The spatial mode is an essential component of an electromagnetic field description, yet it is challenging to characterize it for optical fields with the low average photon number, such as in a squeezed vacuum. We present a method for the reconstruction of the spatial modes of such fields based on the homodyne measurements of their quadrature noise variance performed with a set of structured masks. We show theoretically that under certain conditions, we can recover individual spatial mode distributions by using the weighted sum of the basis masks, where weights are determined using measured variance values and phases. We apply this approach to analyze the spatial structure of a squeezed vacuum field with various amount of excess thermal noise generated in Rb vapor.
REFERENCES
1.
G.
Gibson
,
J.
Courtial
,
M.
Padgett
,
M.
Vasnetsov
,
V.
Pas'ko
,
S.
Barnett
, and
S.
Franke-Arnold
, Opt. Express
12
, 5448
(2004
).2.
A.
Christ
,
C.
Lupo
, and
C.
Silberhorn
, New J. Phys.
14
, 083007
(2012
).3.
V.
Boyer
,
A. M.
Marino
, and
P. D.
Lett
, Phys. Rev. Lett.
100
, 143601
(2008
).4.
G.
Brida
,
M.
Genovese
,
A.
Meda
, and
I. R.
Berchera
, Phys. Rev. A
83
, 033811
(2011
).5.
L.
Zhang
,
V.
Boyer
, and
M. O.
Scully
, Phys. Rev. A
105
, 023725
(2022
).6.
G. C. G.
Berkhout
,
M. P. J.
Lavery
,
J.
Courtial
,
M. W.
Beijersbergen
, and
M. J.
Padgett
, Phys. Rev. Lett.
105
, 153601
(2010
).7.
Y.
Zhou
,
J.
Zhao
,
Z.
Shi
,
S. M. H.
Rafsanjani
,
M.
Mirhosseini
,
Z.
Zhu
,
A. E.
Willner
, and
R. W.
Boyd
, Opt. Lett.
43
, 5263
(2018
).8.
D.
Fu
,
Y.
Zhou
,
R.
Qi
,
S.
Oliver
,
Y.
Wang
,
S. M. H.
Rafsanjani
,
J.
Zhao
,
M.
Mirhosseini
,
Z.
Shi
et al, Opt. Express
26
, 33057
(2018
).9.
R.
Fontaine
,
N. K.
Ryf
,
H.
Chen
,
D. T.
Neilson
,
K.
Kim
, and
J.
Carpenter
, Nat. Commun.
10
, 1865
(2019
).10.
C.
Gerry
and
P.
Knight
, Introductory Quantum Optics
(
Cambridge University Press
, 2005
), p. 152
.11.
R. S.
Bennink
and
R. W.
Boyd
, Phys. Rev. A
66
, 053815
(2002
).12.
V.
Boyer
,
A. M.
Marino
,
R. C.
Pooser
, and
P. D.
Lett
, Science
321
, 544
(2008
).13.
C. S.
Embrey
,
M. T.
Turnbull
,
P. G.
Petrov
, and
V.
Boyer
, Phys. Rev. X
5
, 031004
(2015
).14.
A.
Kumar
,
H.
Nunley
, and
A. M.
Marino
, Phys. Rev. A
98
, 043853
(2018
).15.
A.
Kumar
and
A. M.
Marino
, Phys. Rev. A
100
, 063828
(2019
).16.
A.
Marino
,
J.
Clark
,
Q.
Glorieux
, and
P.
Lett
, Eur. Phys. J. D
66
, 288
(2012
).17.
J. B.
Clark
,
Z.
Zhou
,
Q.
Glorieux
,
A. M.
Marino
, and
P. D.
Lett
, Opt. Express
20
, 17050
(2012
).18.
P. J.
Barge
,
Z.
Niu
,
S. L.
Cuozzo
,
E. E.
Mikhailov
,
I.
Novikova
,
H.
Lee
, and
L.
Cohen
, Opt. Express
30
, 29401
(2022
).19.
S. L.
Cuozzo
,
P. J.
Barge
,
N.
Prajapati
,
N.
Bhusal
,
H.
Lee
,
L.
Cohen
,
I.
Novikova
, and
E. E.
Mikhailov
, Adv. Quantum Technol.
5
, 2100147
(2022
).20.
C.
Bloch
and
A.
Messiah
, Nucl. Phys.
39
, 95
(1962
).21.
S. L.
Braunstein
, Phys. Rev. A
71
, 055801
(2005
).22.
D.
Horoshko
,
L.
La Volpe
,
F.
Arzani
,
N.
Treps
,
C.
Fabre
, and
M.
Kolobov
, Phys. Rev. A
100
, 013837
(2019
).23.
R. M. D.
Araújo
,
J.
Roslund
,
Y.
Cai
,
G.
Ferrini
,
C.
Fabre
, and
N.
Treps
, Phys. Rev. A
89
, 053828
(2014
).24.
Y.
Cai
,
J.
Roslund
,
G.
Ferrini
,
F.
Arzani
,
X.
Xu
,
C.
Fabre
, and
N.
Treps
, Nat. Commun.
8
, 15645
(2017
).25.
P.
Clemente
,
V.
Durán
,
E.
Tajahuerce
,
P.
Andrés
,
V.
Climent
, and
J.
Lancis
, Opt. Lett.
38
, 2524
(2013
).26.
P.
Sidorenko
and
O.
Cohen
, Optica
3
, 9
(2016
).27.
G. M.
Gibson
,
S. D.
Johnson
, and
M. J.
Padgett
, Opt. Express
28
, 28190
(2020
).28.
M.
Li
,
L.
Bian
,
G.
Zheng
,
A.
Maiden
,
Y.
Liu
,
Y.
Li
,
J.
Suo
,
Q.
Dai
, and
J.
Zhang
, Opt. Lett.
46
, 1624
(2021
).29.
B.
Sephton
,
I.
Nape
,
C.
Moodley
,
J.
Francis
, and
A.
Forbes
, Optica
10
, 286
(2023
).30.
S. L.
Cuozzo
,
C.
Gabaldon
,
P. J.
Barge
,
Z.
Niu
,
H.
Lee
,
L.
Cohen
,
I.
Novikova
, and
E. E.
Mikhailov
, Opt. Express
30
, 37938
(2022
).31.
A. B.
Matsko
,
I.
Novikova
,
G. R.
Welch
,
D.
Budker
,
D. F.
Kimball
, and
S. M.
Rochester
, Phys. Rev. A
66
, 043815
(2002
).32.
J.
Ries
,
B.
Brezger
, and
A. I.
Lvovsky
, Phys. Rev. A
68
, 025801
(2003
).33.
E. E.
Mikhailov
and
I.
Novikova
, Opt. Lett.
33
, 1213
(2008
).34.
M. T. L.
Hsu
,
G.
Hetet
,
A.
Peng
,
C. C.
Harb
,
H.-A.
Bachor
,
M. T.
Johnsson
,
J. J.
Hope
,
P. K.
Lam
,
A.
Dantan
et al, Phys. Rev. A
73
, 023806
(2006
).35.
E. E.
Mikhailov
,
A.
Lezama
,
T. W.
Noel
, and
I.
Novikova
, J. Mod. Opt.
56
, 1985
(2009
).36.
M.
Zhang
,
R. N.
Lanning
,
Z.
Xiao
,
J. P.
Dowling
,
I.
Novikova
, and
E. E.
Mikhailov
, Phys. Rev. A
93
, 013853
(2016
).37.
R. N.
Lanning
,
Z.
Xiao
,
M.
Zhang
,
I.
Novikova
,
E. E.
Mikhailov
, and
J. P.
Dowling
, Phys. Rev. A
96
, 013830
(2017
).38.
C.
Weedbrook
,
S.
Pirandola
,
R.
García-Patrón
,
N. J.
Cerf
,
T. C.
Ralph
,
J. H.
Shapiro
, and
S.
Lloyd
, Rev. Mod. Phys.
84
, 621
(2012
).39.
Action of any Gaussian process can be represented by a symplectic matrix S with the transformations
. These transformations connect the moments of the states before and after the process, and since the moments completely describe the state these transformations completely describe the state after the process.
40.
E.
Bolduc
,
N.
Bent
,
E.
Santamato
,
E.
Karimi
, and
R. W.
Boyd
, Opt. Lett.
38
, 3546
(2013
).41.
M.
Zhang
,
M. A.
Guidry
,
R. N.
Lanning
,
Z.
Xiao
,
J. P.
Dowling
,
I.
Novikova
, and
E. E.
Mikhailov
, Phys. Rev. A
96
, 013835
(2017
).42.
T.
Horrom
,
G.
Romanov
,
I.
Novikova
, and
E. E.
Mikhailov
, J. Mod. Opt.
60
, 43
(2013
).43.
T.
Horrom
,
I.
Novikova
, and
E. E.
Mikhailov
, J. Phys. B
45
, 124015
(2012
).2023 Published under an exclusive license by the AVS.
2023
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