Theory of the microwave spectroscopy of a phosphorus-donor charge qubit in silicon: Coherent control in the $\mathrm{Si}:\mathrm{P}$ quantum-computer architecture

We present a theoretical analysis of a microwave spectroscopy experiment on a charge qubit defined by a ${\mathrm{P}}_2^{+}$ donor pair in silicon, for which we calculate Hamiltonian parameters using the effective-mass theory of shallow donors. The master equation of the driven system in a dissipative system is solved to predict experimental outcomes. We describe how to calculate physical parameters of the system from such experimental results, including the dephasing time, $T_2$, and the ratio of the resonant Rabi frequency to the relaxation rate. Finally we calculate probability distributions for experimentally relevant system parameters for a particular fabrication regime.

Figure 1

(Color online) On the top is a schematic of the experimental setup showing the location of the bias gates relative to the phosphorus donors. On the bottom is a diagram showing orientation of the two donors, and the direction of the applied fields, relative to the silicon crystal lattice.

Figure 2

(Color online) A plot of the symmetric-antisymmetric energy gap ${\Delta }_{\mathrm{s}\text{−}\mathrm{as}}$, as a function of donor separation along three high symmetry crystallographic axes. The points denote fcc lattice sites, the lines are a guide to the eye.

Figure 3

(Color online) Donor spectroscopy for several donor separations with a small deviation from the ideal separation of $R=[40a,0,0]$, where $a=0.543\mathrm{nm}$ is the lattice constant of silicon. The data show that the peak heights and positions are not strongly dependent on these positional variations, however the width of the peak changes significantly. All data were calculated with a $40\mathrm{GHz}$ driving field of amplitude $E_{\mathrm{ac}}={10}^{-2}\mathrm{MV}∕\mathrm{m}$, and decoherence rates ${\Gamma }_z=3\mathrm{GHz}$, ${\Gamma }_{-}=1\mathrm{GHz}$, and ${\Gamma }_{+}=100\mathrm{MHz}$.

Figure 4

(Color online) An illustration of the effects of various decoherence channels on the spectroscopic signal height. These channels affect both the peak height and width as described in the text. The lengths are in units of $a=0.543\mathrm{nm}$, the lattice constant of silicon all data in this plot were taken for a donor separation $R=[40a,0,0]$, where $a=0.543\mathrm{nm}$ is the lattice constant of silicon. The frequency of the driving field was $40\mathrm{GHz}$ and its amplitude $E_{\mathrm{ac}}=0.01\mathrm{MV}{\mathrm{m}}^{-1}$. The decoherence rates are (a) ${\Gamma }_z=3\mathrm{GHz}$, ${\Gamma }_{-}=1\mathrm{GHz}$, ${\Gamma }_{+}=100\mathrm{MHz}$, (b) ${\Gamma }_z=3\mathrm{GHz}$, ${\Gamma }_{-}=10\mathrm{MHz}$, ${\Gamma }_{+}=1\mathrm{MHz}$, (c) ${\Gamma }_z=300\mathrm{MHz}$, ${\Gamma }_{-}=10\mathrm{MHz}$, ${\Gamma }_{+}=1\mathrm{MHz}$, (d) ${\Gamma }_z=300\mathrm{MHz}$, ${\Gamma }_{-}=100\mathrm{MHz}$, ${\Gamma }_{+}=1\mathrm{MHz}$.

Figure 5

(Color online) Donor spectroscopy for sample traces for donors of significantly different separations. This data plot is intended to illustrate that increasing the angle between the donor separation and the direction of the applied fields has less effect on the spectroscopic system than increasing the donor separation. All data were calculated with a $40\mathrm{GHz}$ driving field of amplitude $E_{\mathrm{ac}}={10}^{-2}\mathrm{MV}∕\mathrm{m}$, and decoherence rates ${\Gamma }_z=30\mathrm{MHz}$, ${\Gamma }_{-}=10\mathrm{MHz}$, and ${\Gamma }_{+}=1\mathrm{MHz}$.

Figure 6

(Color online) Probability distributions of system parameters for several donor implantation strategies. The first, labeled single, involves the $14\mathrm{keV}$ implantation of two single phosphorus ions in a single aperture of side length $30\mathrm{nm}$. The second strategy, labeled double, involves a $14\mathrm{keV}$ implant of one atom in each of two similar apertures, centered at a distance $50\mathrm{nm}$ apart along a 100 axis. The final strategy involves the $28\mathrm{keV}$ implantation of a single ${\mathrm{P}}_2$ molecule in a single aperture. These data were obtained using an ensemble of $27000$ randomly chosen points. (a) The distribution of donor separations $R$. (b) A log-scale plot of the distribution of the energy gap between the ground and first-excited states of the unperturbed qubit, $\Delta$. (c) A log-scale plot of the distribution of the dc field strength required to bring the system into resonance with a $40\mathrm{GHz}$ field. (d) A log-scale plot of the distribution of $\varphi$, the angle that parametrizes the degree of localization of the ground state. (e) A log-scale plot of the distribution of the magnitude of ${\gamma }_x$, the resonant Rabi frequency.