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

[Hˆ, Tˆ] = 0 Tˆ – translation operator of all particles by the same vector

t =const.

h ψˆ†(x) ψˆ†(x 0) ψˆ(x 0) ψˆ(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 ψˆ†(t) ψˆ†(t0) ψˆ(t0) ψˆ(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 ω 0 the quasi energy difference 0,3 2 ( ωT = 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 ψn(z, t) = En ψn(z, t) ∂t

En – quasi-energy

ψn(z, t) – time periodic function Discrete time crystals Single particle bouncing on an oscillating mirror in 1D

Classically:

2 : 1 resonance Floquet Hamiltonian: 0,3 t=0

  0 ∂ 0,3 Two resonant Floquet states with H(t) − i ψn(z, t) = En ψn(z, t) ∂t ω 0 the quasi energy difference 0,3 2 ωT En – quasi-energy ( = π-difference) 0 2 0,3 ψn(z, t) – time periodic function 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 Bosons with attractive interactions

J |U| N N|U| N ˆ  † †  h † 2 2 † 2 2i X x X z z HF ≈ − aˆ1 aˆ2 +a ˆ2 aˆ1 − (ˆa1 ) (ˆa1) + (ˆa2 ) (ˆa2) = −J si − si sj , LMG model 2 2 i=1 N i,j

N = 104

|N,0i+|0,Ni |ψi ≈ √ 2

KS, Phys. Rev. A 91, 033617 (2015). Discrete time crystals Bosons with attractive interactions

J |U| N N|U| N ˆ  † †  h † 2 2 † 2 2i X x X z z HF ≈ − aˆ1 aˆ2 +a ˆ2 aˆ1 − (ˆa1 ) (ˆa1) + (ˆa2 ) (ˆa2) = −J si − si sj , LMG model 2 2 i=1 N i,j

N = 104

|N,0i+|0,Ni |ψi ≈ √ −→ |N − 1, 0i or |0, 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).

|↑↑...↑i ± |↓↓...↓i |ψi ≈ x √ x −→ | ↑↑ ... ↑i or | ↓↓ ... ↓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 ω Assume s:1 resonance, ω = 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(ωt), 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 ikωt X inθ H1 = f (t) h(x) −→ H1 = fk e hne . k n P2 For example for f (t) = λ cos(ωt), 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 ikωt X inθ H1 = f (t) h(x) −→ H1 = fk e hne . k n

ω Assume s:1 resonance, ω = 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 ikωt X inθ H1 = f (t) h(x) −→ H1 = fk e hne . k n

ω Assume s:1 resonance, ω = 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(ωt), 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

ω Θ= θ − s t x=121 0.6 1 2 3 4 0.4

0.2

0 probability density 0 1 2 3 4 t / T

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

ω Θ= θ − s t

s : 1 resonance (s = 4): 0.1 1 t=0.25T t=0.3T 0.05 4 3 2 0 probability density 0 30 60x 90 120 mirror classical turning point Crystalline structure in time s : 1 resonance

P2 H ≈ + V0 cos(s Θ) 2meff

ω 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

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.) + ε |a |2 F 2 j+1 j j j j=1 j=1

Example for s = 100:

KS, Sci. Rep. 5, 10787 (2015). 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.) + ε |a |2 F 2 j+1 j j j j=1 j=1

Example for s = 100:

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, 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(sωt) + λ1 cos(sωt/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 0 0.05 -2 Quasi - energy -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(sωt) + λ1 cos(sωt/2)

s/2 X ∗ 0 ∗
SSH model: H ≈ − Jb i ai + J ai+1bi i=1 Topological time crystals A particle bouncing on an oscillating mirror

Mirror oscillations ∝ λ cos(sωt) + λ1 cos(sωt/2)

s/2 X ∗ 0 ∗
SSH model: H ≈ − Jb i ai + J ai+1bi i=1

Mirror oscillations ∝ λ cos(sωt) + λ1 cos(sωt/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 0 0.05 -2 Quasi - energy -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). 20:1 resonance

1.0 10 0.5 5 Uij g0 0.0 0 J J -5 -0.5 -10 -1.0 -10-8 -6 -4 -2 0 2 4 6 8 10 π 2 π i-j ωt

K. Giergiel, A. Miroszewski, KS, PRL 120, 140401 (2018).

Exotic Interactions Ultra-cold atoms bouncing on an oscillating mirror Bosons: s s J X 1 X Hˆ = − (ˆa† aˆ + h.c.) + U aˆ†aˆ† aˆ aˆ F 2 j+1 j 2 ij i j j i j=1 i,j=1 Z sT Z 2 2 Uij ∝ dtg 0(t) dx |φi | |φj | , 0 Exotic Interactions Ultra-cold atoms bouncing on an oscillating mirror Bosons: s s J X 1 X Hˆ = − (ˆa† aˆ + h.c.) + U aˆ†aˆ† aˆ aˆ F 2 j+1 j 2 ij i j j i j=1 i,j=1 Z sT Z 2 2 Uij ∝ dtg 0(t) dx |φi | |φj | , 0

20:1 resonance

1.0 10 0.5 5 Uij g0 0.0 0 J J -5 -0.5 -10 -1.0 -10-8 -6 -4 -2 0 2 4 6 8 10 π 2 π i-j ωt

K. Giergiel, A. Miroszewski, KS, PRL 120, 140401 (2018). Many-body localization induced by temporal disorder

For example for bosons:

s s s J X X 1 X Hˆ = − (ˆa† aˆ + h.c.) +  aˆ†aˆ + U nˆ nˆ F 2 j+1 j j j j 2 ij i j j=1 j=1 i,j=1

Many-body localization (MBL): • vanishing of dc transport, • absence of thermalization, • logarithmic growth of the entanglement entropy,

M. Mierzejewski, K. Giergiel, KS, PRB 96, 140201(R) (2017). Time crystals with properties of 2D space crystals

K. Giergiel, A. Miroszewski, KS, PRL 120, 140401 (2018). Time crystals with properties of 2D space crystals 5:1 resonances along x and y directions

J X † 1 X † † Hˆ = − (ˆa aˆ + h.c.) + U aˆ aˆ aˆ aˆ F 2 j i 2 ij i j j i hi,ji 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

2 2 2 2 p1 + p2 P1 + P2 X ik(Θ −Θ ) H = + δ(θ1 − θ2) f (t) −→ H = + f e 1 2 2 eff 2 −2k 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

2 2 2 2 p1 + p2 P1 + P2 X ik(Θ −Θ ) H = + δ(θ1 − θ2) f (t) −→ H = + f e 1 2 2 eff 2 −2k k

K. Giergiel, A. Miroszewski, KS, PRL 120, 140401 (2018). Spontaneous formation of quasi-crystals in time Spontaneous formation of time quasi-crystals

Fibonacci quasi-crystal: LRLLRLRLLR ... LRLLR

t/T

K. Giergiel, A. Kuro´s,KS, ”Discrete Time Quasi-Crystals”, PRB 99, 220303(R) (2019)

Spontaneous formation of time quasi-crystals

H(t + T ) = H(t)

sx : 1 and sy : 1 resonances

√ s 1+ 5 ≈ y = 3 2 sx 2 RLRLRLRLRLRLR

t/T Spontaneous formation of time quasi-crystals

H(t + T ) = H(t)

sx : 1 and sy : 1 resonances

√ s 1+ 5 ≈ y = 3 2 sx 2 RLRLRLRLRLRLR

t/T

LRLLR

t/T

K. Giergiel, A. Kuro´s,KS, ”Discrete Time Quasi-Crystals”, PRB 99, 220303(R) (2019) Spontaneous formation of time quasi-crystals

H(t + T ) = H(t)

sx : 1 and sy : 1 resonances

√ s sy 13 1+ 5 ≈ y = 3 = 2 sx 2 sx 8 RLRLRLRLRLRLR RLRLRLRLRLRLRLRLRL ...

t/T t/T

LRLLR LRLLRLRLLR ...

t/T t/T

K. Giergiel, A. Kuro´s,KS, ”Discrete Time Quasi-Crystals”, PRB 99, 220303(R) (2019) Summary and outlook:

• Condensed matter physics in the time dimension: • Spontaneous breaking of time translation symmetry in periodically driven systems (discrete time crystals, time quasi-crystals, fractional time crystals, DQPT) • Condensed matter phenomena in the time dimension (Anderson localization, many-body localization, topological time crystals, exotic interactions, time lattices with properties of 2D and 3D space crystals).

• Experiments in progress — Peter Hannaford (Melbourne).

• Novel phenomena with the help of time engineering (e.g. Anderson molecule).

K. Giergiel, A. Miroszewski, KS, PRL 120, 140401 (2018). KS, PRA 91, 033617 (2015). A. Kosior, KS, PRA 97, 053621 (2018). KS, Sci. Rep. 5, 10787 (2015). K. Giergiel, A. Kosior, P. Hannaford, KS, PRA 98, 013613 (2018). KS, D. Delande, PRA 94, 023633 (2016). A. Kosior, A. Syrwid, KS, PRA 98, 023612 (2018). K. Giergiel, KS, PRA 95, 063402 (2017). P. Matus, KS, PRA 99, 033626 (2019). M. Mierzejewski, K. Giergiel, KS, PRB 96, 140201(R) (2017). K. Giergiel, A. Dauphin, M. Lewenstein, J. Zakrzewski, KS, D. Delande, L. Morales-Molina, KS, PRL 119, 230404 (2017). New J. Phys. 21, 052003 (2019). A. Syrwid, J. Zakrzewski, KS, PRL 119, 250602 (2017). K. Giergiel, A. Kuro´s,KS, PRB 99, 220303(R) (2019). KS, J. Zakrzewski, Time crystals: a review, Rep. Prog. Phys. 81, 016401 (2018). Physics World, March 2020 Phase diagram for discrete time crystals

h = h0 +  h0 – turning point of the resonant orbit

2 1.0 2 1.0 >

0.8 >

L 0.8 L t t

H 0.6 0.14 H 0.6 Ψ Ψ È

0.4 È symmetry unbroken regime

0 0.4 0 0.12 Φ 0.2 Φ 0.2 < < È 0.0 È 0.0 0.10 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 t T 0.08 25 t T 0

25 h 

gN = 0.01 Ε 20 20 gN = 0.02 0.06

15 15 spectrum  spectrum  0.04 10 10

5 5 0.02 discrete time crystal Fourier Fourier 0 0 0.00 0.496 0.498 0.500 0.502 0.504 0.496 0.498 0.500 0.502 0.504 - 2.0 -1.8 -1.6 -1.4 -1.2 -1.0 - 0.8 f Ω f Ω gN*10- 2

A. Kuro´s,R. Mukherjee, F. Mintert, KS, in preparation,.  

KS, Phys. Rev. A 91, 033617 (2015). Phase transition in Anderson localization in the time domain

p2 + p2 + p2 H = θ ψ φ + V g(θ)g(ψ)g(φ)f (t)f (t)f (t), 2 0 1 2 3 (i) P ikωi t where fi (t + 2π/ωi ) = fi (t) = k fk e .

In the moving frame,Θ= θ − ω1t, Ψ = ψ − ω2t, Φ = φ − ω3t, P2 + P2 + P2 H = Θ Ψ Φ + V h (Θ)h (Ψ)h (Φ), eff 2 0 1 2 3

P (i) ikx where hi (x) = k gk f−k e are disordered potentials.

D. Delande, L. Morales-Molina, KS, PRL 119, 230404 (2017).