Figure 1

(Color online) (a) A schematic potential and energy-level diagram for a single quantum dot in which one electron is confined to the low-energy spectrum of a three-dimensional potential. Only the ground and first excited states, each a Kramer’s doublet, are shown. (b) The lowest orbital state has a spin-$1∕2$ electron interacting with the lattice nuclear spins. (c) Effective magnetic field due to both external field and the nuclear field. When the external field is large, the transverse components of the nuclear field are neglected in a rotating wave approximation.Reuse & Permissions

Figure 2

(Color online) (a) Charge stability diagram for a double-dot system. Double-dot occupation is denoted by $(n_l,n_r)$. The detuning is parametrized by $ϵ$, and the far-detuned regime (light blue) and charge transition (yellow) are shown. (b) Schematic of the double-well potentials along one axis $\left(x\right)$ with tight confinement in the other two axes (i.e., $y$ and $z$). In the far-detuned regime, the (1,1) charge states are the ground state, while in the charge transition regime, (0,2) can be the ground state. Triplet states are indicated in red, while electron charges are indicated in orange. (c) Energy-level structure of the double-dot system as a function of detuning. From left to right, the lowest-energy charge state as a function of $ϵ$ is (2,0), (1,1), and (0,2). The detuning at the middle of the graph corresponds to $ϵ=-E_c∕2$, where $E_c$ is the charging energy of a single dot. The three (1,1) triplet states (shown in red) are split by Zeeman energy.Reuse & Permissions

Figure 3

(Color online) A double quantum dot in the (1,1) configuration. (a) Schematic of the two-electron wave function in the far-detuned regime interacting with lattice nuclear spins. (b) Electron spins in the left and right dots interacting with their respective effective nuclear fields in the quasistatic approximation.Reuse & Permissions

Figure 4

(Color online) (a) Levels in the far-detuned regime, including all couplings of Eq. (20). (b) Levels near the charge transition; the $\mid T_{+}⟩\leftrightarrow \mid \tilde{S}⟩$ is near resonance, with the coupling between $\mid T_{+}⟩$ and $\mid \tilde{S}⟩$ indicated, as per Eq. (22).Reuse & Permissions

Figure 5

(Color online) (a) Energy-level structure as a function of detuning. Coupling to a Fermi sea with $k_{b}T⪢{\gamma }_e\mid B_{\mathrm{ext}}\mid$ leads to equal filling of all four low-energy states in the far-detuned regime (labeled start). Then $ϵ$ is changed rapidly with respect to the tunnel coupling, leading to all four spin states still in the (1,1) charge configuration. The time spent waiting in this configuration results in slow decay of the metastable (1,1) states to the $\mid (0,2)S⟩$ state. (b) Nuclear spins couple between the eigenstates of exchange, and slow inelastic decay at a rate $\Gamma$ proceeds from $\mid S⟩$ to $\mid (0,2)S⟩$. (c) The same process but for the eigenstates of the double-dot Hamiltonian. The $m_s=0$ states are equal superpositions of $\mid S⟩$ and decay rapidly to $\mid (0,2)S⟩$, while the $\mid m_s\mid =1$ states are only weakly mixed at large magnetic field with the $\mid S⟩$, resulting in slow decay to $\mid (0,2)S⟩$.Reuse & Permissions

Figure 6

(Color online) The product $I_l{I}_r$ as a function of external magnetic field $B$ in units of average nuclear field $B_{\mathrm{nuc}}=(B_{\mathrm{nuc},l}+B_{\mathrm{nuc},l})∕2$. Several ratios of dot nuclear fields, $r=B_{\mathrm{nuc},l}∕B_{\mathrm{nuc},r}$, are considered: $r=1$ (black), $r=1∕3$ (red), and $r=0$ (blue). Cusp behavior near zero field is found in the limit of highly inhomogeneous dot field strengths.Reuse & Permissions

Figure 7

(Color online) (a) Rapid-adiabatic passage: starting in the state $\mid (0,2)S⟩$, the detuning is changed from $ϵ⪢T_c$ to $ϵ⪡-T_c$, fast with respect to the nuclear energy scale ${\gamma }_e{B}_{\mathrm{nuc}}$. (b) Slow adiabatic passage: as above, but once the system is past the $S-T_{+}$ degeneracy point, the change of $ϵ$ is made slow with respect to the nuclear energy scale. The zoomed-in section shows the current nuclear energy splitting $\left(\widehat{\omega }\right)$ and the nuclear field eigenstates $\mid s,-s⟩$ and $\mid -s,s⟩$. Both procedures may be reversed to transfer either $\mid S⟩$ (RAP) or $\mid s,-s⟩$ (SAP) to $\mid (0,2)S⟩$ while keeping the other states within the (1,1) charge configuration, allowing for charge-based measurement of the system.Reuse & Permissions

Figure 8

(Color online) Pulse sequences: detuning parameter $ϵ$ versus time for (a) RAP for measurement of the singlet autocorrelation function, (b) SAP for the exchange-gate sequence, and (c) the singlet-triplet echo sequence. Blue is the charge transition region, while yellow is the far-detuned regime. The charge degeneracy point [the crossing from (1,1) to (0,2)] and the degeneracy between $\mid \tilde{S}⟩$ and $\mid T_{+}⟩$ [when $j\left(ϵ\right)={\gamma }_e\mid B_{\mathrm{ext}}\mid$] are shown for reference.Reuse & Permissions

Figure 9

Level structures of the double-dot system in the (a) far-detuned regime, with charge dephasing shown, and (b) near the charge transition with charge-based decay and dephasing.Reuse & Permissions

Figure 10

Decay of exchange oscillations for the four scenarios (clockwise from lower left): white noise, $1∕f$, Ohmic, and super-Ohmic $\left({\omega }^2\right)$. White corresponds to probability 1 of ending in the initial state $\mid s,-s⟩$ after exchange interaction is on for a time ${\tau }_E$ (bottom axis), while black is probability 0 of ending in the initial state. Tunnel coupling is taken to be $10\mu \mathrm{eV}$, and the spectral density $\left(\nu \right)$ is chosen for similar behavior near $ϵ=50\mu \mathrm{eV}$. Note that $1∕f$ terms increase decay in the slow oscillation limit, while increasing powers of $\omega$ (white $\text{noise}=0$, $\text{Ohmic}=1$, and $\text{super}\text{−}\text{Ohmic}=2$) lead to more oscillations for smaller exchange energies.Reuse & Permissions