Time Crystals Krzysztof Sacha Jagiellonian University in Krak´ow. People involved in the research on time crystals: • Alexandre Dauphin, Barcelona • Dominique Delande, Paris • Krzysztof Giergiel, Krak´ow • Peter Hannaford, Melbourne • Arkadiusz Kosior, Dresden • Arkadiusz Kuro´s, Krak´ow • Maciej Lewenstein, Barcelona • Pawe l Matus, Krak´ow • Marcin Mierzejewski, Wroc law • Florian Mintert, London • Artur Miroszewski, Warsaw • Rick Mukherjee, London • Luis Morales-Molina, Santiago • Frederic Sauvage, London • Andrzej Syrwid, Krak´ow • Jakub Zakrzewski, Krak´ow Formation of space crystals [H^; T^] = 0 T^ { translation operator of all particles by the same vector t =const. h ^y(x) ^y(x 0) ^(x 0) ^(x) i Formation of time crystals? Eigenstates of a time-independent Hamiltonian H are also eigenstates of a time translation operator e−iHt ~r is fixed h ^y(t) ^y(t0) ^(t0) ^(t) i F. Wilczek, PRL 109, 160401 (2012). P. Bruno, PRL 111, 070402 (2013). H. Watanabe and M. Oshikawa, PRL 114, 251603 (2015). A. Syrwid, J. Zakrzewski, KS, PRL 119, 250602 (2017). V. K. Kozin and O. Kyriienko, "Quantum Time Crystals from Hamiltonians with Long-Range Interactions", PRL 123, 210602 (2019). Discrete time crystals Spontaneous process 2 : 1 resonance 0,3 t=0 0 0,3 Two resonant Floquet states with ! 0 the quasi energy difference 0,3 2 ( !T = π-difference) 0 2 0,3 probability density 0 0,3 t=T 0 0 10z 20 0 10z 20 A. Buchleitner, D. Delande, J. Zakrzewski, Phys. Rep. 368, 409 (2002). Discrete time crystals Single particle bouncing on an oscillating mirror in 1D Classically: Floquet Hamiltonian: @ H(t) − i n(z; t) = En n(z; t) @t En { quasi-energy n(z; t) { time periodic function 0,3 t=0 0 0,3 0 0,3 0 0,3 probability density 0 0,3 t=T 0 0 10z 20 0 10z 20 Discrete time crystals Single particle bouncing on an oscillating mirror in 1D Classically: 2 : 1 resonance Floquet Hamiltonian: @ Two resonant Floquet states with H(t) − i n(z; t) = En n(z; t) @t ! the quasi energy difference 2 E { quasi-energy !T n ( 2 = π-difference) n(z; t) { time periodic function A. Buchleitner, D. Delande, J. Zakrzewski, Phys. Rep. 368, 409 (2002). Discrete time crystals Bosons with attractive interactions J jUj N NjUj N ^ y y h y 2 2 y 2 2i X x X z z HF ≈ − a^1 a^2 +a ^2 a^1 − (^a1 ) (^a1) + (^a2 ) (^a2) = −J si − si sj ; LMG model 2 2 i=1 N i;j N = 104 jN;0i+j0;Ni j i ≈ p 2 KS, Phys. Rev. A 91, 033617 (2015). Discrete time crystals Bosons with attractive interactions J jUj N NjUj N ^ y y h y 2 2 y 2 2i X x X z z HF ≈ − a^1 a^2 +a ^2 a^1 − (^a1 ) (^a1) + (^a2 ) (^a2) = −J si − si sj ; LMG model 2 2 i=1 N i;j N = 104 jN;0i+j0;Ni j i ≈ p −! jN − 1; 0i or j0; N − 1i 2 KS, Phys. Rev. A 91, 033617 (2015). Discrete time crystals Spin systems V. Khemani, A. Lazarides, R. Moessner, L. S. Sondhi, Phys. Rev. Lett. 116, 250401 (2016). D. V. Else, B. Bauer, C. Nayak, Phys. Rev. Lett. 117, 090402 (2016). Discrete time crystals Spin systems Chain of 10 ions: J. Zhang et al., Nature (2017). j"":::"i ± j##:::#i j i ≈ x p x −! j "" ::: "i or j ## ::: #i 2 x x 6 10 impurities in diamond: S. Choi et al., Nature (2017). P. Matus, KS, "Fractional Time Crystals", Phys. Rev. A 99, 033626 (2019) Condensed matter physics in time crystals ! Assume s:1 resonance, ! = s Ω(I ). In the moving frameΘ= θ − s t 2 P X iksΘ H ≈ + f−k hks e : 2meff k P2 For example for f (t) = λ cos(!t), we get H ≈ + V0 cos(sΘ). 2meff Platform for time crystal research Single particle systems Integrable 1D system: H0(x; p) −! H0(I )= ) I = const; θ = Ω(I ) t + θ0: Time periodic perturbation: ! ! X ik!t X inθ H1 = f (t) h(x) −! H1 = fk e hne : k n P2 For example for f (t) = λ cos(!t), we get H ≈ + V0 cos(sΘ). 2meff Platform for time crystal research Single particle systems Integrable 1D system: H0(x; p) −! H0(I )= ) I = const; θ = Ω(I ) t + θ0: Time periodic perturbation: ! ! X ik!t X inθ H1 = f (t) h(x) −! H1 = fk e hne : k n ! Assume s:1 resonance, ! = s Ω(I ). In the moving frameΘ= θ − s t 2 P X iksΘ H ≈ + f−k hks e : 2meff k Platform for time crystal research Single particle systems Integrable 1D system: H0(x; p) −! H0(I )= ) I = const; θ = Ω(I ) t + θ0: Time periodic perturbation: ! ! X ik!t X inθ H1 = f (t) h(x) −! H1 = fk e hne : k n ! Assume s:1 resonance, ! = s Ω(I ). In the moving frameΘ= θ − s t 2 P X iksΘ H ≈ + f−k hks e : 2meff k P2 For example for f (t) = λ cos(!t), we get H ≈ + V0 cos(sΘ). 2meff s : 1 resonance (s = 4): x=121 0.1 0.6 1 t=0.25T 1 2 3 4 t=0.3T 0.4 0.05 4 0.2 3 2 0 0 probability density 0 30 60 90 120 probability density 0 1 2 3 4 x t / T mirror classical turning point s J X ∗ EF = − (aj+1aj + c:c:) 2 j=1 KS, Sci. Rep. 5, 10787 (2015). L. Guo, M. Marthaler, G. Sch¨on,Phys. Rev. Lett. 111, 205303 (2013). Crystalline structure in time s : 1 resonance P2 H ≈ + V0 cos(s Θ) 2meff ! Θ= θ − s t x=121 0.1 0.6 1 t=0.25T 1 2 3 4 t=0.3T 0.4 0.05 4 0.2 3 2 0 0 probability density 0 30 60 90 120 probability density 0 1 2 3 4 x t / T mirror classical turning point s J X ∗ EF = − (aj+1aj + c:c:) 2 j=1 KS, Sci. Rep. 5, 10787 (2015). L. Guo, M. Marthaler, G. Sch¨on,Phys. Rev. Lett. 111, 205303 (2013). Crystalline structure in time s : 1 resonance P2 H ≈ + V0 cos(s Θ) 2meff ! Θ= θ − s t s : 1 resonance (s = 4): x=121 0.1 0.6 1 t=0.25T 1 2 3 4 t=0.3T 0.4 0.05 4 0.2 3 2 0 0 probability density 0 30 60 90 120 probability density 0 1 2 3 4 x t / T mirror classical turning point Crystalline structure in time s : 1 resonance P2 H ≈ + V0 cos(s Θ) 2meff ! Θ= θ − s t s : 1 resonance (s = 4): s J X ∗ EF = − (aj+1aj + c:c:) 2 j=1 KS, Sci. Rep. 5, 10787 (2015). L. Guo, M. Marthaler, G. Sch¨on,Phys. Rev. Lett. 111, 205303 (2013). Space Crystal Time Crystal t=const z=const 2 2 |ψ(z)| |ψ(t)| (s-1)L 2L (s-1)T 2T L 0 T 0 space time KS, D. Delande, PRA 94, 023633 (2016). K. Giergiel, KS, PRA 95, 063402 (2017). D. Delande, L. Morales-Molina, KS, PRL 119, 230404 (2017). Anderson localization in the time domain H0(t) is a perturbation that fluctuates in time but H0(t + sT ) = H0(t). s s J X X E = − (a∗ a + c:c:) + " ja j2 F 2 j+1 j j j j=1 j=1 Example for s = 100: KS, Sci. Rep. 5, 10787 (2015). Space Crystal Time Crystal t=const z=const 2 2 |ψ(z)| |ψ(t)| (s-1)L 2L (s-1)T 2T L 0 T 0 space time Anderson localization in the time domain H0(t) is a perturbation that fluctuates in time but H0(t + sT ) = H0(t). s s J X X E = − (a∗ a + c:c:) + " ja j2 F 2 j+1 j j j j=1 j=1 Example for s = 100: KS, Sci. Rep. 5, 10787 (2015). KS, D. Delande, PRA 94, 023633 (2016). K. Giergiel, KS, PRA 95, 063402 (2017). D. Delande, L. Morales-Molina, KS, PRL 119, 230404 (2017). Mirror oscillations / λ cos(s!t) + λ1 cos(s!t=2) + f (t), f (t) creates an edge in time: J'/J x ≈ 0 0.6 1. 1.7 2.8 4.7 7.9 6 Edge state 0.1 4 2 energy 0 - 0.05 -2 Quasi -4 Bulk state -6 Probability density 0 -0.013 0. 0.013 0.025 0.038 0.05 0 0.2 0.4 0.6 0.8 1 λ 1 t / T K. Giergiel, A. Dauphin, M. Lewenstein, J. Zakrzewski, KS, New J. Phys. 21, 052003 (2019). E. Lustig, Y. Sharabi, M. Segev, Optica 5, 1390 (2018). Topological time crystals A particle bouncing on an oscillating mirror Mirror oscillations / λ cos(s!t) + λ1 cos(s!t=2) s=2 X ∗ 0 ∗ SSH model: H ≈ − Jb i ai + J ai+1bi i=1 J'/J x ≈ 0 0.6 1. 1.7 2.8 4.7 7.9 6 Edge state 0.1 4 2 energy 0 - 0.05 -2 Quasi -4 Bulk state -6 Probability density 0 -0.013 0. 0.013 0.025 0.038 0.05 0 0.2 0.4 0.6 0.8 1 λ 1 t / T Topological time crystals A particle bouncing on an oscillating mirror Mirror oscillations / λ cos(s!t) + λ1 cos(s!t=2) s=2 X ∗ 0 ∗ SSH model: H ≈ − Jb i ai + J ai+1bi i=1 Mirror oscillations / λ cos(s!t) + λ1 cos(s!t=2) + f (t), f (t) creates an edge in time: K.