9. Exchange interaction theory

The exchange interaction is the main resource for the implementation of two-qubit gates with spin qubits in semiconductors. Indeed, exchange leads to coherent oscillations between the spin states \(|\uparrow\downarrow\rangle\) and \(|\downarrow\uparrow\rangle\) of a system consisting of two electrons stored in neighboring quantum dots. This interaction may be leveraged to implement, e.g., a spin SWAP or an entangling \(\sqrt{\mathrm{SWAP}}\) gate between two Loss–DiVincenzo spin qubits [PJT+05], in which logical qubit states are encoded in the spin state of individual electrons or holes [LD98]. The exchange interaction is also a key computational resource in alternative single-qubit architectures like the singlet–triplet qubit [Lev02, PJT+05], all-exchange [DBK+00, MBT+13a, MBT+13b, RB15, ST16], or hybrid qubit architectures [KSS+14, KGF+12, SSP+12, SSW+14].

In this section, we describe two theoretical techniques that enable to calculate the exchange interaction strength using QTCAD.

Theoretical approaches

Two main methods are considered in this section to calculate the exchange interaction strength [BLP+23].

Perturbative kinetic exchange in the Fermi–Hubbard model — In this approach, we neglect Coulomb interactions between dots (which results in direct exchange), and instead focus on the distinct kinetic exchange interaction, which arises from the combined action of the Pauli exclusion principle and tunneling between the quantum dots [BLP+23]. To treat kinetic exchange, we approximate the two-electron Hamiltonian with the Fermi–Hubbard model, which we diagonalize using perturbation theory. This relates the exchange interaction to two easily computable single-electron quantities: the double dot tunnel coupling and the on-site Coulomb interaction. This computationally light approach will be taken in the Exchange coupling in a double quantum dot in FD-SOI—Part 1: Perturbation theory tutorial.
Exact diagonalization — In this approach, the two-electron system is exactly diagonalized within a truncated single-electron basis set [AGQ+22, GuccluSGH02, JBG+21]. This approach, which is often called full configuration interaction (full CI or FCI), is the most theoretically complete treatment of the two-electron system. However, it is also the most computationally expensive. The exact diagonalization approach will be taken in the Exchange coupling in a double quantum dot in FD-SOI—Part 2: Exact diagonalization tutorial.

Perturbation theory of exchange coupling in a double quantum dot

In this section, we review the theory of kinetic exchange in the Fermi–Hubbard model. This theory was already explored using the same methodology (perturbation theory) by several authors in a wide range of scientific publications (e.g., in [AUMFN21, BLP+23, CL05]). The key steps involved in this standard approach are outlined below.

We start by writing the Fermi–Hubbard Hamiltonian in the basis of localized single-dot ground states. In this basis, the Hamiltonian is:

(9.1)\[H = \sum_{\sigma\in\{\uparrow,\downarrow\}}\left[ \sum_{j\in\{\mathrm L,\mathrm R\}}\epsilon_j\;c^\dagger_{j\sigma}c_{j\sigma} + t_c\left(c^\dagger_{\mathrm L\sigma}c_{\mathrm R\sigma}+\mathrm{H.c.}\right) \right] + U \sum_{j\in\{\mathrm L,\mathrm R\}} c^\dagger_{j\uparrow}c_{j\uparrow}c^\dagger_{j\downarrow}c_{j\downarrow}.\]

Here, the labels \(\mathrm L\) and \(\mathrm R\) refer to the single-electron ground states of the left and right dots, respectively, in the absence of tunneling. Because of finite overlap between these states, a tunneling term appears in Eq. (9.1); this term is proportional to the tunnel coupling strength parameter \(t_c\). In addition, here, we neglect magnetic field, leading to a unique single-dot ground state energy \(\epsilon_j\) which is independent of spin. Finally, \(U\) parameterizes the strength of the on-site Coulomb interaction, which only contributes to total energy when two electrons of opposite spin occupy the ground state of the same quantum dot.

We contrast this version of the Fermi–Hubbard Hamiltonian with the one given by Eq. (8.13) in A simplified many-body Hamiltonian: The Fermi–Hubbard model.

Eq. (8.13) is written in the basis of the single-electron eigenstates of the coupled double quantum dot system. Therefore, while a tunneling term explicitly appears in Eq. (9.1), this term is absent from Eq. (8.13). This does not mean that tunneling is neglected in Eq. (8.13). Instead, in Eq. (8.13), tunneling is accounted for in the single-electron eigenbasis, which diagonalizes the Hamiltonian in the presence of tunneling.
In Eq. (8.13), we include a spin dependence in the single-electron energies \(\epsilon_{j\sigma}\). In Eq. (9.1), we neglect this spin dependence to simplify the theoretical treatment presented here.
In Eq. (8.13), we included both on-site (\(\propto U_i\)) and inter-site (\(\propto V_{ij}\)) Coulomb interactions. Here, in Eq. (9.1), we neglect inter-site interactions, again to simplify the theoretical treatment.
Finally, in Eq. (9.1), we assume that \(U_{\mathrm L} \approx U_{\mathrm R}\), i.e., we use a single on-site Coulomb interaction energy \(U\) for both quantum dots. This on-site Coulomb interaction energy is related to the Coulomb integrals presented in A simplified many-body Hamiltonian: The Fermi–Hubbard model through \(U\approx U_{\mathrm L} \equiv V_\mathrm{LLLL} \approx U_{\mathrm R} \equiv V_\mathrm{RRRR}\).

We remark that, in addition to the approximations presented above, for Eq. (9.1) to be an accurate description of the two-electron system, it is implicitly assumed that excited orbital states may safely be neglected. This is typically possible when the splitting between the ground and first excited single-dot orbital eigenstates is sufficiently large compared with thermal energy, and compared with any term in the Hamiltonian that may couple these states with the ground states.

To derive the exchange interaction strength, the Hamiltonian given by Eq. (9.1) may be written in the basis of spin singlet and triplet states. To do so, we introduce

(9.2)\[ \begin{align}\begin{aligned}|S(1,1)\rangle &\equiv \frac{1}{\sqrt2}\left( c^\dagger_{\mathrm L\uparrow}c^\dagger_{\mathrm R \downarrow} - c^\dagger_{\mathrm L\downarrow}c^\dagger_{\mathrm R \uparrow} \right)|0\rangle,\\|T_0(1,1)\rangle &\equiv \frac{1}{\sqrt2}\left( c^\dagger_{\mathrm L\uparrow}c^\dagger_{\mathrm R \downarrow} + c^\dagger_{\mathrm L\downarrow}c^\dagger_{\mathrm R \uparrow} \right)|0\rangle,\\|T_+(1,1)\rangle &\equiv c^\dagger_{\mathrm L\uparrow}c^\dagger_{\mathrm R\uparrow}|0\rangle,\\|T_-(1,1)\rangle &\equiv c^\dagger_{\mathrm L\downarrow}c^\dagger_{\mathrm R\downarrow}|0\rangle,\\|S(2,0)\rangle &\equiv c^\dagger_{\mathrm L\uparrow}c^\dagger_{\mathrm L\downarrow}|0\rangle,\\|S(0,2)\rangle &\equiv c^\dagger_{\mathrm R\uparrow}c^\dagger_{\mathrm R\downarrow}|0\rangle,\end{aligned}\end{align} \]

where \(|0\rangle\) is the vacuum state, in which no electron is present in the double dot system.

Writing Eq. (9.1) in this singlet-triplet basis results in

(9.3)\[\begin{split}H = (U+\Delta E) |S(2,0)\rangle\langle S(2,0)| + (U - \Delta E) |S(0,2)\rangle\langle S(0,2)|\\ + \sqrt 2 t_c \left( |S(2,0)\rangle\langle S(1,1)| + |S(0,2)\rangle\langle S(1,1)| + \mathrm{H.c.} \right),\end{split}\]

where \(\Delta E \equiv \epsilon_\mathrm L - \epsilon_\mathrm R\), and where we have set the zero of energy such that \(\epsilon_\mathrm L + \epsilon_\mathrm R \equiv 0\), corresponding to the energy of the bare (tunneling-less) singlet and triplet double-dot states.

We may then employ perturbation theory (e.g., through a Schrieffer–Wolff transformation), to eliminate the tunneling term for \(t_c \ll |U\pm\Delta E|\). Writing the full Hamiltonian of Eq. (9.3) as \(H = H_0 + W\), where the unperturbed Hamiltonian \(H_0\) is given by the diagonal terms in Eq. (9.3) and the perturbation \(W\) is given by the off-diagonal terms, we introduce the transformed Hamiltonian

(9.4)\[\tilde H = \mathrm e^S H \mathrm e^{-S},\]

where \(S\) is an anti-Hermitian operator that defines the Schrieffer–Wolff transformation \(\mathrm e^{-S}\). Choosing

(9.5)\[\begin{split}S = \frac{\sqrt 2 t_c}{U+\Delta E}\left( |S(2,0)\rangle\langle S(1,1)| - |S(1,1)\rangle\langle S(2,0)| \right)\\ + \frac{\sqrt 2 t_c}{U-\Delta E}\left( |S(0,2)\rangle\langle S(1,1)| - |S(1,1)\rangle\langle S(0,2)| \right),\end{split}\]

which diagonalizes \(H\) at first order in \(t_c/(U\pm\Delta E)\), then leads to the transformed Hamiltonian

(9.6)\[\begin{split}\tilde H = \left( U+\Delta E + \frac{2t_c^2}{U+\Delta E}|S(2,0)\rangle\langle S(2,0)| \right)\\ + \left( U-\Delta E + \frac{2t_c^2}{U-\Delta E}|S(0,2)\rangle\langle S(0,2)| \right)\\ - J |S(1,1)\rangle\langle S(1,1)| + O\left(\frac{t_c^3}{(U\pm\Delta E)^3}\right),\end{split}\]

where we have introduced the exchange interaction strength

(9.7)\[J = \frac{4 U t_c^2}{U^2-\Delta E^2}.\]

Using \(\Omega = 2 t_c\), where \(\Omega\) is the double-dot tunnel coupling introduced in Tunnel coupling in a double quantum dot in FD-SOI—Part 1: Plunger gate tuning, and setting \(\Delta E\) to zero, we finally arrive at the well-known expression

(9.8)\[J = \frac{4t_c^2}{U} = \frac{\Omega^2}{U}\]

for the exchange interaction strength in a symmetric bias configuration.

We conclude that, within this perturbative approach, the energy of the singlet state of the coupled system is lowered by \(J\) with respect to the energy of the triplet states, which are unaffected by exchange. As can be seen above, the exchange interaction may be tuned electrically either by varying the energy difference between the quantum dots via a plunger gate detuning, or by tuning the tunnel coupling through the barrier gate bias.

Exchange interaction through exact diagonalization

While computationally expensive, the exact diagonalization method is quite straightforward in its theoretical principles and execution. Indeed, in this method, we simply diagonalize the full many-body Hamiltonian given by Eq. (8.5) in a truncated basis set built from the eigenenergies of the single-electron Schrödinger’s equation. For a double quantum dot in a symmetric bias configuration and at zero magnetic field, the first four many-body eigenenergies are then divided into a ground singlet state and a triply-degenerate first excited state that corresponds to the spin triplet states. The exchange interaction strength \(J\) is then simply obtained from the energy difference between the singlet and triplet states, by definition.