Non-Equilibrium Criticality in Driven Open Quantum Systems

Transcription

1 Cold Atoms meet Quantum Field Theory July Bad Honnef, Germany Non-Equilibrium Criticality in Driven Open Quantum Systems Sebastian Diehl Institute for Theoretical Physics, Technical University Dresden Collaboration: L. Sieberer, Innsbruck -> Weizmann E. Altman, Weizmann J. Marino, Dresden

2 Outline Driven Open Quantum Systems: Experimental platforms & Theoretical challenges 3 dimensions: Classical Driven criticality 1 dimension: Quantum Driven Criticality L. Sieberer, S. Huber, E. Altman, SD, PRL 2013; PRB 2014 U. C. Tauber, SD, PRX 2014 J. Marino, S. Diehl, in preparation (2015) Key Questions: Persistence of non-equilibrium conditions: Thermalization? Persistence of quantum dynamics: Decoherence? Relation to equilibrium criticality? Universality class? +

3 Motivation: Driven-dissipative many-body dynamics experimental systems on the interface of quantum optics and many-body physics Driven-open Dicke models Coupled microcavity arrays a R = Ret b R? Ret Koch et al., PRA 2010 Houck, Türeci, Koch, Nat. Phys Baumann et al., Nature 2010 Ritsch et al., RMP 2013 exciton-polariton systems in semiconductor quantum wells Kasprzak et al., Nature 2006 Carusotto, Ciuti RMP other platforms (light-matter): dissipative Rydberg systems polar molecules photon BECs Klaers et al. Nature 2010 trapped ions Barreiro et al. Nature 2011 Carr et al. PRL 2013 Marcuzzi et al. PRL 2014 Zhu et al. PRL 2013

4 Non-Equilibrium Physics with Driven Open Systems Interdisciplinary research area: physics at various length scales Quantum Optics coherent and drivendissipative dynamics on equal footing Many-body physics continuum of spatial degrees of freedom Statistical mechanics Microscopic Questions and Challenges: Novel universal phenomena? Thermodynamic Long wavelength? g c Efficient theoretical tools? Z[J] = Dϕ e i(s[ϕ]+r Jϕ) quantum field theory out of equilibrium Experimental platforms? 1 2 cold atoms, light-driven semiconductors, microcavity arrays, trapped ions...

5 Dynamical Critical Phenomena in Driven Open Quantum Systems L. Sieberer, S. Huber, E. Altman, SD, PRL 2013; PRB 2014 U. C. Tauber, SD, PRX 2014 Microscopic Quantum Optics Thermodynamic Many-body physics Long wavelength Statistical mechanics

6 Motivation: Exciton-Polariton Systems Kasprzak et al., Nature 2006 Imamoglu et al., PRA 1996 E photons relaxation excitons pump lower polaritons loss k phenomenological description: stochastic driven-dissipative Gross-Pitaevskii-Eq i@ t = propagation apple r 2 2m µ + i( p l)+( iapple) 2 + pump & loss rates two-body loss elastic collisions h (t, x) (t 0, x 0 )i = (t t 0 ) (x x 0 ) microscopic derivation and linear fluctuation analysis: Szymanska, Keeling, Littlewood PRL (04, 06); PRB (07)); Wouters, Carusotto PRL (07,10)

7 number of quantum wells in the microcavity, is crucial in maintaining the strong coupling regime of polaritons at high carrier density. The far-field polariton emission pattern was measured to probe the population distribution along the lower polariton branch. The spatially resolved emission and its coherence properties are accessible in a real-space imaging set-up combined with an actively stabilized Bose Condensation of Exciton-Polaritons view Bose condensation seen despite non-equilibrium conditions y physical features: driven-dissipative stochastic Gross-Pitaevskii Equation I ti a t s le stab Kasprzak et al., Nature 2006 stochastic driven-dissipative PDE with Markovian noise:gross-pitaevskii-eq hx (t, x)i = 0 and stochastic i@t = 0 0 h x ( t, x ) x (t, x )i = gd(t 2 r 2m µ + i( p se-einstein condensation phase transition I o ate t s y nar mean-field: neglect noise l) t0 ) d ( x +( x0 ) i ) 2 + Szymanska, Keeling, Littlewood PRL (04, 06); PRB (07)); Wouters, Carusotto PRL (07,10) Figure 2 Far-field emission measured at 5 K for three excitation mean field intensities. Left panels, 0.55 P thr; centre panels, P thr; and right panels, I homogeneous condensate f (t, x) = f P thr; where P thr ¼ 1.67 kw cm22 is the threshold power of condensation. a, Pseudo-3D images of the far-field emission within the neglect noise gpa microcavity gl y dispersion. a, is a angular cone of ^238, with the emission intensity displayed on the vertical axis 2 )ragg fmirrors = for g > g 0 l(in arbitrary at resonance withp solution (x,units). t) =With0increasing excitation power, a sharp and intense peak homogeneous k iton is an optically active dipole is formed in the centre of the emission distribution ðvx ¼ vy ¼ 08Þ; n between an electron in the corresponding to the lowest momentum state k k ¼ 0. b, Same data as in a 2 ) chemical potential µ = l f 0 resolved in energy. For such a measurement, a slice of the far-field e band. In microcavities operating but matter interaction, 2D excitons emission corresponding to vx ¼ 08 is dispersed by a spectrometer and enmodes, called microcavity imaged on a charge-coupled device (CCD) camera. The horizontal axes Q: What is the nature ofthethe critical information in mean of the in-plane wavevector k k in a display emission angle point? (top axis) (no and the in-plane momentum (bottomfield) een exciton and photon modes, axis); the vertical axis displays the emission energy in a false-colour scale s), gives rise to lower and upper (different for each panel; the units for the colour scale are number of counts

8 Universality (Equilibrium) Universality: the art of forgetting about details ' same universality class! <=> same set of critical exponents Bose-Einstein Condensate planar magnets technical tool: renormalization group flow Coarse graining Wilson-Fisher Renormalization group fixed point

9 Universality (Equilibrium) Universality classes: Memory of symmetries kept ' = ' Bose-Einstein Condensate planar magnets trapped ions liquid-gas transition in carbon-dioxide Symmetries: Coarse graining U(1) ' O(2) phase rotations in BEC ~ 80 stable elements => O(10^10) possible compounds ~ 10^23 particles but only a handful universality classes Z 2 O(2) universality class Ising universality class Q: given loss of memory, can non-equilibrium conditions affect universal behavior?

10 Classical vs. Quantum Criticality generic quantum phase diagram T Disordered quantum critical region Ordered, symmetry breaking g c (possibly) ordered, no symmetry breaking g (eg. potential vs. kinetic energy) temperature is a relevant perturbation to the quantum critical point quantum critical scaling for T!! G quantum non-gaussian

11 Theoretical Approach e i [ ] = Z D e is M [ + ]

12 Microscopic Model generic microscopic model: many-body master equation single particle pump H = Z D[ ] t = x ˆ x ( 4 2M µ) ˆx + 2 ( ˆ x ˆx) 2 p Z x i[h, ]+D[ ] L[ ] [ ˆ x ˆx single particle pump 1 2 { ˆx ˆ x, }] + l Z x Z apple x single particle loss [ ˆ2 x ˆ 2 x many-body system single-, two-,... body loss [ ˆx ˆ x 1 2 { ˆ x ˆx, }] { ˆ 2 x two particle loss ˆ2 x, }] cf. Quantum Optics: cf. Many-Body Physics: single mode, H=0, semiclassical approximation: effective laser threshold equations continuum of spatial degrees of freedom: infrared divergence second quantized operator formalism inappropriate need method transfer: develop efficient functional many-body techniques

13 Theoretical Approach generic microscopic model: many-body master equation single particle t = i[h, ]+D[ ] L[ ] many-body system evaluation strategy: single-, two-,... body loss Many-Body Master Equation translation table Microscopic Markovian Dissipative Action RG flow Long Wavelength Effective t = i[h, ]+L[ ] Z,, e i [ ] = D e is M [ + k k = i 2 Tr apple (2) k + R k R k many-body master equation Keldysh real time functional integral Functional Renormalization Group equation closed system Keldysh: Gasenzer, Pawlowski,PLB 08; Berges, Hoffmeister, Nucl. Phys. B, 09 Opens up the powerful toolbox of quantum field theory to driven-dissipative systems derivative expansion including all non-irrelevant operators

14 Open System Functional RG closed system Keldysh: Gasenzer, Pawlowski,PLB 08; Berges, Hoffmeister, Nucl. Phys. B, 09 Evaluation of functional integral via equivalent Functional RG equation adapted to open system Wetterich, k k = i 2 Tr apple (2) k + R k R k Many-Body Master Equation > k k second field variation infrared regulator Markovian dissipative action k= = S translation table Microscopic Markovian Dissipative Action RG flow coarse graining in real space = integrating out high modes in momentum space k=0 = full effective action mode elimination induces RG flow of coupling of effective action Long Wavelength Effective Action solve functional differential equation approximately by systematic derivative expansion truncation ordering principle is power counting

15 Driven Classical Criticality L. Sieberer, S. Huber, E. Altman, SD, PRL 110, (2013) and PRB 89, (2014); U. C. Tauber, SD, PRX 4, (2014)

16 Schematic RG flow Flow in the complex plane of couplings Im Im Im u 3 u K FP action purely dissipative Re non-linear initial flow Re linearized IR flow fixed point Re initial values: k 0 S particles propagate A =Re[K] 1 D =Im[K] coherent collisions ~ two-body loss universal domain encoding universality class decoherence key results (classical): asymptotic thermalization universal decoherence (new independent critical exponent) reveals equilibrium vs. non-equilibrium fine structure

17 Asymptotic Low-Frequency Thermalization global thermal equilibrium: all subparts in equilibrium with each other <=> Temperature is invariant under the partition temperature T B =tr A temperature T

18 Asymptotic Low-Frequency Thermalization global thermal equilibrium: all subparts in equilibrium with each other <=> Temperature is invariant under the partition B 0 =tr A 0 temperature T temperature T

19 Asymptotic Low-Frequency Thermalization global thermal equilibrium: all subparts in equilibrium with each other <=> Temperature is invariant under the partition RG: <=> Temperature is scale invariant q y q x temperature T RG: tracing out momentum shells

20 Equilibrium Symmetry and Asymptotic Thermalization dynamical action functional Z k = t,x c iz t q + c.c. + i k q q +... dynamical term Markovian noise level equilibrium symmetry for c(t, x)! c ( t, x) q(t, x)! q ( t, x)+ 2 Z z@ t c ( t, x) i! i Graham, Tel PRA 1990: Aron et al., J. Stat. Mech. (2010); adapted to real time functional integral (t, x) = (t, x) (t, x) associated Ward identity implies classical FDT with distribution function F = 2T! this symmetry is emergent at criticality Z k Z, k T = 4 Z RG invariant temperature requires Z (g )= (g ) k/z k = const. asymptotic low-frequency thermalization

21 Universal decoherence and exponent fine structure decoherence <=> purely imaginary fixed point action global thermal equilibrium is ensured by symmetry: Im equilibrium dynamics non-equilibrium dynamics Im symmetry protected K u 3 u K u Re initial flow infrared flow Im u 3 u K u K Re no symmetry u 3 Re Re eigenvalue of flow speed lowest eigenvalue R r equilibrium and driven systems are in different universality classes physical reason: independence of coherent and dissipative dynamics formal reason: difference in symmetry

22 Driven Quantum Criticality + J. Marino, SD, in preparation

23 Non-equilibrium analogue of quantum criticality Lindblad Master equation with strong quantum diffusion (1D) d Z x [ra(x) ra (x) 1 2 {ra (x)ra(x), }] possible realization: microcavity arrays cf. D. Marcos et al., NJP (2012) + H c = X i D[ ] = q X i + i (a i [ i + i a i+1 )+h.c. 1 2 { + i i, }] q

24 What is quantum about it? analogy to an equilibrium system: noise level P K (!)! coth! 2T two regimes T!!/2T 1: P K (!) 2T, P K (t t 0 ) (t t 0 ) classical/markovian!/2t 1: P K (!)!, P K (t t 0 ) (t t 0 ) 2 quantum/non-markovian scaling of the noise level

25 Non-equilibrium analogue of quantum criticality strongly momentum dependent noise level markovian non-equilibrium: weak noise at long wavelength P K (q) G equilibrium: weak noise at long timescales P K (!)! coth! 2T G diffusion noise M q T! non-eq variant: cf. Dalla Torre et al., Nat Phys. (2010) identical canonical scaling to quantum problem for z =2 (! q2 ) but spatial vs. temporal noise anomalous scaling regime: two scales Ginzburg scale one-loop perturbative G ' apple d two-body loss non-gaussian critical scaling for M G Markov scale integration of one-loop flow cf. Chiochetta, Mitra, Gambassi, arxiv (2014) rescaled Markov noise at FP M ' G + b 2+a 2+ b 2+a a 0.3 b 0.2! 1 2+a

26 no naive quantum-classical correspondence no asymptotic decoherence no asymptotic thermalization Non-equilibrium analogue of quantum criticality new fixed point with more repulsive directions (fine tuning of loss rate) g 3 thermal-like 1 repulsive direction quantum-like 3 repulsive directions g 2 Gaussian all directions repulsive results for critical exponents mass exponent anomalous dim. dynamic exp. decoherence exp. g 1 QFP (d=1, z=2) CFP (d=3, z=2) Z r new exponent: correction to noise scaling N = 0.26

27 Absence of Asymptotic Decoherence coherent dynamics does not fade out: r i Im(g i ) Re(g i ) 6=0 exponent degeneracy: A = D = 0.03 k!0 = Z!,q c(a + id)q 2 q + c.c A k A, D k D Im ) r kin = D A! r kin 6= 0 Re mixed fixed point with finite dissipative and coherent couplings there is another, purely dissipative fixed point, but with symmetry behind exponent degeneracy? A D = r < 0 less stable!

28 Absence of Asymptotic Thermalization how to detect thermal equilibrium in a quantum system? L. Sieberer, A. Chiochetta, A. Gambassi, U. Tauber, SD, arxiv (2015) symmetry of Schwinger-Keldysh action under transformation T c(!, q) q(!, q) = x cosh(!/2) x sinh(!/2) x sinh(!/2) x cosh(!/2) associated Ward identities are quantum Fluctuation-Dissipation relations to arbitrary order reproduces classical limit for T! composition of time reversal (dynamics) and KMS boundary condition (state) present for any microscopically time translation and time reversal invariant Hamiltonian Im c(!, q) q(!, q) (!, q) = (!, q) (!, q) physical picture: whenever the dynamics is generated microscopically by a time-independent Hamiltonian, the ensuing irreversible dynamics will be thermal Re

29 Absence of Asymptotic Thermalization Im practical benefit: symmetry as straightforward diagnostic tool for Schwinger-Keldysh actions symmetry explicitly violated microscopically due to markovian nature of dynamics not emergent: k!0 = Z!,q driven quantum system: cz! q + qz! c + i (q) q q +... (q) = d q 2 cf. classical driven system (q) = 0 Re Z k Z e i 0 Z log k/, d k d Z =0.08, 0 Z =0.03, d = 0.26 Z k Z, k Z = =0.16 microscopic and universal asymptotic violation of quantum FDR Z cannot be chosen real, limit-cycle like oscillations with (huge!) period k n+1 k n = e 2 0 Z

30 Summary: Driven Classical and Quantum Criticality 3 dimensions Equilibrium fixed point stable 1 dimension New non-equilibrium universality class non-equilibrium fades out: correlations thermalize quantum fades out: decoherence A non-equilibrium persists: no thermalization Z =0 Z 6= 0 quantum persists: no decoherence D = n-eq. r > 0 A D = r =0 responses witness non-equilibrium n-eq r 6= eq r limit cycle for quasiparticle residue Z k Z e i 0 Z log k/ Im Im Re Re

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32 Basic Idea: Keldysh functional integral Compare: Schroedinger equation: evolving a state vector i@ t i(t) =H i(t) ) i(t) =U(t, t 0 ) i(t 0 ) Heisenberg equation: evolving a state (density) matrix U(t, t 0 )=e ih(t t t (t) = i[h, (t)] ) (t) =U(t, t 0 ) (t 0 )U (t, t 0 ) identical for pure (factorizable) states = ih First case: functional integral via Trotterization of time interval and insertion of coherent states: e ih(t t0) = lim (1 + i th) N t = t t 0 N!1 N t t t 0 i(t 0 ) single set of degrees of freedom for vector evolution

33 Basic Idea: Keldysh functional integral Compare: Schroedinger equation: evolving a state vector i@ t i(t) =H i(t) ) i(t) =U(t, t 0 ) i(t 0 ) Heisenberg equation: evolving a state (density) matrix U(t, t 0 )=e ih(t t t (t) = i[h, (t)] ) (t) =U(t, t 0 ) (t 0 )U (t, t 0 ) Second case: Trotterization on both sides: e ih(t t0) = lim (1 + i th) N t = t t 0 N!1 N t (t 0 ) U U t two sets of degrees of freedom for matrix evolution

34 Basic Idea: Keldysh functional integral Compare: Schroedinger equation: evolving a state vector i@ t i(t) =H i(t) ) i(t) =U(t, t 0 ) i(t 0 ) Heisenberg equation: evolving a state (density) matrix U(t, t 0 )=e ih(t t t (t) = i[h, (t)] ) (t) =U(t, t 0 ) (t 0 )U (t, t 0 ) partition function Z =tr (t) =tr (t 0 )=1 + contour t 0! 1,t f! +1 information on all stages t f =+1 program works for Lindblad generator of dynamics (t) =e (t - contour t 0)L 0 = lim N!1 (1 + tl) N 0 t 0 = 1