JTu3A.79.pdf
Frontiers in Optics 2017 © OSA 2017
Interband absorption of topologically structured photon beams by semiconducting quantum dots Maria Solyanik, Andrei Afanasev Department of Physics, The George Washington University, Washington, DC 20052, USA
[email protected],
[email protected]
Abstract: Interaction of topological photon beams with semiconducting quantum dots is described in the QED formalism. Absorption rates are calculated, and distinctive features arising due to the wavefront topology, such as modified selection rules, are discussed. OCIS codes: (270.1670) Coherent optical effects; (020.5580) Quantum electrodynamics
1.
Photo-induced transitions in strongly confined quantum dot
We will consider the case of GaAs-type spherical semiconducting quantum dot (SCG) with parabolic, isotropic, direct band structure. As for the bulk properties, we assume the typical s-like conduction band and p-like valence band character of the Bloch functions. The full solution for the electron and hole confined in a SQD is Ψn`m (~r,t) =
1 r)ψe (~r)bˆ pe ,αe (t) + uv (~r)ψh∗ (~r)cˆ†−ph ,αh (t)}; 3/2 {uc (~ R0
s ψe/h (~r) = Rn` (r)Y`m (Ω); Rn` (r) =
2 j` (kn` r) ; R30 j`+1 (kn` R0 )
(1) (2)
where R0 is the radius of the QD; Y`m (Ω) are the spherical harmonics and Rn` (r) is the normalized radial function; j`m (·) is a spherical Bessel function. Here uc (~r) and uv (~r) are the Bloch functions for conduction and valence bands, describing the bulk properties of the material; and ψe/h (~r) is the solution of single particle Schr¨odinger equation describing the envelope contribution to the electron or the hole state. The operators bˆ pe ,αe (t) and cˆ†−ph ,αh (t) are electron annihilation and hole creation operators, correspondingly. In the QED description of a plane wave interacting with a SQD, the interband photo-absorption amplitude can be written as (Pl)
M f i (0) =
1 R30
Z
u∗c (~r)ψn∗e `e me (~r)(A~ (Pl) ·~p)uv (~r)ψnh `h mh (~r)d 3 r
(3)
Preceding with the formalism, one extracts the following set of selection rules `e − `h = 0, me − mh = 0 for dipole transitions, and `e − `h = ±1, me − mh = 0, ±1 for quadrupole. These selection rules agree with the ones obtained by the authors in [1]. 2.
Interband absorption of twisted laser modes by SQD
The quantum mechanical description of the twisted Bessel beam (BB) was developed, ex., in [2]. The vector potential of the Bessel mode can be expanded in plane waves with the fixed longitudinal momentum ~kz and pitch angle θk = arctan(|~k⊥ |/kz ) as follows r Z κ −iωt dφk µ ~ µ (4) e (−i)mγ eimγ φk ε~ eik·~r Aκmγ kz Λ (~r,t) = kΛ 2π 2π Here Λ = ±1 represents two helicity orientations of a photon and mγ is the projection of the total angular momentum of the photon beam on the direction of propagation. The explicit representation of the twisted photon polarization state is θk µ Λ θk µ µ µ ε~ = e−iΛφk cos2 ηΛ + eiΛφk sin2 η−Λ + √ sin θk η0 (5) kΛ 2 2 2
JTu3A.79.pdf
Frontiers in Optics 2017 © OSA 2017
where η±Λ = √12 {0, ∓1, −i, 0} and η0 = {0, 0, 0, 1} and κ = k sin θk is the transverse part of the wave vector. Photo-absorption of twisted photons by the SQD is obtained in analogy to the hydrogen-like atom photo-excitation, ex. [3]. With the QED interaction hamiltonian Hint = − mqq ~A ·~p, one gets the amplitude in the form r κ (Pl) `h `e |Mn(TW) (b)| = J (θ )d (θ )M (0) (κb) d 0 0 m −m +m k k ∑ (6) γ e h f i ` m Λ m m m m e e e e e h h 2π m0 m0 µ
µ
e h
(Pl) M f i (0),
where given by equation (3), gets conveniently factorized out; the quantum numbers {ne , `e , me } and {nh , `h , mh } correspond to the final electron and initial hole states; and b is the beam impact parameter. Photo-excitation rates are defined as `e
(m )
Γne `γe Λ = 2πδ (Ee − Eh − ωγ )
∑
me =−`e
|Mn(TW) (b)|2 e `e me Λ
(7)
Two main topological effects are identified: the first is the rotational transformation described by the Wigner dfunction, and the second is topological phase factor Jmγ −me +mh (κb). Because Bessel functions form a complete set of orthogonal functions, the laid out approach can be used for the beams of any topological structure. For instance, one can apply a Hankel transform to equation (4) and get the expression for Laguerre-Gaussian mode (LG) p √2 2 j+` 2 2 j+`+1 √ Z ∞ 1 2 (|`| + p)! µ µ j 2π dκ κ 2 j+`+ 2 e−κ /a Aκmγ kz Λ (~r,t) (8) ALG = ∑ (−1) (p − j)!(|`| + j)! j! w a 0 j=0 where p corresponds to the number of nodes and ` is the topological order of the mode. As shown in Fig. 1., near the optical vortex center, the transitions with me − mh = mγ dominate, while the set of plane wave selection rules take over on the beam periphery. These so-called excitations in the dark seems feasible to observe experimentally in SQD in the way, analogous to the experiment, performed on Ca40 ions [4]. BB
LG
0.25
0.25
me =2
me =2 me =1
0.20
me =1
0.20
me =0
me =0 mγ =-1
Rate
0.15
mγ =-1
0.15
mγ =-2
mγ =-2 0.10
0.10
0.05
0.05
0.00
0.00 0
2
4
6 b (μm)
8
10
0
2
4
6
8
10
b (μm)
Fig. 1. Quadrupole transition rates for photo-excitation of a SQD from initial state `h = mh = 0 by LG (left) and BB (right) laser modes λ = 550nm, which corresponds to the spatial dimensions of the order R0 ≈ 10nm. References 1. J. R. Zurita-S´anchez and L. Novotny, “Multipolar interband absorption in a semiconductor quantum dot. i. electric quadrupole enhancement,” JOSA B, vol. 19, no. 6, pp. 1355–1362, 2002. 2. A. Afanasev, C. E. Carlson, and A. Mukherjee, “Off-axis excitation of hydrogenlike atoms by twisted photons,” Physical Review A, vol. 88, no. 3, p. 033841, 2013. 3. A. Afanasev, C. E. Carlson, and A. Mukherjee, “High-multipole excitations of hydrogen-like atoms by twisted photons near a phase singularity,” Journal of Optics, vol. 18, no. 7, p. 074013, 2016. 4. C. T. Schmiegelow, J. Schulz, H. Kaufmann, T. Ruster, U. G. Poschinger, and F. Schmidt-Kaler, “Transfer of optical orbital angular momentum to a bound electron,” Nature communications, vol. 7, no. 12998, 2016.
