Time Crystal Platform Krzysztof Sacha Jagiellonian University in - PowerPoint PPT Presentation

Time Crystal Platform Krzysztof Sacha Jagiellonian University in Krak´ ow.

People Krzysztof Giergiel Artur Miroszewski Topological time crystal part: A. Dauphin M. Lewenstein J. Zakrzewski

Discrete time crystals Theoretical prediction: K. Sacha, PRA 91 , 033617 (2015). | ψ � ∝ | N , 0 � + | 0 , N � | ψ � ∝ | N , 0 � V. Khemani et al. , PRL 116 , 250401 (2016). Peter Hannaford, D. V. Else et al. , PRL 117 , 090402 (2016). Swinburne Univ. of Technology, Melbourne First experiments: J. Zhang et al. , Nature 543 , 217 (2017). S. Choi et al. , Nature 543 , 221 (2017).

Discrete time crystals Theoretical prediction: K. Sacha, PRA 91 , 033617 (2015). | ψ � ∝ | N , 0 � + | 0 , N � | ψ � ∝ | N , 0 � V. Khemani et al. , PRL 116 , 250401 (2016). Peter Hannaford, D. V. Else et al. , PRL 117 , 090402 (2016). Swinburne Univ. of Technology, Melbourne First experiments: J. Zhang et al. , Nature 543 , 217 (2017). S. Choi et al. , Nature 543 , 221 (2017). K. Giergiel, A. Kuro´ s, KS, ”Discrete Time Quasi-Crystals” , arXiv:1807.02105.

Condensed matter physics in time crystals

Platform for time crystal research Single particle systems Integrable 1D system: H 0 ( x , p ) − → H 0 ( I ) = ⇒ I = const , θ = Ω( I ) t + θ 0 . Time periodic perturbation: �� � �� � f k e ik ω t h n e in θ H 1 = f ( t ) h ( x ) − → H 1 = . n k

Platform for time crystal research Single particle systems Integrable 1D system: H 0 ( x , p ) − → H 0 ( I ) = ⇒ I = const , θ = Ω( I ) t + θ 0 . Time periodic perturbation: �� � �� � f k e ik ω t h n e in θ H 1 = f ( t ) h ( x ) − → H 1 = . n k Assume s:1 resonance, ω = s Ω( I ). In the moving frame Θ = θ − ω s t P 2 � f − k h ks e iks Θ . H ≈ + 2 m eff k

Platform for time crystal research Single particle systems Integrable 1D system: H 0 ( x , p ) − → H 0 ( I ) = ⇒ I = const , θ = Ω( I ) t + θ 0 . Time periodic perturbation: �� � �� � f k e ik ω t h n e in θ H 1 = f ( t ) h ( x ) − → H 1 = . n k Assume s:1 resonance, ω = s Ω( I ). In the moving frame Θ = θ − ω s t P 2 � f − k h ks e iks Θ . H ≈ + 2 m eff k P 2 For example for f ( t ) = λ cos( ω t ), we get H ≈ 2 m eff + V 0 cos( s Θ).

Crystalline structure in time A particle bouncing on an oscillating mirror P 2 H ≈ 2 m eff + V 0 cos( s Θ) s : 1 resonance ( s = 4): probability density 0.1 1 t=0.25T t=0.3T 0.05 4 2 3 0 0 30 60 90 120 x mirror classical turning point

Crystalline structure in time A particle bouncing on an oscillating mirror P 2 H ≈ 2 m eff + V 0 cos( s Θ) s : 1 resonance ( s = 4): probability density 0.1 1 t=0.25T t=0.3T 0.05 4 2 3 0 sT 0 30 60 90 120 s J � x � ( a ∗ E F = dt � ψ | H F | ψ � ≈ − j +1 a j + c . c . ) mirror 2 j =1 0 classical turning point x=121 probability density sT 0.6 J = − 2 � dt � φ j +1 | H F | φ j � 1 2 3 4 0 0.4 0.2 KS, Sci. Rep. 5 , 10787 (2015). 0 0 1 2 3 4 t / T

λ Topological time crystals A particle bouncing on an oscillating mirror Mirror oscillations ∝ λ cos( s ω t ) + λ 1 cos( s ω t / 2) s / 2 i a i + J ′ a ∗ � J b ∗ � � SSH model: H ≈ − i +1 b i i =1

Topological time crystals A particle bouncing on an oscillating mirror Mirror oscillations ∝ λ cos( s ω t ) + λ 1 cos( s ω t / 2) s / 2 i a i + J ′ a ∗ � J b ∗ � � SSH model: H ≈ − i +1 b i i =1 Mirror oscillations ∝ λ cos( s ω t ) + λ 1 cos( s ω t / 2) + f ( t ), f ( t ) creates the edge in time: J' / J 0.6 1. 1.7 2.8 4.7 7.9 6 4 Quasi - energy 2 0 - 2 - 4 - 6 - 0.013 0. 0.013 0.025 0.038 0.05 λ 1

Topological time crystals A particle bouncing on an oscillating mirror Mirror oscillations ∝ λ cos( s ω t ) + λ 1 cos( s ω t / 2) s / 2 i a i + J ′ a ∗ � J b ∗ � � SSH model: H ≈ − i +1 b i i =1 Mirror oscillations ∝ λ cos( s ω t ) + λ 1 cos( s ω t / 2) + f ( t ), f ( t ) creates the edge in time: x ≈ 0 t = const. J' / J Probability density 0.6 1. 1.7 2.8 4.7 7.9 6 Edge state 0.1 4 Quasi - energy 2 0 0.05 - 2 - 4 Bulk state - 6 0 - 0.013 0 0.2 0.4 0.6 0.8 1 0. 0.013 0.025 0.038 0.05 t / T λ 1 K. Giergiel, A. Dauphin, M. Lewenstein, J. Zakrzewski, KS, arXiv:1806.10536

π π θ Quasi-crystals in the time domain A particle bouncing on an oscillating mirror Fibonacci quasi-crystal (the inflation rule B → BS and S → B ): B → BS → BSB → BSBBS → BSBBSBSB → . . . P 2 � f − k h ks e iks Θ . H ≈ + 2 m eff k

θ π Quasi-crystals in the time domain A particle bouncing on an oscillating mirror Fibonacci quasi-crystal (the inflation rule B → BS and S → B ): B → BS → BSB → BSBBS → BSBBSBSB → . . . P 2 � f − k h ks e iks Θ . H ≈ + 2 m eff k 0.6 B S B B S B S B B S B B S 0 0.4 c 0.2 f k - 50 V eff s 0.0 f k 100 0 - 0.2 - 100 - 100 - 0.4 5 10 15 20 25 30 0 2 π k K. Giergiel, A. Miroszewski, KS, PRL 120 , 140401 (2018).

ω π π Exotic Interactions Ultra-cold atoms bouncing on an oscillating mirror Bosons: s s H F = − J a j + h . c . ) + 1 ˆ � a † � a † a † (ˆ j +1 ˆ U ij ˆ i ˆ j ˆ a j ˆ a i 2 2 j =1 i , j =1 � sT � dx | φ i | 2 | φ j | 2 , U ij ∝ dt g 0 0

π Exotic Interactions Ultra-cold atoms bouncing on an oscillating mirror Bosons: s s H F = − J a j + h . c . ) + 1 ˆ � a † � a † a † (ˆ j +1 ˆ U ij ˆ i ˆ j ˆ a j ˆ a i 2 2 j =1 i , j =1 � sT � dx | φ i | 2 | φ j | 2 , U ij ∝ dt g 0 0 20:1 resonance 1.0 10 0.5 5 U ij g 0 0 0.0 J J - 5 - 0.5 - 10 - 1.0 - 10 - 8 - 6 - 4 - 2 0 2 4 6 8 10 2 π i - j ω t K. Giergiel, A. Miroszewski, KS, PRL 120 , 140401 (2018).

Time crystals with properties of 2D space crystals

Time crystals with properties of 2D space crystals 5:1 resonances along x and y directions a i + h . c . ) + 1 H F = − J ˆ � a † � a † a † (ˆ j ˆ U ij ˆ i ˆ j ˆ a j ˆ a i 2 2 � i , j � i , j K. Giergiel, A. Miroszewski, KS, PRL 120 , 140401 (2018).

Time engineering Anderson molecule Two atoms bound together not due to attractive interaction but due to destructive interference H = p 2 1 + p 2 H eff = P 2 1 + P 2 2 2 � f − 2 k e ik (Θ 1 − Θ 2 ) + δ ( θ 1 − θ 2 ) f ( t ) − → + 2 2 k K. Giergiel, A. Miroszewski, KS, PRL 120 , 140401 (2018).

Time engineering Anderson molecule Two atoms bound together not due to attractive interaction but due to destructive interference H = p 2 1 + p 2 H eff = P 2 1 + P 2 2 2 � f − 2 k e ik (Θ 1 − Θ 2 ) + δ ( θ 1 − θ 2 ) f ( t ) − → + 2 2 k K. Giergiel, A. Miroszewski, KS, PRL 120 , 140401 (2018).

Summary: 1. Time crystals are analogues of space crystals but in the time domain. 2. Crystalline structures in time can emerge in dynamics of resonantly driven single- and many-particle systems. 3. Periodically driven systems are platform for time crystal research: • topological time crystals, • quasi-crystal structures in time, • many-body systems with exotic interactions, • time crystals with properties of 2D or 3D space crystals, • Anderson localization in the time domain induced by disorder in time, • many-body localization caused by temporal disorder, • dynamical quantum phase transition in time crystals. 4. Time engineering: Anderson molecule. KS, PRA 91 , 033617 (2015). K. Giergiel, A. Miroszewski, KS, PRL 120 , 140401 (2018). KS, Sci. Rep. 5 , 10787 (2015). A. Kosior, KS, PRA 97 , 053621 (2018). KS, D. Delande, PRA 94 , 023633 (2016). K. Giergiel, A. Kosior, P. Hannaford, KS, PRA 98 , 013613 (2018). K. Giergiel, KS, PRA 95, 063402 (2017). A. Kosior, A. Syrwid, KS, arXiv:1806.05597. M. Mierzejewski, K. Giergiel, KS, PRB 96 , 140201 (2017). K. Giergiel, A. Dauphin, M. Lewenstein, J. Zakrzewski, KS, D. Delande, L. Morales-Molina, KS, PRL 119 , 230404 (2017). arXiv:1806.10536. A. Syrwid, J. Zakrzewski, KS, PRL 119 , 250602 (2017). K. Giergiel, A. Kuro´ s, KS, arXiv:1807.02105. KS, J. Zakrzewski, Time crystals: a review , Rep. Prog. Phys. 81, 016401 (2018).

Formation of space crystals [ˆ H , ˆ T ] = 0 ˆ H – solid state system Hamiltonian ˆ T – translation operator of all particles by the same vector 2 � 2 = | ψ | 2 � � � ˆ � e i α ψ � � T ψ = � � � t =const.

Formation of time crystals? Eigenstates of a time-independent Hamiltonian H are also eigenstates of a time translation operator e − iHt � 2 = � 2 = | ψ | 2 � e − iHt ψ � e − iEt ψ � � � � � r is fixed F. Wilczek, PRL 109 , 160401 (2012). P. Bruno, PRL 111 , 070402 (2013). H. Watanabe and M. Oshikawa, Phys. Rev. Lett. 114 , 251603 (2015). A. Syrwid, J. Zakrzewski, KS, ”Time crystal behavior of excited eigenstates” , Phys. Rev. Lett. 119, 250602 (2017).

Discrete time crystals Spontaneous process

Discrete time crystals Single particle bouncing on an oscillating mirror in 1D Classically: ⇐ ⇒ Floquet Hamiltonian: ∂ 2 1 ∂ H F ( t ) = − ∂ z 2 + z + λ z cos(2 π t / T ) − i 2 ∂ t H F ψ n ( z , t ) = E n ψ n ( z , t ) E n – quasi-energy ψ n ( z , t ) – time periodic function