Quantum chaos in Bose-Hubbard circuits

Quasistatic transfer protocols for atomtronic superfluid circuits

Doron Cohen, Ben-Gurion University

Circuits with condensed bosons are the building blocks for quantumAtomtronics. Such circuits will be used as QUBITs (for quantumcomputation) or as SQUIDs (for sensing of acceleration or gravitation).We study the feasibility and the design considerations for devices thatare described by the Bose-Hubbard Hamiltonian. It is essential torealize that the theory involves “Quantum chaos” considerations.

• The Bose-Hubbard Hamiltonian.

• Relevance of chaos for Metastability and Ergodicity.

• Relevance of chaos for Quasistatic transfer protocols.

• Relevance of chaos for Irreversibility and Dissipation.

Bosonic Junction

trimer

The Bose Hubbard Hamiltonian

The system consists of N bosons in M sites. Optionally we add a gauge-field Φ.

HBHH =U

2

M∑j=1

a†ja†jajaj −

Ω

2

M∑j=1

(a†j+1aj + a†jaj+1

)

u ≡NU

Ω[classical, stability, supefluidity, self-trapping]

γ ≡Mu

N2[quantum, Mott-regime]

[Dimer] Maya Chuchem, Smith-Mannschott, Hiller, TK, AV, DC (PRA 2010).

[Zener] Smith-Mannschott, Chuchem, Hiller, TK, DC (PRL 2009).

[Kapitza] Boukobza, Moore, DC, AV (PRL 2010).

[Driven] Christine Khripkov with AV and DC (PRA 2012, JPA 2013, PRE 2013).

[Thermalization] Tikhonenkov, AV, JA, DC (PRL 2013).

[Localization] Christine Khripkov, AV, DC (NJP 2015, PRE 2018, PRA 2020).

[Rings] Geva Arwas et al (PRA 2014, SREP 2015, NJP 2016, PRB 2017, PRA 2017, PRA 2019).

[Stirring] Hiller, TK, DC (EPL 2008, PRA 2008).

[STIRAP] Dey, DC, AV (PRL 2018, PRA 2019).

[Hysteresis] Burkle, AV, DC, JA (SREP 2019, PRL 2019).

[Transfer] Yehoshua Winsten, DC (arXiv 2020).

AV = Ami Vardi; TK = Tsampikos Kottos; JA = James Anglin; DC = Doron Cohen.

Quantum Chaos perspective on Metastability and Ergodicity

Stability of flow-states (I):

• Landau stability of flow-states (“Landau criterion”)

• Bogoliubov perspective of dynamical stability

• KAM perspective of dynamical stability

Stability of flow-states (II):

• Considering high dimensional chaos (M > 3).

• Web of non-linear resonances.

• Irrelevance of the the familiar Beliaev and Landau damping terms.

• Analysis of the quench scenario.

Coherent Rabi oscillations:

• The hallmark of coherence is Rabi oscillation between flow-states.

• Ohmic-bath perspective ; η = (π/√γ)

• Feasibility of Rabi oscillation for M < 6 devices.

• Feasibility of of chaos-assisted Rabi oscillation.

Thermalization:

• Spreading in phase space is similar to Percolation.

• Resistor-Network calculation of the diffusion coefficient.

• Observing regions with Semiclassical Localization.

• Observing regions with Dynamical Localization.

The Bose Hubbard Ring Circuit

In the rotating reference frame we have a Coriolis force,

which is like magntic field B = 2mΩ. Hence is is like having flux

H =

M∑j=1

[U

2a†ja†jajaj −

K

2

(ei(Φ/L)a†j+1aj + e−i(Φ/L)a†jaj+1

)]

H =∑k

εk(Φ) b†kbk +U

2L

′∑b†k4

b†k3bk2

bk1

Φ = 2πR2m Ω

For L=3 sites, using bk =√nke

iϕk , and M = (n1−n2)/2, and n = (n1+n2)/2

H(ϕ, n;φ,M) = H(0)(ϕ, n;M) +[H(+) +H(−)

]

H(0)(ϕ, n;M) = En+ E⊥M −U

3M2 +

2U

3(N − 2n)

[3

4n+

√n2 −M2 cos(ϕ)

]

H(±) =2U

3

√(N−2n)(n±M)(n∓M) cos

(3φ∓ϕ

2

)

Flow-state stability regime diagram

The I of the maximum current state is imaged as a function of (Φ, u)

• solid lines = energetic stability borders (Landau)

• dashed lines = dynamical stability borders (Bogoliubov)

M = 3

Φ/π

u

unstablestable

0.2 0.4 0.6 0.8

−1

0

1

2

3

4

5

0

0.2

0.4

0.6

0.8

1

M = 4

Φ/πu

unstableresonance

stable

−0.5 0 0.5 1

−4

−2

0

2

4

6

0

0.2

0.4

0.6

0.8

1

The traditional paradigm associates flow-states with stationary fixed-points in phase space.

Consequently the Landau criterion, and more generally the Bogoliubov linear-stability-analysis, are

used to determine the viability of superfluidity.

Arwas, Vardi, DC [Scientific Reports 2015]

Monodromy, Chaos, and Metastability of superflow

The swap transition

M = (n1−n2)/2, constant of motion.

n = (n1+n2)/2, conjugate phase ϕ.

Section energy E = E[SP]

Section coordinates (ϕ, n)

Regular separatrix E = Ex(M)

Arwas, DC [PRA 2019]

Quasistatic transfer protocols for atomtronic superfluid circuits

In the rotating reference frame we have a Coriolis force,

which is like magnetic field B = 2mΩ.

which implies an effective flux Φ = area× B

#0

#0

#1

#1

#2#2

Φ < Φmts mtsΦ > Φ

ε ε

n = 12

(n1+n2) = depletion coordinate

M = 12

(n1−n2) = population imbalance

Winsten, DC [arXiv 2020]

Semiclassical simulation of a sweep process

n = (n1+n2)/2 = depletion (color-coded)

M = (n1−n2)/2 = population imbalance

Thresholds: (Φmts,Φstb,Φdyn,Φswp)

Faster sweep

As expected - larger spreading

Slower sweep

Implications on irreversibility and dissipation?

Phase space structure

Dynamic in Phase space

For slow sweep, the central piece of the cloud follows the SP,

and the wings of the cloud spread into the chaotic region.

Efficiency of the sweep process

Optimal rate that maximizes the transfer efficiency

Irreversibility for a quasi-static protocol

H[Q,P ;x(t)]

The Ott-Wilkinson-Kubo picture:

δE2 ∼ 2DE t (diffusion)

DE ∝ x2

E ∝ x2 (dissipation) x(t) = piston position

• Chaos - motion of particles (Q,P ) inside the box is chaotic

• Quasi static limit - the piston position (x) is varied very slowly

The standard dogma: Irreversibility requires chaos and non-adiabaticity.

But in fact: irreversibility can be observed also for a quasi-static driving of a non-chaotic system.