All-optical manipulation of electron spins in carbon-nanotube quantum dots.

Abstract

We demonstrate theoretically that it is possible to manipulate electron or hole spins all optically in semiconducting carbon nanotubes. The scheme that we propose is based on the spin-orbit interaction that was recently measured experimentally; we show that this interaction, together with an external magnetic field, can be used to achieve optical electron-spin state preparation with a fidelity exceeding 99%. Our results also imply that it is possible to implement coherent spin rotation and measurement using laser fields linearly polarized along the nanotube axis, as well as to convert spin qubits into time-bin photonic qubits. We expect that our findings will open up new avenues for exploring spin physics in one-dimensional systems.

Full text

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All-optical manipulation of electron spins in carbon nanotube quantum dots
Christophe Galland and Atac ¸ Imamo˘ glu
Institute of Quantum Electronics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland
(Dated: October 8, 2008)
We demonstrate theoretically that it is possible to manipulate electron or hole spins all optically in semi-
conducting carbon nanotubes. The scheme we propose is based on the spin-orbit interaction that was recently
measured experimentally; we show that this interaction, together with an external magnetic field, can be used
to achieve optical electron-spin state preparation with fidelity exceeding 99 %. Our results also imply that it
is possible to implement coherent spin rotation and measurement using laser fields linearly polarized along the
nanotube axis, as well as to convert spin qubits into time-bin photonic qubits. We expect that our findings will
open up new avenues for exploring spin physics in one-dimensional systems.
Optical manipulation of spins in atoms or semiconductors
relies on the presence of strong spin-orbit interaction (SOI) ei-
ther in the initial or final state of an optical transition. In III-V
semiconductors, it is the large spin-orbit splitting of the va-
lence band states [1] that enables efficient optical pumping of
electron [2] or nuclear spins [3, 4] and leads to a strong cor-
relation between light helicity and electron spin orientation
[5]. In this context, one would argue that optical spin manip-
ulation would be hindered in semiconducting single-wall car-
bon nanotubes (CNTs): due to the weak spin-orbit splitting in
graphene [6] and early experiments suggesting the presence
of electron-hole symmetry [7], it had been assumed that SOI
in CNTs would be small for both electrons and holes. In addi-
tion, the depolarization effect ensures that only electric fields
linearly polarized along the CNT axis couple strongly to elec-
trons and holes [8, 9], ruling out the possibility of obtaining
correlations between electron spin and photon polarization.
In this Letter, we describe a scheme for realizing efficient
optical manipulation of spins in CNTs. Our work is motivated
by the recent experimental observation of SOI-induced zero-
field spin splitting in CNTs [10]. The breakdown of electron-
hole symmetry that is a consequence of finite SOI implies that
a finite external axial magnetic field could be used to cancel
the SOI-induced spin splitting of the hole, while retaining a fi-
nite splitting for the electron spin. The presence of a magnetic
field component perpendicular to the CNT axis then mixes
the hole spin states and allow for a very efficient spin-flip Ra-
man coupling between the electron spin states. In addition to
analyzing the spin pumping efficiency as a function of the ex-
ternal magnetic and laser fields, we discuss applications of the
proposed scheme in quantum information processing.
The band structure of a CNT can be derived from that of
graphene, in which conduction and valence bands are cross-
ing at two inequivalent points in the reciprocal lattice (la-
beled K and K’) with linear dispersions. Since the K and
K’ points are at the boundaries of the first Brillouin zone ,
the states near the energy gap in semiconducting CNTs have
a large azimuthal momentum k⊥. In a semi-classical picture,
these states correspond to electrons having a fast clockwise
or counter-clockwise circular motion around the CNT’s cir-
cumference, therefore exhibiting an orbital magnetic moment
µorb≈ 0.3·d[nm] meV/T pointing along the axis, with oppo-
site signs for states originating from the two different valleys
[10, 11]. The degeneracy between these states is therefore
lifted when a magnetic field B?is applied along the nanotube
axis [11, 12]. The spin of electrons (or holes) also couples to
themagneticfield, yieldingaZeemansplitting∆Z= gµBB?,
with g ≈ 2 for both electrons and holes [10]. In this simple
picture the electronic states should be four-fold degenerate at
zero magnetic field. Here, we focus on a CNT quantum dot
(CNT QD) trapping a single electron, since this is the system
of primary interest from a quantum information perspective
[13].
The SOI leads to a zero-field spin splitting ∆SObetween
states with parallel and anti-parallel spin and orbital magnetic
moments [10, 18] (Fig. 1a). In the following analysis, we rely
on the experimentally measured values for ∆SO[10]. Since
we are interested in optical manipulation of spins, we con-
sider a QD formed on a semiconducting CNT with a diameter
d ∼ 1.2 nm having its lowest optical transition in the near-
infrared (∼ 1500 nm) [19]. Extrapolating measured values
from [10, 11] to this diameter (µorb∝ d and ∆SO∝
expect µorb ≈ 0.36 meV/T, ∆e
and ∆h
We first study the possibility of optical spin pumping using
resonant laser fields. In Fig. 1a we show the energy level dia-
gram of the lowest electron and hole states in a CNT QD un-
der an axial magnetic field B?as confirmed experimentally by
Kuemmeth et al. [10]. We will label U (D) the states having
a positive (negative) orbital magnetic moment. These states
originate from the two different valleys K and K’ and mix
very weakly in clean CNTs (∆KK? ≈ 65µeV in [10]). Al-
lowed optical transitions are of the type U→U or D→D due to
momentum conservation. The up and down arrows represent
the projection of the spin along the CNT axis (↑ for Sz= +?
and the subscripts designate electron or hole states. We now
apply an axial magnetic field B?and a laser field polarized
linearly along the nanotube axis, strongly coupling states from
the same valley with opposite electron and hole spins. In ad-
dition, we assume the presence of a magnetic field component
B⊥orthogonal to the CNT axis that coherently mixes the up
and down electron (and hole) spin states.
Figure 1b shows an energy level diagram equivalent to
that of Fig. 1a in the trion picture [2] where the four low-
est energy spin states of a CNT QD as well as the optically
excited states with two electrons and one hole are depicted
1
d) we
SO≈ 1.5 meV for electrons
SO≈ 0.9 meV for holes [20].
2)
arXiv:0806.3567v2 [cond-mat.other] 8 Oct 2008

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2
||
||
Ee
Eh
B//
D↑e
D↓e
U↑e
U↓e
D↓h
U↓h
U↑h
D↑h
↓
a)
b)
D↓e
D↑e
−
U↓eU↑eU↓h
D↓eU↑eU↓h
D↑eU↑eU↓h
U↑eU↓eU↑h
Bg
B
μ
U↓e
U↑e
//
Bg
B
e
SO
μ+Δ
//
B
orb
e
SO
μ
2
+Δ−
//
Bg
B
e
SO
μΔ
'KK
γ
T
Ω
h
e
SO
Δ
h
SO
Δ
h
SO
B
LLΩ
,
ω
h
Γ
//
h
SO
−Δ
Γ
e
KK'
Δ
↑
B┴
e
SO
B
h ξ
e ξ
FIG. 1: a) Energy diagram of the lowest electron (subscript e) and
hole (subscript h) states in a nanotube quantum dot as a function of
the applied axial magnetic field B//[10, 18]. b) Energy diagram of
a singly charged nanotube quantum dot for large B//showing the
relevant optical transitions coupled by a laser field polarized along
the CNT axis with Rabi frequency ΩL. The orthogonal magnetic
field causes a coherent ↑-↓ coupling with strength ?ΩT = gµBB⊥.
The decay of the excited states is assumed to be spin-conserving and
mono-exponential with rate Γ. Spin relaxation rates for electrons
and holes are denoted by ξeand ξh. K-K’ mixing can cause optically
assisted valley-flip to the state D↓eat an effective rate γKK?.
[21]. We choose the energy of the laser to be resonant with
the U↓e→U↓eU↑eU↓htransition. The optically excited trion
state now couples to U↓eU↑eU↑hbecause the hole spin pre-
cesses around the perpendicular field B⊥. Radiative recom-
bination from U↓eU↑eU↑h leaves a spin-up electron U↑e.
Since the optical transition U↑e→U↓eU↑eU↑his detuned by
∆e
perimentally measured by laser absorption) will vanish [2],
ensuring that the resident electron spin remains in state U↑e.
Conversely, preparation of the spin in the U↓estate can be
achieved by tuning the laser field onto resonance with the
U↑e→ U↓eU↑eU↑htransition.
To assess the efficiency of optical spin pumping as a func-
tion of the applied magnetic and laser fields we have per-
formed numerical simulations using the optical Bloch equa-
SO+ ∆h
SOfrom the applied laser field, light scattering (ex-
tions for the 4-level system shown in the left part of Fig. 1b.
We ignore K-K’ mixing for the time being. The spin-flip
rates ξe(ξh) for electrons (holes) are dominated by phonon-
assisted spin relaxation [18, 23]: these rates are expected to
have magnitudes varying from 1 µs−1to 1 ms−1. We take the
exciton recombination time to be Γ−1≈ 40 ps, correspond-
ing to the photoluminescence (PL) lifetime measured on indi-
vidual CNTs [14, 15]. The narrowest reported nanotube PL
linewidths (0.25−0.5 meV [12, 24]) however, are an order of
magnitude larger than the lifetime broadening. We therefore
include a Markovian dephasing rate of both optical transitions
?γdeph= 0.25 meV.
The resulting spin population imbalance as a function of
the axial and perpendicular components of the magnetic field
is shown in Fig. 2a, when the laser is resonant with the red
(lowest energy) transition. First, we remark that spin prepa-
ration with a fidelity close to 1 is possible at almost any axial
field provided that the perpendicular field is of the order of
a few 100 mT. But the main feature revealed by the simula-
tion is the peculiar behavior of the system when the Zeeman
splitting caused by B?cancels the spin-orbit splitting for ei-
ther electrons or holes: when B?= Bh
states U↓eU↑eU↓hand U↓eU↑eU↑hhave the same energy; as
aconsequencethemixinginducedbyevenavanishinglysmall
B⊥suffices to yield very efficient electron spin pumping. On
the contrary, for B?= Be
occurs between the electronic states and the electron spin re-
mains randomized for all values of B⊥.
Since it is likely that the excited trion states undergo faster
dephasing than the single electron ground states, we per-
formed an additional simulation where the coherence between
the two trion states is also dephased at the rate ?γdeph. The
result, shown in Fig. 2b, is qualitatively similar to the case
depicted in Fig. 2a.For B? ≈ Bh
(inset Fig. 2b) very efficient spin preparation in either state
U↑e or U↓e is achieved upon tuning the laser across the
U↓e→U↓eU↑eU↓hand U↑e→U↓eU↑eU↑htransitions. Pauli
blockade leads to a drop in absorption by more than an or-
der of magnitude: this would be the experimental signature of
spin pumping in differential transmission measurements [2].
Our results demonstrate that selective optical spin preparation
in CNT QDs is in experimental reach. A limitation would
however appear for very small QDs (∼ 10 nm): it was shown
recently that the ohmic coupling of strongly confined exci-
tons to one-dimensional acoustic phonons in CNTs leads to
asymmetric absorption spectrum with a pronounced blue tail,
extending over a few meV [15]. This pure-dephasing process
would nevertheless not alter the efficiency of spin pumping
when driving the lower-energy resonance with a red-detuned
laser.
The presence of K-K’ valley mixing (characterized by the
splitting gµBBKK? = ∆KK?) will result in a finite probabil-
ity that the electron spin leaves the Hilbert space spanned by
U↓eandU↑e. Wedenotetheeffectivespin-flipRamanscatter-
ing rate from state U↓eto U↑ewith γ↑↓and the effective rate
for a laser assisted transition from U↓eto D↓ewith γKK?. Us-
SO= ∆h
SO/(gµB), the
SO= ∆e
SO/(gµB), the resonance
SOand B⊥ = 0.2 T

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-15 -10 -505 1015
0.0
0.1
0.2
0.3
0.4
0.5

0.0
0.1
0.2
0.3
0.4
0.5



a)
b)
h
SO
B
e
SO
B
0
0.2
0.4
0.6
0.8
1.0
Axial magnetic field (T)
Perpendicular magnetic field (T)
-2024
-1
0
1
Detuning (meV)
0.00
0.04
X
X
FIG. 2: The contour plot of the population difference between states
U↑e and U↓e as a function of the magnetic field components when
thelaseriskeptresonantwiththeU↓e→U↓eU↑eU↓htransition(zero
of the detuning scale in the inset). We take ξe = ξh= 0.1 µs−1and
Γ−1= 40 ps. In (a), we assume that the optical transitions are
broadened by Markovian dephasing with rate ?γdeph = 0.25 meV.
In (b), we consider a scenario where the coherence between states
U↓eU↑eU↓h and U↓eU↑eU↑h also undergoes fast dephasing at the
same rate (0.25 meV). In the inset to part (b), we show the laser
absorption (black line) and population difference (red line) between
U↑e and U↓e states for external magnetic fields B? = Bh
B⊥ = 0.2 T. The absorption (right scale) is normalized to its maxi-
mum value in absence of spin pumping (when B⊥= 0 T).
SOand
ing rate equations we obtain:
which is maximum when B?→ Bh
experimentally measured parameters, efficient spin pumping
is possible in large regions of magnetic fields. For example at
point X in Fig. 2, for which B?= 8 T ≈ Bh
B⊥ = 0.2 T, we have
through a valley-flip Raman scattering to state D↓e, the ap-
plied laser field will be detuned from the transition to the state
D↓eU↑eU↓h, due to the exchange terms of Coulomb interac-
tion (see Fig. 1b, right). If BKK? is large it would therefore be
necessary to use a second re-pumping laser on this transition
to reintroduce the electron to the U-valley.
In most experiments it has been observed that the lifetime
of excitons is more than an order of magnitude shorter than
the predicted radiative lifetime [14, 15, 25]. While radiative
broadening can be enforced by embedding CNTs in cavity
structures with a large Purcell factor [26], understanding the
nature of non-radiative relaxation is crucial for identifying the
γ↑↓
γKK?≈
B2
KK?/(∆e
⊥/(∆h
SO−gµBB?)2
SO−2µorbB?)2
B2
SO. We find that using the
SO+ 0.75 T and
γ↑↓
γKK?> 100. Once the system goes
limits of optical spin manipulation. In particular, if this re-
laxation is not spin-conserving, then spin pumping becomes
efficient for an even larger range of applied magnetic field
strengths. Most probable mechanisms for fast non-radiative
decay proposed so far are phonon-assisted relaxation and/or
multi-particle Auger processes [27]. Since these processes
are spin-conserving, they will not alter the efficiency of spin
pumping.
Having demonstrated that it is viable to prepare a single
spin optically, we turn to coherent spin rotation and spin mea-
surement. By using two laser fields satisfying two-photon Ra-
man resonance condition under the same external magnetic
field configurations that allow for efficient spin pumping, we
can implement deterministic spin rotation [5]. To realize all-
optical spin measurement, the field B⊥mixing the electron
(hole) spin states must be turned off. In this limit, presence or
absence of light scattering (or absorption) upon excitation by
a resonant laser conveys information about the spin state [5].
For spin measurements, minimizing spin-flip non-radiative re-
laxation and inter-valley scattering is crucial. We also point
outthatallofourresultswouldapplyforasingle-holecharged
CNT QD as well.
Next, we address the possibility of transferring quantum in-
formation stored in the CNT QD electron spin to a generated
photon. Given that the polarization of the photon is fixed by
the geometry, the logical choice is to use time-bin entangle-
ment [28]. We assume that our CNT QD is coupled to an
optical cavity whose energy ωcavis resonant with the transi-
tion U↓e→U↓eU↑eU↓h(Fig. 3). Using combinations of laser
pulses one can prepare an initial state in the coherent spin su-
perposition: |ψin? = (α|U↑e? + β|U↓e?) ⊗ |0c? where |0c?
is the empty cavity mode. We now send two well separated
π-pulses at time t1and t2with respective energies ωaand ωb
as shown in Fig. 3. We define the two creation operators a†
(b†) for cavity-mode photons emitted immediately after pulse
1 (pulse 2). The optical transition at frequency ωbis allowed
because of the mixing induced by B⊥, and the rates of both
transitions can be made identical by adjusting the pulse in-
tensities. The first pulse excites the trion state if and only if
the spin is initially down. In this case Purcell effect ensures
very fast spontaneous emission and projection onto the state
|U↓e?⊗a†|0c?. Ifthespinisinitiallyup, thetransitionisPauli-
blocked and we are left with |U↑e?⊗|0c?. The initial state has
thus evolved to: |ψ1? = α|U↑e?⊗|0c?+β|U↓e?⊗a†|0c?. We
can do the same analysis for the second pulse and find that
the final state is: |ψf? = |U↓e? ⊗ (αb†+ βa†) ⊗ |0c? where
quantum information has been mapped onto a photon time-bin
qubit. We emphasize that time-bin qubits are promising can-
didates for long range quantum communication using optical
fibers [29] and that CNTs can be chosen to emit in the desired
wavelength window.
We remark that one of the most interesting perspectives en-
abled by the considerations of this letter is the study of nuclear
spin physics. The possibility of electron spin pumping should
allow for the optical manipulation of nuclear spin ensembles,
which has been successfully achieved in GaAs-based struc-

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4
U↓e

U↑e

U↓h
U↑e

U↓e

U↑h
U↓e
cava
ωω =
p
Γ
U↑e
b
ω
t1
t2
a)
b)
FIG. 3:
text. Both laser pulses are polarized along the nanotube axis. Pro-
vided that the pulse separation t2 − t1 is larger than all the other
time scales (inverse Rabi frequency and cavity-enhanced exciton de-
cay rate) quantum information can be efficiently encoded in a photon
time-bin qubit. b) Energy diagram of the nanotube quantum dot with
the relevant transitions and rates used in the scheme.
a) Schematics of the cavity QED setup discussed in the
tures [3, 4]. However, experimental knowledge of the strength
and characteristics of hyperfine interaction in CNTs is still
lacking. Of particular interest in this context would be dy-
namic nuclear spin polarization in a CNT QD where hundreds
or thousands of13C atoms would form an ideal I =
bath. Alternatively, using high-purity12C CNTs, one may re-
alize QDs interacting with only 1 or 2 nuclear spins [30].
This work was supported by a grant from the Swiss Na-
tional Science Foundation (SNSF).
1
2spin
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