doi: 10.1038/nature 07295
SUPPLEMENTARY INFORMATION
A. Effect of radiofrequency and microwave phases
The chosen basis is:
( S, I ) = −
1
2
, −
1
2
,
1
2
, −
1
2
, −
1
2
,
1
2
,
1
2
,
1
2
, (1) where S represents the electron donor spin, and I the 31 P nuclear spin. All pulses are assumed to be selective on a particular electron or nuclear spin transition, as illustrated in
Figure 1 of the main text. The phase of the initial π/ 2 microwave pulse, ϕ e
, determines the phase of the initial electron spin coherence. All other microwave pulses have phase ϕ mw
, while that of the rf pulses is ϕ rf
.
The initial spin density matrix, neglecting any nuclear spin polarisation, is proportional to (
I
+ βS z
), where
I is the Identity matrix and β = − gµ
B
B
0 kT
. This can be rewritten in pseudopure state form, neglecting the majority of the Identity component and omitting the constant factor β :
ρ
0
= ( S z
+
I
/ 2) / 2 = ρ th
=







1 / 2 0 0 0

0 0 0 0
0 0 1 / 2 0




.


0 0 0 0
(2)
After the initial (coherence-generating) π/ 2 microwave pulse:

1 / 4
ρ
1
=





 exp ( iϕ e
) / 4
0
0 exp ( − iϕ e
) / 4 0 0

1 /
0
0
4 0 0
1 / 2 0


 .



0 0
(3)
The next two pulses, π
RF followed by π mw
, transfer this coherence to the nuclear spin:

0 0 0 0

ρ
2
=






0
0
1 /
0
4
1
0
/
0 exp ( i ( ϕ e
+ ϕ
RF
− ϕ mw
)) / 4 0
2 exp ( − i ( ϕ e
+ ϕ
RF
1
0
/ 4
− ϕ mw
)) / 4





.

(4)
The coherences here decay with characteristic time T
2n
, which is typically much longer than T
2e
. Upon applying the reverse of the transfer sequence above ( π mw followed by π rf
), the electron coherence is revived:
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doi: 10.1038/nature0 7295 SUPPLEMENTARY INFORMATION

1 / 4
ρ
3
= ρ
1
=





 exp ( iϕ e
) / 4
0
0 exp ( − iϕ e
) / 4 0 0

1 /
0
0
4 0 0
1 / 2 0




.


0 0
(5)
Thus the relative phase of the microwave and rf sources is cancelled out, though both must remain stable over the course of the experiment.
B. Electron and nuclear spin relaxation
Electron relaxation of the system can be modeled using a standard master equation in
Lindblad form. In order to represent processes that take the system to thermal equilibrium, both raising and lowering terms are included.
ρ ˙ = −
γ
2
( ρS
−
S
+
+ S
−
S
+
ρ − 2 S
+
ρS
−
) −
γe
− β
( ρS
+
S
−
2
+ S
+
S
−
ρ − 2 S
−
ρS
+
) − i [ H , ρ ] (6) where γ is the relaxation rate, S + and S
− are the electron spin raising and lowering operators
( S
±
= S x
± iS y
), and β relates the electron Zeeman splitting to k
B
T , as defined above. This can be simplified by transforming into the rotating frame of the Hamiltonian, taking an
Ising approximation ( H
0
= ω e
S z
− ω
I
I z
+ A · S z
· I z
). In this frame, H goes to 0, and we are left only with the relaxation part of Eq. (7), with S
+ and S
− transformed into the rotating frame. Neglecting direct nuclear relaxation ( T
1n
→ ∞ ) and in the high temperature limit this yields:

ρ
1 , 1
( t ) − ρ
2 , 2
( t )
ρ ˙ ' −
γ
2






ρ
2 , 1
( t )
ρ
3 , 1
( t ) − e iAt ρ
4 , 2
( t )
ρ
4 , 1
( t )
ρ
2 , 2
( t
ρ
ρ
)
1 , 2
−
3 , 2
(
( t t
ρ
)
)
1 , 1
( t )
ρ
4 , 2
( t ) − e
− iAt ρ
3 , 1
( t )
ρ
1 , 3
( t ) − e
− iAt ρ
2 , 4
( t )
ρ
3 , 3
ρ
( t ) − ρ
ρ
2 , 3
4 , 3
(
( t t )
)
4 , 4
( t )

ρ
1 , 4
( t )
ρ
2 , 4
( t ) − e iAt
ρ
1 , 3
( t )
ρ
3 , 4
( t )






ρ
4 , 4
( t ) − ρ
3 , 3
( t )
(7)
The electron relaxation rate (1 /T
1 e
) can be ascertained by observing the appropriate density matrix elements:
ρ ˙
1 , 1
ρ
3 , 3
= −
γ
2
( ρ
1 , 1
+ ρ
3 , 3
) +
γ
2
( ρ
2 , 2
+ ρ
4 , 4
) = −
γ
2
( ρ
1 , 1
+ ρ
3 , 3
) +
γ
2
(1 − ρ
1 , 1
− ρ
3 , 3
)
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doi: 10.1038/nature0 7295 SUPPLEMENTARY INFORMATION
Taking ρ e
= ρ
1 , 1
+ ρ
3 , 3
ρ e
= − γ ( ρ e
− 1 / 2). Solving this gives:
ρ e
= ρ e, 0 e
− γt
+ 1 / 2
Hence, electron relaxation follows e
− γt and the electron relaxation time, T
1 e
=
1
γ
.
(8)
The nuclear coherence is given by ρ nn
= ρ
3 , 1
+ ρ
4 , 2
. Extracting these terms from Eq. 7 yields two coupled differential equations:

ρ
3
˙
, 1

ρ
4
˙
, 2


= −
γ
2

1 − e iAt
 
ρ
3 , 1

− e
− iAt 1
 
ρ
4 , 2


(9)
It is straightforward to get rid of the time dependence in the 2 × 2 matrix in this equation by making a time dependent unitary transformation and solving for new variables ρ
0
3 , 1 and
ρ
0
4 , 2
.

ρ
0
3 , 1

ρ
0
4 , 2
 

= U

ρ
3 , 1
ρ
4 , 2


=

 e
− iAt/ 2
0
 
ρ
3 , 1
0 − e iAt/ 2
 
ρ
4 , 2


(10)
Following this transformation solving the pair of differential equations is a simple eigenvalue problem. In the experiments A = 117 MHz and γ ranges from 1 kHz to less than 1 Hz
(as a function of temperature), hence we can take the limit A γ . In this case, both characteristic eigenvalues have a real part of − γ/ 2, and therefore any nuclear coherence decays with this rate. Thus T
2 n
=
2
γ
= 2 T
1 e as experimentally observed.
C. Density matrix tomography
In this section we describe 1) the preparation and the tomography of the pseudopure initial electron spin states ± X , ± Y , and ± Z , and the Identity, 2) the tomography of the state after transfer to and from the nuclear spin and 3) the measure of fidelity between the starting and recovered states.
We define state ± X as ( ± σ x
+
I
) / 2, and similarly for Y and Z . Figure 1 shows the full set of pulse sequences required for the state preparation and detection. The starting
(thermal) state is +Z, which thus requires no preparation pulse.
− Z is obtained by applying an inversion π pulse, while ± X and ± Y are obtained through a π/ 2 pulse of the appropriate phase. The state
I is obtained by applying an inversion π pulse and waiting some time,
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doi: 10.1038/nature0 7295 SUPPLEMENTARY INFORMATION
+X
+Y
-X
-Y phi
0
90
180
270
+Z
-Z
Identity
π
T = ln2 T1e
Tomography of starting state
π/2(phi)
τ e2
π
τ e2
Sx, Sy preparation
π/2 measurement
π
τ e2
π
τ e2
Sx, Sy preparation
π/2 measurement
π
π
τ e2
π
τ e2
Sx, Sy preparation
π/2 measurement
π
Sz
Sz
Sz preparation
π
τ e2
π/2 π
Sx, Sy measurement
Sz
π
T = ln2 T1e preparation transfer
τ e1
π
τ e1
π
π
τ n preparation transfer
π
τ e1
π
τ e1
π
π
τ n preparation transfer
Tomography of recovered state
τ e1
π/2(phi) π
τ e1
π
π
τ n
π
π
π
τ n
π
π
τ e2
π
τ e2
Sx, Sy transfer
π/2 measurement
π
τ n
π
π
τ e2
π
τ e2
Sx, Sy transfer
π/2 measurement
π
τ n
π
π
τ e2
π
τ e2
Sx, Sy transfer
π/2 measurement
π preparation
π
τ e1
π
π
τ n transfer
π
τ n
π
π
τ e2
π
τ e2
Sx, Sy transfer
π/2 measurement
π
Sz
Sz
Sz
Sz
FIG. 1: Pulse sequences applied to prepare and measure electron spin states.
T = (ln 2) T
1 e
. This is long enough to ensure complete decoherence of the electron spin (off diagonal elements go to zero), while corresponding to the precise point during the relaxation process at which the electron spin populations are equal.
Measurement is performed in the σ x and σ y bases by generating an electron spin echo and observing both in-phase and quadrature components. A measurement in the σ z basis can be performed some short time ( t < T
1e
) later in the pulse sequence by applying a π/ 2 pulse, followed by a π pulse, and then observing the resulting echo. This measurement operation is applied to both the starting states and those recovered after the end of the
process.
Echo traces for the seven states are shown in Figure 2 for both the starting and recovered states. Each corresponds to a measurement in the σ x,y,z bases.
The integrated areas of the electron spin echoes, A x,y,z
, are used to extract the components of σ x,y,z in the spin density matrix. We assume the starting electron spin state is
(pseudo)pure and can thus normalise the areas to extract a density matrix of the starting electron spin state:
ρ =
A x
σ x
+
2 p A 2 x
A
+ y
σ y
A 2 y
+ A z
+ A 2 z
σ z
+
I
/ 2 (11)
We make no such assumptions about the purity of the recovered electron spin state, and
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doi: 10.1038/nature0 7295 SUPPLEMENTARY INFORMATION
1
0.5
0
−0.5
−1
1
0.5
0
−0.5
−1
0
+X -X
1
0.5
0
−0.5
−1
1
0.5
0
−0.5
0.5
0
−0.5
−1 −1
Time (ns) 250 0 Time (ns) 250 0
1
0.5
0
−0.5
−1
1
+Y
Time (ns)
1
0.5
0
−0.5
−1
1
250
0.5
0
−0.5
−1
0
-Y
Time (ns) 250
0.5
0
−0.5
−1
0
1
0.5
0
−0.5
−1
1
+Z
Time (ns) 250
0.5
0
−0.5
−1
0
1
0.5
0
−0.5
−1
1
-Z
Time (ns)
1
0.5
0
−0.5
−1
1
250
0.5
0
−0.5
−1
0
Identity
Time (ns) 250
FIG. 2: Measurements in the σ x,y,z bases of the electron spin state before and after storage in the nuclear spin.
Electron spin echoes (a.u.) are obtained using the pulse sequences shown in Figure 1. The echoes for σ x and σ y occur simultaneously, while that for σ z
(which occurs at a later time) is superimposed here for clarity.
+X +Y +Z Identity -X -Y -Z
1 1
0 0
-1
1
Real
Imag
0
-1
1
0
-1
Fidelity : 0.905
0.887
0.881
1.000
0.905
0.895
0.885
-1
FIG. 3: Density matrices for the initial and recovered states.
normalise the integrated areas of the recovered spin echoes using the areas obtained from the starting state. Density matrices for the initial and recovered state are shown in Figure 3.
One common measure of the difference between two quantum states is the
[ ?
]:
F ( ρ
0
, ρ
1
) = Tr q
√
ρ
1
ρ
0
√
ρ
1
(12)
Here, we use a more aggressive measure of fidelity, F
0
= F
2
, corresponding to the overlap of a pure state and an arbitrary density matrix (rather than its square root). Thus, if our
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doi: 10.1038/nature0 7295 SUPPLEMENTARY INFORMATION initial density matrix ρ
0
= | ψ i h ψ | , the fidelity measure we use is:
F
0
= h ψ | ρ
1
| ψ i .
(13) www.nature.com/nature
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