Fast Incomplete Decoherence of Nuclear Spins in Quantum Hall Ferromagnet

Abstract

A scenario of quantum computing process based on the manipulation of a large number of nuclear spins in Quantum Hall (QH) ferromagnet is presented. It is found that vacuum quantum fluctuations in the QH ferromagnetic ground state at filling factor $\nu =1$, associated with the virtual excitations of spin waves, lead to fast incomplete decoherence of the nuclear spins. A fundamental upper bound on the length of the computer memory is set by this fluctuation effect.

Full text

Page 1

arXiv:cond-mat/0011361v2 [cond-mat.mes-hall] 22 Dec 2000
FAST INCOMPLETE DECOHERENCE OF NUCLEAR SPINS
IN QUANTUM HALL FERROMAGNET
T.Maniv1,2, Yu. A. Bychkov1,3, I.D. Vagner1, and P. Wyder1
1Grenoble High Magnetic Field Laboratory,Max-Planck-
Institute fur Festkorperforschung and CNRS, Grenoble
France.
2Chemistry Department, Technion-Israel Institute of
Technology, Haifa 32000, Israel
3L.D.Landau Institute for Theoretical Physics, Kosygina 2, Moscow,
Russia
(February 1, 2008)
Abstract
A scenario of quantum computing process based on the manipulation of a
large number of nuclear spins in Quantum Hall (QH) ferromagnet is presented.
It is found that vacuum quantum fluctuations in the QH ferromagnetic ground
state at filling factor ν = 1 , associated with the virtual excitations of spin
waves, lead to fast incomplete decoherence of the nuclear spins. A funda-
mental upper bound on the length of the computer memory is set by this
fluctuation effect.
PACS numbers:
A growing number of models for quantum information processing (or Quantum
Computing-QC) has been recently proposed1, some of which were successfully tested ex-
perimentally in devices consisting of a few qbits.The scaling up of these toy devices to the
desired large number of quantum gates seems at present a formidable challenge. Of spe-
cial interest are the models based on the manipulation of nuclear spins in semiconducting
1

Page 2

heterostructures2,3. A scenario for the realization of QC over a large number of qbits in
a model system similar to that proposed in Ref.[2] may be achieved when nuclear spins in
heterojunctions are manipulated via the hyperfine interaction with the electron spins under
the conditions of the odd integer Quantum Hall (QH) effect.
The main idea behind this scenario is motivated by the experimental observation in
optically pumped NMR measurements on GaAS multiple quantum well structure at low
temperatures of a dramatic enhancement of the very small nuclear spin lattice relaxation
rate which charaterizes the QH ferromagnetic state at Landau level filling factor ν = 1 ( i.e.
corresponding to T1> 250 s ), and of a sharp decrease of the Knight shift , as the filling
factor is shifted slightly away from ν = 1 [4]. The prevailing interpretation of these closely
related effects, associates them with the creation of skyrmions (or antiskyrmions) in the
electron spin distribution as the 2D electron system moves away from the QH ferromagnetic
state at ν = 1 [5].
It should be stressed, however, that the nuclear spin dephasing time T2in quantum well
structures based on GaAS/AlGaAs, is expected to be of the order of milliseconds or shorter,
namely much smaller than the shortest value of T1found in this experiments. This drawback
is due to the fact that all elemental components of such structures (i.e. Ga69, Ga71, As75
, all with I = 3/2 , and Al27with I = 5/2 ) have non-zero nuclear spins, which as a result
experience large direct dipolar interactions. An alternative quantum well structure based
on semiconducting host material consisting predominantly of zero nuclear spin isotopes and
small amount of atoms with nonzero nuclear spins (e.g. like Si/Si1−xGexheterojunctions6),
may be fabricated in the future to create a system of virtually noninteracting nuclear spins
in a QH ferromagnet.
In such a system the nuclear dephasing time T2is governed only by the hyperfine inter-
actions with the electron spins. Random impurities, for example, can influence this nuclear
spin dephasing only indirectly through the scattering of electrons by the impurity potential,
leading ,e.g. in typical GaAs quantum well structure to nuclear spin dephasing times of the
order of seconds ( see below )7.
2

Page 3

One therefore expect that after saturating the nuclear spins and tuning the filling factor
at ν = 1 a nearly pure nuclear spin state can be frozen for a relatively long time due to
the very long nuclear spin relaxation times T1 and T2. The nuclear spin system can be
now manipulated by varying the filling factor away from the initial value and then back to
ν = 1 to freeze in a new configuration. During this variation the 2D electron system is in
a condensed state of a large number of spin waves (or spin excitons)8, which are strongly
correlated over a large spatial region. It is thus expected that the proposed manipulation of
the nuclear spin system can be performed in a phase coherent fashion over a spatial region
with size of the order of the skyrmion radius.
In attempting to evaluate the feasibility of this hypothetical scenario several problems of
various degrees of complexity arise. Perhaps, the most fundamental one concerns the mech-
anism in which a single nuclear spin (or qbit) looses phase coherence by the ’manipulating
agent’ itself , i.e. the 2D electron system in the heterostructure. In particular, it is not at
all obvious that even under the ideal conditions of the QH ferromagnetic state at ν = 1 ,
where the nuclear spin relaxation times, T1and T2, are extremely long, an ensemble of large
number of nuclear spins can preserve phase coherence for a sufficiently long time.
Of special interest here is the effect of vacuum quantum fluctuations in the QH ferro-
magnetic state on the decoherence of nuclear spins. As will be shown below, the virtual
excitations of spin waves (or spin excitons10,11), which have a large energy gap (on the scale
of the nuclear Zeeman energy) above the ferromagnetic ground state energy, lead to fast
incomplete decoherence in the nuclear spin system.
To study this effect we exploit an ideal model system consisting of independent nuclear
spins interacting with a 2D electron gas through the Fermi contact hyperfine interaction.
An external stationary magnetic field is applied perpendicular to the 2D layer with strength
corresponding to filling factor ν = 1. The temperature is assumed to be smaller than any
electronic energy scale in the problem , and the influence of electron scattering by impurities
is neglected.
The Hamiltonian of this model system is
? H =? H0+? Hen ,where
3

Page 4

?H0= −γn
?
j
?Ij· B0− γe
?
d2r?S(r) · B0+?Hee
(1)
? Hen= A
?
j
?S(rj) ·?Ij
(2)
Here?Ij is the nuclear spin operator located at rj ,?S(r) is the electronic spin density
operator , B0is the external magnetic field which is assumed to be oriented perpendicular
to the 2D EG ( B0= B0z ) ,? Heeis the electron electron interaction, γnand γethe nuclear
and electronic gyromagnetic ratios respectively,
A = C |u(0)|2/ll2
B
with C =8π
3γnγe, and u(0) the electron wavefunction at a nucleus. The difference between
the Fermi contact hyperfine interaction parameter A in a quantum well at high magnetic
field and the corresponding zero field bulk coupling constant is reflected in the appearance
of the two length parameters; l -the width of the quantum well , and lB=
?c?/eB0- the
magnetic length. The zero field bulk value,
ABulk= C |u(0)|2/Ω
where Ω is the volume of a unit cell , is usually much larger than A given above as long as
ll2
B≫ Ω .
The manipulation of the nuclear spins is carried out through spin flip-flop processes,
associated with the ’transverse’ part of the interaction Hamiltonian? Hen( Eq.(2) ) , i.e.
A
?
j
??Ij,+?S−(rj) +?Ij,−?S+(rj)
are the transverse components of the nuclear
?
where?Ij,+ =?Ij,x+ i?Ij,y,?Ij,− =?Ij,x− i?Ij,y
components of the electron spin density operators. Here?ψσ(r),?ψ†
operators with spin projections σ =↑,↓ .
spin operators, and?S+(r) =?ψ†
↓(r)?ψ↑(r) ,?S−(r) =?ψ†
↑(r)?ψ↓(r) are the corresponding
σ(r) are the electron field
4

Page 5

The ’longitudinal’ part of? Hen, A?
coherence in the nuclear spin system12.
j?Ij,z?Sz(rj) , which commutes with the Hamiltonian
? H0, and so leaves the nuclear spin projections along B0unchanged, can still erodes quantum
To simplify the analysis we assume that the nuclei under study have spin 1/2 . In this
case the transverse components?Ij,+,?Ij,−, are up to a proportionality constant, just the off
diagonal elements (or coherences) of the density matrix of a single nuclear spin (qbit)14,?. The
decay of these elements with time, which determines the rate of decoherence of a single qbit,
can be thus found from the equations of motion for the operators?Ij,±(t) in the Heisenberg
representation?Ij,±(t) = ei? Ht/??Ij,±e−i? Ht/?.
Let us, for the sake of simplicity, consider a single nuclear spin and evaluate its rate of
decoherence due to the coupling with a ’bath’ of spin excitons. Dropping the site index j ,
the corresponding equations for the coherences I+, I− , can be written in the form :
∂
∂t
?I+(t) = −iα?S+(t)?Iz(t) ,
∂
∂t
?I−(t) = iα?S−(t)?Iz(t) (3)
with the supplementary equation for?Iz(t)
∂
∂t
?Iz(t) =i
2α
??S+(t)?I−(t) −?S−(t)?I+(t)
?
(4)
Here α ≡ A/? , and the symbols?I±(t) ,?S±(t) stand for the corresponding spin operators
in the rotating reference of frame with angular velocity ω = γnB0− ASz[13] , i.e. :
?I±(t) ≡ e±iωt?I±(t) ;?S±(t) ≡ e±iωt?S±(t)
Assuming that initially, at time t = 0 , the electronic system is in its ground (QH
ferromagnetic) state |0? , and neglecting the effect of the nuclear spins on the electronic
(bath) states, the average of Eq.(3) over the ’bath’ states reduces to the expectation value
in the ground electronic state |0?. Thus, by integrating Eq.(4) over t , substituting into
Eq.(3), and then averaging over the ’bath’ states, one finds to lowest order in the hyperfine
interaction parameter α :
5

Page 6

∂
∂t
?I+(t) = −1
2α2
?t
0
dτeiωnτχ+−(τ)?I+(t − τ) (5)
where χ+−(τ) ≡ ?0|?S+(t)?S−(t − τ)|0? .
Note that since χ+−(τ) varies on the characteristic electronic time scale 1/γeB0, which is
much shorter than the nuclear time scale ω−1, it is allowed to use the expansion?I+(t − τ) =
?t
?I+(t) − τd
dt?I+(t) + ... under the integral in Eq.(5) to get:
∂
∂t
?I+(t) = −1
2α2
0
dτeiωτχ+−(τ)?I+(t) + O(A4) (6)
The ’bath’ excited states are spin excitons with the well known dispersion relation Eex(k)
[10], which in the long wavelength limit reduces to Eex(k) ≈ εsp+1
4εcl2
Bk2, with εc=
e2
κlB
?π
2
the characteristic Coulomb energy. Here κ is the dielectric constant in the 2D electron gas
region. The energy gap is the ’bare’ Zeeman energy εsp= ?γeB0.
A simple calculation yields for the correlation function χ+−(τ) =?
where?k = lBk . Integrating Eq.(6) over t , and solving for?I+(t) , the time dependence of
the coherence I+is given by
ke−1
2?k2e−iEex(k)τ/?,
I+(t) = I+(0)J (t)
where J (t) = eiΩ(t)−Γ(t), and
Γ(t) =
?
C |u(0)|2
4πl2
Bl
?2?∞
0
?kd?ke−1
2?k21 − cos?1
?Eex(k) − ω?t
[Eex(k) − ?ω]2
(7)
This result is similar to the expression found by Palma et al.12in an artificial model of
pure decoherence, i.e. when energy transfer between the qbit and its environment is not
allowed. The remarkable feature of this expression is due to the presence of the energy gap
εspin the spin exciton spectrum, which is typically much larger than the nuclear Zeeman
energy ?ω . This guarantees that the denominator in the integrand in Eq.(7) is always larger
than or equal to ε2
sp, and that for times t ≫ ?/εsp,
Γ(t) −→?A2
?∞
0
?kd?ke−1
2?k2
?2
??Eex(k)
6

Page 7

with the dimensionless exciton energy?Eex(k) = Eex(k)/εsp≥ 1, and hyperfine coupling
4πl2
constant? A =
Thus , we find that during a short time scale , of the order of ?/εsp, the coherence ,
C|u(0)|2
Blεsp.
I+(t) , of a single nuclear spin diminishes and then saturates for a very long time ( i.e. of the
order of the relaxation time T2, see below) at I+(0)e−κ? A2, where κ =?∞
For GaAs/AlxGa1−xAs heterostructure the coupling constant? A is typically of the order of
0
?kd?ke−1
[?Eex(k)]
2?k2
2 ∼
2εsp
εc
.
10−4
[15] , implying an extremely small deviation , i.e. ∼ κ?A2∼ 10−9, from a pure state
of a single nuclear spin.
It is interesting to compare this effect to the decoherence caused to a nuclear spin as a
result of the scattering of spin excitons by random impurities. This mechanism leads to a
complete decoherence within a time scale [7]
T2∼
1
? A2U2
?/εsp
where U2 ≡
1
Bε2
2πl2
c
?d2r ?Uimp(r)Uimp(r′)? is a dimensionless correlator of the impurity
potential Uimp(r) [16]. In GaAs/AlxGa1−xAs heterostructures ?/εsp∼ 10−12sec and U2is
typically ∼ 0.001[7], so that T2∼ 0.1sec.
Thus, due to the extreme weakness of the hyperfine contact interaction with the 2D
electron gas under high magnetic fields the decoherence of a single qbit arising from both
impurity scattering and quantum fluctuations in such a system is extremely small. A coher-
ent superposition of a great number of qbits can therefore survive in the computer memory
during a very long time period t ≪ T2. To find an upper bound for the length of such a
memory let us consider N independent nuclear spins located at various positions rjin the
quantum well. A number, n , stored in the memory, corresponds to the direct product of
N pure nuclear spin states |n? = |n1? ⊗|n2? ⊗ ... ⊗ |nN? , where |nj? =?
δnj,σis the Kronecker delta , and |j,σ? is a nuclear state with spin projection σ for a nucleus
σ=±1δnj,σ|j,σ? ,
located at rj.
To carry out a quantum computing process, however, a coherent superposition of such
products , i.e. |ψ? =?N
7
n=1αn|n? (see e.g.17), should be prepared at time t = 0.This

Page 8

superposition may be represented more transparently for our purposes by the direct product
of N mixed spin up and spin down states,
|ψ (t = 0)? =
N
?
j=1
⊗(uj|j,↑? + vj|j,↓?)(8)
with the normalization |uj|2+ |vj|2= 1 .
Let us assume that at time t = −t0 < 0 the filling factor was tuned to a fixed value
ν = ν0?= 1 and then kept constant until t = 0 . If t0≫ T2(ν0) then at t = 0 the nuclear spin
system is in the ground state corresponding to the 2D electron system at ν = ν0. Suppose
that at time t = 0 the filling factor is quickly switched ( i.e. on a time scale much shorter
than T2(ν0) ) back to ν = 1 so that the nuclear spin system is suddenly trapped in its
instantaneous configuration corresponding to ν = ν0?= 1 . Thus the nuclear spins will find
themselves for a long time t ( ≫ T2(ν0) ) almost frozen in the ground state corresponding
to the 2D electron system at ν = ν0, since T2(ν = 1) ≫ T2(ν0).
The corresponding state of the nuclear spin system can be found by considering ujand vj
in Eq.(8) as variational parameters , and then minimizing the energy functional E =?ψ|H|ψ?
,
H = −γn
N
?
j=1
?Ij· B0+ A
N
?
j=1
S(rj) ·?Ij
with respect to uj, vj. Note that the effective nuclear spin Hamiltonian H is obtained after
averaging over the electronic (’bath’) states, so that S(rj) =
??S(rj)
?
≈ ?0|?S(rj)|0? is the
expectation value of the electronic spin density at the nuclear position rj. As noted above ,
at ν0?= 1 , this density has nonzero transverse components, associated with the skyrmionic
spin texture, smoothly varying in space.
A simple calculation shows that E =?
ε(uj,vj) =1
jε(uj,vj) ,
2ωj
?|vj|2− |uj|2?+1
2
?Avju∗
jS−(rj) + c.c?
where ωj= γnB0− ASz(rj) is the local nuclear Zeeman energy. The extremum conditions
( subject to the normalization |uj|2+ |vj|2= 1 )
∂ε
∂u∗
j− ǫjuj= 0 ,
∂ε
∂v∗
j− ǫjvj= 0 are readily
solved to yield:
8

Page 9

|uj|2=1
2
?
1 ±
ωj
?2ǫj
?
, |vj|2=1
2
?
1 ∓
ωj
?2ǫj
?
(9)
where
ǫj=1
2
?
ω2
j+ A2S−(rj)S+(rj)
In this state the nuclear spin polarization ?ψ|?Ij|ψ? follows the underlying electronic spin
texture; the transverse component takes the form
?ψ|?Ij,+|ψ? = u∗
jvj= AS+(rj)/2ǫj
whereas the longitudinal component is
?ψ|?Ij,z|ψ? =1
2
?|uj|2− |vj|2?= ±ωj/2ǫj
The topological rigidity of the skyrmionic spin texture thus ensures the rigidity of the
coherences u∗
jvjover a large spatial region.
The key parameter here is the mixing parameter ηj≡ A2|S+(rj)|2/ω2
j, which becomes
significant when the transverse component S+(rj) does not vanish over a large spatial region
, as is the case for skyrmion spin texture. For very small mixing parameter ηjone finds a
pure nuclear ferromagnetic state, namely |uj|2= 1 , |vj|2= 0 , corresponding to the ground
state , or |uj|2= 0 , |vj|2= 1 , corresponding to the fully saturated nuclear spin system. In
the opposite limit of very strong mixing ( ηj≫ 1 ) |uj|2= |vj|2=1
2, which is the desired
state for quantum computing17, all the numbers n are stored in the memory with equal
probability. It should be noted that, since |S+| ≤ 1 , large values of ηjcan be obtained only
when the nuclear Zeeman energy ωjis much smaller than the hyperfine coupling constant
A .
Now, during the long time following t = 0 , when the electronic system is set at filling
factor ν = 1 so that its ground state is a uniform quantum ferromagnet and the elementary
spin excitations are the spin waves discussed above , the nuclear state |ψ (t)? evolving from
|ψ(0)? after time t can be readily calculated in terms of the operators?Ij,±(t),?Ij,z(t) . Using
9

Page 10

the solutions?Ij,+(t) = J (t)?Ij,+(0),?Ij,z(t) = I (t)?Ij,z(0) , derived above ( Eq.(7) ), it is easy
to show that the probability that after time t the memory remains in the coherent state
ψ (0):
Pψ= |?ψ (0) | ψ (t)?|2
?
(10)
=
1
2N
N
j=1
{[1 + I (t)] + 4[I (t) − ReJ (t)]?|uj|4− |uj|2?}
Let us further assume that the mixing parameter ηjare large so that in the initial state
ψ (0) , |uj|2= 1/2 as required. Under these circumstances we easily find that:
?1 + ReJ (t)
Pψ(t) =
2
?N
≈ exp
?
−1
2N [1 − ReJ (t)]
?
≈ e−1
2NΓ(t)
It is evident that due to the even distribution of nuclear spins in the initial state ψ (0)
the survival probability Pψ(t) depends only on the decoherence factor J (t). The decay of
Pψ(t) therefore follows e−1
2NΓ(t), saturating at e−1
2Nκ? A2for t ≫ ?/εsp. Note that, despite
the much larger drop in the level of coherence in the many qbit system , the time scale over
which the coherence diminishes is the same as for a single qbit.
An upper bound on the length of possible memories in future quantum computers based
on the proposed model can be now estimated by the requirement Pψ(t ≫ ?/εsp) ∼ 1/e ,
which yields
Nmax∼ 2/? A2
The restriction imposed by this condition is inherent to the manipulation mechanism of
qbits via the hyperfine interaction with the electron spins, and so can not be removed or
even relaxed by any technical improvement.
In conclusion we have found that a system of many nuclear spins, coupled to the electronic
spins in the 2D electron gas through the Fermi contact hyperfine interaction, partially looses
its phase coherence during the short (electronic) time ?/εsp, even under the ideal conditions
of the QHE, where both T1, and T2are infinitely long. The effect arises as a result of vacuum
10

Page 11

quantum fluctuations associated with virtual excitations of spin waves (or spin excitons )
by the nuclear spins. The incompleteness of the resulting decoherence is due to the large
energy gap (on the scale of the nuclear Zeeman energy) of these excitations whereas the
extreme weakness of the hyperfine interaction with the 2D electron gas under high magnetic
fields guarantees that the loss of coherence of a single nuclear spin is extremely small. The
memory of a quantum computer to be constructed in such a system is therefore limited in
principle to lengths of the order of Nmax, which is found to be about 109for GaAS multiple
quantum well structure.
This research was supported by the German-Israeli Foundation for Scientific Research
and Development, Grant No. G-0456-220.07/95, and by the fund for the promotion of
research at the Technion. Yu.A.B. wishes to acknowledge support from grants RFFI-00-02-
17292, IR-97-0076, and INTAS-99-01146. I.V. acknowledges the support by the Caesarea
Edmond Benjamin de Rotschild Foundation.
11

Page 12

REFERENCES
1For a recent review see: A. Steane, Rep. Prog. Phys., 61, 117 (1998).
2V. Privman, I.D. Vagner and G. Kventsel, Phys. Let. A 236, 141 (1998).
3B. E. Kane, Nature, 393, 133 (1998)
4R. Tycko, S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, and K. W. West, Science 268 , 1460
(1995).
5For a review see S. M. Girvin, cond-mat/9907002 (1999)
6I. S. Shlimak et al., Sov. Techn. Phys. Lett., 9, 377 (1983)
7D. Mozyrsky, V. Privman and I. D. Vagner, cond-mat/0002350
8Yu. A. Bychkov, T. Maniv and I.D. Vagner, Phys. Rev. B53, 10148 (1996).
9Yu. A. Bychkov, A. V. Kolesnikov, T. Maniv and I.D. Vagner, J. Phys.: Cond. Matt.10,
2029 (1998).
10Yu. A. Bychkov, S. V. Iordanskii and G. M. Eliashberg, JETP Lett. 33, 143 (1981)
11C. Kallin and B. I. Halperin, Phys. Rev. B 31, 3635 (1984)
12G. M. Palma, K. A. Suominen and A. K. Ekert, Proc. Royal Soc. London, A 452, 567
(1996)
13C. P. Slichter, Principles of Magnetic Resonance, 3rd Ed. (Springer-Verlag, Berlin, 1990)
14C. Cohen-Tannoudji, B. Diu and F. Laloe , Quantum Mechanics ( Hermann , Paris 1977)
15Yu. A. Bychkov, T. Maniv and I.D. Vagner, Solid State Commun. 94, 61 (1995)
16S. V. Iordanskii, S. V. Meshkov, and I. D. Vagner, Phys. Rev. B44 , 6554 (1991).
17W. G. Unruh, Phys. Rev. B 51 , 992 (1995)
12