Non-Markovian dynamics of double quantum dot charge qubits due to acoustic phonons

We investigate the dynamics of a double quantum dot charge qubit which is coupled to piezoelectric acoustic phonons, appropriate for GaAs heterostructures. At low temperatures, the phonon bath induces the non-Markovian dynamical behavior of the oscillations between the two charge states of the double quantum dot. Upon applying the numerically exact quasiadiabatic propagator path-integral scheme, the reduced density matrix of the charge qubit is calculated, thereby avoiding the Born-Markov approximation. This allows a systematic study of the dependence of the $Q$ factor on the lattice temperature, on the size of the quantum dots, as well as on the interdot coupling. We calculate the $Q$ factor for a recently realized experimental setup and find that it is two orders of magnitudes larger than the measured value, indicating that the decoherence due to phonons is a subordinate mechanism.

Figure 1

Sketch of the geometry of the double quantum dot charge qubit and the various angles of the phonon propagation.

Figure 2

The bath autocorrelation function (response function) $L\left(t\right)=L_R\left(t\right)+i{L}_I\left(t\right)$ for the spectral density $J\left(\omega \right)$ [inset] of the phonon bath for the case of GaAs with $g_{\mathrm{ph}}=0.05$, $s=5\times {10}^3\mathrm{m}∕\mathrm{s}$, a dot radius of $a=60\mathrm{nm}$, and interdot distance $d=180\mathrm{nm}$. Temperature is $T=15\mathrm{mK}$. The dashed lines in the inset mark the superohmic limit $J(\omega \to 0)={\alpha }_{\mathrm{SO}}{\omega }_c^{-2}{\omega }^3$ at low frequencies and the high-frequency limit $J(\omega \to \infty )\propto 1∕\omega$.

Figure 3

Time evolution of the charge population of the left dot for the case $\Delta =27\mu \mathrm{eV}$ and $T=15\mathrm{mK}$. Shown are the exact results obtained from QUAPI (◆) and the result of a fit of an exponentially decaying cosine to ${\rho }_{11}\left(t\right)$ (solid line). The dashed line indicates the result of a weak-coupling (Born-Markov) approximation using Eqs. (11, 12) for the same parameters. Deviations from the exact results are striking. The remaining parameters are the same as in Fig. 2.

Figure 4

The quality factor $Q$ of the coherent charge oscillations as a function of the tunneling amplitude $\Delta$ in natural units (lower scale) and scaled with respect to ${\omega }_c$ (upper scale). The data shown are obtained with QUAPI (+) using the phonon spectral density [Eq. (8)] and with the weak-coupling approximation (wca) of Eqs. (11, 12). The parameters are the same as in Fig. 2.

Figure 5

Zoom into the small $\Delta$ region. Shown are the QUAPI results for the phonon (+) and the pure Ohmic (▾) spectral density. In addition, they are compared with the approximate analytical, weak-coupling results of Eqs. (11, 12) for the superohmic (solid line) and the ohmic (dashed line) cases. The parameters are for the Ohmic case, $\alpha =0.05$, and for the superohmic case, ${\alpha }_{\mathrm{SO}}=0.075$. For all cases, $T=15\mathrm{mK}$.

Figure 6

$Q$ factor as a function of temperature for two different tunneling amplitudes $\Delta$. The remaining parameters are the same as in Fig. 2.

Figure 7

$Q$ factor as a function of the dot radius $a$ for two different tunneling amplitudes $\Delta$ and for a fixed ratio $d∕a=3$. The remaining parameters are the same as in Fig. 2.