Intrinsic phonon decoherence and quantum gates in coupled lateral quantum-dot charge qubits

Recent experiments by [Hayashi et al., Phys. Rev. Lett. 91, 226804 (2003)] demonstrate coherent oscillations of a charge quantum bit in laterally defined quantum dots. We study the intrinsic electron-phonon decoherence and gate performance for the next step: a system of two coupled charge qubits. The effective decoherence model contains properties of local as well as collective decoherence. Decoherence channels can be classified by their multipole moments, which leads to different low-energy spectra. It is shown that due to the super-Ohmic spectrum, the gate quality is limited by the single-qubit Hadamard gates. It can be significantly improved, by using double dots with weak tunnel coupling.

Figure 1

(Color online) Sketch of the two coupled identical charge qubits realized in a lateral quantum-dot structure. $d=100\mathrm{nm}$ is the distance of the dot centers in one qubit, $l=200\mathrm{nm}$ is the distance between the right dot center of qubit 1, and the left dot center of qubit 2. The width of the Gaussian wave function of an electron in each dot is $\sigma =5\mathrm{nm}$. The values chosen for the distances $d$ and $l$ are slightly smaller than in experimental realizations (Refs. 2, 4) in order to provide a lower bound for the decoherence times. Tunneling processes between both qubits, i.e., the QDs two and three in the chain, are quenched by applying appropriate gate voltages, as indicated by the gray box between the qubits.

Figure 2

Illustration of the even (top) and odd (bottom) contributions to the total rates. Filled circles indicate occupied dots. For long-wavelength modes, the energy shifts induced by underlying phonons in the two dots add up coherently in the even case but cancel in the odd case. Note that moving charges from the black to the white dots changes the dipole moment in the even but not in the odd case.

Figure 3

(Color online) Spectral functions $J_{e,o}\left(\omega \right)$ in the case of one common phonon bath for the fixed parameters $c_s=5000\mathrm{m}∕\mathrm{s}$, $g=0.05$, $d=100\mathrm{nm}$, $l=200\mathrm{nm}$, and $\sigma =5\mathrm{nm}$. Inset: zoom for small frequencies.

Figure 4

(Color online) Spectral functions $J_{e,o}\left(\omega \right)$, in the case of two independent phonon baths for the fixed parameters $c_s=5000\mathrm{m}∕\mathrm{s}$, $g=0.05$, $d=100\mathrm{nm}$, $l=200\mathrm{nm}$, and $\sigma =5\mathrm{nm}$. Inset: magnification for small frequencies.

Figure 5

(Color online) Temperature dependence of the relaxation and dephasing rates normalized by the single-qubit relaxation and dephasing rates. The two-qubit relaxation rate is given by the trace of the relaxation part of the Redfield tensor in secular approximation. The energy scales for the two-qubit transitions, $1\leftrightarrow 3$ and $2\leftrightarrow 4$, are comparable to the single qubit energy scale, the characteristic qubit energies are $E_s=(1∕8)\mathrm{GHz}$. The different cases are (a) ${\epsilon }_1={\Delta }_1=(1∕40)E_s$, ${\epsilon }_2={\Delta }_2=-(21∕40)E_s$, and coupling energy $K=0$; (b) ${\epsilon }_1={\Delta }_1=-(1∕2)E_s$, ${\epsilon }_2={\Delta }_2=-(21∕40)E_s$, and $K=0$); and (c) ${\epsilon }_1={\Delta }_1=-(1∕2)E_s$, ${\epsilon }_2={\Delta }_2=-(1∕2)E_s$, and $K=10E_s$. Note that cases (a) and (b) model uncoupled qubits and, especially for case (a), the overall relaxation rate for the two-qubit system is approximately twice that of the single-qubit relaxation rate when calculated for the dominating larger energy scale of the two-qubit system $[{\epsilon }_2={\Delta }_2=-(21∕40)E_s]$.

Figure 6

(Color online) Temperature dependence of the deviation of the four gate quality factors from their ideal values for the CNOT gate. The decoherence due to phonons is taken into account. The solid line shows the results for a single phonon bath and the dashed line is for two phononic baths. The characteristic qubit energies are $E_s=1∕4\mathrm{GHz}$ and the tunnel amplitudes are ${\Delta }_i=E_s$ $(i=1,2)$ due to the spacing of the double dots. In the curves for the deviation of the purity, we included lines for the analytical expressions Eq. (34) and Eq. (39), which agree perfectly with the numerical results.

Figure 7

(Color online) Temperature dependence of the deviation of the four gate quality factors from their ideal values for the CNOT gate. The decoherence due to phonons is taken into account. The solid line shows the results for a single phonon bath and the dashed line is for two phonon baths. The characteristic qubit energies are $E_s=1∕4\mathrm{GHz}$ and the tunnel amplitude during the Hadamard operation on the second qubit is ${\Delta }_2=1∕4E_s$, i.e., a factor 4 smaller than in Fig. 6. In the curves for the deviation of the purity, we included lines for the analytical expressions Eq. (34) and Eq. (39), which agree perfectly with the numerical results.