Integrable models in atomtronics: Atomtronic toolbox

Karin Wittmann Wilsmann1, Leandro Ymai2, Daniel S. Grün1, Arlei P. Tonel2, Jon Links3, Angela Foerster11Instituto de Física da UFRGS, Brazil, 2Universidade Federal do Pampa, Brazil, 3School of Mathematics and Physics, The University of Queensland, Australia

Integrable Multi-Well Hamiltonian [1]A bipartite model obtained through the Quantum Inverse Scattering Method

Hn,m = U(NA −NB)2 + µ(NA −NB) + Σ

ni=1Σ

mi=1ti,j(aib

†j + a

†ibj) (1)

Some multi-well geometric models (NA, NB):(2,1) (3,1) (2,2) (4,2) (5,1)

Switching Device: 2+1 modes [2, 3]

H0 = U(N1 −N2 +N3)2 + J1(a†1a2 + a1a

†2) + J3(a

†2a3 + a2a

†3)

[H,N] = [H,Q] = [N,Q] = 0, Q = J21N3 + J23N1 − J1J3(a

†1a3 + a

†3a1)

The additional conserved quantity Q provides an Heff which, in the resonant regime,yield analytical frequency and amplitude equations.

J1 3J

Uy

By applying an external field, ε(N3 − N1), the second-order tunneling amplitudebetween wells 1 and 3 can be controlled while N2 remains constant.

Abstract: The precise control of quantum systems will play a major role in therealization of atomtronic devices. Here we study models of dipolar bosons confinedto three and four wells. The analysis considers both integrable and non-integrableregimes within the models. Through variation of the external field, we demonstratehow the triple well system can be controlled between various “switched-on” and“switched-off” configurations and how the four well system can be controlled toencode a phase into a NOON state

NOON states: 2+2 modes [4, 5]

Integrable Extended Bose-Hubbard

H =U0

2

4∑i=1

Ni(Ni − 1) +4∑i=1

4∑j=1,j 6=i

Uij

2NiNj −

J

2

∑i=1

(a†iai+1 + a

†i+1ai). (2)

where aj, a†j : j = 1, 2, 3, 4. Eq.(1) could be derived from EBHM as long as it com-

plies with the integrability condition U0 = U13 = U24, which provides four conservedquantities, [H,N] = [H,Qk] = [N,Qk] = 0, k = 1, 2, as many as number modes.

NOON ProtocolBelow, we describe two protocols that enable the generation of NOON states. ForProtocol I the outcomes are probabilistic while Protocol II are deterministic.

Protocol IConsidering U(t, µ, ν) = exp

−ith[H+ µ(N2 −N4) + ν(N1 −N3)]

,here we employ breaking of integrability through an applied field µ(N4 − N2) tosubsystem B = 2, 4 and a measurement process M:(i) |ΨI

1〉 = U(tm − tµ, 0, 0) |Ψ0〉;(ii) |ΨI

2〉 = U(tµ, µ, 0) |ΨI1〉;

(iii) |ΨI3〉 = M |ΨI

2〉,where tm = hπ/(2Ω) and M = 0,M represents a projective measurement of thenumber of bosons at site 3 which heralds a high-fidelity NOON state in subs. B.

In an idealized limit, with β = (−1)(N+1)/2 and the initial state |Ψ0〉 = |M,P, 0, 0〉,

|ΨI3〉 =

1√2

β |M,P, 0, 0〉 + eiPθ |M,0, 0, P〉 , r = 0,

1√2

|0, P,M, 0〉 − βeiPθ |0, 0,M, P〉 , r =M,

(3)

These states are recognized as products of a NOON state for subsystem B with Fockbasis states for subsystem A = 1, 3.

Protocol IIEmploying the same initial state |Ψ0〉, the following sequence of steps are implementedto arrive at a NOON state in subsystem B and deterministic state in A:(i) |ΨII

1〉 = U(tm − tν, 0, 0) |Ψ0〉;(ii) |ΨII

2〉 = U(tν, 0, ν) |ΨII1〉;

(iii) |ΨII3〉 = U(tm − tµ, 0, 0) |Ψ

II2〉;

(iv) |ΨII4〉 = U(tµ, µ, 0) |Ψ

II3〉.

where ν represent the breaking of integrability through an applied field ν(N3 −N1)to subsystem A = 1, 3 . In an idealized limit, with Υ = β exp(i(Pθ− π/2)),

|ΨII4〉 =

1√2

|M,P, 0, 0〉 + Υ |M,0, 0, P〉 (4)

Fidelity and ProbabilityThe fidelities are computed for Pθ ranging from 0 to π, achieved by varying tµ.

FI = | 〈ΨI3|Φ

I3〉 | > 0.9 FII = | 〈ΨII

4|ΦII4〉 | > 0.9

where |Ψ〉 denotes the analytical states and |Φ〉 the numerically stateobtained by EBHM (2) time evolution. For physically realistic settings,with M=4 and P=11, the NOON state fidelities for Pθ ε [0, π] aregreater than 0.9 with probabilities varying as:

References[1] Leandro H. Ymai, et al. J. Phys. A, 50, 2017.[2] Karin Wittmann W., et al. Communications Physics, 1(1), 2018.[3] Arlei P. Tonel, et al. SciPost Phys. Core 2(003) (2020).[4] Daniel S. Grun, et al. arXiv preprint arXiv:2004.11987 (2020).[5] Daniel S. Grun, Karin Wittmann W., et al. arXiv preprint arXiv:2102.02944 (2021).

Contact Information: angela@if.ufrgs.br – karin.wittmann@ufrgs.brIntegrable models in atomtronics: Atomtronic toolbox