Orbital and spin dynamics of intraband electrons in quantum rings driven by twisted light

doi:10.1364/OE.19.026733

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Orbital and spin dynamics of intraband electrons in quantum rings driven by twisted light

G. F. Quinteiro, P. I. Tamborenea, and J. Berakdar »View Author Affiliations

Optics Express, Vol. 19, Issue 27, pp. 26733-26741 (2011)
http://dx.doi.org/10.1364/OE.19.026733

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▸ Article Outline
– Abstract
1. Introduction
2. Twisted light and quantum rings
3. Interaction between twisted light and quantum rings
4. Evolution of single-particle states
5. Conclusions
– References and links

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Abstract

We theoretically investigate the effect that twisted light has on the orbital and spin dynamics of electrons in quantum rings possessing sizable Rashba spin-orbit interaction. The system Hamiltonian for such a strongly inhomogeneous light field exhibits terms which induce both spin-conserving and spin-flip processes. We analyze the dynamics in terms of the perturbation introduced by a weak light field on the Rasha electronic states, and describe the effects that the orbital angular momentum as well as the inhomogeneous character of the beam have on the orbital and the spin dynamics.

1. Introduction

In a recent series of articles we studied the interaction of twisted light (TL) [1

D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Academic Press, 2008). [PubMed]

] with semiconductor nanostructures, and showed that interesting new effects are produced because of the orbital angular momentum (OAM) and the inhomogeneous character of the TL beam. In particular, we demonstrated that circulating electric currents are generated in interband transitions in semiconductor quantum rings (QR) [2

G. F. Quinteiro and J. Berakdar, “Electric currents induced by twisted light in quantum rings,” Opt. Express 17, 20465 (2009). [CrossRef] [PubMed]

], and that new electronic transitions become optically allowed in semiconductor quantum dots (QD) [3

G. F. Quinteiro and P. I. Tamborenea, “Electronic transitions in disk-shaped quantum dots induced by twisted light,” Phys. Rev. B 79, 155450 (2009). [CrossRef]

, 4

G. F. Quinteiro, A. O. Lucero, and P. I. Tamborenea, “Electronic transitions in quantum dots and rings induced by inhomogeneous off-centered light beams,” J. Phys.: Condens. Matter 22, 505802 (2010). [CrossRef]

]. Nevertheless, the topic of spin dynamics driven by TL has not been, to the best of our knowledge, addressed so far in the literature. Detailed descriptions of spin dynamics are of the utmost importance in condensed matter physics, from first principles to applications in spintronics. At the same time, the control of nanosystems by optical means is a very active field of research, for it proves to be an efficient, fast technique to manipulate quantum states.

In this article, we report our theoretical predictions on the orbital and the spin dynamics of conduction-band electrons in QR illuminated with TL, when the Rashba spin-orbit interaction (SOI) in the QR is taken into account. By comparison with the canonical case of irradiation with plane waves, we show that a variety of new effects arise, some connected to the orbital angular momentum of the TL beam, and others to its inhomogeneous character.

In Sec. 2 the theoretical model is introduced, the matrix elements of the TL-QR interaction are obtained in Sec. 3, Sec. 4 studies the quantum evolution of the photoexcited QR, and conclusions are presented in Sec. 5.

2. Twisted light and quantum rings

The system under investigation is a phase-coherent mesoscopic quantum ring of radius a, thickness d, and height h (with a ≫ d,h), illuminated at normal incidence (z-axis) by THz twisted light radiation. We consider the situation where the QR and the TL symmetry axes coincide, which we believe poses no technological difficulty since large QRs are currently manufactured [5

A. Fuhrer, S. Luscher, T. Ihn, T. Heinzel, K. Ensslin, W. Wegscheider, and M. Bichler, “Energy spectra and broken symmetry in quantum rings,” Nature 413, 822–825 (2001). [CrossRef] [PubMed]

]; otherwise, our previous findings on off-centered beams [4

] may serve to clarify experimental results. The electronic states of the QR are described in the envelope-function approximation, and will be given after the Hamiltonian is introduced. While studying spin and orbital dynamics, we will assume that there is one electron occupying initially (t = 0⁻) a conduction-band QR eigenstate. Experimentally, if phase-coherence is desired, net charge in the conduction-band states is injected via modulation doping, in order to reduce impurity scattering. Electrons can also be promoted to the conduction-band states by means of photo-excitation with ultrashort optical pulses, although in this case their lifetime prior to recombination is of the order of nanoseconds. THz TL radiation will then induce intraband transitions.

The TL beam is represented by its vector potential in cylindrical coordinates (keeping only its transverse components) [6

M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. Dávila Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. 89, 143601 (2002). [CrossRef] [PubMed]

]

\begin{array}{l} A (r, t) = ɛ_{σ} F_{l} (q_{r} r) e^{i (q_{z} z - ω t)} e^{i l ϕ} + c . c . \\ = A^{(+)} (r, t) + A^{(-)} (r, t), \end{array}

(1)

with the polarization vectors given by ɛ_σ = x̂+σiŷ = e^σiϕ (r̂ + σiϕ̂), σ = ±1, and c.c. denoting the complex conjugate. The radial function F_l(q_rr) is left, for the moment, unspecified. Disregarding the longitudinal component of A is justified in usual experimental conditions, in which q_z ≫ q_r.

The Hamiltonian of the system of QR plus TL, including the Rashba spin-orbit coupling in the QR [7

Z.-G. Zhu and J. Berakdar, “Photoinduced nonequilibrium spin and charge polarization in quantum rings,” Phys. Rev. B 77, 235438 (2008). [CrossRef]

], is decomposed into

H = H_{SOI} + H_{1},

with

H_{SOI} = \frac{{\hat{p}}^{2}}{2 m_{e}^{*}} + V (r) + \frac{α_{R}}{\bar{h}} {[\hat{σ} \times \hat{p}]}_{z}

(2)

H_{1} = - \frac{q}{m_{e}} A (r, t) \cdot \hat{p} - \frac{q α_{R}}{\bar{h}} {[\hat{σ} \times A (r, t)]}_{z}

(3)

where q = −e and

m_{e} (m_{e}^{*})

are the charge and mass (effective mass) of the electron, V (r) the QR confinement potential, and σ̂ is the vector of Pauli matrices. The Rashba coupling constant is α_R. The perturbation H₁ introduced by the light beam has been deduced from minimal coupling up to first order in A(r,t). We leave out the Dresselhaus spin-orbit coupling since, as shown in Ref. [8

C. L. Romano, S. E. Ulloa, and P. I. Tamborenea, “Level structure and spin-orbit effects in quasi-one-dimensional semiconductor nanostructures,” Phys. Rev. B 71, 035336 (2005). [CrossRef]

], in quasi-one-dimensional structures it can be eliminated by an adequate choice of the lateral confinement.

In principle, Eq. (3) should include a Zeeman term coming from the magnetic component of the TL field. A simple calculation of ∇×A(r,t) shows that this magnetic field has transverse as well as longitudinal components, which are proportional to q_z and q_r, respectively. Therefore, both are small compared to the other terms in the Hamiltonian and can be safely neglected.

Eq. (3) exhibits the unfamiliar term (−qα_R/h̄)[σ̂ × A(r,t)]_z coupling linearly the light electric field to the spin operators via the Rashba-type spin-orbit coupling. Note that the final expression of the Hamiltonian in Ref. [7

Z.-G. Zhu and J. Berakdar, “Photoinduced nonequilibrium spin and charge polarization in quantum rings,” Phys. Rev. B 77, 235438 (2008). [CrossRef]

] does not contain an analogous term. That simplification was possible due to the applicability of the dipole approximation in that work. The Göppert-Mayer transformation, leading to the dipole-moment Hamiltonian, is not possible in our problem due to the inhomogeneous nature of the TL beam. Therefore, we continue our analysis within the Coulomb gauge which leads to the Hamiltonian of Eqs. (2,3).

As has been shown in previous works [9

J. Splettstoesser, M. Governale, and U. Zülicke, “Persistent current in ballistic mesoscopic rings with Rashba spin-orbit coupling,” Phys. Rev. B 68, 165341 (2003). [CrossRef]

, 7

Z.-G. Zhu and J. Berakdar, “Photoinduced nonequilibrium spin and charge polarization in quantum rings,” Phys. Rev. B 77, 235438 (2008). [CrossRef]

] the Hamiltonian H_SOI can be exactly diagonalized, having envelope eigenfunctions Φ_ns(r) = ψ_ns(φ)R(r)Z(z) with

ψ_{n s} (φ) = \frac{1}{\sqrt{2 π}} e^{i (n + 1 / 2) φ} ν_{s} (γ, φ),

(4)

with the z-projection of the total angular momentum n + 1/2, angular coordinate φ, and angle-dependent spinor

ν_{1} (γ, φ) = (\begin{matrix} cos (γ / 2) e^{- i φ / 2} \\ sin (γ / 2) e^{i φ / 2} \end{matrix})

(5)

ν_{- 1} (γ, φ) = (\begin{matrix} - sin (γ / 2) e^{- i φ / 2} \\ cos (γ / 2) e^{i φ / 2} \end{matrix}),

(6)

where γ is the angle between the z-axis and the spin; this angle depends on the quantum number n, but for negligible Zeeman energy, it becomes independent of n: tanγ = −ω_R/ω₀, with h̄ω_R = 2α_R/a and

\bar{h} ω_{0} = {\bar{h}}^{2} / (m_{e}^{*} a^{2})

. The energies of the states are

ɛ_{n s} = \frac{\bar{h} ω_{0}}{2} [{(n - x_{s})}^{2} - \frac{Q_{R}^{2}}{4}],

(7)

with x_s = −(1 − sw)/2,

w = \sqrt{1 + Q_{R}^{2}} = 1 / cos (γ)

, and s = ±1 for respectively spin up and down in the (φ-dependent) local frame. (The notation is summarized in Table 1.)

Table 1 Notation

tanγ = −ω_R/ω₀

h̄ω_R = 2α_R/a

\bar{h} ω_{0} = {\bar{h}}^{2} / (m_{e}^{*} a^{2})

x_s = −(1 – sw)/2

w = \sqrt{1 + Q_{R}^{2}} = 1 / cos (γ)

3. Interaction between twisted light and quantum rings

We now obtain the matrix elements of the light-matter interaction given by Eq. (3), in the case of TL beams (Eq. (1)) applied on the QR described in the previous Section. The matrix elements are thus calculated in the basis set of the QR states, Eq. (4), and we will treat separately the two terms of H₁ = H₁₁ + H₁₂ given in Eq. (3). In Sec. 4 we will employ the matrix elements obtained here in order to analyze the orbital and spin dynamics of the QR in time-dependent perturbation theory, considering H₁ as a perturbation to H_SOI.

3.1. Hamiltonian H₁₁ = −(q/m_e)A(r,t) ·p̂

As customary, we separate the Hamiltonian into positive and negative parts

\begin{array}{l} H_{11} & = & H_{11}^{(+)} + H_{11}^{(-)} \\ = & - \frac{q}{m_{e}} [A^{(+)} (r, t) + A^{(-)} (r, t)] \cdot \hat{p} . \end{array}

(8)

Let us calculate the matrix element

〈 n^{'} s^{'} | H_{11}^{(+)} | n s 〉

of the positive term, between initial |ns〉 and final 〈n′s′| states, where 〈r|ns〉 = Φ_ns(r)u_λ (r), u_λ (r) is the microscopic (with lattice periodicity) wave-function. A series of simplifications are possible thanks to the assumption that all processes occur in the same subband: i) p̂ acting on the microscopic wave-function yields a vanishing matrix element; ii) p̂ acting on the envelope wave functions Z(z) and R(r) would induce transitions between different z/r-subbands, and thus are disregarded. In addition, since the QR is thin, the smooth function F_l(q_rr) can be taken as constant at the value of r = a and pulled out of the matrix element. Finally, we separate the integral over the whole QR into an integral on the cell and a sum (that can be taken as an integral) over all cells [10

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors , 4th ed. (World Scientific Publishing Co., 2004).

]. After these simplifications, we are left with the element

\begin{array}{l} 〈 n^{'} s^{'} | H_{11}^{(+)} | n s 〉 = - σ \frac{1}{\sqrt{2}} \frac{\bar{h} q}{m_{e}} e^{i q_{z} z_{0}} F_{l} (q_{r} a) e^{- i ω t} \\ \times \int_{V} d^{3} r Φ_{n^{'} s^{'}}^{*} (r) e^{i (l + σ) φ} [\frac{1}{r} \partial_{φ} Φ_{n s} (r)] . \end{array}

(9)

Using Eq. (4)

\partial_{φ} ψ_{n s} (φ) = i (n + 1 / 2) ψ_{n s} (φ) - \frac{i s}{2 \sqrt{2 π}} e^{i (n + 1 / 2) φ} ν_{s} (- γ, φ) .

(10)

Then,

〈 n^{'} s^{'} | H_{11}^{(+)} | n s 〉 = ξ_{σ} e^{- i ω t} δ_{l + σ, n^{'} - n} [δ_{s, s^{'}} (n + 1 / 2 - s cos γ) + δ_{s, - s^{'}} sin γ],

(11)

where

ξ_{σ} = - \frac{i σ}{\sqrt{2}} \frac{\bar{h} q}{m_{e} a} e^{i q_{z} z_{0}} F_{l} (q_{r} a)

. The product of spinors does not depend on φ, and was pulled out of the integral. For the negative part

H_{11}^{(-)}

we simply use

〈 n^{'} s^{'} | H_{11}^{(-)} | n s 〉 = {〈 n s | H_{11}^{(+)} | n^{'} s^{'} 〉}^{*}

. We see that the Hamiltonian H₁₁, though not a spin-orbit interaction, produces spin flips due to the last term of Eq. (11). This is possible because H₁₁ acts upon eigenstates of the Rashba Hamiltonian H_SOI, which are not eigenstates of σ̂_z. In the case of weak Rashba effect, the spin-flip term is proportional to the angle γ ∝ α_R; for vanishing spin-orbit coupling, we recover the spin-conserving interaction.

3.2. Hamiltonian H₁₂ = −(qα_R/h̄)[σ̂ × A(r,t)]_z

As mentioned previously,

H_{12} = - \frac{q α_{R}}{\bar{h}} {[\hat{σ} \times A (r, t)]}_{z}

(12)

appears as a consequence of the inhomogeneous nature of the light field and, in contrast to H₁₁, is a spin-light coupling which is linear in the electric field strength. This dependence stems from the functional form of the Rashba-type spin-orbit coupling.

Again, for the positive part of the vector potential

\begin{array}{l} 〈 n^{'} s^{'} | H_{12}^{(+)} | n s 〉 = - \frac{q α_{R}}{2 π \sqrt{2} \bar{h}} e^{i q_{z} z_{0}} F_{l} (q_{r} a) e^{- i ω t} \\ \times \int_{V} d φ e^{- i (n^{'} - n - l) φ} ν_{s^{'}}^{†} (γ, φ) (σ i {\hat{σ}}_{x} - {\hat{σ}}_{y}) ν_{s} (γ, φ) . \end{array}

The products

ν_{s^{'}}^{†} (γ, φ) (σ i {\hat{σ}}_{x} - {\hat{σ}}_{y}) ν_{s} (γ, φ)

can be evaluated, and brought into the matrix form

i σ e^{i σ φ} (\begin{matrix} sin γ & cos γ + σ \\ cos γ - σ & - sin γ \end{matrix}) = i σ e^{i σ φ} M_{σ} .

(13)

Note how the polarization of the light couples to the spin degree of freedom, through the off-diagonal terms. With these expressions, the matrix element reads

〈 n^{'} s^{'} | H_{12}^{(+)} | n s 〉 = η_{σ} e^{- i ω t} δ_{n^{'} - n, l + σ} M_{σ s^{'} s},

(14)

where

η_{σ} = - i σ \frac{q α_{R}}{\bar{h}} e^{i q_{z} z_{0}} F_{l} (q_{r} a)

3.3. The total perturbation H₁

A compact matrix form of the total perturbation, in a representation where

ν_{1}^{†} = (1, 0)

and

ν_{- 1}^{†} = (0, 1)

, is

\begin{array}{l} H_{1, n^{'} n} = e^{- i ω t} δ_{n^{'} - n, l + σ} \\ \times [ξ_{σ} (\begin{matrix} n + \frac{1}{2} - cos γ & sin γ \\ sin γ & n + \frac{1}{2} + cos γ \end{matrix}) + η_{σ} (\begin{matrix} sin γ & cos γ + σ \\ cos γ - σ & - sin γ \end{matrix})] + H . c ., \end{array}

(15)

where the H.c. implies not only the conjugate and transposition of the spin matrix, but also the transposition n ↔ n′. We remind the reader that the terms proportional to ξ_σ (η_σ) come from H₁₁ (H₁₂).

4. Evolution of single-particle states

We use standard time-dependent perturbation theory, for the case of a harmonic perturbation. The general formula for the wave function is

Ψ (r, t) = \sum a_{n s} (t) e^{- i ɛ_{n s} t / \bar{h}} ψ_{n s} (φ)

(16)

where the coefficients are given by

\begin{array}{l} a_{n s}^{(1)} (t) = \frac{1}{\bar{h}} 〈 n s | H_{1}^{(+)} (0) | n_{0} s_{0} 〉 \frac{1 - e^{i (ω_{f i} - ω) t}}{ω_{f i} - ω} \\ + \frac{1}{\bar{h}} 〈 n s | H_{1}^{(-)} (0) | n_{0} s_{0} 〉 \frac{1 - e^{i (ω_{f i} + ω) t}}{ω_{f i} + ω} \end{array}

(17)

where {n₀, s₀} are the quantum numbers of the initial state, and h̄ω_fi = ɛ_ns – ɛ_n₀s₀.

In contrast to inter-band transitions, where the Rotating Wave Approximation (RWA) is usually applied in expressions such as Eq. (17)—e.g. by neglecting the term with ω_fi + ω for absorption—, in intra-band transitions this approximation is not justified. In fact, the resonant |ω_fi| – ω and non-resonant |ω_fi| + ω terms are of the same order of magnitude, and should both in principle be included.

Some comments on the topic of gauge invariance as related to twisted light are in order. As previously mentioned, we work in the Coulomb gauge, where the vector potential appears as the main quantity, in contrast to other treatments of light-matter interaction based on the light’s electric and magnetic fields. In the standard—and simplest—case of plane waves, the Göppert-Mayer transformation on a fully homogeneous field yields a Hamiltonian whose p²/2m is gauge invariant, thus representing a physical quantity, i.e. the kinetic energy; in contrast, the same term in the Coulomb gauge is not, by itself, gauge invariant. As a consequence, for example, the eigenvectors of p²/2m + V(r), though the same mathematical functions, do not have the same meaning in both gauges. In general, the correspondence between the predictions obtained in both gauges is achieved via a transformation of the wave function. In the special case of the calculation of transition probabilities, these coincide in both gauges without the need of a transformation of the wave function if the vector potential is zero at the initial and final times, as explicitly demonstrated by Lamb et al. [11

W. E. Lamb, R. R. Schlicher, and M. O. Scully, “Matter-field interaction in atomic physics and quantum optics,” Phys. Rev. A 36, 2763–2772 (1987). [CrossRef] [PubMed]

]. In addition, the discrepancies between the results of a E(t)·d and a A(t)·p approach are reported to be smaller the closer to resonance excitation one is [11

W. E. Lamb, R. R. Schlicher, and M. O. Scully, “Matter-field interaction in atomic physics and quantum optics,” Phys. Rev. A 36, 2763–2772 (1987). [CrossRef] [PubMed]

, 12

K. Rzazewski and R. W. Boyd, “Equivalence of interaction Hamiltonians in the eElectric dipole approximation,” J. Mod. Opt. 51, 1137–1147 (2004).

]. This suggests that consideration of the gauge invariance problem is more important in our present work, where we deal with intraband excitation. Furthermore, the case of twisted light is a more delicate one, since the inhomogeneous character of the beam precludes our use of the Göppert-Mayer transformation. In any case, our results, obtained within a given gauge, are perfectly valid even though they may require further work in order to be used to interpret particular experimental situations.

4.1. General considerations

For arbitrary OAM l and polarization σ of the light, Eq. (15) shows that transitions to nearby and distant states are possible. We mention that a particular situation happens for the value l = −σ: the total angular momentum of the electron is unchanged by the light.

In general, we can say that in this system there will be two different time scales present in the quantum evolution: one associated with the evolution of the orbital, and another one with the spin, degrees of freedom. In the limit of zero Rashba SOI, there would be no spin evolution. By the same token, since the SOI is in general a weak interaction, the spin evolution will be slow compared to the orbital one.

The rate of spin conversion achieved by TL irradiation is not simply proportional to the OAM of the beam, but it is rather related to the beam function F_l(q_rr) for the radial profile via the constants ξ and η. The orbital motion is dictated by the constant ξ, while the spin evolution (spin-flip) by both constants ξ and η.

The case of plane waves can be deduced from our formalism by taking l = 0. Although in this case a gauge transformation could be made towards a E(t) · d Hamiltonian, our present Hamiltonian can still be used. Then, Eq. (15) shows that spin flips occur even when l = 0 due to the presence of the Rashba SOI.

In what follows we consider in more detail the evolution under small spin-orbit coupling, since it is the usual situation in real materials.

4.2. Small spin-orbit coupling

The condition ω_R ≪ ω₀ allows for the simplification of the Hamiltonian Eq. (15) to

\begin{array}{l} H_{1, n^{'} n} = e^{- i ω t} δ_{n^{'} - n, l + σ} \\ \times [ξ σ (\begin{matrix} n - \frac{1}{2} & γ \\ γ & n + \frac{3}{2} \end{matrix}) + η_{σ} (\begin{matrix} 0 & 1 + σ \\ 1 - σ & 0 \end{matrix})] + H . c . \end{array}

(18)

We exemplify by studying the evolution of a particle initially in state |n₀, 1〉. A TL field, having l = 1 and σ = 1, is turned on at time t = 0⁺ and turned off at time t = T. Then, the wave function is that of Eq. (16) with coefficients

\begin{array}{l} a_{n_{0} - 2, - 1}^{(1)} (T) & = & \frac{1 - e^{i (ω_{f i} + ω) T}}{\bar{h} (ω_{f i} + ω)} (ξ_{1}^{*} γ + 2 η_{1}^{*}) \\ a_{n_{0} + 2, - 1}^{(1)} (T) & = & \frac{1 - e^{i (ω_{f i} - ω) T}}{\bar{h} (ω_{f i} - ω)} ξ_{1} γ \\ a_{n_{0} + 2, 1}^{(1)} (T) & = & \frac{1 - e^{i (ω_{f i} - ω) T}}{\bar{h} (ω_{f i} - ω)} ξ_{1} (n_{0} - 1 / 2) \\ a_{n_{0} - 2, 1}^{(1)} (T) & = & \frac{1 - e^{i (ω_{f i} + ω) T}}{\bar{h} (ω_{f i} + ω)} ξ_{1}^{*} (n_{0} - 5 / 2), \end{array}

(19)

where ω_fi is understood as having the right indices for each case; while in all cases i = {n₀, 1}, for example, for the third line of Eq. (19), f = {n₀ + 2,1}.

A pictorial representation of the process is given in Fig. 1. The initial state evolves into a superposition of neighboring states having the same and the opposite spin states and differing in their total angular momentum by l + σ = 2. The perturbation H₁₁ is responsible for spin-conserving and spin-flip terms, while H₁₂ causes only spin flips. As seen from Eq. (18), the ratio

η / (ξ γ) = 1 / \sqrt{2}

determines the relative contribution for spin conversion between H₁₁ and H₁₂. On the other hand, the rate of change of orbital and spin degrees of freedom can be estimated from the corresponding matrix element of the interaction Hamiltonian. The rates are then estimated by Γ_s = ξγ/h̄ for the spin and Γ_o = ξ/h̄ for the orbital degrees of freedom, as can also be easily seen from the expressions for

a_{n_{0} + 2, - 1}^{(1)} (T)

and

a_{n_{0} + 2, 1}^{(1)} (T)

in Eq. (19). In the case of a GaAs QR of radius a = 10⁻⁶ m having α_R = 10⁻¹⁴ V m and pulsed laser parameters typical for experiments: power 10⁶ J Hz, repetition rate 10⁶ Hz and wavelength 10⁻⁵ m, we obtain Γ_s = 0.1 ps⁻¹ and Γ_o = 0.5 ps⁻¹.

Fig. 1 Pictorial representation of the electronic bands and a transition induced by TL having OAM l = 1 and σ = 1. According to time-dependent perturbation theory: an electron initially in the state {n₀, ↑} evolves into a superposition of neighboring states having the same and the opposite spin states. Transitions, indicated by enclosed numbers (blue), correspond to the coefficients: (1)

a_{n_{0} - 2, - 1}^{(1)} (t)

, (2)

a_{n_{0} + 2, - 1}^{(1)} (t)

, (3)

a_{n_{0} + 2, 1}^{(1)} (t)

, (4)

a_{n_{0} - 2, 1}^{(1)} (t)

, of Eqs. (19).

One can notice that, even without applying the RWA, only one of the terms in Eq. (17) survives for each final state f. This is due to the fact that we keep track of the selection rule for the momentum conservation (encoded in the matrix element of the interaction Hamiltonian), which is usually missing in the standard treatment of plane-waves excitation. In fact, the four terms are in Eq. (19) correspond to resonant processes, since ω_fi < 0 for transitions # 1 and # 4. Moreover, the processes are real, in the sense that an electronic transition to a higher (lower) energy state, occurs simultaneosly with an annihilation (creation) of a photon as signaled by the negative (positive) sign of the complex exponential exp(−iωT) (exp(iωT))—the exponential stems from the positive (negative) part of the vector potential Eq. (1), which in a second-quantization formalism, will be accompanied by an annihilation (creation) photon operator.

Let us imagine a different situation from that illustrated in Fig. 1, that of an initial state n₀ < 0, i.e. lying to the left of the band minimum. The selection rule on the total angular momentum δ_n–n₀,2 in the first term of Eq. (18) tells us that this is a transition to a state n such that n > n₀, to the right of n₀. Since this term is H⁽⁺⁾, it corresponds to a photon annihilation; however, the transition leaves the electron in a lower energy state, giving rise to a virtual process. Virtual processes serve as intermediate states in a sequence of transitions ending in energy conservation, or they may happen in a time scale compatible with the Heisenberg’s uncertainty principle. In general, the situation is somewhat complicated, and the type of transition depends on the values of n₀, σ and l. Using large values of the OAM can create a large difference between transitions, and one effectively may neglect some of them: non-resonant terms can be disregarded.

4.3. Induced polarization and current

Since both resonant and non-resonant terms ought to be considered, the current in intraband transitions is much smaller than expected for interband transitions. This is easily seen for the case of TL acting upon a QR having no SOI. When the SOI is not present, we can replace the wave functions by ψ_m(φ) = e^imφ—for a discussion on the difference between this and Eq. (4) see Ref. [7

Z.-G. Zhu and J. Berakdar, “Photoinduced nonequilibrium spin and charge polarization in quantum rings,” Phys. Rev. B 77, 235438 (2008). [CrossRef]

]. It is a simple matter to derive the matrix elements for the interaction with TL

〈 m^{'} | H_{11}^{(+)} | m 〉 = ξ_{σ} e^{- i ω t} δ_{l + σ, m^{'} - m} m .

(20)

To exemplify, let us take l = 1 and σ = 1, and study the evolution from an initial state |m₀〉. Then, the only non-vanishing terms are {a_m₀, a_m₀−2, a_m₀+2}. A calculation of the current density, after the light is turned off at t = T, yields, after integrating the current density in the whole ring

J ≃ \frac{q \bar{h}}{m} {m_{0} + 2 {| ξ_{-} |}^{2} D^{2} m_{0}^{3}},

(21)

where D is a constant that depends on the sign of m₀, but for intraband transitions we expect D(−|m₀|) ≃ D(|m₀|). The perturbation increases the rotation in the same direction the particle was originally moving, either right or left. Then, we conclude that there is no significant current for a balanced population of electrons in the QR.

5. Conclusions

We have studied the dynamics of electrons confined to semiconductor-based quantum rings under irradiation by twisted light, in the presence of Rashba spin-orbit coupling. We worked out the matrix elements of the light-matter interaction in the Coulomb gauge on the Rashba states of the quantum ring. Because the unperturbed states encode the SOI, the common light-matter interaction p̂ · A, though not a spin-orbit coupling, can produce spin flips. We pointed out that the dynamics of orbital motion and spin exhibit two distinct time scales, and we have estimated these two in the case of small spin-orbit interaction. We point out that, for strong spin-orbit interaction as reported by Zhu et al [7

Z.-G. Zhu and J. Berakdar, “Photoinduced nonequilibrium spin and charge polarization in quantum rings,” Phys. Rev. B 77, 235438 (2008). [CrossRef]

], the spin dynamics can be faster than the orbital one. In view of the impossibility to apply the Rotating Wave Approximation, we studied the differences between optical excitation of inter- and intraband transitions. Thanks to the freedom to choose the value of OAM, we observe that resonant and non-resonant terms may be manipulated in order to gain control of specific transitions.

Acknowledgments

We acknowledge support from the Cooperation Program ANPCyT–Max-Planck Society, through grant PICT-2006-02134, and from the University of Buenos Aires, through grant UBA-CyT X495. G.F.Q. gratefully acknowledges the financial support of the DAAD.

References and links

1.	D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Academic Press, 2008). [PubMed]
2.	G. F. Quinteiro and J. Berakdar, “Electric currents induced by twisted light in quantum rings,” Opt. Express 17, 20465 (2009). [CrossRef] [PubMed]
3.	G. F. Quinteiro and P. I. Tamborenea, “Electronic transitions in disk-shaped quantum dots induced by twisted light,” Phys. Rev. B 79, 155450 (2009). [CrossRef]
4.	G. F. Quinteiro, A. O. Lucero, and P. I. Tamborenea, “Electronic transitions in quantum dots and rings induced by inhomogeneous off-centered light beams,” J. Phys.: Condens. Matter 22, 505802 (2010). [CrossRef]
5.	A. Fuhrer, S. Luscher, T. Ihn, T. Heinzel, K. Ensslin, W. Wegscheider, and M. Bichler, “Energy spectra and broken symmetry in quantum rings,” Nature 413, 822–825 (2001). [CrossRef] [PubMed]
6.	M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. Dávila Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. 89, 143601 (2002). [CrossRef] [PubMed]
7.	Z.-G. Zhu and J. Berakdar, “Photoinduced nonequilibrium spin and charge polarization in quantum rings,” Phys. Rev. B 77, 235438 (2008). [CrossRef]
8.	C. L. Romano, S. E. Ulloa, and P. I. Tamborenea, “Level structure and spin-orbit effects in quasi-one-dimensional semiconductor nanostructures,” Phys. Rev. B 71, 035336 (2005). [CrossRef]
9.	J. Splettstoesser, M. Governale, and U. Zülicke, “Persistent current in ballistic mesoscopic rings with Rashba spin-orbit coupling,” Phys. Rev. B 68, 165341 (2003). [CrossRef]
10.	H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors , 4th ed. (World Scientific Publishing Co., 2004).
11.	W. E. Lamb, R. R. Schlicher, and M. O. Scully, “Matter-field interaction in atomic physics and quantum optics,” Phys. Rev. A 36, 2763–2772 (1987). [CrossRef] [PubMed]
12.	K. Rzazewski and R. W. Boyd, “Equivalence of interaction Hamiltonians in the eElectric dipole approximation,” J. Mod. Opt. 51, 1137–1147 (2004).

OCIS Codes
(250.0250) Optoelectronics : Optoelectronics
(320.7130) Ultrafast optics : Ultrafast processes in condensed matter, including semiconductors

ToC Category:
Optoelectronics

History
Original Manuscript: September 2, 2011
Revised Manuscript: October 26, 2011
Manuscript Accepted: November 3, 2011
Published: December 14, 2011

Citation
G. F. Quinteiro, P. I. Tamborenea, and J. Berakdar, "Orbital and spin dynamics of intraband electrons in quantum rings driven by twisted light," Opt. Express 19, 26733-26741 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26733

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References

D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Academic Press, 2008). [PubMed]
G. F. Quinteiro and J. Berakdar, “Electric currents induced by twisted light in quantum rings,” Opt. Express17, 20465 (2009). [CrossRef] [PubMed]
G. F. Quinteiro and P. I. Tamborenea, “Electronic transitions in disk-shaped quantum dots induced by twisted light,” Phys. Rev. B79, 155450 (2009). [CrossRef]
G. F. Quinteiro, A. O. Lucero, and P. I. Tamborenea, “Electronic transitions in quantum dots and rings induced by inhomogeneous off-centered light beams,” J. Phys.: Condens. Matter22, 505802 (2010). [CrossRef]
A. Fuhrer, S. Luscher, T. Ihn, T. Heinzel, K. Ensslin, W. Wegscheider, and M. Bichler, “Energy spectra and broken symmetry in quantum rings,” Nature413, 822–825 (2001). [CrossRef] [PubMed]
M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. Dávila Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett.89, 143601 (2002). [CrossRef] [PubMed]
Z.-G. Zhu and J. Berakdar, “Photoinduced nonequilibrium spin and charge polarization in quantum rings,” Phys. Rev. B77, 235438 (2008). [CrossRef]
C. L. Romano, S. E. Ulloa, and P. I. Tamborenea, “Level structure and spin-orbit effects in quasi-one-dimensional semiconductor nanostructures,” Phys. Rev. B71, 035336 (2005). [CrossRef]
J. Splettstoesser, M. Governale, and U. Zülicke, “Persistent current in ballistic mesoscopic rings with Rashba spin-orbit coupling,” Phys. Rev. B68, 165341 (2003). [CrossRef]
H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th ed. (World Scientific Publishing Co., 2004).
W. E. Lamb, R. R. Schlicher, and M. O. Scully, “Matter-field interaction in atomic physics and quantum optics,” Phys. Rev. A36, 2763–2772 (1987). [CrossRef] [PubMed]
K. Rzazewski and R. W. Boyd, “Equivalence of interaction Hamiltonians in the eElectric dipole approximation,” J. Mod. Opt.51, 1137–1147 (2004).

Andrews, D. L.
Babiker, M.
Bennett, C. R.
Berakdar, J.
Bichler, M.
Boyd, R. W.
Dávila Romero, L. C.
Ensslin, K.
Fuhrer, A.
Governale, M.
Haug, H.
Heinzel, T.
Ihn, T.
Koch, S. W.
Lamb, W. E.
Lucero, A. O.
Luscher, S.
Quinteiro, G. F.
Romano, C. L.
Rzazewski, K.
Schlicher, R. R.
Scully, M. O.
Splettstoesser, J.
Tamborenea, P. I.
Ulloa, S. E.
Wegscheider, W.
Zhu, Z.-G.
Zülicke, U.

J. Mod. Opt.

K. Rzazewski and R. W. Boyd, “Equivalence of interaction Hamiltonians in the eElectric dipole approximation,” J. Mod. Opt.51, 1137–1147 (2004).

J. Phys.: Condens. Matter

G. F. Quinteiro, A. O. Lucero, and P. I. Tamborenea, “Electronic transitions in quantum dots and rings induced by inhomogeneous off-centered light beams,” J. Phys.: Condens. Matter22, 505802 (2010). [CrossRef]

Nature

A. Fuhrer, S. Luscher, T. Ihn, T. Heinzel, K. Ensslin, W. Wegscheider, and M. Bichler, “Energy spectra and broken symmetry in quantum rings,” Nature413, 822–825 (2001). [CrossRef] [PubMed]

Opt. Express

G. F. Quinteiro and J. Berakdar, “Electric currents induced by twisted light in quantum rings,” Opt. Express17, 20465 (2009). [CrossRef] [PubMed]

Phys. Rev. A

W. E. Lamb, R. R. Schlicher, and M. O. Scully, “Matter-field interaction in atomic physics and quantum optics,” Phys. Rev. A36, 2763–2772 (1987). [CrossRef] [PubMed]

Phys. Rev. B

G. F. Quinteiro and P. I. Tamborenea, “Electronic transitions in disk-shaped quantum dots induced by twisted light,” Phys. Rev. B79, 155450 (2009). [CrossRef]
Z.-G. Zhu and J. Berakdar, “Photoinduced nonequilibrium spin and charge polarization in quantum rings,” Phys. Rev. B77, 235438 (2008). [CrossRef]
C. L. Romano, S. E. Ulloa, and P. I. Tamborenea, “Level structure and spin-orbit effects in quasi-one-dimensional semiconductor nanostructures,” Phys. Rev. B71, 035336 (2005). [CrossRef]
J. Splettstoesser, M. Governale, and U. Zülicke, “Persistent current in ballistic mesoscopic rings with Rashba spin-orbit coupling,” Phys. Rev. B68, 165341 (2003). [CrossRef]

Phys. Rev. Lett.

M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. Dávila Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett.89, 143601 (2002). [CrossRef] [PubMed]

Other

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th ed. (World Scientific Publishing Co., 2004).
D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Academic Press, 2008). [PubMed]

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Orbital and spin dynamics of intraband electrons in quantum rings driven by twisted light

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Abstract

1. Introduction

2. Twisted light and quantum rings

3. Interaction between twisted light and quantum rings

3.1. Hamiltonian H11 = −(q/me)A(r,t) ·p̂

3.2. Hamiltonian H12 = −(qαR/h̄)[σ̂ × A(r,t)]z

3.3. The total perturbation H1

4. Evolution of single-particle states

4.1. General considerations

4.2. Small spin-orbit coupling

4.3. Induced polarization and current

5. Conclusions

Acknowledgments

References and links

References

J. Mod. Opt.

J. Phys.: Condens. Matter

Nature

Opt. Express

Phys. Rev. A

Phys. Rev. B

Phys. Rev. Lett.

Other

Cited By

Figures

3.1. Hamiltonian H₁₁ = −(q/m_e)A(r,t) ·p̂

3.2. Hamiltonian H₁₂ = −(qα_R/h̄)[σ̂ × A(r,t)]_z

3.3. The total perturbation H₁