# A Quantum Network of Silicon Qubits using Mid-Infrared Graphene Plasmons

M. J. Gullans  Joint Quantum Institute, National Institute of Standards and
Technology, Gaithersburg, MD 20899, USA  J. M. Taylor  Joint Quantum
Institute, National Institute of Standards and Technology, Gaithersburg, MD
20899, USA

###### Abstract

We consider a quantum network of mid-infrared, graphene plasmons coupled to
the hydrogen-like excited states of group-V donors in silicon. First, we show
how to use plasmon-enhanced light-matter interactions to achieve single-shot
spin readout of the donor qubits via optical excitation and electrical
detection of the emitted plasmons. We then show how plasmons in high mobility
graphene nanoribbons can be used to achieve high-fidelity, two-qubit gates and
entanglement of distant Si donor qubits. The proposed device is readily
compatible with existing technology and fabrication methods.

###### pacs:

03.67.-a, 81.05.ue, 73.20.Mf, 78.55.Ap

The spins of Group-V donors (P, Bi, As, Sb) in silicon have long been
intriguing as potential quantum bits Zwanenburg et al. (2013). Dramatic
advances in the implementation of these ideas continue apace, including
demonstration of state initialization and readout, robust single qubit
operations Zwanenburg et al. (2013); Yang et al. (2006, 2009); Sekiguchi et
al. (2010); Wolfowicz et al. (2013), and quantum memory exceeding an hour
Saeedi et al. (2013). However, localized control, single-shot readout Morello
et al. (2010); Pla et al. (2012), and coherent coupling between two qubits
Koiller et al. (2001); Weber et al. (2014) remain challenging. One approach to
overcome these challenges, which has been successful in other solid-state
impurity systems such as nitrogen-vacancy centers in diamond Jel  and quantum
dots Hen , is to use higher orbital states to control the spin optically Vinh
et al. (2008); Greenland et al. (2010). This allows mapping of the spin-state
into light, which can be used to both readout the spin and induce entanglement
between distant impurities Bernien et al. (2013). In the case of the group-V
donors in Si, such higher orbital excitations lie in the mid- to far-infrared,
making it challenging to use conventional optics to achieve this goal.

Recently, graphene has emerged as a powerful platform for plasmonics in this
mid-infrared frequency regime Jablan et al. (2009); García de Abajo (2014).
Graphene’s high mobility Mayorov et al. (2011); Elias et al. (2011);
Baringhaus et al. (2014), versatile fabrication, and the ability to tune the
plasmon properties via external gate voltages has already suggested many
potential applications in this challenging portion of the electromagnetic
spectrum Low and Avouris (2014). Furthermore, Si is a natural base substrate
for graphene plasmonics due to its ubiquitousness in fabrication technology
and its low loss in the mid-infrared. This coincidence of properties suggests
that graphene plasmons may be a powerful resource for orbital control of Si
impurity qubits.

In this Letter, we propose to achieve this control by coupling the Si
impurities to mid-infrared plasmons in a graphene nanoribbon. First, we show
how to use the plasmon enhanced light-matter interaction to achieve single-
shot spin readout. We then show how plasmons in high mobility nanoribbons can
be used to achieve high-fidelity, two-qubit gates and entanglement of distant
Si donor qubits.

Figure 1: (a) Two Si dopant qubits are coupled via plasmons in a graphene
nanoribbon (GNR) that is encapsulated in hexagonal-BN (h-BN). The plasmons are
detected by a field effect transitor (FET), allowing single-shot readout of
the spin state. $d$ is the distance from the qubit to the nanoribbon. (b) Top
view of the device. The color scale shows the electric field intensity of the
plasmon mode. $W$ is the width of the nano ribbon, $\lambda_{sp}$ is the
wavelength of the plasmon, and $r=n\,\lambda_{sp}/2$ ($n$=1 for the case
shown) is the qubit separation. (c) Spectrum of the group-V donors in Si with
the pathway for the plasmon decay (red) and phonon decay (black). We assume a
strain is applied to make the $2s$ and $2p$ state degenerate, which allows
spin selective excitation from the 1$s$ ground states to the $2p$ states
through the exciton state $D^{0}X$ by controlling the polarization of the
excitation light. We work with the reduced set of states: the two electron
spin ground state labeled $|g,s\rangle$ and a Kramer’s pair in the 2s-2p
manifold $|e_{s}\rangle$. (d) Dispersion of the first three plasmon modes in
the nanoribbon ($W=50$ nm and $n_{e}=10^{12}$ cm-2) and Si:Sb, P, As 1$s$
ground to $2p_{0}$ transitions and Si:Bi 1$s$ to $2p_{\pm}$ transition.

The device we consider is illustrated in Fig. 1(a-b). Shallow dopant atoms in
Si are coupled via plasmons in a graphene nanoribbon (GNR). The excited states
of the impurity have a hydrogen-like spectrum with a manifold of $1s$ ground
states and excited $s,~{}p,~{}\ldots$ states [Fig. 1(c)] Ramdas and Rodriguez
(1981). As illustrated in Fig. 1(d), these transitions have a strong overlap
with the plasmonic modes of the nanoribbons. This gives rise to a strong
Purcell enhancement of the radiation from the excited states into plasmons
Chang et al. (2007); Koppens et al. (2011). This is the crucial feature of
this system, which, when combined with spin-selective excitation, allows
control, readout and entanglement of the impurity spins.

In our analysis, we restrict ourselves to the reduced level-scheme labeled in
Fig. 1(c) and detailed in the supplemental material sup . There are two
electron spin ground states $|g,s\rangle$ [$1s(A_{1})$ state] coupled via
optical, spin selective excitation to a Kramer’s doublet of excited states
$|e_{s}\rangle$ (hybridized $2p$-$2s$ state). The excited state can decay back
to the ground state either through phonons or plasmons. Spin-selective
excitation is achieved with two methods that take advantage of optical
excitation to the spin-3/2, donor bound, exciton state $D^{0}X$ as the first
leg in a two photon transition from the $1s$ ground state to the $2p$ states.
The first, demonstrated in Fig. 1(c), uses circularly polarized light on one
leg of the lambda transition and linearly polarized light on the second. We
refer to this as the $\sigma_{\pm}\pi$-excitation scheme. This will only
excite one spin state, e.g., spin-down to spin-up as in Fig. 1(c), while
leaving the other spin state unaffected. In the second approach, we apply
opposite circular polarization on each leg, which excites both spin states,
but at different rates due to the difference in Clebsch-Gordon coefficients.
We refer to this as the $\sigma_{\mp}\sigma_{\pm}$-excitation scheme.

We first show how the strong Purcell enhancement can be combined with
electrical detection to achieve single-shot readout. Plasmonic excitations in
graphene nanoribbons have been systematically characterized in Ref.
Christensen et al. (2012). Their properties are similar to optical waveguides
with a discrete spectrum of transverse modes that reduces to a single mode as
the width decreases to zero. In the single mode regime of a nanoribbon with
width $W$ and length $L$, the emission rate into the plasmons can be
approximated by Fermi’s golden rule $\gamma_{p}=2\pi
g^{2}_{k}(\omega)D(\omega)$ where $D(\omega)=L/2\pi v_{g}$ is the density of
states and $v_{g}=d\omega/dk=\omega W\left\lvert\eta^{\prime}\right\rvert/2$
is the group velocity. Here $\eta^{\prime}$ (dimensionless and $\sim 1$) is
the derivative of a universal function $\eta(kW)$ associated with the lowest
order mode (defined below) and $g_{k}$, analyzed in detail later, can be
interpreted as the quantized electric field per plasmon times the dipole
moment of the impurity. The small mode volume and reduced group velocity of
the plasmon compared to free space gives the Purcell enhancement for a single
impurity

$\frac{\gamma_{p}}{\gamma_{r}}=\frac{3\,\chi}{16}\frac{c^{3}}{v_{g}^{3}}\,\left\lvert\eta^{\prime}(kW)\right\rvert^{2}\,e^{-4\pi
d/\lambda_{sp}},$ (1)

where $d$ is the distance from the impurity to the nanoribbon, $\chi$ is a
factor due to the surrounding dielectric environment, $c$ is the speed of
light, $\gamma_{r}\sim 1$ kHz is the radiative decay rate of the $e_{s}$
states, and $\lambda_{sp}=2\pi/k$ is the plasmon wavelength. As $v_{g}$
approaches speeds as low as the Fermi velocity $v_{F}\approx 10^{6}$ m/s, the
Purcell enhancement approaches $10^{8}$ compared to free space Koppens et al.
(2011). The enhancement relative to the phonon broadening $\gamma$ is shown
for the Si:P $1s(A_{1})-2p_{0}$ transition in Fig. 2(a) and for the Si:Bi
$1s(A_{1})$-2$p_{\pm}$ transition. Crucially, near the nanoribbon, the plasmon
emission exceeds the phonon decay by several orders of magnitude, enabling
deterministic conversion of the excited state population into a propagating
plasmon in the nanoribbon.

To achieve single shot-readout we use the $\sigma_{-}\sigma_{+}$-excitation
scheme. In this case, the spin-down state will emit plasmons at nine times the
rate of the spin-up state, enabling one to distinguish the spin-states via the
difference in the number of emitted plasmons. However, when the excited state
relaxes via phonon decay, the spin will become depolarized by the
1$s(T_{2}/E)$ valley states which have a large spin-orbit coupling. As a
result, the optimal excitation time scales as $\sqrt{\gamma/\gamma_{p}}$ and,
thus, the excess in the emitted plasmon number will scale with the same factor
sup . For large Purcell enhancement, one could achieve single-shot readout of
the impurity spin by detecting the plasmons electrically with a field effect
transistor (FET) [Fig. 1(b)] as was demonstrated for optical plasmons in Ref.
Falk et al. (2009). Figure 2(b) shows the distribution of the emitted plasmon
number for $\gamma_{p}/\gamma=300$, where the two distributions can be
distinguished with a fidelity of $96\%$, demonstrating the potential for
single-shot readout.

Figure 2: (a) Enhancement of plasmon decay compared to phonon decay with
vertical distance of the qubit from the nanoribbon for Si:P/Bi (solid/dashed).
Here $W=10/5$ nm and $n_{e}=2\cdot 10^{11}/7\cdot 10^{11}$ cm-2 for P/Bi.
(inset) Reduced level diagram with the $\sigma_{-}\sigma_{+}$-excitation
scheme. Plasmon decay (red) occurs at an enhanced rate compared to phonon
decay (black). (b) Finite time fluorescence distribution conditioned on the
initial spin state (here $\gamma_{p}/\gamma=300$). Readout fidelity is
determined by the distinguishability of the two distributions. In the case
shown the spin-up state (red) can be distinguished from the spin-down state
(blue) with a fidelity as high as 96$\%$ by assigning events with more than 10
detected plasmons as spin-down.

We have now established the necessary ingredients to build a quantum network
of the impurities. However, before we can give a rigorous analysis of impurity
entanglement and two-qubit gates, we need to explore the details of the
plasmonic modes and the spin-impurity coupling.

Focusing on these details, we define the universal function $\eta$ for the
plasmons Christensen et al. (2012). An important difference from conventional
optics is that the plasmons are near field excitations and can be accurately
described by electrostatics. As a result, the free space wavelength does not
set an absolute scale and the dispersion relation $\omega_{j}(k)$ of each
transverse mode $j$ is determined by the width of the nanoribbon $W$ and the
universal function $\eta_{j}(x)$

$\omega_{j}(k)=\frac{\chi}{4\pi\epsilon_{0}}\frac{\mathrm{Im}[\sigma(\omega_{j})]}{\eta_{j}(kW)\,W}.$
(2)

Here $\epsilon_{0}$ is the dielectric constant and the conductivity is well
approximated by the Drude formula
$\sigma(\omega)=\frac{ie^{2}}{\pi\hbar}\frac{\omega_{F}}{\omega+i\kappa}$ for
frequencies far below twice the Fermi frequency $\omega_{F}$ ($\kappa$ is the
inverse scattering time, $\hbar$ is Planck’s constant and $e$ is the
elementary charge) Thongrattanasiri et al. (2012). In Fig. 1(d) we show the
dispersion for the first three modes using the universal functions found in
Ref. Christensen et al. (2012).

The coupling $g_{k}$ between a plasmon and the orbital state of the impurity
is determined by the interaction Hamiltonian $H_{I}=-\bm{\mu}_{I}\cdot\bm{E}$,
where $\bm{\mu}_{I}$ is the dipole moment of the impurity and $\bm{E}$ is the
electric field of the graphene plasmon. For small enough widths that the
impurity is only resonant with the first mode in Fig. 1(d), we can write the
Hamiltonian for a pair of dopant atoms interacting with the plasmons as

$\displaystyle H$
$\displaystyle=\omega_{d}\sum_{i,s}\mbox{$|e_{is}\rangle$}\mbox{$\langle
e_{is}|$}+\sum_{k}\omega(k)a_{k}^{\dagger}a_{k}$
$\displaystyle+\sum_{k,s}g_{k}a_{k}(\sigma_{1s}^{+}+e^{ikr}\,\sigma_{2s}^{+})+h.c.,$
(3)

where we take the dipole polarized perpendicular to the nanoribbon. Here
$\sigma_{is}^{+}=\mbox{$|e_{is}\rangle$}\mbox{$\langle g_{i},s|$}$,
$\sigma_{is}^{-}=(\sigma_{is}^{+})^{\dagger}$, $a_{k}$ are the bosonic
operators for the plasmon modes, $\omega_{d}$ is the transition frequency of
the donor, $\omega_{k}$ is the plasmon frequency, and $g_{k}$ is plasmon-
impurity coupling. The lowest order mode is a monopole with the charge density
approximately constant across the nanoribbon. Knowing the mode function and
the dispersion allows us to derive an analytical expression for
$g_{k}=\frac{e\,\chi\mu_{I}}{4\pi\epsilon_{0}\eta
W}\sqrt{\frac{\omega_{F}}{\omega(k)\pi LW}}e^{-kz}$ by quantizing the plasmon
energy, which has contributions from both kinetic and electromagnetic energy
Gullans et al. (2013).

The system we have described falls into a well known class of quantum optics
models where several emitters are strongly coupled to a common, infinite range
(without losses) optical mode. It is analogous to a cavity quantum
electrodynamics (cQED) model in the weak coupling, but large Purcell factor
limit. As a result, we can draw on several established techniques for
achieving two-qubit gates Pellizzari et al. (1995); Domokos et al. (1995) and
entanglement Reiter et al. (2012) in these models. The enabling feature is the
emergence of super-radiance due to quantum interference of the emitted light
into a single mode Gross and Haroche (1982). In particular, when $kr=n\pi$ for
integers $n$ the plasmon emission from the two impurities maximally interferes
and the system is super-radiant because there are two states $|ee\rangle$ and
$\mbox{$|ge\rangle$}+(-1)^{n}\mbox{$|ge\rangle$}$ with collectively enhanced
decay $2\gamma_{p}$ and one sub-radiant state
$\mbox{$|ge\rangle$}-(-1)^{n}\mbox{$|eg\rangle$}$, which decays at the rate
$\delta\gamma=\gamma+\gamma_{p}\,\big{(}1-e^{-r/L_{p}}\big{)}$, where $r$ is
the distance between the impurities and
$L_{p}=2\mu\,v_{g}\omega_{F}/e\,v_{F}^{2}$ is the plasmon propagation length,
which increases with the mobility $\mu$ Jablan et al. (2009). Room temperature
mobilities approaching 106 cm2/Vs have been demonstrated for bulk graphene
surrounded by h-BN Mayorov et al. (2011); Elias et al. (2011), which would
allow mid-infrared plasmons to coherently propagate for several microns. The
narrowing of the sub-radiant state is measured by the ratio
$f=\delta\gamma/\gamma_{p}$. Below we show how to utilize the sub-radiant
state to achieve a two-qubit phase gate and entanglement via dissipation. For
these protocols, the error probability scales with $f$, making it crucially
important that $f\ll 1$. Figure 3(a) shows $f$ for $n=1$ with increasing
mobility, where we see that it can be below $1\%$.

Figure 3:  (a) Scaling of the sub radiance narrowing $f$ for $n$=1 with
increasing mobility for (red) a fixed width W=20(10) nm and a carrier doping
of $n_{e}=$2$\cdot 10^{12}$(7$\cdot 10^{12}$) cm-2 for P(Bi) and (blue) for an
optimized nanoribbon width and carrier doping. For the optimized case $W$
decreases with the mobility from 25 nm to 3 nm and the doping decreases from
$10^{13}$-1011 cm-2. The vertical line is the highest observed mobility for
graphene encapsulated in h-BN. (inset) Reduced level diagram with the
$\sigma_{-}\pi$-excitation scheme. (b) Error probability for two-qubit
entanglement preparation and the C-$\pi$ gate for Si:P and Si:Bi. We take a
graphene mobility of $6\cdot 10^{4}$ cm${}^{2}/$V$\cdot s$ and optimize the
doping and nanoribbon width at each point in the same range as in (a).

To implement a C-Phase gate we use the $\sigma_{-}\sigma_{+}$-excitation
scheme, where the driving field takes the form

$\displaystyle H_{c}$
$\displaystyle=\frac{2\Omega^{\prime}(t)}{3}\sum_{s}\big{(}\sigma_{1s}^{+}-e^{i\theta}\,\sigma_{2s}^{+}+h.c.)(1-s)$
(4)

here $s=\pm 1/2$, $\Omega^{\prime}(t)$ is the Raman Rabi frequency of the two-
photon transition from $g$ to $e$ and $e^{i\theta}$ is the relative phase of
the drive, which we set to $\textrm{sign}(\cos kr)=(-1)^{n}$.

All of the two-qubit states are excited to sub-radiant states by this drive,
but each at different rates because of a combination spin-selective excitation
and the inability to excite the super-radiant states due to a strong quantum
Zeno effect. For a far off resonant, weak drive we can imprint a different
phase on each state with minimal dephasing. The resulting unitary operation is
locally equivalent to a C-Phase gate with the phase $e^{i\phi(t)}$ where
$\phi(t)=4\int
dt^{\prime}\left\lvert\Omega^{\prime}(t^{\prime})\right\rvert^{2}/9\delta$
Pellizzari et al. (1995); Domokos et al. (1995). Such a gate is universal for
quantum computation. To prevent errors during the gate operation, we require
$\delta\gg\delta\gamma$ to prevent dephasing from the sub-radiant state. In
addition, we require $\delta\ll\gamma_{c}$ because the phase picked up from
the super-radiant states cancels out the desired phase on
$|\uparrow\uparrow\rangle$. These two constraints lead to the optimal, average
gate fidelity when $\delta\sim\sqrt{\delta\gamma\,\gamma_{c}}$ as
$\mathcal{F}_{\phi}\approx 1-3\sqrt{3}\,\sqrt{{f}}\,\phi/4$ Fig. 3(b) shows
the minimized error probability $1-\mathcal{F}_{\pi}$ for a C-$\pi$ gate with
increasing dopant separation.

The two-qubit gate described above is coherent and deterministic; as a result,
any dephasing from phonons or plasmon loss will lead to errors. To create an
entangled state between the two qubits, one can overcome this limitation by
engineering the desired state to be a unique dark state of some driven, open
system dynamics. For our system, where two-qubits are coupled via an excited
state to a common bosonic mode, it has been well established that one can
drive the qubits towards a maximally entangled state with an error probability
that scales linearly in $f$ instead of as the square root Reiter et al.
(2012). We can achieve this in our case with the $\sigma_{-}\pi$-excitation
scheme that excites spin-up electrons to spin-down electrons in the excited
state

$H_{c}=\Omega(t)\big{(}\mbox{$|e_{1\downarrow}\rangle$}\mbox{$\langle
g_{1},\uparrow|$}-e^{i\theta}\,\mbox{$|e_{2\downarrow}\rangle$}\mbox{$\langle
g_{2},\uparrow|$}+h.c.\big{)}$ (5)

where $\Omega$ is the Raman rabi frequency and $\theta$ is defined as above.

With the addition of transverse magnetic field, the singlet state becomes the
unique dark state because it is not coupled to any sub-radiant states or
triplet states. As a result the system is optically pumped into the singlet
state. We give only a brief description here, since similar schemes based on
cQED and plasmons have been extensively analyzed Reiter et al. (2012). The
crucial feature is that when the triplet states are excited to the sub-radiant
states, at rate $\Omega^{2}/\delta\gamma$, the phonon decay occurs
predominantly through the excited 1$s(T_{2},E)$ states. These states have a
relatively large spin-orbit coupling due to a large overlap with the donor
nucleus [$\sim 0.02(1)$ meV in Si:P(Bi) Ramdas and Rodriguez (1981)], which
will mix the singlet and triplet states before the electron decays back to the
ground state. Thus, there is an optical pumping rate into the singlet state at
rate $\sim\Omega^{2}/\delta\gamma$, while the rate out, $\sim
f\Omega^{2}/\delta\gamma$, is much slower. For large transverse fields, the
dynamics result in the fidelity for singlet state preparation
$\mathcal{F}_{s}\approx 1-3f$. In Fig. 3(b) we show the error probability for
singlet state preparation $1-\mathcal{F}_{s}$ with dopant separation, where it
goes below a few percent at the shortest distances.

In addition to phonons and plasmon loss, there are several potential sources
of error in both the C-Phase gate and the entanglement generation: (i)
disorder in the position of the dopants, (ii) enhanced plasmon emission to the
excited $1s(T_{2}/E)$ valley states, and (iii) dephasing and decay of the
ground state spins during gate operation. (i) The precision with which the
dopant atoms must be placed with respect to the nanoribbon and each other is
determined primarily by Eq. (1) (to achieve sub-radiance their distance from
each other must be a half integer multiple of the plasmon wavelength, but this
can be tuned with the Fermi energy). It follows that the position error of the
two dopants should be less than $f/2k$ relative to the nanoribbon. Since
$k=W/\eta^{-1}(x)$, this implies that for $W\approx 50$ nm and $f\lessapprox
0.1$ the donor placement precision should be $\lessapprox 5$ nm for the value
of $\eta^{-1}\approx 0.6$ used in Fig. 3(b). Such precision is achievable with
current technology. (ii) In addition, to the enhanced plasmon decay to the
ground state, there will also be enhanced decay to the $1s(T_{2}/E)$ states
which cannot be made sub-radiant. However, when the system is optimized for
the ground state transition, the enhanced emission to the other $1s$ states
scales as $\gamma_{p}(\omega_{T,E}/\omega_{d})^{9}$. For Si:P(Bi) the ratio
$(\omega_{T,E}/\omega_{d})^{9}\approx 10^{-2}(10^{-4})$ so for P it is a small
correction and for Bi it is negligible. (iii) For Si:P(Bi) qubits electron
spin coherence times as long as a few ms have been observed Zwanenburg et al.
(2013). This sets a lower limit on the gate speed which is proportional to
$\Omega^{2}/\gamma$. At 1 $e\cdot$pm, the dipole moment of the $D^{0}X$
transition is extremely weak; thus, a strong laser is required to achieve
sufficiently fast gates, e.g., at a laser intensity of 10 W/cm2 the gate time
will be roughly 10 $\mu$s, which is slow, but still much faster than the
decoherence rate of the electron spins. A key advantage of the group-V donor
qubits in Si is that, once the gate or entanglement operation has been
performed, the electron spin states can be mapped to the donor nuclear spin,
where coherence times over an hour have been demonstrated Saeedi et al.
(2013).

We have shown that mid-infrared graphene plasmonics may be a powerful resource
for optical manipulation and inducing interactions of group-V donor spin
qubits in Si. In addition to the applications described here, such a system
could be a powerful resource for photonic applications, e.g., an electrically
pumped THz laser Apalkov and Stockman (2013). The main experimental challenges
will be combining high-quality graphene nanoribbons and the precise placement
of impurities in the Si lattice. The already dramatic progress along these
fronts suggests that graphene plasmons may enable a new route towards spin
based metrology and quantum computation in Si.

_Acknowledgments –_ We thank G. Wolfowicz, J. J. L. Morton, M. Keller and D.
Newell for helpful discussions. Funding is provided by NIST and the NSF
Physics Frontier at the JQI.

## References

  * Zwanenburg et al. (2013) F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y. Simmons, L. C. L. Hollenberg, G. Klimeck, S. Rogge, S. N. Coppersmith, and M. A. Eriksson, Rev. Mod. Phys 85, 961 (2013).
  * Yang et al. (2006) A. Yang, M. Steger, D. Karaiskaj, M. Thewalt, M. Cardona, K. Itoh, H. Riemann, N. Abrosimov, M. Churbanov, A. Gusev, et al., Phys. Rev. Lett. 97, 227401 (2006).
  * Yang et al. (2009) A. Yang, M. Steger, T. Sekiguchi, M. Thewalt, T. Ladd, K. Itoh, H. Riemann, N. Abrosimov, P. Becker, and H. J. Pohl, Phys. Rev. Lett. 102, 257401 (2009).
  * Sekiguchi et al. (2010) T. Sekiguchi, M. Steger, K. Saeedi, M. L. W. Thewalt, H. Riemann, N. V. Abrosimov, and N. Nötzel, Phys. Rev. Lett. 104, 137402 (2010).
  * Wolfowicz et al. (2013) G. Wolfowicz, A. M. Tyryshkin, R. E. George, H. Riemann, N. V. Abrosimov, P. Becker, H.-J. Pohl, M. L. W. Thewalt, S. A. Lyon, and J. J. L. Morton, Nat. Nanotechnol. 8, 561 (2013).
  * Saeedi et al. (2013) K. Saeedi, S. Simmons, J. Z. Salvail, P. Dluhy, H. Riemann, N. V. Abrosimov, P. Becker, H.-J. Pohl, J. J. Morton, and M. L. Thewalt, Science 342, 830 (2013).
  * Morello et al. (2010) A. Morello, J. J. Pla, F. A. Zwanenburg, K. W. Chan, K. Y. Tan, H. Huebl, M. Möttönen, C. D. Nugroho, C. Yang, J. A. van Donkelaar, et al., Nature 467, 687 (2010).
  * Pla et al. (2012) J. J. Pla, K. Y. Tan, J. P. Dehollain, W. H. Lim, J. J. L. Morton, D. N. Jamieson, A. S. Dzurak, and A. Morello, Nature 489, 541 (2012).
  * Koiller et al. (2001) B. Koiller, X. Hu, and S. Das Sarma, Phys. Rev. Lett. 88, 027903 (2001).
  * Weber et al. (2014) B. Weber, Y. H. M. Tan, S. Mahapatra, T. F. Watson, H. Ryu, R. Rahman, L. C. L. Hollenberg, G. Klimeck, and M. Y. Simmons, Nat. Nanotechnol. 9, 430 (2014).
  * (11) F. Jelezko and J. Wrachtrup, J. Phys.: Condens. Mat. 18, 807 (2006).
  * (12) F. Henneberger and O. Benson, Semiconductor Quantum Bits, (Pan Stanford Publishing – World Scientific, Singapore), 2008.
  * Vinh et al. (2008) N. Q. Vinh, P. T. Greenland, K. Litvinenko, B. Redlich, A. F. G. van der Meer, S. A. Lynch, M. Warner, A. M. Stoneham, G. Aeppli, D. J. Paul, et al., Proc. Natl. Acad. Sci. 105, 10649 (2008).
  * Greenland et al. (2010) P. T. Greenland, S. A. Lynch, A. F. G. van der Meer, B. N. Murdin, C. R. Pidgeon, B. Redlich, N. Q. Vinh, and G. Aeppli, Nature 465, 1057 (2010).
  * Bernien et al. (2013) H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo, T. H. Taminiau, M. Markham, D. J. Twitchen, L. Childress, et al., Nature 497, 86 (2013).
  * Jablan et al. (2009) M. Jablan, H. Buljan, and M. Soljačić, Phys. Rev. B 80, 245435 (2009).
  * García de Abajo (2014) F. J. García de Abajo, ACS Photonics 1, 135 (2014).
  * Mayorov et al. (2011) A. S. Mayorov, R. V. Gorbachev, S. V. Morozov, L. Britnell, R. Jalil, L. A. Ponomarenko, P. Blake, K. S. Novoselov, K. Watanabe, T. Taniguchi, et al., Nano Lett. 11, 2396 (2011).
  * Elias et al. (2011) D. C. Elias, R. V. Gorbachev, A. S. Mayorov, S. V. Morozov, A. A. Zhukov, P. Blake, L. A. Ponomarenko, I. V. Grigorieva, K. S. Novoselov, F. Guinea, et al., Nat. Phys. 7, 701 (2011).
  * Baringhaus et al. (2014) J. Baringhaus, M. Ruan, F. Edler, A. Tejeda, M. Sicot, AminaTaleb-Ibrahimi, A.-P. Li, Z. Jiang, E. H. Conrad, C. Berger, et al., Nature 506, 349 (2014).
  * Low and Avouris (2014) T. Low and P. Avouris, ACS Nano 8, 1086 (2014).
  * Ramdas and Rodriguez (1981) A. K. Ramdas and S. Rodriguez, Rep. Prog. Phys. 44, 1297 (1981).
  * Chang et al. (2007) D. Chang, A. Sorensen, P. Hemmer, and M. Lukin, Phys. Rev. B 76 (2007).
  * Koppens et al. (2011) F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, Nano Lett. 11, 3370 (2011).
  * (25) See supplemental material for further information on device properties, fidelity of single-shot readout, hyperfine interaction in Si:Bi, and the plasmon induced interactions between the impurities.
  * Christensen et al. (2012) J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, ACS Nano 6, 431 (2012).
  * Falk et al. (2009) A. L. Falk, F. H. L. Koppens, C. L. Yu, K. Kang, N. de Leon Snapp, A. V. Akimov, M.-H. Jo, M. D. Lukin, and H. Park, Nat. Phys. 5, 475 (2009).
  * Thongrattanasiri et al. (2012) S. Thongrattanasiri, A. Manjavacas, and F. J. García de Abajo, ACS Nano 6, 1766 (2012).
  * Gullans et al. (2013) M. Gullans, D. E. Chang, F. H. L. Koppens, F. J. G. de Abajo, and M. D. Lukin, Phys. Rev. Lett. 111, 247401 (2013).
  * Pellizzari et al. (1995) T. Pellizzari, S. A. Gardiner, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 75, 3788 (1995).
  * Domokos et al. (1995) P. Domokos, J. M. Raimond, M. Brune, and S. Haroche, Phys. Rev. A 52, 3554 (1995).
  * Reiter et al. (2012) F. Reiter, M. J. Kastoryano, and A. S. Sorensen, New J. Phys. 14, 053022 (2012).
  * Gross and Haroche (1982) M. Gross and S. Haroche, Phys. Rep. 93, 301 (1982).
  * Apalkov and Stockman (2013) V. Apalkov and M. I. Stockman, arXiv:1303.0220 (2013).
  * Tyuterev et al. (2010) V. Tyuterev, J. Sjakste, and N. Vast, Phys. Rev. B 81, 245212 (2010).
  * Mohammady et al. (2012) M. H. Mohammady, G. W. Morley, A. Nazir, and T. S. Monteiro, Phys. Rev. B 85, 094404 (2012).
  * Kimble and Mandel (1976) H. J. Kimble and L. Mandel, Phys. Rev. A 13, 2123 (1976).

##  Appendix A Supplemental Material

Figure S1: (a) Reduced level diagram for the $\sigma_{-}\sigma_{+}$-excitation
scheme. (b) Effective ground state evolution for the single-shot readout. Nine
times as many plasmons are emitted in the spin-down state. (c) Hyperfine
states for Si:Bi. There is an electron(nuclear) Zeeman energy $\omega_{0(n)}$
and a Hyperfine coupling between pairs of states in the submanifolds. For the
protocols in Si:Bi we use the pseudo-spin-1/2 subspace of
$\mbox{$|g,\downarrow\rangle$}=\mbox{$|m_{s}=-1/2,m_{I}=-9/2\rangle$}$ and
$\mbox{$|g,\uparrow\rangle$}=\mbox{$|+,-4\rangle$}$. At large magnetic field
these two states correspond to up/down electron spin and the fully polarized
nuclear spin $m_{I}=-9/2$.

_Device Properties –_ The graphene could be encapsulated between two thin
layers of hexagonal-BN (h-BN), while the doping of the nanoribbon is
controlled by either direct chemical doping or a metal top gate. Room
temperature mobilities approaching 106 cm2/Vs have been demonstrated for bulk
graphene surrounded by hBN Mayorov et al. (2011); Elias et al. (2011), which
would allow mid-infrared plasmons to coherently propagate for several microns
Jablan et al. (2009). The $1s$ states of the donor have a six-fold valley
degeneracy, which splits into a single orbital ground state $A_{1}$ and five
excited states lying 10-30 meV higher in energy: a doublet $E$ and triplet
$T_{2}$. Due to their large overlap with the donor nucleus they have a
relatively large spin-orbit coupling [$\sim 0.02(1)$ meV in Si:P(Bi)]. The
excited state transitions from $1s(A_{1})$ to $2p$ states lie in the mid-
infrared between 30-70 meV. The $2p$ states primarily decay via phonons in a
two step process through the $1s(T_{2}/E)$ valley states, but have long
lifetimes: observed as long as 250 ps Vinh et al. (2008) and predicted to be
as long as 1.3 ns Tyuterev et al. (2010). The magnitude of the dipole moment
for the $1s$ to $2p$ transition is $\left\lvert\mbox{$\langle
e_{s}|$}\mu_{I}\mbox{$|g,s\rangle$}\right\rvert\approx 0.3~{}e\cdot$nm. In
addition to the valley and spin fine structure, there is a hyperfine
interaction with the donor nucleus, which has been used to achieve long term
quantum memory in these systems Mohammady et al. (2012); Wolfowicz et al.
(2013); however, for all the donors except Bi (see below), the hyperfine
interaction can be neglected at moderate magnetic fields $\gtrsim 10$ mT.

The spin-$3/2$ character of the $D^{0}X$ state has been measured and used for
spin state preparation and detection in ensembles of Si:P and Si:Bi Yang et
al. (2006); Sekiguchi et al. (2010). The $D^{0}X$ states lie $1.1$ eV above
the $1s(A_{1})$ ground state and have a lifetime of 272 ns, limited by Auger
recombination. In principle, a lambda transition from the ground state to the
to the 2$p$ states, through the $D^{0}X$ states, is allowed, but it is parity
non-conserving, which is only weakly broken in this system. Instead we propose
to hybridize the $2s$ states with the $2p$ states through application of an
external stress, to make them degenerate, and an external electric field to
mix them. This requires a pressure of $\sim 25$ MP along one of Si principal
axes Ramdas and Rodriguez (1981). To apply suitable electric fields, one could
either use an external gate or the nanoribbon itself, which will induce a
2s-2p splitting as large as $\approx 1$ meV for doping levels of $n_{e}\approx
10^{12}$ cm-2.

_Fidelity of Single-Shot Readout_ – To achieve single-shot readout we use the
$\sigma_{-}\sigma_{+}$-excitation scheme discussed in the main text and shown
in Fig. S1(a). The Hamiltonian for the Raman Rabi driving field takes the form

$\displaystyle H_{c}$
$\displaystyle=\Omega^{\prime}(t)\bigg{(}\frac{1}{3}\mbox{$|e_{\uparrow}\rangle$}\mbox{$\langle
g,\uparrow|$}+\mbox{$|e_{\downarrow}\rangle$}\mbox{$\langle
g,\downarrow|$}+h.c.\bigg{)}$ (S1)

The factor of $1/3$ follows from the ratio of Clebsch-Gordon coefficients in
the two excitation pathways through the spin-3/2 exciton state. When the
impurity is driven resonantly below saturation, the fluorescence distribution
of the emitted plasmons follows a Poisson distribution Kimble and Mandel
(1976), which is different for the two spin states

$p_{s}(n,t)=\frac{(R_{s}t)^{n}}{n!}e^{-Rt}$ (S2)

where $R_{\downarrow}=\frac{2\Omega^{\prime
2}}{(\gamma+\gamma_{p})^{2}}\gamma_{p}$ and
$R_{\uparrow}=\frac{2\Omega^{\prime 2}}{9(\gamma+\gamma_{p})^{2}}\gamma_{p}$.
When the excited state emits a phonon instead of a photon it decays through
the $1s(T_{2}/E)$ valley states where it can flip its spin due to the large
spin orbit coupling in these states Ramdas and Rodriguez (1981). As a worst
case scenario, we assume that each time a phonon decay occurs, there is an
electron spin flip with probability 1/2. This leads to the evolution for the
ground state spin populations $n_{\uparrow/\downarrow}$

$\dot{n}_{\uparrow}=-\frac{\gamma}{2\gamma_{p}}R_{\uparrow}\,n_{\uparrow}+\frac{\gamma}{2\gamma_{p}}R_{\downarrow}\,n_{\downarrow}$
(S3)

and $n_{\downarrow}\approx 1-n_{\uparrow}$ since we are far below saturation.
The dynamics are illustrated in Fig. S1(b) The emitted plasmon number
distribution $P(n,t)$ is

$\displaystyle P(n,t)$
$\displaystyle=n_{\uparrow}(t)p_{\uparrow}(n,t)+n_{\downarrow}(t)p_{\downarrow}(n,t),$
(S4) $\displaystyle n_{\uparrow}(t)$
$\displaystyle=n_{\uparrow}(0)e^{-\gamma(R_{\uparrow}+R_{\downarrow})t/\gamma_{p}}+\frac{1}{10}\big{[}1-e^{-\gamma(R_{\uparrow}+R_{\downarrow})t/\gamma_{p}}\big{]}.$
(S5)

In Fig. 2(b) of the main text we show an example of the distribution for the
two initial conditions $p_{\uparrow}=1$ and 0. The distribution for the two
initial spin states will be disjoint when $\gamma_{p}/\gamma\gg
R_{\downarrow}\,t\gg 1$ which guarantees that many plasmons are emitted before
the spin is depolarized. The optimum occurs when
$R_{\downarrow}\,t\sim\sqrt{2\gamma_{p}/\gamma}$.

_Hyperfine Interaction in Si:Bi_ – The Hamiltonian for the ground state spin
interacting with the donor nucleus is

$H=\omega_{0}s_{z}-\omega_{n}I_{z}+A\,\bm{s}\cdot\bm{I}$ (S6)

where $I_{z}$ is the donor nuclear spin operator, $I=9/2$ in the case of
Si:Bi, $\omega_{0(n)}$ is the Zeeman energy of the electron(nuclear) spin, and
$A$ is the hyperfine coupling constant. For Si:Bi the hyperfine coupling is
rather large $A/\hbar=1.47$ GHz. To find the eigenstates, we first notice that
$H$ has the two decoupled eigenstates $|m_{s}=\pm 1/2,m_{I}=\pm I\rangle$,
where $m=m_{s}+m_{I}$ is the total z-angular momentum. The other eigenstates
are formed from superpositions of pairs of states $|m_{s}=\pm 1/2,m_{I}=m\mp
1/2\rangle$ with $\left\lvert m\right\rvert<I+1/2$ as shown in Fig. S1(c). We
can define Pauli operators within each of these two-state submanifolds, then
we can rewrite the Hamiltonian for these submanifolds as Mohammady et al.
(2012)

$\displaystyle H^{m}_{sub}$
$\displaystyle=\frac{\Delta_{m}}{2}\sigma_{z}+\frac{\Omega_{m}}{2}\sigma_{x}-\epsilon_{m}$
(S7) $\displaystyle\Delta_{m}$ $\displaystyle=A\,m+\omega_{0}+\omega_{n}$ (S8)
$\displaystyle\Omega_{m}$ $\displaystyle=A\sqrt{I(I+1)-m^{2}}$ (S9)
$\displaystyle\epsilon_{m}$ $\displaystyle=\frac{A}{4}+\omega_{n}m$ (S10)

Defining $\theta_{m}=\tan^{-1}(\Omega_{m}/\Delta_{m})$, the eigenstates are

$\begin{split}\mbox{$|\pm,m\rangle$}&=\cos(\theta_{m}/2)\mbox{$|m_{s}=\pm\frac{1}{2},m_{I}=m\mp\frac{1}{2}\rangle$}\\\
&\pm\sin(\theta_{m}/2)\mbox{$|m_{s}=\mp\frac{1}{2},m_{I}=m\pm\frac{1}{2}\rangle$}\end{split}$
(S11)

To apply the results discussed in the main text to case of Si:Bi we work in
the pseudo-spin 1/2 subspace with
$\mbox{$|g,\downarrow\rangle$}=\mbox{$|m_{s}=-1/2,m_{I}=-9/2\rangle$}$ and
$\mbox{$|g,\uparrow\rangle$}=\mbox{$|+,-4\rangle$}$. At large magnetic field
these two states correspond to up/down electron spin and the fully polarized
nuclear spin $m_{z}=-9/2$. The $|g,\downarrow\rangle$ state can be prepared by
optically pumping with the $\sigma_{-}\pi$-excitation scheme. In addition, we
note that both the two-qubit C-Phase gate and the entanglement preparation
carry through unaltered in this pseudo-spin-1/2 subspace.

_Master Equation –_ When the propagation time between impurities $r/v_{g}$ is
much faster than the other time scales we can integrate out the plasmon modes
in a Markov approximation and the super-radiance can be clearly seen in the
master equation for the density matrix $\rho$ of the two impurities

$\displaystyle\dot{\rho}$
$\displaystyle=-\frac{i\omega_{d}}{2}\left[\sigma_{1}^{z}+\sigma_{2}^{z},\rho\right]-\frac{i}{2}v\left[(\sigma_{1}^{+}\sigma_{2}^{-}+\sigma_{1}^{-}\sigma_{2}^{+}),\rho\right]$
$\displaystyle-\gamma_{c}\mathcal{D}[\sigma_{1}^{-}+\alpha\,\sigma_{2}^{-}]\rho-{\delta\gamma}(\mathcal{D}[\sigma_{1}^{-}]+\mathcal{D}[\sigma_{2}^{-}])\rho$
(S12) $\displaystyle v$ $\displaystyle=\gamma_{p}\,\sin
kr\,e^{-r/L_{p}},~{}~{}~{}~{}~{}~{}\gamma_{c}=\gamma_{p}\,\left\lvert\cos
kr\right\rvert\,e^{-r/L_{p}}$ (S13) $\displaystyle\delta\gamma$
$\displaystyle=\gamma+\gamma_{p}-\gamma_{c},~{}~{}~{}~{}L_{p}^{-1}=\frac{e\,v_{F}^{2}}{2\mu\,v_{g}\omega_{F}}$
(S14)

where $\mathcal{D}[c]\rho=1/2\\{c^{\dagger}c,\rho\\}-c\rho c^{\dagger}$ and
$\alpha=\textrm{sign}(\cos kr)$. We have included the losses in the plasmon
propagation in terms of the plasmon propagation length $L_{p}$ defined in the
main text.