Single electron spin decoherence due to the hyperfine interaction 27

heavy hole spin and nuclear spin is given by [86]:

H_QDhf^h =X

k

A^h_kh

S_z^hI_z^k+O(η) S_x^hI_x^k+S_y^hI_y^k

+O(η²) S_x^hI_z^k+S_y^hI_z^ki , (1.4.29) where the last term of Eq. (1.4.29) is referred to as non-collinear hyperfine interaction [87]. The magnitude of the mixing η ranges between 0.2−0.7 [86].

1.5 Single electron spin decoherence due to the hyperfine interaction

The main source for the electron spin decoherence in semiconductor quantum dots is the hyperfine interaction with nuclear spins [9,29,78]. Therefore, the effect of nuclear spins on the free induction decay and spin echo decay of the electron spin was extensively studied. Electron spin state dynamics differs strongly depending on the initial state of the nuclear bath and strength of the magnetic field applied. To use the electron spin in a quantum dot as a qubit, one applies a magnetic field to lift the degeneracy of the spin states. The Hamiltonian describing the electron spin in a magnetic field in z direction (B=Bˆz) and interacting with nuclear spins is,

H =b Sz+S·h= (b+hz)

| {z }

H0

+¹₂(h+S−+h−S+)

| {z }

V

(1.5.1)

where b = gµBB is the Zeeman splitting of the electron. The Hamiltonian can be divided into the longitudinal partH₀and the transverse partV, which is expressed through the ladder operators for the electron spin,S± =S_x±iS_y, and nuclear field ladder operators,h±=hx±ihy. The transverse term of the Hamiltonian (1.5.1) describes flip-flop processes between the electron spin and nuclear spin ensemble: i. e. exchange of angular momenta between the two systems. The total angular momentum of the complete system is conserved. The flip-flop terms can be used to prepare a polarized nuclear spin state. It was also shown that this interaction leads to nuclear spin diffusion via coupling to the electron spin, which is responsible for the fluctuation of the nuclear spin polarization on timescales on the order of 100µs.

The time evolution of the electron spin depends strongly on the initial nuclear state. For describing the electron-nuclear system in a single quantum dot, one can apply the “sudden” approximation [88]. The electron and nuclear states are decoupled at times t < 0. After the electron is injected into the

quantum dot and it is brought into the contact with nuclear spins, the initial density matrix can be written asρ(0) =ρ_e(0)⊗ρ_nuc(0). The electron state is given by the density operatorρ_e and the nuclear density operator is denoted byρ_nuc.

The decoherence time of the electron spin can be found by averaging over the ensemble of nuclear spins. As in Refs. [9, 88] we consider three types of initial nuclear states, which describe systems with N nuclear spins 1/2. First, we choose a pure initial nuclear state which can be written asρ⁽¹⁾nuc(0) =

|ψ_nuci hψ_nuc|, where

The value f↑ defines the nuclear polarization p = 2f↑ − 1. The spin-up and spin-down states of the kth nuclei are denoted by |↑i_k and |↓i_k. The site-dependent phase is given by φ_k. The nuclear state |ψ_nuci is chosen in such a way, that the z component of its spin is translationally invariant:

hψ_nuc|I_z^k|ψ_nuci=p/2.

Secondly, assuming a nuclear product state of the form |↓↑↑↓ · · · i, which describes N↑ nuclear spins in spin-up state and N −N↑ in spin-down state, we can write a mixed initial nuclear state as:

ρ⁽²⁾_nuc(0) =X As third and last special type of initial nuclear state, we can consider a pure state |ni, which is an eigenstate of h_z with eigenvalue pA/2:

ρ⁽³⁾_nuc(0) =|ni hn|, (1.5.4) with A=P

kAk.

The transverse part of the Hamiltonian (1.5.1) can be omitted at large mag-netic fields, because the electron Zeeman energy is much larger than the energy difference between the nuclear spin-up and spin-down states. The flip-flop transitions are suppressed in this case. However, these transitions can be energetically allowed by employing second-order processes like simul-taneous absorption/emission of a phonon [62, 65] with energy equal to the mismatch of the electronic and nuclear spin splittings. Here, we assume that there is no such process favoring the flip-flop transitions. Thus, we evaluate the electron spin dynamics only under the action of H0. For simplicity we assume also a uniform coupling constant: A_k = A/N. We have a constant hS_zi_t, because S_z commutes with H₀. However, the transverse components

1.5. Single electron spin decoherence due to the hyperfine interaction 29 hS±i_t =hS_xi_t±ihS_yi_tundergo a nontrivial time evolution, since[H₀, S±]6= 0. For the initial nuclear conditions (1.5.2) and (1.5.3) it can be found [88]:

hS±i^(1,2)_t =hS±i₀X

N↑

N N↑

f_↑^N↑(1−f↑)^N−N↑e^{±i(b+A(2N↑−N)/2N)t}. (1.5.5) This expression can be averaged over the binomial distribution, by applying the central limit theorem, one obtains:

hS±i^(1,2)_t =hS±i₀e^−t2/2τc2±i(b+pA/2)t

. (1.5.6)

The transverse dephasing time is given by τ_c = 2~p

N/(1−p²)/A = 1/σ, where σ is the variance of the distribution for the eigenvalues of hz. As one can see, the reduction of the decoherence time for the electron spin follows straightforwardly for a high degree of nuclear polarization or for quantum dots of larger size. For a GaAs quantum dot with N = 10⁵,A = 90µeV and p² 1, the decoherence time is τ_c ≈5 ns.

For the initial nuclear state in the form of the eigenstate |ni (see Eq. 1.5.4) there is no decay for the transverse components:

hS±i⁽³⁾_t =hS±i0e±i(b+pA/2)t

. (1.5.7)

A generalized master equation approach provides a reliable method to include the flip-flop terms in a perturbative way for high magnetic fields. Moreover, one can extend the decay analysis by including isotropic hyperfine couplings:

A_k =A₀exph

−(k/N)^m/di

, (1.5.8)

where d is the dimension of the confinement and m is defined by the form of the electron envelope function ψ₀(r) (see Section 1.4.2). Here, A₀ is the coupling constant atr₀. The asymptotic behavior of the longitudinal electron spin component to the leading order of δ = N/ω_n² with ω_n = b+hn|h_z|ni can be written as:

hS_zi_t=hS_zi_∞+σ_z^dec(t), (1.5.9) wherehSzi_∞ is a constant term and is a sum of the initial expectation value hS_zi₀ and a small correction of orderδ. The decaying part σ_z^dec(t)is found to be ∼ δ/t^d/m (d/m < 2) for long-time limit τ 1, where the characteristic timescale is given by τ = A0t/2~ [9, 88]. For a Gaussian electron envelope wave functionψ₀(r)the factormis equal to 2 and in the case of the parabolic confinement d= 2.

The asymptotic solution for the transverse part can also be written as a sum of two terms:

hS₊i_t=σ^osc₊ (t) +σ^dec₊ (t). (1.5.10)

The oscillating term σ₊^osc(t) ∼ hS₊i₀e^iωnt has a constant absolute value like hS_zi_∞. The decaying part has similar behavior forτ 1as the longitudinal and it can be shown σ^dec₊ (t 2~/A₀)∼δhS₊i₀/t^d/m (d/m <2).

The perturbative treatment of the Hamiltonian (1.5.1) reveals a long-time decay for the electron spin even for a nuclear system in an eigenstate of h_z. This irreversible decay, which is not observed in the zero-order solution (1.5.7), is caused by the inhomogeneity of the hyperfine coupling constant.

An exact solution for the electron spin dynamics can be defined for the ini-tially fully polarized nuclear bath (p= 1). For large magnetic field (b A), and after long time τ 1, the transverse electron component saturates at the constant value A²/b²N [29, 88, 89, 90]. For zero magnetic field (b = 0) the long-time decay for the transverse electron polarization is given by∼1/ln^3/2τ [89, 90].

The power-law decays obtained in [88, 89, 90] indicate non-Markovian dy-namics for the electron spin. An exponential decay can be observed in the Markovian limit, when the perturbative treatment cannot be applied any-more. It is justified for large time scales, when the virtual flip-flops be-tween the electron spin and the nuclear spins determinate the time evolution [91, 92]. The complete Hamiltonian consists of the Zeeman nuclear term in addition to the Hamiltonian (1.5.1). An effective second-order Hamiltonian can be obtained by applying a Schrieffer-Wolff transformation [91, 93]:

H_M= b+h_z+1 Here, ω = b +h_z and I_k^+/− denote the raising/lowering nuclear operators for spin at the kth site. After transformation, the diagonal terms of order

∼ A²/N b can be omitted, however, the corrections of the same order to the non-diagonal part of H_M cannot be neglected. This assumption can be applied for short correlation times τ_c ∼ N/A N b/A² compared to the timescale, when the diagonal corrections become relevant for b A [9, 91,94].

The perturbation term in the effective Hamiltonian H_M is XS_z with X = 1/2P

k6=lA_kA_lI_k⁻I_l⁺/ω. The Markov approximation can be used to obtain a compact expression for the decoherence rate defined by the dynamics of the non-diagonal term X(t) = e^−iωtXe^iωt: The average represents the expectation value taken with respect to the ini-tial nuclear state. The iniini-tial nuclear state is assumed to be prepared in an

1.5. Single electron spin decoherence due to the hyperfine interaction 31 eigenstate or “narrowed” state: ω|ni=ω_n|ni. The shift of the precession fre-quency∆ωis defined self-consistently as shown in [91]: hS₊i_t= ^x₂^t e^{i(ωn+∆ω)t}, wherex_tis a slow varying envelope function. For initially unpolarized (it can be at the same time narrowed) nuclear state the expression (1.5.12) can be evaluated to a simple form:

1

where the geometrical factor is given by [91], f This result for the decoherence rate is valid only for d/m >1/2. Otherwise, 1/T2 diverges and the decoherence time approaches zero. This indicates the range of parameters (here, the properties of the confinement potential and the electron wave function) where the Markov approximation cannot be applied anymore. The expression (1.5.13) reveals an important dependence of the decoherence time on the magnetic moment of the nucleus: 1/T₂ ∼I⁴. This means, that electron spin confined in a quantum dot containing In (I = 9/2) will decay faster than in a pure GaAs (I = 3/2) quantum dot.

At low magnetic field (b ∼A), the dipolar interaction between nuclear spins [95] is an important mechanism for the decoherence besides the virtual flip-flops between the electron spin and the nuclear spins. The flip-flip-flops between nuclear spins mediated by the dipolar interaction redistribute nuclear spin polarisation and the nuclear spin state cannot be assumed static anymore.

The nuclear dipolar interaction together with the hyperfine interaction is the origin of the spectral diffusion or randomised precession frequency for the electron spin [78, 96, 97]. The spectral diffusion problem was solved for low magnetic field by using resummation techniques, which provide a controlled perturbative solution [98,99, 100].

Another nuclear-nuclear interaction can influence the decoherence of an elec-tron spin in a quantum dot. The quadrupole interaction [95] is relevant for nuclear spin states I >1/2, because they have a finite electric moment that couples to the electric field gradients. Crystal strain originating from defects, dopants, and/or semiconductor heterostructures produces finite electric field gradients, which cause nuclear quadrupole splittings [78, 101]. One of the effects of the quadrupole interaction is the shift of the nuclear Zeeman levels in an applied magnetic field ∼ m²_z, where mz is the nuclear spin projection along the magnetic field. If the nuclear Zeeman levels are shifted inhomoge-neously through the quantum dot, the dipolar flip-flops between the nuclear

spins are suppressed, because they become energetically forbidden. There-fore, the quadrupole interaction preserves the nuclear spin polarisation and suppresses the nuclear spin diffusion. However, depending on the orienta-tion of the magnetic field relative to the anisotropy axis and the strength of the quadrupole coupling, it can also cause electron spin decoherence with an exponential decay [102].

1.6 Fidelity

In Chapters2, 3, and 4 we define the errors in optically induced operations for electron spin qubits in quantum dots originating from the hyperfine inter-action. For this purpose we find and calculate the fidelity between the ideal operation and the operation involving the nuclear spin effects. The fidelity can be defined as a difference between two states as well as between two unitary operators. Since quantum gates are unitary, it is a suitable measure for difference between an ideal quantum gate and a gate containing an error

“pathogen”.

The Uhlmann fidelity can be used as a measure for a difference between two states ρ and σ [103, 104]:

F(ρ, σ) =

Trq√ ρ σ√

σ 2

. (1.6.1)

The general properties of the fidelity function are:

• 0≤F(ρ, σ)≤1. For orthonormal states F = 0and for identical states F = 1.

• Symmetry: F(ρ, σ) = F(σ, ρ).

• Convexity: ifσ₁, σ₂ >0 and p₁+p₂ = 1, then F(ρ, p1σ1+p2σ2)≥p1F(ρ, σ1) +p2F(ρ, σ2).

• Multiplicativity: F(ρ₁⊗ρ₂, σ₁⊗σ₂) =F(ρ₁, σ₁)F(ρ₂, σ₂)

• Non-decreasing: the fidelity is not changed by unitary evolution. If ρ, σ are mapped into ρ⁰, σ⁰ by a measurement, it holds that F(ρ⁰, σ⁰)≥ F(ρ, σ).

If one of the states is pure (let ρ = |ψihψ|), then the definition (1.6.1) can be reduced by using the cyclic property of the trace:

F(|ψihψ|, σ) =hψ|σ|ψi. (1.6.2)

1.6. Fidelity 33 An unitary transformation U maps pure states to pure states, therefore

F(U|ψihψ|U^†, σ) =hψ|U^†σU|ψi. (1.6.3) Thereby, it is shown that the fidelity of a quantum operation in the presence of noise or decoherence can be defined as the Uhlmann fidelity between the ideal U_ideal and actual U_actual unitary operator. For this, one can assume a random state vector |χi on a Hilbert space and apply the operators on it.

The transformed statesU_ideal|χiand U_actual|χican be identified with|ψiand σ from Eq. (1.6.2), so that the fidelity depends only on the state |χi[105]:

F(U_ideal, U_actual) =hχ|U_ideal^†

| {z }

hψ|

U_actual|χihχ|U_actual^†

| {z }

σ

U_ideal|χi

| {z }

|ψi

. (1.6.4)

An average fidelity is defined by averaging over all random states|χi: F_average(U_ideal, U_actual) = 1

N(|χi) X

|χi

F(U_ideal, U_actual), (1.6.5) whereN(|χi)is the total number of the states|χi. The sum can be changed to an integral:

F_average(U_real, U_actual) = Z

S²ⁿ⁻¹

|hχ|M|χi|²dV, (1.6.6) where M =U_ideal^† U_actual and dV is a normalized measure. In Refs. [106, 107]

it was shown that for a linear operator M on a complex Hilbert space Cⁿ and for state vectors |ψi defined on the unit sphere S²ⁿ⁻¹ the Eq. (1.6.6) is given by:

Z

S²ⁿ⁻¹

|hχ|M|χi)|²dV = Tr(M M^†) +|Tr(M)|²

n(n+ 1) . (1.6.7)

For a unitary operatorM the value of Tr(M M^†)is equal tonand|Tr(M)|² 6 n².

Im Dokument Nuclear Spin Phenomena in Optically Active Semiconductor Quantum Dots (Seite 39-47)