Ultrashort electromagnetic pulse control of intersubband quantum well transitions
© Paspalakis and Boviatsis; licensee Springer. 2012
Received: 18 July 2012
Accepted: 3 August 2012
Published: 23 August 2012
We study the creation of high-efficiency controlled population transfer in intersubband transitions of semiconductor quantum wells. We give emphasis to the case of interaction of the semiconductor quantum well with electromagnetic pulses with a duration of few cycles and even a single cycle. We numerically solve the effective nonlinear Bloch equations for a specific double GaAs/AlGaAs quantum well structure, taking into account the ultrashort nature of the applied field, and show that high-efficiency population inversion is possible for specific pulse areas. The dependence of the efficiency of population transfer on the electron sheet density and the carrier envelope phase of the pulse is also explored. For electromagnetic pulses with a duration of several cycles, we find that the change in the electron sheet density leads to a very different response of the population in the two subbands to pulse area. However, for pulses with a duration equal to or shorter than 3 cycles, we show that efficient population transfer between the two subbands is possible, independent of the value of electron sheet density, if the pulse area is Π.
KeywordsCoherent control Semiconductor quantum well Intersubband transition Ultrashort electromagnetic pulse
The coherent interaction of electromagnetic fields with intersubband transitions in semiconductor quantum wells has led to the experimental observation of several interesting and potentially useful effects, such as tunneling-induced transparency[1, 2], electromagnetically induced transparency, Rabi oscillations[4, 5], self-induced transparency, pulsed-induced quantum interference, Autler-Townes splitting[7, 8], gain without inversion, and Fano signatures in the optical response. In most of these studies, atomic-like multi-level theoretical approaches have been used for the description of the optical properties and the electron dynamics of the intersubband transitions.
Many-body effects arising from the macroscopic carrier density have also been included in a large number of theoretical and experimental studies of intersubband excitation in semiconductor quantum wells[6, 10–38]. These studies have shown that the linear and nonlinear optical responses and the electron dynamics of intersubband quantum well transitions can be significantly influenced by changing the electron sheet density.
An interesting problem in this area is the creation of controlled population transfer between two quantum well subbands[23–27, 29, 30]. This problem was first studied by Batista and Citrin including the many-body effects arising from the macroscopic carrier density of the system. They showed that the inclusion of the electron-electron interactions makes the system behave quite differently from an atomic-like two-level system. To have a successful high-efficiency population transfer in a two-subband, n-type, modulation-doped semiconductor quantum well, they used the interaction with a specific chirped electromagnetic field, i.e., a field with time-dependent frequency. They showed that a combination of Π pulses with time-dependent frequency that follow the population inversion can lead to high-efficiency population inversion. Their method was refined in a following publication where only linearly chirped pulses were used for high-efficiency population transfer and was also applied to three-subband quantum well systems.
Different approaches for creating high-efficiency intersubband population transfer were also proposed by our group[24, 25, 29, 30]. Using analytical solutions of the effective nonlinear Bloch equations, under the rotating wave approximation, we presented closed-form analytical solutions for the electric field amplitude of the electromagnetic field that leads to high-efficiency population transfer[24, 25]. In addition, closed-form conditions for high-efficiency transfer were also presented[24, 29]. Moreover, efficient population transfer is found when a two-subband system interacts with a strong chirped electromagnetic pulse, for several values of the chirp rate and the electric field amplitude.
In this article, we continue our work on the creation of high-efficiency controlled population transfer in intersubband transitions of semiconductor quantum wells. We give emphasis to the case of interaction of the semiconductor quantum well with electromagnetic pulses with a duration of few cycles and even a single cycle. We numerically solve the effective nonlinear Bloch equations for a specific double GaAs/AlGaAs quantum well structure, taking into account the ultrashort nature of the applied field, and show that high-efficiency population inversion is possible for specific pulse areas. The dependence of the efficiency of population transfer on the electron sheet density and the carrier envelope phase of the pulse is also explored. More specifically, we find that for electromagnetic pulses with duration of several cycles, the change in the electron sheet density leads to a very different response of the population in the two subbands to pulse area. However, a Π pulse with a duration equal to or shorter than 3 cycles can lead to efficient population transfer between the two subbands independent of the value of electron sheet density.
We note that the interaction of ultrashort electromagnetic pulses with atoms has been studied in the past decade, giving emphasis either to ionization effects[39–41] or to population dynamics in bound two-level and multi-level systems[40, 42–46]. Also, the interaction of ultrashort electromagnetic pulses with intersubband transitions of semiconductor quantum wells has been recently studied[47, 48], but without taking into account the effects of electron-electron interactions in the system dynamics.
Here, N is the electron sheet density, ε is the relative dielectric constant, e is the electron charge, E0 and E1 are the eigenvalues of energy for the ground and excited states in the well, respectively, and L ijkl = ∫∫ dzd z ′ ξ i (z)ξ j (z ′ )|z−z ′ |ξ k (z ′ )ξ l (z), with i,j,k,l = 0,1. Also, ξ i (z) is the envelope wavefunction for the i th subband along the growth direction (z-axis). Finally, in Equations 1 to 3, the terms containing the population decay time T1 and the dephasing time T2 describe relaxation processes in the quantum well and have been added phenomenologically in the effective nonlinear Bloch equations. If there is no relaxation in the system T1,T2→∞, then.
In comparison with the atomic (regular) optical Bloch equations, we note that in the effective nonlinear Bloch equations, the electron-electron interactions renormalize the transition frequency by a time-independent term (see Equation 4). The parameter γ consists of two compensating terms: the self-energy term and the vertex term. In addition, the applied field contribution is screened by the induced polarization term with coefficient β. The screening is due to exchange correction. Surprisingly, the exchange corrections appear with terms which are linearly dependent on the electron sheet density, as all exchange terms which present a nonlinear dependence on the electron sheet density are exactly canceled out due to the interplay of self-energy and vertex corrections.
In the above formula, the first term corresponds to an electromagnetic pulse with a sine-oscillating carrier field, while the second term arises because of the finite pulse duration. This second term can be neglected for pulses with a duration of several cycles, but has an important effect in the single-cycle regime[39–43, 47, 48].
where Λ(t) is the time-dependent pulse area. At the end of the pulse, Λ(t) takes a constant value that is known as pulse area θ. Equation 9 clearly shows how important pulse area can be. If θ is an odd multiple of Π, then complete inversion between the two states is found at the end of the pulse, while if θ is an even multiple of Π, then the population returns to the lower state at the end of the pulse.
Results and discussion
This system has been studied in several previous works[20, 24, 25, 28, 35–38]. The electron sheet density takes values between 109and 7 × 1011cm−2. These values ensure that the system is initially in the lowest subband, so the initial conditions can be taken as S1(0) = S2(0) = 0 and S3(0) = −1. The relevant parameters are calculated to be E1 − E0 = 44.955 meV and z01 = −3.29 nm. Also, for electron sheet density N = 5 × 1011cm−2, we obtain Π e2N(L1111−L0000)/2ε = 1.03 meV, ℏγ = 0.2375 meV, and meV. In all calculations, we include the population decay and dephasing rates with values T1= 10 ps and T2= 1 ps. Also, in all calculations, the angular frequency of the field is at exact resonance with the modified frequency ω10, i.e., ω = ω10.
A quite different behavior is found in Figure2b,c,d for pulses with smaller number of cycles. In Figure2b, we see that essentially the inversion dynamics differs slightly for N = 109cm−2, N = 3 × 1011cm−2, and N = 5 × 1011cm−2 and all of these values lead to essentially the same final inversion. There is only a small difference in the inversion dynamics for the case of N = 7 × 1011cm−2 that leads to slightly smaller inversion. For even smaller number of cycles, Figure2c,d, the inversion dynamics differs slightly for all the values of electron sheet density, and the final value of inversion is practically the same, independent of the value of electron sheet density. We note that the largest values of inversion are obtained for np = 2 and np = 3 and not for np = 1, as one may expect, as in the latter case the influence of the decay mechanisms will be weaker. However, the second term on the right-hand side of the electric field of Equation 8 influences the dynamics for np = 1, and in this case, the pulse area θ = Π does not lead to the largest inversion.
In this work, we have studied the electron dynamics of intersubband transitions of a symmetric double quantum well, in the two-subband approximation, that is coupled by a strong pulsed electromagnetic field. We have used the effective nonlinear Bloch equations for the description of the system dynamics, giving specific emphasis to the interaction of the quantum well structure with few-cycle pulses. We have found that high-efficiency population inversion is possible for specific pulse areas. The dependence of the efficiency of population transfer on the electron sheet density and the carrier envelope phase of the pulse has also been explored. More specifically, we have shown that for electromagnetic pulses with a duration of several cycles, the change in the electron sheet density leads to a very different response of the population in the two subbands to pulse area. However, electromagnetic pulses with pulse area Π or close to Π and with duration equal to or shorter than 3 cycles can lead to efficient population transfer between the two subbands independent of the value of electron sheet density.
EP holds a PhD degree from the Physics Department of Imperial College from 1999. In 2001, he joined the Materials Science Department of the University of Patras, where he is currently an assistant professor. JB holds a PhD degree from the Department of Physics of the University of Patras from 1982. From 1991 to 2001, he was an assistant professor of Physics at the Department of Applied Sciences of the Technological Educational Institute of Chalkis. In 2001, he joined the Department of Renovation and Restoration of Buildings of the Technological Educational Institute of Patras, where he is currently a professor of Physics.
This research has been co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the operational program ‘Education and Lifelong Learning’ of the National Strategic Reference Framework (NSRF) - Research Funding Program: Archimedes III. Discussions with AF Terzis and CH Keitel are gratefully acknowledged.
- Faist J, Capasso F, Sirtori C, West KW, Pfeiffer LN: Controlling the sign of quantum interference by tunnelling from quantum wells. Nature 1997, 390: 589–591. 10.1038/37562View ArticleGoogle Scholar
- Schmidt H, Campman KL, Gossard AC, Imamoglu A: Tunneling induced transparency: Fano interference in intersubband transitions. Appl Phys Lett 1997, 70: 3455–3457. 10.1063/1.119199View ArticleGoogle Scholar
- Serapiglia GB, Paspalakis E, Sirtori C, Vodopyanov KL, Phillips CC: Laser-induced quantum coherence in a semiconductor quantum well. Phys Rev Lett 2000, 84: 1019–1022. 10.1103/PhysRevLett.84.1019View ArticleGoogle Scholar
- Luo CW, Reimann K, Woerner M, Elsaesser T, Hey R, Ploog KH: Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation. Phys Rev Lett 2004, 92: 047402.View ArticleGoogle Scholar
- Choi H, Gkortsas VM, Diehl L, Bour D, Corzine S, Zhu J, Höfler G, Capasso F, Kärtner FX, Norris TB: Ultrafast Rabi flopping and coherent pulse propagation in a quantum cascade laser. Nat Photonics 2010, 4: 706–710. 10.1038/nphoton.2010.205View ArticleGoogle Scholar
- Müller T, Parz W, Strasser G, Unterrainer K: Influence of carrier-carrier interaction on time-dependent intersubband absorption in a semiconductor quantum well. Phys Rev B 2004, 70: 155324.View ArticleGoogle Scholar
- Dynes JF, Frogley MD, Beck M, Faist J, Phillips CC: ac Stark splitting and quantum interference with intersubband transitions in quantum wells. Phys Rev Lett 2005, 94: 157403.View ArticleGoogle Scholar
- Wagner M, Schneider H, Stehr D, Winnerl S, Andrews AM, Schartner S, Strasser G, Helm M: Observation of the intraexciton Autler-Townes effect in GaAs/AlGaAs semiconductor quantum wells. Phys Rev Lett 2010, 105: 167401.View ArticleGoogle Scholar
- Frogley MD, Dynes JF, Beck M, Faist J, Phillips CC: Gain without inversion in semiconductor nanostructures. Nat Mater 2006, 5: 175–178. 10.1038/nmat1586View ArticleGoogle Scholar
- Golde D, Wagner M, Stehr D, Schneider H, Helm M, Andrews AM, Roch T, Strasser G, Kira M, Koch SW: Fano signatures in the intersubband terahertz response of optically excited semiconductor quantum wells. Phys Rev Lett 2009, 102: 127403.View ArticleGoogle Scholar
- Chuang SL, Luo MSC, Schmitt-Rink S, Pinczuk A: Many-body effects on intersubband transitions in semiconductor quantum-well structures. Phys Rev B 1992, 46: 1897–1900. 10.1103/PhysRevB.46.1897View ArticleGoogle Scholar
- ZaluŻny M: Influence of the depolarization effect on the nonlinear intersubband absorption spectra of quantum wells. Phys Rev B 1993, 47: 3995–3998. 10.1103/PhysRevB.47.3995View ArticleGoogle Scholar
- ZaluŻny M: Saturation of intersubband absorption and optical rectification in asymmetric quantum wells. J Appl Phys 1993, 74: 4716–4722. 10.1063/1.354339View ArticleGoogle Scholar
- Heyman JN, Craig K, Galdrikian B, Sherwin MS, Campman K, Hopkins PF, Fafard S, Gossard AC: Resonant harmonic generation and dynamic screening in a double quantum well. Phys Rev Lett 1994, 72: 2183–2186. 10.1103/PhysRevLett.72.2183View ArticleGoogle Scholar
- Sherwin MS, Craig K, Galdrikian B, Heyman JN, Markelz AG, Campman K, Fafard S, Hopkins PF, Gossard AC: Nonlinear quantum dynamics in semiconductor quantum wells. Physica D 1995, 83: 229–242. 10.1016/0167-2789(94)00266-SView ArticleGoogle Scholar
- Craig K, Galdrikian B, Heyman JN, Markelz AG, Williams JB, Sherwin MS, Campman K, Hopkins PF, Gossard AC: Undressing a collective intersubband excitation in a quantum well. Phys Rev Lett 1996, 76: 2382–2385. 10.1103/PhysRevLett.76.2382View ArticleGoogle Scholar
- Nikonov DE, Imamoglu A, Butov LV, Schmidt H: Collective intersubband excitations in quantum wells: Coulomb interaction versus subband dispersion. Phys Rev Lett 1997, 79: 4633–4636. 10.1103/PhysRevLett.79.4633View ArticleGoogle Scholar
- Batista AA, Tamborenea PI, Birnir B, Sherwin MS, Citrin DS: Nonlinear dynamics in far-infrared driven quantum-well intersubband transitions. Phys Rev B 2002, 66: 195325.View ArticleGoogle Scholar
- Li JZ, Ning CZ: Interplay of collective excitations in quantum-well intersubband resonances. Phys Rev Lett 2003, 91: 097401.View ArticleGoogle Scholar
- Olaya-Castro A, Korkusinski M, Hawrylak P, Ivanov MYu: Effective Bloch equations for strongly driven modulation-doped quantum wells. Phys Rev B 2003, 68: 155305.View ArticleGoogle Scholar
- Haljan P, Fortier T, Hawrylak P, Corkum PB, Ivanov MYu: High harmonic generation and level bifurcation in strongly driven quantum wells. Laser Phys 2003, 13: 452.Google Scholar
- Wijewardane HO, Ullrich CA: Coherent control of intersubband optical bistability in quantum wells. Appl Phys Lett 2004, 84: 3984–3987. 10.1063/1.1751611View ArticleGoogle Scholar
- Batista AA, Citrin DS: Rabi flopping in a two-level system with a time-dependent energy renormalization: intersubband transitions in quantum wells. Phys Rev Lett 2004, 92: 127404.View ArticleGoogle Scholar
- Paspalakis E, Tsaousidou M, Terzis AF: Coherent manipulation of a strongly driven semiconductor quantum well. Phys Rev B 2006, 73: 125344.View ArticleGoogle Scholar
- Paspalakis E, Tsaousidou M, Terzis AF: Rabi oscillations in a strongly driven semiconductor quantum well. J Appl Phys 2006, 100: 044312. 10.1063/1.2227647View ArticleGoogle Scholar
- Batista AA: Pulse-driven interwell carrier transfer in n-type doped asymmetric double quantum wells. Phys Rev B 2006, 73: 075305.View ArticleGoogle Scholar
- Batista AA, Citrin DS: Quantum control with linear chirp in two-subband n-type doped quantum wells. Phys Rev B 2006, 74: 195318.View ArticleGoogle Scholar
- Cui N, Niu Y-P, Sun H, Gong S-Q: Self-induced transmission on intersubband resonance in multiple quantum wells. Phys Rev B 2008, 78: 075323.View ArticleGoogle Scholar
- Paspalakis E, Simserides C, Baskoutas S, Terzis AF: Electromagnetically induced population transfer between two quantum well subbands. Physica E 2008, 40: 1301–1304. 10.1016/j.physe.2007.08.078View ArticleGoogle Scholar
- Paspalakis E, Simserides C, Terzis AF: Control of intersubband quantum well transitions with chirped electromagnetic pulses. J Appl Phys 2010, 107: 064306. 10.1063/1.3329377View ArticleGoogle Scholar
- Cui N, Xiang Y, Niu YP, Gong SQ: Coherent control of terahertz harmonic generation by a chirped few-cycle pulse in a quantum well. New J Phys 2010, 12: 013009. 10.1088/1367-2630/12/1/013009View ArticleGoogle Scholar
- Kosionis SG, Terzis AF, Simserides C, Paspalakis E: Linear and nonlinear optical properties of a two-subband system in a symmetric semiconductor quantum well. J Appl Phys 2010, 108: 034316. 10.1063/1.3457855View ArticleGoogle Scholar
- Karabulut I: Effect of Coulomb interaction on nonlinear (intensity-dependent) optical processes and intrinsic bistability in a quantum well under the electric and magnetic fields. J Appl Phys 2011, 109: 053101. 10.1063/1.3553641View ArticleGoogle Scholar
- Kosionis SG, Terzis AF, Simserides C, Paspalakis E: Intrinsic optical bistability in a two-subband system in a semiconductor quantum well: analytical results. J Appl Phys 2011, 109: 063109. 10.1063/1.3553871View ArticleGoogle Scholar
- Kosionis SG, Terzis AF, Paspalakis E: Pump-probe optical response and four-wave mixing in intersubband transitions of a semiconductor quantum well. Appl Phys B 2011, 104: 33–43. 10.1007/s00340-010-4365-3View ArticleGoogle Scholar
- Kosionis SG, Terzis AF, Paspalakis E: Kerr nonlinearity in a driven two-subband system in a semiconductor quantum well. J Appl Phys 2011, 109: 084312. 10.1063/1.3561484View ArticleGoogle Scholar
- Evangelou S, Paspalakis E: Pulsed four-wave mixing in intersubband transitions of a symmetric semiconductor quantum well. Photon Nanostr Fund Appl 2011, 9: 168–173. 10.1016/j.photonics.2010.09.004View ArticleGoogle Scholar
- Yao HF, Niu Y-P, Peng Y, Gong S-Q: Carrier-envelope phase dependence of the duration of generated solitons for few-cycle rectangular laser pulses propagation. Opt Commun 2011, 284: 4059–4063. 10.1016/j.optcom.2011.04.018View ArticleGoogle Scholar
- Chelkowski S, Bandrauk AD: Sensitivity of spatial photoelectron distributions to the absolute phase of an ultrashort intense laser pulse. Phys Rev A 2002, 65: 061802.View ArticleGoogle Scholar
- Nakajima T, Watanabe S: Effects of the carrier-envelope phase in the multiphoton ionization regime. Phys Rev Lett 2006, 96: 213001.View ArticleGoogle Scholar
- Liu CP, Nakajima T: Anomalous ionization efficiency by few-cycle pulses in the multiphoton ionization regime. Phys Rev A 2007, 76: 023416.View ArticleGoogle Scholar
- Doslic N: Generalization of the Rabi population inversion dynamics in the sub-one-cycle pulse limit. Phys Rev A 2006, 74: 013402.View ArticleGoogle Scholar
- Wu Y, Yang X-X: Carrier-envelope phase-dependent atomic coherence and quantum beats. Phys Rev A 2007, 76: 013832.View ArticleGoogle Scholar
- Huang P, Xie X-T, Lo X, Li J-H, Yang X-X: Carrier-envelope-phase-dependent effects of high-order harmonic generation in a strongly driven two-level atom. Phys Rev A 2009, 79: 043806.View ArticleGoogle Scholar
- Xie X-T, Macovei M, Kiffner M, Keitel CH: Probing quantum superposition states with few-cycle laser pulses. J Opt Soc Am B 2009, 26: 1912–1917. 10.1364/JOSAB.26.001912View ArticleGoogle Scholar
- Li H, Sautenkov VA, Rostovtsev YV, Kash MM, Anisimov PM, Welch GR, Scully MO: Carrier-envelope phase effect on atomic excitation by few-cycle rf pulses. Phys Rev Lett 2010, 104: 103001.View ArticleGoogle Scholar
- Luo J, Niu Y-P, Sun H, Cui N, Jin S-Q, Gong S-Q, Zhang H-J: Third harmonic enhancement due to Fano interference in semiconductor quantum well. Eur Phys J D 2008, 50: 87–90. 10.1140/epjd/e2008-00198-7View ArticleGoogle Scholar
- Yang W-X, Yang X-X, Lee R-K: Carrier-envelope-phase dependent coherence in double quantum wells. Opt Express 2009, 17: 15402–15407. 10.1364/OE.17.015402View ArticleGoogle Scholar
- Allen L, Eberly JH: Optical Resonance and Two-Level Atoms. Dover, Toronto; 1987.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.