Nonadiabatic dynamics and coherent control of nonequilibrium superconductors
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1 Nonadiabatic dynamics and coherent control of nonequilibrium superconductors Andreas Schnyder Workshop on strongly correlated electron systems Schloss Ringberg, November 13 k F 1 t (ps 3 4 in collaboration with: Holger Krull, Adolfo Avella, Dirk Manske Götz Uhrig, Alireza Akbari, Nikolaj Bittner, Ilya Eremin
2 Outline 1. Introduction - Recent pump-probe experiments on NbN films - How does a quantum system relax?. Nonadiabatic dynamics of superconductors - Superconductor coupled to laser field and optical phonons - Density matrix formalism 3. Results & Discussion - Response of superconductor: Damped order parameter oscillations - Generation of coherent phonons in non-equilibrium SC state: beating phenomena, resonant generation of coherent phonons - Experimental signatures: Oscillations of pump-probe conductivity 4. Conclusions & Outlook
3 Motivation I: Recent pump-probe experiments Matsunaga, Shimano, et al. PRL 111, 57 (13 (b Pump delay time WGP THz-pump THz-probe spectroscopy: Probe t pp sample y polarizer WGP polarizer SC gap : - THz-pump pulse: Excite with intense femto-second pulse, induce dynamics - THz-probe pulse: After delay time, measure with second, less intense pulse t Optical conductivity / transmitivity as a function of and t gives information about: x z 1 1 mev 1 terahertz 4.1 mev 1 ps cm -1 Dynamics of SC condensate, order parameter oscillations Cooper pair recombination / recovery dynamics Coherent phonon oscillations
4 Motivation I: Recent pump-probe experiments Matsunaga, Shimano, et al. PRL 19, 187 (1;PRL 111, 57 (13 THz-pump THz-probe spectroscopy on NbN films 3 mev - pump-pulse duration: 9 fs non-adiabatic excitation of SC: p h/( - Measure change in transmission of probe field Observation: E - algebraically damped oscillations in E as a function of delay time t E probe (t gate =t (arb. units 1 (a τ nj/cm pump/ τ =.57 f (THz b (b (c f - frequency changes with laser intensity -4 - t pp (ps Pump Energy (nj/cm Interpretation: order parameter amplitude oscillations
5 Motivation II: How does a quantum system relax?? How does a quantum system thermalize? As t 1, generic observables Ô become time independent True relaxation vs. decoherence: -- Decoherence: Only averaged observables become time-independent non-interacting systems, integrable systems only show decoherence -- True relaxation: Generic local (unaveraged observables become time-independent Open questions: Ō = lim t1 (t Ô (t time independent (i systems coupled to bath, (ii closed systems with certain interactions (infinite closed systems; otherwise recurrence -- Which type of interactions lead to thermalization? -- Which observables Ô to consider? -- How to describe thermalized sate? (is there a density matrix (ensemble such that:? Ō =Tr[Ô ]
6 SC coupled to laser field and optical phonons Goal: simulate non-adiabatic dynamics of superconductor coupled to (i pump laser field and (ii optical phonons Microscopic model: with gap equation: Gaussian pump and probe pulse: pump pulse: ω A q (t = A e (t/τ ( δq,q e iω t + δ q, q e +iω t probe pulse: ω
7 Theoretical methods to compute non-equilibirum dynamics in SCs Goal: simulate non-adiabatic dynamics of superconductor coupled to optical phonons = considered hierarchies of time-scales: le subsystems evo τ p τ ph,τ τ ϵ pproaches for com time-dependent Ginzburg Landau theory, mu*-, and T*- models: - quasi-equilibrium is assumed at all times - time evolution of a single collective order parameter - only valid for: (i.e., close to Tc, very dirty SCs τ τ ϵ Boltzmann kinetic equation: - requires adiabaticity on all time-scales - does not capture coherent evolution of quasi-particle distributions - only valid for: τ p τ, τ ph Phenomenological Rothwarf-Taylor models: - rate equations for quasi-particle and phonon occupations - only valid for: τ p τ, τ ph = g ph t/~ instead use density matrix formalism (expansion in d dt O = i ~ [H, O]
8 Density matrix formalism: Equations of motion α ( k = u kc k + [ v kc k ] β ( k = u kc k v k c k k = [ ( ] W k,k u k v k 1 α k α k β k β k + u k β k α k v α k k β k k Superconducting state: Bogoliubov transformation All quantities of interest can be expressed in terms of these dynamical variables α k α k (t, β k β k (t, α k β k (t, α k β k (t bp (t, b p (t Current density: j(q, ω e mv Lattice displacement: Density-matrix theory: k [ k α k α k+q U(r,t = β k+q β k h Mω ph V + k q 1 ( = m (Ek D p (te +ip r, with the coherent phonon is amplitude: the reduced mass of the lattice ions a d dt O = i ~ p t (r isconnected D p (t = b p + b p [H, O] α k β k+q yields equations of motions for the above expectation values + α k+q β k ]
9 Density matrix formalism: Equations of motion Density-matrix theory: α k α k ( [ ] ( (t, d dt O = i ~ yields equations of motions for β k β k (t, [H, O] α k β k (t, α k β k (t bp (t, b p (t For example i d α k dt β k+q = (R k + R k+q a k β k+q + C k+q α k α k+q + Ck ( β k+q β k δ q, + e { k A q (t L + k,q α m k+q β k+q + L+ k+q, q α k β k+q q q =±q } M k+q, q α k α k+q q + M k,q ( β k+q β k+q δ q,q + Up to order (A q n, laser field only couples (k, k to (k, k + nq leads to an effectively one-dimensional system of equations interaction with laser field can be computed essentially exactly. [Papenkort, Kuhn, Axt, PRB 8]
10 Response of superconductor (w/o phonons Two regimes: - Adiabatic behavior for - Non-adiabatic behavior for Algebraically damped order parameter oscillations after short pump pulse ( p : ( Higgs amplitude mode [Volkov, Kogan, JETP 74] [Yuzbashyan, Altshuler PRL 6] (t (mev Order parameter oscillations -1 1 t (ps p p p = 4 fs p = 15 fs p = 1 fs 3 (mev p =.5 ps p =.5 ps p = 1. ps 1.38 ps p =. ps A p (arb. units Quasiparticle occupations p p =. ps p = 5. ps (c 1 [APS, Manske, Avella, PRB 84, (11]
11 Algebraically damped oscillations due to decoherence Algebraically damped oscillations in averaged quantities due to decoherence X = W hc k c +k i k W j q = e~ mv But no true relaxation X (k + q hc k, c k+q, i k, (t (mev 1.6 Order parameter oscillations p = 4 fs p = 15 fs p = 1 fs 1 t (ps 3 Quasiparticle occupations Quasiparticle occupations h k k i h k k i k F k 1 t (ps 3 4
12 Gap oscillations: Comparison with experiment Qualitative agreement between theory and experiment APS, Manske, Avella, PRB 84, (11 Numerical simulations: (mev p =.5 ps p =.5 ps p = 1. ps 1.38 ps p (c p =. ps p = 5. ps p =. ps A p (arb. units E probe (t gate =t (arb. units 1 THz-pump THz-probe spectroscopy NbN: -4 p = 6 fs (a τ nj/cm pump/ τ =.57 t pp (ps f (THz b Matsunaga, Shimano, et al. PRL 111, 57 ( (b (c f 5 1 Pump Energy (nj/cm
13 Response of superconductor (w/o phonons Pump-probe conductivity shows signatures of non-adiabatic dynamics ( t, =j( /[i A( ] oscillations in pump-probe response as a function of delay time t Numerical simulation: Re[σ] [arb. units] δt [ps] (a arxiv: ( ω/ω 1.1 delay time THz-pump THz-probe spectroscopy NbN: Matsunaga, Shimano, et al. PRL 111, 57 (13
14 Superconductor coupled to optical phonons Linear coupling to optical phonons: H el-ph = g ph X p,k, - perform expansion in using h i (b p + b p c k+p, c k, +c.c. d g dt O = i ph [H, O] ~ infinite hierarchy of equations of motion Study generation of coherent phonons: - Break hierarchy at first order: - Non-vanishing b p leads to finite lattice displacement: α k β k b p α k β k b p U(r,t = with D p =(b p + b p h Mω ph V = D p (te +ip r, is the reduced mass of the forced lattice harmonic ions aoscillator [ d ] - Equation of motion for coherent phonon amplitude: dt + ω ph D p (t = F p (t, with forcing term: F p (t = ω ph g [ ] ph M + k,p ( α k+p β k α k β k+p + L k,p ( α k α k+p + β k+p β k, k p
15 Equation of motion for coherent phonon amplitude - Equation of motion for coherent phonon amplitude: forced harmonic oscillator [ d dt + ω ph ] D p (t = F p (t, F p (t = ph ~ g ph X k h M + k,p k+p k k k+p i + Hierarchy of time scales: D p (t A p ω ph p [1 cos(ω ph t] ph Forcing term can be approximated by F p (t A p (t. erent-phonon am step-function Displacive excitation of coherent phonons: (similar to semiconductors - abrupt change in quasiparticle states leads to jump in equilibrium position of lattice - cosine oscillations in lattice displacement - extrema at integer and half-integer values of ph
16 Generation of coherent phonons: Numerical results Hierarchy of time scales: U(,t (arb. units p ph D p (t A p Numerical results for displacive excitation of coherent phonons: ω ph 1 p =.5 ps, ph =.5 mev 8 p =.5 ps, ph =.1 mev 5 p =. ps, ph =.5 mev 6 5 p =. ps, ph =.1 mev 4 p = 1. ps, ph =.5 mev 4 4 p = 1. ps, ph =.1 mev [1 cos(ω ph t] t (ps cosine oscillations in lattice displacement U(R,t - extrema at integer and half-integer values of τ ph APS, Manske, Avella, PRB 84, (11
17 Generation of coherent phonons: Quantum beats Hierarchy of time scales: forcing term can be approximated by F p (t (t[a p + B p cos ( t/ h / t], quantum beating when D p (t B p ω ph π ω d [cos(tω ph S (tω d + sin(tω ph C (tω d ], 1 9 p ph d = 1 ph ph (8 6 U(,t (arb. units p =.5 ps p = 5. ps p = 15. ps t (ps APS, Manske, Avella, PRB 84, (11
18 Generation of coherent phonons: Pump-probe conductivity Hierarchy of time scales: p ph Signatures of non-adiabatic dynamics in pump-probe conductivity Re(σ [arb. units] 1.5 (a (c 1. Re[ ( t, = ph ] δt [ps] ω/ω Re[σ(ω=ω ph ] [arb. units] δt [ps] = beating phenomenon in pump-probe conductivity as a function of delay time t ( t, = ph arxiv: (13
19 Generation of coherent phonons: Resonance Hierarchy of time scales: p = ph resonant generation of coherent phonons for D p (t B p ω ph t sin(ωph t+ ph = 1 (= 1/~ U(,t (arb. units Lattice displacement: (a A A A 5 p = 54 p = 415 p = 561 const t 1 15 t (ps Pump-probe conductivity: Re(σ [arb. units] (a δt [ps] (e.85 resonantly enhanced oscillations in pump-probe conductivity as a function of delay time = t ω/ω arxiv: (
20 Conclusions & Outlook Microscopic simulation of ultrafast dynamics in superconductors Non-adiabatic regime - order parameter oscillations qualitative agreement with experiment - generation of coherent phonos - for ω ph = : resonant enhancement of coherent phonons Pump-probe conductivity: ( t, p, ph : - oscillations in as a function of delay time with frequencies = 1 1/~ and ph - strong enhancement of oscillation amplitude when frequencies are in resonance: 1 = ph Outlook: - consider higher order in correlation expansion: incoherent phonons, feedback on SC condensate [mev] U(,t [arb. units] (d t [ps] (e Re(σ [arb. units] (a t [ps] δt [ps] ω/ω PRB 84, (11; EPL 11, 17 (1; arxiv: (13
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