Over all, the time modulated electric field strength is 10−5 order in unit of kvolt/cm.
Actually this magnitude is such small. This limitation comes from one side band approx-imation which must ignore two photon scattering process and high order correction to wavefunction.
Second, Fig. 5.7 shows the SD standard deviations of spin density σ is always larger
0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.05
0.1 0.15 0.2 0.25 0.3 0.35
L=2.9(µm) E0=2.74×10−5(kvolt/cm) E=0 (b 0)
| ψ +ω−σ| 2
0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3
0.2 0.4 0.6 0.8 1
| ψ −ω+σ| 2
frequcncy ω (b0) (a)
(b)
Figure 5.5: Incident wave comes from σ branch and the magnitude square of ψ±ω−σ is the probability of changing branch(spin flip).
Figure 5.6: Incident wave comes from σ branch and the magnitude square of ψ±ω−σ is the probability of changing branch(spin flip). The inserted plot in (b) is a re-scale view of small peak at ω = 1.35.
0 1 2 3 4 5 1.2
1.4 1.6 1.8 2
SD
z/ SD
n0 1 2 3 4 5
0 0.005 0.01 0.015 0.02 0.025
ω /b
0
=1.125 E
0
=2.74 × 10
−5(kvolt/cm) E=0(b
0
)
L ( µ m)
SD
SDn
SDz
Figure 5.7: Standard deviations show that SDz > SDn to guarantee that it is not nec-essary to consider screening correction. At length of a.c. electric field is L = 2.9µm, the ratio between them is almost twice.
condition that SDz almost is twice of SDn. The result indicates that it is not necessary to consider about screening correction.
Third, all fields including external static magnetic, time modulated electric field and induced magnetic field due to Rashba type SOI are in-plane fields. By intuition, spin precession axis is in-plain and the expectation value of σz should be zero. But Fig. 5.3 does not agree and show the evidence of spin flip. It can be interpreted by a semi-classical picture. In conventional NMR configuration and in rotating frame with frequency which is the same with external rotating magnetic field Brot, the spin performs precession about B instead of static magnetic field B , see Fig. 5.8.
Figure 5.8: The resonance occurs and spin flip that can be interpreted by rotating frame with semi-classical picture. Where BSis static magnetic field and Brotis rotating magnetic field. The dashed line indicates spin and precession.
elastic scattering process. If incident wave comes from branch σ then it is impossible to t±ω−σ 6= 0 unless the spin flip due to EDSR.
Future work
In the future, two incident waves coming from two different branches condition will be considered and the one sideband approximation will be extend to exact numerical method mentioned in section 4.2 in order to simulate realistic physics and explore new possible to enhance effects.
The analysis and physical interpretation of fano profile shown in the Fig. 5.6 are in progress.
Appendix A
Review of nuclear magnetic resonance and sideband
Let we review nuclear magnetic resonance(NMR). The configuration of fields of NMR is composed of B1 static magnetic field along z axis and B2(t) rotating field in x − y plane.
The time-dependent Schr¨odinger equation of this configuration is described as
Hχ = i∂
∂tχ (A.1)
where H = B2cos(ωt)σx + B2sin(ωt)σy + B1σz and χ is a spin state. The analytically general solution can be solved as following
χ(t) =
exp(−iB1t)£
c1exp¡i
2β+t¢
+ c2exp¡i
2β−t¢¤
−exp(iB1t) exp(−iβt) 2B2
£β+c1exp¡i
2β+t¢
+ β−c2exp¡i
2β−t¢¤
(A.2)
where c1 and c2 are two undeterminate coefficients which can be determinanted by an initial condition, β± = β ±p
β2+ 4B22, β = 2B1 − ω and ω is angular frequency of B2
rotating field.
If the initial state of χ(0) = [1 0]T is employed then the evaluation in time of proba-bility of the spin-down state [0 1]T, called P↓(t), is
0 5 10 15 20 25 30 35 40 45 50
Probability of spin down
ω=2
Figure A.1: When ω = 2(unit of energy), the resonance occurs, i.e the max of probability of spin down is one. symmetrically, the max of probability of spin down decay with the fluctuation of the resonance frequency ω = 2.
P↓(t) = B22
The numerical result of Eq. (A.3) is shown in Fig. A.1. The strength of rotating field, B2, is relative about the coupling between two states of spin up and spin down. In other word, The ability of spin flip resonance is proportional to the value of B2. For example, the value of B2 is more small, the full width at half maximum of resonance profile is shaper. It indicates that the frequency of B (t) is necessary to closely match the Lamor
SIDEBAND
0 0.5 1 1.5 2 2.5 3 3.5 4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Max probability pf spin down
B2=0.1 B2=0.5 B2=0.25
B
1=1
Frequency ω
Figure A.2: The strength of rotating field, B2, is proportional to the coupling between two states of spin up and spin down. The full width at half maximum of resonance profile is larger when the strength of rotating field, B2, is larger.
Figure A.3: The relation between NMR and sideband.
The relation between NMR and sideband is introduced in Fig. A.3. First, The sideband of rotating field B2 offers a energy ~ω is not match with the energy difference between the two states of spin up and spin down. The spin flip happens but the probability is small, see in the Fig. A.1. On the other hand, the spin resonance occurs while the sideband match the state with large density of state. Second, although the sideband does not closely match the spin-down state the probability is enough large unless the coupling between the two spin states is strength enough, i.e. the value of B2 is large enough, see Fig. A.2.
Appendix B
Derivation of particle current
We start the derivation from the Hamiltonian Eq. (B.1) which just is considered along longitudinal direction without subband mixing term αi∂y∂σx
µ
Taking dagger to Eq. (B.1) and product −ψ(x, t) on right hand side of it, we can obtain
The summation of Eq. (B.2) and Eq. (B.3) is
∂
∂t
£ψ†(x, t)ψ(x, t)¤
= − ∂
∂x
·
ψ†(x, t)1 i
∂
∂xψ(x, t) −1 i
∂
∂xψ†(x, t)ψ(x, t) − αψ†(x, t)σyψ(x, t)
¸ (B.4)
According to continuity equation,
∂
∂tρ + ∂
∂xJ = 0
the L.H.S. of Eq. (B.4) is regarded as time derivative of particle density and R.H.S of Eq. (B.4) is regarded as spatial derivative of particle current with respect to x. The compact format of particle current is shown as following
J(x, t) = ψ†(x, t) µ
−i ∂
∂x −α 2σy
¶
ψ(x, t) + c.c. (B.5)
where the abbreviation, c.c., is standing for complex conjugate of the former terms.
QED
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