PHYSICAL REVIEW B 76, 064301 共2007兲
Spin echo without an external permanent magnetic field
Joakim Bergli1,* and Leonid Glazman2,†
1Physics
Department, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, Norway
Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA
共Received 19 September 2006; revised manuscript received 14 January 2007; published 2 August 2007兲
2W.I.
We suggest a sequence of instantaneous spin rotations, that reverses the time evolution of a spin subject to
an arbitrary built-in magnetic field. The sequence does not require the knowledge of the direction or strength
of the built-in field, and is applicable as long as that field is constant in time. The advantage of the suggested
sequence compared to the known spin-echo sequences is that it does not involve the application of a permanent
strong magnetic field. The spin rotations may be achieved by application of short magnetic field pulses.
DOI: 10.1103/PhysRevB.76.064301
PACS number共s兲: 76.60.Lz
I. INTRODUCTION
The use of echoes is a standard technique in spinresonance experiments1 with macroscopic samples. Echo sequences typically consist of a number of high-frequency
pluses that induce controlled rotations on the precessing spin.
The objective of an echo technique is to remove the reversible dephasing in the ensemble of spins caused by the inhomogeneity of the built-in random magnetic field. That field
adds to the external permanent field and causes variation of
the spin precession frequencies across the sample 共inhomogeneous broadening of magnetic resonance兲. Application of
the echo technique allows one to compensate for the inhomogeneities of the static magnetic field and to reveal the
effects of irreversibility in the spin evolution. The irreversible relaxation of the spin coherent precession is due to the
interaction of spins with the dynamic fluctuations of the medium. The corresponding homogeneous broadening of the
magnetic resonance is determined by the spectrum of the
medium fluctuations.
Recent work on the implementation of the idea of quantum computing has brought a new perspective to the physics
of spin echo. To implement quantum computing, one needs
the ability to coherently manipulate a single two-level system. The role of such a system may be played by a single
spin or by a small number of coupled spins. The spin state
can be detected electrically.2,4 Charge transport measurements enabled one to observe the electron spin resonance
共ESR兲 of a single electron trapped in a quantum dot.5 Rabi
flops of spins of electrons occupying the 31P donors in a Si
host were also detected recently by means of photoconductivity measurements.4 In another quantum dots experiment,
oscillations were observed between the singlet and triplet
states of two spins residing in two quantum dots.3 The
singlet-triplet splitting there was due to the tunnel coupling
between the dots.
Observation of the ESR or the singlet-triplet oscillations
required cycling the electrons through the dots. That enabled
one to repeat the measurements with varying delay between
the preparation and read-out of a state. The statistics of the
spin-flip probabilities 共or the probabilities of the singlettriplet transitions in a double-dot system兲 as a function of the
delay time P共t兲 reveals the oscillations between the spin
states. The corresponding Fourier transform P共␻兲 is similar
1098-0121/2007/76共6兲/064301共6兲
to the ESR absorption line: P共␻兲 has a well-defined peak at
the frequency defined by the level splitting.
The peak width does depend on the irreversible processes
of electron spin-relaxation, but apparently a major contribution comes from the processes of the inhomogeneous broadening type. Indeed, the electron spin in a GaAs quantum dot
interacts with the nuclear spins; in the simplest approximation, those act as some random static field on the electron
spin. Similarly, an electron in a donor state created by a 31P
atom in a Si host is exposed to the effective field of the
nuclei of 29Si impurities.6 By cycling electrons through the
localized state one explores the realizations of that random
field. In GaAs quantum dots, the estimate of the random field
agrees reasonably well with the width of the peak in P共␻兲 for
singlet-triplet transitions3 and for the single-spin evolution.5
The effect of a random static field, in principle, may be
eliminated in an electron spin-echo experiment.
Nuclear spins are not fully static, and may evolve at least
because of their interaction with the electron in the quantum
dot 共this interaction may be described by a central spin
problem.7兲 This causes some fluctuations of the effective random magnetic field ␦B acting on the electron spin. Moreover, these fluctuations depend on the strength of the external
field. In a strong external field, fluctuations of the effective
field induced by the nuclei are weak.7 The fluctuations may
increase once the external field is of the order of ␦B. It is still
debated, how strong are the fluctuations in the absence of an
external field.8
The advantage of the single-spin read-out techniques2 is
that the detection of an echo does not involve a measurement
of a free-induction signal. That allows one to investigate
echo at an arbitrarily weak external field B. The existing
echo techniques, however, work only if the condition B
Ⰷ ␦B is satisfied. The purpose of the present work is to explore the possibility of an echo without that constraint.
We will explore the possibility of finding an echo sequence that is applicable in the case of a constant effective
field with an arbitrary and unknown magnitude and direction.
The goal of this sequence is to return the spin to its starting
position at the end of the sequence, independently of the
effective magnetic field. In the case of a single electron confined to a quantum dot, the control rotations would have to
be effected by short pulses of a sufficiently strong magnetic
field. In order for the control rotations to be independent of
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©2007 The American Physical Society
PHYSICAL REVIEW B 76, 064301 共2007兲
JOAKIM BERGLI AND LEONID GLAZMAN
the nuclear field, the applied magnetic fields would have to
significantly exceed the nuclear field. With a typical5 nuclear
field of 1 mT it means that a ␲ rotation would require the
application of a magnetic field of 10– 100 mT with a pulse
length of the order of 1 – 10 ns 共assuming a g factor of the
order 0.3兲. In the experiments5 magnetic pulses at GHz frequencies were applied 共though it is not clear how short single
pulses are achievable at present兲, but the strength of the field
was limited to about 2 mT. Thus, it seems that the requirements for the pulse parameters are out of reach of present
experiments, but not so impossibly far.
The motivation for this work comes from the problem of
spin dynamics of an electron confined to a quantum dot. The
pulse sequence which we analyze, however, is quite general
and applicable for controlling any two-level quantum system. In fact, it has recently been discussed in the context of
dynamical decoupling of a qubit11 where it is known as concatenated dynamical decoupling. In this context the present
work is an analysis of higher-order decoupling efficiency.
II. FORMULATION OF THE PROBLEM
Consider a spin- 21 particle precessing in a constant magnetic field. Let n be the unit vector along the precession axis
and ␻ be the precession angular frequency. The evolution of
the density matrix of the spin during the time ␶ is then given
by ␳共␶兲 = U−1␳共0兲U with
U = cos ␹I + i sin ␹ni␴i ,
共1兲
where I and ␴i are the identity and Pauli matrices, and ␹
= ␻␶ / 2. The ␲ rotations about the coordinate axes are denoted X = i␴x, Y = i␴y, and Z = i␴z.
The simplest form of a spin echo is for the situation where
the spin is precessing in a field with known direction, say
along the z axis, but with unknown magnitude. Then nx = ny
= 0 and nz = 1, while ␹ is arbitrary. The usual echo sequence
then consists in waiting for the time ␶, applying an X rotation, waiting time ␶ and applying a final X rotation. The
success of the procedure is expressed by the fact that for this
sequence the total evolution operator Utot = XUXU = −I is the
identity 共up to a global sign, which is unimportant兲, independently of U. This means that the final density matrix ␳共t兲
−1
= Utot
␳共0兲Utot = ␳共0兲 is equal to the initial. Readers familiar
with spin echoes will know that an echo is seen after an
evolution UXU. The second X pulse is needed to obtain the
identity 共or its negative兲 for the overall evolution operator.
By symmetry, the same is true if X is replaced by a ␲ rotation around any axis in the xy-plane 共perpendicular to the
precession axis兲. In particular, we have YUYU = −I.
We want to extend this to the case where n is not known.
That is, we want to find a sequence of control rotations
A , B , C , . . . , F such that
FU ¯ CUBUAU = I
共2兲
for any U. We make the following assumptions: 共i兲 The control pulses are effectively instantaneous, meaning they can be
performed in a time much less than the precession period. 共ii兲
The magnetic field is unknown but constant, so that the precession operator U does not change in time. Note that we
have written Eq. 共2兲 as if the time ␶ between the pulses is
fixed. Different time intervals between the pulses are
achieved by choosing some of the A , B , C , . . . , F to be the
identity. This is sufficient if all intervals are integer multiples
of a smallest unit. Intervals with irrational ratios will require
a more general form than Eq. 共2兲. In this paper we only use
equal intervals. This problem was also discussed recently in
Ref. 9 but only in the limit where the time ␶ between the
pulses is small compared to the period of the rotation of the
system under the rotation U 共i.e., when the angle ␹ Ⰶ 1兲. The
echo sequences that we discuss in this work are not limited
by this condition.
III. ITERATED MAPPINGS
As was explained above, if we know the direction n of the
magnetic field we can create an echo by applying two ␲
pulses around any axis perpendicular to n. The sequence
XUXU can be used if nx = 0, and similarly we may use YUYU
if we know that ny = 0. Consider the longer sequence Utot
= XUYUXUYU for which, using the general U of Eq. 共1兲, we
get
XUYUXUYU = 共8n2x n2y sin4 ␹ − 1兲I
− 8inxn2y sin3 ␹ cos ␹␴x − 8inxn2y sin4 ␹␴y
− 4inxny sin2 ␹共1 − 2n2y sin2 ␹兲␴z .
共3兲
We see that XUYUXUYU = −I for either nx = 0 or ny = 0. This
may seem like a small gain, but this sequence is the key to
the full solution. Let us construct the mapping
U → T共U兲 = XUYUXUYU.
共4兲
The idea is that T共U兲, being composed of the arbitrary U and
the fixed control rotations, will be “less arbitrary” than the
original U. Iterating this mapping we then construct the set
U共1兲 = T共U兲, U共2兲 = T共U共1兲兲 , . . . of pulse sequences that will be
better and better approximations to the identity. Since T共U共n兲兲
contains U共n兲 four times the time needed for the sequence
U共n兲 is 4n␶. That is, the length of the sequence grows exponentially in n. This is of course unfortunate as the sequences
quickly will become impractically long. However, we will
see below that for a large portion of the space of parameters
determining U, a few iterations n are sufficient to reach a
good approximation to the identity. The domain of “bad”
parameters shrinks exponentially with the increase of n.
To illustrate this we represent a rotation by the polar angle
␪ and azimuth angle ␾ of the rotation axis 共so that nx
= sin ␪ cos ␾, nx = sin ␪ sin ␾ and nz = cos ␪兲 and the rotation
angle ␹. Figure 1 shows the evolution of the parameters
共␪ , ␾ , ␹兲 upon successive transformations given by Eq. 共4兲.
The set of initial rotations fills this space uniformly 关see Fig.
1共a兲兴. Figures 1共b兲–1共f兲 show the successive iterations U共1兲–
U共5兲 of these points, illustrating the convergence of the
mapping.
From Eq. 共3兲 we can write down explicit formulas for the
parameters 共n⬘ , ␹⬘兲 of U⬘ = T共U兲 in terms of 共n , ␹兲,
064301-2
sin ␹⬘ = 4nxny sin2 ␹冑1 − 4n2x n2y sin4 ␹ ,
PHYSICAL REVIEW B 76, 064301 共2007兲
SPIN ECHO WITHOUT AN EXTERNAL PERMANENT…
(a) U
(d) U(3)
(b) U(1)
(e) U(4)
(c) U(2)
(f) U(5)
FIG. 1. 共a兲 A set of rotations U filling uniformly the space 共␪ , ␾ , ␹兲 of parameters. Each point corresponds to a certain rotation U. 共b兲–共f兲
successive iterations of the mapping T共U兲 showing the convergence to the fixed point U = −I.
nx⬘ = −
n⬘y = −
2ny sin ␹ cos ␹
冑1 − 4n2x n2y sin4 ␹
2nynz sin2 ␹
nz⬘ = −
,
冑1 − 4n2x n2y sin4 ␹ ,
1 − 2n2y sin2 ␹
冑1 − 4n2x n2y sin4 ␹ .
We see that sin ␹ = nx = ny = 0, nz = 1 is a fixed point, and this
represents the identity operator. The stability of the fixed
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PHYSICAL REVIEW B 76, 064301 共2007兲
JOAKIM BERGLI AND LEONID GLAZMAN
(a) χ = 0.5
(b) χ = 1.0
(c) χ = 1.25
(d) χ = 1.5
0
1
2
3
4
5
6
7
8
9
10
11
12
n
FIG. 2. The levels of gray indicate the number n of iterations that is needed to reach a rotation U共n兲 with the rotation angle ␹共n兲
⬍ 10−3. The darker shade means faster convergence according to the scale at the bottom.
point can be analyzed by expanding close to the fixed point
in the independent small quantities ⑀s = sin ␹ / 冑2, ⑀x = nx / 冑2,
and ⑀y = ny / 2. We get
⑀s⬘ , ⑀x⬘ , ⑀⬘y are even smaller and the fixed point is locally
stable.
⑀s⬘ = ⑀x⑀y⑀s2 ,
IV. CONVERGENCE PROPERTIES
⑀x⬘ = − ⑀y⑀s ,
⑀⬘y = − ⑀y⑀s2 .
where the primed quantities refer to the transformed rotation
U⬘. It is clear that if ⑀s , ⑀x , ⑀y are small quantities, then
We studied numerically the convergence of the mapping
T共U兲 for points that are not close to the fixed point. Looking
at Fig. 1 we see that although most points converge to the
vicinity of the fixed point in a few iterations, there are some
points that do not converge fast. To get a better understanding we do as follows. Let us choose some initial rotation
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PHYSICAL REVIEW B 76, 064301 共2007兲
SPIN ECHO WITHOUT AN EXTERNAL PERMANENT…
ln pn
0
-5
-10
-15
-20
5
10
15
20
n
FIG. 3. The fraction pn of the initial random set of rotations,
which did not reach the condition ␹ ⬍ 10−3 until the nth iteration.
angle ␹ and for each point in the 共␪ , ␾兲 plane construct the
sequence U共1兲, U共2兲 , . . . stopping when U共n兲 is within a specified distance from the identity 共we used the stopping criterion ␹共n兲 ⬍ 10−3兲.
Figure 2 shows the 共␪ , ␾兲 plane for various ␹, the shades
of gray representing the number n of iterations needed for
convergence. We see that for small ␹ convergence is fast and
the pattern is simple, but for larger initial ␹ the pattern of
convergence time is quite complex, and that there exist
“hard” points, i.e., initial rotations 共n , ␹兲 that need a large
number n of iterations to converge.
To estimate the fraction of initial rotations that need a
certain number of iterations to converge to the fixed point we
do the following. We start with an ensemble of random initial
rotations U characterized by a unit vector n distributed uniformly on a sphere and by an angle ␹ distributed uniformly
from in the interval 关0 , ␲兴. We proceed with the iterations,
Eq. 共4兲, until ␹共n兲 ⬍ 10−3 is reached. The logarithm of the
fraction pn of initial rotations U that needs n iterations to
converge is shown in Fig. 3 as a function of n.
We find that about 74% of the initial rotations will converge after applying the third iterate U共3兲 and about 91% do
so after the iterate U共4兲. Observe that except for the first
points, all points fall on a straight line, which means that the
fraction pn of “difficult” initial rotations decays exponentially with the number of iterations,
p n ⬀ e −␣n,
␣ ⬇ 1.1.
共5兲
If k is the number of repetitions of U in T共U兲 关for the mapping 共4兲 we have k = 4兴, the total time of the sequence U共n兲 is
t = ␶kn, where ␶ is the time between the pulses. The relation
共5兲 can then also be written as the fraction p共t兲 of the initial
rotations which did not converge till time t,
p共t兲 ⬀
冉冊
t
␶
−␤
,
␤=
␣
.
ln k
共6兲
The relations Eqs. 共5兲 and 共6兲 are established by running a
simulation with the use of a specific convergence criterion
FIG. 4. The set M 9 of initial rotations that do not converge in
nine iterations, according to the criterion ␹ ⬍ 10−3.
␹共n兲 ⬍ 10−3. Changing the criterion affects the proportionality
coefficients in these relations, but does not change the values
of exponents ␣ and ␤.
Let M n denote the set of initial rotations that do not converge in n iterations. 关As an example, the domain M 9 of
“difficult” initial rotations 共according to the criterion ␹
⬍ 10−3兲 is presented in Fig. 4兴. The sequence of sets M n is
such that each set is contained in the previous one,
M n 傺 M n−1, and the fraction of points pn in Fig. 3 is proportional to the difference in the volumes of M n+1 and M n. The
⬁
M n of “infinitely hard” points appears to be a
set M = 艚n=0
fractal with fractal dimension D ⬇ 1.5 according to the boxcounting algorithm we used.
V. DISCUSSION
We considered the possibility of constructing an echo
pulse sequence that does not require applying a high permanent field to the system. The advantage of such a method is
that between the pulses the dynamics of the system is not
influenced by external perturbations. In the context of ESR
of a quantum dot, this method may help distinguish between
different effects of the hyperfine interaction: this interaction
creates some random effective magnetic field acting on the
electron spin, but may also lead to electron spin-relaxation.
The absence of the external permanent field in this problem
is crucial, as its application definitely suppresses the spinrelaxation part of the hyperfine interaction effects.7
We have provided a solution to the general echo problem
in terms of a set of longer and longer pulse sequences that
give successively better approximations to the identity operator. We have tested several longer sequences, e.g., T共U兲
= ZUXUYUXUZUXUYUXU. In all cases we found that the
exponent ␤ in Eq. 共6兲 is independent of the particular mapping chosen. Whether this represents some inherent property
of the problem or only is the case for the limited class of
mappings we have studied is not known. For example, we
have not studied sequences with control pulses other than ␲
rotations about the coordinate axes. We have also not ruled
out the possibility of a solution that will yield an ideal echo
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PHYSICAL REVIEW B 76, 064301 共2007兲
JOAKIM BERGLI AND LEONID GLAZMAN
with the help of a finite number of control pulses.
The idea of using iterated mappings to generate pulse sequences has been used in NMR applications10 but as far as
we know this particular problem or the mapping we study
was never discussed. The mapping we have used was also
proposed in Ref. 11 in the context of dynamical decoupling
of a qubit.
*Electronic address: jbergli@fys.uio.no
†Electronic
address: glazman@umn.edu
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ACKNOWLEDGMENTS
This work was supported by the Norwegian Research
Council via a StorForsk program and by DOE Grant No.
DE-FG02-06ER46310, at the University of Minnesota. The
hospitality of MPIPKS-Dresden 共L.I.G.兲 is gratefully acknowledged.
Loss, J. Phys.: Condens. Matter 15, R1809 共2003兲; The nuclear
spin diffusion apparently plays a smaller role because of the
hyperfine effective field exerted by the electron on the nuclei in
the volume of the quantum dot; see C. Deng and X. Hu, Phys.
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B. N. Harmon, Phys. Rev. Lett. 97, 037204 共2006兲; W. A. Coish,
E. A. Yuzbashyan, B. L. Altshuler, and Daniel Loss, J. Appl.
Phys. 101, 081715 共2007兲.
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