Using driven cold atoms as quantum simulators Charles Creffield 1 , Germ´ an Sierra 2 , and Fernando Sols 1 c.creffield@fis.ucm.es 1. Universidad Complutense de Madrid ısica Te´ 2. Instituto de F´ orica, UAM-CSIC Using driven cold atoms as quantum simulators – p.1/21
Outline How to control tunneling by shaking • shaking optical lattices: how and why • control of tunneling amplitude ⇒ simulation of Riemann function 1 • control of tunneling phase ⇒ synthetic gauge potentials 2 1. CEC & G. Sierra, PRA 91, 063608 (2015) 2. CEC & F . Sols, PRA 90, 023536 (2014) Using driven cold atoms as quantum simulators – p.2/21
Optical lattices Expose atoms to far-detuned laser field ⇒ conservative potential ∝ laser intensity ⇒ standing wave produces a lattice potential Model with the Bose-Hubbard Hamiltonian + U + U � � � � � � a † � � a † H BH = − J i a j + H.c. n i ( n i − 1) H BH = − J eff i a j + H.c. n i ( n i − 1) 2 2 i i � i,j � � i,j � Concentrate on controlling J Using driven cold atoms as quantum simulators – p.3/21
Coherent destruction of tunneling Is it possible to control J without altering the optical lattice? As the system is quantum coherent, we can use quantum interference effects to control motion. A time-periodic driving potential can be applied � H ( t ) = H BH + K cos ωt x j n j j by piezo-actuating a mirror, or from a phase modulator. Interference arises from dynamical phases acquired during tunneling, and can produce cancellation of tunneling. Using driven cold atoms as quantum simulators – p.4/21
Floquet theory Seek eigensystem of Floquet operator H ( t ) = H ( t ) − i∂ t ⇒ quasienergies and Floquet states High-frequency limit, ω ≫ J , ⇒ perturbative expansion 1st order: � T � t J eff /J = 1 dt e − iF ( t ) , where F ( t ) = dt ′ f ( t ′ ) T 0 0 e.g. for harmonic driving [ f ( t ) = K cos ωt ] J eff = J J 0 ( K/ω ) and so tunneling destroyed at zeros of J 0 : K/ω = 2 . 4048 , 5 . 5201 , . . . Using driven cold atoms as quantum simulators – p.5/21
Experiment Lignier et al , PRL 99, 220403 (2007) Tunneling suppressed at K/ω ≃ 2 . 40 1.0 1.0 0.8 | J eff / J | 0.6 0.4 0.8 0.2 0 2 4 6 8 10 | J eff / J | 0.6 h w / J Assuming J eff ∝ expansion 0.4 gives reasonable agreement 0.2 0.0 0 1 2 3 4 5 6 K 0 Using driven cold atoms as quantum simulators – p.6/21
Riemann zeta function CRITICAL t STRIP s NON-TRIVIAL ZEROS 10 trivial zeros 8 100 � Ζ � z �� 6 σ 4 80 1 2 − 6 − 4 − 2 60 0 0 0 pole y 0.2 0.2 40 0.4 0.4 20 0.6 0.6 x x 0.8 0.8 0 1 CRITICAL LINE complex continuation original domain • “trivial zeros” at x = − 2 , − 4 , · · · • non-trivial zeros along critical line x = 1 / 2 + iy Using driven cold atoms as quantum simulators – p.7/21
Riemann hypothesis Riemann hypothesis: All non-trivial zeros lie on the critical line “probably the most important unresolved problem in pure mathematics” • Hilbert’s eighth problem (1900) • one of the Millennium Prize problems P´ olya-Hilbert approach: Find a Hermitian operator whose eigenvalues are E n , where the Riemann zeros are 1 / 2 + iE n . Using driven cold atoms as quantum simulators – p.8/21
Inverse problem Can we drive the system so that J eff ∝ Ξ(1 / 2 + iE ) ? ⇒ inverse problem : Given J eff , what is f ( t ) ? 10 15 f 0 -10 10 E 0.5 0.5 5 1 1 Time Time 1.5 1.5 0 2 Using driven cold atoms as quantum simulators – p.9/21
Improvements 2 f(t) 1 0 0 1 2 3 4 t / T Using driven cold atoms as quantum simulators – p.10/21
Improvements 2 f(t) 1 0 0 1 2 3 4 t / T • avoid discontinuities • zero time-average (avoid heating) • well-defined parity: x → − x, t → t + T/ 2 • high frequency: f ( t ) → Ω f (Ω t ) Using driven cold atoms as quantum simulators – p.10/21
Improvements 2 2 f(t) f(t) 1 1 0 0 0 1 2 3 4 0 1 2 3 4 t / T t / T • avoid discontinuities • zero time-average (avoid heating) • well-defined parity: x → − x, t → t + T/ 2 • high frequency: f ( t ) → Ω f (Ω t ) Using driven cold atoms as quantum simulators – p.10/21
Improvements 2 2 2 f(t) f(t) f(t) 0 1 1 -2 0 0 0 0 1 1 2 2 3 3 4 4 0 1 2 3 4 t / T t / T t / T • avoid discontinuities • zero time-average (avoid heating) • well-defined parity: x → − x, t → t + T/ 2 • high frequency: f ( t ) → Ω f (Ω t ) Using driven cold atoms as quantum simulators – p.10/21
Improvements 8 2 2 2 4 f(t) f(t) f(t) f(t) 0 0 1 1 -4 -2 0 0 -8 0 0 1 1 2 2 3 3 4 4 0 1 2 3 4 0 1 2 3 4 t / T t / T t / T t / T • avoid discontinuities • zero time-average (avoid heating) • well-defined parity: x → − x, t → t + T/ 2 • high frequency: f ( t ) → Ω f (Ω t ) Using driven cold atoms as quantum simulators – p.10/21
Results: quasienergies 1 Quasienergy 0 -1 0 4 8 12 16 20 24 1 0.01 Quasienergy 0.0001 1e-06 0 4 8 12 16 20 24 E Using driven cold atoms as quantum simulators – p.11/21
Results: tunneling Recall: the quasienergies directly govern J eff ⇒ Ξ( E ) directly observable in experiment Using driven cold atoms as quantum simulators – p.12/21
Summary I • we have successfully linked a physical system to the Riemann zeros • the spectrum is given by a periodically-driven system (not static) • accessible to current experiment Using driven cold atoms as quantum simulators – p.13/21
Phase control We can control the amplitude of J eff , what about its phase ? Most general driving K sin( ωt + φ ) has three parameters: K 0 = K/ω sets the condition for CDT, What is the effect of φ ? Perturbation theory for resonant driving: J eff = J e − iK 0 cos φ e in ( φ + π/ 2) J n ( K 0 ) where V ( t ) = nω + K sin( ωt + φ ) • for cosine driving: J eff = ( − 1) n J n ( K 0 ) • but otherwise J eff is complex! (gauge potential) CEC & F . Sols, PRA 84, 023630 (2011) Using driven cold atoms as quantum simulators – p.14/21
Gauge fields A (charged) particle hopping around a closed loop acquires an Aharonov-Bohm phase: � φ AB = φ i,j � Engineering hopping phases ⇒ applying a B -field Landau gauge requires a phase gradient 3φ • y-hoppings have no induced phases 2φ • x-hoppings have a y -dependent phase φ Using driven cold atoms as quantum simulators – p.15/21
Simple approach † 16 16 14 14 8 12 12 10 10 y 8 8 6 6 0 4 4 2 2 -4 0 4 x 2 4 6 8 10 12 14 2 4 6 8 10 12 14 16 16 14 14 8 12 12 10 10 y 8 8 6 6 0 4 4 2 2 -4 0 4 x 2 4 6 8 10 12 14 2 4 6 8 10 12 14 † A.R. Kolovsky, EPL 93, 20003 (2011) Using driven cold atoms as quantum simulators – p.16/21
What went wrong? • adjacent rungs of the lattice are out of phase • ⇒ there is also a periodic driving in the y direction • these “accidental phases” cancel † the effective B Solution: “digital Hamiltonian simulation” Decompose H into x and y , and apply sequentially: e − iH ∆ t ≃ e − iH x ∆ t e − iH y ∆ t (Trotter decomposition) Interval 1: x hopping suppressed, evolution under H y Interval 2: drive the system in x , with H y suppressed † CEC & F . Sols, EPL Comment 101, 40001 (2013) Using driven cold atoms as quantum simulators – p.17/21
Weak fields I We initialise the system as a narrow Gaussian in the centre, and kick it in the + y direction 14 φ=0.05 • centre-of-mass makes a circular orbit φ=0.1 • higher flux ⇒ tighter orbit 12 • cyclotron orbits φ=0.15 y 10 φ=0.2 8 4 5 6 x Using driven cold atoms as quantum simulators – p.18/21
Weak fields II • add a weak, parabolic trap potential, V ( r ) = kr 2 / 2 • initialise system in ground state of trap • displace potential to excite condensate into motion 12 11 Foucault 10 9 8 7 6 5 5 6 7 8 9 10 11 12 Using driven cold atoms as quantum simulators – p.19/21
Strong fields High flux ⇒ dynamics is not semiclassical 0.2 bulk states edge states 0.1 Quasienergy 0 0.04 8 0.02 6 0 -0.1 2 2 4 4 4 2 6 6 -0.2 π 0 2 π 8 Flux, Φ • fractal quasienergy spectrum: Hofstadter’s butterfly • chiral edge states • topologically protected (Chern number) Using driven cold atoms as quantum simulators – p.20/21
Summary Shaking the lattice can provide fine control over the coherent dynamics of BECs • CDT can regulate the amplitude of J • the phase can also be controlled • allows quantum simulation of physical and mathematical systems Using driven cold atoms as quantum simulators – p.21/21
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