Immunity of intersubband polaritons to inhomogeneous broadening
J-M. Manceau, G. Biasiol, N. L. Tran, I. Carusotto, R. Colombelli
IImmunity of intersubband polaritons to inhomogeneous broadening
J-M. Manceau , † , G. Biasiol , N.L. Tran , I. Carusotto , R. Colombelli , ‡ Centre de Nanosciences et de Nanotechnologies, CNRS UMR 9001,Univ. Paris-Sud, Universit´e Paris-Saclay, C2N - Orsay, 91405 Orsay cedex, France Laboratorio TASC, CNR-IOM, Area Science Park,S.S. 14 km 163.5 Basovizza, I-34149 Trieste, Italy INO-CNR BEC Center and Dipartimento di Fisica, Universit´a di Trento, I-38123 Povo, Italy
We demonstrate that intersubband (ISB) polaritons are robust to inhomogeneous effects origi-nating from the presence of multiple quantum wells (MQWs). In a series of samples that exhibitmid-infrared ISB absorption transitions with broadenings varying by a factor of 5 (from 4 meV to 20meV), we have observed polariton linewidths always lying in the 4 - 7 meV range only. We have ex-perimentally verified the dominantly inhomogeneous origin of the broadening of the ISB transition,and that the linewidth reduction effect of the polariton modes persists up to room-temperature. Thisimmunity to inhomogeneous broadening is a direct consequence of the coupling of the large numberof ISB oscillators to a single photonic mode. It is a precious tool to gauge the natural linewidth ofthe ISB plasmon , that is otherwise masked in such MQWs system , and is also beneficial in viewof perspective applications such as intersubband polariton lasers.
PACS numbers:
I. INTRODUCTION
The mechanisms responsible for broadening of opticaltransition lines are typically classified in two classes: ho-mogeneous - related to the dynamics of each emitter -and inhomogeneous - inherent to the presence of multi-ple emitters. In the textbook example of an atomic gas,the former class contains the radiative broadening due tospontaneous emission. The latter class includes Dopplerbroadening due to the wide thermal distribution of ve-locities, that randomly shifts each emitter resonance .A similar physics is found in solid-state systems. Forinstance, optical transitions in assemblies of quantumdots suffer from a strong inhomogeneous broadening .This comes on top of the radiative linewidth and deco-herence due to interaction with other degrees of freedomsuch as phonons. As a first step towards observing thenatural linewidth of quantum dots, the inhomogeneousbroadening due to the slightly different sizes and shapesof the various dots can be overcome by restricting to asingle object .In the last decades, ISB transitions in semiconduc-tor quantum wells have been attracting a growing in-terest from the fundamental and applied physics com-munity. This is due to their very strong coupling tothe electromagnetic field, and their easily tunable emis-sion/absorption frequency across a wide spectral range(from terahertz up to mid-infrared frequencies), wherefew other compact emitters are available. As the fre-quency of the ISB transition is dependent on the QWthickness , and on the doping level , slight fabricationinhomogeneities are a major source of broadening of ISBplasmons in MQWs systems.In addition to this, even in the single QW case thephysics of ISB transitions is complicated by the homoge-neous and inhomogeneous broadening mechanisms thataffect the transition linewidth. On one hand, the ho- mogeneous linewidth is governed by the emission rateof photons and phonons, and by the in-plane disorderwhich opens additional decay channels for ISB plasmons.On the other-hand, the non-parabolicity of the electronicbands and the in-plane spatial variations of the QW dop-ing level and thickness are responsible for inhomogeneousbroadening . Since the mid ’90’s this physics has beenwidely studied for interband excitons in the IR domain,where several peculiar line-narrowing effects can beat play to reduce the transition linewidth in strong light-matter coupling regimes (see also a review in Ref. 14).In the ISB context, all these broadening mechanismscan have dramatic consequences for fundamental stud-ies of many body effects in 2D electronic plasmas , butalso for optoelectronic devices. For instance, the thresh-old current density of quantum cascade (QC) lasers isaffected by the inhomogeneous broadening of the mate-rial gain and can only be partially mitigated in the socalled broadband QC lasers .In the last decade, exciting new avenues are beingopened by devices operating in the strong light-mattercoupling regime, where ISB transitions coupled to acavity photon mode form new bosonic excitations ofmixed nature called ISB polaritons . Such a regime isat the heart of intriguing proposed applications to low-threshold lasers and quantum photon sources in thefar-infrared . Transferring into the realm of polaritonsthe design flexibility of ISB transitions - the key ingredi-ent for QC lasers - has been one of the motivations behindthe development of ISB polaritonics . Still, most ofthese applications require a narrow polariton linewidth,which may seem incompatible with the wide absorptionlinewidth typically observed in devices containing a rel-atively large number of QWs.In this article, we experimentally show how the strongcoupling regime permits to largely suppress the inhomo-geneous broadening originating from the presence of a a r X i v : . [ c ond - m a t . o t h e r] N ov Λ t X z k // R=1-A θ E
105 120 135 1500,00,51,01,52,02,53,0 105 120 135 1500,00,51,01,52,02,53,0 T r an s m i tt an c e e no r m . f Energye4meVf cm -2 cm -2 cm -2 cm -2 R e f l e c t an c e e no r m . f Energye4meVf cm -2 cm -2 cm -2 cm -2 FIG. 1: (a) Transmission measurements of the 4 studied sam-ples performed at 78 K in a 45 o , multipass waveguide con-figuration. The data are offset for clarity. (b) Schematic ofthe polaritonic device and experimental probing conditions.(c) Reflectivity measurements performed on the 4 polaritonicdevices at 15 o incidence angle and at 78 K. The reflectivitydips correspond to the lower and upper ISB polaritons. large number of QWs. The underlying mechanism is orig-inally discussed in the atomic cavity-quantum electro-dynamics (QED) context , and it was theoreticallytranslated to the solid-state context in Ref. 25. The coreidea is that by coupling the oscillators to a single cavitymode, one effectively singles out a single state out of theinhomogeneous broadened spectral distribution. The re-sulting linewidth reduction turns out to be quantitativelyimportant and can be beneficial in the development ofISB polaritonics.In a series of samples with increasing absorptionlinewidth, we measure polariton linewidths consistentlymuch narrower than the average of cavity mode and ISBplasmon taken independently. While hints of this phe-nomenon have been observed in Ref. 26, here we pro-vide a full and quantitative characterization of it. Weexperimentally demonstrate the dominantly inhomoge-neous origin of the ISB plasmon absorption linewidth inMQWs system, and we quantitatively explain our ob-servations using a theoretical model inspired from theseminal work Ref. 25. II. THE EXPERIMENTS
The four samples investigated are GaAs/AlGaAs mul-tiple quantum-wells exhibiting ISB absorption resonancesaround 10 µ m wavelength. They have been grown bymolecular beam epitaxy, and consist in 36-period repe- n c m - 2 ) ( b ) I S B L P U PC a v i t y I S B L P U PC a v i t y
Linewidth (meV) ( a )
FIG. 2: Summary of the experimental linewidth of ISB tran-sitions (blue dots), empty grating resonator (red dots), upperpolaritons (green up triangles) and lower polaritons (greendown triangles). Panel (a) reports the results at 78 K.Panel (b) at 300 K. The dashed lines in the two panels rep-resent the average of cavity and ISB transition linewidths,(Γ
ISB + Γ cav ) / expected polaritonlinewidth at zero detuning, i.e. when the Hopfield coefficientsare equal to 0.5. titions of 8.3-nm-thick GaAs QWs separated by 20-nm-thick Al . Ga . As barriers. The substrate is undopedGaAs, and δ − doping is introduced in the center of thebarriers. The 4 samples differ in the nominal modulationdoping level n : 0.6 × cm -2 (HM3821), 1.7 × cm -2 (HM3820), 4.4 × cm -2 (HM3872), and 7.6 × cm -2 (HM3875).All the absorption measurements are done usinga Fourier transform infrared spectrometer (FTIR)equipped with a Globar thermal source and a deuter-ated triglycine sulfate (DTGS) detector operating atroom temperature. The samples are shaped in multi-passwaveguide configuration and mounted on a continuousflow cryostat having anti-reflection coated Zinc-Selenide(ZnSe) windows implemented on the shroud. Resolutionwas set at 0.5 meV (4 cm -1 ) for all the measurements andthe polarization of the incoming radiation is selected witha wire grid holographic Thallium Bromoiodide (KRS-5)in order to get onto the sample the desired electric-fieldprojection in the direction perpendicular to the growthplane.The low temperature (78 K) results are reported in Fig.1a (results at 300 K are in Supplemental Information ,Fig. S1). The peak absorption frequency blue-shifts withincreasing doping because of the depolarization shift .Most notably, the ISB transitions dramatically broadenswith increasing doping.We have then inserted the four samples in disper-sive, grating-based metal-dielectric-metal resonators - assketched in Fig. 1b - following the procedure describedin Ref.28. The tight electromagnetic confinement in-duced by the two metallic surfaces places the system Transmittance (norm.)
E n e r g y ( m e V ) ( b )
Linewidth (meV)
N u m b e r o f Q W s( a )
FIG. 3: (a) Transmission measurements at 78 K of sampleHM3872 after removal of 0, 9, 15, and 26 wells respectively.The data are stacked for clarity and the number of remain-ing QW is reported for each measurement. The dotted lineserves as guide to the eye to mark the central energy of the36 QWs sample.(b) Extracted linewidths, using a Voigt fittingprocedure, as a function of the number of remaining QWs. in the regime of strong light-matter coupling, and ISBpolaritons can be probed with surface-reflectivity mea-surements. The cryostat is placed within a fixed anglereflection unit (15 degree) inserted within the FTIR andthe beam size is selected to cover the entire sample sur-face (2.5 mm x 2.5 mm). Care is taken to suppress anylight that would impinge outside the sample surface.Figure 1c reports the reflectivity measurements on allthe samples at 78 K and 15 o incidence. For each sample,we have adjusted the grating period Λ (filling factor iskept constant at 80%) in order to obtain the minimumpolaritonic splitting at θ ∼ o . The measurements revealthe two polaritonic modes as reflectivity dips, whose en-ergy distance - the Rabi splitting - increases roughly asthe square root of the doping (Fig. S2 of Supplemen-tal Information ), i.e. proportionally to the electronicplasma frequency .The important observation concerns instead the po-lariton linewidths that appear insensitive to the massivebroadening of the bare ISB transition with doping. Fig. 2summarizes the linewidths of ISB absorption transitions(Γ ISB , blue dots), upper and lower polaritons (Γ UP andΓ LP , green triangles), and bare cavity resonators (Γ cav ,red dots) at 78 K and 300 K. The dashed line repre-sents the average of cavity and ISB transition linewidths,(Γ ISB +Γ cav )/2, that corresponds to the expected polari-ton linewidth at zero detuning, i.e. when the Hopfieldcoefficients are equal to 0.5 . The experimental evi-dence is instead that both Γ UP and Γ LP are much smallerthan the expected average cavity and ISB transitionlinewidths. The phenomenon is particularly evident at78 K and is not compatible with the expected polaritonlinewidth being the average of Γ ISB and Γ cav weighted bythe Hopfield coefficients.As a final piece of experimental information, we demonstrated the dominantly inhomogeneous origin ofthe bare ISB transition broadening observed in Fig. 1ain the absence of the cavity. The presence of a low en-ergy shoulder in the absorption spectra of the two high-est doped samples is already a strong indication. Togain further insight, we measured the ISB absorption ofthree pieces of sample HM3872 ( n =4.4 × cm -2 ) af-ter removal by sulphuric-acid-based wet chemical etchof 9, 15, and 26 wells respectively. The low-temperature(T=78 K) multipass waveguide absorption measurementsare presented in Fig. 3a: the peak absorption energyred-shifts and the low-energy shoulder disappears, thusproving that the 36 QWs composing the sample are notall identical. This leads to a linewidth reduction, at both78 K and room temperature, when a reduced number ofQWs is probed (Fig. 3b). These measurements provethat a large fraction of the ISB linewidth in these sam-ples indeed stems from inhomogeneous mechanisms dueto the different parameters of the various wells. III. THE THEORETICAL MODEL
Based on this experimental input and taking inspira-tion from the seminal prediction made in Ref.25, we con-jecture that ISB polaritons are not affected by inhomo-geneous broadening of the MQWs bare ISB transition,but only by the homogeneous one. To substantiate ourclaim, we generalize the temporal coupled-mode theory ofRef.28 by including the spatial periodicity of the grating-based resonator and the inhomogeneous broadening ofthe multiple matter oscillators, and we show that thismodel is able to fully explain in a quantitative way theexperimental findings.
A. Simplified model: planar cavity
As a first step, we neglect the lateral patterning of thecavity and we build a simplified theory that does not in-clude Bragg scattering processes. This simplification willbe useful to fully appreciate the role of the inhomoge-neous vs. homogeneous broadenings, before proceedingwith the development of a complete theory in the nextSubsection.Under the approximation that the cavity is spatiallyhomogeneous along the xy plane, the in-plane wavevec-tor k is a good quantum number. The system’s dynam-ics can then be written in terms of the following motionequations for the components at in-plane wavevector k of the cavity field and of the ISB oscillator amplitude inthe j -th quantum well (with j =1,..., N QW ) : i da k dt = ω cav k a k − i ( γ rad + γ nr )2 a k ++ Ω (cid:88) j b j, k + E inc ( t ) (1) i db j, k dt = ω ISBj b j, k + Ω a k − i γ hom b j, k . (2)Here, ω cavk denotes the cavity mode dispersion, and thefrequencies ω ISBj of the ISB transition in each well areassumed to be independent of the in-plane wavevector k and distributed around their central frequency ω ISB ac-cording to a Gaussian distribution of standard deviation σ inh . Furthermore, Ω is the Rabi frequency of each ISBplasmon coupling to the cavity mode. E inc is the incidentfield. γ rad and γ nr are the radiative and non-radiativelinewidths of the cavity mode, and γ hom is the homoge-neous linewidth of the ISB plasmon, resulting from allnon-radiative decay and decoherence mechanisms due,e.g., to electron scattering on interface roughness . Thereflected field results then from the interference of the di-rectly reflected incident field and the cavity emission , E refl = E inc − iγ rad a k . (3)Reflection spectra are then straightforwardly obtained byinverting the linear set of equations describing the steady-state of the motion equations (1)-(2) and inserting theresult into (3). An intuitive physical understanding ofthis model can be summarized as follows along the linesof Ref.25.In the absence of inhomogeneous broadening, all wellsare identical ω ISBj = ω ISB , so the light-matter couplingsingles out the fully symmetric combination of ISB plas-mons b B, k = Σ j b j, k / (cid:112) N QW . This single bright combina-tion couples to the cavity mode with a collective Rabi fre-quency Ω R = (cid:112) N QW Ω. Coherent mixing of the brightISB and the cavity modes gives rise to the ISB polari-tons, whose linewidth results from a weighted average ofthe cavity and ISB homogeneous linewidths. All othercombinations of ISB’s remain at ω ISB but are dark andtherefore do not appear in the optical spectra.The situation is more interesting in the presence ofsome inhomogeneous broadening. As long as its stan-dard deviation σ inh does not exceed the collective Ω R ,the inhomogeneous broadening of the ISB plasmons isonly responsible for a corresponding spectral broadeningof the the dark states and a weak mixing of them withthe bright states. As a consequence, dark states trans-form into a wide band of weakly optically active stateslocated in between the polaritons, which however remainspectrally well separated and almost unaffected by theinhomogeneous broadening. Since large Ω R values are apeculiar character of ISB polariton systems, this mech-anism explains why ISB polaritons are extremely robustagainst inhomogeneous broadening.The behaviour suddenly changes when the inhomoge-neous broadening σ inh becomes comparable to the collec-tive Ω R : in this case, the energy range over which darkand bright states are mixed by the inhomogeneous broad-ening reaches the spectral position of polaritons. Seen inthe polariton basis, the magnitude of the mixing termscan cross the polariton gap and effectively contaminatethe polaritons with the dark states. As a result, the two polaritons lose their character of spectrally isolated statesand their linewidth suddenly increases washing out thecorresponding spectral features. B. Complete theory including Bragg processes
While the simplified is able to account for the mainfeatures of the inhomogeneous broadening, it does notinclude the spatial periodicity of period a of the top mir-ror and the subsequent Bragg scattering processes thatare responsible for the folding of the photonic and po-laritonic bands and the consequent Bragg gaps that openbetween them.In this Subsection we develop a complete theory that isable to quantitatively reproduce the experimental results.Restricting for simplicity our attention to the k y = 0line that is addressed in the experiments, the photon andISB bands can be labeled by the k x component along thespatial periodicity, denoted for brevity k and belonging tothe first Brillouin zone k ∈ [ − k Br / , k Br /
2] with k Br =2 π/a , and by the band index n = 1 , , , . . . .The equation of motion for the amplitudes in the pho-tonic and ISB modes read: i da n,k dt = ω cavn,k a n,k + Ω (cid:88) j b j,n,k + Ω Br [ a n +1 ,k + a n − ,k ] − i γ nr a n,k − i γ rad (cid:88) n (cid:48) √ η n,k η n (cid:48) ,k a n (cid:48) ,k ++ √ η n,k E inc ( t ) (4) i db j,n,k dt = ω ISBj b j,n,k + Ω a n,k − i γ hom b j,n,k (5)The momentum-independence of the ISB transitions forwavevectors much smaller than the Fermi momentum ofthe electron gas reflects in the independence of ω ISBj fromthe in-plane wavevector k and the band index n . The in-homogeneous broadening of the ISB of the different wells(labeled by j ) has the same shape as discussed in theprevious section for the simplified model.The periodic patterning of the mirror is described byletting the frequencies ω cavn,k of the photonic bands to fol-low the folded dispersion ω cavn,k = cn (cid:20) k + ( n − k Br (cid:21) for n odd (6) ω cavn,k = cn (cid:20) n k Br − k (cid:21) for n even (7)for k >
0, and a symmetric one for k <
0. The coefficient n is the effective refractive index of the cavity, deter-mined by a combination of the bulk material index andthe penetration into the mirrors. The amplitude of theBragg processes opening gaps at the center and at theedges of the Brillouin zone is quantified by Ω Br .The η n,k coefficients account for the different radiativecoupling of each band to the external radiative modesand are responsible for the peculiar radiative couplings that appear in the equations of motion (4) for the fieldamplitudes. In our numerics, we have considered the sim-plest case where this coupling coefficient vanishes for thelowest n = 1 band (which falls out of the light cone), isfull and equal to 1 for the n = 2 , . n behaviorare actually not important to match the experimental re-sults, the light-cone condition on the lowest n = 1 bandand, even more, the simultaneous and comparable radia-tive coupling of the n = 2 , n and on the wavevector k .Finally, the reflection spectra can be obtained by nu-merically inverting the linear set of equations describingthe steady-state of the motion equations (4)-(5) and in-serting the result into the generalized reflection ampli-tude, E refl = E inc − iγ rad (cid:88) n √ η n,k a n, k . (8)This approach straightforwardly lead to reflection spectrasuch as the ones shown in Fig.4(b,c). C. Comparison with experiments
We are now going to show how this theoreticalmodel provides a quantitative explanation of the exper-imental observations. Figure 4(a) reports the exper-imental polaritonic band-structure of sample HM3872( n =4.4 × cm -2 ) experimentally acquired at room-temperature. It is obtained by measuring the zero-orderreflectivity of the sample between θ =13 o and θ =61 o , andthen applying the transformation k = ωc sin θ to obtainthe in-plane wave-vector k (cid:107) . For this measurement, thesample is placed in a motorized angular-reflection unitlocated within the chamber of the FTIR. The entire sur-face of the sample is probed and care is taken to suppressany light impinging outside the sample surface, especiallyat high angles where the beam spread with a cos( θ ) de-pendency.The corresponding numerically calculated polaritonicdispersions are shown in Figs. 4(b,c), respectively inthe presence and in the absence of an inhomogeneousbroadening of magnitude σ inh =5.1 meV (that is FWHM w inh = 12 meV) comparable to the experimental onearound the central frequency ω ISB = 110 meV. Thehomogeneous broadening is taken as γ hom = 6 meV.The cavity radiative and non-radiative linewidths are γ rad =2.5 meV and γ nr =3.5 meV, respectively, while theeffective refractive index is n = 3 . Br = 5 meV. The collective Rabi frequency istaken as Ω R = 11 meV. (a)(b) (c) FIG. 4: (a) Experimental band-diagram of the polariton sam-ple HM3872. The measurements are performed at 300 K. Thedispersion is obtained by measuring the zero-order reflectivityof the sample between θ = 13 o and θ = 61 o . The reflectiv-ity minima of the polaritonic branches are marked with bluedots. (b,c) Theoretically calculated reflectivity as a functionof the in-plane wavevector and the frequency for no inhomo-geneous broadening w inh =0 (b) and for the experimentallyrealistic value w inh =12 meV (c). All other parameters arechosen to match the experiment and are given in the text andin the SM. The black dashed lines indicate the edges of theexperimentally accessible region. The dotted line indicatesthe θ = 15 o line at which the spectra in Fig.5 are taken. The quantitative agreement between the theoreticalmodel and the experimental data is very good: thetwo strongest lines corresponding to the upper andlower polaritons are almost unaffected by the inhomoge-neous broadening and show linewidths of approximately γ LP,UP =4 meV each. This correctly reproduces the ex-perimental observation of polaritonic linewidths that aremuch smaller than the absorption linewidth of bare ISBtransition shown in Fig.1(a).The faint features visible in Fig.4(b) within the polari-ton gap between 105 and 120 meV are due to the foldingof the polaritons by the Bragg periodicity. As it is shownin Fig.4(c), their structure is washed out by the inhomo-geneous broadening and they transform into an almoststructureless band stemming from the weak mixing ofthe dark states with the bright ones. A weak trace ofthese features is anyway still visible in the experimentalband-diagram of Fig.4(a) and in the reflection spectra ofFig.1(c).To corroborate the findings, and to better illustratethe consequences of this phenomenon, the different im-pact of inhomogeneous or homogeneous broadenings istheoretically illustrated in Fig.5. The two (a,b) panelsrespectively show the polariton peak positions and thelinewidth of the LP mode, extracted from the reflection ( b )
H o m o g e n e o u s I n h o m o g e n e o u s H o m o g e n e o u s I n h o m o g e n e o u s
T o t a l I S B b r o a d e n i n g ( m e V )
Polariton Energies (meV) ( a )
LP linewidth (meV)
T o t a l I S B b r o a d e n i n g ( m e V )
FIG. 5: Polaritons peaks positions and lower polaritonlinewidth as extracted from theoretically calculated reflectiv-ity spectra at 15 o for growing values of the total ISB broad-ening. Black dots correspond to the homogeneous case, whileblue dots to the inhomogeneous one. In this latter case, thehomogeneous broadening is kept constant at an experimentalvalue of γ hom = 6 meV). In (a), the polariton peaks positionsare reported. A rapid reduction of the splitting is observedin the homogeneous case, while it is absent in the inhomoge-neous one. In (b), a similar effect is observed in the linewidthof the lower polariton state. spectra calculated at a given incidence angle of 15 o , asa function of the total ISB linewidth γ tot . In the homo-geneous case, γ tot = γ hom . In the inhomogeneous case, γ tot = γ hom + σ inh , and the homogeneous contribution iskept constant at a value γ hom = 6 meV.Fig. 5(a) shows that the Rabi splitting immediatelystarts to quench with increasing homogenous broaden-ing, while it is much more stable against an equivalentincrease of inhomogeneous one. For instance, a purelyhomogeneous ISB linewdith of 22 meV significantly re-duces the Rabi splitting to 2Ω Rabi = 17 meV, at theonset of weak coupling. On the other hand, in the inho-mogeneous case the Rabi splitting remains stable arounda value 2Ω
Rabi ≈
25 meV well within the strong cou-pling regime and is even slightly reinforced by the spec-tral broadening as compared to the purely homogeneouscase.Fig. 5(b) highlights the different effect of homoge-neous and inhomogeneous broadening on the polaritonlinewdith. In the homogeneous case, the LP FWHM in-creases approximately linearly with increasing ISB broad-ening, as expected. In the inhomogeneous case, instead, the LP FWHM is essentially unaffected until γ tot is of theorder of the initial vacuum field Rabi splitting, ≈
20 meV.At that point it starts increasing, but –importantly– it isalways smaller than in the homogeneous case.These data highlight the qualitative difference betweenthe two cases: the homogeneous broadening is responsi-ble for a rapid reduction of the Rabi splitting with si-multaneous broadening of the polaritonic states, whichgradually morph into a single bare cavity photon lineas typical of weak coupling. On the other hand, theinhomogeneous broadening does not initially affect theRabi splitting nor the polariton linewidths, which re-main approximately constant. Only when the inhomoge-neously broadened ISB transition exceeds the polaritonsplitting, the polaritonic states suddenly collapse into asingle broad photon line.
IV. CONCLUSIONS
In conclusion, we have experimentally demonstratedhow the strong light-matter coupling regime permits tostrongly reduce the inhomogeneous broadening of ISBtransitions in a multiple semiconductor QW system, atleast its component stemming from the presence of a largenumber of slightly different QWs. The mechanism under-lying the observed line narrowing effect is a direct conse-quence of (i) the coupling between a large number of os-cillators and a single photonic mode and (ii) the elevatedcoupling constants typical of ISB polariton systems.In addition to offering a clear illustration of a gen-eral physical mechanism, the large line narrowing achiev-able makes this result an important step in the directionof extending solid-state quantum optical experiments tointermediate wavelengths between the visible/IR rangeof interband devices and the microwaves of circuit-QEDdevices. Furthermore, the elucidated mechanism couldbe useful to disentangle the homogeneous and inhomoge-neous contributions of in-plane disorder to the ISB tran-sition linewidth in single quantum wells.We thank F. Julien, C. Ciuti, G. C. La Rocca, S. DeLiberato for useful discussions. This work was partlysupported by the French RENATECH network. We ac-knowledge financial support from European Union FET-Open grant MIR-BOSE 737017. † E-mail: [email protected] ‡ E-mail: raff[email protected] A. Siegman,
Lasers (University Science Books, 1986). D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer,and D. Park, Science , 87 (1996). Alternative advanced spectroscopic techniques wereadopted in P. Borri et al., Phys. Rev. Lett. , 157401(2001) to get rid of inhomogeneous effects but they arehardly applied in practical optoelectronic devices. H. Liu and F. Capasso,
Intersubband transitions in quan- tum wells: physics and device applications , vol. 5 (Aca-demic press, 1999). T. Ando, A. B. Fowler, and F. Stern, Reviews of ModernPhysics , 437 (1982). E. B. Dupont, D. Delacourt, D. Papillon, J. P. Schnell, andM. Papuchon, Applied Physics Letters , 2121 (1992). K. L. Campman, H. Schmidt, A. Imamoglu, and A. C.Gossard, Applied Physics Letters , 2554 (1996). R. J. Warburton, K. Weilhammer, J. P. Kotthaus,M. Thomas, and H. Kroemer, Physical Review Letters , J. B. Khurgin, Applied Physics Letters , 091104 (2008). D. M. Whittaker, P. Kinsler, T. A. Fisher, M. S. Skol-nick, A. Armitage, A. M. Afshar, M. D. Sturge, and J. S.Roberts, Physical Review Letters , 4792 (1996). V. Savona, C. Piermarocchi, A. Quattropani, F. Tassone,and P. Schwendimann, Physical Review Letters , 4470(1997). C. Ell, J. Prineas, T. R. Nelson, S. Park, H. M. Gibbs,G. Khitrova, S. W. Koch, and R. Houdr´e, Physical ReviewLetters , 4795 (1998). D. M. Whittaker, Physical Review Letters , 4791 (1998). M. Litinskaia, G. C. La Rocca, and V. M. Agranovich,Physical Review B , 165316 (2001). J. Faist,
Quantum Cascade Lasers (OUP Oxford, 2013). C. Gmachl, D. L. Sivco, R. Colombelli, F. Capasso, andA. Y. Cho, Nature , 883 (2002). M. R¨osch, G. Scalari, M. Beck, and J. Faist, Nat Photon , 42 (2015). S. De Liberato and C. Ciuti, Physical Review Letters ,136403 (2009). R. Colombelli and J.-M. Manceau, Physical Review X ,011031 (2015). C. Ciuti and I. Carusotto, Physical Review A , 033811(2006). D. Dini, R. Khler, A. Tredicucci, G. Biasiol, and L. Sorba,Physical Review Letters , 116401 (2003). R. Colombelli, C. Ciuti, Y. Chassagneux, and C. Sirtori,Semiconductor Science and Technology , 985 (2005). J. Dalibard, J.-M. Raimond, and J. Zinn-Justin,
Systmesfondamentaux en optique quantique (North-Holland Ams-terdam etc., 1992). J. Ningyuan, A. Georgakopoulos, A. Ryou, N. Schine,A. Sommer, and J. Simon, Physical Review A , 041802(2016). R. Houdr´e, R. P. Stanley, and M. Ilegems, Physical ReviewA , 2711 (1996). F. J. Murphy, A. O. Bak, M. Matthews, E. Dupont, H. Am-rania, and C. C. Phillips, Physical Review B , 205319(2014). Supplemental Material can be found at... J.-M. Manceau, S. Zanotto, T. Ongarello, L. Sorba,A. Tredicucci, G. Biasiol, and R. Colombelli, AppliedPhysics Letters , 081105 (2014). C. Ciuti, G. Bastard, and I. Carusotto, Physical Review B , 115303 (2005). Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liberato,C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, PhysicalReview Letters , 196402 (2010). M. Ghulinyan, F. Ramiro Manzano, N. Prtljaga, M.Bernard, L. Pavesi, G. Pucker, I. Carusotto, Phys. Rev.A90