Hyperfine interaction induced decoherence of electron spins in quantum dots

Abstract

We investigate in detail, using both analytical and numerical tools, the decoherence of electron spins in quantum dots (QDs) coupled to a bath of nuclear spins in magnetic fields or with various initial bath polarizations, focusing on the longitudinal relaxation in low and moderate field/polarization regimes. An increase of the initial polarization of nuclear spin bath has the same effect on the decoherence process as an increase of the external magnetic field, namely, the decoherence dynamics changes from smooth decay to damped oscillations. This change can be observed experimentally for a single QD and for a double-QD setup. Our results indicate that substantial increase of the decoherence time requires very large bath polarizations, and the use of other methods (dynamical decoupling or control of the nuclear spins distribution) may be more practical for suppressing decoherence of QD-based qubits. Comment: Rev. Tex, 5 pages, 3 eps color figures, submitted to Phys. Rev. B

Full text

Page 1

arXiv:cond-mat/0609185v2 [cond-mat.mes-hall] 15 Nov 2006
Hyperfine interaction induced decoherence of electron spins in quantum dots
Wenxian Zhang,1V. V. Dobrovitski,1K. A. Al-Hassanieh,2,3E. Dagotto,2,3and B. N. Harmon1
1Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA
2Department of Physics, University of Tennessee, Knoxville, TN 37831, USA and
3Condensed Matter Science Division, Oak Ridge National Laboratory, Oak Ridge, TN 37996, USA
(Dated: February 6, 2008)
We investigate in detail, using both analytical and numerical tools, the decoherence of electron
spins in quantum dots (QDs) coupled to a bath of nuclear spins in magnetic fields or with various ini-
tial bath polarizations, focusing on the longitudinal relaxation in low and moderate field/polarization
regimes. An increase of the initial polarization of nuclear spin bath has the same effect on the de-
coherence process as an increase of the external magnetic field, namely, the decoherence dynamics
changes from smooth decay to damped oscillations. This change can be observed experimentally
for a single QD and for a double-QD setup. Our results indicate that substantial increase of the
decoherence time requires very large bath polarizations, and the use of other methods (dynami-
cal decoupling or control of the nuclear spins distribution) may be more practical for suppressing
decoherence of QD-based qubits.
PACS numbers: 75.10.Jm, 03.65.Yz, 03.67.-a, 02.60.Cb
I.INTRODUCTION
Quantum dots (QDs) are very promising candidates
for future implementation of quantum computations: an
electron spin in a QD is a natural two-state quantum
system, which can be efficiently manipulated by the ex-
ternal magnetic fields and gate voltages1,2.
QD-based architectures are potentially scalable, and rely
on well-developed semiconductor technology3. However,
due to interaction with the environment, the electron spin
loses coherence very quickly, on a time scale of order of
nanoseconds for typical GaAs QDs1. It is vitally impor-
tant for realization of QD-based quantum computing to
understand the decoherence dynamics in detail, in or-
der to find practical ways of decoherence suppression.
Moreover, decoherence of open systems is of fundamen-
tal interest for understanding of the quantum phenomena
taking place in mesoscopic systems4,5,6, and therefore at-
tracts much attention from scientists working in the areas
of nanoscience, spintronics, and quantum control.
Among different sources of decoherence relevant for an
electron spin in a QD, the decoherence by the bath of nu-
clear spins (spin bath) is dominant for magnetic fields less
than a few Tesla, and experimentally relevant tempera-
tures of tens or hundreds of milliKelvin. Much research
has been focused on the case of a large external magnetic
field or large bath polarizations, where the perturbation
theory allows an extensive analysis7,8,9,10. But the inter-
esting and experimentally relevant regime of moderate
magnetic fields and/or moderate bath polarization has
received much less attention.
Below, we study in detail the influence of moderate
magnetic fields and bath polarization on the longitudinal
decoherence of a single spin in a quantum dot, and two
spins located in neighboring QDs, where the perturba-
tion theory is not applicable. In this regime, as the mag-
netic field or the bath polarization increase, the dynamics
of the electron spins undergoes a transition from sim-
Also, the
ple overdamped decay to underdamped oscillations, and
these oscillations can be detected using existing experi-
mental schemes. In this transition, the increase of bath
polarization affects the decoherence dynamics in exactly
the same manner as the increase of the magnetic field.
Our results also show that suppression of decoherence re-
quires very large bath polarizations, suggesting that the
use of such methods as dynamical decoupling1,11,12, or
control of the nuclear spins distribution13, may be more
practical.
Theoretical description of the spin-bath decoherence is
a very complex problem14, where strong correlation and
essentially non-Markovian bath dynamics play an impor-
tant role7,15,16. In contrast with the well studied boson-
bath decoherence4,6, decoherence by a spin bath has not
yet been understood in detail, especially in moderate ex-
ternal magnetic fields.
II. SINGLE ELECTRON SPIN IN A QD
An electron spin in a QD interacting with a bath of
nuclear spins is described by the Hamiltonian which in-
cludes the Zeeman energy of the electron spin in the
external magnetic field B0 and the contact hyperfine
coupling17,18: H = H0Sz +?N
is the operator of the electron spin, Ik is the opera-
tor of the k-th bath spin (k = 1,2,...,N), and Ak =
(8π/3)g∗
eµBgnµnu(xk) is the contact hyperfine coupling
which is determined by the electron density u(xk) at the
site xkof the k-th nuclear spin and by the Land´ e factors
of the electron g∗
eand of the nuclei gn. The terms omit-
ted in the above Hamiltonian, such as the Zeeman energy
of the nuclear spins, the anisotropic part of the hyperfine
coupling, etc., are small, and can be safely neglected at
the nanosecond timescale, which is considered here.
In spite of apparent simplicity of the Hamiltonian,
it is very difficult to determine the dynamics of the
k=1AkS · Ik, where S

Page 2

2
central spin S(t). Previously, the quasi-static approxi-
mation (QSA) for the nuclear spin bath, which treats
?
Analytical calculations beyond the QSA have been car-
ried out only for large magnetic field and/or large bath
polarizations7,8,9,10, where the perturbative approach is
valid.Although many qualitative arguments support
QSA, its validity has not yet been checked in detail. Be-
low, along with explicit analytical solutions, we provide
direct verification of our analytics by employing the exact
numerical simulations for medium-size (N = 20) baths,
and approximate simulations for large (N = 2000) baths.
The agreement between all three methods ensures us that
our findings do not depend on the approximations in-
volved.
The analytical calculations within the QSA can be
performed in different ways which all give the same re-
sults. The conceptually simplest way is to assume a uni-
form electron density in the QD, so that all Ak are the
same, Ak = A, and the Hamiltonian can be written as
H = H0Sz+AS·I, where I =?
spin of the bath. This Hamiltonian can be analyzed ex-
actly, since I2and Iz+ Szare the integrals of motion.
First, we consider an unpolarized bath, with the den-
sity matrix ρb(0) = (1/2N)11⊗ ··· ⊗ 1N, where 1k is
the unity matrix for k-th bath spin. This density matrix
can be re-written in the eigenbasis of I2, Iz as ρb(0) =
?N/2
suming that the initial state of the electron is | ↑?, the
quantum-mechanical average of the z-component of the
electron spin at time t is
kAkIkas a constant, has often been invoked19,20,21,22.
kIkis the total nuclear
I=0
?I
M=−IP(I,M)|I,M??I,M| where M = Iz. As-
σz(t) ≡ 2?Sz(t)? =
?
I,M
P(I,M)B2+ C2cosΩt
Ω2
(1)
where C = A?(I − M)(I + M + 1), B = H0+ A(M +
1/2), and Ω2= B2+ C2. Due to the symmetry of the
problem and the initial condition, σx(t) = σy(t) = 0,
so we omit these components. The distribution function
P(I,M) can be calculated23and for large N and I it
is approximated by a Gaussian distribution P(I,M) ≈
(I/D√2πD )e−I2/2Dwhere D = N/4. Replacing the
summation by integration in Eq. (1), we find
σz(t) = 1 − 2W(λ,D;t), (2)
and the function W(λ,D;t) has the form
W =
D
λ2−D
?
λ2e−Dt2/2cosλt + i
Φ(Dt − iλ
√2D
?π
2
D3/2
2λ3e−λ2/2D
iλ
√2D)
×
) − Φ(Dt + iλ
√2D
) + 2Φ(
?
(3)
where Φ(x) is the error function. In this equation, we
took A = 1, and introduced the notation λ = H0/A;
this corresponds to normalized energy and time scales,
so that the time t is measured in the units of 1/A. The
dynamics of σz(t) for several values of λ are shown in
Figs. 1 and 2.
0
1
0.5
1
(a)
λ =0
0.92
0.96
(b)
λ =15
0123
0.96
0.98
1
(c)
Time
λ =25
0
1
0.5
1
(d)
β =−0.1
0.92
0.96
(e)
β =−1.9
0123
0.96
0.98
1
(f)
Time
β =−2.3
FIG. 1: (Color online) Electron spin decoherence in various
magnetic fields [(a), (b), (c)] and with polarized initial nuclear
spin baths [(d), (e), (f)] for N = 20. The red dashed curves
denote the exact numerical simulations results for N = 20,
and the blue solid curves correspond to the analytical results.
The numerical results agree well with the analytical predic-
tions, and the underdamped oscillations appear once λ or κ
is larger than
√N.
Before discussing the above results, we consider next a
polarized bath, assuming that its initial state is described
by a density matrix ρb(0) = (1/Zβ)exp(−βM), where β
is the inverse spin temperature, and Zβ= [2cosh(β/2)]N
is statistical sum, and the initial bath polarization is p ≡
?M?/(N/2) = −tanh(β/2). For large spin temperatures,
the polarization is small, p ≈ −β/2, and we approximate
P(I,M) ≈ (I/D√2πD )e−Dβ2/2e−I2/2De−βM, so that
σz(t) = 1 − 2W(κ,D;t),
where κ = Dβ, and the function W(κ,D;t) is defined by
Eq. (3). The dynamics of σz(t) for several values of β is
shown in Fig. 1.
The functional form of Eqs. (2) and (4) is identical, up
to replacing κ by λ, so the small nonzero bath polariza-
tion p affects the central spin in exactly the same way as
the external field of the magnitude H0= −pAN/2, equal
to the average Overhauser field exerted on the central
spin by the nuclear bath. Indeed, the noticeable average
magnetization ?M? = pN/2 of the polarized bath leads to
a noticeable average Overhauser field, but the variation
of the magnetization ?(∆M)2? = (1 − p2)N/4 changes
very little at small p, and, correspondingly, the spread in
the Overhauser fields is almost unchanged.
From Eqs. (2) and (4), we see that σz(t) always de-
cays with characteristic time T∗
for H0 = 0 (or p = 0), we reproduce a well-known re-
sult σz(t) = (1/3) + (2/3)(1 − Dt2)exp(−Dt2/2), i.e.
σz(t) first falls to the value ≈ 0.036, then increases and
saturates at 1/3.However, for H0 ?= 0 (or p ?= 0),
due to the oscillatory terms in Eq. (3), σz(t) can ex-
hibit oscillations, provided that λ (or κ) is compara-
(4)
2=
?8/(NA2). E.g.,

Page 3

3
0 0.020.04
Time
0.06
0
0.5
1
(a)
λ = 0
0 0.020.04
Time
0.06
0.8
0.9
1
(b)
λ = 2N1/2
FIG. 2: (Color online) The same as Fig. 1 but for N = 2000
nuclear bath spins in zero magnetic field (a) and in λ = 2√N
(b) with initially unpolarized bath.
ble or larger than
smooth decay at small field/polarization, to the oscilla-
tions at larger field/polarization, is similar to the well-
known transition in the dynamics of a damped oscillator:
the evolution of the central spin is overdamped (or un-
derdamped) depending on whether the decay time T∗
larger (or smaller) than the “bare” oscillation frequency
determined by λ or κ. Note that the crossover value for
magnetic field (polarization) is of order of
the range of applicability of the perturbation theory7,10.
For a typical GaAs QD with the electron delocalized
over N = 106nuclear spins, A ∼ A0/N ∼ 10−4µeV
(where A0 ≈ 0.1 meV is the hyperfine coupling for an
electron localized on a single nucleus24). The correspond-
ing decoherence time T∗
2∼ 10 ns (taking into account the
I = 3/2 spins of69Ga,71Ga and75As), as confirmed by
recent experiments1,19. To observe oscillations for a sin-
gle electron spin, a very modest external field of order of
3 mT, or polarization of order of 0.5% is needed.
It is noteworthy that the decoherence time remains
practically constant in the course of transition from
smooth to oscillatory decay. The decoherence time is de-
termined by the spread in the Overhauser fields, which is
proportional to the variation of the bath magnetization
?(∆M)2? = (1−p2)N/4, and it is little affected by small
bath polarizations p7,25. For a moderate bath polariza-
tion, the equivalence between the external field and the
nonzero bath polarization disappears, and the analytical
expression for σz(t) becomes rather complex, involving
hypergeometric functions2F1(a,b,c,z) of complex argu-
ment, so we do not present it here. In this regime, the
decoherence time does not increase much, as shown in
Fig. 1(e) and (f).
In order to substantially increase the decoherence
time, an extremely large bath polarization is required,
as suggested by the results from the perturbation
method7,8,9,10. Such a large bath polarization is beyond
the scope of our paper. Currently, strongly polarized
baths are difficult to achieve experimentally, and such
methods as narrowing the nuclear spin distribution13, dy-
namical decoupling11and spin echo techniques1,12, may
be more practical for suppression of decoherence.
The quasistatic bath approximation is far from reality.
E.g., for H0= 0 and p = 0, the QSA predicts saturation
of σz(t) at 1/3 for t → ∞, which is just an artifact of
the approximation: the detailed analysis shows that σz
√D ∼
√N. This transition, from
2is
√N, beyond
slowly decays (as 1/lnt) to zero7,17,20. However, we ex-
pect QSA to be valid at times of order of T∗
bath’s internal dynamics is not yet important.
For verification, we perform exact numerical simula-
tion with medium-size baths of N = 20 spins. A real
QD is approximated by taking nuclear spins located at
the sites of the 4 × 5 piece of a square lattice, with the
lattice constant a = 1. Assuming a parabolic confining
potential, the electron density is approximated as 2-D
Gaussian with the widths of wx = wy = 1.5, and with
the center shifted by dx= 0.1 and dy = 0.29 along the
x- and y-axis respectively. For comparison with the an-
alytical results above, the effective coupling constant is
defined as A =??
ics is simulated by directly solving the time dependent
Schr¨ odinger equation for the wave function of the full
many-spin system (central spin plus the bath), using the
Chebyshev polynomial expansion of the evolution opera-
tor as described in Ref. 16.
Fig. 1(a), (b), and (c) show analytical [obtained from
Eq. (2)] and numerical results for σz(t), for different mag-
netic fields. The panels (d), (e), and (f) present σz(t) for
polarized baths with different initial polarizations. For
the analytical curve in panel (d) the small-polarization
formula Eq. (4) has been used, while the analytical curves
in panels (e) and (f) have been calculated from the for-
mula for moderate initial polarization. The agreement of
the analytics and the exact numerics is good. The dif-
ference is caused only by the modest number N = 20 of
the bath spins used for numerical simulations, but exact
simulations even for N = 50 are beyond the capabilities
of modern computers (since the computation time and
memory grow exponentially with N).
To study large baths with N = 2000, we use the co-
herent state P-representation described in Ref. 17. Al-
though approximate, this numerical approach demon-
strates excellent accuracy. Figure 2 presents the results
for σz(t) obtained analytically, from Eq. (2), and numer-
ically, from P-representation simulations with N = 2000
spins. For N = 2000, the agreement between the ana-
lytics and the numerics is very good. Overall, the data
presented in Figs. 1 and 2 justify the use of QSA, so that
we expect our predictions to be valid for real QDs with
106nuclear spins.
2, when the
kA2
k/N?1/2. The decoherence dynam-
III. TWO ELECTRON SPINS IN DOUBLE QD
Measurement of the Rabi oscillations for a single elec-
tron spin in a QD has not yet been achieved26. Recent
experiments use two electron spins in two neighboring
QDs1,19.The spins are prepared in the singlet state,
and the measured quantity is the probability PS(t) to
stay in the singlet state after time t. The oscillations
described above can be also detected in this double-dot
setup. When the coupling between the two electron spins
is negligible, the Hamiltonian of the double-QD system is
H = H01S1z+H02S2z+?N1
j=1AjS1·Ij+?N2
k=1AkS2·Ik

Page 4

4
02468
x 10
−3
0.25
0.5
0.75
1
Time
λ = 10 N1/2
λ = 0
(a)
02468
x 10
−3
0
0.5
1
Time
(b)
FIG. 3: (Color online) Time evolution of the probability of the
singlet state of the two electrons in a double QD by applying
(a) parallel magnetic fields λ1 = λ2 = λ and (b) antiparallel
magnetic fields λ1 = −λ2 = 10√N with N1 = N2 = N = 106.
Underdamped Rabi oscillations appear once the difference in
the magnetic fields applied onto each QD is large enough.
where indices 1 and 2 denote the quantities describing
left and right QD, respectively (e.g., H01 and H02 are
the magnetic fields acting on the left and right spins,
respectively). Note that the evolution of QD 1 and 2 is
not independent because of the initially entangled singlet
state, although the Hamiltonian is separable. Using the
quasistatic approximation, we get
PS =
1
2U(λ1,D1)U(λ2,D2) +1
+ F(λ1,D1)F(λ2,D2) + G(λ1,D1)G(λ2,D2)
2W(λ1,D1)W(λ2,D2)
(5)
where the function W(λ,D) is defined by Eq. (3),
and F(λ,D) = 1/2
?
G(λ,D)=1 − F(λ,D) − W(λ,D),
(1/λ2)[Dλtcosλt + (λ2− D)sinλt]e−Dt2/2. Figure 3(a)
illustrates dynamics of PS(t) for unpolarized baths, in the
case of uniform magnetic field H0= H01= H02. PS(t)
decays in the beginning, and saturates at non-zero value
at long times. The saturation value is 1/3 for zero field,
and increases with the magnetic field, reaching 1/2 for
strong fields19.
However, if the difference between H01and H02is com-
parable or larger than A?N1,2(which corresponds to few
milliTesla for realistic GaAs QD), the probability PS(t)
exhibit oscillations [see Fig. 3(b)], analogous to the oscil-
lations of σz(t) in the single-QD case above. In experi-
ments, a non-uniform magnetic field can be created e.g.
by micromagnets27. Another opportunity is using non-
uniformly polarized nuclear spin baths in the left and
1 + [cosλt − (Dt/λ)sinλt]e−Dt2/2?
,
U(λ,D)=
right QDs. In analogy to the single-QD calculations, it
can be shown that PS(t) in the case of nonzero initial
polarization is still given by Eqs. (5), with replacement
of λ1,2by κ1,2. The difference in polarization should be
about 0.5% for the oscillations to appear.
IV.CONCLUSION
In summary, we study in detail the influence of mag-
netic fields and bath polarization on the decoherence of
a single spin in a quantum dot, and two spins located
in neighboring QDs. We focus on the regime of mod-
erate fields and polarizations, where the perturbation
theory is not yet applicable, using both analytical tools
(the quasi-static bath approximation) and the numerical
simulations (exact, for medium-size baths with 20 spins,
and approximate, for large baths with 2000 spins). The
agreement between all three approaches is good, so we
believe our results are applicable to real QDs with mil-
lions of nuclear spins. The nonzero bath polarization and
the external magnetic field influence the decoherence dy-
namics in exactly the same way, and lead to a transition
from smooth decay to oscillations once the field (polariza-
tion) exceeds a certain crossover value. This transition
can be observed in experiments with a single QD, and
with two quantum dots. Our results show that substan-
tial increase of the decoherence time requires extremely
large bath polarizations, so that such methods as dy-
namical decoupling1,11,12or control of the nuclear spins
distribution13may be more practical for controlling de-
coherence of QD-based qubits.
V. ACKNOWLEDGEMENT
This work was supported by the NSA and ARDA un-
der Army Research Office (ARO) contract DAAD 19-03-
1-0132. This work was partially carried out at the Ames
Laboratory, which is operated for the U. S. Department
of Energy by Iowa State University under contract No.
W-7405-82 and was supported by the Director of the Of-
fice of Science, Office of Basic Energy Research of the U.
S. Department of Energy. K.A.A. and E.D. are supported
by the NSF grant No. DMR-0454504.
1A. C. Johnson et al., Nature (London) 435, 925 (2005);
F. H. L. Koppens et al., Science 309, 1346 (2005); J. R.
Petta et al., Science 309, 2180 (2005).
2M. A. Nielsen and I. L. Chuang, Quantum Computations
and Quantum Information (Cambridge University Press,
Cambridge, 2002).
3D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120
(1998).
4C. W. Gardiner and P. Zoller, Quantum Noise (Springer-
Verlag, Berlin, Heidelberg, New York, 2000).
5T. Yu and J. H. Eberly, Phys. Rev. Lett. 97, 140403 (2006).
6A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987).
7W. A. Coish and D. Loss, Phys. Rev. B 70, 195340 (2004).
8J. Schliemann, A. Khaeskii, and D. Loss, J. Phys.: Con-
dens. Matter 15, R1809 (2003).
9A. V. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. Lett.
88, 186802 (2002).
10C. Deng and X. Hu, Phys. Rev. B 73, 241303(R) (2006).

Page 5

5
11L. Viola and E. Knill, Phys. Rev. Lett. 94, 060502 (2005);
K. Khodjasteh and D. A. Lidar, Phys. Rev. Lett. 95,
180501 (2005).
12N. Shenvi, R. de Sousa, and K. B. Whaley, Phys. Rev. B
71, 224411 (2005); W. Yao, R. Liu, and L. J. Sham, eprint
cond-mat/0508441; W. M. Witzel, R. de Sousa, and S. Das
Sarma, Phys. Rev. B 72, 161306(R) (2005).
13D. Stepanenko et al., Phys. Rev. Lett. 96, 136401 (2006).
14A. Garg, Phys. Rev. Lett. 74, 1458 (1995); J. Shao and P.
H¨ anggi, Phys. Rev. Lett. 81, 5710 (1998); N. V. Prokof’ev
and P. C. E. Stamp, Rep. Prog. Phys. 63, 669 (2000).
15J. Lages et al., Phys. Rev. E 72, 026225 (2005).
16V. V. Dobrovitski and H. A. De Raedt, Phys. Rev. E 67,
056702 (2003).
17K. A. Al-Hassanieh et al., Phys. Rev. Lett. 97, 037204
(2006).
18V. V. Dobrovitski, J. M. Taylor, and M. D. Lukin, Phys.
Rev. B 73, 245318 (2006).
19J. M. Taylor et al., eprint cond-mat/0602470; Y. A. Sere-
brennikov, Phys. Rev. B 74, 035325 (2006); W. A. Coish
and D. Loss, Phys. Rev. B 72, 125337 (2005); K. Schulten
and P. G. Wolynes, J. Chem. Phys. 68, 3292 (1978).
20S. I. Erlingsson and Y. V. Nazarov, Phys. Rev. B 70,
205327 (2004).
21I. A. Merkulov, Al. L. Efros, and M. Rosen, Phys. Rev. B
65, 205309 (2002).
22Y. G. Semenov and K. W. Kim, Phys. Rev. B 67, 073301
(2003).
23A. Melikidze et al., Phys. Rev. B 70, 014435 (2004).
24D. Paget et al., Phys. Rev. B 15, 5780 (1977).
25V. Cerletti et al., Nanotechnology 16, R27 (2005).
26The driven Rabi oscillations of an electron spin in a quan-
tum dot have been reported after our manuscript submis-
sion, F. H. L. Koppens et al., Nature (London) 442, 766
(2006). The Rabi oscillations without external drive have
not been reported yet.
27J. Wr´ obel et al., Phys. Rev. Lett. 93, 246601 (2004); Y.
Tokura et al., Phys. Rev. Lett. 96, 047202 (2006).