In this short computational essay i reproduce the simulated results from https://arxiv.org/pdf/2404.00740, Chen, Tao and Huang, Chenxi and Gadway, Bryce and Covey, Jacob P. (2024), Quantum walks and correlated dynamics in an interacting synthetic Rydberg lattice, Phys. Rev. Lett. 133, 120604. https://doi.org/10.1103/PhysRevLett.133.120604, a work that explores interesting dynamics of interacting atoms with some control parameters of the Hamiltonian. Phenomena such as Bloch oscillations and confinement of particles are shown in correlated quantum walks, this sets up foundations of exploring quantum many-body dynamics in Rydberg optical lattices.
Single particle dynamics
Single particle dynamics
Install and load the Wolfram QuantumFramework:
PacletInstall["https://www.wolfr.am/DevWQCF",ForceVersionInstall->True]<<Wolfram`QuantumFramework`
9-site Rydberg lattice
9-site Rydberg lattice
Single particle Hamiltonian, where Ω/2 is the hopping rate and the on-site potential energy x :
j
Δ
Nine site atom level structure diagram (taken from the reference):
In[]:=
tmax=3;(*Simulationtime(μs)*)
Defining the 9 site lattice basis:
In[]:=
sitesBasis=QuantumBasis[AssociationThread[Range[-4,4]->IdentityMatrix[9]]];
Initial state is |0〉:
In[]:=
psinit=sitesBasis["BasisStates"][[5]]
Out[]=
|0〉
In[]:=
H=QuantumOperator[DiagonalMatrix[Range[-4,4]Δ]+(Ω/2)(DiagonalMatrix[Table[1,{sitesBasis["Dimension"]-1}],1]+DiagonalMatrix[Table[1,{sitesBasis["Dimension"]-1}],-1]),sitesBasis,"Parameters"->{Δ,Ω}];
Quantum walk for Δ = 0 and Ω/h = 0.45:
In[]:=
evolved=QuantumEvolve[H[0,2π0.45],psinit,{t,0,tmax}];MatrixPlottab=Table[evolved[t]["ProbabilityList"],{t,Subdivide[0,tmax,15]}],
Out[]=
Quantum walk for Δ/h = 0.4 and Ω/h = 0.45:
In[]:=
evolved=QuantumEvolve[H[2π0.4,2π0.45],psinit,{t,0,tmax}];MatrixPlotTable[evolved[t]["ProbabilityList"],{t,Subdivide[0,tmax,15]}],
Out[]=
Calculate evolution of the states:
In[]:=
evolvedStates=Table[QuantumEvolve[H[2πx,2π0.45],psinit,{t,0,tmax}],{x,{0,0.4,0.8}}];
In[]:=
P0spreadings=Table[{evolvedStates[[i]][t]["ProbabilityList"][[5]],Dot[Abs[Range[-4,4]],evolvedStates[[i]][t]["ProbabilityList"]]},{i,3},{t,0,3,0.1}];
P
0
MapIndexedListLinePlot#1[[All,1]],&,P0spreadings//Column
Out[]=
Spreading width as function of time:
In[]:=
MapIndexedListLinePlot#1All,2,&,P0spreadings//Column
Out[]=
8 site lattice with Escher-type structure
8 site lattice with Escher-type structure
Level structure with Escher structure (continuous staircase):
The Escher interaction term is included:
Simulated population dynamics for different detunings Δ, we can observe refocusing in this regime:
Interacting atom pairs
Interacting atom pairs
Interacting Hamiltonian for two atoms labeled A and B:
Initial state |00〉, both atoms in state 0:
Bloch oscillations for an interacting pair for different V/Δ ratios, breakdown and revival of oscillations:
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Quantum walks in Rydberg optical lattices
by Bruno Tenorio
Wolfram Community, STAFF PICKS, January 20, 2025
https://community.wolfram.com/groups/-/m/t/3358936
by Bruno Tenorio
Wolfram Community, STAFF PICKS, January 20, 2025
https://community.wolfram.com/groups/-/m/t/3358936