Anderson localization and Mott insulator phase in the time domain - PowerPoint PPT Presentation

Anderson localization and Mott insulator phase in the time domain Krzysztof Sacha Marian Smoluchowski Institute of Physics, Jagiellonian University in Krak´ ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Outline: • Formation of time crystals • Modeling time crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 110 , 118901 (2013). T. Li et al. , PRL 109 , 163001 (2012). F. Wilczek, PRL 110 , 118902 (2013). J. Zakrzewski, Physics 5 , 116 (2012). P. Bruno, PRL 111 , 029301 (2013). P. Coleman, Nature 493 , 166 (2013). T. Li et al. , arXiv:1212.6959. KS, PRA 91 , 033617 (2015). P. Bruno, PRL 111 , 070402 (2013). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Modeling time crystals Single particle systems Space periodic potential: H ( x + L ) = H ( x ) Periodic driving: H ( t + T ) = H ( t ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Modeling time crystals Single particle systems Space periodic potential: H ( x + L ) = H ( x ) Periodic driving: H ( t + T ) = H ( t ) Example: single particle bouncing on an oscillating mirror in 1D: A. Buchleitner, D. Delande, J. Zakrzewski, Phys. Rep. 368 , 409 (2002). 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Modeling time crystals Single particle systems Space periodic potential: H ( x + L ) = H ( x ) Periodic driving: H ( t + T ) = H ( t ) Example: single particle bouncing on an oscillating mirror in 1D: A. Buchleitner, D. Delande, J. Zakrzewski, Phys. Rep. 368 , 409 (2002). 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 t=0 0,3 0.5T ψ n ( z , t ) – time periodic function 0 0,3 0.6T probability density 0.1T 0 0,3 0.2T 0.7T 2 : 1 resonance 0 0,3 0.3T 0.8T λ = 0 . 06, T = 5 . 7 0 0,3 0.4T T 0 0 10 20 0 10 20 z z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Modeling time crystals Single particle systems In the s : 1 resonance case: • There are s Floquet eigenstates with quasi-energies E j ≈ E j ′ . These quasi-energies form a band when s → ∞ . • s individual wavepackets, φ j ( z , t ), can be prepared by superposing s Floquet eigenstates. For s → ∞ , the wavepackets φ j ( z , t ) become analogues of Wannier states but in the time domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Modeling time crystals Single particle systems In the s : 1 resonance case: • There are s Floquet eigenstates with quasi-energies E j ≈ E j ′ . These quasi-energies form a band when s → ∞ . • s individual wavepackets, φ j ( z , t ), can be prepared by superposing s Floquet eigenstates. For s → ∞ , the wavepackets φ j ( z , t ) become analogues of Wannier states but in the time domain. Example for s = 4: s Restricting to the Hilbert subspace ψ = ∑ a j φ j , probability density 0,1 j =1 1 t=0.25T t=0.3T 0,05 sT s 4 ∫ J 2 ∑ ( a ∗ 3 E F = dt ⟨ ψ | H F | ψ ⟩ ≈ − j +1 a j + c . c . ) 2 0 j =1 0 30 60 90 120 0 z sT z=121 ∫ J = − 2 dt ⟨ φ j +1 | H F | φ j ⟩ probability density 0,6 0 1 2 3 4 0,4 The lowest and higher quasi-energy bands can be 0,2 considered. 0 0 1 2 3 4 t / T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Anderson localization in the time domain s s E F = − J ∑ ∑ ( a ∗ ε j | a j | 2 j +1 a j + c . c . ) + 2 j =1 j =1 sT dt ⟨ φ j | H ′ ( t ) | φ j ⟩ , with ε j = ∫ 0 where H ′ ( t ) is a perturbation that fluctuates in time but H ′ ( t + sT ) = H ′ ( t ). Example for s = 4: probability density 0 10 z=121 -2 10 -4 10 -6 10 0 1 2 3 4 t / T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Anderson localization in the time domain s s E F = − J ∑ ∑ ( a ∗ ε j | a j | 2 j +1 a j + c . c . ) + 2 j =1 j =1 sT dt ⟨ φ j | H ′ ( t ) | φ j ⟩ , with ε j = ∫ 0 where H ′ ( t ) is a perturbation that fluctuates in time but H ′ ( t + sT ) = H ′ ( t ). Space Crystal Time Crystal Example for s = 4: t=const z=const probability density 0 10 z=121 2 2 -2 |ψ (t) | 10 |ψ (z) | -4 10 -6 2L 2T 10 (s-1)L (s-1)T L T 0 0 space 0 1 2 3 4 time t / T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Mott insulator-like phase in the time domain Many-body systems Bosons: s s H F = − J a j + h . c . ) + 1 ˆ ∑ a † ∑ (ˆ j +1 ˆ U ij ˆ n i ˆ n j 2 2 j =1 i , j =1 sT ∞ dz | φ i | 2 | φ j | 2 , where | U ii | > | U ij | for i ̸ = j . ∫ ∫ with U ij = g 0 dt 0 0 • For g 0 → 0, the ground state is a superfluid state with long-time phase coherence. • For strong repulsion, U ii ≫ NJ / s , the ground state becomes a Fock state | N / s , N / s , . . . , N / s ⟩ and long-time phase coherence is lost. Time Crystal z=const (s-1)T 2T 0 time T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Summary: • Time crystals are analogues of space crystals but in the time domain. • Space periodic potentials are often used to model properties of space crystals. Crystalline structures in the time domain can be modeled by periodically driven systems. • We show that Anderson localization and Mott insulator-like phase can be observed in the time domain. • Possible experimental realization: • electronic motion of Rydberg atoms in microwave fields, • ultra-cold atoms bouncing on an oscillating mirror. KS, Phys. Rev. A 91 , 033617 (2015). KS, Sci. Rep. 5 , 10787 (2015). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Anderson localization in the time domain s s J ε j | a j | 2 ∑ ( a ∗ ∑ E F = − j +1 a j + c . c . ) + 2 j =1 j =1 where H ′ ( t ) is a perturbation that fluctuates in time but H ′ ( t + sT ) = H ′ ( t ). 4 { [ t ( t )]} H ′ ( t ) = z ∑ α n cos 2 π − sin 2 π . n 4 T 4 T n =1 ε j are chosen randomly, e.g. according to a Lorentzian distribution (Lloyd model), sT ∫ dt ⟨ φ j | H ′ ( t ) | φ j ⟩ . then α n are chosen so that ε j = 0 Space Crystal Time Crystal Example for s = 4: t=const z=const probability density 0 10 z=121 2 2 -2 |ψ (t) | |ψ (z) | 10 -4 10 -6 2L 2T 10 (s-1)L (s-1)T L T 0 0 space 0 1 2 3 4 time t / T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Maximal number of states localized in a s -resonant island: √ n max ≈ s 8 λ ω 3 . Resonant action I s = s 3 π 2 3 ω 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .