Spectral properties and magneto-optical excitations in semiconductor double rings under Rashba
spin-orbit interaction
Wen-Hsuan Kuan,1Chi-Shung Tang,2and Cheng-Hung Chang1,3
1Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan, Republic of China 2Research Center for Applied Sciences, Academia Sinica, Taipei 11529, Taiwan, Republic of China
3Institute of Physics, National Chiao Tung University, Hsinchu 30013, Taiwan, Republic of China
共Received 13 September 2006; revised manuscript received 6 January 2007; published 23 April 2007兲
We have numerically solved the Hamiltonian of an electron in a semiconductor double ring subjected to the magnetic flux and Rashba spin-orbit interaction. It is found that the Aharonov-Bohm energy spectrum reveals multizigzag periodic structures. The investigations of spin-dependent electron dynamics via Rabi oscillations in two-level and three-level systems demonstrate the possibility of manipulating quantum states. Our results show that the optimal control of photon-assisted inter-ring transitions can be achieved by employing cascade-type and ⌳-type transition mechanisms. Under chirped pulse impulsions, a robust and complete transfer of an electron to the final state is shown to coincide with the estimation of the Landau-Zener formula.
DOI:10.1103/PhysRevB.75.155326 PACS number共s兲: 74.25.Gz, 71.70.Ej, 74.25.Nf
I. INTRODUCTION
The progress in epitaxial growth promotes the use of low-dimensional semiconductor nanostructures in optoelectronic devices. Investigations on fundamental physical properties such as the electronic structure and the carrier population can be directly measured and estimated from the photo-luminescence spectrum. Theoretically, several proposals on magneto-optical studies for quantum dot systems have been put forward in the last decade.1Nowadays, coherent optical manipulations of single quantum systems have attracted fur-ther attention. The mature technologies in optical control and measurements provide a great opportunity to realize quantum qubits as logical gates2 in storage and quantum information processing.3
In the 1990s, the progress of technology has enabled the experimental study of a mesoscopic ring threaded by a static magnetic-field display persistent currents,4,5 which oscillate as a function of magnetic flux⌽ with a period ⌽0= hc / e. Recently, applications on spin-orbit interaction 共SOI兲 origi-nated from the breaking of inversion symmetry that gives rise to intrinsic spin splitting in semiconductor systems open a field of spintronics. It was pointed out that the quantum transport of electrons in a spin-polarized system differs greatly from that in a spin-degenerate device.6The utilization of the spin degree of freedom offers the mechanism to speed up quantum information processing. In nature electronic de-vices such as the Datta-Das transistor,7spin waveguide8and spin filter9were proposed.
Several theoretical works associated with Rashba SOI due to structural inversion asymmetry in quantum dot systems were studied.10More recently, the success in self-assembled formation of concentric quantum double rings11 provides a new system to explore electron dynamics by magneto-optical excitations on the basis of fully analyzed signature of Aharonov-Bohm共AB兲 spectrum within the effect of Rashba SOI. The radius of flat double rings is about 100 nm with thickness of approximately 3 nm. Therefore, carriers are co-herent all throughout these small geometries. Within a time
scale shorter than the dephasing time,12the Rabi oscillation 共RO兲 can provide a direct control of excited state population especially in strong excitation regime. It was proposed to be a good optical implement in quantum dot systems.13 How-ever, a simple two-level system involving Rabi oscillations with Rashba SOI in a coaxial double quantum ring has not yet been studied. Therefore, in this paper, we consider two-level and three-two-level models to explore spin-dependent elec-tron dynamics assisted by RO processes. Under the influence of magnetic flux, the spin feature of the system is demon-strated only through the effects of Rashba SOI. The presence of Rashba SOI also plays an important role in the mixture of neighboring angular momenta as well as spins that build up a new selection rule, and it opens more dipole-allowed transi-tion routes.
In view of quantum algorithm realization, the two-level Rabi oscillators are often the prototype of the quibit genera-tors. However, it is also important to establish coherent con-trol in realistic multilevel quantum systems.14,15 Hence, we explore the multilevel dynamical system involving the cascade-type and the ⌳-type three-level schemes driven by either sinusoidal impulsions or chirped laser pulses. To achieve efficient transfers, we employ adiabatic rapid pas-sage method共ARP兲,16,17 namely, that the excitation process rapid compared with the natural lifetime of an excited state in the limit of slowly varying detuning field, to simulate complete transfer processes. We will show that the probabil-ity of an electron that occupies the final state coincides with the estimation of the Landau-Zener formula in the adiabatic limit.18
The paper is organized as follows. In Sec. II, the single-particle Hamiltonian is derived and numerically solved and SOI accompanied AB energy spectrum is analyzed. In Sec. III, ROs between two levels selected from the double-ring spectrum are studied. In Sec. IV, photon-assisted electron transitions in the three-level systems of the cascade and ⌳ schemes are investigated. Finally, the paper ends with a con-clusion in Sec. V.
II. THE ENERGY SPECTRA OF THE DOUBLE-RING SYSTEM
The system of a double-ring two-dimensional electron gas is enclosed by a magnetic flux in the presence of Rashba SOI. The electron is confined in an axial-symmetric potential
Vc, shown in Fig. 1, which is exposed in a monochromatic
electromagnetic共EM兲 field. In semiclassical description, the Hamiltonian is given by H = 1 2m*
冋
pជ− e cA共rជ,t兲册
2 + Vc共r兲 + ␣ ប再
ជ⫻冋
pជ− e cA共rជ,t兲册
冎
z , 共1兲 where A共rជ, t兲 contains contributions from the magnetic flux and the EM field and␣ is the coupling constant of Rashba SOI. The double-ring potential is modeled asVc共r兲 = 1 2m * 0 2共r − r 0兲2+
兺
i=1 3 Vie−共r − ri兲 2/ i 2 , 共2兲where 0 is a factor defining the characteristic length lc
=
冑
ប/m*0 of the system and i is the Gaussian spatial
width. The magnetic flux applied through the central region of the inner ring within r⌽is described by the vector poten-tial Aជ⌽= Br / 2ˆ for r艋r⌽ and Aជ⌽= Br⌽2/ 2rˆ for r⬎r⌽. Therein, the unit vector in the angular direction
ˆ =−sin共兲xˆ+cos共兲yˆ has been used. The effect of the lin-early polarized EM wave is simply expressed as AជEM共t兲
= A0sin共kz−⍀t兲xˆ, where k and ⍀ are the wave vector and the frequency of the wave.
The Hamiltonian can be divided as H = H0+ Hint, where H0 and Hint correspond to the unperturbed and time-dependent Hamiltonians, respectively. From energy conservation, the rapidly oscillating quadratic term inAជEM共t兲 is omitted, and
hence Hint⯝ −
冋
eAជEM m*c冉
pជ− e cAជ⌽冊
+ ␣e បc关ជ⫻ AជEM兴z册
. 共3兲It is convenient to rewrite Hint= HD+ HB+ HSO, indicating
three different types of interaction. The first term is the elec-tron dipole interaction, given by
HD=
− e
m*cAជEM· pជ⯝ − exជ· Eជ, 共4兲
where the time-dependent electric field, Eជ=E0cos共⍀t兲xˆ, is polarized along the x direction. The second contribution HB
is due to the applied magnetic flux,
HB=
e2
m*c2AជEM· Aជ⌽= e2E0
m*c⍀A⌽共r兲sin共兲sin共⍀t兲. 共5兲
The third term HSOdenotes the SO coupling mechanism,
HSO= −
␣e
បc共ជ⫻ AជEM兲z=
−␣eE0
ប⍀ ysin共⍀t兲. 共6兲
The electron dynamics can be derived based on the knowledge of eigenfunctions of H0. At t = 0, the normalized two-component wave function is⌿=共⌿↑,⌿↓兲T, where
⌿=共rជ兲丢, 共7兲 with=↑ or ↓ indicating two spin branches. Since total an-gular momentum Jzcommutes with time-independent
Hamil-tonian in the presence of SOI, the spatial wave function can be expressed in the form
↑共rជ兲 =l↑共r兲eil↑,
↓共rជ兲 =l↓共r兲eil↓, 共8兲
where the orbital angular momenta l↑= mj− 1 / 2 and l↓= mj
+ 1 / 2 follow the relation l↓= l↑+ 1, with mjcorresponding to
the eigenvalue of Jz. While dipole interaction does not flip
spin directly, the spin flipping is possibly achieved in the presence of the SOI, and therefore we can investigate the spin-dependent charge dynamics. To characterize this fea-ture, we define the net spin polarizability
P =具⌿兩z兩⌿典
具⌿兩⌿典 =
具⌿↑兩⌿↑典 − 具⌿↓兩⌿↓典
具⌿兩⌿典 . 共9兲
For the case of兩P兩=1, this indicates that the system is totally polarized into spin↑ 共spin ↓兲 if P= +1 共P=−1兲. Otherwise, the spin polarizability can be generally specified by the no-tationP↑ifP⬎0 and P↓ ifP⬍0.
Specifically, we consider an InAs-based double quantum ring system, of which the quantum structures are appropriate to investigate some spin-related phenomena.19 Below, we have selected the InAs material parameters m*/ m
0= 0.042 and␣⬃40 meV nm. Correspondingly, the characteristic en-ergy E =ប0= 5 meV and for the EM fieldប⍀=1 meV. The dimensionless parameters of the double-ring potential are
FIG. 1. 共Color online兲 The diagram of part of the double-ring potential depicted in 0艋艋3/2. The radius of the ring is about 160 nm and effective range of the magnetic flux r⌽is about 7 nm.
rc= 0, V1= 70, V2= 20, V3= −20, 1= 1.825, 2= 1.0, 3 = 2.236, r1= 0, r2= 5.0, and r3= 6.0. In the numerical calcula-tion, we shall present the magnetic flux in units of the flux quantum⌽0= hc / e.
By choosing the above typical physical quantities and di-agonalizing the time-independent Hamiltonian, it is easy to obtain the single-particle energy spectrum, as shown in Fig.
2共a兲. As compared with the energy spectra in usual quantum dots, a genuine effect of SOI can be revealed in the reduction of degeneracies from fourfold to at most twofold. However, for rings there are only twofold degeneracies at zero mag-netic flux whether the SOI exists or not. The presence of the
magnetic flux breaks the time reversal symmetry but Kram-er’s degeneracy is not lifted due to the absence of the Zee-man effect. In Fig. 2共a兲, the solid curves indicate energy levels of positive mj, whereas those of negative mj are
de-picted by dashed curves. The lowest five pairs of energy levels belong to兩mj兩=2.5, 3.5, 1.5, 4.5, and 0.5. The ordering
of these levels could change if the coupling constant␣of the Rashba SOI is varied. For instance, when␣= 5 meV nm, the lowest five pairs of these levels are with兩mj兩=0.5, 1.5, 0.5,
2.5, and 1.5. In the concern of varying ␣, one can drive coherent ROs and the idea has been realized in quantum dot systems.20
In the absence of the magnetic flux, the second and the third共as well as the fourth, the fifth, etc.兲 levels in Fig.2共a兲 are close to each other. The gap between these adjacent lev-els arises from the zero-field splitting of the Rashba SOI, which will disappear when the SOI coupling constant␣ ap-proaches zero. In the limit␣→0, not only the adjacent levels at⌽/⌽0= 0 mentioned above but also the curves split from these levels in the region⌽/⌽0⬎0 will merge together. An-other decisive feature distinguishing quantum rings from quantum dots stands out that for the former the ground state will periodically shift to that of higher total angular momen-tum. However, it always corresponds to the state with the lowest angular momentum in quantum dots.
An energy level E in Fig. 2共a兲 is a piecewise smooth function of⌽/⌽0with singular crossing points. The zigzag thick curve shows an example of the sixth lowest level, which has five crossing points around⌽/⌽0= 0.5, 0.6, 1.0, 1.4, and 1.5 within the unit interval 0.5⬍⌽/⌽0⬍1.5 at␣ = 40 meV nm. If ␣→0, the pairs of curves merge as dis-cussed above and the set of crossing points reduce to ⌽/⌽0= 0.5 and 1.5 for the ground state, and reduce to ⌽/⌽0= 0.5, 1.0, and 1.5 for other levels. It turns out that at
␣= 0, the electron in the double-ring reveals a similar oscil-lation pattern, with the same osciloscil-lation period one, as the AB oscillations in a single ring without SOI.21,22 For␣⬎0, the splitting of local spin branches in AB oscillating spec-trum of a single ring can be identified.23For the double ring, spectral patterns become more complicated, but patterns with regular oscillations are still rather apparent. Moreover, the spin polarizability P defined in Eq. 共9兲 varies in ⌽/⌽0. It seems that the crossing points of the lowest level in Fig.2共a兲, i.e., at ⌽/⌽0=兵0.5,1,1.5,2, ...其, are exactly the positions whereP changes its sign, at least in the range we studied.
To explain the location of the crossing points in Fig.2共a兲, let us consider an ideal one-dimensional ring of radius R enclosing a magnetic flux. The spectral property of this sys-tem reflects some key features of a radial subband of the double ring关Fig.2共b兲兴. The flux-dependent energy spectrum can be derived in the analytical form,
E =បa 2
再
冉
l↑− ⌽ ⌽0冊
2 +冉
l↓− ⌽ ⌽0冊
2 ±冑
冋
冉
l↑− ⌽ ⌽0冊
2 −冉
l↓− ⌽ ⌽0冊
2册
2 +4␣ 2 R2 1 ប2 a 2冉
l↑− ⌽ ⌽0冊冉
l↓− ⌽ ⌽0冊
冎
, 共10兲FIG. 2. 共Color online兲 共a兲 The Aharonov-Bohm oscillations in the energy spectrum of the double ring in the presence of the Rashba SOI with␣=40 meV nm and a static magnetic flux. The spectrum of states with positive共negative兲 mjis depicted in solid 共dashed兲 curves. The lowest five pairs of the states are specified by 兩mj兩=2.5, 3.5, 1.5, 4.5, and 0.5. In each 1/2−⌽/⌽0region, up or
down arrows denote net spin orientations of ground states. The zigzag curve shows the sixth lowest eigenenergy.共b兲 Energy levels within lowest four subbands, in which energies of subband bottoms line close by 30.6, 31.2, 52.1, and 53.6 meV, respectively.共c兲 Near the second subband bottom, anticrossing levels are depicted in thick curves. From left to right, these states are of mj= 0.5, 2.5, 1.5, and 3.5.
whereបa=ប2/ 2m*R2. Due to the relation l↓= l↑+ 1, the
en-ergy E can be expressed as a function of the variables l↓and ⌽/⌽0. A crossing point will come up at a certain ⌽/⌽0 where different integers l↓have the same energy E. Accord-ing to Eq.共10兲, this happens when ⌽ increases from zero to
a period⌽0 if the Rashba SOI is absent. In the presence of the Rashba SOI, additional crossing points appear before⌽ reaches⌽0. As compared to the Fock-Darwin spectrum in a quantum dot where the energy is in linear proportion to the trapping frequency and the cyclotron frequency in weak and strong magnetic fields, respectively. However, while SOI can be regarded as the perturbation, Eq. 共10兲 tells well a
qua-dratic relation with the magnetic flux.
Level crossing points can be easily seen in Fig.2共a兲. An-ticrossing levels also appear, for instance, in the example of Fig. 2共c兲, and even in high-energy regimes, as seen in the two dashed lines at 共⌽/⌽0, E兲=共1.7,53.7兲 in Fig. 4共a兲. While the splitting of the accidental level degeneracy in quantum dots has been demonstrated both theoretically and experimentally,24the repulsions in the avoiding levels due to the interplay between Zeeman and Rashba terms are also reported recently.25In our case, anticrossings near the second subband bottom arise in the presence of strong SOI. How-ever, for high-energy pairs, the repulsions here are mainly attributed to the geometric effect of the double ring under the influence of magnetic flux. In other words, the repulsion lev-els in the double ring will not disappear, even when Rashba effect is turned off. In the vicinity of the minimal splitting points, wave functions vary acutely and cannot be specified by a set of good quantum numbers. The double-ring Hamil-tonian thus typically manifests the signature of quantum chaos. A comparison between the spectra of ␣= 0 and 20 meV nm shows that the Rashba SOI will increase the level splitting in each energy pair mentioned above but de-crease the gap of the repulsion levels from 0.32 to 0.28 meV. Moreover, by adiabatically modulating the gate voltage to change SOI, the double-ring system dis-cussed here serves as a candidate for testing the Berry phase.26
III. RABI OSCILLATIONS IN TWO-LEVEL TRANSITIONS Since an electron in the inner共outer兲 ring has a definite angular momentum, its tunneling probability to the neighbor ring is suppressed under the constraint of the angular mo-mentum conservation. Therefore, an eigenfunction in the double ring may be localized only in the inner or the outer ring, if its corresponding energy is lower than the barrier. In the following, we are going to investigate the dynamics of an electron under irradiations of an external EM field in the presence of SOI. The transition between two such kind of quantized levels in the energy space corresponds to an inter-ring transition in the spatial space.
We consider arbitrarily two levels in the double ring. Sup-pose the electron initially occupies state兩b典 with eigenfunc-tion ub共rជ兲 in the outer ring, the irradiation process is designed
to pump it to state兩a典 with eigenfunction ua共rជ兲 in the inner
ring. For convenience, the time-dependent wave function can be written as
共rជ,t兲 = ca共t兲ei共␦/2−a兲tua共rជ兲 + cb共t兲ei共−␦/2−b兲tub共rជ兲, 共11兲
where ua共rជ兲 and ub共rជ兲 associated with Ej=បjfor j = a and b,
related to the two-component wave function⌿ in Eq. 共7兲, as
eigenfunctions and eigenenergies of H0. Moreover, an optical transition takes place between two states that correspond to dipole-allowed eigenstates conforming to the relation ⌬mj
= ± 1. As usual,␦⬅共⌬E/ប−⍀兲 is the detuning defined as the frequency difference between the level spacing and the laser field, as shown in the inset of Fig. 3共a兲. Inserting Eq. 共11兲
into the time-dependent Schrödinger equation, the time evo-lution of an electron can be expressed as
冋
c˙a共t兲 c˙b共t兲册
= i 2冋
−␦ RD+ R˜ RD * + R˜* ␦册
冋
ca共t兲 cb共t兲册
, 共12兲with R˜ =RA+ RSO, which after some calculations have the
re-lations
FIG. 3. 共Color online兲 共a兲 The energy spectrum including two states:兩a典=兩−2.5典P
↑and 兩b典=兩−1.5典P↓. The sketch in the inset de-picts inter-ring transitions under EM wave stimulations.共b兲 Popu-lation inversion as a function of time. When ignoring the spontane-ous emission, W共t兲 with large detuning is demonstrated by dotted symbol. If the spontaneous emission of an excited state is consid-ered, then in small detuning regime, where we set ␥=0.03 and
␦= 0.02, W共t兲 manifests an underdamped oscillation, as depicted by the solid curve, and the decay behavior is fitted by an exponential decay function, as shown by the dashed line.
RD= eE0具ua共r兲兩r兩ub共r兲典/4ប,
RB= e2E0具ua共r兲兩A⌽共r兲兩ub共r兲典/4បm*c⍀,
RSO=␣eE0具ua共r兲兩ub共r兲典/2ប2⍀.
Therein, RDis the dipole-induced Rabi frequency as usually
discussed and RB and RSOdenote the couplings of the laser
field with the magnetic flux and the Rashba SOI, respec-tively. In calculating RSO, only inner products between partial
waves with different spin orientations would be taken into account. In deriving Eq.共12兲, we utilized the rotating-wave
approximation共RWA兲 and ignored the counter-rotating terms proportional to exp关±i共⍀+⌬E/ប兲t兴.
For an electron initially occupying the low-energy state 兩b典, the time-dependent population probability can be exactly written as 兩ca兩2= R ˜ eff 2 ⍀d 2 sin 2
冉
⍀dt 2冊
, 兩cb兩2= 1 − R ˜ eff 2 ⍀d 2 sin 2冉
⍀dt 2冊
, 共13兲 where⍀d 2= R˜eff2 +␦2 and R˜eff= RD+ R˜ can be regarded as the
effective Rabi frequency in the presence of the external fields
and the SOI. The on-resonance transitions occur when␦= 0; in the case, the population probability can be simply reduced to兩ca兩2= sin2共R˜efft / 2兲 and 兩cb兩2= cos2共R˜efft / 2兲.
If the energy dissipation to the environment is considered as the interaction between an electron with continuous vacuum modes, a phenomenological decay parameter␥ will be introduced to the first element of the matrix in Eq.共12兲,
which opens a decay path from the excited state to its sur-rounding. Thus, the coefficients describing the system will be changed in the more complicated form,
兩ca兩2= R ˜ eff 2 ⍀˜2e −␥te−⍀˜ tsin2
冉
⍀˜t 2冊
, 兩cb兩2= e−␥t ⍀˜2e −⍀˜ t冋
共␥2+␦2兲sin2冉
⍀˜t 2冊
+⍀ ˜2cos2冉
⍀˜t 2冊
+ 2⍀˜ 共␥−兩␦兩兲sin共⍀˜t兲册
, 共14兲 where= cos共1/ 2兲 and= sin共2/ 2兲, with1= cos−1关共␥2−⍀d 2兲/⍀˜2兴, 2= sin−1关2兩␦兩␥/⍀˜2兴, ⍀˜2=
冑
共␥2−⍀ d 2兲2+共2␦␥兲2,in which1and2should be taken from the same quadrant. For convenience, we introduce the notation 兩mj典P to
specify a state in terms of the total angular momentum mj
and its spin polarizability P. We choose two states 兩a典=兩−2.5典P↑ and 兩b典=兩−1.5典P↓, respectively, at ⌽=1.3⌽0 with Ea= 37.03 meV and Eb= 30.64 meV, as depicted in Fig. 3共a兲. The corresponding population inversion
W共t兲 = 兩cb共t兲兩2−兩ca共t兲兩2 共15兲
of this system is demonstrated in Fig. 3共b兲. If the vacuum fluctuation is absent 共␥= 0兲, the total probability is con-served, i.e.,兩cb共t兲兩2+兩ca共t兲兩2= 1 at any time. In this case, W共t兲
manifests the oscillating behavior within the interval 关1−2共R˜eff
2 /⍀
d
2兲,1兴 without any dissipation. For ␦→0, this interval approaches to its maximum values关−1,1兴, namely, that ⍀d→R˜eff. For large detuning, e.g., ␦= 0.2, this interval will shrink to its 70%, and the blueshifted inversion curve is plotted in the dotted line in Fig.3共b兲.
If the spontaneous emission of an excited state is consid-ered, W共t兲 will decay with time, as shown by the solid curve in Fig.3共b兲, in which␥= 0.03 and␦= 0.02. In this case, W共t兲 oscillates underdamped. The damping behavior which is con-sistent with the Weisskopf-Wigner theory27 is well fitted by
Wfit= e−⌫t, where the Fermi’s rate ⌫⬃0.03. When the Rabi relaxation time is defined asR= 1 /⌫, our calculation shows
that a cycle of inter-ring transition accomplishes inR for a
given␥. While the spontaneous decay can be experimentally controlled and suppressed,28 in a cavity with a limited num-ber of modes at the transition frequency, a long decoherence time is permitted. In assumption of weak system-environment coupling, efficient population transfers are fea-sible. So, in conclusion, under SOI we can simultaneously manipulate electron transitions associated with its spin orien-tations in either rings via the RO processes of a two-level model.
IV. THE PHOTON-ASSISTED TRANSITIONS IN THREE-LEVEL SCHEMES
In this section, we apply the Rabi model to several inter-esting cases. We show how inter-ring transition are achieved via the photon-assisted processes. The processes are demon-strated by considering three-level systems in cascade-type and⌳-type schemes that are shown in Figs4共a兲and4共b兲. For clarity, we rewrite the time-dependent wave function as
共rជ, t兲=兺j=13 cj共t兲e−ijtuj共rជ兲, where Ej=បj and we define
E12= E2− E1, E13= E3− E1 and E12+␦= E13−␦=ប⍀. The in-volved states construct two dipole-allowed transitions 兵兩1典↔兩2典其 and 兵兩1典↔兩3典其 and a 兵兩2典↔兩3典其 dipole-forbidden paths. For a clear demonstration, we shall ignore spontane-ous emission processes.
A. Cascade type
It is well known that the quantum-beat spectroscopic method permits the resolution of closely neighboring levels.29 Earlier experiments demonstrate that quantum-beat spectroscopy is a useful technique in the measurement of Zeeman splittings and hyperfine intervals in atomic and mo-lecular systems.30It is then interesting to study spectroscopic dynamics involving the direct inter-ring transitions in the
cascade-type scheme nearby the avoided crossing points. Again, we use the notation兩mj典Pto specify the involving
states. Here, we consider an up-spin state兩1典=兩−1.5典P ↑, and down-spin doublets 兩2典=兩−2.5典P
↓ and 兩3典=兩−2.5典P↓ at ⌽=1.7⌽0. For state兩1典, the electron is localized in the inner ring. The electron wave functions, however, extend over the double ring for two higher states. The energy spectrum is depicted in Fig.4共a兲, and in Fig.4共b兲we show the sketch of the cascade model. Similar to solving Eq.共12兲 for the
two-level system, transitions among different spin states can be investigated through c˙1共t兲 = i 2关e −i␦t R+12c2共t兲 + ei␦tR−13c3共t兲兴, c˙2共t兲 = i 2e i␦t R+*21c1共t兲, c˙3共t兲 = i 2e −i␦tR − *31c 1共t兲, 共16兲 where R±ij= RD ij ± R˜ij. Adopting transformations c 2共t兲 = C2共t兲ei␦t and c3共t兲=C3共t兲e−i␦t, Eq.共16兲 can be reexpressed as a homogeneous autonomous equation. For an electron
ini-tially occupying state兩1典, the population probability in small detuning regime can be expressed approximately as
兩c1共t兲兩2= cos2
冉
MRt 2冊
, 兩c2共t兲兩2=冉
R+12 MR冊
2 sin2冉
MRt 2冊
, 兩c3共t兲兩2=冉
R−13 MR冊
2 sin2冉
MRt 2冊
, 共17兲 where MR=冑
共R+ 12兲2+共R −13兲2. Beyond the small detuning ap-proximation, Eq.共16兲 is numerically solved and the result for
␦= 0.086 is shown in Fig.4共c兲. Since transition probabilities come up differently between two paths, namely, R+12⫽R−13, occupations with time on each states turn out to be aperiodic and less regular compared with probabilities obtained from Eq. 共17兲. Discrepancies between Figs. 4共c兲 and 4共d兲 are clearly illustrated. However, in the case ␦→0, the electron tends to oscillate between 兩3典 and 兩1典 and the transition 兩2典↔兩3典 is less efficient, as compared with the off-resonance transitions.
This drawback can be removed by another interesting ma-nipulation in the cascade scheme, namely, by transferring electrons ladder by ladder with chirped laser pulses. The idea originates from the electron transfer in molecules, in which stepwise excitations are applied for a rapid and efficient dis-sociation of specific chemical bonds.31Under ARP condition, electrons that are resonantly pumped to state 兩2典 could be efficiently transited to the final state 兩3典 in a long trapping time.
To this end, we select a chirped laser pulse with time-dependent electric field along the radial direction, given by
Eជch共t兲 = Echexp
冉
−t2
22− i⍀cht − i
t2
2
冊
rˆ, 共18兲 where is the pulse duration,⍀ch is the central frequency,andis the temporal chirp. The equation of motion in RWA modified from Eq.共12兲 becomes
冋
c˙2共t兲 c˙3共t兲册
= i 2冋
0 Rch共t兲 Rch*共t兲 2␦ch共t兲册
冋
c2共t兲 c3共t兲册
, 共19兲in which ␦ch共t兲=t is linearly chirped detuning, and Rch共t兲
stands for the time-dependent Rabi frequency related to the pulse envelope ofEជch共t兲. By choosing proper parameters for
a pulse with peak Rabi frequency R0= 0.25, = 0.01, and = 10, a complete transfer is demonstrated by the dash-dotted line in Fig. 4共d兲. The numerical result coincides with the estimation of the Landau-Zener formula,18
P⯝ 1 − exp
冉
−R0 22
冊
. 共20兲In the adiabatic limit兩兩ⰆR02, electrons have great probabil-ity to occupy an excited state in the long-time limit. Distin-guished time-evolution populations between stimulated tran-sitions by cws and steady transfer by a chirp pulse are
FIG. 4. 共Color online兲 共a兲 The energy spectrum including the chosen three levels, an up-spin state兩1典=兩−1.5典P
↑ and down-spin doublets兩2典=兩−2.5典P↓ and兩3典=兩−2.5典P↓at⌽=1.7⌽0. Level
repul-sion occurs between the doublets, and the gap is about 0.17 meV for␣=40 meV nm. 共b兲 A sketch of a cascade-type model in the double ring population as a function of time for the cases of共c兲
␦= 0.086 and 共d兲␦→0. Dash-dotted curve in 共d兲 shows the ARP estimation of the transition from兩2典 to 兩3典.
depicted by curves in Fig.4共d兲. Therefore, in addition to the manipulation of electron transitions between a single ring and a double ring via the subjection to cw irradiations, we also arrive at the optimal control on stably selective excita-tions with chirped pulses in cascade-type systems.
B.⌳-type transition
Finally, we shall also investigate an interesting and impor-tant phenomenon: Whenever there are level crossings for two energy states that belong to either one ring, ⌳-type scheme of indirect inter-ring transitions among which and one higher energy state can be switched on. Applications of this model has been proposed both in superconducting quantum interfer-ence device33 and semiconductor double quantum dots,34 in which multilevel ROs as a target toward coherent control have been demonstrated.
As Fig.5共a兲 shows that for an electron occupying state 兩2典, the photon-assisted quantum transition is initiated from the inter-ring surpassing an intermediate-state兩1典 and finally reaching the outer-ring of state 兩3典. On-resonance solutions of the mediated-indirect-transition system with initial condi-tions c1共0兲=0, c2共0兲=1, and c3共0兲=0 are
兩c1共t兲兩2=
冉
R+12 MR冊
2 sin2冉
MRt 2冊
, 兩c2共t兲兩2=冉
R+12 MR冊
4 cos2冉
MRt 2冊
+冉
R−13 MR冊
4 +2共R+ 12R − 13兲2 MR 4 cos冉
MRt 2冊
, 兩c3共t兲兩2= 共R+ 12 R−13兲2 MR 4冋
cos冉
MRt 2冊
− 1册
2 , 共21兲and the transition probabilities are shown in Fig.4共b兲. The incomplete transfer and occupation are restricted due to un-equal effective Rabi frequencies in two paths. Once the ex-ternal field optimizes the efficiency of one path, the effi-ciency of the other is not optimal.
The formulas in Eq.共21兲 clearly show that the probability
兩c3共t兲兩2 has a maximum value at t = m/ MR with an odd
in-teger m and a minimum value at t = n/ MR with an even
integer n. At these extreme points, we have the ratio 兩c2共t兲兩2/兩c3共t兲兩2=关共R+ 12兲2−共R − 13兲2兴2/ 4共R + 12兲2共R − 13兲2, which indi-cates the optimal transfer at 兩R+12兩/兩R−13兩=1. Away from this ratio, the transfer efficiency will decrease. In Fig.4共b兲, tran-sition among a doublet of mj= 0.5 and 2.5 and an auxiliary
level of mj= 1.5 at⌽=0.8⌽0is considered. In this case, there is only 45% occupation in兩3典 for R+12/ R−13⬃0.38. If the ef-fective Rabi frequencies of the paths 2↔1 and 1↔3 are the same, the optimal control of electron dynamics is feasible. To show this, we calculate the time-dependent occupation prob-ability in each state. In the inset, it is clear that at extreme times, occupation in兩3典 based on a pseudo-⌳-transition pro-cess has a maximum value. Meanwhile, states兩1典 and 兩2典 are left empty. Moreover, we also investigate the transition pro-cess stimulated by chirped pulse irradiation. By setting R0
= 0.25,= 0.001, and= 14.15, we obtain a long-time occu-pation of the final state, obeying estimation of Landau-Zener’s formula. The result is shown by dash-dotted line in the inset.
Within proper controls of the pulse width against the typical level spacings and the dephasing time of electrons, it is easy to manipulate electronic states in ⌳-scheme system by successive application of pulses.32Using the same param-eters as in Fig.4共b兲, we simulate the pulse-induced periodic oscillations. Here, the Gaussian pulse duration= 7.3 and the pulse interval is about 7. Different from the sinusoidal ROs, an alternative square wave is shown in Fig.4共c兲. Apparently, the level occupation time in both inner and outer rings is prolonged. Moreover, since the duration of the pulse is prop-erly tuned, it can be expected that time evolution of the oc-cupation should be complete in the pseudo-⌳-transition pro-cess. Otherwise, underexcitation or overexcitation takes place corresponding to the duration being too short or too long, respectively. Level occupation will never be or just be
FIG. 5.共Color online兲 共a兲 A sketch of the ⌳-type scheme. States here are all with effective up-spin orientations. An electron initially occupies state兩2典 can be optically pumped to an outer-ring state 兩3典 mediated by state兩1典. 共b兲 The population probabilities among three levels. The long-time occupation of an excited state is feasible by applying a short and intense pulse. In the inset, we show the optimal transfer is feasible provided that there is common Rabi frequency in the two paths.共c兲 An alternative output signal can be obtained under successive pulse stimulations.
transiently complete. The well controled pulse delay time also shows the flexibility of manipulations in quantum states.
V. CONCLUSION
In this work, we have studied magneto-optical transitions in a semiconductor double ring in the presence of Rashba spin-orbit coupling and magnetic flux. First, based on accu-rate numerical calculations, we obtain SOI-accompanied AB energy spectra and corresponding eigenstates. The presence of the SOI has important influence on the occurrence of level crossings, showing the evidence both for the periodic orbital motion and for spin flips. In addition, there are anticrossing levels playing the role of the magnetic-resonant extraditions of electrons between inner and outer rings. In high-energy regime, occurrence of avoided crossings indicates a chaotic signature in its classical analogy. To facilitate the peculiar features of the double-ring system, we have designed some interesting dynamic processes such that the system can be easily explored experimentally in the near future.
We have studied the temporal evolution processes in two-level and two three-two-level models. The interaction between external fields and electron results in the successive stimu-lating absorption and emission of a photon and turns out as the effective Rabi oscillators. In the two-level model, we demonstrate an alternative manipulation of electrons transit-ing between two rtransit-ings. In cascade scheme, aperiodic and incomplete population transfers are revealed under the sinu-soidal field excitations. By appropriate tuning SOI strength, the gap between avoided crossing levels can be reduced such that the rectified output signals are measurable. Moreover, by
short pulse excitations, we also demonstrate the possibility of optimal control of selective and direct signals. In this work, only one ladder transition is demonstrated, which, however, can be extended to multiladder transitions following the same principle. Finally, we have explored the photon-assisted tunneling in the⌳-type model. In addition to gen-eration of ROs also, we give the criterion of the most effi-cient transfer via the mediated-indirect-tunneling paths. Further, by successive pulse irradiations, the well control on pulse delay results in the time prolongation on state popula-tions. We should also emphasize that in the⌳-type scheme, the minimization of the intermediate-level population is achieved which is an important and practical strategy in de-vice realization.
The presence of SOI allows the manipulation of spin de-gree of freedom and it is timely to examine the spin-dependent optical response. Since similar features could be found in double-dot systems, the above theoretical results in a double ring might shed light on future experimental find-ings in these burgeoning quantum systems. While there are few works on optically induced and SOI-driven spin dynam-ics in quantum systems,20,35,36we believe that the theoretical and experimental works of related spin readout information by optical pumping in ringlike systems could be carried out in the near future.
ACKNOWLEDGMENTS
This work was supported by National Science Council and Academia Sinica in Taiwan. The authors are grateful for valuable discussions with V. Gudmundsson, Y. N. Chen, G. Y. Chen, and W. Xu.
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