PHYSICAL REVIEW B 96, 039905(E) (2017)
Erratum: State-selective intersystem crossing in nitrogen-vacancy
centers [Phys. Rev. B 91, 165201 (2015)]
M. L. Goldman, M. W. Doherty, A. Sipahigil, N. Y. Yao, S. D. Bennett, N. B. Manson, A. Kubanek, and M. D. Lukin
(Received 6 July 2017; published 25 July 2017)
DOI: 10.1103/PhysRevB.96.039905
We have become aware that the vibrational overlap function F (), which is defined in Eq. (5), was incorrectly normalized.
This quantity factors into the intersystem crossing (ISC) rates Ŵ
A
1
and Ŵ
E
1,2
from the states |A
1
〉 and |E
1,2
〉 [Eqs. (6) and (13),
respectively]. As a result, the theoretical value of Ŵ
A
1
that is plotted in Fig. 5 is too large by a factor of 2π . We thank A. Gali for
bringing this error to our attention.
We have also reexamined an assumption regarding the transverse spin-orbit coupling rate λ
⊥
and found it not to be valid.
The dynamic Jahn-Teller effect in the
3
E excited state manifold causes the strengths of the spin-orbit interaction and the excited
states’ response to externally applied fields to be reduced [1] by a quantity known as the Ham reduction factor [2]. This factor
changes the value of λ
⊥
that also enters into both intersystem crossing (ISC) rates. We had previously assumed that this effect is
negligible, but we discovered this not to be the case when we explicitly calculated the Ham reduction factor. The magnitude of
this effect for the NV center and its implications for the ISC are topics of current theoretical interest, so we present our explicit
calculation of the Ham reduction factor here.
I. INTRODUCTION TO DYNAMIC JAHN-TELLER EFFECT
The
3
E excited state manifold exhibits the dynamic Jahn-Teller (JT) effect, which means that E-symmetric lattice vibration
modes couple to the two degenerate, E-symmetric electronic orbital states [1,3]. We now calculate explicitly how this coupling
impacts the transverse spin-orbit coupling rate λ
⊥
, which is a crucial input to our model of the ISC mechanism.
Let {|X〉, |Y 〉} and (x,y ) be the orbital electronic states of the triplet
3
E manifold and the generalized displacement coordinates
of a set of pseudolocal E-symmetric phonon modes, respectively. In the absence of the JT interaction, the vibrational equations
for the phonon modes corresponding to each orbital state are identical and separable, and have the form
−
¯ h
2M
∇
2
+
1
2
k(x
2
+ y
2
)
|χ
i
(x,y )〉= ǫ
i
|χ
i
(x,y )〉, (E1)
where M is the carbon atomic mass, k is the harmonic spring constant of the phonon modes, and |χ
i
〉 is the i th vibrational
solution with energy ǫ
i
= (ν
i
+ 1/2)¯ h(k/M)
1/2
determined by the total number of quantum excitations ν
i
; this is simply the
standard solution to the two-dimensional harmonic oscillator. It follows that the vibronic states of the
3
E manifold have well
defined electronic and vibrational quantum numbers,
X,i
=|X〉⊗|χ
i
〉,
Y,i
=|Y 〉⊗|χ
i
〉. (E2)
In the presence of the JT interaction, however, the vibrational equations of the |X〉 and |Y 〉 orbital states become linearly
coupled and cannot be solved separately. Consequently, the vibronic states no longer have well-defined electronic and vibrational
quantum numbers. The i th vibronic state is now
|
i
〉=|X〉⊗|χ
i,x
〉+|Y 〉⊗|χ
i,y
〉 (E3)
and satisfies the equation
−
¯ h
2M
∇
2
+
1
2
k(x
2
+ y
2
)
+ f (xσ
z
− yσ
x
) + g[(x
2
− y
2
)σ
z
+ 2xyσ
x
]
|
i
(x,y )〉= ǫ
i
|
i
(x,y )〉, (E4)
where
σ
x
=|X〉〈Y |+|Y 〉〈X|,σ
z
=|X〉〈X|−|Y 〉〈Y |, (E5)
and f and g are the linear and quadratic JT interaction parameters, respectively [1].
2469-9950/2017/96(3)/039905(4) 039905-1 ©2017 American Physical Society