Quantum decoherence of the central spin in a sparse system of dipolar coupled spins

Wayne M. Witzel, Malcolm S. Carroll, Łukasz Cywiński, and S. Das Sarma

Phys. Rev. B 86, 035452 – Published 30 July 2012

Abstract

The central spin decoherence problem has been researched for over 50 years in the context of both nuclear magnetic resonance and electron spin resonance. Until recently, theoretical models have employed phenomenological stochastic descriptions of the bath-induced noise. During the last few years, cluster expansion methods have provided a microscopic, quantum theory to study the spectral diffusion of a central spin. These methods have proven to be very accurate and efficient for problems of nuclear-induced electron spin decoherence in which hyperfine interactions with the central electron spin are much stronger than dipolar interactions among the nuclei. We provide an in-depth study of central spin decoherence for a canonical scale-invariant all-dipolar spin system. We show how cluster methods may be adapted to treat this problem in which central and bath spin interactions are of comparable strength. Our extensive numerical work shows that a properly modified cluster theory is convergent for this problem even as simple perturbative arguments begin to break down. By treating clusters in the presence of energy detunings due to the long-range (diagonal) dipolar interactions of the surrounding environment and carefully averaging the effects over different spin states, we find that the nontrivial flip-flop dynamics among the spins becomes effectively localized by disorder in the energy splittings of the spins. This localization effect allows for a robust calculation of the spin echo signal in a dipolarly coupled bath of spins of the same kind, while considering clusters of no more than six spins. We connect these microscopic calculation results to the existing stochastic models. We, furthermore, present calculations for a series of related problems of interest for candidate solid state quantum bits including donors and quantum dots in silicon as well as nitrogen-vacancy centers in diamond.

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Received 16 April 2012

DOI:https://doi.org/10.1103/PhysRevB.86.035452

Authors & Affiliations

Wayne M. Witzel^1,*, Malcolm S. Carroll¹, Łukasz Cywiński², and S. Das Sarma³

¹Sandia National Laboratories, New Mexico 87185 USA
²Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, PL 02-668 Warszawa, Poland
³Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA

^*wwitzel@sandia.gov

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Vol. 86, Iss. 3 — 15 July 2012

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Images

Figure 1
“Celestial map” representations of six randomly chosen spatial configuration instances of our canonical problem. Each show left and right hemispheres with the applied magnetic field in the up (north) direction.Reuse & Permissions
Figure 2
Example of a connected cluster of Lemma 1 where edges represent the existence of $b_{i, j}$ factors of a given term of ${\tilde{L}}_{C}$ .Reuse & Permissions
Figure 3
(Top) Depiction of a 2-cluster whose flip flops are suppressed by the magnetic field gradient generated by a third spin. (Bottom) Spin echo results from the CCE applied to spatial configuration instance A (see Fig. 1) when we do not define a cluster contributions to be “externally aware.” Each “spaghetti” strand is the result for a different random initial spin state (a product state of up or down for each spin) with the mean of $L_{CCE}^{(2)}$ [ $L_{CCE}^{(3)}$ ] as encircled $+$ 's [triangles]. The 3-cluster contributions must compensate for the lack of “external awareness” which leads to numerical instability at later times.Reuse & Permissions
Figure 4
$L_{CCE}^{(2)}$ (left) and $L_{CCE}^{(3)}$ spin echo results corresponding to spatial configuration instances A-F (see Fig. 1) of our canonical problem using the “externally aware” treatment of clusters. Each “spaghetti” strand is the result for a different random initial spin state (a product state of up or down for each spin) with the mean of $L_{CCE}^{(2)}$ [ $L_{CCE}^{(3)}$ ] as encircled $+$ 's [triangles].Reuse & Permissions
Figure 5
Illustration of $K (J, C, Γ)$ [see Eq. (10)]. Bath spins are denoted by the circles; up/down arrows denote the spin state “template” $J$ . Clusters in $Γ$ (the included clusters) are denoted by ovals that encompass multiple bath spins, $C$ being the one with the solid, thick outline. Superclusters have dashed outlines and all other clusters have dotted outlines. $K (J, C, Γ)$ are all spins states, including $J$ , that only differ from $J$ in spins within superclusters of $C$ , shown with the thicker outline and $↑ \leftrightarrow ↓$ or $↓ \leftrightarrow ↑$ inscriptions (the up/down arrow on the left, say, denotes the state of the $J$ template, but all combinations of these are included in $K$ ).Reuse & Permissions
Figure 6
$L_{CCE}^{(2)}$ (left), $L_{CCE}^{(3)}$ (middle), and $L_{CCE}^{(4)}$ (right) spin echo results corresponding to spatial configuration instances A-F (Fig. 1) of our canonical problem using “external awareness” and “interlaced spin averaging” in our implementation of the CCE as expressed in Eq. (9). Each “spaghetti” strand is the result for a different random spin state template $J$ [see Eq. (9)]. The mean of $L_{CCE}^{(2)}$ , $L_{CCE}^{(3)}$ , and $L_{CCE}^{(4)}$ are represented as encircled $+$ 's, triangles, and squares respectively. The brown (color online) dashed curves of the form $\exp (- t^{3})$ on the right panels are presented for comparison.Reuse & Permissions
Figure 7
Depiction of important 3-cluster (top) and 4-cluster (bottom) contributions in our adapted CCE with “external awareness” and “interlaced spin averaging.” At the 2-cluster level, all flip-flopping pairs are treated independently. The 3-cluster level must compensate for overlapping flip-flopping pairs that cannot be independent because they share a bath spin. The 4-cluster level must compensate for the fact that two separate flip-flopping pairs in close proximity influence each other magnetically (through the Ising-like ${\hat{S}}_{i}^{z} {\hat{S}}_{j}^{z}$ iteractions) as they flip-flop. This can enhance spectral diffusion, for example, if the two pairs are off resonance independently, but on resonance (conserve energy) together as a simultaneous process.Reuse & Permissions
Figure 8
$L_{CCE}^{(2)}$ (left) and $L_{CCE}^{(4)}$ (right) spin echo results of our canonical problem using “external awareness” and “interlaced spin averaging” in our implementation of the CCE as expressed in Eq. (9). We use the scaling parameter $ζ$ : time is scaled inversely with $ζ$ and $g^{2} C_{E} = ζ \times 4 \times 10^{13} / {cm}^{3}$ (i.e., $ζ = 1$ is the result for $g = 2$ and $C_{E} = 10^{13} / {cm}^{3}$ ). Each “spaghetti” strand is the result for a different random spatial configuration averaged over a large number of spin state templates. The mean of $L_{CCE}^{(2)}$ and $L_{CCE}^{(4)}$ are represented as encircled $+$ 's and squares respectively. The median of $L_{CCE}^{(2)}$ and $L_{CCE}^{(4)}$ are represented as encircled x's, and diamonds, respectively. The bottom figures show the same spin echo results on a logarithmic scale for the decay.Reuse & Permissions
Figure 9
Ensemble-averaged spin echo results and corrections for our canonical problem within various approximations using our adaptation of the CCE. The upper panel gives $L_{CCE}^{(2)}$ for various cutoffs in our cluster sampling heuristics (see Appendix ). The $ζ$ scaling parameter is used as in Fig. 8. Relative to corresponding solid line curves, dotted lines include twice as many clusters ( $N_{k}$ ), dashed lines increase the radial cutoff ( $R_{C}$ ) by $30 %$ , and dot-dashed lines do both. The blue curves (color online) in the top panel demonstrate cutoffs that are insufficient at later time scales ( $N_{2} = 200$ and $R_{C} = 600 nm$ for the solid blue line). The black curves use good cutoff values ( $N_{k} \sim 30 000$ and $R_{C} \sim 1000 nm$ ); for this reason, it is difficult to distinguish corresponding curves of differing line patterns. The lower panel shows successive corrections as we increase the maximum cluster size.Reuse & Permissions
Figure 10
Comparison of Hahn echo results for different $g_{0}$ values (the central spin $g$ -factor) in the large magnetic field limit and off-resonant central spin treatment (e.g., the canonical problem otherwise). The $ζ$ scaling parameter is used as in Fig. 8 with $g_{i > 0}^{2} C_{E} = ζ \times 4 \times 10^{13} / {cm}^{3}$ . The upper panels display $L_{CCE}^{(2)}$ (+'s), $L_{CCE}^{(3)}$ (triangles) and $L_{CCE}^{(4)}$ (squares) mean value results. The lower panels display $L_{CCE}^{(4)}$ median (diamonds) as well as mean (squares) value results on a logarithmic scale.Reuse & Permissions
Figure 11
Comparison of Hahn echo results for different degrees of thermal polarization: zero polarization (left), $B / T = 1$ (center), and $2 T / K$ (right). The zero polarization, infinite temperature results are shown in blue, provided for comparization along the polarization columns. The upper panels display $L_{CCE}^{(2)}$ (+'s), $L_{CCE}^{(3)}$ (triangles), and $L_{CCE}^{(4)}$ (squares) mean value results. The lower panels display $L_{CCE}^{(4)}$ median (diamonds) as well as mean (squares) value results on a logarithmic scale. The $ζ$ scaling parameter is used as in Fig. 8: $g^{2} C_{E} = ζ \times 4 \times 10^{13} / {cm}^{3}$ .Reuse & Permissions
Figure 12
(Top) Comparison of Hahn echo results for different resonance scenarios. We compare one bath species (left) and two bath species (right) with a 50/50 random mixture [see Eq. (14)]. Furthermore, results when the central spin is resonant with bath spins [see Eq. (13)] are shown in blue. The upper panels display $L_{CCE}^{(1)}$ (filled circles) $L_{CCE}^{(2)}$ (+'s), and $L_{CCE}^{(4)}$ (squares) mean value results. Below those, the panels display $L_{CCE}^{(4)}$ median (diamonds) as well as mean (squares) value results on a logarithmic scale. The $ζ$ scaling parameter is used as in Fig. 8: $g^{2} C_{E} = ζ \times 4 \times 10^{13} / {cm}^{3}$ . (Bottom) Depiction of a 1-cluster process when the bath spins are resonant with the central spin (left) and of an important type of 4-cluster process when there are two bath species (right).Reuse & Permissions
Figure 13
(Top) Hahn echo results on logarithmic scales for a central electron spin at various distances from a sheet of random electron spins at a density of $ζ^{2 / 3} \times 10^{11} / {cm}^{2}$ with a magnetic field parallel (left) and perpendicular (right) to the sheet. The distances are $ζ^{- 1 / 3} \times 10$ (black), $ζ^{- 1 / 3} \times 20$ (red), $ζ^{- 1 / 3} \times 40$ (green), and $ζ^{- 1 / 3} \times 80 nm$ (blue), as labeled and generally from left to right (central spins more distant from the surface have longer coherence times). The upper panels display $L_{CCE}^{(2)}$ (+'s), $L_{CCE}^{(3)}$ (triangles), and $L_{CCE}^{(4)}$ (squares) mean value results. Below those, panels display $L_{CCE}^{(4)}$ median (diamonds) as well as mean (squares) value results. (Bottom) Contour plots show relative strengths of dipolar couplings to the central spin from points on the sheet of bath spins (darker regions have larger relative coupling strengths).Reuse & Permissions
Figure 14
Color map representation of the nonzero dipolar tensor elements of Eq. (17) in the $x_{2} = 0$ plane for interactions with a quantum dot electron geometrically defined by Eq. (15). By the $x_{1}$ vs $x_{2}$ symmetry, the $x_{2} = 0$ plane is sufficient to express the dipolar information. Furthermore, due to the reflection symmetry about the $x_{1}$ and $x_{3}$ axes, one quadrant is sufficient for each of the nonzero tensor elements. We multiply the tensorial values by $x^{3} = {(x_{1}^{2} + x_{3}^{2})}^{3 / 2}$ for convenience. The probability density of the electron in the $x_{2} = 0$ plane is represented in a grayscale image (black for high probability ranging to white for low probability) between the upper and lower quadrants.Reuse & Permissions
Figure 15
(Top) Comparison of Hahn echo results affected by the spatial extent of the central spin electron to various degrees. Given a central spin “quantum dot” geometry defined by Eq. (15), we show results for various electron spin bath concentrations, $C_{E}$ , and compare with the canonical results in which the central spin has zero extent. The applied magnetic field is parallel (left) or perpendicular (right) to the surface whose normal is the $x_{3}$ direction of Eq. (15). We use $g = 2$ ( $g^{2} C_{E} = ζ \times 4 \times 10^{13} {cm}^{- 3}$ ). Using the scaling parameter $ζ$ , time is rescaled inversely with the bath concentration so that the “zero extent” curves are universal for all concentrations. The upper panels display $L_{CCE}^{(2)}$ (+'s), and $L_{CCE}^{(4)}$ (squares) mean value results. Below those, panels display $L_{CCE}^{(4)}$ median (dashed line with diamonds) as well as mean (solid line with squares) value results. (Bottom) Corresponding results when we use an artificial isotropic $1 / R^{3}$ interaction with the central spin. $L_{CCE}^{(2)}$ mean value (solid $+$ 's) and median value (dashed x's) results are displayed. Showing only 2-cluster results is sufficient to make our point: without the anisotropy of the interactions there is a consistent trend, monotonic in $ζ$ .Reuse & Permissions
Figure 16
(Top) Hahn echo result of a Gaussian-shaped quantum dot defined by Eq. (15) with decoherence induced from a two-dimensional bath, in the $x_{1}, x_{2}$ plane, 5 Å from its edge ( $x_{3} = 3$ nm). This was chosen as somewhat of a limiting case in terms of the closeness of the bath to the central spin quantum dot. Dashed lines show results for a point-like central spin for comparison. We show mean value $L_{CCE}^{(2)}$ (+'s), $L_{CCE}^{(3)}$ (triangles), and $L_{CCE}^{(4)}$ (squares) results, as well as median value $L_{CCE}^{(4)}$ (diamonds) results. Different colors are used to help make the curves more easily distinguishable. Bath densities are $10^{11} {cm}^{- 2}$ (left) and $10^{12} {cm}^{- 2}$ (right), and the applied magnetic field is perpendicular to the bath and dot (upper) or parallel to the bath and dot (lower). (Bottom) Depiction of a laterally extended quantum dot in the presence of sheet of bath spins.Reuse & Permissions
Figure 17
Hahn echo result of a Gaussian-shaped quantum dot defined by Eq. (15) with decoherence induced from a two-dimensional bath, in the $x_{1}, x_{2}$ plane, at various depth distances along $x_{3}$ . Dotted lines show results for a zero extent (point-like) central spin for comparison. Bath densities are $10^{11} {cm}^{- 2}$ (left) and $10^{12} {cm}^{- 2}$ (right), and the applied magnetic field is perpendicular to the bath and dot (top) or parallel to the bath and dot (bottom). As labeled, the depths are defined in correspondence to Fig. 13: $ζ^{- 1 / 3} \times 10$ (black), $ζ^{- 1 / 3} \times 20$ (red), $ζ^{- 1 / 3} \times 40$ (green), $ζ^{- 1 / 3} \times 80$ (blue), $ζ^{- 1 / 3} \times 160$ (orange), and $ζ^{- 1 / 3} \times 320$ (purple) nm, where $ζ = 1$ for $10^{11} {cm}^{- 2}$ and $ζ = 10^{3 / 2} \approx 32$ for $10^{12} {cm}^{- 2}$ ( $ζ^{- 1 / 3} = 10^{- 1 / 2} \approx 0.32$ ). Additionally, the 3 nm depth results from Fig. 16 are displayed in thick brown. These are all results of $L_{CCE}^{(2)}$ mean values (+'s) as a reasonable short time approximation to understand trends.Reuse & Permissions
Figure 18
Top left: probability distribution of magnetic field shifts due to background electron spins in the infinite temperature, point dipole limit. The distribution is over different random spatial realizations and spin states of the background spins. Top right: coherence versus time for ensemble FID (black), FID with characterization (red), Hahn spin echo (green), and CPMG (blue), with a left to right trend, respectively. We show mean values of $L_{CCE}^{(2)}$ (solid $+$ 's) and $L_{CCE}^{(4)}$ (solid squares). The bottom of the right panels displays results on a logarithmic scale and includes median values of $L_{CCE}^{(4)}$ (dashed diamonds). Bottom: coherence versus $τ$ , the time between $π$ pulse for the same pulse sequences. $τ = t / 2$ for the Hahn spin echo and $τ = t / 4$ for the two-pulse CPMG. The $ζ$ scaling parameter is used as in Fig. 8: $g^{2} C_{E} = ζ \times 4 \times 10^{13} {cm}^{- 3}$ .Reuse & Permissions
Figure 19
Hahn spin echo $T_{2}$ times, when the Hahn echo reaches a value of $\exp (- 1)$ , for phosphorus donors as a function of the fraction of $^{29}$ Si donors, or corresponding concentration $C_{N}$ , as well as the concentration of background donors $C_{E}$ . At high $C_{N}$ , contact hyperfine interactions dominate and $T_{2}$ is dependent upon the magnetic field direction relative to the lattice orientation. At low $C_{N}$ , $T_{2}$ is dependent upon $C_{E}$ , and eventually dominated only by dipolar interactions (which includes dipolar-approximated electronuclear interactions). Experimental results are shown as square symbols, from Ref. 15, and a star symbol at 50 ppm $^{29}$ Si, from Ref. 25. This figure is slightly revised from that of Ref. 23, updating the $50 ppm$ $^{29}$ Si experimental value to the published $T_{2} = 450 ms$ .Reuse & Permissions
Figure 20
Hahn spin echo vs time corresponding with the 50 ppm $^{29}$ Si and $C_{E} = 1.2 \times 10^{14} {cm}^{- 3}$ experiment of Ref. 25. Time is shown on a logarithmic scale in the main plot and a linear scale in the inset. The theory exhibits a $T_{2} \approx 600$ ms, reasonably close to the 450 ms $T_{2}$ fit of the experiment.[25] The short-time (first few percent of the decay) behavior is dominated by 1-cluster contributions shown with filled circle symbols; these are direct flip flops between the central spin and other donors. Indirect flip-flop processes then dominate the decay, 2-clusters (+'s) then 4-clusters (squares). In the $T_{2}$ regime near 600 ms, the decoherence is a combined effect from $^{29}$ Si flip flops as well as donor flip flops.Reuse & Permissions
Figure 21
Statistical dependence of the Hahn spin echo upon various random instances of spin baths, $^{29}$ Si (left) or donors (right), in the low-concentration limit. (Top) Hahn echo error (one minus the echo) for various percentiles, computed at each time point independently, and the ensemble averaged spin echo. Bottom: fits of the error to $1 - \exp [- {(t / T_{q})}^{n_{q}}] \approx {(t / T_{q})}^{n_{q}}$ in the $10^{- 4}$ error regime (motivated by common fault-tolerance thresholds) for various percentiles. The $ζ$ scaling parameter applies simultaneously to the time scale and concentration scale, having an inverse relationship in the low-concentration limit.Reuse & Permissions
Figure 22
$L_{CCE}^{(2)}$ Hahn spin echo results for a phosphorus nuclear spin in a bath of $^{29}$ Si. Mean values are encircled $+$ 's and median values are encircled x's. The left panels are for natural Si with $4.67 %$ $^{29}$ Si and includes lattice effects. The right panels are for varied low concentrations of $^{29}$ Si in the continuum limit. The bottom panels present the same data as respective top panels but in a logarithmic scale.Reuse & Permissions
Figure 23
Computed Hahn spin echo $T_{2}$ times, when the Hahn echo reaches a values of $\exp (- 1)$ as well as $T_{2}^{*}$ times, when the ensemble FID decay reaches $\exp (- 1)$ , for P donors as well as quantum dots (q-dots) defined with the probability density of Eq. (15) in Si. We consider decoherence induced by various concentrations of $^{29}$ Si. For our quantum dot model, we chose the lattice orientation such that [100] corresponds with the $x_{3}$ direction (normal to the confinement well) as depicted.Reuse & Permissions
Figure 24
Analogous to Fig. 8 but with the spin echo results for an NV center in a bath of P1 centers at a concentration of $C_{E} = ζ \times 10^{19} {cm}^{- 3}$ . Each “spaghetti” strand is the result for a different random spatial configuration averaged over a large number of spin state templates. The mean of $L_{CCE}^{(2)}$ and $L_{CCE}^{(4)}$ are represented as encircled $+$ 's and squares, respectively. The median of $L_{CCE}^{(2)}$ and $L_{CCE}^{(4)}$ are represented as encircled x's and diamonds, respectively. The bottom figures show the same spin echo results on a logarithmic scale for the decay.Reuse & Permissions
Figure 25
Illustration of the constructive proof for Lemma 2, showing a path with numbered edges and vertices that traverses the entire graph $G$ . A connected subgraph $H$ is formed from the edges and vertices of the path with the exclusion of the final edge and vertex. The edges and vertex that are excluded from $H$ are in gray.Reuse & Permissions
Figure 26
Schematic illustration of our algorithm to find the strongest clusters, the set of desired clusters $D$ , with $N_{k}$ quotas (in this example, 13 2-clusters, seven 3-clusters, and two 4-clusters). The dashed circle denotes the $R_{C}$ cutoff. The black dot in the center represent the central spin and other dots represent bath spins. (a) shows the initial population of 1-clusters within $R_{C}$ that must seed all selected clusters. (a)–(f) shows a progression of cluster selections, skipping a few selections at a time. The clusters grow from previously considered clusters (which may are may not be in $D$ depending on the $N_{k}$ quotas) based upon the interactions strengths of the spin system being represented.Reuse & Permissions

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