Spin transport

Lecture given at the International School on Quantum Information

Cord Muller, MPIPKS Dresden, September 11, 2005

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Outline

Classical information processing uses charge encoding, represented for instanceby states |0〉 and |1〉 of a supplementary electron present or absent in a quantumdot. Since they interact via long-range Coulomb forces, charge states sufferstrongly from decoherence. A natural degree of freedom to encode informationis the spin. The spin 1

2 states, typically noted | ± 1〉 or | ↑, ↓〉 are the naturalrealization of a qubit. Information needs physical carriers. Candidates here areelectrons (massive particle of spin s = 1

2 ) and photons (massless, spin s = 1).Spint(r)onics: Spin-based information transport and processing with elec-

trons and photons.The Leitmotiv of this lecture: Spin is a geometrical quantity. Use irreducible

representations of the rotation group to take advantage of the symmetries of thesystem.

Recommended reading for a rapid introduction: Chap. 1 in the book byChaichian and Hagedorn [1], chaps. 4 and 8 in the book by Blum [2], the recentreview on “Spintronics” by Zutic, Fabian, and Das Sarma [3].

1. Irreducible spin superoperators

1.1. Spin

General references on spin: good angular momentum theory books, for example[1, 4]

1.1.1. Historical background

“Spin” is internal angular momentum. The name was coined by Uhlenbeckand Goudsmith [5] following W. Pauli [6] who postulated the existence of a

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fourth quantum number to explain fine-structure features of atomic spectra.The clearest experimental manifestation of quantized spin is arguably the Stern-Gerlach experiment [7] where silver atoms are deviated by an inhomogeneousmagnetic field into two distinct spots on a detector screen.

P.A.M. Dirac showed that bispinors (vectors of four components, a spinorand an anti-spinor) appear naturally when one looks for a Schrodinger-typewave equation in a relativistic kinetic framework (Poincare group of translations,rotations and Lorentz transformation of the four-dimensional Minkowski space-time). E. Wigner [8] showed that spin is one of the fundamental quantumnumbers that permits to identify an elementary particle in the first place: itcharacterises the particle’s properties under rotations in its proper rest frame.

1.1.2. Rotation group

Physical objects are described by coordinates x = (x1, x2, x3) ∈ R3 with respectto a reference frame in configuration space. Rotations (called “active” whenthe object is turned and “passive” when the reference frame is turned) arerepresented by 3×3 matrices: x′ = Rx. Proper rotations conserve the euclideanscalar product x · y =

∑i xiyi and the orientation of the frame. The rotation

matrices are therefore members of SO(3), the set of orthogonal matrices RRt =RtR = 13 of unit determinant det R = +1. With the usual matrix multiplicationas an internal composition law, theses matrices form a group, satisfying thegroup axioms:

1. Internal composition: ∀R1,2 ∈ S0(3) : R21 = R2R1 ∈ S0(3);

2. Existence of the identity: ∃E : RE = ER = R ∀R ∈ S0(3) with E = 13;

3. Existence of the inverse: ∀R∃R−1 : RR−1 = R−1R = E.

The group is nonabelian because the matrices do not commute: R2R1 6= R1R2.An exception are rotations of the plane around one and the same axis, formingthe group SO(2).

A possible parametrization of a rotation is the polar description (n, θ) =: θwith n the unit vector along the rotation axis and θ ∈ [0, π] the rotation angle.The following two rotations of configuration space are identical:

R(n, θ = π) = R(−n, θ = π). (1)

These opposite points must be identified such that there are closed parametercurves that can not be reduced to a single point. This means that SO(3) is adoubly connected manifold. Instead of studying this “projective group”, onemay also turn to its “universal covering group” SU(2), the group of all unitary2 × 2 matrices over C with unit determinant. SU(2) is simply connected, andthere is a homomorphism (a mapping preserving the group structure) fromevery element U ∈ SU(2) to the rotation R ∈ SO(3): the rotation x′ = R(θ)xis described by

x′ · σ = U(θ)x · σU(θ)†. (2)

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Here, σ = (σ1, σ2, σ3) are the Pauli matrices. The unitary rotation matrixacting from the left and from the right is given by

U(θ) = 12 cosθ

2− in · σ sin

θ

2= exp(−i

θ · σ2

). (3)

Note that this mapping is two-to-one (−U and U yield the same rotation). Notethat it takes a 4π-angle to recover the identity transformation: U(θ = 2π) =−12, but U(θ = 4π) = +12. This is not a mysterious quantum property asis sometimes stated, but reflects the non-connectedness of SO(3)-rotations ofreference frames. Dirac’s construction of a solid body connected by strings to areference frame is supposed to convey an “experimental” idea of this property[4].

1.1.3. Representations

A group G can act in many different disguises that share the same abstractgroup structure, as defined by the multiplication law or group table. Thesedifferent transformations are called representations: they are homomorphismsDi : G → GL(Vi) into (linear) regular transformations D : Vi → Vi of a vectorspace Vi into itself (D standing for the German word “Darstellung”, in anglo-saxon literature, often the symbol Γ is used). “Homomorphism” means thatthe representation has the same group structure as G. Notably, for all elementsg1, g2 ∈ G (with the product g2g1 ∈ G) one has D(g2g1) = D(g2)D(g1).

dim Vi is called the dimension of the representation Di. Finite-dimensionallinear representations (all that we will see in this lecture) are given in terms ofquadratic matrices of size dim Vi. A representation is called reducible if thereis a basis of V = V1 ⊕ V2 such that all transformations D ∈ D are written

D =(

D1 ∗0 D2

). (4)

In other words: the transformations D1 : V1 → V1 are already a representationD1 of its own. If the matrix is block-diagonal (∗ = 0), the representation iscompletely reducible. If it is not reducible, it is called irreducible, meaningthat one has achieved to work in the smallest possible subspace.

The rotation group SU(2) is a manifold that depends on a continuous set ofparameters θ = (θ1, θ2, θ3) with respect to which it is infinitely differentiable.This structure is called a Lie group. The group multiplication law is completelydetermined by the commutation relation of its generators:

[Sj , Sk] := SjSk − SkSj = ihεjklSl. (5)

These generators are said to form a Lie algebra. In the so-called natural rep-resentation of SU(2) by itself [eq. (3)], the generators are S = hσ

2 . Otherrepresentations will feature different generators, but all group representationsshare the same commutation relation! Finite rotations are generated by expo-nentiation: U(θ) = exp(−iθ · S/h). Topologically speaking, SU(2) is compact.

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According to a general theorem, all representions of a compact Lie group arecompletely reducible to finite-dimensional irreducible representations.

Including the orbital part L of angular momentum, one defines the totalangular momentum J = L⊗ 1S + 1L ⊗ S =: L + S. For the remainder of thissection, we will consider only the spin part.

What is quantum about spin?

As pointed out above, reference frames for solid bodies introduces half-integerspin (E. Cartan already in 1913 [9]). Half-integer spin does not require specialrelativity, it arises also in Galilean relativity. But there are genuine quantumfeatures:

1. If the Hamiltonian is invariant under infinitesimal transformations, thegenerators T of rotations U = exp(−iθ · T ) are conserved quantities(Noether theorem):

H ′ = UHU† ⇔ [H,U ] = 0 ⇔ [H,S] = 0 ⇔ S = 0. (6)

In quantum mechanics, these generators are actually observables withdimension of angular momentum (S = hT ).

2. The observables S generate irreducible representations D(s) of dimensionds = 2s+1 with s = 0, 1

2 , . . . and discrete magnetic quantum numbers m =−s,−s + 1, . . . , s. States are noted |sm〉. The so-called Casimir operatorS2 tells us which irreducible representation: S2|sm〉 = h2s(s + 1)|sm〉,whereas the magnetic quantum number gives the projection of the spinonto the quantization axis (usually called the z-axis): Sz|sm〉 = hm|sm〉.

3. States can be classified regarding their transformation properties (cf. themultipole expansion for charge distributions in classical electrodynamics),but here with a more general importance due to the superposition princi-ple. Example: an atomic s-orbital (“superposition of all possible Keplerorbits”) is invariant under all rotations, i.e., transforms under D(0).

1.1.4. Irreducible tensor operators

An irreducible tensor operator by definition is a set of 2K +1 components T(K)q ,

q = −K,−K + 1, . . . ,K, that transform under irreducible representations ofrank K (i.e., whose transformation does not mix different K):

(T (K)q )′ = UT (K)

q U† =K∑

q=−K

D(K)qq′ TK

q′ (7)

Equivalently, one specifies the infinitesimal rotation properties by requiring

[J±, T (K)q ] = h

√K(K + 1)− q(q ± 1)T (K)

q±1 (8)

[J0, T(K)q ] = hqT (K)

q (9)

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Here, the angular momentum raising and lowering operators are J± = Jx± iJy.Examples:(i) Scalar operator: a single quantity T

(0)0 that commutes with all compo-

nents of the total angular momentum J . For instance J2, the Casimir operatorindexing the irreducible representations.

(ii) Vector operator: Its components A = (A1, A1, A3) satisfy

[Jj , Ak] = ihεjklAl (10)

(iii) Tensor operator of rank 2: direct product of 2 vectors Tij = AiBj canbe decomposed into irreducible representations of rank K = 0, 1, 2.

Exercise 1: Is the Hamiltonian H = p2

2m − µS ·B of a free massive particlecoupled via the magnetic moment µ = µS of its spin S to an external magneticfield B an irreducible tensor operator? If so, determine its rank.

1.2. Spin superoperators

1.2.1. Unitary spin dynamics

As a simple mode for spin dynamics, we shall study the Hamiltonian

H = −µ

hS ·B . (11)

It describes the coupling of the magnetic moment µ = µ(S/h) to a magnetic fieldB. For electrons, µ = −gµB in terms of the Bohr magneton µB = |e|h/(2mec)and the gyromagnetic ratio g = 2.003... in vacuum. The density matrix orstatistical operator ρ of a spin S is a positive linear operator of trace unity onthe state Hilbert space Hs = Cds of dimension dimHs ==: ds = 2s + 1.

According to one of the fundamental axioms of quantum theory, any closedsystem shows unitary dynamics governed by the Liouville-von Neumann equa-tion ih∂tρ = [H, ρ].

1.2.2. Superoperators

Rewriting the LvN-equation as i∂tρ = Lρ defines the Liouvillian

L =1h

[H, ·] (12)

The linear operators on Hs are themselves elements of a linear vector space (wecan add operators and multiply them by complex numbers). This vector spaceis called Liouville space L(Hs) and is spanned, for example, by the basis ofdyadics

|m〉〈n| =: |mn), n,m = 1, . . . , ds (13)

induced by the basis vectors |n〉 of Hs. As a operator between operators, theLiouvillian L : L(Hs) → L(Hs) is called a superoperator [10, 11, 12, 13].

The matrix elements of the Liouvilian (12) in the dyadic basis are

(mn|L|m′n′) = Lmn,m′n′ = Hmm′δnn′ −Hn′nδmm′ (14)

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Exercise 2: Show that the eigenvalues of the Liouvillian in the basis |mn)induced by the energy basis H|m〉 = εm|m〉 are the possible transition frequen-cies ωmn = (εm − εn)/h. These are experimentally accessible quantities, incontrast to the absolute energy eigenvalues of the Hamiltonian H.

The superoperator formalism is a convenient starting point for projection op-erator techniques in statistical dynamics [11], effective dynamics of open quan-tum systems [14] and time-dependent perturbation theory [15].

1.2.3. Non-unitary spin dynamics

“Relaxation, damping, dephasing, decoherence,...” have in common irreversibledynamics (“arrow of time” [16]) due to the irrevocable loss of information intoinobservable degrees of freedom (“bath, environment”).

As an introductory model, we consider H = −µS ·B(t) with a fluctuatingclassical magnetic field. Predictions about the spin will involve an averageover the uncontrolled B-field which we will consider as a classical noise (inprinciple, its dynamics can of course be traced back to a quantum description ofmoving charges somewhere, but for the present purpose this is not necessary).As a stochastic process [17], the field is completely specified by its correlationfunctions

Ci1,i2,...,in(t1, t2, . . . , tn) := Bi1(t1)Bi2(t2) . . . Bin(tn) . (15)

The fluctuating field is taken to be Gaussian such that only its first two momentsneed to be specified, Bi(t) = 0 (no static field) and Bi(t1)Bj(t2) =: B2cij(t1, t2).We assume a stationary process that depends only on the time difference t1− t2and has a very short internal correlation time τc such that cij(t) = cijτcδ(t).This last assumption of white noise assumes that the noise correlation time τc ismuch shorter than the relevant time scale on which we want to study the systemdynamics. Lastly, we assume that the fluctuations are isotropic, cij = 1

3δij .To obtain the effective dynamics, we develop the time-propagated density

matrix ρ(t + ∆t) = U(t + ∆t)ρ(t)U(t + ∆t)† to second order in the interactingHamiltonian (11), 1

ρ(t + ∆t) = ρ(t)− i∫ t+∆t

t

[H(t′), ρ(t′)]dt′

− 12T

∫∫ t+∆t

t

(H(t1)H(t2)ρ(t) + ρ(t)H(t1)H(t2))dt1dt2

+∫∫ t+∆t

t

H(t1)ρ(t)H(t2)dt1dt2

(16)

Now we perform the average over the field fluctuations. The term linear in Hdisappears because B is zero on average. In the second order terms, one ofthe time integrations is contracted by the correlation function δ(t1 − t2); theremaining integrand is time independent such that the integrals gives just a

1In 2.2.2, we will formulate an equivalent way of obtaining an effective master equation

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factor ∆t. The effective equation of motion for the averaged density matrix ona coarse-grained time scale ∆t τc is then given by

lim∆→0

ρ(t + ∆t)− ρ(t)∆t

=: ρ(t) =: Lρ(t) (17)

with the Lindbladian [13]

Lρ = − 12τs

∑i

[Si, [Si, ρ]] (18)

Its prefactor, setting the new time scale of evolution, is the spin relaxation rate1/τs := ω2

0τc given in terms of the effective Larmor frequency ω0 = µ√

B2/3/h.The Lindbladian is a pure relaxation superoperator in the Lindblad form Lρ =1

2τs

∑i[Siρ, Si] + [Si, ρSi] which assures the complete positivity of the time-

evolved density matrix.Exercise 3: Show that the superoperator matrix elements of the Lindbladian

are given by

Lmn,m′n′ =1τs

(Smm′ · Sn′n − s(s + 1)δmm′δnn′) (19)

and verify the trace-preserving property∑

m Lmm,m′n′ = 0 from this expression.From now on, we omit the overline and write the Lindbladian L as before.

1.3. Irreducible spin superoperators

Formally, the solution of (17) is very simple: ρ(t) = exp[Lt]ρ(t). In Liouvillespace, this a matrix equation |ρ(t)) = expLt|ρ) of dimension d2

s × d2s. For spin

12 , diagonalizing a 4× 4 matix is elementary, but already for spin 1 with d2

s = 9this becomes cumbersome. The task therefore is to diagonalize L intelligently,i.e., using the rotational symmetries.

1.3.1. Scalar relaxation process: what results we should expect

The Lindbladian was obtained by an isotropic average and is thus a scalar ob-ject, i.e., invariant under rotations (“transforms under the trivial representationD(0)”). We may anticipate that the statistical operator can be decomposedρ =

∑K ρ(K) into parts that transform under the irreducible representations

D(K) of the rotation group. In a adapted basis of Liouville space, the Lindbla-dian as a scalar object can only connect subspaces of equal rank K. Further-more, inside each subspace it can not distinguish between different orientationsand must be diagonal as well. Thus, it will be written as a purely diagonalmatrix

L =

λ0 0 . . . 00 λ1 . . . 0...

.... . .

...0 0 . . . λd2

s

(20)

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How many different eigenvalues may we expect? The total number is the di-mension of the Liouville space d2

s. Each subspace of rank K will have dimensiondK = 2K + 1. Therefore, we will have to find only 2s + 1 different eigenvaluesλK , each of which has degeneracy 2K + 1:

particle s ds d2s eigenvalues degeneracy

electron 12 2 4 λ0, λ1 1, 3

photon 1 3 9 λ0, λ1, λ2 1, 3, 5

1.3.2. State multipoles

In the usual Hilbert space basis, the statistical operator reads

ρ =∑

m,m′

ρmm′ |sm〉〈sm′| (21)

Here, the ket |sm〉 transforms under the irreducible representation D(s), whereasthe bra 〈sm′|, as its complex conjugate, transforms under (D(s))∗, the contra-gredient representation. The ket-bra |sm〉〈sm′| transforms under the directproduct D(s) ⊗ (D(s))∗ which is reducible. One introduces therefore an ensem-ble of elements that do transform under the irreducible representation D(K),

T (K)q := T (K)

q (s, s) :=∑

m,m′

(−)s−m〈ssm′−m|Kq〉|sm′〉〈sm| (22)

with K = 0, 1, . . . , 2s and q = −K,−K + 1, . . . ,K. The Hermitian conjugate is(T (K)

q )† = (−)qT(K)−q .

The Clebsch-Gordan coefficients 〈s1s2m1m2|Kq〉 are the coefficients of theunitary basis change from the direct product Hs1 ⊗ Hs1 towards the Hilbertsubspace HK of spin K that should be familiar from the addition of two spins.The CG coefficients are non-zero only if two selection rules are satisfied: (i)the two magnetic quantum numbers on the left add up to the one on the right,m1+m2 = q; and (ii) if the angular momentum on the right satisfies the triangleinquality |s1 − s2| ≤ K ≤ s1 + s2.

In our case, we do not couple two spins, but a spin and its complex conjugate.Since there is a contragredient representation in the game, the CG coefficientsin (22) feature a characteristic minus sign in front of −m. The triangle selectionrule implies for us 0 ≤ K ≤ 2s.

The irreducible tensor operators T(K)q =: |Kq) form a basis of Liouville

space that is properly orthonormal with respect to the trace scalar product ofmatrices:

(Kq|K ′q′) := tr(T (K)q )†T (K′)

q′ = δKK′δqq′ (23)

Any linear operator A can be decomposed in this basis,

A =∑Kq

AKqT(K)q with AKq := (Kq|A) = tr(T (K)

q )†A. (24)

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The irreducible components

ρKq = tr(T (K)q )†ρ =

⟨(T (K)

q )†⟩

(25)

of the density matrix are called state multipoles or statistical tensors, and havebeen introduced already in the 1950’s by Fano and Racah [2, 18].

Exercise 4: (0) Show that all T(K)q except T

(0)0 have zero trace and calculate

the state monopole moment ρ00 (use√

2s + 1〈ssm′−m|00〉 = (−)s−mδmm′).(1) Show that the irreducible vector operator is proportional to the spin

operator, T(1)q = c

−1/2s Sq. Hint: consider the action of T

(1)0 (

√cs〈ssm−m|10〉 =

(−)s−mm), argue with rotational invariance and fix the proportionality constantcs by computing tr[(T (1))2].

1.3.3. Irreducible spin superoperators

By inserting the decomposition (24) on the left and right side of an arbitrarysuperoperator acting like LA, one obtains

LA =∑

KqK′q′

|K ′q′) (K ′q′|L|Kq) (Kq|A) (26)

such that with the notation LK′q′,Kq := (K ′q′|L|Kq) the superoperator reads

L =∑

KqK′q′

LK′q′,Kq|K ′q′)(Kq| (27)

where the tetradics on the right hand side transform under D(K′) ⊗ (D(K))∗,in an analogous manner to (21). Following the same strategy as previously,we recouple the elements again using the appropriate CG-coefficients to getirreducible superoperators [19]

T Lm (K, K ′) :=

∑q,q′

(−)K−q〈K ′Kq′−q|Lm〉 |K ′q′)(Kq| . (28)

Finally, any superoperator in completely decomposed form reads

L =∑Lm

∑K,K′

LLm(K, K ′)T Lm (K, K ′) (29)

with coefficients

LLm(K, K ′) =∑q,q′

(−)K−q〈K ′Kq′−q|Lm〉LK′q′,Kq . (30)

This decomposition is completely general and applies to arbitrary superopera-tors. It is only worth the effort, however, if the superoperator has rotationalsymmetries. The greatest gain in computational speed and conceptual clarity

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is obtained if the superoperator is a scalar such that only its only nonzero com-ponent is L = 0,m = 0. In that case, which applies to our Lindbladian (18),one finds by virtue of the triangle rule that K = K ′: as promised, L indeedconnects subspaces of equal rank. We can choose the decomposition

L =2s∑

K=0

λKT (K) . (31)

In this form, the T (K) are orthogonal projectors onto the subspaces L(Hs)(K)

of irreducible tensor operators of rank K:

T (K) =√

2K + 1T 00 (K, K) =

∑q

|Kq)(Kq| . (32)

They are indeed orthogonal,

T (K)T (K′) = δKK′T (K) (33)

by virtue of (23) and resolve the identity in Liouville space,∑

K T (K) = 1, byvirtue of a completeness relation of CG-coefficients.

1.3.4. Calculation of scalar eigenvalues

Once the invariant subspaces are known, the eigenvalues are obtained by pro-jecting the superoperator onto an arbitrary basis element:

λK = (Kq|L|Kq) = tr(T (K)q )†LT (K)

q . (34)

For the spin relaxation Lindbladian (18), one must calculate commutators ofthe form [Si, [Si, T

(K)q ]]. In order to use the commutation relations (8) and (9),

one writes the scalar product of spin operators∑i

SiSi =∑

p=0,±1

(Sp)†Sp =∑

p=0,±1

(−)pSpS−p (35)

in terms of the standard components S0 = Sz, S±1 = ∓ 1√2(Sx± iSy) such that

λK = − 12τs

∑p=0,±1

(−)ptr(T (K)q )†[S−p, [Sp, T

(K)q ]] (36)

It is now a simple exercise to show with the help of (8) and (9) that thedouble commutator gives back the tensor operator itself, [S−p, [Sp, T

(K)q ]] =

c(p, q,K)T (K)q . Summing all three terms gives remarkably simple eigenvalues,

λK = −K(K + 1)2τs

, K = 0, 1, . . . , 2s. (37)

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1.3.5. Isotropic spin relaxation

We can now solve the equation of motion

ρ(t) =∑Kq

ρKq(t)T (K)q =

∑Kq

ρKq(t)LT (K)q =

∑Kq

ρKq(t)λKT (K)q (38)

in each irreducible mode, ρKq(t) = λKρKq(t), to obtain the decay of statemultipoles

ρKq(t) = e−t/τK ρKq(0) (39)

with their relaxation rates τ−1K = |λK | = K(K + 1)/2τs.

The scalar mode relaxation rate λ0 = 0 assures the trace preservation ofρ(t).

The vector mode relaxation rate |λ1| = τ−1s describes the relaxation of the

orientation or average spin vector:

〈S(t)〉 = tr(ρ(t)S) = e−t/τs 〈S(0)〉 (40)

since tr(T (K)q Sq′) projects onto K = 1 (remember the exercise 4.(1) and the

orthogonality relation (23)).

2. Mesoscopic spin transport

2.1. General remarks

“Transport” is a movement from a point r to a point r′ induced by an externalcause. In a regular medium, propagation is ballistic: the average square of thedistance covered after a time t scales like

⟨x2(t)

⟩∼ v2t2 where v is the particle’s

velocity. A disordered medium contains impurities that interrupt the ballisticmovement. Quenched disorder is fixed for each realization of an experiment, butvaries from experiment to experiment when samples are changed. Predictionsabout observables will involve an average (·) that may be done over a classicaldisorder distribution or by tracing out uncontrolled quantum degrees of freedom.Generically, the averaged expectation value 〈·〉 of travelled distance behaves asa diffusive quantity: 〈x2(t)〉 ∼ Dt with D the diffusion constant.

In a classical transport setting (think of a soccer ball shot inot a forest) onepropagates probabilities. In quantum transport, one propagates amplitudes, al-lowing for interference phenomena if phase coherence is preserved. This requiressmall system sizes, low temperatures, etc. Examples are

1. Aharonov-Bohm effect: the resistivity of a metallic ring as function of amagnetic flux oscillates with period of one flux quantum Φ0 = h/e.

2. Weak localization (WL): The resistance of weakly disorderd metallic sam-ples show negative magnetoresistance dρ

dB > 0 at small field. Explana-tion: The quantum return probablility to a point includes the construc-tive interference of counterpropagating amplitudes and is therefore larger

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than the classical probability. This interference effect is destroyed by alarge enough magnetic field since loops of different sizes pick up differentAharonov-Bohm phases.

2.2. Diffusive spin transport

Imagine that we can inject spin-polarized particles (p+(0) = 1) on one end of adiffusive medium of length L. What is the probability p+(L) of retaining the spinpolarization at the other end (assuming that we have spin-sensitive detection)?Need two quantities: the momentum diffusion constant D that determines thetransmission time t ∼ L2/2D and the spin relaxation time τd as function of themicroscopic parameters of the scattering mechanisms for momentum and spin.

2.2.1. Classical diffusion

In a hydrodynamic description, diffusion is a direct consequence of linear re-sponse j = −D∇n and a local conservation law ∂tn + ∇ · j = σ. Solvingthe diffusion equation (∂t − D∇2)n = σ in Fourier components (q, ω), we seethat a small fluctuation δn(x, t) from a homogenous equilibrium density decaysexponentially as δn(q, t) = exp(−Dq2t)δn(q). A kinetic description in termsof collisions of carrier particles with obstacles permits to derive the diffusionconstant D associated with this process as function of microscopic parameters.

2.2.2. Master equation approach to diffusion

A microscopic quantum derivation of diffusive behaviour typically uses a systemHamiltonian

H = H0 + V . (41)

The one-particle Hamiltonian H0 describes ballistic propagation in momentumand spin eigenstates |kσ〉 := |k〉 ⊗ |sσ〉 with eigenenergies ε0(k, σ). The impu-rity potential V describes momentum scattering, spin-flip scattering, spin-orbitcoupling, and the like, by randomly distributed scatterers. Deriving observablequantities will involve averages over all possible realizations of the disorder.Typically, averages over the random positions ri of impurities will be done as-suming a uniform distribution over the entire volume Ld,

F (ri) =∫

(Πjddrj

Ld)F (ri) (42)

Starting from the Liouville equation ˙ρ(t) = 1i [V (t), ρ(t)] in the interaction

representation A(t) = U†0 (t)AU0(t) and developing to second order in the inter-

action leads to the master equation for the effective density matrix ρeff = ρ, 2

˙ρeff(t) = −∫ t

0

[V (t), [V (t− t′), ρeff(t− t′)]dt′ . (43)

2Remember the perturbation theory we carried out in 1.2.3.

12

Here, one uses the fact that V = 0 (always achievable by redefining the zero ofenergies in H0). Furthermore, one assumes a Gaussian model of disorder suchthat only impurity pair-correlations V V need to be retained.

By selecting a typical term of type V (t)ρeff(t− t′)V (t− t′) under the integralin (43), and inserting the identity (write |k1σ1〉 =: |1〉 for brevity), we naturallyfind the so-called four-point scattering vertex,

V23,14 := 〈2|Vimp|1〉〈4|Vimp|3〉 =

21

34(44)

The arrows on the propagation lines indicate whether the state appears as a ket|·〉 or a bra 〈·| in the amplitude; one speaks of “incident” momentum and spinfor |·〉 and “scattered” momentum and spin for 〈·|.

The standard way of proceeding would now be to evaluated the average,use a Markov approximation for coarse-grained evolution on time scales longerthan the bath correlation time. In the end, it should be possible to derivea diffusion equation for the probability density in real space leading to thehydrodynamic description of 2.2.1, thus determining the diffusion constant fromthe microscopic scattering parameters. The derivation is very involved, and wetake a short-cut.

2.2.3. Elastic scattering mean free time

Let’s look at a momentum scattering impurity potential of the type

Vel =Nel∑i=1

v0(r − ri) (45)

with v0(r) centered at random locations ri without action on the spin:

〈k2σ2|Vel|k1σ1〉 = δσ1σ2

Nel∑i=1

∫ddr

Ldei(k1−k2)·rv0(r − ri) (46)

An individual scattering process |k1σ1〉 =: |1〉 7→ |k2σ2〉 =: |2〉 induced by Vel

has the amplitude 〈2|Vel|1〉 (to first order in V : Born approximation). Accordingto Fermi’s Golden Rule, the total scattering rate out of state |1〉 is

1τel

= 2π∑k2σ2

|〈2|Vel|1〉|2δ(ε1 − ε2) (47)

which actually can be seen to correspond to (the imaginary part of) the partiallycontracted scattering vertex

∑2 V22,11 defined in (44). The elastic scattering

mean free time is an experimentally accessible parameter (for instance through

13

conductivity measurements for electrons) and will be taken as given in the fol-lowing. It describes the average life-time on a momentum eigenstate in thepresence of scattering impurities.

The weak disorder limit in which the Born approximation and Fermi’s goldenrule are valid corresponds to τ−1

el ε. For massive particles with kinetic energyε = k2/2m and velocity v = k/m, one introduces the elastic scattering mean freepath `el = vτel. The weak disorder limit can thus also be stated as 1/(k`el) 1which indicates that successive scatterers are placed in the far field of scatteredwaves (distance `el wavelength λ). This is essentially a low-density argument:In terms of the density nel of elastic scatterers and their total scattering crosssection σel one has `el = 1/(nelσel).

The relevant momentum diffusion constant is then D = v`el/d. The numer-ical factor of the dimension d in the denominator is conventional. The scalingin v and `el stems for the diffusion equation in Fourier space iω + Dq2 wherethe relevant scales are ω ∼ τ−1

el and q ∼ `−1el .

2.2.4. Spin-flip scattering vertex

Consider now a spin-flip interaction with a collection of freely orientable mag-netic impurities centered at sites ri,

Vsf =Nsf∑i=1

g(r − ri)S · J i (48)

The relevant spin-flip vertex is V14,23 = (2πρτsf)−1Vσ1σ4,σ2σ3 in terms of thenormalized spin-flip vertex (or spin-flip superoperator)

Vσ1σ4,σ2σ3 =Sσ2σ1 · Sσ4σ3

s(s + 1)(49)

The scalar product of spin operators originates from the isotropic average (Ji)αJ(j)β =δij

13δαβJ2 over all possible orientations of the impurity spin. The position-

dependent potential part g(r) induces momentum scattering such that the totalmomentum scattering rate reads

1τ

=1τel

+1τsf

(50)

In order to determine the spin relaxation rates, we need to diagonalize thespin superoperator Vσ1σ4,σ2σ3 . But here, we can make use of our results for theLindbladian L treated in 1.2.3: by comparing the scattering approach to themaster equation formulation, we can identify L = τ−1

sf (V − I). Thus, the spindecay rates of the state multipoles in the present case read

λK = − K(K + 1)2s(s + 1)τsf

. (51)

We have of course λ0 = 0 (the trace is preserved), and the spin decay rate ofthe spin orientation (K = 1) is |λ1| = 1/(s(s + 1)τsf) =: 1/τd.

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2.2.5. Diffusive spin relaxation

During a propagation time t, the spin density matrix will therefore relax as

ρ(t) =∑Kq

eλKtρKq(0)T (K)q (52)

Starting from an initial polarized state |+〉 := |s,m = +s〉 with p+(0) =〈+|ρ0|+〉 = 1, we find (exercise !)

p+(t) =1

2s + 1+ e−t/τd

3s2

s(s + 1)(2s + 1)+ . . .

=12(1 + e−t/τd)

(53)

with the last line valid for electrons (spin s = 12 ). Naturally, the probability

relaxes towards its equilibrium value peqm = 1

2s+1 to have any magnetic quantumnumber m = −s, . . . , s.

2.3. Dephasing of weak localization

Weak localization is observed as a decrease of diffusivity or conductivity. Thereason is the constructive interference of counterpropagating amplitudes alongidentical scattering paths. The interference correction to the classical diffusionconstant D0 can be written by integrating over all closed amplitude loops,

δD(ω)D0

∝∫

ddQ

D0Q2 + iω(54)

Including spin-flip scattering into the picture can be done by diagonalizing thespin-flip vertex appropriately. The irreducible subspaces turn out to be the sin-glet and triplet state subspaces |Kq〉 (not operators |Kq) as for the propagatedintensity: the difference is due to the fact that the two amplitudes in weak lo-calisation loops propagate in opposite directions). To sketch the result, the WLcorrection is written

δD

D0∝

∫ddQ

∑K

wK

D0Q2 + τc(K)−1(55)

where each spin channel comes with a certain weight

wK = (−)2s+K 2K + 12s + 1

(56)

For electons, w0 = − 12 and w1 = 3

2 . More importantly, each spin channel isdamped with its coherence time τc(K). They are given in terms of the spinrelaxation eigenvalues (51):

1τc(K)

=2τsf

+ λK =2τsf

(1− K(K + 1)

4s(s + 1)

)(57)

15

such that for electrons τc(0) = τsf/2 and τc(1) = 3τsf/2.Magnetic impurites are a very effcient source of decoherence/dephasing for

interference phenomena of spin-carrying particles!

2.4. Spin-orbit scattering

Spin-orbit interaction is a coupling between the momentum degree of freedom ofitinerant spins and their spin orientation. This mechanism is of high relevancefor spintronics applications [3] and mesoscopic physics.

2.4.1. Elliott-Yafet mechanism

A coupling between orbital motion and spin arises already when an itinerantelectron is scattered by a scalar impurity potential V (r): the electrostatic fieldE = −e∇V (r) deflecting the electron gives rise to a magnetic field in the elec-tron’s frame of reference. Transforming back to the impurity frame gives thepotential

Vso =h

4mc2σ · (∇V (r)× k) . (58)

This potential is still a linear coupling to the spin S ·Beff with a k-dependenteffective magnetic field. As far as spin transformations are concerned, wecan therefore use all the machinery developed in the previous sections. No-tably, the spin-orbit scattering mean free time for the momentum relaxationis 1/τso = 2πρNso

⟨v20

⟩where 〈·〉 is an angular average over possible wavevec-

tor orientations k with modulus fixed by the energy conservation δ(εk − εk′).Furthermore, the spin relaxation time is given by (51): τd = 3

4τso

In principle, all processes with momentum change will induce a spin-orbitcoupling of this type, e.g., scattering by impurities and boundaries (dominantat low temperatures) or phonons (dominant at higher temperatures).

2.4.2. Dyakonov-Perel mechanism

If H0 is time-reversal invariant, the energy levels are two-fold degenerate, ε(k, σ) =ε(−k,−σ) (Kramers degeneracy). If H0 is space-inversion invariant (under par-ity operation Pr = −r,Pσ = σ), then one has ε(k, σ) = ε(−k, σ).

A necessary ingredient for the DP spin-coupling is the absence of inversionsymmetry which implies ε(k, σ) = ε(−k,−σ) 6= ε(k,−σ): different spin statesare no longer degenerate and can be coupled, which induces a precession ofthe electronic spin around a k-dependent effective vector Ω(k) during ballisticmotion: Vso = 1

2σ ·Ω(k). The effective field Ω depends on the crystal structureand the band structure of the conduction band.

The Dyakonov-Perel spin-orbit precession can be pictured as a diffusion pro-cess or random walk on the Bloch sphere. Between scattering events separatedin time by the total momentum scattering mean free time τ , the spin will onaverage precess by an angle δφ ∼ Ωτ . For small precession frequencies such thatΩτ 1, the spin will cover only a small angle before the field vectors changes

16

its direction after the next momentum scattering event. Then, a new precessionsets in, and so forth. After a time t (corresponding to t/τ such steps) the totalangle covered will be φ2 ∼ δφ t/τ (remember the diffusion relation L2 ∼ Dtwith D = `2/τ ; now, the step-length ` is δφ). The relevant spin coherence timeτs may be estimated by the time t it takes to arrive at φ = 1 (or 2π, numericalfactors are of no relevance in this simple argument), which gives 1/τs ∼ Ω

2τ . In

contrast to the Elliott-Yafet mechanism, the spin decoherence rate gets smallerwith smaller τ (“motional narrowing”) because the spin does not have the timeto precess a lot if the momentum scattering events follow too fast.

2.5. Spintronics - some characters of the play

2.5.1. Spin-resolved density of states

The spin-up and spin-down components ν↑↓(ω) of the total density of states met-als or semiconductors are degenerate at thermal equilibrium and in the absenceof polarizing fields. For ferromagnetic metals (F) however, the spontaneousalignment of spins leads to a finite spin polarization (in the direction oppositeto the magnetization) even at equlibrium. If the net spin polarization

ns =N↑ −N↓

N↑ + N↓(59)

is positive, the ↑-spins are called majority spins, the ↓-spins are minority spins.After spin injection from a ferromagnet, the spin polarization in a normal metalcan be of order 15%.

2.5.2. Spin field-effect transistor

This propotypical spintronics device was proposed in 1990 by Datta and Das[23]. It consists of a ferromagnet-normal-ferromagnet (F/N/F) junction of con-ductors. The electrons in the normal metal between the two ferromagnets (with,say, both ↑ spin orientation) are subjected to a spin-orbit precession that can beprecisely controlled with an external gate voltage. If the spin ↑ does not precess,then conduction will be normal. For a precession such that spin ↑ has rotatedto ↓, the entrance to the second ferromagnet is prohibited, and no current willflow.

This very nice device has a small disadvantage: it does not exist yet. Thepurity required for the normal metal (of linear size of order 1 µm) as well asthe control fields are not yet met by actual experimental techniques. But otherinteresting things are under study.

2.5.3. Tunneling magneto resistance

A rather old scheme of charge current control through spin dates back to the70’s [24]: a ferromagnet–insulator–ferromagnet junction (F/I/F) where spin-polarized electrons can tunnel between the two contacts. A simple model as-sumes that the tunneling matrix elements do not involve spin-flip scattering and

17

are the same for all spins. Then, the conductance G = 1/R of the junction withparallel or anti-parallel orientation will be given by

G↑↑ ∼ M1M2 + m1m2

G↑↓ ∼ M1m2 + m1M2

(60)

where Mi is the density of states (at the Fermi energy) of majority spins in theferromagnet i, and mi the minority density, since ↑-spins on the left will tunnelinto the ↑-subband on the right and vice versa for ↓-spins. A simple exercisepermits to verify that the tunneling magnetoresistance ratio is given by

TMR =R↑↑ −R↑↓

R↑↑ + R↑↓=

2P1P2

1− P1P2(61)

and depends therefore on the spin polarisation Pi = (Mi−mi)/(Mi +mi) of theleft and right ferromagnets. The technological advantage of magnetoresistancejuntions is that the magentization persists even after the driving currents areswitched off (non-volatile memory). This is used in many current data storingdevices.

2.5.4. Mesoscopic spin filter

A spin polarizer-analyzer involving a quantum dot and magnetic fields has beenpresented experimentally by Folk et al., [25]:

figure taken from [3].

A spin polarization of 70% is reached

2.5.5. Spin Hall effect

After an early theoretical proposition by Dyakonov and Perel [26] that spin-orbit coupling in 2D-semiconductors can lead to a spatial separation of spindensities, the spin Hall effect has been recently experimentally realized by Katoet al. [27]:

18

3. Quantum dot spin computation

3.1. Interacting spins

The original proposal by D. Loss and D. DiVincenzo [29] (see also [30]) suggestedtwo excess electron spins S1 and S2 in neighboring quantum dots with a time-dependent exchange coupling J(t) that can be controlled through an externalgate voltage:

H = J(t)S1 · S2 (62)

Also, individual spin control gi(t)Bi(t) · Si via a local effective field should bepossible. This allows the realization of the universal

√swap gate on realistic

time scales taking into account the decohering effect of the environment. Lossand DiVincenzo set up a master equation approach using spin superoperatorsbut do not make explicit use of irreducible representations.

3.2. Interaction with nuclear spins

Today’s major limitation to a coherent functioning of spin quantum dots comesfrom the hyperfine interaction with nuclear spins from the semiconductor ma-trix, the so-called “central spin problem” [31]. Master equation treatments ofthis problem have been developed, showing a rich variety of non-markoviandynamics [32].

19

3.3. Spin-relaxation measurements

Recent experiments by Johnson et al. [33] with two coupled spin quantumdots allowed to determine the spin dephasing time from all-electrical high fre-quency measurements. The inferred time is of order 10 ns, much too short forgate operations, but naturally, new schemes and propositions are devised.

20

4. Closing remarks on a sunny Friday afternoon

Taken from: M. Ziese and M.J. Thornton, “Spin Electronics” (Spinger 2001)... leads to ...

21

On the other hand, respectable theoreticians (card-carrying?) arrive at [34]

or even

22

So here you are, the Spin TV repairman, and you have to judge the worthi-ness of different curved manifolds [35]:

23

References

[1] M. Chaichian and R. Hagedorn, Symmetries in Quantum Mechanics (IOPPublishing, Bristol, 1998)

[2] K. Blum, Density Matrix Theory and Applications (Plenum Press, 1996)

[3] I. Zutic, J. Fabian, and S. Das Sarma, “Spintronics: Fundamentals andapplications”, Rev. Mod. Phys. 76, 323 (2004)

[4] L.C. Biedenharn and J. D. Louck, “Angular Momentum in Quantum Me-chanics”, vol. 8 of the Encyclopedia of Mathematics (Addison Wesley, 1981)

[5] G.E. Uhlenbeck and S. Goudsmith, Naturwiss. 13, 953 (1925)

[6] W. Pauli, Z. Phys. 31, 765 (1925)

[7] W. Gerlach and O. Stern, Z. Phys. 9, 349 (1922)

[8] D. J. Gross, Physics Today (Dezember 1995) and references therein. pdf

[9] E. Cartan, Bull. Soc. Math. France 41, 513 (1913)

[10] E. Fick and G. Sauermann, The Quantum Statistics of Dynamic Processes(Spinger Series in Solid-State Science vol. 86, 1990)

[11] R. Zwanzig, Physica 30, 1109 (1964)

[12] H. Gabriel, “Theory of the influence of environment on the angular dis-tribution of nuclear radiation”, Phys. Rev. 181, 506 (1969), especially theappendix.

[13] J. Preskill, Lecture Notes for Physics 229: Quantum Information and Com-puting, chap. 3 (1998)

[14] C.W. Gardiner and P. Zoller, Quantum Noise (Springer, 2004)

[15] S. Mukamel, “Superoperator representation of nonlinear response: Unifyingquantum field and mode coupling theories”, Phys. Rev. E 68, 021111 (2004)

[16] H.-D. Zeh, The Physical Basis of the Direction of Time, chap. 3 (Springer)

[17] N. van Kampen, Stochastic Processes in Physics and Chemistry (NorthHolland, Amsterdam, 1990)

[18] U. Fano and G. Racah, Irreducible Tensorial sets (Academic, New York);U. Fano, Rev. Mod. Phys. 29, 74 (1957). A bit old-fashioned presentation,references given for completeness.

[19] C.A. Muller, C. Miniatura, E. Akkermans, and G. Montambaux, “Meso-scopic scattering of spin s particles”, J. Phys. A: Math. Gen. 38, 7807(2005). pdf

24

[20] S. Datta, Electronic transport in mesoscopic systems (Cambridge Univer-sity Press)

[21] J. Rammer, “Quantum transport theory” (Perseus Books, Reading, 1998)

[22] J. Fabian and S. Das Sarma, “Spin relaxation of conduction electrons”, J.Vac. Sci. Technol. B (17)4, 1708 (Jul/Aug. 1999)

[23] S. Datta and D. Das, Appl. Phys. Lett. 56, 665 (1990)

[24] M. Julliere, Phys. Lett. A 54, 225 (1975)

[25] J.A. Folk, R.M.Potok, C.M. Marcus, V. Umansky, Science 299, 679 (Jan2003)

[26] M.I. Dyakonov and V.I. Perel, Phys. Lett. A 35, 459 (1971)

[27] Y.K. Kato, R.C. Myers, A.C. Gossad, D.D. Awshalom, Science 306, 1910(Dec 2004)

[28] D.D. Awschalom, D. Loss, and N. Samarth (eds.), Semiconductor Spin-tronics and Quantum Computation (Spinger 2002)

[29] D. Loss and D.P. DiVincenzo, “Quantum computation with quantum dots”,Phys. Rev. A 57, 120 (1998)

[30] G. Burkard and D. Loss, in: “Semiconductor Spintronics and QuantumComputation”, eds. D.D. Awshalom, D. Loss, N. Samarth (Spinger 2002)

[31] A.V. Khaetskii, D. Loss, L. Glazman, Phys. Rev. Lett. 88, 186802 (2002)

[32] W.A. Coish and D. Loss, Phys. Rev. B 70, 195340 (2004). See also talk byD. Loss on Thursday 29/9 during the ISSQUI workshop.

[33] A.C. Johnson et al, Nature , (2005)

[34] D. M. Basko, I. L. Aleiner, B. L. Altshuler, cond-mat/0506617

[35] R. Hudson, oral communication, MPIPKS Colloquium on 5.9.2005

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