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Density-matrix theory of the optical dynamics and transport in quantum cascadestructures: The role of coherence

Weber, Carsten; Wacker, Andreas; Knorr, A.

Published in:Physical Review B (Condensed Matter and Materials Physics)

DOI:10.1103/PhysRevB.79.165322

2009

Link to publication

Citation for published version (APA):Weber, C., Wacker, A., & Knorr, A. (2009). Density-matrix theory of the optical dynamics and transport inquantum cascade structures: The role of coherence. Physical Review B (Condensed Matter and MaterialsPhysics), 79(16), [165322]. https://doi.org/10.1103/PhysRevB.79.165322

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Density-matrix theory of the optical dynamics and transport in quantum cascade structures:The role of coherence

C. Weber,1,* A. Wacker,1 and A. Knorr2

1Mathematical Physics, Lund University, P.O. Box 118, 22100 Lund, Sweden2Institut für Theoretische Physik, Nichtlineare Optik und Quantenelektronik, Technische Universität Berlin,

Hardenbergstr. 36, 10623 Berlin, GermanyReceived 23 November 2008; revised manuscript received 3 April 2009; published 30 April 2009

The impact of coherence on the nonlinear optical response and stationary transport is studied in quantumcascade laser structures. Nonequilibrium effects such as the pump-probe signals, the spatiotemporally resolvedelectron density evolution, and the subband population dynamics Rabi flopping as well as the stationarycurrent characteristics are investigated within a microscopic density-matrix approach. Focusing on the station-ary current and the recently observed gain oscillations, it is found that the inclusion of coherence leads toobservable coherent effects in opposite parameter regimes regarding the relation between the level broadeningand the tunnel coupling across the main injection barrier. This shows that coherence plays a complementaryrole in stationary transport and nonlinear optical dynamics in the sense that it leads to measurable effects inopposite regimes. For this reason, a fully coherent consideration of such nonequilibrium structures is necessaryto describe the combined optical and transport properties.

DOI: 10.1103/PhysRevB.79.165322 PACS numbers: 78.20.Bh, 74.78.Fk, 42.65.k, 73.63.b

I. INTRODUCTION

Quantum cascade lasers QCLs Ref. 1 are semiconduc-tor heterostructure lasers employing the transitions betweenquantized intersubband levels in quantum well structures2–6

and act as a source of radiation in the terahertz THz/mid-infrared regime. The laser consists of multiquantum well pe-riods comprising the electron injector and the optically activeregion. These periods are repeated tens or even hundreds oftimes over the length of the structure.7 To drive the electronsthrough the sample, an external bias is applied. Scatteringprocesses and optical recombination between the conductionsubbands in the doped structure within as well as betweenperiods lead to stationary electronic occupations out of equi-librium. While on the technological side, this light sourcecan be used for spectroscopy in the fields of environmentaldetection or medicine,7,8 it offers on a fundamental groundan interesting model system to study intersubband chargedynamics in a structure where the optical and the transportproperties are closely interrelated.

The first semiconductor heterostructure laser operating inthe intersubband regime was realized by Faist et al.1 in 1994.Since then, many types of QCLs of different design havebeen built and optimized; see, e.g., Refs. 7 and 9 for anoverview. The QCL has been the subject of extensive theo-retical research. The stationary properties of QCLs werestudied by a rate equation10,11 and a Boltzmann-typeapproach12–17 as well as a quantum theory employing bothnonequilibrium Green’s functions18–22 and density-matrixtheory.23–25 Here, the gain, the current-voltage characteris-tics, and the stationary charge distributions have been estab-lished. First results regarding the nonlinear optical propertiessuch as optically induced subband population dynamics havebeen presented by us within a density-matrix theory.26

One central result of these studies is that coherence canplay an important role in the determination of the stationarycurrent, requiring a fully microscopic theory of the current

including nondiagonal density-matrix elements.20,27,28 Here,the often applied rate and Boltzmann equation approaches,based on Wannier-Stark hopping WSH, fail. Experimen-tally, indications of coherent charge transport have been ob-served in the oscillatory gain recovery in pump-probe experi-ments of mid-infrared QCLs Refs. 29–32 as well asrecently in THz structures.33 This oscillatory behavior hasbeen attributed to resonant tunneling through the injectionbarrier of the laser. These observations suggest that coherenteffects might also become visible in optics in the time regimebeyond the light-matter interaction. However, differentstudies34,35 reveal a simple relaxation in the gain recovery,showing that not all samples exhibit this coherent effect.

Motivated by these investigations, this paper is focusedon the role and importance of coherence in the interplaybetween ultrafast optical dynamics and stationary transportin quantum cascade structures. We investigate the regimeswhere coherence is of relevance in the combined optics-transport system. To this end, we present a fully microscopictheory describing the dephasing and tunneling processes in aQCL structure. In order to systematically investigate the op-tical and transport regimes, we focus on two quantities de-scribing the coherent and the incoherent evolution within thesystem: the tunnel coupling 2 between the two statesacross the main injection barrier and the level broadening of the states. We systematically vary the width of the maintunneling barrier in the QCL structure, thus establishing arelation between these two central quantities. Using this re-lation, we investigate the importance of coherence in thecalculated signals: In the transport regime, we compare a rateequation WSH approach with a fully microscopic currenttheory to investigate the regime of validity of both models.In the optical regime, we consider pump-probe calculations,complemented by the spatiotemporally resolved electrondensity evolution and the subband population dynamics dueto strong ultrafast excitation. Using these nonlinear results,aspects of the nature of the optically induced charge dynam-

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ics and the importance of coherence can be addressed.23

Comparing the optical and transport results, we find thatthe influence of coherence is observable in opposite param-eter regimes: the inclusion of coherence in stationary trans-port calculations becomes important for 2, where thecoherence between subbands limits the current flowingthrough the structure, while in the nonlinear optical gain dy-namics, the inclusion of coherence drives the observed oscil-lations in the gain recovery for 2. Combining the tworesults, we thus find that it is necessary to consider a fullycoherent theory in order to understand the combined optics-transport problem independent of the parameter range of op-eration.

II. THEORY

The QCL heterostructure is modeled as a multiconductionsubband system, with each period comprising an injector andan active region. Within the effective mass and envelopefunction approximations,36 the wave functions, assumed tobe separable and infinitely extended in the quantum wellplane, are given by

ikx,z =1

Aeikxizuk0x , 1

where A is the in-plane quantization area, x and k are thein-plane spatial and momentum vector, respectively, iz isthe envelope function in growth direction, and uk0x is theBloch function taken at the band edge. There are severalnatural possibilities to choose the wave functions iz todescribe the system. Since we consider the optical responseof the system as well as the WSH model in the current cal-culation, we choose the Wannier-Stark WS basis which di-agonalizes the heterostructure potential as well as the exter-nally applied bias. This offers an intuitive physicalinterpretation of the optical transitions and the scattering be-tween single-particle WS states as well as of the electroncounting between approximate system eigenstates. It shouldbe noted that, even though the physical observables are inprinciple independent of the choice of basis, different resultsare expected for different basis choices due to the necessaryapproximations applied in many-particle problems. In thefollowing, the theory is applied to the THz QCL discussed inRef. 37. The band structure of this QCL in the original de-sign is shown in Fig. 1 with the important subbands marked.This structure is considered throughout this paper, consider-ing different injection barrier widths b. The parameters usedin the calculations are found in Table I.

In order to describe the ultrafast nonlinear optical andstationary transport properties of the laser, it is preferable toconsider a fully microscopic treatment of the scatteringmechanisms in the structure. In this work, we consider theinteraction of the electronic system with longitudinal opticalLO phonons via the polar Fröhlich coupling as well aswith the ionized doping centers in the laser. Even thoughCoulomb scattering can be treated in the same way,38,39 wedo not consider this interaction here. For the relatively lowlydoped THz laser considered here, it was shown that scatter-

ing with impurities typically dominates over Coulombscattering.40 Acoustic phonons are excluded since they in-duce only very long scattering times in quantum wells andact mainly as a low-energy intrasubband thermalizationmechanism.41 Interface roughness scattering is excluded dueto the difficulty of quantifying it microscopically, while alloyscattering is small in structures with a binary material for thewells GaAs and therefore neglected.42 The included scat-tering mechanisms are treated in a density-matrix correlationexpansion approach within a second-order Born-Markovapproximation.43,44 We do not include renormalizations ofthe subband energies due to scattering in the form of princi-pal values see Sec. II B 3 for a discussion of this approxi-mation; it is assumed that these are small and approximatelyconstant for the different transitions, leading only to an ab-solute energy shift without physical consequences see Ref.45 for a discussion of scattering-induced energy renormaliza-tions in quantum wells.

A. Hamiltonian

To derive the dynamical equations, we divide the Hamil-tonian of the system into three parts,

H = H0 + Hel−light + Hscatt. 2

The first part,

H0 = HSL + H + H0,ph = ik

ikaik† aik +

qqbq

†bq, 3

describes the kinetics of the electrons in the heterostructurepotential HSL as well as the externally applied bias H and

-100

-50

0

-20 0 20 40

ener

gy(m

eV)

growth position (nm)

5

4

2’

1’

b

FIG. 1. Color online Band structure of the THz QCL from Ref.37 for the injection barrier width b=5.5 nm under resonance con-dition. The subbands involved in the laser transition 4,1 as wellas the injector subbands 5,2 are marked.

TABLE I. Structural and material parameters used in thecalculations.

material system GaAs /Ga0.15Al0.85As

doping density n2d 2.9451010 cm−2

well electron mass m 0.067 m0

LO phonon energy LO 36.7 meV

high-frequency permittivity 10.9

static permittivity s 12.9

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the kinetics of the phonons H0,ph. The electronic part is di-agonal due to the choice of basis. aik

† aik denotes the cre-ation annihilation operator of an electron in subband i withquasimomentum k and energy ik =i+2k2 / 2mi and bq

†bqthe creation annihilation operator of an LO phonon withthree-dimensional quasimomentum q and energy qLO. Subband nonparabolicity is neglected, and the sub-band effective masses are assumed to be constant, mim,where m is the effective mass of the well material.

Hel−light describes the interaction of the system with a co-herent classical light source. The polarization of the lightfield is chosen in the direction of the dipole moment, i.e., inthe growth z- direction, and the field is assumed to be aspatially homogeneous Gaussian pulse,

At = Atez, At = A0 exp− t/ 2cosLt , 4

with the laser frequency L and Gaussian pulse duration .The pulse area is defined via the envelope of the pulse.46

Under the assumption of a homogeneous excitation, theHamiltonian is momentum diagonal and reads as

Hel−light = eAtij

k

Mijaik† ajk , 5

with the elementary charge e0, the coupling elementsMij =

12 ipz /mz+eAt / 2mz+H.c.j , and the momen-

tum operator pz= / iz with the space-dependent effectivemass mz.

Finally, Hscatt describes the scattering processes:electron-LO phonon interaction Hel−ph as well as scatteringwith ionized doping centers Hel−imp.

1. Electron-LO phonon interaction

The electron-LO phonon coupling Hamiltonian is givenby

Hel−ph = ij

k,q

gijqaik

† bqajk−q + gijqajk−q

† bq†aik , 6

where q is the in-plane projection of q. The coupling matrixelement is given by the Fröhlich coupling,

gijq = − i e2LO

20V 1

−1

s1/2eq · q

q2 ieiqz j . 7

Here, s and are the static and high-frequency permittiv-ity, respectively, V is the quantization volume, eq is thedisplacement unit vector, and q is the projection of the mo-mentum q in growth direction.

2. Interaction with ionized doping centers

Since the quantum cascade structure is doped, it is neces-sary to take into account the interaction of the electrons withthe ionized doping centers. Typically, either a barrier or awell in the QCL is doped, making it necessary to distributethe ions in this layer. Here, we treat the interaction followingRef. 19, where the dopant density is distributed on several sheets located at zl in the doped barrier/well. The ionizeddoping centers are considered as classical scattering centers.

The Hamiltonian for the interaction is given by

Hel−imp = ij

k,q

l

Vijl qaik+q

† ajk , 8

with the screened electron-impurity interaction potential

Vijl q =

1

A

− e2

20sq2 + 2

ie−iqxle−q2+2z−zl j . 9

Here, is the screening constant and l labels the position ofthe randomly distributed individual ion distributions in thegrowth direction. For the screening, we use the static limit ofthe Lindhard formula for a homogeneous electron gas.47

B. Dynamical equations

1. Equation structure and approximations

We derive the dynamical equations using a correlationexpansion within a density-matrix approach in second-orderBorn-Markov approximation, applying a bath approximationfor the LO phonons.43,44 This leads to equations of motionfor the microscopic polarizations f ij,k = aik

† ajk i j and the

subband occupations nik = aik† aik . On the level of the

phonon-impurity-assisted coherences aik† bq

†ajk aik

† Vmnl qajk , respectively, we neglect the interaction

with the optical field since these terms are of higher order inthe coupling.48 We assume the system to be homogeneous inthe plane perpendicular to the growth direction z; then, wecan restrict to a density-matrix diagonal in the in-plane mo-mentum: aik

† ajk = f ij,kkk.The general dynamical equations are found in the Appen-

dix. While for the interaction with the doping centers, thecomplete equation structure is considered, only the terms lin-ear in the density matrix are taken along for the electron-phonon interaction due to numerical reasons. This approxi-mation is justified if we assume that we are working in theregime of nondegenerate electron gases, where nik 1 whichis typically fulfilled in these QCL structures.

We should stress at this point that, due to the spatial ex-tension of the Wannier-Stark wave functions, it is essential toconsider the whole set of matrix elements in the calculations.Testbed calculations considered only certain sets of matrixelements, e.g., we restricted to the diagonal/nondiagonalscattering terms gij

qgijq, considered typically in nonbiased

equilibrium quantum well systems,5,38,39,49 or alternativelyto terms where the overlap of the wave functions is the mainargument, i.e., terms such as gii

qgijq along with the above set

of terms. Both versions lead to nonstable results in someparameter regimes where the coherences are important. It isthus essential to consider the full set of matrix elementswhen considering the properties of the QCL at resonancecondition and for large injection barrier widths, i.e., smalltunnel coupling across the injection barrier.

When considering the WSH model, the relaxation pro-cesses are restricted to the Boltzmann scattering given by

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nik0 = − ik

outnik + ikin1 − nik . 10

The in- and out-scattering rates ikin and ik

out are functions ofthe subband occupations nlk and are given for both theelectron-phonon and electron-impurity interactions in theAppendix.

The system temperature enters the calculations throughthe scattering rates which include the phonon distribution,assumed to be given by an equilibrium Bose-Einstein distri-bution nq= expq /kBT−1−1, and the screening param-eter for the interaction of the electrons with the dopingcenters see also Ref. 50. Here, the approximation is madethat the lattice and the electronic temperature are the same.

2. Periodic boundary conditions

In order to describe the extension of the periodicallycoupled structure, it is necessary to apply appropriate bound-ary conditions in the growth direction. Since the inclusion ofcoherence between states of different periods is important todescribe the nonlinear optical dynamics due to the spatialextension of the wave functions,26 we consider a nearest-neighbor approach where the coherence and wave functionoverlap between two adjacent periods n ,n are taken intoaccount. We checked this approximation by showing thatf injn,k, ginjn

q ,Vinjnl q0 for n−n2, where

f injn,kginjnq ,Vinjn

l q denotes the coherence coupling ele-ment between the states i and j in the periods n and n,respectively. We apply translational invariance of the densitymatrix between periods,51

f injn,k = f in+1jn+1,k, nin,k = nin+1,k . 11

For the wave functions and the band edge energies i, weapply a coordinate shift and a bias drop, respectively,

inz = in+1z + Lper, in = in+1 − eELper, 12

where Lper is the period length and E the applied electricfield. In the following, the index ii ,n is taken as a com-posite index.

3. Shortcomings of the model

Even taking along the whole set of equations within theapproximations discussed above, we partially encounternegative occupations nik see Fig. 2 which is known fromthe treatment of Redfield-type equations such as considered

here.52–54 However, this typically does not lead to unphysicalresults if we focus on averaged observables. For all dynami-cal calculations, we obtain physical values for the currentdensity, the total electron densities in each subband, and thegain spectrum. However, the broadening in Eq. 13 maybecome negative for regimes where coherences play an im-portant role due to the negative occupations. This may bedue to the fact that i the scattering rates as such are notobservables and ii the simple formula for the broadeningused here, which is typically applied in the literature, doesnot take into account stationary coherence. In order to guar-antee strictly positive values for the averaged scattering ratesto calculate the broadening in Eq. 13, which we can use asa measure of the lifetime of the Wannier-Stark states, weperform a Gaussian smoothening of the stationary distribu-tions and use these in the calculations of the broadening .Figure 2 shows an example of this smoothening for two dif-ferent barrier widths: while for small barrier widths and lowtemperatures the negative occupations are negligible, andthus the corresponding averaged scattering rates are alwayspositive, the smoothening becomes important for large bar-rier widths where the negative occupations can lead to nega-tive values. Thus, the values obtained for the broadening should be viewed as approximative values. Since we are con-cerned with qualitative results only, this is justified. For thecases where the negative occupations are negligible, and thusthe averaged scattering rates strictly positive, the smoothen-ing induces changes in the broadening of 2%–3%. We wouldlike to stress again, however, that all dynamical calculationsand all results except for Fig. 5 are performed without arti-ficial modifications of the calculated data, such as any kindof smoothening. In addition, it should be noted that the cal-culations here are performed at the point of “maximal coher-ence” with respect to the tunnel coupling across the injectionbarrier, i.e., at resonance cf. Sec. III A. Away from reso-nance, the influence of coherence and the correspondinglyrelated problems such as nonpositivity are strongly reduced.

The reason for the quite large negative values for the oc-cupations partially encountered may be explained by the factthat the system under consideration is a very complicatedone: for the nearest-neighbor coupling considered in the cal-culations, we have 35 independent coherences, and thusthere is a very strong interplay between the different frequen-cies of the coherences which can destroy the strict positivityof the occupations. For small barrier widths and low tem-peratures, the general dynamical Eqs. A1 and A5 reduceto the Boltzmann dynamics given in Eq. 10. Here, anynegative occupations observed in our full calculations arenegligible nik−10−8 see, e.g., Fig. 2a. In the case ofthe Boltzmann scattering dynamics, i.e., including only thediagonal elements of the density matrix, the stationary occu-pations remain strictly positive for all barrier widths andtemperatures.

The specific oscillation energies of the peaks can be ex-plained by the energetic structure of the system: the peakscorrespond to different combinations of the bias drop perperiod eFLper and the LO phonon energy ELO. In Fig. 2b,e.g., the large peaks correspond to the energy eFLper−ELO=56.40−36.7 meV=19.7 meV, while the smaller peak po-sitions are given by 2eFLper−3ELO=2.7 meV. The latter are

0

0.01

0.02

0.03

0 10 20 30 40

n ik

−h2 k2/(2 m) (meV)

subb. 1subb. 5

-0.01

0

0.01

0.02

0 10 20 30 40

n ik

−h2 k2/(2 m) (meV)

subb. 1subb. 5

(b)(a)

FIG. 2. Color online Gaussian smoothening of the subbandoccupations nik for the two main subbands injector 5, upper laserstate 1 at T=10 K: a b=4.5 nm; b b=11.0 nm.

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weaker since they represent a higher order process of itera-tive transport and scattering. For the two subbands, the peaksare shifted by E0.1 meV, corresponding to the tunnelsplitting between the two states, E1−E50.1 meV.

As stated above, there is a strong interplay between dif-ferent coherences of sharp energy. For T=50 K and a certainbarrier width regime, we need to incorporate additionaldamping in order to obtain convergence. In the calculations,we do this by adding a phenomenological, low-energy intra-subband scattering mechanism with an energy which is un-commensurable with the LO phonon energy E0.5 meVand use the LO phonon coupling element with an increasedstrength. We should note that this damping is of a purelyphenomenological character and not a physical scatteringmechanism as implemented here. The concerned barrierwidths are marked in Figs. 5 and 6. The problem with con-vergence may be a consequence of the combination of ne-glecting low-energy scattering channels and applying theBorn-Markov approximation within the density-matrixtheory. Another explanation of this convergence problem isthat scattering-induced energy renormalizations are ne-glected at this order of the perturbation expansion. This is anapproximation often applied within the Born-Markovapproximation.25,38 Since the renormalizations, like the scat-tering rates, are temperature dependent, this could explainwhy the convergence problems only appear for certain tem-peratures. On the other hand, since the resonances depend onthe specific energy structure, and thus on the barrier width b,problems are expected to appear only for certain barrierwidths, as witnessed in the results.

Including an artificial damping as discussed above, wefind that the value of the stationary current is approximatelyindependent of the scattering strength. This has already beenobserved in earlier studies of the QCL Refs. 15 and 41 andattributed to the fact that a low-energy dissipative scatteringmechanism such as LA phonons leads mainly to a thermali-zation within the subbands but has no strong effect on theactual value of the stationary current. We adopt the validityof this statement here and take the obtained values of thecurrent as physical results. On the other hand, it is clear thatthe inclusion of additional large damping leads to a strongoverestimation of the total broadening as calculated in Eq.13. Since the relation between the broadening and thetunnel coupling 2 is used to discriminate qualitatively dif-ferent regimes, the inclusion of additional scattering mecha-nisms for the concerned barrier widths which typically leadsto a larger broadening does not lead to qualitatively newregimes of interest see Fig. 5. It is for this reason that onecan safely disregard these values in the discussion of thebroadening and the tunnel coupling as is done in the follow-ing.

III. NUMERICAL RESULTS

In this section, we apply the theory of Sec. II to the THzlaser from Ref. 37. This structure has been investigated withrespect to its transport in the stationary laser regime as wellas its gain properties.19 In the regime of ultrafast nonlinearoptics, Rabi oscillations were considered recently.26

First, the systematic setup is discussed, along with thecalculation of the central quantities used for the further dis-cussion of the transport and optical properties. The stationarynonequilibrium due to the different scattering mechanisms isdetermined. This is necessary for the determination of thestationary value of the current but also as a starting point forthe consideration of the optical response. Then, the WSHapproach is compared with a full current calculation to in-vestigate the importance of the inclusion of coherence in thestationary transport. Finally, the nonlinear optical response ofthe structure is considered, focusing on pump-probe signalsas well as the spatiotemporally resolved electron density andthe optically induced population dynamics, to study the roleof coherence in the optical regime.

A. Systematic setup and stationary state

In order to systematically investigate the importance ofcoherence, we vary the width of the main tunneling barrierconnecting the injector and the active region of the laser.When resonant tunneling between the injector and the upperlaser state is small, the WS states are approximately localizedin the injector and in the active region. For a certain appliedbias, the WS states become delocalized across the injectionbarrier and an anticrossing occurs, where the injector and theupper laser level form a pair of binding/antibinding states.The system is assumed to be in resonance at the center of theanticrossing, i.e., for the bias at which the splitting energybetween the two respective levels has a minimum. Here, thelevel splitting E equals twice the tunnel coupling 2 be-tween the localized injector and the upper laser state seeFig. 3.

The QCL as a multisubband nonequilibrium system is avery complex structure, and thus a straightforward analyticaldetermination of the stationary population and coherence dis-tributions due to scattering is not possible. We thus numeri-cally determine the initial conditions to be stationary solu-tions for f ij,k and nik without the optical field by solving allscattering contributions from an arbitrary state of fixed totalpopulation until a steady state is reached. Figure 4 shows theprocess of determining the initial distributions by startingfrom subband populations ni=2 /Aknik given by single-subband Fermi distributions with equal populations in eachsubband. An approximate stationarity is reached on a time

-150

-100

-50

-40 -20 0 20 40 60

ener

gy(m

eV)

growth position (nm)

b5 4

1’

∆E

0

0.1

0.2

0.3

56 56.5 57 57.5

∆E(m

eV)

bias drop per period (mV)

b = 11.0 nm b = 10.0 nm

∆E=2Ω

(b)(a)

FIG. 3. Color online a Injector 5 and upper laser state 1showing an anticrossing at the main tunneling barrier at exact reso-nance of the corresponding Wannier states. Lower laser state 4 isalso shown. b Tunnel splitting energy E= E5−E1 for varyingbias and different barrier widths b determining the resonance bias atthe center of the anticrossing and the tunnel coupling 2E.

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scale of a few picoseconds, while long time intraband redis-tributions can last up to several nanoseconds not shown.For small barrier widths, i.e., large tunnel coupling , therelaxation is described well by a Boltzmann relaxation nik

0.For larger barrier widths, the influence of the coherence onthe stationary populations cannot be neglected any longerand becomes important which is discussed in detail in thenext section.

Having determined the stationary distributions, we cancalculate the scattering-induced broadening of the states. Thelevel broadening of the considered states is approximatelygiven by the mean of the two tunnel-split Wannier-Starkstates, = 1+5 /2, where the broadening is determinedby the stationary Boltzmann out-scattering rate,

i =

2

Ak

nik0

ikoutnlk

0

ni0 , 13

for the stationary occupations nlk0. At these low tempera-

tures, the broadening is typically dominated by the impurityscattering, which is strongly temperature dependent due tothe screening length50 and thus increases strongly from 10 to50 K, while the interaction with phonons is essentially givenby spontaneous emission.

Figure 5 shows the calculated values of and as afunction of the barrier width b. As is expected, the broaden-ing remains roughly constant for varying barrier widths sincethe scattering rates are not strongly influenced by the width,

while the tunnel coupling across the barrier approaches zerofor b→. For T=10 K, the two energies are roughly thesame for b6.0. This is the regime which is physically in-teresting, and thus barrier widths slightly larger and less thanthis value are the focus in the following investigations. ForT=50 K, the broadening of the states is much larger then thetunnel coupling for all considered barrier widths, and thus notransition between different regimes is expected.

B. Stationary results: current calculations

The stationary distributions far from equilibrium deter-mined in Sec. III A are the starting point for the optical dy-namics discussed later in this chapter. Here, we consider theaccompanying evolution to the steady state value of the cur-rent flowing through the structure. In order to investigate theinfluence of coherence on the system, we focus on two dif-ferent approaches to the current calculation: the first restrictsto the occupations nik to consider a rate equation approach—the so-called Wannier-Stark hopping model—and the secondtakes the coherences f ij,k into account.

We start with the rate equation approach, where in effectthe electrons are counted as they cross an interface at a fixedpoint in the growth direction of the structure due to scatter-ing between states which are localized at different spatialparts of the structure. The WSH current is just given by theapplication of Fermi’s golden rule to each set of states. Forthe electron-phonon interaction, it is determined via n−q=nq,

JWSH t = − 2

2

ALij

k,q

gijq 2ik − jk+q LO

nq +1

2

1

2dii − djjnik1 − njk+q , 14

where J denotes the absorption/emission of a phonon, dii=−ezii is the expectation value of z for the WS state i , andthe sum i , j is carried out over all states. The factor 2 arisesdue to spin degeneracy and L=NLper is the length of thestructure with N periods of length Lper. The current is fullydetermined by the diagonal elements of the density matrixnik, considering jumping of electrons from one state to thenext, and reaches its steady state value as determined by therelaxation in Fig. 4 when restricting to Boltzmann dynamics.An analogous expression is found for the elastic impurityscattering. While Eq. 14 can be motivated by hopping ofelectrons between different positions zii and zjj, it is in factan approximation for the full current driven by coherencesEq. 15; see Ref. 56.

When deriving the current microscopically from the cur-

rent operator J=−e / 2m0†x , tp+eAtx , t+H.c.,one arrives at another picture of the current, where it is thecoherence between states which carries the current flowingthrough the structure. Here, the current is given completelyvia the nondiagonal elements of the density matrix f ij,k,which in growth direction yields

0

0.4

0.8

1.2

0 20 40 60

subb

and

pop.

(10-4

nm-2

)

time (ps)

T = 10 K

12

34

5

0 1 2 3

0

0.4

0.8

1.2

0 20 40 60

subb

and

pop.

(10-4

nm-2

)

time (ps)

T = 50 K

1

2

34

5

(b)(a)

FIG. 4. Color online Population relaxation dynamics using thefull set of dynamical equations for a barrier width of b=5.5 nm atT=10 and 50 K, determining the stationary nonequilibrium of thesystem. Inset: Short-time evolution of the subband populations.

0

2

4

6

5 6 7 8 9 10 11 12 13

Γ(m

eV)

2Ω

(meV

)

barrier width (nm)

2 ΩΓ 10 KΓ 50 K

FIG. 5. Color online Tunnel coupling 2 and mean broaden-ing = 5+1 /2 of the injector and the upper laser state fordifferent barrier widths b and at T=10 and 50 K see remark inRef. 55.

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Jfullt = − 2e

ALij

k

vij f ij,k , 15

where vij =12 ipz /mz+eAt /mz+H.c.j are the veloc-

ity matrix elements and the factor 2 again arises due to spindegeneracy. For the calculation, the polarizations f ij,k are ini-tially set to zero. The finite occupations in the subbands leadto a stationary current which is fully determined by the scat-tering in the structure.

We now carry out current calculations using both theWSH and the full approach for varying barrier widths. Theresult is shown in Fig. 6. The WSH current remains approxi-mately constant for varying barrier widths. This is due to thefact that at resonance, the injector and the upper laser levelform a pair of binding/antibinding states whose nature hardlychanges with b. Thus, the hopping current is not affected bythe barrier width; see also Ref. 27. The slight increase in thecurrent may be attributed to the impact of the other levelswhich change slightly with decreasing bias, which is neededto keep the two tunnel-coupled states in resonance.

In contrast, the full calculation Eq. 15 based on thecoherences shows the expected drop of the current with bar-rier thickness. We first focus on the case T=10 K. For smallbarrier widths where 2, the two stationary currentresults are approximately the same and show a qualitativelysimilar behavior, i.e., the slope of the two curves are almostparallel. In this regime, the WSH and the full model bothdescribe the stationary current well. The slightly smallervalue for the full model stems from the fact that only themain injection barrier is investigated systematically here.Resonant tunneling at the other barriers in the structure is notconsidered, so that a similar argumentation as applied in thefollowing can also be applied to them. Thus, it is expectedthat a constant offset can occur, probably due to the extrac-tion barrier where the two delocalized states lower laserlevel, extraction level are also close to resonance.

For large barrier widths where 2, the full currentdecreases until it almost vanishes in the limit of large barrier

widths. This decrease is the physically expected behavior, asthe growing injection barrier width restricts the current flow-ing across the barrier and thus through the whole structure.For larger barrier widths, the growing localization of thecharge in the injection region leads to the fact that theWannier-Stark states which are delocalized across the injec-tion barrier no longer constitute a “good” basis to describethe states of the system. For this reason, the coherence be-tween these two states becomes important in the stationarystate, whereas it is negligible for smaller barrier widths. Thiscan be seen in the relaxation dynamics as well, where forlarge barrier widths, the influence of the coherences on thepopulations, mainly of the injector and the upper laser state,becomes very important, and thus a pure Boltzmann relax-ation description fails for the stationary populations.

For the case of T=50 K, the situation is different. Forsmall barrier widths, the drop of the full current is morepronounced than for T=10 K, showing that even for thosebarrier widths the WSH model for the current fails. Weshould note that this difference is not as pronounced aswould be expected; however, since we are interested inqualitative results, it suffices to find that the onset of devia-tion between the WSH and the full calculation is shifted tolower barrier widths. It should be noted that for all barrierwidths here, the relation 2 is fulfilled.

It is thus found that in the stationary current calculations,the inclusion of coherence becomes important if 2 un-der resonance. For 2, the WSH approach to the currentcalculation is a good approximation, and thus the inclusionof coherence is negligible. For 2, only the full modelshows the physically expected decrease in the current as thetunnel coupling decreases since the coherence between theinjector and the upper laser state becomes important to de-scribe the localization of the eigenstates. The decrease in thetunnel coupling across the main tunneling barrier limits thecurrent flowing through the structure. This limitation is notreproduced by the WSH approach. It should be noted thatthis result has already been discussed before in Ref. 27 forthe QCL and in Ref. 57 for superlattices.

C. Ultrafast optical dynamics

We now turn to the nonlinear optical response due to anultrafast external perturbation of the laser structure from sta-tionarity. Here, the situation is not as clear as in the transportcase. Obviously, the light-matter interaction as such is a co-herent process as the vector potential couples to the nondi-agonal elements of the density matrix f ij,k. However, it ismore interesting to consider the dynamics after the passageof the pulse, where the signal is typically dominated by in-coherent scattering.

As discussed in Sec. III B, coherences between Wannier-Stark states are important in the regime of large barrierwidths and high temperatures, where the stationary states arecoherent superpositions of the WS states; they describe thetunneling through the barriers and thus the return to the sta-tionary state. This is expected to remain important in theoptical dynamics in order to describe the return to the steadystate after optical excitation. However, the comparatively

0

0.2

0.4

0.6

5 6 7 8 9 10 11 12 13

curr

entd

ensi

ty(k

A/c

m2 )

barrier width (nm)

T = 50 K

WSHfull

0

0.2

0.4cu

rren

tden

sity

(kA

/cm

2 )

T = 10 K

WSHfull

FIG. 6. Color online Stationary current for both the WSH andthe full calculation at resonance for varying barrier widths at T=10 and 50 K see remark in Ref. 55.

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large scattering with respect to the tunneling see Fig. 5 isexpected to destroy all observable coherent effects in thisregime. The question arises whether coherent optical effectsafter ultrafast optical excitation can be observed in the oppo-site regime, where coherences were shown not to be impor-tant in the stationary transport. Recently, optical pump-probemeasurements in mid-infrared29–32 and THz QCLs Ref. 33have shown an oscillatory behavior attributed to coherentresonant tunneling through the injection barrier. In this sec-tion, this effect is investigated with respect to the relationbetween the tunnel coupling and the level broadening.

We begin by considering the linear response of the systemto characterize the laser. Then, we analyze the nonlinear op-tical response in form of the experimentally studied pump-probe signals to study the regime of importance and the rolecoherence plays in the optical signal. We complement thisstudy by looking at the optically induced charge as well asthe subband population dynamics.

The linear absorption of the structure is calculated via thecomplex susceptibility

=1

0

J2A

. 16

Jt=Jt−J0 denotes the change in the current density in-duced by the optical field with vector potential A, whereJ0 is the stationary value of the current density determined inthe previous section and Jt is given by Eq. 15. From this,the absorption Im is calculated.

In Fig. 7, the absorption spectrum of the gain transition isshown for different injection barrier widths. Due to the anti-crossing at resonance, the spectrum shows a double-peakstructure which can be resolved for small barrier widthswhere the tunnel splitting is sufficiently large. For larger bar-rier widths, the two peaks merge to form a single resonancefor sufficiently small tunnel splitting energies compared tothe dephasing of the transition. Due to this splitting, it is apriori unclear which of the transitions is to be considered theoptical gain transition. In the following considerations, thesystem is excited resonantly on the transition yielding thelarger peak gain which here is the lower gain transition fre-quency, and the accompanying optical dynamics is investi-gated. The second peak appearing for b=9.0 nm at E16 meV arises from an enhanced dipole moment betweenthe two resonant states 1 ,5 and extractor state 3.

When increasing the barrier width further, the coherencebetween the injector and the upper laser state becomes im-portant so that the stationary state of the system is in a linearsuperposition of the two WS states, localized in the injectorregion. This coherence can lead to a strong decrease in oreven a vanishing of the gain. To illustrate this, it is helpful toconsider a simple three-level system consisting of the tworesonant states injector 5 and upper laser state 1 and lowerlaser state 4. We assume that the coherences between thelower laser state and other two states is small compared tothe stationary tunneling coherence at the time of the probepulse t0, which is taken as a pulse, i.e., f41t0 , f45t0 f15t0. Solving the semiconductor Bloch equations, thegain at the laser frequency is then given by under the as-sumption E5−E1 E5−E4

− d452 n5 − n4 + d41

2 n1 − n4 + 2d45d41 Ref15t0 ,

17

with the constant level densities ni. If the system is now inthe state 5 + 1 d45−d41 which is localized inthe injection region, as is the case for large barrier widths,we have n5n1. The gain is then given by the stationaryinversion of the states n5−n4 as well as a further term−2 Ref15t0 containing the coherence between the tworesonant states,

− d452 2n5 − n4 − 2 Ref15t0 . 18

Typically, the first term containing the inversion determinesthe optical spectrum. In the presence of a strong stationarycoherence f15t0, the gain can be strongly reduced or evenvanish, even in the presence of a strong inversion, which isthe case for all barrier widths considered. The stationary stateas a superposition of the two WS states is fully localized inthe injector, leading to a very small dipole overlap with thelower laser state which is witnessed in the vanishing gain.

For the nonlinear optical dynamics, we focus on thepump-probe signals which have been experimentally mea-sured recently.29–35 An ultrafast nonlinear pump pulse reso-nant on the laser transition excites the sample, leading to again saturation at the laser energy, and a subsequent weakprobe pulse tests the laser transition as a function of thedelay time between the two pulses see Fig. 8. Correspond-ing to the experiments, a 170 fs 1 pump pulse P and a incomparison to all optical transitions of interest spectrallybroad probe pulse Pr, with T=50 fs are used. In contrast toa standard differential transmission spectrum where the com-plete energy range is recorded,47 the focus is on a monochro-matic measurement, yielding the absorption at a fixed energyfor different delay times. The experimental separation ofpump and probe pulses which is done via directional filteringof the response is carried out here in such a way that theresponse to only the pump pulse JP is subtracted from theresponse to both the pump and probe pulses PP+Pr, yieldingthe purely linear response after the influence of the pumppulse whose direct response is filtered out. Thus, we considerfor the absorption

0

8 10 12 14 16 18

abso

rptio

n(a

rb.u

nits

)

energy (meV)

increasingbarrier width

4.5nm5.5 nm6.5 nm7.0 nm9.0 nm

11.0 nm13.0 nm

FIG. 7. Color online Dependence of the absorption spectrumon the injection barrier width at T=10 K.

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td, ImJP+Prtd, − JPAPrtd,

, 19

with the delay time td, the fixed pump and probe energy which is equal to the laser energy here, and the probe fieldAPr. Ji is the Fourier transform of the correspondingsimulated time signal Jit taken at the fixed frequency .In Ref. 58, it is shown that the nonlinear absorption accord-ing to Eq. 19 describes the signal contribution measured inthe probe direction. While the signal contains all coherenteffects also in the limit of delay times shorter than or com-parable with the dephasing times, additional coherent effectsare included for short delay times which are not measuredexperimentally due to directional filtering; these effects areexpected to be small. Our result reproduces well-knownstudies in the probe direction, for instance, Ref. 59. The dif-ferential pump-probe response is then given by exp−td−0tdL−1−td+0tdPPtd, where 0td de-notes the linear response without the pump pulse and it isassumed that L1 where L is the length of the structure.For reasons of presentation, we plot the negative differentialpump-probe response. Then, positive signals correspond to adecreased gain compared to the stationary reference gainvalue.

The pump-probe response for different barrier widths andat T=10 and 50 K is shown in Fig. 8 see also Fig. 11 for thecorresponding population dynamics caused by the pumppulse for b=4.5 and 7.0 nm at T=10 K. The ultrafast pumppulse saturates the gain transition, leading to a strong absorp-tion at this energy positive pump-probe signal. The follow-ing return to the steady state shows two main features: i adecay of the signal on a time scale of a few picoseconds.This feature is due to the incoherent scattering which leads toa return to the steady state, yielding the original gain for thelaser transition. ii For smaller barrier widths, an oscillatory

modulation of the signal whose amplitude decreases forgrowing barrier widths, while its period of the order of pico-seconds increases correspondingly. In addition, fast oscilla-tions with a period on the order of 0.1 ps are visible whichare addressed briefly below but which are not the focus ofthe paper.

The oscillatory feature is a coherent effect in the dynam-ics after the passage of the pulse: The pump pulse depopu-lates the upper laser state locally in the excited region. Atresonance condition, such a locality is represented by a co-herent superposition of levels 1 and 5. This causes coherentcharge oscillations between the two superpositions which arelocalized in the injector and in the active region,60 leading toa modulation of the coherence between the two laser statesand thus to an oscillation of the gain signal. The oscillationhardly causes any changes in the WS subband populationsand no oscillating inversion as will be discussed later in thissection. Thus, the oscillation is a coherent effect. The chargeoscillations are visible for the case that the tunneling periodTosc1 /E is less than the scattering period Tscatt1 /,i.e., ToscTscatt. Let us first focus on T=10 K. Here, gainoscillations are strongly visible for b=4.5 and 5.5 nm where2, i.e., ToscTscatt. Already for b=6.5 nm, the oscilla-tions become hardly visible and fully vanish for d7.0 nm, where ToscTscatt. For very large barrier widths,it is not possible to invert the gain transition due to the strongcoherence between the injector and the upper laser state. Thetwo states are now in a superposition state localized in theinjector, and thus a dipole interaction via the electric fieldbetween the lower laser state and the upper state is not pos-sible since the charge is localized outside the active region.Thus, no charge inversion occurs compare the discussion ofthe gain for large barrier widths in Fig. 7. At T=50 K,2, i.e., ToscTscatt is fulfilled for all barrier widths cf.Fig. 5, and thus only very weak gain oscillations are foundfor b=4.5 nm. Again, for large barrier widths, no inversionoccurs, leading to an almost vanishing pump-probe signal. Itshould be noted that the fast oscillations on a time scale ofaround 300 fs seen in the insets of the pump-probe signalsfor small delay times td1 ps can be attributed to tunnelingdue to the coherence between the lower laser state and thetwo resonant states which show an energy splitting of E13 meV, corresponding to an oscillation period of Tosc300 fs. Due to the changing applied bias and the corre-spondingly changing laser energy, the oscillation periodchanges slightly between Tosc200–300 fs.

Experimentally, gain oscillations have been observed in amid-infrared laser in Refs. 29–32 and in a THz laser in Ref.33. In the case of the mid-infrared laser, pronounced gainoscillations were found up to relatively high temperatures,including a gain overshoot when probing close to the reso-nance. This effect depends strongly on the strength of thescattering mechanisms, specifically on the depletion of thelower laser subband and the strong coupling through the in-jection barrier, and has not been observed in our calculations.According to Ref. 31, the lifetime of the superposition of thetwo resonant states Tscatt1 ps which is much longer thanthe observed oscillation of Tosc500 fs, and thus 2which is the regime where gain oscillations are expected. InRef. 35, time-resolved pump-probe differential transmission

FIG. 8. Color online Pump-probe signals after ultrafast strongpumping and subsequent linear probing at the laser energy for dif-ferent delay times td and barrier widths at T=10 and 50 K. Bothpump and probe pulses are chosen to be resonant on the laser tran-sition. Insets: Pump-probe signals for small delay times.

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measurements did not show signs of gain oscillations. As theauthors argue in the paper, this is due to the very short scat-tering rates due to Coulomb scattering Tscatt100 fs whichis much smaller than the tunneling oscillation period Tosc=517 fs, and thus no gain oscillations are expected.

The interpretation of the oscillations in the pump-probesignals being caused by coherent charge transfer across theinjection barrier is strengthened by considering the spatioen-ergetic structure of the stationary state obtained from ourGreen’s function model.20 In Figs. 9a and 9c, the spectralfunction AE ,k=0,z which describes the density of statesfor vanishing in-plane momentum is shown. The peaksroughly correspond to the WS wave functions iz2 with anenergetic width i. For b=4.5 nm, the binding and antibind-ing combinations of the injector and the upper laser level canbe resolved around E36 meV, while this is not the casefor b=7.0 nm as E. The right figures show the lesserGreen’s function ImGE ,k=0,z describing the corre-sponding carrier density. For b=4.5 nm, the carrier densityqualitatively follows the spectral function corresponding totwo occupied levels in the stationary state. Thus, when per-turbing the stationarity on an ultrashort time scale, a super-position of the two states leads to an oscillatory modulationof the gain. For b=7.0 nm, the carrier density does not fol-low the density of states; only the superposition of the twostates, localized in the injector, is occupied at stationarity,and thus no superposition can be excited and no oscillationsoccur. The stationary distribution is already a superpositionof the two WS states, which is also the reason why the in-clusion of coherence is important to correctly describe thestationary transport of the system see Fig. 6.

In addition to the spatioenergetic resolution of the station-ary charge density, it is insightful to consider its spatiotem-poral evolution during and especially after the excitationwith an ultrafast laser pulse. To do this, we consider thespatiotemporally resolved electron density which is given by

nz,t = 21

Aij

iz jz

kf ij,kt , 20

where the time dependence is included in the density matrixf ij,kt. In order to simplify the representation and to focus onthe optically induced dynamics, the difference betweennz , t and the stationary electron density n0z, nz , t=nz , t−n0z, is considered in the following. Similar calcu-lations have been presented in Ref. 23, where however nodetailed modeling of the pump pulse was done.

In Fig. 10, the electron density evolution is shown for thesame parameters of the pump pulse as used for the calcula-tion of the pump-probe signal discussed above. Shortly afterthe pulse has passed at t0.3 ps, the density is depleted inthe active region due to the fast nonradiative extraction fromthe lower laser level to the extractor state and into the injec-tor.

After this, two qualitatively different behaviors are seenfor the two barrier widths. For b=4.5 nm, the resonant tun-neling across the injection barrier leads to a fast refilling ofthe upper laser state and thus to an increased density in theactive region at t1 ps. At t2 ps, the density in the ac-tive region again decreases. Thus, the oscillation of chargewhich occurs on a time scale of 1.8 ps as detected by theprobe pulse see Fig. 8 left can be directly seen in the

(b)(a)

(c) (d)

FIG. 9. Color online a,c Spectral function and b,d the imaginary part of the lesser Green’s function for two different barrierwidths at T=10 K.

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electron density evolution. When considering the electrondensity without the polarizations, i.e., restricting to the diag-onal elements of the density matrix for the calculation, oreven just without including the coherences between differentperiods not shown, this oscillation is not observed at all.

For the case of b=7.0 nm, a roughly monotonous returnof the density depletion to the steady state is found, and thusno gain oscillations are expected, as verified in the pump-probe signals. Due to the weaker tunnel coupling betweenthe injector and the active region, the charge transfer is slowso that the oscillation period induced by the tunnel couplingTosc is larger than the scattering period Tscatt, and thus chargeoscillations cannot be observed. Additionally, there are fastoscillations in the active region on a time scale of around 300fs which we already addressed in the discussion of the pump-probe signals. The oscillations in the injector region can beexplained by coherences between the injector states whereE2−E55 meV=850 fs.

It has already been mentioned that the charge oscillationsare not seen in the electron density evolution when only thediagonal elements of the density matrix are used in Eq. 20.Thus they constitute a coherent effect. This is better exem-plified by considering directly the subband population evo-lution caused by the strong pump pulse excitation. We thusconsider the dynamics of the WS populations nit=2 /Aknikt under ultrafast optical excitation. It should bestressed here that the dynamical calculations are carried outwith the full equation structure including both the diagonaland nondiagonal density-matrix elements. Again, the dynam-ics is considered for the same nonlinear excitation used inthe pump-probe calculations. The results are shown in Fig.11.

During the excitation with the pump pulse, the popula-tions of the laser levels undergo Rabi flopping, while theother levels follow adiabatically.26,61–63 After the excitation,the populations return to the steady state due to the incoher-ent scattering processes. While for b=4.5 nm, the Rabiflopping is restricted mainly to the two resonantly excitedsubbands 1 ,4; the subband flopping for b=7.0 nm in-cludes both the two laser states 4,1 and injector level 5since both transitions from the lower laser state are energeti-cally very close due to the very small tunnel splitting. Thisweakens the oscillator strength of each transition, reducingthe effective pulse area compared to the full population in-version seen for a 1 pump pulse in an ideal two-level sys-

tem. Still, in both cases, the gain transition is saturated, lead-ing to absorption at this energy. An oscillating inversion ofthe upper und lower laser states corresponding to the oscil-lations in the pump-probe signals Fig. 8 or in the spa-tiotemporal electron density evolution on a time scale of afew picoseconds Fig. 10 is found neither for b=4.5 nm norfor b=7.0 nm. As a matter of fact, if the dynamics are con-sidered without scattering not shown, no further populationdynamics are found after the passage of the pulse, while thecorresponding pump-probe signals show an undamped oscil-lation in the case of b=4.5 nm. This illustrates again that theoscillations in the gain recovery are a coherent effect.

In summary, it is found that gain oscillations are observedwhere the tunnel oscillation period is sufficiently less thanthe scattering period, where the relation 2 is valid. Thetunnel coupling leads to pronounced coherent charge transferbetween the injector and the active region as witnessed in thepump-probe signals and the spatiotemporally revolved elec-tron density evolution. Considerations of quantities which donot directly relate to the coherence, such as the dynamics ofthe WS populations, show that the oscillations observed inthe pump-probe signals constitute a coherent effect whichrequires a fully coherent theory to be observed. It should benoted here that compared to the current calculations, where astationary localization of charge leads to the drop of thecurrent signal, it is here the dynamical transfer of chargebetween different locations in the laser structure which con-stitutes the coherent effect. Thus, the inclusion of coherencebecomes essential in the nonlinear optical dynamics for theregime where 2 to describe the observed coherent ef-fects. Here, the coherence in the system drives the gain os-cillations, while for 2, the scattering destroys the co-herent effects.

IV. CONCLUSION

We have established a microscopic quantum theory ofscattering for quantum cascade lasers to treat stationarytransport as well as nonlinear optical dynamics. A key issueis the importance of coherence, i.e., the nondiagonal ele-ments of the density matrix, and the role which it plays indifferent parameter regimes of the tunnel coupling be-tween the two states across the main injection barrier and the

FIG. 10. Color online Spatiotemporal evolution of the electrondensity of the QCL after ultrafast nonlinear optical excitation at T=10 K. The pump pulse parameters are the same as for the pump-probe results.

0

0.4

0.8

1.2

0 1 2 3 4

subb

and

pop.

(10-4

nm-2

)

time (ps)

1

25

43

b = 4.5 nm

0

0.4

0.8

1.2

0 1 2 3 4

subb

and

pop.

(10-4

nm-2

)

time (ps)

1

2

5

43

b = 7.0 nm

(b)(a)

FIG. 11. Color online Optically induced population dynamicsnit for the barrier widths b=4.5 and 7.0 nm at T=10 K, showingcoherent Rabi oscillations as well as subsequent scattering-inducedrelaxation back into the steady state. The pump pulse parameters arethe same as for the pump-probe results the pump pulse is shown bythe dashed line in the figures.

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mean level broadening of the states. In the transport re-gime, we have compared a rate equation WSH approach tothe current with our microscopic theory at resonance condi-tion and found that for 2, the WSH approach is a goodapproximation. For 2, only the full approach shows thephysically expected decrease of the current for decreasingtunneling across the main injection barrier, whereby the in-cluded coherence limits the current flowing through thestructure via a stationary localization of charge in the injec-tion region. On the other hand, the consideration of ultrafastnonlinear optical pump-probe signals has shown that the in-clusion of coherence is important to describe the experimen-tally observed coherent effects for 2, where it drivesthe dynamical charge transfer between the injector and theactive region, resulting in oscillations in the gain recovery.For 2, only the typical decrease in the pump-probesignal due to incoherent scattering is observed. It should benoted again that, in general, coherences are necessary to de-scribe the return to the stationary state after optical excitationdue to the strong interplay between transport and optics inthis system. The spatially resolved electron density evolutionin time shows the density oscillating between the injectorand the active region for 2. The consideration of thepopulation dynamics of the levels does not show these oscil-lations, revealing that they are an inherent coherent effectresulting from the nondiagonal elements of the density ma-trix.

Thus, the inclusion of coherence is important in oppositeparameter regimes in stationary transport and nonlinear op-tical dynamics: in the former, it becomes important for2, while in the latter, it allows the observability ofcoherent effects for 2. In this sense, the coherence actsin complementary ways: in the transport regime, it leads to alimitation of the current which is not reproduced by the rateequation approach. In the optical regime, the coherencedrives the oscillatory modulation of the gain recovery whichis not destroyed by the scattering. Thus, it is necessary toconsider a fully coherent theory in order to describe the com-bined system of optics and transport in this nonequilibriumstructure in the entire parameter range.

ACKNOWLEDGMENTS

We acknowledge fruitful discussions with RikardNelander, Marten Richter, Stefan Butscher, MichaelWoerner, and Klaus Reimann. This work was supported bythe Deutsche Forschungsgemeinschaft DFG and the Swed-ish research council VR.

APPENDIX: SCATTERING EQUATIONS

1. Electron-LO phonon interaction

The general form of the scattering equations in Born-Markov approximation is given by for nq=n−q

f i j,k =

lmn

q

Nmnli qnj − fnj,kf lm,k+qnq + 1nk

− mk+q + LO + nqnk − mk+q − LO − lm

− f lm,k+qfnj,knq + 1nk − mk+q − LO + nqnk

− mk+q + LO +

lmn

q

Nmnljq in

− f in,kfml,k+qnq + 1nk − mk+q + LO + nqnk

− mk+q − LO − ml − fml,k+qf in,knq + 1nk

− mk+q − LO + nqnk − mk+q + LO , A1

where Nmnij q=gij

qgmnq . In the calculations, the considerations

are restricted to terms linear in the density matrix since in thesystems considered here, f ij,k1 so that f ij,k f lm,k f ij,k.

The Boltzmann relaxation is determined via

nik0 = − ik

outnik + ikin1 − nik , A2

with the semiclassical Boltzmann in- and out-scattering ratesgiven by

ikin =

2

q,l

gilq2nlk+qnq + 1ik − lk+q + LO

+ nqik − lk+q − LO , A3

ikout =

2

q,l

gilq21 − nlk+qnqik − lk+q + LO

+ nq + 1ik − lk+q − LO . A4

2. Ionized doping centers—impurity scattering

The general form of the scattering equations is given by

f i j,k = −

mn

qMnm

mi qfnj,knk − mk+q

− Mjnmiqfmn,k+q jk − nk+q + Mnm

mj qf in,knk

− mk+q − Minmjqfnm,k+qik − nk+q , A5

where the average Mmnij q= l,lVij

l qVmnl −q S

c over thestatistically distributed doping atoms is introduced. Since thesemiclassical interaction is linear in the density matrix, allterms are taken along for this interaction.

The semiclassical Boltzmann in- and out-scattering ratesare given by

ikin =

2

q ,l

Mililik − lk+qnlk+q , A6

ikout =

2

q ,l

Mililik − lk+q1 − nlk+q . A7

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