A minimal double quantum dot

Double quantum dots (DQDs) are a versatile platform for solid-state physics, quantum computation and nanotechnology. The micro-fabrication techniques commonly used to fabricate DQDs are difficult to extend to the atomic scale. Using an alternative approach, which relies on scanning tunneling microscopy and spectroscopy, we prepared a minimal DQD in a wide band-gap semiconductor matrix. It is comprised of a pair of strongly coupled donor atoms that can each be doubly charged. The donor excitation diagram of this system mimicks the charge stability diagram observed in transport measurements of DQDs. We furthermore illustrate how the charge and spin degrees of freedom of the minimal DQD may be used to obtain a single quantum bit and to prepare a Bell state. The results open an intriguing perspective for quantum electronics with atomic-scale structures.

H C describes the inter-donor coupling, which comprises three items: 1) H m , the Coulomb interaction between two donors; 2) H s , the indirect chemical potential shift; 3) H T , the coherent tunneling. The indirect chemical potential shift reflects the effect that the TIBB at one donor may induce a redistribution of the background charge around it and consequently change the local chemical potential at another donor.
In our STM experiment, the two donor atoms are spatially separated about 6.5nm, direct coherent tunneling is suppressed by the barrier, so the coherent tunneling term H T is negligible. (n 1 , n 2 ) are good quantum numbers and the system is in the semi-classical regime. To simulate our donor excitation diagram, position-dependent tip-induced band bending effect should be taken into account. This effect is mainly characterized by the shifted effective local chemical potential that is assumed to be proportional to V − V F (V F is the flat band voltage). It is modified by a tip-sample-distance-dependent factor β(x) describing the effect of the tip, bulk screening and the environment. It has been found that a squared Lorentzian factor β(x) = β 0 σ 2 σ 2 + x 2 2 (2) describes the experimental observations well. Therefore, the chemical potential µ t in Eqn. 1 becomes voltage (V ) and position (x) dependent.
To simulate a symmetric DQD, we set γ ij = γβ j , where β j is the TIBB form factor of site j and γ is a small dimensionless constant (about 0.1) describing the the strength of the inter-donor chemical potential shift due to TIBB. Figure S1 displays fits to two stronglycoupled symmetric DQD excitation diagrams from the experiments. The fits serve to derive the key parameters in Table 1 of the main text, such as the on-site binding energy d , the on-site Coulomb repulsion U and inter-site Coulomb interaction U m .

Quantum regime -Realization of entangled quantum states
Next we discuss the model in the presence of coherent tunneling, i.e. t 12 = 0. The occupation numbers of each site are no longer good quantum numbers, the semiclassical picture breaks down, and the system can be in states that are superpositions of semiclassical states. For simplicity, we assume that the microwave used for generating coherent tunneling couples to the donor pair through the electric dipole moment. Therefore, t ij is independent of the spin. We also assume that the frequency of the microwave is near-resonant to the transitions between the relevant states. The effective Hamiltonian for the light-coupled transitions can be written in a simple form with the rotating-wave approximation 1,2 . This approximation more clearly reveals the relevant physics as it leads to closed form solutions an avoids tedious mathematics.
To initialize the system to a desired state, e.g. in (0, 1) or (0, 2), the tip is moved to a suitable position and a bias voltage is applied.
1) Starting from the (0, 1) state, in order to operate the donor pair as a qubit, coherent tunneling must be enabled by adjusting the microwave frequency close to resonance with the (0, 1) − (1, 0) transition. The effective total Hamiltonian of the two-level system is given by where δ is often called detuning and Ω is denoted Rabi frequency in atomic physics. Here the detuning is defined as δ = E (0,1) + ω L − E (1,0) , where ω L is the frequency of the microwave.
The Rabi frequency Ω is proportional to the complex amplitude of the microwave. The qubit may be manipulated by tuning the amplitude and frequency of the microwaves.
In contrast to atomic or molecular systems, the spontaneous decay of the quantum entanglement in our DQD system is weak. The main source of decoherence here is thermal agitation. However, at 5 K, the temperature of the experiment, the thermal energy is much smaller than the smallest of all interactions, namely the inter-site Coulomb interaction U m ≈ 30 meV, which was determined from the spectroscopic data. Therefore, the single charge qubit implemented in the DQD should be stable.
2) The DQD may also be used to prepare a Bell state. We assume that all conditions identical to those in the previous example, except that the system is initialized to the (0,2) state, where two electrons occupy the same orbital of the same donor, thus forming a spin singlet. Then the near-resonant microwave couples the (0,2) state to the (1, 1) state that is fourfold degenerate due to the spin degrees of freedom. The effective total Hamiltonian is  1916 (1996).