Using driven cold atoms as quantum simulators Charles Creffield 1 , - - PowerPoint PPT Presentation

Using driven cold atoms as quantum simulators

Charles Creffield1, Germ´ an Sierra2, and Fernando Sols1

c.creffield@fis.ucm.es

• 1. Universidad Complutense de Madrid
• 2. Instituto de F´

ısica Te´

• rica, 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 HBH = −J

• i,j
• a†

iaj + H.c.

• + U

2

• i

ni (ni − 1) Concentrate on controlling J HBH = −Jeff

• i,j
• a†

iaj + H.c.

• +U

2

• i

ni (ni − 1)

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) = HBH + K cos ωt

• j

xjnj 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: Jeff/J = 1 T T dt e−iF(t) , where F(t) = t dt′ f(t′) e.g. for harmonic driving [f(t) = K cos ωt] Jeff = JJ0(K/ω) and so tunneling destroyed at zeros of J0: 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 0.8 0.6 0.4 0.2

| Jeff / J |

10 8 6 4 2

hw / J

1.0 0.8 0.6 0.4 0.2 0.0

| Jeff / J |

6 5 4 3 2 1

K0

Assuming Jeff ∝ expansion gives reasonable agreement

Using driven cold atoms as quantum simulators – p.6/21

Riemann zeta function

0.2 0.4 0.6 0.8 1 x 20 40 60 80 100 y 2 4 6 8 10 Ζz 0.2 0.4 0.6 0.8 x

s

σ t

−2 −4 −6 trivial zeros 1 pole CRITICAL STRIP CRITICAL LINE NON-TRIVIAL ZEROS complex continuation

• riginal 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´

• lya-Hilbert approach:

Find a Hermitian operator whose eigenvalues are En, where the Riemann zeros are 1/2 + iEn.

Using driven cold atoms as quantum simulators – p.8/21

Inverse problem

Can we drive the system so that Jeff ∝ Ξ(1/2 + iE)? ⇒ inverse problem: Given Jeff, what is f(t)?

0.5 1 1.5 2

Time

5 10 15

E

• 10

10

f

0.5 1 1.5

Time

Using driven cold atoms as quantum simulators – p.9/21

Improvements

1 2 3 4

t / T

1 2

f(t)

Using driven cold atoms as quantum simulators – p.10/21

Improvements

1 2 3 4

t / T

1 2

f(t)

• avoid discontinuities
• zero time-average (avoid heating)
• well-defined parity: x → −x, t → t + T/2
• high frequency: f(t) → Ωf(Ωt)

Using driven cold atoms as quantum simulators – p.10/21

Improvements

1 2 3 4

t / T

1 2

f(t)

• avoid discontinuities
• zero time-average (avoid heating)
• well-defined parity: x → −x, t → t + T/2
• high frequency: f(t) → Ωf(Ωt)

1 2 3 4

t / T

1 2

f(t)

Using driven cold atoms as quantum simulators – p.10/21

Improvements

1 2 3 4

t / T

1 2

f(t)

• avoid discontinuities
• zero time-average (avoid heating)
• well-defined parity: x → −x, t → t + T/2
• high frequency: f(t) → Ωf(Ωt)

1 2 3 4

t / T

1 2

f(t)

1 2 3 4

t / T

• 2

2

f(t)

Using driven cold atoms as quantum simulators – p.10/21

Improvements

1 2 3 4

t / T

1 2

f(t)

• avoid discontinuities
• zero time-average (avoid heating)
• well-defined parity: x → −x, t → t + T/2
• high frequency: f(t) → Ωf(Ωt)

1 2 3 4

t / T

1 2

f(t)

1 2 3 4

t / T

• 2

2

f(t)

1 2 3 4

t / T

• 8
• 4

4 8

f(t)

Using driven cold atoms as quantum simulators – p.10/21

Results: quasienergies

4 8 12 16 20 24

• 1

1

Quasienergy

4 8 12 16 20 24

E

1e-06 0.0001 0.01 1

Quasienergy

Using driven cold atoms as quantum simulators – p.11/21

Results: tunneling

Recall: the quasienergies directly govern Jeff ⇒ 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 Jeff, what about its phase? Most general driving K sin(ωt + φ) has three parameters: K0 = K/ω sets the condition for CDT, What is the effect of φ? Perturbation theory for resonant driving: Jeff = J e−iK0 cos φ ein(φ+π/2) Jn (K0) where V (t) = nω + K sin(ωt + φ)

• for cosine driving: Jeff = (−1)nJn (K0)
• but otherwise Jeff 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: φAB =

• φi,j

Engineering hopping phases ⇒ applying a B-field

Landau gauge requires a phase gradient

• y-hoppings have no induced phases
• x-hoppings have a y-dependent phase

3φ

φ 2φ

Using driven cold atoms as quantum simulators – p.15/21

Simple approach†

2 4 6 8 10 12 14 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 2 4 6 8 10 12 14 16

• 4

4

x

8

y

2 4 6 8 10 12 14 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 2 4 6 8 10 12 14 16

• 4

4

x

8

y

† 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−iHx∆te−iHy∆t (Trotter decomposition) Interval 1: x hopping suppressed, evolution under Hy Interval 2: drive the system in x, with Hy 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

• centre-of-mass makes a circular orbit
• higher flux ⇒ tighter orbit
• cyclotron orbits

4 5 6 x 8 10 12 14 y φ=0.05 φ=0.1 φ=0.15 φ=0.2

Using driven cold atoms as quantum simulators – p.18/21

Weak fields II

• add a weak, parabolic trap potential, V (r) = kr2/2
• initialise system in ground state of trap
• displace potential to excite condensate into motion

Foucault

5 6 7 8 9 10 11 12 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

π 2π Flux, Φ

• 0.2
• 0.1

0.1 0.2 Quasienergy

bulk states edge states

2 4 6 8 2 4 6 8 0.02 0.04 2 4 6

• 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