spin manifolds, and qubit encodings

Circular double quantum dots support several distinct few-electron encodings. In the hybrid silicon double-dot qubit, the logical subspace is a three-electron $$V_p(\mathbf{r}) \sim \frac{1}{2} m^* \omega_0^2 r^2,$$4 manifold. Two electrons occupy one dot and form either a singlet or a triplet, while the third electron occupies the other dot. The logical states are chosen in the $$V_p(\mathbf{r}) \sim \frac{1}{2} m^* \omega_0^2 r^2,$$5, $$V_p(\mathbf{r}) \sim \frac{1}{2} m^* \omega_0^2 r^2,$$6 subspace: $$V_p(\mathbf{r}) \sim \frac{1}{2} m^* \omega_0^2 r^2,$$7

$$V_p(\mathbf{r}) \sim \frac{1}{2} m^* \omega_0^2 r^2,$$8

Both states have the same total spin quantum numbers but differ in the internal spin configuration of the doubly occupied dot. The work explicitly describes this as a “decoherence-free subspace” protected against uniform magnetic-field fluctuations (Shi et al., 2011).

A pulse-gated formulation of the same three-electron hybrid qubit emphasizes a three-state basis

$$V_p(\mathbf{r}) \sim \frac{1}{2} m^* \omega_0^2 r^2,$$9

where $$V(x,y) = \frac{1}{2} m^* \omega^2 \min\left[(x-L)^2 + \frac{\epsilon}{2},\; (x+L)^2 - \frac{\epsilon}{2}\right] + \frac{1}{2} m^* \omega^2 y^2 .$$0 and $$V(x,y) = \frac{1}{2} m^* \omega^2 \min\left[(x-L)^2 + \frac{\epsilon}{2},\; (x+L)^2 - \frac{\epsilon}{2}\right] + \frac{1}{2} m^* \omega^2 y^2 .$$1 are predominantly $$V(x,y) = \frac{1}{2} m^* \omega^2 \min\left[(x-L)^2 + \frac{\epsilon}{2},\; (x+L)^2 - \frac{\epsilon}{2}\right] + \frac{1}{2} m^* \omega^2 y^2 .$$2 spin-like states and $$V(x,y) = \frac{1}{2} m^* \omega^2 \min\left[(x-L)^2 + \frac{\epsilon}{2},\; (x+L)^2 - \frac{\epsilon}{2}\right] + \frac{1}{2} m^* \omega^2 y^2 .$$3 is a $$V(x,y) = \frac{1}{2} m^* \omega^2 \min\left[(x-L)^2 + \frac{\epsilon}{2},\; (x+L)^2 - \frac{\epsilon}{2}\right] + \frac{1}{2} m^* \omega^2 y^2 .$$4 state with the same total $$V(x,y) = \frac{1}{2} m^* \omega^2 \min\left[(x-L)^2 + \frac{\epsilon}{2},\; (x+L)^2 - \frac{\epsilon}{2}\right] + \frac{1}{2} m^* \omega^2 y^2 .$$5, $$V(x,y) = \frac{1}{2} m^* \omega^2 \min\left[(x-L)^2 + \frac{\epsilon}{2},\; (x+L)^2 - \frac{\epsilon}{2}\right] + \frac{1}{2} m^* \omega^2 y^2 .$$6. The effective Hamiltonian is

$$V(x,y) = \frac{1}{2} m^* \omega^2 \min\left[(x-L)^2 + \frac{\epsilon}{2},\; (x+L)^2 - \frac{\epsilon}{2}\right] + \frac{1}{2} m^* \omega^2 y^2 .$$7

with $$V(x,y) = \frac{1}{2} m^* \omega^2 \min\left[(x-L)^2 + \frac{\epsilon}{2},\; (x+L)^2 - \frac{\epsilon}{2}\right] + \frac{1}{2} m^* \omega^2 y^2 .$$8 the singlet–triplet splitting of the doubly occupied dot, $$V(x,y) = \frac{1}{2} m^* \omega^2 \min\left[(x-L)^2 + \frac{\epsilon}{2},\; (x+L)^2 - \frac{\epsilon}{2}\right] + \frac{1}{2} m^* \omega^2 y^2 .$$9 the detuning, and $R$0 the effective tunnel couplings to the $R$1 state. This representation makes the hybrid nature explicit: the logical states are spin-encoded, but gate operations are mediated by a charge-like intermediate state (Koh et al., 2012).

In two-electron double quantum dots, the relevant basis is instead the familiar singlet–triplet manifold. The singlet and triplets are

$R$2

$R$3

Projecting the Hubbard model into the two-electron subspace yields an effective spin Hamiltonian

$R$4

where $R$5 depends on tunnel coupling, detuning, and Coulomb energies. This description is stated to be generic to any double-dot geometry, including circular confinement (Barthel et al., 2010).

The six-electron circular double quantum dot realizes a different singlet–triplet qubit in which four electrons form closed shells and two valence electrons define the low-energy manifold. The qubit is encoded in the lowest $R$6 and $R$7, $R$8 states, denoted $R$9 and $\varphi$0. The effective two-level Hamiltonian is

$\varphi$1

where $\varphi$2 is the singlet–triplet splitting and $\varphi$3 is the coupling produced by a magnetic-field gradient across the dots. The significance of the circular shell structure is that closed shells screen charge disorder while leaving a low-lying valence singlet–triplet manifold analogous to the two-electron case (Nielsen et al., 2013).

A separate conceptual point arises in the ring geometry. There, even and odd parity orbital states can cross as detuning is varied, and spin–orbit coupling mixes them through nonzero matrix elements of $\varphi$4. In the $\varphi$5 subspace, the angular momentum acts as

$\varphi$6

so the mixed eigenstates become “ring states” with finite $\varphi$7. This is not a singlet–triplet encoding, but it defines a circular double-dot orbital basis in which angular momentum, rather than left–right charge localization alone, organizes the low-energy physics (Omlor et al., 18 Sep 2025).

4. Electrical control, readout, and sensing

In lateral circular or approximately circular double dots, the primary control knobs are detuning and tunnel coupling. The hybrid silicon architecture states that all control is electrical: gate voltages tune the singlet–triplet splitting within the doubly occupied dot and the interdot exchange couplings. In the projected logical basis, the effective two-level Hamiltonian is

$\varphi$8

which is equivalent to a generic qubit Hamiltonian of the form

$\varphi$9

The longitudinal splitting is set mainly by the singlet–triplet splitting $\varphi=\pm \pi/2$0, while coherent mixing is mediated by tunneling-induced exchange $\varphi=\pm \pi/2$1 (Shi et al., 2011).

The pulse-gated hybrid-qubit protocol exploits two avoided crossings and a charge-like intermediate state. Near any avoided crossing, the dynamics reduce to

$\varphi=\pm \pi/2$2

with Rabi frequency $\varphi=\pm \pi/2$3 at exact resonance. The paper constructs primitive $\varphi=\pm \pi/2$4, $\varphi=\pm \pi/2$5, and $\varphi=\pm \pi/2$6 operations by pulsing detuning to the $\varphi=\pm \pi/2$7-$\varphi=\pm \pi/2$8 crossing, the $\varphi=\pm \pi/2$9-$V_0(\varphi) = \beta \big[\delta(\varphi - \tfrac{\pi}{2}) + \delta(\varphi + \tfrac{\pi}{2})\big] +\begin{cases} -\dfrac{\Delta}{2}, & -\pi/2 < \varphi \le \pi/2,\[4pt] +\dfrac{\Delta}{2}, & \text{otherwise}. \end{cases}$0 crossing, and an intermediate phase point. An arbitrary single-qubit rotation is then decomposed as

$V_0(\varphi) = \beta \big[\delta(\varphi - \tfrac{\pi}{2}) + \delta(\varphi + \tfrac{\pi}{2})\big] +\begin{cases} -\dfrac{\Delta}{2}, & -\pi/2 < \varphi \le \pi/2,\[4pt] +\dfrac{\Delta}{2}, & \text{otherwise}. \end{cases}$1

The significance for circular double dots is that the construction assumes the standard lateral double-dot detuning/tunneling picture associated with approximately harmonic confinement (Koh et al., 2012).

Charge and spin readout in double dots can be performed by spin-to-charge conversion and proximal sensing. In the GaAs experiment, a gate-defined lateral double quantum dot is capacitively coupled to a proximal sensor quantum dot operated on the side of a Coulomb-blockade peak and measured by rf reflectometry. The sensor yields measurement times down to $V_0(\varphi) = \beta \big[\delta(\varphi - \tfrac{\pi}{2}) + \delta(\varphi + \tfrac{\pi}{2})\big] +\begin{cases} -\dfrac{\Delta}{2}, & -\pi/2 < \varphi \le \pi/2,\[4pt] +\dfrac{\Delta}{2}, & \text{otherwise}. \end{cases}$2 ns, a conductance response $V_0(\varphi) = \beta \big[\delta(\varphi - \tfrac{\pi}{2}) + \delta(\varphi + \tfrac{\pi}{2})\big] +\begin{cases} -\dfrac{\Delta}{2}, & -\pi/2 < \varphi \le \pi/2,\[4pt] +\dfrac{\Delta}{2}, & \text{otherwise}. \end{cases}$3 at the $V_0(\varphi) = \beta \big[\delta(\varphi - \tfrac{\pi}{2}) + \delta(\varphi + \tfrac{\pi}{2})\big] +\begin{cases} -\dfrac{\Delta}{2}, & -\pi/2 < \varphi \le \pi/2,\[4pt] +\dfrac{\Delta}{2}, & \text{otherwise}. \end{cases}$4 transition, and a $V_0(\varphi) = \beta \big[\delta(\varphi - \tfrac{\pi}{2}) + \delta(\varphi + \tfrac{\pi}{2})\big] +\begin{cases} -\dfrac{\Delta}{2}, & -\pi/2 < \varphi \le \pi/2,\[4pt] +\dfrac{\Delta}{2}, & \text{otherwise}. \end{cases}$5-fold larger dc sensitivity than a comparable quantum point contact sensor (Barthel et al., 2010).

The same work emphasizes that the sensing principle is geometry-independent: the charge stability diagram remains a honeycomb in gate-voltage space, and rf sensor-dot readout requires only capacitive coupling between the double quantum dot and the sensor. A plausible implication is that circular confinement changes lever arms and orbital structure, but not the conceptual structure of charge-state detection, Pauli-blockade-based spin-to-charge conversion, or reflectometric readout (Barthel et al., 2010).

Mechanical probing provides another route. In a capacitively coupled double quantum dot, cantilever motion modulates detuning and thereby the coherent eigenstates. In the one-electron regime, the effective two-level Hamiltonian is

$V_0(\varphi) = \beta \big[\delta(\varphi - \tfrac{\pi}{2}) + \delta(\varphi + \tfrac{\pi}{2})\big] +\begin{cases} -\dfrac{\Delta}{2}, & -\pi/2 < \varphi \le \pi/2,\[4pt] +\dfrac{\Delta}{2}, & \text{otherwise}. \end{cases}$6

with $V_0(\varphi) = \beta \big[\delta(\varphi - \tfrac{\pi}{2}) + \delta(\varphi + \tfrac{\pi}{2})\big] +\begin{cases} -\dfrac{\Delta}{2}, & -\pi/2 < \varphi \le \pi/2,\[4pt] +\dfrac{\Delta}{2}, & \text{otherwise}. \end{cases}$7 the detuning and $V_0(\varphi) = \beta \big[\delta(\varphi - \tfrac{\pi}{2}) + \delta(\varphi + \tfrac{\pi}{2})\big] +\begin{cases} -\dfrac{\Delta}{2}, & -\pi/2 < \varphi \le \pi/2,\[4pt] +\dfrac{\Delta}{2}, & \text{otherwise}. \end{cases}$8 the coherent interdot tunnelling amplitude. The resulting cantilever frequency shift and damping can be used to extract the magnitude of coherent interdot tunnelling and, in some cases, the value of the double-dot $V_0(\varphi) = \beta \big[\delta(\varphi - \tfrac{\pi}{2}) + \delta(\varphi + \tfrac{\pi}{2})\big] +\begin{cases} -\dfrac{\Delta}{2}, & -\pi/2 < \varphi \le \pi/2,\[4pt] +\dfrac{\Delta}{2}, & \text{otherwise}. \end{cases}$9 time. This treatment is presented as general enough to apply to a circular double quantum dot geometry as long as the system reduces to two localized orbitals with capacitive coupling to the mechanical resonator (Gardner et al., 2011).

5. Decoherence, noise suppression, and the role of symmetry

For lateral circular double-dot qubits, one of the central consequences of symmetric confinement is suppression of charge-noise sensitivity. In the hybrid silicon architecture, the paper argues that because the two logical states differ mainly by an internal singlet–triplet rearrangement within one dot, rather than by moving an electron between dots, charge noise is significantly suppressed compared with a conventional charge qubit. In the harmonic limit, the dipole moment between singlet and triplet states vanishes. The work further states that the dominant dephasing mechanism is intervalley electron–phonon coupling, giving an estimated

$7.3$0

while the triplet-to-singlet relaxation time in a single Si/SiGe dot is reported as

$7.3$1

This combination underpins the claim of fast control with potentially long decoherence times (Shi et al., 2011).

The six-electron circular double quantum dot addresses noise through a different mechanism: filled shells screen charge impurities. The paper states that previous work and the present calculations show that the exchange energy in multi-electron double quantum dots is less sensitive to nearby charge defects than in two-electron double quantum dots. A plausible implication is that the circular shell structure is not merely a modeling convenience; it is part of the rationale for using several electrons per dot, since clear core–valence separation depends on the Fock–Darwin spectrum of circular confinement (Nielsen et al., 2013).

In the ring-based circular double quantum dot, noise enters through detuning fluctuations and parity-breaking disorder. Around an even–odd crossing, the effective Hamiltonian is

$7.3$2

Here $7.3$3 is the odd-parity disorder matrix element, and the level splitting within a doublet can exhibit a charge-noise sweet spot. The paper states that at a specific magnetic-field angle the system exhibits a second-order charge-noise sweet spot, lowering sensitivity to dephasing while retaining substantial spin–photon coupling strength (Omlor et al., 18 Sep 2025).

More specifically, the sweet-spot condition is obtained by setting the detuning and disorder derivatives of the doublet splitting to zero. At $7.3$4 and $7.3$5, the splitting obeys

$7.3$6

and if $7.3$7 as well, the splitting is protected up to second order against both kinds of charge noise. This places symmetry and geometry at the center of the circular ring-based design: the orbital angular momentum that the geometry introduces is precisely what allows simultaneous tunability of effective $7.3$8-factors, spin–charge hybridization, and noise protection (Omlor et al., 18 Sep 2025).

6. Extensions, material platforms, and competing design logics

The circular double quantum dot is not tied to a single material system. In Si/SiGe, the architecture of interest is a gate-defined lateral double dot with electrical control of exchange and singlet–triplet splitting (Shi et al., 2011). In GaAs, circular or near-circular lateral double dots support standard Hubbard-like charge and spin physics as well as rf sensor-dot readout (Barthel et al., 2010). In explicitly modeled GaAs circular double dots, isotropic harmonic confinement and Fock–Darwin shell structure motivate multi-electron singlet–triplet encodings (Nielsen et al., 2013).

PbTe nanowire double quantum dots illustrate a different regime. These are electrostatically defined double dots in semiconductor PbTe nanowires, not circular lateral dots, but they show how double-dot phenomenology changes when Coulomb charging is heavily screened by a huge dielectric constant. In Device A, negligible separation between paired triple points implies that the mutual capacitance is effectively quenched, and the stability diagrams show single points where paired triple points would normally appear. Spin degeneracy at zero magnetic field and fourfold splitting of triangles at finite field are observed. A plausible implication is that any attempt to realize a PbTe-based circular double quantum dot would need to confront the same screened-interaction regime, even if the lateral geometry were made rotationally symmetric (Byard et al., 3 Sep 2025).

Graphene magnetic circular quantum dots provide a single-dot counterpart relevant to double-dot generalization. In that case, a circular quantum dot of radius $7.3$9 is defined by a magnetic profile and electrostatic potential in an infinite graphene sheet, and the energy spectrum is obtained by matching inner Landau-like solutions to outer Bessel solutions. The paper shows that the energy levels depend on radius, magnetic field, and electrostatic potential, and that interface states and an energy gap emerge. This suggests that a graphene circular double quantum dot could be built by coupling two such circular magnetic dots, with interdot tunneling controlled by overlap of interface-localized states and detuning of the individual dot spectra (Belouad et al., 2019).

The vertically stacked quantum-dot molecule extends the double-dot concept toward photonic quantum technologies. The coupled dots are separated by a $$\epsilon_{\alpha i} = \int d\mathbf{r} \, \phi_{\alpha i}^*(\mathbf{r})\left[ \frac{\hat{\mathbf{p}}^2}{2m^*} + V_p(\mathbf{r}) \right]\phi_{\alpha i}(\mathbf{r}),$$0 nm spacer and embedded in a p-i-n diode beneath a circular Bragg grating. For the positively charged exciton manifold, the work uses a $$\epsilon_{\alpha i} = \int d\mathbf{r} \, \phi_{\alpha i}^*(\mathbf{r})\left[ \frac{\hat{\mathbf{p}}^2}{2m^*} + V_p(\mathbf{r}) \right]\phi_{\alpha i}(\mathbf{r}),$$1 Hamiltonian with hole tunnel coupling $$\epsilon_{\alpha i} = \int d\mathbf{r} \, \phi_{\alpha i}^*(\mathbf{r})\left[ \frac{\hat{\mathbf{p}}^2}{2m^*} + V_p(\mathbf{r}) \right]\phi_{\alpha i}(\mathbf{r}),$$2, electron–hole exchange $$\epsilon_{\alpha i} = \int d\mathbf{r} \, \phi_{\alpha i}^*(\mathbf{r})\left[ \frac{\hat{\mathbf{p}}^2}{2m^*} + V_p(\mathbf{r}) \right]\phi_{\alpha i}(\mathbf{r}),$$3, and electric-field-controlled detuning. Experimentally, it reports an anticrossing splitting $$\epsilon_{\alpha i} = \int d\mathbf{r} \, \phi_{\alpha i}^*(\mathbf{r})\left[ \frac{\hat{\mathbf{p}}^2}{2m^*} + V_p(\mathbf{r}) \right]\phi_{\alpha i}(\mathbf{r}),$$4, extraction efficiency up to $$\epsilon_{\alpha i} = \int d\mathbf{r} \, \phi_{\alpha i}^*(\mathbf{r})\left[ \frac{\hat{\mathbf{p}}^2}{2m^*} + V_p(\mathbf{r}) \right]\phi_{\alpha i}(\mathbf{r}),$$5, and single-photon purity $$\epsilon_{\alpha i} = \int d\mathbf{r} \, \phi_{\alpha i}^*(\mathbf{r})\left[ \frac{\hat{\mathbf{p}}^2}{2m^*} + V_p(\mathbf{r}) \right]\phi_{\alpha i}(\mathbf{r}),$$6. This platform is electronically different from gate-defined lateral circular double dots, but it demonstrates that coupled-dot physics can be embedded in a circular photonic environment while retaining precise electrical control (Schall et al., 2021).

A recurring design tension across these platforms concerns simplicity, robustness, and spectral isolation. The hybrid silicon double-dot qubit emphasizes that only a double dot is needed, not a triple dot, and that all control is electrical (Shi et al., 2011). The six-electron circular double dot emphasizes improved screening but also notes denser many-body spectra in larger dots (Nielsen et al., 2013). The ring-based circular double dot emphasizes tunable angular momentum and a switchable spin–photon interface, but introduces disorder-sensitive crossover behavior between ring-state physics and the flopping-mode mechanism familiar from conventional double dots (Omlor et al., 18 Sep 2025).

7. Conceptual unification and scope of the term

The literature does not use the expression “circular double quantum dot” in a single, exclusive sense. In the most common lateral-semiconductor usage, it denotes a double quantum dot whose individual confinement minima are smooth, nearly harmonic, and approximately rotationally symmetric, so that circular-dot orbitals, Fock–Darwin shell structure, and symmetry-based simplifications of Coulomb integrals are appropriate. This is the sense most directly connected to exchange-based spin qubits, singlet–triplet physics, hybrid qubits, and rf charge sensing (Shi et al., 2011).

In a stricter theoretical sense, it can denote a model in which each constituent dot is explicitly a two-dimensional isotropic harmonic well, with shell filling and many-body states constructed in Fock–Darwin or Gaussian bases. That usage is especially clear in multi-electron singlet–triplet qubits, where local circular symmetry is central to the core–valence picture (Nielsen et al., 2013).

In a more specialized mesoscopic sense, it can denote a ring-shaped double quantum dot, where the two dots are defined along a circular path, detuning acts azimuthally, and the relevant orbital states may carry finite angular momentum. There the circular geometry is not an approximation to two lateral puddles; it is the defining topological feature of the device (Omlor et al., 18 Sep 2025).

Finally, in photonic quantum-dot-molecule devices, “circular” may refer to the surrounding circular Bragg grating rather than to the electronic dot shape. The coupled-dot degrees of freedom remain those of a double quantum dot, but the significance of circularity shifts from electronic symmetry to optical extraction and mode shaping (Schall et al., 2021).

Taken together, these usages show that circularity in double quantum dots can signify local rotational symmetry of each dot, exact isotropic harmonic confinement, ring topology with azimuthal tunneling, or circular photonic integration. What unifies them is the same structural core: two coupled quantum confinement regions whose low-energy properties are governed by interdot tunneling, detuning, Coulomb interactions, and symmetry. The specific meaning of “circular” determines which additional degree of freedom becomes important—dipole suppression in harmonic dots, shell filling in isotropic wells, angular momentum in rings, or directional emission in circular photonic structures.