Selection Rules for Optical Vortex Absorption by Landau-quantized Electrons

Selection Rules for Optical Vortex Absorption by Landau-quantized Electrons

Hirohisa T. Takahashi    1517000122@campus.ouj.ac.jp    Igor Proskurin       and Jun-ichiro Kishine
\abst

An optical vortex beam carries orbital angular momentum in addition to spin angular momentum . We demonstrate that a Landau-quantized two dimensional electron system absorbs the optical vortex beam through modified selection rules, reflecting two kinds of angular momenta. The lowest Landau level electron absorbs the optical vortex beams with (positive helicity) and or (negative helicity) and in the electric dipole transition. The induced electric currents survive only along the edge of the sample, due to cancellation of the bulk currents. Thus, the magnetization can be induced by only the edge current. It is shown that the induced orbital magnetization disappears when the dark ring of the beam coincides with the disk edge. This scheme may provide a helicity-dependent absorption using the optical vortex beam.

1 Introduction

Originally, it was suggested by Poynting that circularly polarized light carries spin angular momentum (SAM) equal to per photon, which can be transferred to medium and produce a mechanical torque in light-matter interactions. [1] Later, Beth experimentally confirmed angular momentum transfer from light in 1935. [2, 3] After about a century, it was suggested that lights can also carry an orbital angular momentum (OAM) in addition to SAM by Allen et al..[4] This part of angular momentum appears as a modulation of a phase front, so it was dubbed an optical vortex (OV) or twisted light. It was experimentally demonstrated that a single photon is able to carry quantized OAM. [5] Theoretical and experimental techniques were developed to generate OVs in various forms such as the Laguerre-Gaussian (LG) and the Bessel light beams. [4, 6, 7] These unique forms of light beams have triggered much interest on the transfer of optical OAM to material particles and atoms via light-matter interactions. [8, 9]

Mathematically, OV is described by a constant phase profile given by , where is the azimuthal angle in the cylindrical coordinate system for a light beam propagating in the direction with the wavenumber . It carries an intrinsic OAM equal to per photon (), which is independent of the polarization state of light. [4] Geometrically the phase front of OV is a helix with the winding number determined by . The radial dependence of the beam amplitude is typically given in terms of either Laguerre-Gaussian or Bessel modes. The former has the property of gradually expanding as the beam propagates, while the latter is diffraction free, or propagation invariant.[6, 10] Experimentally, Bessel OVs can be created in the back focal plane of a convergent lens by a plane wave,[10] by an axicon lens from a Gaussian beam,[11] by the use of computer-generated holograms, [12] or by a Fabry-Perot resonator. [13]

From the point of classical mechanics, exerting a torque by transferring angular momentum from OV has been actively studied, for example, with particles rotating in an optical tweezers,[14, 15, 16] and the laser ablation technique. [17] In recent years, coupling of twisted light with condensed matter also saw a considerable development, including such topics as generation of atomic vortex states by coherent transfer of OAM from photons to the Bose-Einstein condensate [18], photocurrents excited by the OV beam-absorption in semiconductors and graphene,[19, 20, 21] excitation of multipole plasmons in metal nanodisks,[22] spin and charge transport on a surface of topological insulator,[23] generation of skyrmionic defects in chiral magnets.[24]

However, whether OAM affects any spectroscopic selection rules via optically induced electronic transitions is still an open question. Although transferring of the OAM to atomic electrons from the OV beam via the electric quadrupole transition was reported,[25] for electric dipole transitions in atoms, it has been proved that the optical OAM can be transferred only to the center-of-mass motion of the atom or molecule,[26, 27] thus the electric dipole selection rules remain unchanged. Similar in the coupling of OV with the exciton, the optical OAM can be transferred only to the exciton center-of-mass motion.[28] These phenomena are analogous to the fact that the cyclotron resonance frequency is independent of short-range electron-electron interactions.[29]

In this case, it interesting to see whether these concepts are applicable to a degenerated two-dimensional electron gas (2DEG) in magnetic field. To our best knowledge, such a system has not been considered before our previous letter,[30] where we discussed the optical conductivity and the selection rules in 2DEG exposed to OV with optical OAM. By applying the magnetic field, 2DEG is characterized by discrete energy levels with localized semi-classical electron orbits. It was demonstrated that the bulk current induced by OV disappears, and only the edge current survives when the 2DEG is irradiated by a Bessel beam.[30] This situation is similar to the picture of orbital magnetization, [31] which is known to appear due to the existence of the edge currents. Therefore, in 2DEG we can anticipate an orbital Edelstein effect [32] where additional magnetization is induced by the OV, which is the central issue of this paper. In this paper, we extend discussions on the results we shortly presented in our previous letter to present theoretical details including the induced orbital magnetization.[30]

This paper is organized as follows. We briefly review the derivation of a circularly-polarized Bessel-mode OV in Section II and 2DEG on circular disc in Sec. III. We calculate the induced photocurrent in 2DEG in Sec. IV using the Kubo linear response theory. In Sec. V, we discuss cancellation of bulk currents in the semi-classical picture. It is demonstrated how the magnetization is induced by the OV light beam in Sec. VI. Sec. VII is reserved for conclusions.

2 Circularly Polarized Bessel-mode Optical Vortex

It is crucial for studying quantum mechanical properties of light to separate the total angular momentum (TAM) into spin and orbital parts, since they can be conserved separately for light interacting with particles. In the paraxial approximation, this separation can be done explicitly, and the light beam has a well-defined SAM related to its polarization state and OAM determined by the phase modulation. In this paper, we adopt the paraxial approximation, which is a usual situation in real-world experiments.

We briefly review derivation of the Bessel-mode OV within the paraxial approximation following Matula et al.[33] The wave equation for the vector potential of a monochromatic light with the frequency in vacuum in the Coulomb gauge is given by the Helmholtz equation:

(1)

where is a Laplace operator, and with a speed of light in a vacuum . In order to obtain twisted solutions, we have to take account of additional requirements. First is that is a propagating wave along -axis, so it is the eigenvector of the linear momentum operator , . Second is that should also be the eigenvector of -component of the TAM operator

(2)

where the operator is given by the corresponding components of the orbital and spin angular momentum operators:

(3)

where we define the modulus of the transverse linear momentum .

The normalized scalar solution of the Helmholtz equation in cylindrical coordinates can be written in the form

(4)

where determines the OAM of light which is the eigenvalue of the OAM operator (3), and is the -th order Bessel function of the first kind. The normalization condition is

(5)

Expansion over plane waves of the scalar function  is

(6)

with and . Each plane wave component is written as

(7)

These expressions show that can be viewed as a superposition of plane waves with fixed whose direction belongs to the cone with the cone angle .

When the scalar solution of the Helmholtz equation is considered as a superposition of plane waves, it is important to study the polarization structure of the plane wave with the propagation vector . The vector potential of the plane wave has to be an eigenvector of the SAM operator, . For the plane wave traveling along , the spin angular momentum operators has the following eigenvectors:

(8)

and the vector potential is given by , where is a constant.

When the plane wave travels in arbitrary direction , which does not necessary coincide with the -axis, , its polarization vector can be found from original polarization vectors by rotating them with rotation matrix

(9)

which gives

(10)

Then the vector potential for the plane wave traveling along is given by

(11)

where the Coulomb gauge is used and the polarization vector then describes photon carrying a helicity . We can expand over the orthonormal basis of the eigenvectors of the SAM operator :

(12)

where the expansion coefficients are given by

(13)

Now we can find the expression for the vector potential for OV based on the expansion over the plane waves in Eq. (2) and taking into account that each plane wave is characterized by its own polarization vector :

(14)

where we introduced as the eigenvalue of the TAM operator . Integrating over , we finally obtain the vector potential of the OV with Bessel mode

(15)

In the paraxial approximation, we assume that the longitudinal momentum of the photon is much greater than its transverse momentum, , so the expansion coefficients become , and we get the vector potential in the form:

(16)

where we introduced a OAM quantum number, . Moreover, if we take the limit with being fixed, then the Bessel function gives , and we recover a plane wave solution with propagating along the -axis.

The Bessel-mode OV exhibits a feature of being diffraction free and has a phase singularity. The first feature can easily be seen by using Eq. (16). The intensity of the vector potential, , is independent of . The phase singularity is located on the beam axis where the intensity becomes zero. To demonstrate a transfer of OAM, the target particles are usually located in non-zero intensity region off the beam axis and dark rings. The radius of -th dark ring of the higher-order Bessel beam is given by

(17)

which is determined by . In particular, the central core size of the zero-order Bessel beam is given by . We exhibit some examples of the radial profile of the Bessel-mode OV, and the definition of the dark ring radius and the central core spot size in Figure 1.

Figure 1: (Color online) Some examples of the radial profile of the Bessel-mode optical vortex, . (a) . The central core spot size is given by the fisrt zeros of . (b) , The dark ring radius is given by -th zeros of . (c) , (d) .

We note that the Bessel-mode OV even with has the dark rings corresponding to transversely traveling wave, This feature is also the crucial difference with the plane wave.

3 Landau-quantized Electron

The quantized energy levels of 2DEG in the magnetic field are given by [34] , which usually appear by solving the Schrödinger equation in the Landau’s gauge, where is the Landau level index, and is the cyclotron frequency with the elementary charge , and the bare electron mass . We here note that the electron mass  should be interpreted as an effective mass  for GaAs. However, when we consider 2DEG interacting with the Bessel OV light beam, the symmetric gauge in the cylindrical coordinates becomes a natural choice. Hence, we consider 2DEG on with a circularly shaped disk geometry and take the cylindrical coordinates as shown in Fig.2.

The non-perturbative Hamiltonian for 2DEG under the magnetic field is given by

Figure 2: (Color online) Schematic picture showing 2D electron distributions in the lowest LLs in the circular disc geometry. The OV beam is vertically irradiated to 2DEG. The direction of propagation of OV is taken the -axis. Directions of the induced photocurrents are indicated by arrows. The azimuthal angle is on the 2D electron system.
(18)

where , and the magnetic field is along the -axis direction. The energy spectrum is obtained by solving the Schrödinger equation which gives[35]

(19)

where is the magnetic quantum number related to the angular momentum of the electron. The eigenfunction is also obtained as

(20)

where is the magnetic length, is the normalization constant, and is the associated Laguerre polynomials. In this picture, we call the principal quantum number. Its relation to ordinary Landau index is . This leads to for states with . Each Landau level with given is multiply degenerated with respect to and due the finite system size with the degeneracy factor given by , where is the area of 2DEG.

For example, the lowest Landau level (LLL) is obtained by the condition , which leads to and . The probability density for the electron with the wave function (3) has the maximal value at . This means that the electron is distributed on the circle with the radius . Because the expectation value of is given by , we find that the electron state covers the area . Then, the maximum for the disk geometry is given by[36]

(21)

which allows us to define the filling factor as

(22)

where is the total number of electrons on the disk. Throughout this paper, we concentrate on the system with the filling factor , where the Fermi energy lies in the gap between the LLL and the second Landau level (2LL). We display the energy diagram of the axial symmetric 2DEG system as shown in Fig.3.

Figure 3: Allowed transitions from the lowest LL () to the second LL () are indicated by the gray thick allows. The opened circles denote unoccupied states, whereas closed ones are occupied.

4 Photocurrent Induced by the Optical Vortex

Here, we investigate the interaction of a Landau-quantized 2DEG with the Bessel OV by applying the linear response theory. We start with the following total Hamiltonian, which contains the non-perturbative Hamiltonian (18) interacting with the vector potential of the OV:

(23)

where is given by Eq. (16), and the electric current is determined by . We neglect the electron spin.

The Kubo formula for -component of the induced current is written as [37, 38]

(24)

where is the Fermi distribution with a chemical potential and an inverse temperature , and is the electron wavefunction in Eq.(3). From now on, we assume zero-temperature limit and keeping the chemical potential lie between th LLL () and the second LL ().

It should be mentioned that, although we work in the cylindrical coordinates, which manifest the symmetry of the OV, our final results, of course, are not specific to a particular coordinate system. Alternatively, we can consider the spherical coordinates and examine the multipole expansion by the vector spherical harmonics (VSH) of currents in Eq. (24) as discussed in Appendix B where we obtain the general expression in Eq. (69). We also show that the selection rules for the dipole transitions in Eq. (78) are consistent with the results obtained without multipole expansion in Eq. (31).

Let us now return to discussion without multipole expansion. To investigate the OV-induced photocurrent, we adopt the chiral basis . First, we consider the matrix element of photocurrent that can be written as

(25)

where is the thickness of 2DEG and we denote the radial integral as

(26)

Here, we obtain the selection rule  from the azimuthal integral . After calculating the radial integral and the energy factor by Eq.(19), we can obtain the matrix elements of the photocurrent as

(27)
(28)

For the filling factor , these matrix elements reduce to

(29)
(30)

Therefore, we find that the transition is allowed only [38]. We summarize possible transitions from the LLL () to the second LL () as follows,

(31)

Next, we consider the matrix element of the minimal coupling of 2DEG with OV. As shown in Appendix A, the dipole approximation is justified in our model. Then the matrix element for the photon absorption in this approximation is obtained as

(32)

where we denoted the radial integral as

(33)

We also obtain the selection rule from the azimuthal integral , where . This means that the OV can transfer its TAM to the electron via the dipole interaction.

We note that, when we fix the filling factor (the chemical potential lies between and ), the left-handed current is not induced. Therefore, only the right-handed current arises by transferring the optical TAM, . Because the OV carries the SAM , the OAM and SAM must be , , or , , respectively, with the other transitions being prohibited.

On the other hand, if we apply the external magnetic field anti-parallel to the light traveling, since it corresponds to the time inverse, the electron in the LLL carries positive value angular momentum. Then, to excite the electron in the LLL, the electron can absorb the optical TAM . As a result, the possible absorptions are reduced to , , and , .

Next, we calculate the photocurrent using the Kubo formula. For the transition from to with , the OV-induced current (24) reduces to

(34)

where or and the factors are given by

(35)

with . In the summation with respect to , by using the explicit form, , only one term corresponding to an edge current along the circle with the radius survives. The other terms corresponding to the bulk currents cancel each other. After some algebra, we obtain

(36)

where is  or , , , which has an order of magnitude of unity. is the flux quantum, and is the electron Compton wavelength.

5 Physical Meaning of Cancellation of Bulk Currents

In this section, we present a physical interpretation on the reason why the bulk currents are cancelled out, based on the coherent state representation. Introducing the Larmor radius vector and the guiding center vector satisfying, we can rewrite the 2DEG Hamiltonian as

(37)

where we note the relation . We can then define the non-commuting operators which satisfy

(38)

We find that one electron occupies the area determined by the uncertainty principle:

(39)

Then we can define the ladder operators

(40)

with , and . The eigenstates are thus determined by the two integer quantum numbers, and , associated with the two ladder operators,

(41)
(42)

Then in terms of the two ladder operators, the Hamiltonian and the angular momentum operator are written by

(43)
(44)

Here, comparing above eigenvalues with Eq.(19) and , we can determine the relation between , and , as

(45)

The average value of the guiding center operator gives

(46)

but its absolute value leads to

(47)

Similarly, the average value of Larmor radius operator gives

(48)

but its absolute value is given by

(49)

Therefore, the arbitrary state distributes at the center of the Larmor motion with radius  is located at the position of guiding center

(50)

The geometric meaning of this is illustrated in Fig. 4. When we focus on the LLL, that is, and, we see and . Therefore, the guiding center in the LLL is , and the Larmor radius in it is .

Figure 4: Schematics of the classical orbits of LLs. The quantum number assigns the radius of the guiding center , whereas is the radius of Larmor orbit .

This kind of distribution can semi-classically be described by the coherent states. We now introduce the displacement operators

(51)

The first displacement operator  generates a displacement to the position at the guiding center . Since commutes with the Hamiltonian , the guiding center is the constant of motion. Therefore, the Hamiltonian  does not depend on quantum number . On the other hand, the second displacement operator generates a displacement to the position . Since does not commute with the Hamiltonian , the coherent state is not an eigenstate of the Hamiltonian.

Applying these displacement operators to the ground state , we can thus construct the coherent state,

(52)

where and are eigenvalues of the annihilation operators and of the eigenstate . That is, these eigenstates satisfy

(53)
(54)

and the eigenvalues and are given by

(55)
(56)

To see the absence of bulk currents, we pay our attention to one coherent state at the guiding center , which produces the circular current by the Larmor motion with radius . Because of the uncertainty (39), it seems that the circular current flows the edge of the area . When the LLs state can be constructed by the superposition of the coherent states, the superposition produces contact points of the circular current at the center with the surrounding circular currents. Thus, the circular current at the center  is canceled by the surrounding circular current. Such the cancellation occurs on whole system except to the edge, we can say the bulk currents are all cancelled out, i.e.,

(57)

6 Magnetization Induced by Edge Current

Now, we naturally expect that the edge currents induce an orbital magnetization, which can be observed experimentally. The magnetic vector potential at the position induced by the magnetization at the guiding center is given by

(58)

where  represents the edge of the 2D system , and is a normal vector with respect to the edge , and indicate that the integration is done with respect to a variable . The first term can be regarded as the vector potential induced by the bulk current, , whereas the second term is due to the edge current at the system size ,

(59)

However, since the bulk currents cancel out by Eq. (57) as mentioned in the previous section, only the edge current contributes to the magnetization in Eq. (58).

In Eq. (59), since the normal vector with respect to the edge of circular disk is given by , and the edge current flows along the edge, , the magnetization points along the -direction, , where

(60)

The magnetization in Eq. (60) can be regarded as a manifestation of the magneto-electric effect, since it is induced by the electric field of OV.

The magnetization obviously depends on the external magnetic field. Here, we imply that the frequency of the OV is always kept in resonance with the transition from the LLL to 2LL, so that when we apply the external magnetic field , the excitation energy from the LLL to 2LL is given by [T]eV. To make the transitions possible, the wavelength of OV must be controlled to satisfy the energy conservation, . Then the wavelength of OV and wavenumber should be [T]m and [T]m, respectively. As a consequence, when the magnetic field increases, the transverse wavenumber should be increased to hold the ratio,