Influence of Pulse width and Rabi frequency on the Population dynamics

Transcription

1 Influence of Pulse width and Rabi frequency on the Poulation dynamics of three-level system in two-hoton absortion rocess Nam-Chol Kim,,* Myong-Chol Ko, Song-Jin Im, Zhong-Hua Hao Deartment of Physics, Kim Il Sung University, Pyongyang, D. P. R. Korea Deartment of Physics, Key Laboratory of Acoustic and Photonic Materials and Devices of Ministry of Education, Wuhan University, Wuhan 437, P. R. China Abstract : We investigate the oulation dynamics of the three-level system in the two-hoton absortion (TPA) rocess, mainly focusing the influence of ulse width and Rabi frequency on the oulation dynamics of the system. We observe the deendency of the oulation with the Rabi frequency and the ulse width. We also show that the arbitrary suerosition state consisted in two states, uer state and lower state, is ossible by controlling the ulse width and Rabi frequency. The results obtained can be used to the case of more comlex multilevel system and they can be valuable for coherent quantum control in quantum information rocessing. Keywords: Two-hoton absortion; Poulation dynamics; Pulse width * address: elib.rns@hotmail.com

2 . Introduction Two-hoton absortion (TPA) in nanostructures has been the focus of attention of many researchers because of their otential alication in a wide range of otical devices[-3]. TPA is one of the most basic radiation-matter interaction mechanics. It consists in the excitation of an atom or molecule from a lower quantum state(ground state) > to an uer quantum state (excited state) > of the same arity as > in a single ste. In this case the initial and final states cannot be connected through an electric-diole transition. Thus arity conservation imlies that two light quanta must be absorbed simultaneously. The theory of TPA was first develoed by Maria Göert-Mayer in 93[4]. As a multihoton rocess, TPA is closely related to Raman scattering. In the latter rocess, one hoton is absorbed while the other is simultaneously emitted, the energy difference being retained by the molecule. While sontaneous Raman scattering was observed as early as 98[5], TPA was not observed until 9[] after the advent of the laser. The reason for that delay in the observation of the two multihoton rocesses lies in the fact that while in sontaneous Raman scattering the scattered light intensity is roortional to the intensity of the incoming radiation, in TPA the ower absorbed is roortional to the square of the intensity of the incoming field and thus higher excitation energy is required for TPA. One of the many alications of TPA is in sectroscoy. The use of two-hoton absortion for sectroscoy studies has advantages over the single-hoton rocess because of the restriction on selection rules; e.g., when a single hoton absortion is forbidden by selection rules, a higher-order absortion rocess may be allowed [7]. Another advantage of two-hoton absortion sectroscoy is the ability to study the roerties truly characteristic of the crystalline volume because of the small values of the nonlinear absortion coefficient [8]. Nowadays, coherent control of quantum states with light has attracted great interest as a means to influence the outcome of a quantum-mechanical interaction. In rincile, the quantum system can be controlled towards a desired state by its interaction with light [9]. Steering quantum rocesses in atoms and molecules through the maniulation of the roerties of otical fields is the goal of coherent quantum control [-]. In articular, control of oulation transfer in multihoton transitions with laser ulses has been achieved

3 through several methods. A common strategy for ultrashort ulses is the maniulation of the sectral hases and amlitudes of the frequency comonents of the fields exciting the medium, resulting in ulses or sequences of ulses with very diverse temoral shaes which enhance or suress the couling between selected states by exloiting quantum interference effects [3 9]. Other techniques using longer ulses such as adiabatic methods [ ] and π -ulse olychromatic control [3-5] can be imlemented to induce comlete oulation transfer between a air of quantum states. Multihoton control techniques are valuable in several areas, including nonlinear sectroscoy [], in femtochemistry and biology [7], or in quantum information and in the quantum engineering of light states [8]. In recent work [9] it was shown analytically and numerically that the oulation of the quantum system can be controlled by changing the duration, shae and intensity of the inut ulse for a TPA rocess. Recently, it was also shown that the two-hoton transition rate of quantum dots couled to nanocavities are enhanced by u to several orders of magnitude relative to quantum dots in bulk host and roosed a simle cavity design to achieve QD degenerate two-hoton absortion with low ower, which allows us to emloy enhanced TPA to coherently excite QDs and enable generation of indistinguishable single hotons on demand [3]. So far, most reorted multi-hoton studies on quantum systems were focused on two-hoton absortion and excitation rocesses with changing of such arameters as ulse shae, ulse chir, ulse area, detuning and the size of quantum systems like quantum dots(qds). In this aer, we consider the oulation dynamics of a three-level system mainly focusing the influence of ulse width and the Rabi frequency to the occuation robabilities of the quantum system.. Theoretical Model Now, we are considering the two-hoton absortion in a three-level system (e.g., semiconductor quantum dot; QD), the schematic of which is shown in Fig.. 3

4 Fig.. Energy diagram of two-hoton three-level model. We start from the Hamiltonian in the Schrödinger icture. In the Schrödinger icture, the Hamiltonian of the QD system can be written in the form of the sum of the unerturbed art and interaction art as follow, Ĥ s = Ĥ + Ĥ. () The unerturbed Hamiltonian of the QD three-level system is given as follow, where = ( E E )/ h ij i j Ĥ = h ω + h, () ω ω are the angular frequencies of the transitions, with E i being the energies of the different quantum state, the intermediate state of which is taken to be the origin of energies. ω is the angular frequency of the incoming laser field and Γ, Γ are the decay rate of the states and, resectively. The allowed electric-diole transitions are and, because levels (excited state)and (ground state) have the same arity, and contrary to that of intermidiate state. Thus the diole-moment oerator can be written as μ = μ + μ + μ + μ, (3) where μij = i μ j, that can be taken to be real without loss of generality through roer choice of the basis states hases( μij = μ ji ). The interaction Hamiltonian of an atom located at z = reads Ĥ ( t) = μ Ε, (4) where we consider a classical monochromatic electromagnetic field of the form ( t) = E( t) Ε e. (5) In order to transform the Hamiltonian from the Schrödinger icture to the interaction 4

5 icture, we introduce a unitary oerator as ihˆ t / h iωt iωt () = = + + Uˆ t e e e. () The interaction Hamiltonian in the interaction icture is defined by ˆ ˆ ˆ ˆ I H = U HU. (7) The interaction Hamiltonian in the Schrödinger icture is as follow, (8) Ĥ μ Ε () t = Ω ( t) +Ω ( t) +Ω ( t) +Ω () t = where Ω () t = ( μ e) E( t) h h, and Ω () t = ( μ e) E( t) transitions and, resectively. The interaction Hamiltonian in the interaction icture is given by h are the Rabi frequencies of the ˆ ˆ ˆ ˆ I = H U HU () i () () i () = h ˆ σ ˆ Ω t + σ Ω t + H.. c (9) where ( ) e iω t ˆ i ˆ i e iω σ = and ( ) σ = t are the diole oerators of the transition in the interaction icture. The oulation dynamics of the system is described by the master equation in the Lindblad form where the dissiative term d = i [ H ˆ, ] + Lˆ( ) I dt ρ h ρ ρ, () L( ˆ ρ ) is ˆ Γ Γ L( ρ ) = ( ˆ σρσˆ ˆ σρ ρσˆ ) + ( ˆ σρσˆ ˆ σρ ρσˆ ) () Combining Eqs. () - (), the otical Bloch equations for the oulation of the system are obtained: [ ] & ρ = iω() t ρ ρ + Γ ρ, (b) & ρ = iω ()( t ρ ρ ) Γ ρ, (c) Γ & ρ = iω()( t ρ ρ) + iω () t ρ iωρ ρ, (d) Γ & ρ = iω()( t ρ ρ) iω () t ρ + iωρ ρ, (e) & ρ = iω() t ρ iω ()( t ρ ρ ) + iω ρ ( Γ + Γ) ρ, (f) 5

6 & ρ = iω() t ρ + iω ()( t ρ ρ) iωρ ( Γ + Γ) ρ, (g) Γ & ρ = iω() t ρ + iω () t ρ iωρ ρ, (h) Γ & ρ = iω() t ρ iω () t ρ + iωρ ρ (k) In Eqs. (a) - (k), ρ, ρ and ρ are the oulations of levels, and, resectively, ρ, ρ and ρ are the coherences between the energy levels. 3. The numerical evaluation of the oulation dynamics In order to get the values of numerical integration of the above master equations (Eqs. (a) - (k)), we use the secant hyerbolic ulse laser, the ulse electric field of which can be written as () t ee sec h[ t / ] cosωt τ E =, (3) where E is the constant amlitude of electric field, τ is the ulse width and ω is the laser field frequency. We can evaluate the value of the quantities as () t = ( e) E( t) / h () t = ( e) E() t / h Ω Ω μ, μ on condition that the diole couling coefficients μ = μ e, μ = μ e and the ulse electric field E(t) are given. In our numerical calculations, we choose the arameters as ω =. GHz, ω =.4 GHz, ω = GHz, 9 μ E μ = μ =.85 Cm, thus Ω ( t) = Ω ( t) = Ω sech[ t /τ ] cosωt, where Ω =, h which is deendent with the amlitude of electric field and the diole moments, as well as the angle between the diole moments and the direction of the electric field. At first, we set Γ = Γ for simlicity. = Figure shows the oulation of three-level system of QD as function of time; we kee the ulse width constant while changing the Rabi frequency. Fig. (a) shows the comlete deoulation of the excited state with the Rabi frequency Ω =. GHz, where we set the ulse width t = s. Fig. (b) shows that setting the Rabi frequency as Ω =.53 GHz results the suerosition state with about 73% ground state and % excited state. One can see from Fig. (c) that equal occuation robabilities of the states are

7 also ossible by setting such suitable arameter as Ω =. 499 GHz, in which the state of the QD is the suerosition of the 5% ground state and the 5% excited state. We can also found from Fig. (d) that oulation of the uer state can also be nearly, but not the same. These numerical calculations show that coherent control of the oulation transfer can be achieved by setting roer arameters such as the amlitudes of the electric field and the diole couling coefficients. Fig.. Poulations of QD versus time with different Rabi frequencies. (a) Ω =. GHz, (b) Ω =.53 GHz, (c) Ω =.499 GHz, (d) Ω =.785 GHz, where t = s, Γ = Γ. = We can find the deendency of the oulation with the Rabi frequencies of the system. Figure 3 shows the oulation as a function of Rabi frequency while keeing the ulse width constant as t = s. One can see from the Fig. 3 easily the eriodic deendency of the oulation with the Rabi frequency, however, the height of the eaks becomes lower as the Rabi frequency increases. ( ) 7

8 Fig. 3. Poulation as a function of Rabi frequency Ω ( t = s, Γ = Γ ). = Figure 4 shows the oulation of three-level system of QD as function of ulse width, where we set the Rabi frequency as Ω =. 4 GHz. From Fig. 4 one can see that the width of the oulation eaks is increased with increasing of the ulse width. We can also find that the maximum value of the eaks does not change monotonously. Fig. 4. Poulation as a function of ulse width t ( Ω =.4 GHz, Γ = Γ ). = Finally, we can also consider the decoherence rocess of the QD system in the TPA rocess. Figure 5 shows the rate of the excitation state oulation to the ground state oulation versus decay rate of the states. Here, we set Γ = Γ = Γ for simlicity. As we can see easily from Fig. 5, the rate of the oulations of the QD system is decreased raidly with increasing decay rate and saturated about from the decay rate.5s -. This result gives an insight of the range of decay rate of the system considered. 8 ( ) Fig. 5. Rate of Poulations ρ / ρ versus decay rate Γ. The arameters used are t = s, Ω =.785 GHz.

9 4. Conclusion In conclusion, we have investigated the influence of ulse width and Rabi frequency on the oulation dynamics of a three-level system in the two-hoton absortion rocess. Our study gives valuable ranges of arameters such as ulse width, amlitude of the electric field and diole couling coefficients for ractical alications. We confirmed that controlling ulse width and intensity of the electric field can be used to achieve oulation transfer on demand. The results can be extended to the case of more comlex system such as, e. g., multilevel systems and they can be valuable for coherent quantum control schemes and quantum information rocessing. Acknowledgments This work was suorted by Key Project for Frontier Research on Quantum Information and Quantum Otics of Ministry of Education of D. P. R of Korea. References. F. Boitier, A. Godard, A. Ryasnyanskiy, N. Dubreuil, P. Delaye, C. Fabre, and E. Rosencher; Ot. Exress, 8(9), 4 ().. G.-S. He, K.-T. Yong, Q.-D. Zheng, Y. Sahoo, A. Baev, A. I. Ryasnyanskiy, and P. N. Prasad, Ot. Exress, 5(), 88 (7). 3. S. Xiao, H.- M. Gong, X.-R. Su, J.-B. Han, Y.-B. Han, M.-T.C, and Q.-Q. Wang, J. Phys. Chem. C., (7). 4. M. Göert - Mayer, Ann. Physik 9, 73 (93). 5. C. V. Raman, Indian J. Phys., 387 (98).. W. Kaiser, and C. G. B. Garret, Phys. Rev. Lett. 7, 9 (9). 7. V. Nathan, A. H. Guenther, and S. S. Mitra, J. Ot. Soc. Am. B, 94 (985). 8. H. Garcia, Phys. Rev. B 74, 35 (). 9. D. L. Silva, L. Misoguti, and C. R. Mendonca, J. Phys. Chem. A. 3, (9).. J. L. Herek, J. Photochem. Photobiol. A: Chem. 8, 5 ().. A. Monmayrant, B. Chatel, B. Girard, Ot. Comm. 4, 5-3 (). 9

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