High harmonic spectroscopy of disorder-induced Anderson localization
Adhip Pattanayak, ?. Jiménez-Galán, Misha Ivanov, Gopal Dixit
HHigh harmonic spectroscopy of disorder-induced Anderson localization
Adhip Pattanayak, ´A. Jim´enez-Gal´an, Misha Ivanov,
2, 3, 4 and Gopal Dixit ∗ Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India Max-Born Straße 2A, D-12489 Berlin, Germany Department of Physics, Humboldt University, Newtonstraße 15, D-12489 Berlin, Germany. Blackett Laboratory, Imperial College London, South Kensington Campus, SW7 2AZ London, United Kingdom.
Exponential localization of wavefunctions in lattices, whether in real or synthetic dimensions, is afundamental wave interference phenomenon. Localization of Bloch-type functions in space-periodiclattice, triggered by spatial disorder, is known as Anderson localization and arrests diffusion of clas-sical particles in disordered potentials. In time-periodic Floquet lattices, exponential localization ina periodically driven quantum system similarly arrests diffusion of its classically chaotic counterpartin the action-angle space. Here we demonstrate that nonlinear optical response allows for cleardetection of the disorder-induced phase transition between delocalized and localized states. Theoptical signature of the transition is the emergence of symmetry-forbidden even-order harmonics:these harmonics are enabled by Anderson-type localization and arise for sufficiently strong disordereven when the overall charge distribution in the field-free system spatially symmetric. The ratio ofeven to odd harmonic intensities as a function of disorder maps out the phase transition even whenthe associated changes in the band structure are negligibly small.
Disorder is an ubiquitous effect in crystals [1]. Theseminal work by Anderson [2] predicted that above acritical disorder value, the electronic wavefunction willchange from being delocalized across the lattice to ex-ponentially localized (insulating state) due to the inter-ference of multiple quantum paths originating from thescattering with random impurities and defects. Andersonlocalization is a fundamental wave phenomenon and thuspermeates many branches of physics; it has been observedin matter waves [3], light waves [4], and microwaves [5].Anderson localization also finds direct analogues in pe-riodically driven systems, with time-periodic dynamicstaking the role of space-periodic structure. While peri-odically driven classical systems can develop chaotic be-haviour for sufficiently strong driving fields, leading todelocalization of the original ensemble across the wholephase space, their quantum counterpart shows exponen-tial localization of the light-dressed states [6].Dramatic changes in a wavefunction during transi-tion from a delocalized to a localized state may lead tochanges in the nonlinear optical response of the system.In this context, symmetry–forbidden harmonics of thedriving field are an appealing bellwether candidate. In-deed, while even-order harmonics are known to be for-bidden in systems with inversion symmetry [7], they arealso known to arise in such systems if and when chargeslocalize [8, 9]. The required symmetry breaking can thenbe triggered by even a small asymmetry in the oscillatingelectric field of the driving laser pulse. Such asymmetry isnatural in short laser pulses and is controlled by the phaseof the electric field oscillations under the pulse envelope,i.e., the carrier-envelope phase (CEP) [10]. Exponen-tially localized states in symmetric multiple well poten-tials appear to be particularly sensitive to even small fieldasymmetries, leading to even harmonics in the nonlinearresponse even for pulses encompassing tens of cycles [8]. High harmonic generation (HHG) is a powerful toolfor ultrafast spectroscopy [11, 12]. Extremely largecoherent bandwidth of harmonic spectra enables sub-femtosecond resolution. HHG is a sensitive probe ofCooper minima [13], Auger decay [14], attosecond dy-namics of optical tunnelling [15, 16] and the dynamicsof electron exchange [17] in atoms, ultrafast hole dynam-ics [11, 18–20], nuclear motion [21–23] in small molecules,and enantio-sensitive electronic response in more com-plex chiral molecules [24–27]. In solids, high harmonicspectroscopy has allowed observation of dynamical Blochoscillations, band structure tomography [28], probingof defects in solids [29, 30], sub-fs monitoring of coreexcitons [31], optical measurement of the valley pseu-dospin [32–34], tracking of van Hove singularities [35],picometer resolution of valence band electrons [36], imag-ing internal structures of a unit cell [37], monitoring oflight-driven insulator-to-metal transitions [38], and prob-ing of topological effects [39, 40].In this work, we employe HHG to track phase tran-sition between delocalized and localized states in theAubry-Andr´e (AA) system [41] (similar to that pro-posed in Ref. [42]), where localization occurs only abovea critical value of disorder, already in one dimension.This model captures the metal-to-insulator transition,is a workhorse to study non-trivial topology, and hasbeen realized in optical lattices and photonic quasi-crystals [43, 44].The model system is described by the following tight-binding Hamiltonian,ˆ H = − t L − (cid:88) j =1 (cid:16) c † j c j +1 + h.c. (cid:17) + V L (cid:88) j =1 cos(2 πσj ) c † j c j , (1)where t is the nearest neighbour hopping term, V is thestrength of the potential, c † j and c j are, respectively, thefermionic creation and annihilation operators at site j , a r X i v : . [ c ond - m a t . d i s - nn ] J a n h.c. stands for the hermitian conjugate, L is the totalnumber of lattice sites and σ determines the periodic-ity of the potential. A rational value of σ correspondsto a periodic potential and consequently to delocalizedelectronic wavefunctions. If σ is irrational, the potentialbecomes quasi-periodically disordered (for finite systems,this may also happen for rational σ ). For a disorderedpotential, the system undergoes the localization phasetransition at V /t = 2. For V /t > V /t < σ = √ , with 100 lattice sites, in the de-localized ( V /t = 1 .
9) and localized (
V /t = 2 .
1) phases,respectively. The hopping term t = 0 .
26 eV and the lat-tice constant a = 7 .
56 atomic unit of length ( ∼ E F = − . V /t = 1 . V /t = 2 .
1. These differences be-tween the charge distribution in individual eigenstatesdisappear completely when we consider the fully-filledvalence band, see Figs. 1(e) and (f): the differences be-tween both phases are barely visible.The question is: will the non-linear optical response besensitive to the phase transition? To address this ques-tion, we consider a non-resonant, low-frequency field po-larized along the 1D chain, F ( t ) = F f ( t ) cos( ωt + φ ) , (2)with φ the carrier-envelope phase and f ( t ) the sine-squared envelope with a full duration of 10 optical cy-cles. We include the laser-matter interaction via thetime-dependent Peierls phase: t → t e ia eA ( t ) , where A ( t ) is the field vector potential, F ( t ) = − dA ( t ) /dt , and e is the electron charge. The carrier ω = 0 .
136 eV is setwell below the bandgap ∆ (cid:39) . F a approaches and/or exceeds the characteristic en-ergy gap ∆, F a (cid:39) ∆. This regime enables rapid en-ergy gain by the system within a few laser cycles, allow-ing it to climb to the top of the energy scale [45]. Inour case, its signature would be the emission of harmon-ics up to the maximum transition frequency of the sys- FIG. 1. Electronic structure of the model system, for σ = √ . Left panels (a,c,e) show the delocalized phase( V /t = 1 . V /t = 2 . m = 10), (e,f) lattice site occu-pation numbers in a fully-filled valence band. Red and bluecurves in (a) and (b) represent valence and conduction states,respectively. tem (harmonic 10), for all fields enabling resonant tun-nelling (RT). The onset of this regime corresponds to F , RT ∼ ∆ /a (cid:39) . F a ω ≥ ∆ [46], i.e., when F ≥ F (cid:39) m nor-malized eigenstates | ψ m ( t = t i ) (cid:105) of the field-freeHamiltonian Eq. (1) that lie below E F , | ψ m ( t ) (cid:105) = e − i (cid:82) tti ˆ H ( t (cid:48) ) dt (cid:48) | ψ m ( t = t i ) (cid:105) , where ˆ H ( t ) is the Hamiltonianin Eq. (1) with the time-dependent Peierls substitution.The current from a single eigenstate m is calculated as j m ( t ) = (cid:68) ψ m ( t ) | ˆ J ( t ) | ψ m ( t ) (cid:69) (3)where the current operator is defined asˆ J ( t ) = − iea t L (cid:88) j =1 (cid:16) e − ia eA ( t ) c † j c j +1 − e ia eA ( t ) c † j +1 c j (cid:17) . (4)The total current from all the valence band eigenstates m , i.e., the fully-filled valence band, is, j ( t ) = (cid:88) m j m ( t ) . (5)The harmonic spectra are then calculated from theFourier transform of the time derivative of the total cur-rent. Prior to the Fourier transform, we apply an en-velope to the current that coincides with the laser pulseenvelope [47], to filter out emission after the end of thelaser pulse. We consider 100 lattice sites and 0.01 atomicunit of time-step ( ∼ m = 10,shown in Fig. 1(c,d)). The charge distribution of the ini-tial state is strongly asymmetric in both phases, whichbreaks the left-right symmetry of the chain in the HHGprocess and leads to the appearance of even harmonics,both in the delocalized [Fig.2(a)] and localized [Fig.2(b)]phases.However, the equilibrium initial state corresponds tothe fully-filled valence band, with the charge distributedrelatively evenly between all sites; the distribution is vir-tually identical between the two phases [Figs. 1(e) and(f)]. With this initial condition, the HHG spectra of thetwo phases, shown in Figs. 2(c) and (d) for a field strengthof F = 0 . F = 0 . F , RT = 0 . FIG. 2. (a,b) High harmonic spectra for the system in the (a,c, e, g, i) delocalized and (b, d, f, h, j) localized phase: (a, b)HHG from the 10 th eigenstate only for a field strength F =0 . F = 0 . F = 2 . F = 0 . F = 1 . FIG. 3. High harmonic spectra in the delocalized (top pan-els, red curve) and the localized (lower panels, green curve)phase calculated for different carrier-envelope phases (CEP)of the field: (a,d) CEP=+ π/
2, (b,e) CEP= − π/ π/ − π/ symmetry breaking and the appearance of even harmon-ics in the localized phase. We find that even harmon-ics emerge already at F (cid:39) . F ∼ F = 2 . π , and their coherent superposition inFigs. 3(c) and (f). In the delocalized case, even harmon-ics are absent regardless of the CEP [Figs. 3(a) and (b)].In the localized case, they are identical in both cases,Figs. 3(d) and (e), but with opposite phase: upon coher-ent addition even harmonics are completely washed out[Fig. 3(f)]. The reason is that the laser-induced asym-metry in the electron charge distribution at CEP=+ π/ − π/
2, as graphicallyillustrated by the cancellation of the emission upon in-
Even-odd ratio (R)
FIG. 4. Even-to-odd harmonic peak ratio (R) for different
V /t ranging from 1.7 to 2.3. The ratio R is calculatedas R = log ( ¯ H even ) / log ( ¯ H odd ), where ¯ H = H peak /H min .Blue tiangles (orange circles) represent the R value correspondto 4 th (6 th ) harmonic and 5 th (7 th ) harmonic. The incrementof R value at V /t > terference.Figure 4 shows that the relative intensity of even har-monics does indeed track the metal to insulator (delocal-ized – localized) phase transition in the system. The Fig-ure plots the ratio of even to odd harmonics for differentvalues of V /t , at a field strength of F = 0 . V /t = 2, showing thatHHG, driven by a phase-stable CEP pulse, is able tomap disorder-induced electron localization in a solid.In conclusion, we have used high harmonic spec-troscopy to track delocalized to localized phase transi-tion in the Aubry-Andr´e system. For aperiodic potentials( σ = √ ), the localized and delocalized phases showalmost identical eigenspectra and site occupation num-bers for a state with fully-filled valence band. Yet, highharmonic spectra between the two phases show strik-ing differences, especially in the appearance of forbid-den (even) harmonics. This effect is a consequence ofdynamical symmetry breaking induced by the field butenabled by initial electron localization, which promotesresonant tunneling and leads to CEP-dependent symme-try breaking even in the low-frequency regime. Our workshows that the localisation-delocalisation phase transi-tion, which can be driven by a very small modificationof the on-site energy, can be effectively traced by HHGspectroscopy.A. P. acknowledges sandwich doctoral fellowship fromDeutscher Akademischer Austauschdienst (DAAD, ref-erence no. 57440919). M.I. and A. J-G acknowledgesupport of the FET-Open Optologic grant. 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