ORIGINAL ARTICLE: Verde, M., Schmiegelow, C.T., Poschinger, U. et al. Trapped atoms in spatially-structured vector light fields. Sci Rep 13, 21283 (2023). https://doi.org/10.1038/s41598-023-48589-1
Article Abstract: Spatially-structured laser beams, eventually carrying orbital angular momentum, affect electronic transitions of atoms and their motional states in a complex way. We present a general framework, based on the spherical tensor decomposition of the interaction Hamiltonian, for computing atomic transition matrix elements for light fields of arbitrary spatial mode and polarization structures. We study both the bare electronic matrix elements, corresponding to transitions with no coupling to the atomic center-of-mass motion, as well as the matrix elements describing the coupling to the quantized atomic motion in the resolved side-band regime. We calculate the spatial dependence of electronic and motional matrix elements for tightly focused Hermite–Gaussian, Laguerre–Gaussian and for radially and azimuthally polarized beams. We show that near the diffraction limit, all these beams exhibit longitudinal fields and field gradients, which strongly affect the selection rules and could be used to tailor the light-matter interaction. The presented framework is useful for describing trapped atoms or ions in spatially-structured light fields and therefore for designing new protocols and setups in quantum optics, -sensing and -information processing. We provide open code to reproduce our results or to evaluate interaction matrix elements for different transition types, beam structures and interaction geometries.
We provide the Mathematica code for reproducing the results presented in the article: “Trapped atoms in spatially-structured vector light fields” Sci Rep 13, 21283 (2023), https://doi.org/10.1038/s41598-023-48589-1.​The code is intended to be used in close connection to the article. It is organized through the following sections:​1: HG and LG beams, with various polarization structures, and propagating along the z direction are defined. Non-paraxial vector fields

α
(x,y,z) are calculated by means of equations (2),(3) and (4) for different beam structures α.2: Longitudinal
E
z
(x, y) and circular
E
σ=±1
(x, y) electric field components of focal fields are calculated by means of equation (5). Their modulus is rescaled and compared within Fig. 1. 3: From

α
(x,y,z) we compute the tensor field
\[Laplacian]J=2
T
\[Laplacian]m
(x, y, z) via equation (12) for an arbitrary angle
θ
x
between z and the rotated magnetic quantization (the rotation being performed around the x-axis).4: Relative strengths |
µ
α
(x,y,z0;
J
2
,
J
1
,
m
2
,
m
1
)| are computed from equation (15) for different \[Laplacian]m electric quadrupole transition for
θ
x
= 0. The calculations refer to the transverse section at the plane z=z0. The results are shown in Fig. 2.5: Relative strengths |
µ
α
(x,y,z0;
J
2
,
J
1
,
m
2
,
m
1
)| are computed from equation (16) for a particular \[Laplacian]m electric quadrupole transition and different angles
θ
x
. The calculations refer to the transverse section at the plane z=z0. Results are shown in Fig. 3.6: Transverse cross sections of relative strengths |
µ
bsb,α
(x,y,z)| for motional blue sidebands of quadrupole transition are calculated via equation (20). Results are shown in Fig. 4.

1: Light beams definition and focal fields evaluation

Input: λ and w0 (wavelength and beam waist [μm] )
Output: non-paraxial electric vector field

α
(x,y,z)
​
Description:
We define the paraxial electric field solutions Hermite-Gaussian and Laguerre-Gaussian: respectively ElHG and ElLG (expressed in [V/μm]).
By means of equation (2), we evaluate the electric vector field

α
(x,y,z) for tightly focused beams by considering the non-paraxial corrections (up to the first order in the spatial derivatives) to the paraxial ElHG and ElLG solutions for different paradigmatic cases α:
​
- α = 1 : HG(0,0) circularly polarized
- α = 2 : HG(0,1) circularly polarized
- α = 3 : LG(0,1) circularly polarized
- α = 4 : LG(0,-1) circularly polarized
- α = 5 : LG(0,1) radially polarized
- α = 6 : LG(0,1) azimuthally polarized
(*Inputparameters*)​​λ=0.729;(*lightwavelenght[μm]*)​​w0=1.0;(*beamwaistatthefocusofElHGandElLGsolutions[µm]*)​​​​(*Physicalconstants*)​​ϵ0=8.8542
-18
10
;(*ϵ0[J
2
V
µm]*)​​c=2.9979
14
10
;(*[µm/s]*)​​ℏ=1.055
-34
10
;(*[J*s]*)​​kz=2π/λ;(*wavevector[
-1
μm
]*)​​ω=ckz;(*2πfrequency[Hz]*)​​​​(*Electricfieldparameters*)​​P0=
-3
10
;(*power[W]*)​​σ0=1;(*circularpolarization*)​​​​(*Opticalfunctions*)​​zR[w_]=π
2
w
/λ;(*Rayleighlength*)​​R[z_,w_]=z(1+
2
(zR[w]/z)
);(*curvatureradius*)​​wz[z_,w_]=w
(1+
2
(z/zR[w])
)
;​​El0[P_,w_]=
4P/(cϵ0
2
w
)
;(*[V/μm]*)​​ψ[z_,w_]=ArcTan
z
zR[w]
;(*Gouyphase*)​​ElHG[x_,y_,z_,p_,l_]=
El0[P0,w0]
π
w0
wz[z,w0]
HermiteHp,
2
x
wz[z,w0]
HermiteHl,
2
y
wz[z,w0]

-
2
x
+
2
y
2
wz[z,w0]

Ikz
2
x
+
2
y
2R[z,w0]

Iψ[z,w0]

Ikzz

;​​ψLG[z_,p_,l_]=-(2p+Abs[l]+1)ArcTan
z
zR[w0]
;(*GeneralizedGouyphase*)​​ElLG[r_,ϕ_,z_,p_,l_]=
El0[P0,w0]
2
2p!
π(p+Abs[l])!
w0
wz[z,w0]
Abs[l]
r
2
wz[z,w0]
-
2
r
2
wz[z,w0]

LaguerreLp,Abs[l],
2
2
r
2
wz[z,w0]

Ikz
2
r
2R[z,w0]

Ilϕ

IψLG[z,p,l]

Ikzz

;
Forα=1,α<7,α++,​​Whichα1,(*vectorbeamTEM(0,0)circularlypolarizedσ=+1*)​​u[x_,y_,z_,p_,l_]=ElHG[x,y,z,p,l];​​ElField[x_,y_,z_][α,1]=u[x,y,z,0,0];​​ElField[x_,y_,z_][α,2]=Iσ0u[x,y,z,0,0];​​ElField[x_,y_,z_][α,3]=Ikz(D[u[x,y,z,0,0],x]+Iσ0D[u[x,y,z,0,0],y]);,​​​​α2,(*vectorbeamHG(0,1)circularlypolarizedσ=+1*)​​u[x_,y_,z_,p_,l_]=ElHG[x,y,z,p,l];​​ElField[x_,y_,z_][α,1]=u[x,y,z,0,1];​​ElField[x_,y_,z_][α,2]=Iσ0u[x,y,z,0,1];​​ElField[x_,y_,z_][α,3]=Ikz(D[u[x,y,z,0,1],x]+Iσ0D[u[x,y,z,0,1],y]);,​​​​​​α3,(*vectorbeamLG(0,1)circularlypolarizedσ=+1*)​​u[x_,y_,z_,p_,l_]=ElLG
2
x
+
2
y
,ArcTan[x,y],z,p,l;​​ElField[x_,y_,z_][α,1]=u[x,y,z,0,1];​​ElField[x_,y_,z_][α,2]=Iσ0u[x,y,z,0,1];​​ElField[x_,y_,z_][α,3]=Ikz(D[u[x,y,z,0,1],x]+Iσ0D[u[x,y,z,0,1],y]);,​​​​α4,(*vectorbeamLG(0,-1)circularlypolarizedσ=+1*)​​u[x_,y_,z_,p_,l_]=ElLG
2
x
+
2
y
,ArcTan[x,y],z,p,l;​​ElField[x_,y_,z_][α,1]=u[x,y,z,0,-1];​​ElField[x_,y_,z_][α,2]=Iσ0u[x,y,z,0,-1];​​ElField[x_,y_,z_][α,3]=Ikz(D[u[x,y,z,0,-1],x]+Iσ0D[u[x,y,z,0,-1],y]);,​​​​α5,(*vectorbeamLG(0,-1)radiallypolarized*)​​u[x_,y_,z_,p_,l_]=
1
2
ElLG
2
x
+
2
y
,ArcTan[x,y],z,p,l;​​ElField[x_,y_,z_][α,1]=u[x,y,z,0,-1]+u[x,y,z,0,1];​​ElField[x_,y_,z_][α,2]=Iσ0u[x,y,z,0,-1]-Iσ0u[x,y,z,0,1];​​ElField[x_,y_,z_][α,3]=Ikz(D[u[x,y,z,0,-1],x]+D[u[x,y,z,0,1],x]+Iσ0D[u[x,y,z,0,-1],y]-Iσ0D[u[x,y,z,0,1],y]);,​​​​α6,(*vectorbeamLG(p,l)azimuthallypolarized*)​​u[x_,y_,z_,p_,l_]=
I
2
ElLG
2
x
+
2
y
,ArcTan[x,y],z,p,l;​​ElField[x_,y_,z_][α,1]=u[x,y,z,0,-1]-u[x,y,z,0,1];​​ElField[x_,y_,z_][α,2]=Iσ0u[x,y,z,0,-1]+Iσ0u[x,y,z,0,1];​​ElField[x_,y_,z_][α,3]=Ikz(D[u[x,y,z,0,-1],x]-D[u[x,y,z,0,1],x]+Iσ0D[u[x,y,z,0,-1],y]+Iσ0D[u[x,y,z,0,1],y]);​​;​​;

2: Longitudinal and circular components evaluation of focal fields (Fig. 1)

Input: z0 and

α
(x,y,z)
Output: |
E
z
(x, y)| and |
E
σ=±1
(x, y) | (rescaled)
​
Description:
We choose the particular transverse plane of interest z = z0.
We calculate the profiles |
E
z
(x, y) and |
E
σ=±1
(x, y)| of the focal fields calculated in Section 1.
We multiply each profile by a scale factor corresponding to the ratio between the overall maximum value MaxAllEl (calculated over all the 3 components of the 6 focal fields) and its maximum value MaxEl. When MaxEl=0 the scale factor is set to 0.
The different profiles are graphically visualized to show and compare the different beam structures contributions: the higher the scale the lower is the contribution from one particular component (scale = 0 corresponds to vanishing contribution).
(*Inputparameter:*)​​z0=
-10
10
;(*transverseplaneofinterest[µm];Numericalinfinitiesattheexactz=0*)​​​​For[α=1,α<7,α++,​​SqrtInt[x_,y_,z_][α,1]=Abs[ElField[x,y,z][α,1]-IElField[x,y,z][α,2]];(*|
E
σ=1
|*)​​SqrtInt[x_,y_,z_][α,2]=Abs[ElField[x,y,z][α,3]];(*|
E
z
|*)​​SqrtInt[x_,y_,z_][α,3]=Abs[ElField[x,y,z][α,1]+IElField[x,y,z][α,2]];](*|
E
σ=-1
|*)​​scale=ConstantArray[1,{6,3}];​​​​For[α=1,α<7,α++,​​For[n=1,n<4,n++,​​ElMax[α,n]=FindMaximum[{SqrtInt[x,y,z0][α,n],{x,y}∈Disk[{x,y},2w0]},{{x,0.5},{y,0.5}}][[1]];​​If[ElMax[α,n]<
-5
10
,ElMax[α,n]=
-10
10
;SqrtInt[x_,y_,z_][α,n]=0;scale[[α,n]]=0;];​​]]​​​​MaxAllEl=Max@Table[ElMax[αi,ni],{αi,1,6},{ni,1,3}];​​numPoints=5;​​TableContourPlot
1
ElMax[αi,ni]
SqrtInt[x,y,z0][αi,ni],{x,-2w0,2w0},{y,-2w0,2w0},ImageSize270,PlotPointsnumPoints,LabelStyleDirective[Black,25],PlotRangeAll,Contours10,ColorFunctionColorData[{"M10DefaultDensityGradient",{0,1}}],ColorFunctionScalingFalse,PlotLabelStyleText"scale: "<>ToStringRound
MaxAllEl
ElMax[αi,ni]
*scale[[αi,ni]],0.1<>"x"1,15,Black,{ni,1,3},{αi,1,6}​​

3: Tensor field evaluation

4: Calculation of relative strengths for different quadrupole transitions (Fig. 2)

5: Calculation of relative strengths for a quadrupole transition at different interaction angles (Fig. 3)

6: Calculation of relative strengths for motional sidebands of quadrupole transitions (Fig. 4)

CITE THIS NOTEBOOK

Trapped atoms in spatially-structured vector light fields​
by Maurizio Verde, Christian T. Schmiegelow, Ulrich Poschinger & Ferdinand Schmidt-Kaler
Wolfram Community, STAFF PICKS, January 10, 2024
​https://community.wolfram.com/groups/-/m/t/3099859