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

Optics Express, Vol. 19, Issue 27, pp. 26733-26741 (2011)

http://dx.doi.org/10.1364/OE.19.026733

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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.

© 2011 OSA

## 1. Introduction

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]

## 2. Twisted light and quantum rings

*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

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]

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]

*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.

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]

*ɛ*=

_{σ}*x̂*+

*σiŷ*=

*e*(

^{σiϕ}*r̂*+

*σiϕ*̂),

*σ*= ±1, and

*c.c*. denoting the complex conjugate. The radial function

*F*(

_{l}*q*) is left, for the moment, unspecified. Disregarding the longitudinal component of

_{r}r**A**is justified in usual experimental conditions, in which

*q*≫

_{z}*q*.

_{r}7. Z.-G. Zhu and J. Berakdar, “Photoinduced nonequilibrium spin and charge polarization in quantum rings,” Phys. Rev. B **77**, 235438 (2008). [CrossRef]

*q*= −

*e*and

*V*(

**r**) the QR confinement potential, and

*σ*̂ is the vector of Pauli matrices. The Rashba coupling constant is

*α*. The perturbation

_{R}*H*

_{1}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

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]

**A**(

**r**,

*t*) shows that this magnetic field has transverse as well as longitudinal components, which are proportional to

*q*and

_{z}*q*, respectively. Therefore, both are small compared to the other terms in the Hamiltonian and can be safely neglected.

_{r}*qα*/

_{R}*h̄*)[

*σ*̂ ×

**A**(

**r**,

*t*)]

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

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

*H*

_{SOI}can be exactly diagonalized, having envelope eigenfunctions Φ

*(*

_{ns}**r**) =

*ψ*(

_{ns}*φ*)

*R*(

*r*)

*Z*(

*z*) with with the z-projection of the total angular momentum

*n*+ 1/2, angular coordinate

*φ*, and angle-dependent spinor 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*: tan

*γ*= −

*ω*/

_{R}*ω*

_{0}, with

*h̄ω*= 2

_{R}*α*/

_{R}*a*and

*x*= −(1 −

_{s}*sw*)/2,

*s*= ±1 for respectively spin up and down in the (

*φ*-dependent) local frame. (The notation is summarized in Table 1.)

## 3. Interaction between twisted light and quantum rings

*H*

_{1}=

*H*

_{11}+

*H*

_{12}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

*H*

_{1}as a perturbation to

*H*

_{SOI}.

### 3.1. Hamiltonian H_{11} = −(q/m_{e})**A**(**r**,t) ·**p**̂

*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*) can be taken as constant at the value of

_{r}r*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]. After these simplifications, we are left with the element Using Eq. (4) Then, where

*φ*, and was pulled out of the integral. For the negative part

*H*

_{11}, though not a spin-orbit interaction, produces spin flips due to the last term of Eq. (11). This is possible because

*H*

_{11}acts upon eigenstates of the Rashba Hamiltonian

*H*

_{SOI}, which are not eigenstates of

*σ*̂

*. In the case of weak Rashba effect, the spin-flip term is proportional to the angle*

_{z}*γ*∝

*α*; for vanishing spin-orbit coupling, we recover the spin-conserving interaction.

_{R}### 3.2. Hamiltonian H_{12} = −(qα_{R}/h̄)[σ̂ × **A**(**r**,t)]_{z}

*H*

_{11}, 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.

### 3.3. The total perturbation H_{1}

*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*) come from

_{σ}*H*

_{11}(

*H*

_{12}).

## 4. Evolution of single-particle states

*n*

_{0},

*s*

_{0}} are the quantum numbers of the initial state, and

*h̄ω*=

_{fi}*ɛ*–

_{ns}*ɛ*

_{n0s0}.

*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.

### 4.1. General considerations

*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.

*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.

### 4.2. Small spin-orbit coupling

*ω*≪

_{R}*ω*

_{0}allows for the simplification of the Hamiltonian Eq. (15) to We exemplify by studying the evolution of a particle initially in state |

*n*

_{0}, 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

*ω*is understood as having the right indices for each case; while in all cases

_{fi}*i*= {

*n*

_{0}, 1}, for example, for the third line of Eq. (19),

*f*= {

*n*

_{0}+ 2,1}.

*l*+

*σ*= 2. The perturbation

*H*

_{11}is responsible for spin-conserving and spin-flip terms, while

*H*

_{12}causes only spin flips. As seen from Eq. (18), the ratio

*H*

_{11}and

*H*

_{12}. 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*= 10

^{−6}m having

*α*= 10

_{R}^{−14}V m and pulsed laser parameters typical for experiments: power 10

^{6}J Hz, repetition rate 10

^{6}Hz and wavelength 10

^{−5}m, we obtain Γ

*= 0.1 ps*

_{s}^{−1}and Γ

*= 0.5 ps*

_{o}^{−1}.

*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*< 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(−

_{fi}*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.

*n*

_{0}< 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*>

*n*

_{0}, to the right of

*n*

_{0}. 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

*n*

_{0},

*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

*ψ*(

_{m}*φ*) =

*e*—for a discussion on the difference between this and Eq. (4) see Ref. [7

^{imφ}**77**, 235438 (2008). [CrossRef]

*l*= 1 and

*σ*= 1, and study the evolution from an initial state |

*m*

_{0}〉. 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 where

*D*is a constant that depends on the sign of

*m*

_{0}, but for intraband transitions we expect

*D*(−|

*m*

_{0}|) ≃

*D*(|

*m*

_{0}|). 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

**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

**77**, 235438 (2008). [CrossRef]

## References and links

1. | D. L. Andrews, |

2. | G. F. Quinteiro and J. Berakdar, “Electric currents induced by twisted light in quantum rings,” Opt. Express |

3. | G. F. Quinteiro and P. I. Tamborenea, “Electronic transitions in disk-shaped quantum dots induced by twisted light,” Phys. Rev. B |

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 |

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 |

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. |

7. | Z.-G. Zhu and J. Berakdar, “Photoinduced nonequilibrium spin and charge polarization in quantum rings,” Phys. Rev. B |

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 |

9. | J. Splettstoesser, M. Governale, and U. Zülicke, “Persistent current in ballistic mesoscopic rings with Rashba spin-orbit coupling,” Phys. Rev. B |

10. | H. Haug and S. W. Koch, |

11. | W. E. Lamb, R. R. Schlicher, and M. O. Scully, “Matter-field interaction in atomic physics and quantum optics,” Phys. Rev. A |

12. | K. Rzazewski and R. W. Boyd, “Equivalence of interaction Hamiltonians in the eElectric dipole approximation,” J. Mod. Opt. |

**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).

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