Efficient excitation of nonlinear phonons via chirped mid-infrared pulses: induced structural phase transitions
aa r X i v : . [ c ond - m a t . o t h e r] A ug Efficient excitation of nonlinear phonons via chirped mid-infrared pulses: inducedstructural phase transitions
A.P. Itin , and M.I. Katsnelson Radboud University, Nijmegen, The Netherlands, Space Research Institute, Moscow, Russia
Nonlinear phononics play important role in strong laser-solid interactions. We discuss a dynamicalprotocol for efficient phonon excitation, considering recent inspiring proposals: inducing ferroelec-tricity in paraelectric material such as KTaO , and inducing structural deformations in cuprates(e.g La CuO ) [A. Subedi et.al, Phys.Rev. B 89, 220301 (2014), Phys.Rev. B 95, 134113 (2017)].High-frequency phonon modes are driven by mid-infrared pulses, and coupled to lower-frequencymodes those indirect excitation causes structural deformations. We study in a more detail the caseof KTaO without strain, where (at first glance) it was not possible to excite the needed low fre-quency phonon mode by resonant driving of the higher frequency one. Behaviour of the system isexplained using a reduced model of coupled driven nonlinear oscillators. We find a dynamical mech-anism which prevents effective excitation at resonance driving. In order to induce ferroelectricitywe employ driving with sweeping frequency, realizing so called capture into resonance. The methodworks for realistic femtosecond pulses and can be applied to many other related systems. Research in ultrafast light-control of materials haveattracted a lot of interest recently [1–9]. Intense mid-infrared pulses have been used to directly control thedynamical degrees of freedom of the crystal lattice [5–9], in particular, inducing melting of orbital and mag-netic orders. In many recent suggestions and experimentsphonon modes are driven indirectly: a laser drives a high-frequency infrared-active phonon mode which then ex-cites required modes not easily accessible by direct driveby means of nonlinear couplings. Here we suggest to adda new useful tool to the arsenal of nonlinear phononics:capture into a resonance. Such phenomenon is encoun-tered in classical and celestial mechanics [10–12] and wasemployed, e.g. in plasma and accelerator physics [13–17].In a nonlinear system with near-resonant driving, as am-plitude of perturbation grows, frequency of the systemvaries, and it stops to absorb energy efficiently. Drivingwith changing frequency enables to sustain resonance re-lation between the drive and the system [18].We find that in perovskite paraelectric KTaO (be-ing described below), as well as in other systems (likeLa CuO cuprate (LCO)), it is possible to considerablyexcite a high-frequency phonon mode using a protocolbased on capture into resonance, which requires a pulsewith chirped frequency. This excitation makes a cou-pled low-frequency phonon mode dynamically unstable.Pulses with frequency chirps on picosecond timescalehave been generated e.g. in FELIX [19].In all considered examples below, a phonon mode with(at least) quartic nonlinearity is driven by a laser pulse,creating effective potential for a coupled lower-frequencyphonon mode those excitation triggers structural phasetransition. Neglecting dissipation, our starting Hamil-tonian H = P + Ω Q + c Q + c .. − QF sin Φ( t ) , where Φ( t ) is the phase of the driving ˙Φ( t ) = Ω( t ), Ω is linear frequency of the driven phonon mode, c , c .. are anharmonic coefficients, F ( t ) is the amplitude ofthe external field. Introducing symplectic coordinates FIG. 1: Top panel: steady states of the driven Duffing oscilla-tor (2) as a function of ω ≡ Ω / Ω at several values of the driv-ing force amplitude F ( A is the amplitude of Q ). (a) dampedDuffing oscillator (b) undamped Duffing oscillator. Curvesfrom bottom to top correspond to: E = 5 , , , , , − . Bottom panel: phase portraits of the effectiveHamiltonian (1) (c). λ < λ ∗ = µ / (d) λ > λ ∗ = µ / . P = √ I Ω cos φ, Q = q I Ω sin φ we then make atransformation to the resonance phase γ = φ − Φ us-ing the time-dependent generating function W = J ( φ − Φ). The new Hamiltonian H ′ = H − J Ω can be av-eraged over the fast phase, with the result (neglect-ing nonlinearities higher than the quartic for a while) H = δ Ω J + c J Ω − F p J/ Ω cos γ. Introducing now x = √ J sin γ, y = √ J cos γ, we get an effective Hamil- FIG. 2: Dynamics of the system under resonant driving. (a-d): Q z , Q x , Q hx modes (in units of ˚ A √ amu [2]), and E ( t )pulse, correspondingly. (in units of MV cm − ). Time is inpicoseconds. Parameters are: E = 30 MV cm − , damp-ing coefficients γ h , γ are 4% of Ω h , Ω , correspondingly. Thewidth of the pulse is σ =2ps. Similar to [2], Q z was givena small initial excitation to initiate its dynamics (simulatingtemperature fluctuations). Strong driving of Q hx and corre-sponding excitation of Q x modes almost do not affect decayof the transverse mode Q z , and leaves it unexcited. tonian H = c Ω ( x + y ) + δ Ω2 ( x + y ) − F √ Ω y . Uponrescaling H → H/ c Ω , t → t c Ω and introduction of λ = − δ Ω2 / c Ω , µ = − F √ Ω we bring the Hamiltonian tothe form H = ( x + y ) − λ ( t )( x + y ) + µ ( t ) y. (1)This Hamiltonian is often encountered in problems ofcelestial mechanics and plasma physics [12]. Under slowchange of frequency and/or amplitude of driving, pa-rameters of (1) are changing (increasing frequency cor-responds to ˙ λ > λ ∗ = µ / .Below this value, there is a single equilibrium ( A ), whileat higher values of λ there are two stable ( A, B ) and oneunstable equilibrium ( C ). Provided certain conditionsare met, a phase particle can follow the initial equilib-rium point ( A ) which moves away from origin (corre-spondingly, in the original system amplitude of oscilla-tion grows, while its frequency remains approximatelyequal to the instanteneous frequency of the drive, so thisregime is called capture into the resonance ). Under in-fluence of a gaussian pulse with fixed frequency, the point A is shifted by a certain amount and then returns backto the origin (see Fig. 6b, upper curve). In contrast,a pulse with sweeping frequency can shift the equilib-rium far away (Fig. 6b, bottom curve). In the adiabaticapproximation, dynamics can be described in a great de-tail [12, 18]. E.g., as the parameters of the system arechanging, phase space area within the trajectory remainsapproximate adiabatic invariant, and from behaviour ofthe areas inside separatrix loops it can be predicted whenthe phase point will be thrown away from the resonance. FIG. 3: Dynamics of the system under off-resonant driv-ing with frequency Ω = 1 . h . From top to bottom: Q z , Q x , Q hx modes, and the instanteneous coefficient G ( t )of the ( Ω + G ( t )) Q z term in the potential energy. The formof the electric field pulse is the same as in Fig.[2] (but withshifted base frequency), and Q z mode is now remarkablyexcited. When G ( t ) exceeds − Ω ≈ − .
06, the effective po-tential for the mode Q z becomes unstable. Due to violentbeatings in Q hx mode only for a short fraction of the pulselength the mode Q z experiences the inverted parabolic po-tential. In our realistic system non-adiabaticity and dissipationbecome very important, nevertheless qualitative under-standing of dynamics allows to construct simple and ef-fective protocol for phonon excitation. With dissipation,the equation of motion becomes¨ Q + γ ˙ Q + Ω Q + 4 c Q = F sin Φ( t ) , (2)which is the same as in a damped and driven Duffingoscillator. At fixed frequency, searching for a periodicsolution Q = A sin(Ω t + φ ), one gets a cubic equationfor the amplitude of a steady solution: A (cid:16) γ Ω + (Ω − Ω + 3 c A ) (cid:17) = F (3)Solutions of Eq.(3) are shown on Fig. (1a) for differentvalues of the driving amplitude F . They are dissipativecounterparts of fixed points in phase portraits of (1). Onecan see that driving frequency higher than the resonantone allows to achieve higher steady-state amplitudes. Todemonstrate induced ferroelectricity, consider a specificexample of KTaO : a perovskite oxide with cubic struc-ture, posessing a paraelectric phase. Such materials havefour triply degenerate optical phonon modes at the zonecenter. Three of these modes are infrared active (havethe irreducible representation T u [2]). The remainingone is optically inactive ( [2]). Ferroelectricity is relatedto dynamical instability of an infrared-active transverseoptic phonon mode: most ferroelectric materials showa characteristic softening of an infrared transverse opticmode as the transition temperature is approached. In[2] it was investigated if a similar softening and insta-bility of the lowest frequency T u mode can be achieved FIG. 4: Dynamics of the KTa O model system under drivingwith sweeping frequency. Instanteneous frequency in the cen-ter of the pulse is Ω = 1 . h . (a-d) Q z , Q x , Q hx modes, andthe instanteneous coefficient G ( t ) of the ( Ω + G ( t )) Q z termin the potential energy. (e,f) Maximal amplitude of (e) Q hx and (f) Q z modes under driving with sweeping frequency atdifferent sweeping rates a . Curves from top to bottom cor-respond to amplitudes of driving F = 30 , ,
10 MV cm − .Damping rates are 5% of the linear frequencies. Instanteneousfrequency of the drive is linearly increased, reaching the linearfrequency of the phonon mode Ω h at t ∗ = 0 . σ before themaximum of the pulse. σ = 2 ps. a = 0 corresponds to thedrive with the constant frequency Ω = Ω h .FIG. 5: Excitation by a 350-femtosecond pulse. (a,b) Maxi-mal amplitude of modes Q hx , Q z under driving with sweep-ing frequency at different sweeping rates a . (c) Mode Q z ata=13.2 (d) Mode Q hx at a=13.2. The pulse width is σ = 0 . Q IR , Q R in thedriven LCO system under driving with sweeping frequencyat different sweeping rates a (for clarity Q IR / σ = 2 ps. by an intense laser-induced excitation of the highest fre-quency T u mode. In the case of cubic structure itwas not achieved due to certain dynamical reasons (be-ing discussed below), however with addition of internalstress which modifies the crystal lattice such mechanismworked. Let us consider in a more detail the cubic struc-ture case (without stress). The calculated in [2] phononfrequencies are Ω = 85 cm − and Ω h = 533 cm − forthe lowest and highest frequency T u modes, respectively(Ω from previous discussion plays the role of Ω h now).Following [2] we simplify analysis by considering a casewhere the x component of the highest frequency mode Q hx is pumped by an intense light source and influencesthe dynamics of the lowest frequency T u modes along thelongitudinal Q x and transverse Q z coordinates. Dy-namics along the second transverse coordinate Q y is ne-glected. The energy surface V ( Q hx , Q z , Q x ) has a com-plicated form with many kinds of nonlinear couplings andanharmonicities: V ( Q hx , Q z , Q x ) + Ω ( Q z + Q z ) + Ω h Q hx + V nl ( Q hx , Q z , Q x ) , where V nl is the nonlinearpart of the energy, obtained in state-of-the-art calcula-tions of [2], and given in [20] for convenience. Equationsof motions are:¨ Q hx + γ h ˙ Q hx + Ω hx Q hx = − ∂V nl ( Q hx , Q z , Q x ) ∂Q hx + F ( t ) , ¨ Q j + γ ˙ Q j + Ω Q j = − ∂V nl ( Q hx , Q z , Q x ) ∂Q j , (4)where j = x, z ; γ h and γ are damping coefficients (typi-cally few percents of the corresponding harmonic frequen-cies), external force F ( t ) = Z ∗ hx E sin(Ω t ) e − t / σ , Z ∗ hx is the effective charge and Ω is the driving frequency.Qualitative understanding of possible dynamics can becaptured by drawing a projection of the potential en-ergy V ( Q z , Q x , Q hx ) by the plane Q x = 0 (Fig. 3of [2]). Resulting curves V ( Q z ) at fixed values of Q hx have single-well form (at small absolute values of Q hx ),or a double well form (at larger Q hx ). In the lattercase, induced ferroelectricity is possible, as finite valueof Q z at equilibrium corresponds to the ferroelectricphase. However, to reach such a state dynamically isa nontrivial issue. Excitation of Q hx mode should bedone with a pulse of limited power and duration. Whenthe driving frequency Ω is chosen close to the phononmode frequency Ω h [2], the transverse mode Q z remainsalmost unaffected (Fig. (2) due to insufficient ampli-tude of the Q hx mode. In [2], the amplitude of driv-ing was varied in a large range, up to pump amplitudesof 100 M V cm − , with no signs of dynamical instabilityin Q z . Increase of the pump amplitude makes the dy-namics of Q hx , Q x chaotic, but does not result in no-ticeable response of the Q z mode: at strong drivingthe ’axuilary’ longitudinal mode Q x becomes excited([20]), and chaotic dynamics prevents efficient excitationof Q hx . There is a range of driving frequencies (awayfrom the exact resonance with Ω h ), where a pulse of thesame amplitude can effectively excite all three modes, in-ducing transient dynamical instability in Q z . Indeed,shifting driving frequency about 15-20% from the ex-act resonance with Ω h leads to considerable excitationof the modes Q z , Q hx (Fig. (3) ). The mode Q hx experiences beatings which create transient double-wellpotential for the Q z mode (Fig. (3)d). From the fullpotential energy of the system, we can single out theterm quadratic in Q z : (cid:16) Ω + G ( t ) (cid:17) Q z , where G ( t ) = m Q hx + dQ hx Q x + lQ hx + pQ x + e Q hx Q x + e Q hx Q hx ( [20] ). Due to strong coupling between Q hx and Q x modes, they oscillate synchronously although their linearfrequencies are very different. When the average value ofthe coefficient h G ( t ) i exceeds − Ω /
2, the effective poten-tial for the mode Q z becomes unstable. Due to violentbeatings in the Q hx mode, it does not happen smoothly,and only for a short fraction of the pulse excitation timethe mode Q z experiences the inverted parabolic poten-tial (at the first maxima of the beatings, see Fig. (3)d).There is a way to create the needed effective potential ina more robust way. Consider driving with sweeping fre-quency. A chirped pulse has the form F = F ( t ) sin Φ( t ),where Φ( t ) = Ω t + αt , F ( t ) = exp (cid:0) − t / σ/ (cid:1) . Time-dependence of the instantenious frequency and theamplitude translates into the dependence of parameters µ, λ of the Hamiltonian (1) on time. Corresponding phaseportraits are slowly deformed, and, if our phase point isnot thrown away from the region where the initial equilib-rium is located (see [18] for details), it oscillates aroundthe equilibrium point moving away from the origin. Suchregime, illustrated in Fig.(4), is not only effective forexcitation of the Q hx mode, but also provides smoothgeneration of the effective potential for the Q z mode(Fig.(4)d). The important feature of the dynamics is thatthe axillary longitudinal mode Q x also gets excited con-siderably. Unlike the case of very strong resonant driving([20]), where chaotic dynamics happens after excitationof the Q x mode, here the longitudinal modes oscillatesynchronously (in 1:1 resonance), and the resulting dy-namics is regular. Regular dynamics happens becausethe pulse with sweeping frequency excites the system in such a way that a phase point remains not far from theinstanteneous equilibrium point.We note that damping play important role in dynam-ics. Damping parameters typically are 5-10 % of corre-sponding linear frequencies. In case we assume higherdamping for low-frequency phonons, the axillary longi-tudinal mode Q x is not excited considerably, and dy-namics can be understood from the driven Duffing oscil-lator model for the Q hx mode alone (with small correc-tions from Q x ). While in the Hamiltonian model thereare three equilibria above the critical frequency detuning,and a phase point captured into resonance can reach largeamplitudes of Q moving near one of them, damping leadsto termination of this process at the tip of the resonance’tongue’, where stable and unstable equilibria collide andannihilate. At weak driving amplitudes, the tips lie at a’backbone’ defined as A tip = F γ Ω , ω tip = 1 + c F γ Ω . Fora rough estimate of the optimal sweep, assume that apulse starts with the linear resonance frequency Ω , andreaches the tip of the resonance ’tongue’ at its maximum.Then, the estimate for the sweeping rate is α = c F γ Ω . We make numerical experiments with various sweepingrates and amplitudes of driving (see Fig.4e,f), and find aremarkable improvement in efficiency of excitation com-pared to pulses with constant frequency. Most exciting,the protocol with sweeping frequency works also for muchshorter, sub-picosecond pulses. We show in Fig.5 an ex-ample with σ =350fs, which corresponds to laser param-eters used in A.Cavalleri group.The same approach applies not only to ferroelectrics,but to many other systems, e.g. laser driven LCO [1].There, a relevant reduced model consists of infrared-active Q IR mode (described by a driven Duffing oscil-lator) coupled to a Raman mode Q R by a quadratic-quadratic term. Q R ( B g (18)) mode describes in-planerotations of CuO octahedra, whereas Q IR mode de-scribes in-plane stretching of Cu-O bond. The energysurface has the form V = Ω Q IR + Ω Q R + c Q IR + b Q R − g Q IR Q R . Values of the coefficients were derivedin elaborate calculations of [1]. There is a single-potentialwell around the equilibrium value for Q R mode at smallamplitudes of Q IR , which becomes double-well potentialat larger amplitudes of Q IR (instanteneous quadratic po-tential felt by the slow mode is Q R (cid:0) Ω − gQ IR (cid:1) , whichbecomes inverted parabolic potential for sufficiently highamplitudes of the driven IR mode. The critical valueof driving force F c depends on detuning δ Ω ≡ Ω − Ω and can be made smaller than its value on the reso-nance (being used in [1] ). Indeed, to the first approx-imation, the averaged potential for the slow mode is Q R (cid:0) Ω − gQ IR,max / (cid:1) and becomes unstable at criticalvalue of the fast mode amplitude Q IR,max = Ω p /g .This can be achieved at sufficiently smaller driving forceamplitudes provided sweeping frequency pulse is used.We show corresponding results of numerical calculationsin Fig.(6). Fig.(6)c shows also instanteneous locations ofstable equilibrium of the effective potential as a functionof time for different values of sweeping rates.Excitation of the in-plane rotations associated with the Q R mode can be used to modulate superexchange cou-pling in this cuprate [1]. Recently there has been a lotof interest in effective models arising from periodic driv-ing [22–27], and the suggested method can be useful forthis area of research as well. The proposed method canbe useful also for recent proposals and experiments ondriven orthorombic perovskites (like ErFeO , see [28, 29]), where three-linear phonon coupling is realized: twohigh-frequency infrared-active modes are coupled to thethird, Raman mode.To conclude, we demonstrate that drastic improvementin efficiency of excitation of nonlinear phonons can beachieved using chirped pulses. In terms of nonlinear dy- namics of reduced classical models, capture into the res-onance happens and the driven mode is transferred toa higher amplitude state efficiently, which triggers in-stabilities in the coupled low-frequency modes, and cor-responding phase transitions. The method is especiallyremarkable in cases where a system cannot be excited bybare increase of the power of drive, like in KTaO . Theapproach can be useful in many recent proposals on laser-induced phase transitions, including induced ferroelec-tricity in perovskites, induced structural transitions inLCO cuprates, and excitation of orthorombic perovskites.We are grateful to A.I.Neishtadt and A.Cavalleri forinsightful discussions. The work was supported by NWOvia Spinoza prize and by European Research Council(ERC) Advanced Grant No. 338957 FEMTO/NANO. [1] A. Subedi, A. Cavalleri, and A. Georges, Phys. Rev. B 89,R220301 (2014).[2] A. Subedi, Phys. Rev. B 95, 134113 (2017).[3] K. Miyano, T. Tanaka, Y. 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A. Nonlinear part of the potential energy surface for KTaO The nonlinear part of the potential energy surface is: V nl ( Q hx , Q z , Q x ) = (S1)= X k =2 (cid:16) a k ( Q k z + Q k x ) + c k Q khx (cid:17) + lQ z Q hx + m Q z Q hx + m Q z Q hx + n Q z Q hx + n Q z Q hx + n Q z Q hx + t Q x Q hx + t Q x Q hx + t Q x Q hx + pQ z Q x + q Q z Q x + q Q z Q x + r Q z Q x + r Q z Q x + r Q z Q x + dQ z Q x Q hx + gQ z Q x Q hx + X k =1 e k Q z Q − k x Q khx + X k =1 u k Q − k x Q khx + X k =1 f k Q z Q − k x Q khx + X k =1 h k Q z Q − k x Q khx Values of the main coefficients are:
TABLE I: The values of the coefficients of the polynomial for energy surfaces of Raman and IR modes obtained in [S1], [S2].The units of a Q m Q p Q n term are meV A − ( m + p + n ) Coefficient Term KTaO Coefficient Term LCOΩ h Q hx Q IR z Q z Q R x Q x g − Q R Q IR /2 46.98 a Q z a Q R c Q hx c Q IR B. Strong resonant driving: chaotic dynamics
FIG. S1: Resonant driving, the same as in Fig.[2], but with a larger driving amplitude. From left to right: Q z , Q x , Q hx modes, and E ( t ) pulse. E = 60 MV cm − . Amplitude of driving is so large that dynamics become chaotic, nevertheless themode Q z is unexcited. C. Two coupled modes with periodic driving
Although the linear frequency of the Q x mode is much smaller than that of Q hx , strong coupling between themcauses these modes to oscillate synchronously. One can therefore introduce an effective system combining these modestogether and averaging over the driving frequency. The third mode, Q z can be neglected until it become excited dueto creation of the effective double-well potential. Consider firstly a conservative system, without dissipation.The reduced two-mode Hamiltonian is (we denote Q ≡ Q x , Q h ≡ Q hx ) H = P P h Q h Q h a Q + c Q h ++ t Q Q h + t Q Q h + t Q Q h + X k =1 u k Q − k Q kh + − Q h F sin Ω t (S2)Moreover, we neglect the u − terms in the analytical considerations below, as values of t − coefficients are considerablyhigher.Introducing radial coordinates ( I h , φ h , I , φ ) Q hx = r I h Ω h sin φ h , P h = p I h Ω h cos φ h ,Q x = r I Ω sin φ , P = p I Ω cos φ , (S3)we then switch to rotating phases γ h − Ω t , γ − Ω t by means of a generating function W = ρ h ( φ h − Φ) + ρ ( φ − Φ) , Φ ≡ Ω t (S4)Averaging the resulting Hamiltonian over the explicit time dependence, we get as an effective reduced two-modeHamiltonian H = ρ h (Ω h − Ω) + ρ (Ω − Ω) + 32 c ρ h Ω h + 32 a ρ Ω + F r ρ h Ω h sin γ h + 32 t (cid:16) ρ Ω (cid:17) / (cid:16) ρ h Ω h (cid:17) / cos( γ − γ h )+ 32 t (cid:16) ρ Ω (cid:17) / (cid:16) ρ h Ω h (cid:17) / cos( γ − γ h ) (S5)+ 12 t (cid:16) ρ Ω (cid:17)(cid:16) ρ h Ω h (cid:17)(cid:16) γ − γ h ) (cid:17) + u-terms + ... We then return to cartesian coordinates via x k = q ρ k Ω k cos φ k , y k = √ ρ k Ω k sin φ k , and search for equilibria of theresulted two-mode system.We get two coupled algebraic equations for coordinates of the equilibrium:32 c X h + X h (cid:16) Ω h (1 − x ) + 34 t X (cid:17) + 98 t X X h (S6) − t X = F ,X (cid:16) Ω (1 − x Ω h Ω ) + 34 t X h (cid:17) + 98 t X X h + 32 a X − t X h = 0 (S7)As amplitude of X h grows beyond certain critical value, deviation of X from 0 becomes considerable. Then,dynamics of the system becomes complicated and the first mode stops to absorb the energy from the drive. Note FIG. S2: Steady states of the mode X h of the driven reduced two-mode system (S6) without dissipation as a function offrequency x ≡ Ω / Ω hx at several values of the driving force amplitude E . From bottom to top: E = 5 , , , , ,
30 MeVcm − . (b) Steady values of amplitude of the mode Q x (i.e., X ) as a function of X h , x . White areas correspond to highestvalues of X that the steady state value of X h in Eq. S6 is a function of F and depends on X , while X is determined fromthe second equation which do not depends on F . So we could depict a value of X as a density plot on the plane(Ω , X h ): see Fig. S2b. It is clearly seen that as one tries to increase steady amplitude X h , above certain curve on(Ω , X h ) plane the mode Q x becomes excited. Importantly, detuning from the exact resonance to higher frequenciesallows to reach higher values of X h without considerable excitation of the Q x mode. This important qualitativeresult remains valid in the full system. When we take into account dissipation and the remaining nonlinear couplingterms, the critical curve goes higher, and it become possible to reach sufficiently high values of X h (again, shiftingfrom the exact resonant to the higher frequencies allows higher values of X h ).).