Photon echoes retrieved from semiconductor spins: access to basis for long-term optical memories
L. Langer, S.V. Poltavtsev, I.A. Yugova, M. Salewski, D.R. Yakovlev, G. Karczewski, T. Wojtowicz, I.A. Akimov, M. Bayer
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Photon echoes retrieved from semiconductor spins:access to basis for long-term optical memories
L. Langer, S.V. Poltavtsev, , I.A. Yugova, M. Salewski, D.R. Yakovlev, , G. Karczewski, T. Wojtowicz, I.A. Akimov, , and M. Bayer Experimentelle Physik 2, Technische Universit¨at Dortmund, 44221 Dortmund, Germany Spin Optics Laboratory, Saint Petersburg State University, 198504 St. Petersburg, Russia A.F. Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia Institute of Physics, Polish Academy of Sciences, PL-02668 Warsaw, Poland
The possibility to store optical information is important for classical and quantum com-munication. Atoms or ions as well as color centers in crystals offer suitable two-levelsystems for absorbing incoming photons. To obtain a reliable transfer of coherence,strong enough light-matter interaction is required, which may enforce use of ensem-bles of absorbers, but has the disadvantage of unavoidable inhomogeneities leading tofast dephasing. This obstacle can be overcome by echo techniques that allow recoveryof the information as long as the coherence is preserved. Albeit semiconductor quan-tum structures appear appealing for information storage due to the large oscillatorstrength of optical transitions, inhomogeneity typically is even more pronounced forthem and most importantly the optical coherence is limited to nanoseconds or shorter.Here we show that by transferring the information to electron spins the storage timesfor the optical coherence can be extended by orders of magnitude up to the spin re-laxation time. From the spin reservoir it can be retrieved on purpose by inducing astimulated photon echo. We demonstrate this for an n-doped CdTe/(Cd,Mg)Te quan-tum well for which the storage time thereby could be increased by more than threeorders of magnitude from the picosecond-range up to tens of nanoseconds.
Photon echoes are amazing optical phenomena in which resonant excitation of a medium by shortoptical pulses results in a delayed response in form of a coherent optical flash. Since their first ob-servation in ruby in 1964 [1], photon echoes were reported for atom vapors [2], rare earth crystals [3]and semiconductors [4, 5]. Spontaneous (two-pulse) and stimulated (three-pulse) photon echoes weredemonstrated and used for studying the involved energy levels and the coherence evolution of the op-tical transitions [6, 7, 8]. Currently there is great interest in application of photon echoes for quantummemories [9, 10]. Photon echoes occur in an ensemble of oscillators with an inhomogeneous distributionof optical transitions. Such an ensemble provides high efficiency and large bandwidth allowing one to1tore multiple photons with high capacity. Current activities on photon echoes have mainly concentratedon rare earth crystals and atomic vapors with long storage times, that are crucial for implementation ofrobust light-matter interfaces.Already at the early stage of photon echo experiments the spin level structure of ground and ex-cited states was recognized to contribute to the formation of spontaneous and stimulated photon echosignals [11, 12, 13]. If optically addressed states possess orbital and/or spin angular momenta thenthe splitting of these states by a magnetic field (the Zeeman effect) provides an additional degree offreedom for the control of photon echo through optical selection rules [14, 15, 16]. Moreover, trans-fer of coherence from optical to spin excitations has been suggested to considerably extend the storagetimes as demonstrated for quasi-atomic systems having optical and spin coherences with comparablelifetimes [17]. Here we demonstrate that a transverse magnetic field applied to a semiconductor leadsto transfer of short-lived optical coherence into long-lived electron spin coherence. This allows one toinduce stimulated photon echoes on sub- µ s time scales, exceeding the radiative lifetime of the opticalexcitations by more than three orders of magnitude. We reveal two mechanisms leading to this extensionof stimulated echo revival - coherence transfer and spin fringes, and show that depending on the polar-ization configuration of the three involved laser pulses we are able to shuffle the optical coherence intoa spin component that is directed either along or perpendicular to the magnetic field. The spins directedalong magnetic field are free of any dephasing and are affected little by spin relaxation, which makes thisconfiguration highly appealing for future applications in memory devices.For demonstration of magnetic-field-induced stimulated photon echoes, we study a semiconductorCdTe/(Cd,Mg)Te quantum well (QW) which serves as model system, that can be tailored for the targetedapplication on a detailed level by nanotechnology. The fundamental optical excitations in semiconduc-tors, the excitons, possess large oscillator strength so that resonant absorption may be achieved withclose to unity efficiency even for structure thicknesses smaller than the light wavelength. Thereforepropagation effects are not as important as in atomic vapors and rare-earth crystals.Ultrafast coherent spectroscopy of excitons employing laser pulses is well established for semicon-ductor nanostructures [7]. However, for storage applications excitons have been scarcely consideredbecause of their limited optical coherence time T due to complex many body interactions and their shortradiative lifetime ( T ≤ ns) being the downside of the large oscillator strength. In nanostructures suchas quantum dots (QDs) the optical decoherence is weak but still limited by radiative decay. Thereforeapproaches to involve the long-lasting coherence of electron spins have been pursued recently wheremost of the studies were focused on optical control of the spin [18, 19, 20, 21]. The storage and retrievalof optical coherence by its encoding in an ensemble of electron spins has not yet been addressed.Figure 1 summarizes the experimental approach and the main results on the optical properties of thestudied QW in zero magnetic field. We use a sequence of three excitation pulses with variable delays τ between pulses 1 and 2 and τ between pulses 2 and 3. The duration of the pulses τ p ≈ ÷ ps.Pulses 2 and 3 are propagating along the same direction, so that their wave vectors are equal k = k .Both the spontaneous (PE) and stimulated (SPE) photon echoes are then directed along the k − k direction. The transients are measured by taking the cross-correlation of the resulting four-wave mixingsignal E FWM ( t ) with the reference pulse E ref ( t ) using heterodyne detection as schematically shown in2ig. 1A [see appendix A]. This allows us to distinguish between the PE and SPE signals because of theirdifferent arrival times at the detector.Essential for our experiment is selection of a well-defined spin level system, optically excitableaccording to clean selection rules. That is why we did not select the neutral exciton but the chargedexciton consisting of two electrons and a hole, which requires a resident electron population. The studiedsample comprises 20-nm thick CdTe QWs separated by 110 nm Cd . Mg . Te barriers. The barriersare doped with donors which provide resident electrons for the QWs with density n e ≈ cm − .At the cryogenic temperature of 2 K in experiment these electrons become localized in QW potentialfluctuations due to well width and composition variations [22]. In the photoluminescence spectrum boththe neutral ( X ) and charged exciton (in short, trion T − ) are observed, separated by the trion bindingenergy [23]. In the T − ground state the two electrons form a spin singlet state. The narrow widths of thespectral lines indicate a high structural quality (see Fig. 1B).To address solely the optical transition from electron to trion, the photon energy of the laser is tunedto the lower energy flank of the trion emission line, namely to ~ ω = 1 . eV. Thereby we also selec-tively excite T − complexes with enhanced localization, having longer optical coherence time. The laserspectrum is shown by the dashed line in Fig. 1B. Figure 1C gives the characteristic four-wave mixingsignal taken for τ = 23 ps and τ = 39 ps, while the delay τ ref of the reference pulse is scannedrelative to pulse 1. Signatures of spontaneous (PE) and stimulated (SPE) photon echoes are clearly seenin the transients: the PE signal appears at τ ref = 2 τ and the SPE shows up at τ ref = 2 τ + τ [6]. Thedecay of the PE and SPE peak amplitudes with increasing τ and τ , respectively, are shown in Fig. 1D.From these data we evaluate decay times of T = 72 ps and T = 45 ps so that T ≈ T indicating thatthe loss of coherence for the trions is mainly due to radiative decay with lifetime τ r = T .The investigated electron-trion transition can be considered as four-level system, as shown schemat-ically in Fig. 2 [24]. We excite optical transitions between the doubly degenerate electron states withspin projections S z = ± / (ground states | i and | i ) and the doubly degenerate trion states with spinprojections J z = ± / (excited states | i and | i ). The excited state spin projections are determined bythe heavy-hole total angular momentum because the trion electrons form a singlet state ( S = 0 ). Theselection rules for optical excitation follow from angular momentum conservation, i.e. | i + σ + → | i and | i + σ − → | i [25]. Here σ + and σ − denote the corresponding circular photon polarizations. Atzero magnetic field the transitions are decoupled. In addition, the spin relaxation time of hole T h andelectron T e are long compared to the radiative lifetime ( T e , T h ≫ τ r ) [22]. Therefore the SPE signal isexpected to decay with T = τ r when the delay τ is increased, in full accord with the experimentaldata in Fig. 1D.Application of an external magnetic field in the QW plane, B k x , leads to Larmor precession of theelectron spin in the ground state. In that way the transfer of optical coherence into long-lived electronspin coherence can be achieved and a dramatic increase of the SPE decay time by several orders ofmagnitude may be accomplished. There are two different mechanisms which contribute to magnetic-field-induced signal (see Fig. 2). The first one (A) is based on direct transfer of optical coherence intoelectron spin coherence, and the second one (B) is due to de-synchronization of the spectral gratings forelectron and trion spins. 3or simplicity, let us consider a situation when the pulse duration τ p is significantly shorter than theperiod of Larmor precession T L = 2 π/ω L = 2 π ~ /gµ B B , where g is the electron g-factor and µ B isthe Bohr magneton. Under these conditions the selection rules for optical transitions remain unchanged.For the mechanisms (A) and (B) it is important that the Larmor precession frequencies of the trion andelectron spins are different. This is perfectly well the case in QW structures where the confinementalong z direction splits the heavy-hole and light-hole bands and therefore the optically excited trionstates do not become coupled by the weak transverse magnetic field because the transverse heavy-holeg-factor g hh ≈ [26]. This feature simplifies analysis of the system’s time evolution because one hasto account only for Larmor precession of the electron spin, i.e. a periodic exchange between S y and S z with frequency ω L . Also for simplicity, we consider the following relations between the time constantsrepresenting realistically the situation for the studied electron-trion system and corresponding as well tothe most interesting case of long-lived echoes: T L ≤ τ ≪ T , (1) T ≪ τ ≪ T e and T ≪ T h . (2)The first relation requires fast Larmor spin precession and conservation of optical coherence beforearrival of pulse 2. The second relation limits our consideration to arrival times of pulse 3 after trionrecombination, i.e. the system is in the ground state at t = τ and all required information is stored inthe electron spin. Here we remind that τ r ≈ T ≈ T / . If relations 1 and 2 hold the solutions of theLindblad equation of motion for the ( × ) density matrix ρ ij , which describes the four-level electron-trion system, can be written in a compact form. For the electron spin components containing the requiredterms with exp( − iω τ ) time evolution (see appendix B) we obtain: S x = ρ + ρ ∗ ∝ K Σ sin (cid:16) ω L τ (cid:17) exp (cid:18) − τ T e (cid:19) , (3) S y = ρ − ρ ∗ i ∝ K ∆ sin (cid:16) ω L τ ω L τ (cid:17) exp (cid:18) − τ T e (cid:19) , (4) S z = ρ − ρ ∝ − K ∆ cos (cid:16) ω L τ ω L τ (cid:17) exp (cid:18) − τ T e (cid:19) . (5)Here K = exp[ − i ( ω τ − k r + k r )] exp( − τ /T ) + c.c. is the term that carries the information onthe optical phase, ∆ = θ θ − θ − θ − and Σ = θ θ + θ − θ − account for σ ± light polarization ofthe excitation pulse n with pulse area θ n ± ≪ . T e and T e correspond to the longitudinal and transverseelectron spin relaxation times, respectively. ω is the trion resonance frequency. Here we assume thatinitially, before the arrival of pulse 1 ( t < ), the electron spins are unpolarized, i.e., ρ = ρ = 1 / ,while all other elements of the density matrix vanish. From equations 3-5 it follows that at B > all spincomponents are finite with magnitudes depending critically on the polarization of the exciting pulses.Qualitatively the magnetic-field-induced SPE evolution is easy to follow for a circular polarizedpulse sequence as shown in Fig. 2. At t = 0 pulse 1 creates a coherent superposition between the states | i and | i . This is an optical coherence associated with the ρ element of the density matrix. Due4o inhomogeneous broadening of optical transitions, this coherence ρ disappears due to dephasing.Each dipole in the ensemble with a particular optical frequency ω acquires an additional phase φ ( t ) =( ω − ω ) t = δω t before arrival of the second pulse at t = τ (see term K in Eqs. 3-5). This is indicatedby a set of arrows with different lengths, symbolizing the phase distribution of the dipoles with differentfrequencies.The first mechanism (A) (see Fig. 2A) is most efficient when the second pulse arrives at τ = (2 m +1) π/ω L , where m is an integer, i.e. the optical coherence ρ is shuffled into a ρ coherence betweenthe optically inaccessible states | i and | i . Pulse 2 transfers this coherence into a superposition of states | i and | i , corresponding to spin coherence ρ = S x − iS y (see Eqs. 3 and 4). There the coherence isfrozen in the ground state without any further optical dephasing and can survive for much longer timesthan the zero-field coherence, even after radiative trion recombination τ r . Note, however, that the Larmorprecession of the electron spin is continuing then. Therefore dephasing of the S y component may occur,while S x remains constant as it is directed along the magnetic field (see Eqs. 3 and 4). Finally, pulse3 retrieves the coherence ρ by converting it back into the optical frequency domain and starting therephasing process. Again, the rephasing will be most efficient if ρ ( t = τ + τ ) = ρ ∗ ( t = τ ) , i.e.for τ = (2 l + 1) π/ω L , where l is an integer.The second mechanism (B) (see Fig. 2B) can be considered as an incoherent one. In contrastto the first one it relies on population rather than coherence. Here, the accumulated phase of eachdipole φ is projected by pulse 2 into population interference fringes ρ ∝ sin ( δω τ / and ρ ∝ cos ( δω τ / , i.e. into spectral population gratings in the excited and ground states with oppositephase [6]. They are equivalent to spectral spin gratings for the electrons S z = ( ρ − ρ ) / andthe trions J z = ( ρ − ρ ) / . In contrast to the previous mechanism the interference fringes havethe highest contrast when the second pulse arrives after an integer number of electron spin revolutions t = τ = 2 mπ/ω L (see Eq. 5). For t > τ the Larmor precession of the electron spin de-synchronizesthe spin gratings in ground and excited state. Therefore even after trion recombination the electron spingrating does not disappear. The spin fringes S z ( δω ) and J z ( δω ) are schematically shown in Fig. 2C,Dfor two different times: the time of their creation t = τ and the time after trion recombination beforearrival of pulse 3. Accordingly a long-lived electron spin grating is present, allowing one to retrieve thephase information φ and observe the SPE pulse with maximum signal at τ = lπ/ω L .In practice, the initial condition of zero electron spin polarization is hard to match for circularlypolarized pulse sequences. This is because σ + excitation induces a macroscopic spin polarization. Theelectron spin relaxation T e ∼ ns is longer than the pulse repetition period T R = 13 . ns in ourexperiment and S ( t = 0) = 0 [22]. Moreover, for arbitrary τ , both mechanisms are present andtherefore all spin components contribute to the SPE signal. Therefore circularly polarized pulses arenot optimal for demonstration of magnetic-field-induced SPE. From Eqs. 3-5 it follows that a linearlypolarized pulse sequence is much more attractive. For linearly H = ( σ + + σ − ) / √ or V = ( σ + − σ − ) / √ co-polarized pulses, ∆ = 0 and Σ = 2 θ θ so that only the dephasing-free S x component isinvolved. Here we exploit only the first mechanism (A) of coherence transfer. If the pulses 1 and 2 arecross-polarized the opposite situation with ∆ = 2 θ θ and Σ = 0 is obtained. Here, the S y and S z P HHHH ∝ sin (cid:16) ω L τ (cid:17) exp (cid:18) − τ T e (cid:19) (6) P HVVH ∝ cos [ ω L ( τ + τ )] exp (cid:18) − τ T e (cid:19) , (7)where the polarization sequence is denoted by the subscript ABCD with A, B, C corresponding to thepolarizations of pulses 1, 2, 3, respectively, and D is the polarization of the resulting SPE pulse (seeappendix B). Note that the SPE echo in HVVH configuration oscillates with the Larmor precessionfrequency in dependence of τ + τ , while in HHHH configuration it varies only with the τ delaytime. Also the decay of the signals is different. For HVVH the decay occurs according to the transversespin relaxation time T e and additional dephasing due to the electron g-factor inhomogeneity ∆ g mayplay a role. In HHHH the signal decays with the longitudinal spin relaxation time T e .The experimental data for the SPE amplitude as function of delay time τ and magnetic field B ,measured in the HHHH and HVVH polarization configurations, are summarized in Fig. 3. The delaytime between pulses 1 and 2 is set to τ = 27 ps, which corresponds to T L /τ = 0 . at B = 0 . T and T L /τ = 0 . at B = 0 . T. For comparison the SPE decay measured at B = 0 is shown in Fig. 3A.The magnetic field dependence of the SPE amplitude in Fig. 3B and 3D is measured for τ = 1 . nswhich is significantly longer than the radiative lifetime T . Note that the experimentally measured signalcorresponds to the absolute value of amplitude | P | . In full accord with our expectations we observe along-lived SPE signal when applying the magnetic field. The theoretical curves are in good agreementwith the data. The calculations take into account the full dynamics of the system, including the excitedstates, and therefore reproduce also the short decay component of the SPE due to trion recombination aswell as the signals at low magnetic fields with T L > τ (see appendix B).In case of HHHH no oscillations appear when τ is varied (see Fig. 3A), which is in line withEq. 6. Particularly fascinating is the observation of SPE signal at negative delays ( τ ∼ − ps)at a 75% level of the SPE amplitude at positive delays ( τ ∼ ps), see Fig. 3A. This means thatwithin the pulse repetition period of 13 ns the SPE amplitude reduces by 25% only. From these data weestimate T e ∼ ns, which allows us to observe SPE signals in magnetic field on a sub- µ s timescale.The short dynamics at negative delays −
200 ps < τ < is due to excitation of trions by pulse 3,which influences the initial conditions at t = 0 . Note that for the linearly polarized pulse sequence nomacroscopic spin polarization becomes involved in the PE experiment due to inhomogeneous broadeningof the optical transitions. Therefore we obtain information about the intrinsic longitudinal spin relaxationof the electron spin. The experimental data demonstrate that transfer of coherence to S x is feasible andattractive because this spin component being parallel to B is robust against relaxation and is not sensitiveto dephasing. In contrast, the oscillatory signal in the HVVH polarization configuration decays muchfaster due to electron g-factor inhomogeneity and consequently no long-lived SPE signal is observed atnegative delays (see Fig. 3C). The reduction of SPE signal at B > . T in Fig. 3D is also related to thisfact. From fitting the data we evaluate ∆ g = 0 . .In conclusion, we have demonstrated magnetic-field-induced long-lived stimulated photon echoes in6he electron-trion system. By a proper choice of polarization pulse sequence optical coherence can betransferred into spin directed along magnetic field. Although no metastable states are involved, due to thelong-lived electron spin coherence, the timescale of echo stimulation can be extended by more than 3 or-ders of magnitude over the optical coherence time in the QW system. Note that the electron-trion energylevel structure is identical in QWs and self-assembled quantum dots (QDs). We used QWs for demon-stration purpose because the trion transitions are well isolated spectrally. As a downside, we had to keepthe excitation power low in order to suppress many-body effects. In case of singly charged QD structures π/ and π pulses can be efficiently used for coherent manipulation [27]. In addition the longitudinalspin relaxation times in, e.g., (In,Ga)As QDs may be as long as 0.1 s [28]. Further exploiting hyperfineinteraction between electrons and nuclei might enable storage time of seconds or longer [29]. Thereforeour findings open a new avenue for realization of optical memories in semiconductor nanostructures. Acknowledgements
The Dortmund team would like to acknowledge financial support of this work by the DeutscheForschungsgemeinschaft, the Bundesministeriuum f¨ur Bildung und Forschung (project Q.com-H). Theproject ”SPANGL4Q” acknowledges the financial support of the Future and Emerging Technologies(FET) programme within the Seventh Framework Programme for Research of the European Commis-sion, under FET-Open grant number: FP7-284743. S.V.P. and I.A.Yu. acknowledge partial financialsupport from the Russian Ministry of Science and Education (contract No.11.G34.31.0067). The re-search in Poland was partially supported by the National Science Center (Poland) under the GrantsDEC-2012/06/A/ST3/00247 and DEC-2013/ST3/229756.7
Methods
The semiconductor quantum well (QW) structure was grown by molecular-beam epitaxy. It comprises5 electronically decoupled 20 nm thick CdTe QWs embedded in 110 nm Cd . Mg . Te barriers. Thebarriers are doped by iodine donors, which provide the QW layers with conduction band electrons of lowdensity n e ∼ cm − [22]. The sample was mounted into a liquid He bath cryostat at a temperature of2 K. Magnetic fields up to 0.7 T were applied in Voigt geometry using an electromagnet. The directionof the magnetic field was parallel to the QW plane ( B k x ).We used tunable self mode-locked Ti:Sa laser as source of the optical pulses with durations of ÷ psat a repetition rate of 75.75 MHz (repetition period T R = 13 . ns). The laser was split into four beams.Three of them were used for the three pulse sequence required for stimulating the photon echo. Thefourth beam was used as reference pulse in the heterodyne detection. The delay between all four pulsescould be scanned by reflectors mounted on mechanical translation stages. The three-pulse four-wavemixing (FWM) experiment was performed in reflection geometry. Pulse 1 with wavevector k hit thesample under an incidence angle of about 7 ◦ . Pulses 2 and 3, both traveled along the same direction( k = k ) different from that of the first beam, hit the QW structure under an incidence angle of about ◦ . The beams were focused onto the sample in a spot of about µ m in diameter. The intensities ofeach pulse were selected such as to remain in the linear excitation regime for each of the beams (pulseenergy around 10-100 nJ/cm ). The FWM signal was collected along the k − k direction. We usedinterferometric heterodyne detection where the FWM signal and the reference beam are overlapped at abalanced detector [30]. The optical frequencies of pulse 1 and reference pulse were shifted by 40 MHzand 41 MHz with acousto-optical modulators. The resulting interference signal at the photodiode wasfiltered by a high frequency lock-in amplifier selecting | ω − ω − ω ref | = 1 MHz. This provided ahigh sensitivity measurement of the absolute value of the FWM electric field amplitude in real time whenscanning the reference pulse delay time τ ref , which was taken relative to the pulse 1 time arrival. Thepolarization of the first and the second pulses, as well as the detection polarization, were controlled withretardation plates in conjunction with polarizers. B Theoretical description of magnetic-field-induced SPE
B.1 Stimulated photon echo in magnetic field
Let us consider optical excitation of the negatively charged trion by a short laser pulse with frequency ω close to the trion resonance frequency ω . The incident electromagnetic field induces optical transitionsbetween the electron state and the trion state creating a coherent superposition of these states. In accor-dance with the selection rules, σ + circularly polarized light creates a superposition of the +1 / electronand +3 / trion states, while σ − polarized light creates a superposition of the − / electron and − / trion states. In order to describe these superpositions and the resulting dynamics in a magnetic field weuse a 4x4 time-dependent density matrix, comprising the two electron spin projections ( ± / ) (index 1and 2) and the two hole spin projections ( ± / ) (index 3 and 4).8he temporal evolution of the density matrix is described by the Lindblad equation: ˙ ρ = − i ~ [ ˆ H, ρ ] + Γ . (8)Here ˆ H is the Hamiltonian of the system and Γ describes relaxation processes phenomenologically. Inour case the Hamiltonian contains three contributions: ˆ H = ˆ H + ˆ H B + ˆ V , where ˆ H is the Hamiltonianof the unperturbed spin system, ˆ H B gives the interaction with magnetic field and ˆ V describes the interac-tion with light. In the calculations we use the short pulse approximation assuming that the pulse durationis significantly shorter than the trion lifetime, the decoherence times and the electron spin precessionperiod in transverse magnetic field. This assumption is justified for our experimental conditions. Un-der these circumstances, we can separate and consider consistently the interaction of the electron-trionsystem with light and its dynamics in magnetic field. B.2 Electron-trion system under action of short light pulse
The interaction with the electromagnetic wave in the electric-dipole approximation is described by theHamiltonian: ˆ V ( t ) = − Z [ ˆ d + ( r ) E σ + ( r , t ) + ˆ d − ( r ) E σ − ( r , t )]d r , (9)where ˆ d ± ( r ) are the circularly polarized components of the dipole moment density operator, and E σ ± ( r , t ) are the correspondingly polarized components of the electric field of a quasi-monochromatic electromag-netic wave. The electric field of this wave is given by E ( r , t ) = E σ + ( r , t ) o + + E σ − ( r , t ) o − + c . c . , (10)where o ± are the circularly polarized unit vectors that are related to the unit vectors o x k x and o y k y through o ± = ( o x ± i o y ) / √ . Here the components E σ + and E σ − contain temporal phase factors e − i ωt .The strength of the light interaction with the electron-trion system is characterized by the correspond-ing transition matrix element of the operators ˆ d ± ( r ) calculated with the wave functions of the valenceband, | ± / i , and the conduction band, | ± / i : [31] d ( r ) = h / | ˆ d − ( r ) | / i = h− / | ˆ d + ( r ) | − / i . (11)We assume, that the trion recombination time is considerably shorter than the laser repetition period.Therefore, before the first pulse only elements of the density matrix describing the electron are unequalzero. We also assume, that these are only the populations ρ and ρ . Thus, the initial conditions are: ρ (0) = ρ (0) 0 0 00 ρ (0) 0 00 0 0 00 0 0 0 . (12)9he Hamiltonian ˆ H = ˆ H + ˆ V in our basis is given by: f ∗ + e i ωt ~
00 0 0 f ∗− e i ωt ~ f + e − i ωt ~ ~ ω f − e − i ωt ~ ~ ω . (13)Here f ± ( t ) is proportional to the smooth envelopes of the circular polarized components σ + and σ − ofthe excitation pulse, given by f ± ( t ) = − i ωt ~ Z d ( r ) E σ ± ( r , t )d r . For simplicity we hereafter consider pulses with rectangular shape. This means that the pulse areas forthe σ ± polarized components are equal to f ± t p . In the experiment we investigate localized electronsand trions. We suppose, that the light wavelength is much larger than the length scale of localization.This allows one to extract the electric field E σ ± from the integral and write the pulse areas in the form f ± t p = θ ± e i kr , where r is the position of the localized resident electron [32]. For calculations of thelight-induced optical polarization and, therefore, the photon echo amplitude, we have to integrated thefinal expressions over the electron positions [32]. In experiment we operate in the linear regime of opticalexcitation of the QW, so that | f ± t p | ≪ [33].With these assumptions, the solution of the von Neumann equation i ~ ˙ ρ = [ ˆ H + ˆ V , ρ ] after the pulseaction gives: ρ a = ρ b + i t p f + ρ b − f ∗ + ρ b ) + ( ρ b − ρ b ) | f + t p | ,ρ a = ρ b − i t p f + ρ b − f ∗ + ρ b ) − ( ρ b − ρ b ) | f + t p | ,ρ a = ρ b + i t p f − ρ b − f ∗− ρ b ) + ( ρ b − ρ b ) | f − t p | ,ρ a = ρ b − i t p f − ρ b − f ∗− ρ b ) − ( ρ b − ρ b ) | f − t p | , (14a) ρ a = e i ωt p [ ρ b − i f ∗ + t p ρ b − ρ b ) + f ∗ + t p f + ρ b + f ∗ + ρ b )] ,ρ a = e i ωt p [ ρ b − i f ∗− t p ρ b − ρ b ) + f ∗− t p f − ρ b + f ∗− ρ b )] , (14b) ρ a = e i ωt p [ ρ b + ρ b f ∗ + f ∗− t p f ∗− t p ρ b − i f ∗ + t p ρ b ] ,ρ a = e − i ωt p [ ρ b + ρ b f + f − t p f − t p ρ b − i f + t p ρ b ] , (14c)10 a = ρ b + ρ b f ∗ + f − t p f − t p ρ b − i f ∗ + t p ρ b ,ρ a = ρ b + ρ b f + f ∗− t p f ∗− t p ρ b − i f + t p ρ b (14d)The superscripts ‘b’ and ‘a’ denote the matrix elements before and after pulse arrival, respectively. B.3 Precession and relaxation in magnetic field
Next, we have to consider the dynamics in a transverse magnetic field. The magnetic field is appliedperpendicular to the propagation direction of the incident light and to the structure growth axis. Thecorresponding Hamiltonian is: ~ ω L ~ ω L ~ ω ~ ω TL ~ ω TL ~ ω , (15)where ω L and ω TL are the electron and trion Larmor precession frequencies. For simplicity we neglectthe trion spin precession in magnetic field, i.e. ω TL = 0 , which is justified by the hole g-factor being closeto zero.Relaxation processes Γ are taken into account in the following way: − ρ − ρ T ez + ρ τ r − ρ T es − ρ T es − ρ T − ρ T − ρ T es − ρ T es − ρ − ρ T ez + ρ τ r − ρ T − ρ T − ρ T − ρ T − ρ − ρ T hs − ρ τ r − ρ T hs − ρ τ r − ρ T − ρ T − ρ T hs − ρ τ r − ρ − ρ T hs − ρ τ r , (16)Here T is the decay time of the optical coherence and τ r is the trion recombination time. T es = T ex + T ey , T es = T ex − T ey . T ex,y,z are the electron spin relaxation times. Because the magnetic field pointsalong the x axis, we assume T ex ≡ T e (longitudinal spin relaxation time) and T ez = T ey ≡ T e (transversespin relaxation time). For the trion spin relaxation one can write down similar decay terms, but becauseof ω TL = 0 we introduce only one the spin relaxation time T hs .First, let us consider the dynamics of the non-diagonal terms of the density matrix in magnetic field.After pulse action these elements are: ρ ( t ) = [ ρ a cos( ω L ( t − t ) / − i ρ a sin( ω L ( t − t ) / ( t − t )(i ω − /T ) ,ρ ( t ) = [ ρ a cos( ω L ( t − t ) / − i ρ a sin( ω L ( t − t ) / ( t − t )(i ω − /T ) ,ρ ( t ) = [ ρ a cos( ω L ( t − t ) / − i ρ a sin( ω L ( t − t ) / ( t − t )(i ω − /T ) ,ρ ( t ) = [ ρ a cos( ω L ( t − t ) / − i ρ a sin( ω L ( t − t ) / ( t − t )(i ω − /T ) . (17)Here t is the time after end of the pulse action. 11t is convenient to describe the evolution of the other elements of the density matrix through theelectron and trion spin dynamics. Obviously the following relations hold: S z = ( ρ − ρ ) / , S y = i( ρ − ρ ) / , S x = ( ρ + ρ ) / ,J z = ( ρ − ρ ) / , J y = i( ρ − ρ ) / , J x = ( ρ + ρ ) / ,n e = ( ρ + ρ ) / , n T = ( ρ + ρ ) / , (18)where S x,y,z and J x,y,z are the components of the electron and trion spin polarization. n e and n T are thepopulations of the electron and trion states.After the excitation pulse the spin dynamics of the trion in magnetic field is given by[33]: J z ( t ) = J az e − ( t − t ) /τ T , J y ( t ) = J ay e − ( t − t ) /τ T , J x ( t ) = J ax e − ( t − t ) /τ T , . (19)Here τ T is the trion spin lifetime, /τ T ≡ /T hs + 1 /τ r .The electron spin components after the pulse are: S z ( t ) = e − ( t − t ) /T e (cid:2) ( S az + ξ J az ) cos( ω L ( t − t )) + ( S ay + ξ J az ) sin( ω L ( t − t )) (cid:3) − J az ξ e − ( t − t ) /τ T S y ( t ) = e − ( t − t ) /T e (cid:2) − ( S az + ξ J az ) sin( ω L ( t − t )) + ( S ay + ξ J az ) cos( ω L ( t − t )) (cid:3) − J az ξ e − ( t − t ) /τ T S x ( t ) = S ax e − ( t − t ) /T e . (20)Here the superscript a denotes the spin components at time t , when the excitation pulse has passed. ξ + i ξ = 1 τ r ( γ − i ω ) , (21)and γ = 1 /τ T − /T e > . The populations n e and n T are given by: n T ( t ) = n aT e − ( t − t ) /τ r , (22) n e ( t ) = n ae + n aT (1 − e − ( t − t ) /τ r ) . Equations (14) as well as (19), (20), (17) are basis for the following calculations and discussion.
B.4 Optical polarization after the third pulse
In our experiment we measure the amplitude of the electromagnetic wave propagating along the direction k − k at delay τ after the third pulse. This amplitude is determined by the optical polarizationcreated in the sample by all three pulses. The polarization in turn is proportional to the correspondingelements of the density matrix, ρ , ρ and c.c. averaged over the ensemble of excited electron-trionsystems. At τ after the third pulse, ρ and ρ for the individual systems are given by: ρ (3 t p + 2 τ + τ ) = [ ρ a cos( ω L τ / − i ρ a sin( ω L τ / τ (i ω − /T ) ,ρ (3 t p + 2 τ + τ ) = [ ρ a cos( ω L τ / − i ρ a sin( ω L τ / τ (i ω − /T ) . (23)12o obtain the macroscopic polarization one has to average these expressions for an individual electron-trion system over the trion resonance frequency ω . It is obvious, that the condition for observing astimulated photon echo signal is excluding the factor e i ω τ from Eq. (23). As it will be shown below,this is possible because some contributions to ρ a , ρ a , ρ a , ρ a contain e − i ω τ . We turn now to step-by-step calculations of ρ and ρ from Eq. (23).Before excitation with the first pulse only elements of the density matrix describing the electron areunequal to zero (see Eq. 12). We also assume, that these non-zero elements are only populations ρ and ρ , ρ = ρ = 1 / .1. The first pulse changes electron and trion populations and creates optical polarization, that isproportional to the elements ρ , ρ and c.c. ρ a = i θ e i( ωt p − k r ) / , ρ a = − i θ e − i( ωt p − k r ) / , (24) ρ a = i θ − e i( ωt p − k r ) / , ρ a = − i θ − e − i( ωt p − k r ) / . Here the superscript ‘a1’ indicates the values after the first pulse action.2. At the moment of the second pulse arrival these elements, ρ , ρ and c.c., (and also those arisingin magnetic field ρ , ρ and c.c.) - in contrast to other elements - contain the phase factor exp( ± i ω τ ) ,see Eqs. (14). For distinctness, the nondiagonal elements, which are proportional to exp( − i ω τ ) , aregiven by: ρ b = − i θ e − i( ωt p + ω τ − k r ) cos( ω L τ / − τ /T / , (25) ρ b = θ e − i( ωt p + ω τ − k r ) sin( ω L τ / − τ /T / ,ρ b = − i θ − e − i( ωt p + ω τ − k r ) cos( ω L τ / − τ /T / ,ρ b = θ − e − i( ωt p + ω τ − k r ) sin( ω L τ / − τ /T / . The superscript ‘b2’ indicates that these values are before the second pulse coming in.3. The action of the second pulse , together with the already existing optical polarization of theelectron-trion system, leads to additions to populations ρ a , ρ a , ρ a , ρ a and spin coherences ρ a , ρ a ,which are proportional to exp( − i ω τ ) . For convenience we rewrite these elements of the densitymatrix through the components of the electron and trion spin polarizations and populations. S a z ∼ − K ∆ cos( ω L τ / , S a y ∼ K ∆ sin( ω L τ / , S a x ∼ − i K Σ sin( ω L τ / , (26) J a z ∼ K ∆ cos( ω L τ / , J a y ∼ − K ∆ T sin( ω L τ / , J a x ∼ i K Σ T sin( ω L τ / ,n a e ∼ − K Σ cos( ω L τ / , n a T ∼ K Σ cos( ω L τ / , where K = 116 e − i( ωt p + ω τ − k r + k r ) e − τ /T , (27) ∆ = θ θ − θ − θ − , Σ = θ θ + θ − θ − , ∆ T = θ θ − − θ − θ , Σ T = θ θ − + θ − θ .
13. Before the third pulse arrival the elements ρ b , ρ b , ρ b , ρ b , ρ b , ρ b , which correspond tospin populations and spin coherences, contain only a phase factor with optical frequency exp( ± i ω τ ) ,which arose from the second pulse action. Spin dynamics and relaxation after the second pulse resultin Larmor precession and decay (see Eqs. (19), (20)). The other elements contain phase factors suchas exp( ± i ω τ ) , exp( ± i ω ( τ ± τ )) . It should be noted, that if τ ≫ T and τ ≫ τ r then onlyelectron spin coherence and populations can contribute to the echo signal.The trion spin polarization shortly before the third pulse is given by: J b z ∼ K ∆e − τ /τ T cos( ω L τ / , J b y ∼ − K ∆ T e − τ /τ T sin( ω L τ / , (28) J b x ∼ i K Σ T e − τ /τ T sin( ω L τ / , where as the electron spin polarization is given by: S b z ∼ K ∆ h e − τ /T e [( − ξ ) cos( ω L τ /
2) cos( ω L τ )+ (29) +(sin( ω L τ /
2) + ξ cos( ω L τ / ω L τ )] − e − τ /τ T ξ cos( ω L τ / i ,S b y ∼ K ∆ h e − τ /T e [(1 − ξ ) cos( ω L τ /
2) sin( ω L τ )++(sin( ω L τ /
2) + ξ cos( ω L τ / ω L τ )] − e − τ /τ T ξ cos( ω L τ / i ,S b x ∼ − i K Σe − τ /T e sin( ω L τ / . Further, the populations n e and n T read: n b e ∼ − K Σe − τ /τ r cos( ω L τ / , (30) n b T ∼ K Σe − τ /τ r cos( ω L τ /
5. The amplitude of the stimulated photon echo is proportional to the optical polarizations terms ρ and ρ after the third pulse arrival at t = 3 t p + 2 τ + τ . ρ (3 t p + 2 τ + τ ) = − i2 e i( ωt p + ω τ − k r ) e − τ /T × (31) (cid:20) θ ( n b T − n b e J b z − S b z ) cos( ω L τ /
2) + ( θ − ( J b y − i J b x ) − θ ( S b y − i S b x )) sin( ω L τ / (cid:21) ρ (3 t p + 2 τ + τ ) = − i2 e i( ωt p + ω τ − k r ) e − τ /T × (cid:20) θ − ( n b T − n b e − J b z + S b z ) cos( ω L τ /
2) + ( − θ ( J b y + i J b x ) + θ ( S b y + i S b x )) sin( ω L τ / (cid:21) B.5 Stimulated photon echo amplitude
Our analysis shows that the choice of the polarization configuration for the excitation pulses gives usflexibility for observing different mechanisms of photon echo generation. There are two radically differ-ent configurations: (i) all pulses are linearly co-polarized, for example, horizontally (HHH) and (ii) thefirst and the second pulses are linearly cross-polarized. For clarity we will discuss configuration HVV inthe following. One can rewrite Eqs. (31) for these two configurations.14 .5.1 Configuration HHH
In this case θ = θ − ≡ θ , θ = θ − ≡ θ , and θ = θ − ≡ θ . This leads to ∆ = ∆ T = 0 , Σ = Σ T = 2 θ θ , and, therefore, S z = S y = J z = J y = 0 . The polarization of the stimulated echosignal is also horizontal: P HHHH ∼ − ie i( ωt p + ω τ − k r ) e − τ /T θ × (32) h n b T − n b e cos( ω L τ / − i( J b x − S b x ) sin( ω L τ / i + c.c. From this equation it is obvious, that only one term of this expression, namely the one which is pro-portional to S b x decays exponentially with the long relaxation time of the electron spin, T e , when weincrease the delay between the second and the third pulses τ . The other terms, which are proportionalto n b T , n b e and J b x , decay on the shorter time scales τ r or τ T . One can also clearly see that the long-livedpart appears exclusively in magnetic field.If we rewrite the previous equation with substitution of all terms, we obtain: P HHHH ∼ − i8 e i( k − k − k ) r e − τ /T θ θ θ × (33) (cid:2) e − τ /τ r (2 cos ( ω L τ /
2) + e − τ /T h sin ( ω L τ / − τ /T e sin ( ω L τ / (cid:3) + c.c. The long-lived part is expected to show up as a constant background at fixed magnetic field, when thestimulated echo is measured as function of τ , or as slowly oscillating background when the echo ismeasured as function of magnetic field. B.5.2 Configuration HVV
In this configuration θ = θ − ≡ θ , θ = − θ − ≡ θ , θ = − θ − ≡ θ . This leads to ∆ = − ∆ T = 2 θ θ and Σ = Σ T = 0 .The polarization of the stimulated echo signal is horizontal: P HV V H ∼ − ie i( ωt p + ω τ − k r ) e − τ /T θ × (34) (cid:2) ( J b z − S b z ) cos( ω L τ / − ( J b y + S b y ) sin( ω L τ / (cid:3) + c.c. Clearly this signal contains long-lived and short-lived parts. After substitution of S b z,y and J b z,y , weobtain: P HV V H ∼ − i8 e i( k − k − k ) r e − τ /T θ θ θ × (35) (cid:2) e − τ /τ T (cos( ω L τ ) + cos( ω L τ / ξ cos( ω L τ /
2) + ξ sin( ω L τ / − τ /T e [cos( ω L ( τ + τ )) − cos( ω L τ / ξ cos( ω L ( τ / τ )) + ξ sin( ω L ( τ / τ ))]] (cid:3) + c.c. If trion spin relaxes before trion recombination, T h ≪ τ r , then ξ = ξ = 0 and P HV V H : P HV V H ∼ − i8 e i( k − k − k ) r e − τ /T θ θ θ × (36) (cid:2) e − τ /τ T cos( ω L τ ) + e − τ /T e cos( ω L ( τ + τ )) (cid:3) + c.c. P HV V H over all ω L to take into account a possible spread of Larmor frequencies, then the oscillating signal in magnetic fieldcan decay faster than the signal in zero magnetic field.If the trion spin relaxes slowly, T h ≫ τ r , then the amplitude of the long-lived signal depends stronglyon magnetic field. In zero magnetic field P HV V H is: P HV V H ∼ − i8 e i( k − k − k ) r e − τ /T θ θ θ × h e − τ /τ T (1 + ξ ) +e − τ /T e (1 − ξ ) i + c.c. (37)The long-lived part of the signal is proportional to − ξ ≈ τ r / ( T h + τ r ) , therefore if T h ≫ τ r ,then the signal at B = 0 vanishes. This is due to the fact that after the second pulse a long-livedspin polarization S , which is responsible for the long-lived echo signal, does not appear: after trionrecombination the spin state of the resident carrier does not differ from that before trion formation.In a transverse magnetic field the change of the relative orientation of the electron spin and the trionspin leads to appearance of spin polarization of the resident electrons, even if the spins of the carriersin the trions do not relax [33]. With increasing magnetic field this imbalance leads to an increase of thestimulated photon echo amplitude. B.5.3 Stimulated photon echo before the first pulse arrival
It is worth recalling that all pulses in our experiment arrive periodically with the laser repetition period T R ≈ ns. The stimulated photon echo signal before the first pulse arrival (see Fig. 3) is in factthe signal created by the preceding pair of pulses 1 and 2. At τ close to T R changes of the initialconditions for ρ by the third pulse before the arrival of the first pulse have to be taken into account. Forthe configurations with linearly polarized pulses we have to consider only changes of n e and n T inducedby the previous three pulses, because the other components of spin or polarization contain phase factorsand do not contribute to the signal. If τ close to T R , n e and n T are changed only by the third pulsethen. As a result, the density matrix before the subsequent first pulse arrival is given by: ρ b = 12 n b e n b e n b T
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Optical Spectroscopy of Semiconductor Nanostructures (Alpha Science, HarrowUK, 2005).[32] I. A. Yugova, M. M. Glazov, E. L. Ivchenko, and A. L. Efros, Phys. Rev. B , 104436 (2009).[33] I. A. Yugova, M. M. Glazov, D. R. Yakovlev, A. A. Sokolova, and M. Bayer, Phys.Rev. B ,125304 (2012). 19 z xy Fig. 1. Scheme of photon echo experiment and optical properties of investigated structure. (A)The CdTe/(Cd,Mg)Te quantum well (QW) is optically excited with a sequence of three laser pulses withvariable delays τ and τ relative to each other. The resulting four-wave mixing transients | E FWM ( t ) | are detected in k − k direction using heterodyne detection. All measurements are performed attemperature of 2 K. (B) Top: schematic presentation of exciton ( X ) and trion ( T − ) complexes in QW.The QW potential of conduction (CB) and valence (VB) bands leads to spatial trapping of electronsand holes. Bottom: Photoluminescence (PL) spectrum (solid line) measured for above-barrier excitationwith photon energy 2.33 eV, demonstrating X and T − emission. The laser spectrum (dashed line) usedin photon echo experiment is tuned to the low energy flank of T − emission line. (C) Four-wave mixingtransients for τ = 23 ps and τ = 39 ps. Spontaneous (PE) and stimulated (SPE) photon echo signalsappear at τ ref = 2 τ and τ ref = 2 τ + τ , respectively. (D) Decay of PE and SPE peak amplitudes.From exponential fits (dashed lines) we evaluate T = 72 ps and T = 45 ps.20 .00.80.60.40.20.0 1.00.80.60.40.20.0 -1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0 t >> + T S z , J z ( a r b . un i t s ) Detuning laser S z J z -1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0 t = = 2 / L S z , J z ( a r b . un i t s ) Detuning laser S z J z DCB ++++ t = 2 + t = + t = = / L t = / L t=0 g B B/ ++++ t = 2 + t = + t = / L t = / L t = 0 g B B/ A Fig. 2. Schematic presentation of the main mechanisms responsible for magnetic-field-inducedstimulated photon echoes (SPE).
Optical pulses are circularly polarized. (A) Transfer of optical coher-ence into electron spin coherence ( S x and S y components). The efficiency is maximum for τ = π/ω L .(B) Creation of spectral spin fringes for electrons and trions ( S z and J z components). This mechanismis most efficient for τ = 2 π/ω L . The spectral spin gratings for electrons and trions are shown in (C) atthe moment of creation by the second pulse ( t = τ = 2 π/ω L ) and in (D) after trion recombination andbefore arrival of pulse 3 ( t ≫ τ + T ). 21
400 0 400 800 12000200400600 DC B = 0.7 TB = 0 T A HHHH B τ = 27 ps HHHH τ = 1.27 ns τ = 27 ps SPE a m p li t ude ( a r b . un i t s ) -400 0 400 800 12000200400600 B = 0.15 T
Delay time τ (ps) HVVH
HVVH E c ho a m p li t ude ( a r b . un i t s ) Magnetic field B (T)
Fig. 3. Experimental demonstration of magnetic-field-induced long-lived stimulated photon echo(SPE).