Quantum theory for electron spin decoherence induced by nuclear spin dynamics in semiconductor quantum computer architectures: Spectral diffusion of localized electron spins in the nuclear solid-state environment

We consider the decoherence of a single localized electron spin due to its coupling to the lattice nuclear spin bath in a semiconductor quantum computer architecture. In the presence of an external magnetic field and at low temperatures, the dominant decoherence mechanism is the spectral diffusion of the electron spin resonance frequency due to the temporally fluctuating random magnetic field associated with the dipolar interaction induced flip-flops of nuclear spin pairs. The electron spin dephasing due to this random magnetic field depends intricately on the quantum dynamics of the nuclear spin bath, making the coupled decoherence problem difficult to solve. We provide a formally exact solution of this non-Markovian quantum decoherence problem which numerically calculates accurate spin decoherence at short times, which is of particular relevance in solid-state spin quantum computer architectures. A quantum cluster expansion method is developed, motivated, and tested for the problem of localized electron spin decoherence due to dipolar fluctuations of lattice nuclear spins. The method is presented with enough generality for possible application to other types of spin decoherence problems. We present numerical results which are in quantitative agreement with electron spin echo measurements in phosphorus doped silicon. We also present spin echo decay results for quantum dots in GaAs which differ qualitatively from that of the phosphorus doped silicon system. Our theoretical results provide the ultimate limit on the spin coherence (at least, as characterized by Hahn spin echo measurements) of localized electrons in semiconductors in the low temperature and the moderate to high magnetic field regime of interest in scalable semiconductor quantum computer architectures.

Figure 1

(Color online) The electron of a P donor in Si experiences spectral diffusion due to the spin dynamics of the eveloped bath of Si nuclei. Of the naturally occuring isotopes of Si, only ${}^{29}\mathrm{Si}$ has a net nuclear spin which may contribute to spectral diffusion by flip-flopping with nearby ${}^{29}\mathrm{Si}$. Natural Si contains about 5% ${}^{29}\mathrm{Si}$ or less through isotopic purification. Isotopic purification or nuclear polarization will suppress spectral diffusion.

Figure 2

(Color online) Some possible nuclear “processes” where the numbered two-sided arrows represent a sequence of flip-flops between pairs of nuclei. (a) Depicts a two-nuclei process and (b) depicts a three-nuclei process.

Figure 3

(Color online) $L$ is loosely defined as the average number of neighbors in a near neighbor approximation that converges to the exact answer. For example, we can include neighbors up to a distance $r_0$ away such that increasing this maximum neighbor distance in the near neighbor approximation (where non-neighbor interactions are neglected) does not significantly change the solution.

Figure 4

(Color online) (a) The set of nuclei in a contributing cluster must form a connected graph. Edges represent neighbor connections. The set of blue nuclei on the left form a valid cluster. The set of green nuclei on the upper right are not fully connected so they do not form a valid cluster. (b) A set of $b_{nm}$ factors in a term of an expansion of $v_E\left(\tau \right)$ determines a set of disjoint clusters (the connected subgraphs formed from $b_{nm}$ edges).

Figure 5

(Color online) Set of all possible sets, $\left\{C_i\right\}$, of disjoint contributing clusters contained in a set, $\mathcal{S}$, of four nuclei as an example. Contributing clusters are of size 2 or greater (a single nucleus gives no contribution on its own). The cases on the left involve 2-nuclei, middle ones involve 3-nuclei, and the ones on the right are the trivial cases of $\left\{C_i\right\}=\varnothing$ or $\left\{C_i\right\}=\left\{\mathcal{S}\right\}$. Such possibilities are iterated over in the summation of Eq. (20). Equation (21) removes the $\left\{C_i\right\}=\varnothing$ case and Eq. (23) removes the $\left\{C_i\right\}=\left\{\mathcal{S}\right\}=\left\{\mathcal{C}\right\}$ case from their respective summations.

Figure 6

(Color online) One possible combination of simultaneously included pair contributions. The red/dark circles are the nuclei whose processes are being considered.

Figure 7

(Color online) The practical implementation of the cluster expansion approximates the ideal cluster expansion up to errors resulting from overlapping clusters. At the lowest order, such errors involve overlapping pairs. (a) A single pair multiplied by itself (i.e., a pair overlapping itself) and (b) two pairs overlapping by sharing a nucleus.

Figure 8

(Color online) Hahn echo decay $v_E(\tau ,\theta )$ of a phosphorus donor electron spin in silicon due to the dipolar nuclear spin bath dynamics. (a) Theory (solid lines) and experiment (Ref. 10) is shown for several orientation angles of the magnetic field with respect to the crystal lattice, ranging from the [100] to the [110] direction $(\theta =0,10,20,\dots ,90)$. (b) Here we plot $-\ln v_E(\tau ,\theta )+\ln v_E(\tau ,\theta =0)$, allowing for the removal of any decoherence mechanism which is independent of $\theta$. The qualitative and quantitative agreement between theory and experiment is remarkable, in contrast to the stochastic approach (dashed).

Figure 9

(Color online) Lowest order results for the natural log of the Hahn echo, $\ln \mathbf{\left(}{v}_E\left(\tau \right)\mathbf{\right)}\approx ⟨{\Sigma }_2\left(\tau \right)⟩\propto f^2$, for Si:P in a log-log plot. The solid lines give ${\Sigma }_2\left(\tau \right)$ with $f=1$. Dotted lines give $⟨{\Sigma }_2\left(\tau \right)⟩$ for natural Si $(f=4.67\%)$. In this log-log plot, multiplying by $f^2$ simply shifts the curves vertically. Isotopic purifucation would shift these curves up further. The two magnetic field angles shown give extremal results. Corresponding to $\theta$ angles in Fig. 8, $B\Vert \left[100\right]$ is $\theta =0°$ and $B\Vert \left[111\right]$ is $\theta \approx 54.7°$. Dashed lines fit the natural Si curves near their $-1$ values (where $v_E\sim 1∕e$) with ${\tau }^{2.3}$ power law curves (linear in the log-log plot).

Figure 10

(Color online) Relative errors (with scales on the right) to the log of the Hahn echo due to overlapping pairs for both 100% ${}^{29}\mathrm{Si}$ (top graph) and natural Si (bottom graph). In these examples, $B\Vert \left[100\right]$. The Hahn echoes themselves are shown, as well, with the 0 to 1 scales on the left. All curves share the same logarithmic time $\left(\tau \right)$ scale. It is apparent that these relative corrections are very small up to the tail of their respective echo decays.

Figure 11

(Color online) Successive orders of the cluster expansion for the natural log of the Hahn echo [Eq. (72)], computed for Si:P and $B\Vert \left[100\right]$, for the 100% ${}^{29}\mathrm{Si}$ theoretical scenario. The thick black line gives the lowest order result, ${\Sigma }_2\left(\tau \right)$, and other solid lines give higher order ${\Sigma }_k\left(\tau \right)$ results. The dotted lines give the negative of their corresponding functions provided to assist in the absolute value comparison of these higher order corrections. A failure of convergence occurs near $\tau \sim 1\mathrm{ms}$ where all of the curves are the same order of magnitude. This occurs well into the tail of the decay, however, and therefore has no practical consequence.

Figure 12

(Color online) Low order corrections to the Hahn echo, $v_E\left(\tau \right)$, computed for Si:P in natural Si $(f=0.0467\%)$ with $B\Vert \left[100\right]$. (top) Log-log plot of low order contributions to the natural log of the Hahn echo, $\ln \mathbf{\left(}{v}_E\left(\tau \right)\mathbf{\right)}$. Ordinate axis is negative as in the bottom graph of Fig. 11; however, dashed lines indicate negated curves (and thus represent positive values). (bottom) Conservative estimate of the absolute error of the lowest order Hahn echo result (scale on the right) due to 3-cluster contributions, $⟨{\Sigma }_3\left(\tau \right)⟩$. The lowest order Hahn echo result is shown as a reference (scale on the left). The logarithmic time scale is the same for all plots (top and bottom).

Figure 13

(Color online) This compares the $\tau$-expansion results with the lowest order cluster-expansion result, $\exp \left[{\Sigma }_2\left(\tau \right)\right]$ [Eq. (32)], for an arbitrary set of nuclei (ranging from 15 to 20 lattice constants away from the center) for the phosphorus donor in natural silicon. Shown are the $\tau$-expansion results at the highest two orders of $\tau$ that we have computed. Note the agreement with the cluster expansion results before the two $\tau$-expansion results diverge. Also shown is $1+{\Sigma }_2\left(\tau \right)$; its disagreement indicates the importance of accounting for simultaneous pairs in the cluster expansion.

Figure 14

(Color online) For GaAs quantum dots, $t_0$ (circles), the characteristic initial decay time, and $t_{1∕e}$ (diamonds), the $e^{-1}$ decay time, vs the Fock-Darwin radius $\ell$ for various quantum well thicknesses, $z_0=5$, 10, and $20\mathrm{nm}$. The orientation of the magnetic field is (a) along the $z_0$ confinement of the quantum dot and [100] lattice direction, or (b) perpendicular to the $z_0$ confinement direction and along [110] of the lattice.

Figure 15

(Color online) For GaAs quantum dots, $t_0$, the characteristic initial decay time, and $t_{1∕e}$, the $e^{-1}$ decay time, vs the Fock-Darwin radius $\ell$ for a quantum well thicknesses of $z_0=2.8\mathrm{nm}$. The orientation of the magnetic field is parallel to [110] of the lattice; this is perpendicular to the $z_0$ confinement of the quantum dot. This shows results both when including or excluding the indirect exchange coupling.

Figure 16

(Color online) Low order corrections to the Hahn echo, $v_E\left(\tau \right)$, computed for a GaAs quantum dot with $B\Vert \left[110\right]$, $z_0=10\mathrm{nm}$, and $\ell =50\mathrm{nm}$. (top) Log-log plot of low order contributions to the natural log of the Hahn echo, $\ln \mathbf{\left(}{v}_E\left(\tau \right)\mathbf{\right)}$. Ordinate axis is negative; however, dashed lines indicate negated curves (and thus represent positive values). (bottom) Conservative estimate of the absolute error of the lowest order Hahn echo result (scale on the right) due to 3-cluster contributions, $⟨{\Sigma }_3\left(\tau \right)⟩$. The lowest order Hahn echo result is shown as a reference (scale on the left). The logarithmic time scale is the same for all plots (top and bottom).