Long-baseline interferometry (LBI) is used to reconstruct the image of faint
thermal objects. The image quality, for a given exposure time, is in general
limited by a low signal-to-noise ratio (SNR). We show theoretically that a
significant increase of the SNR, in a LBI, is possible by adding or subtracting
photons to the thermal beam. At low photon counts, photon addition-subtraction
technology strongly enhances the image quality. We have experimentally realized
a nondeterministic physical protocol for photon subtraction. Our theoretical
predictions are supported by experimental results.Comment: 4 pages, 5 figure

Birrittella, Richard

Boyd, Robert

Capron, Barbara

Dowling, Jonathan

Gard, Bryan

Gerry, Christofer

Gerwe, David

Idell, Paul

Koltenbah, Benjamin

Magan-Loiza, O S

Mirhosseini, Mohammad

Parazzoli, Claudio

Rafsanjani, S M Hashemi

English

arXiv

Long-baseline interferometry (LBI) is used to reconstruct the image of faint thermal objects. The image quality, for a given exposure time, is in general limited by a low signal-to-noise ratio (SNR). We show theoretically that a significant increase of the SNR, in a LBI, is possible by adding or subtracting photons to the thermal beam. At low photon counts, photon addition-subtraction technology strongly enhances the image quality. We have experimentally realized a nondeterministic physical protocol for photon subtraction. Our theoretical predictions are supported by experimental results

Parazzoli, Claudio G.

Koltenbah, Benjamin E.

Gerwe, David R.

Idell, Paul S.

Gard, Bryan T.

Hashemi Rafsanjani, S. M.

Magaña-Loaiza, O. S.

Dowling, Jonathan P.

Gerry, Christofer C.

Boyd, Robert W.

Capron, Barbara A.

Caltech Authors - Main

1 
 
Enhanced Thermal Object Imaging  
by Photon Addition or Subtraction  
Claudio G Parazzoli1, Benjamin E. Koltenbah1, David R. Gerwe1, Paul S. Idell1, Bryan T. Gard2, Richard Birrittella3,  
S. M. Hashemi Rafsanjani4, Mohammad Mirhosseini4, O. S. Magana-Loaiza4, Jonathan P. Dowling2, Christopher C. 
Gerry3, Robert W. Boyd4, Barbara A Capron1 
 
1The Boeing Company, P.O. Box 3707 Seattle, WA. 98124, 2Hearne Institute for Theoretical Physics and Department of Physics &  Astronomy 
Louisiana State University, Baton Rouge LA 70803,  3Lehman College, The City University of New York,  4The University of Rochester, 
Rochester, NY 14627 
 
Abstract: Long-baseline interferometry (LBI) is used to reconstruct the image of faint thermal 
objects. The image quality, for a given exposure time, is in general limited by a low signal-to-noise 
ratio (SNR). We show theoretically that a significant increase of the SNR, in a LBI, is possible by 
adding or subtracting photons to the thermal beam. At low photon counts, photon addition-
subtraction technology strongly enhances the image quality. We have experimentally realized a 
nondeterministic physical protocol for photon subtraction. Our theoretical predictions are supported 
by experimental results. 
 
Imaging of faint objects, illuminated by thermal 
light from natural sources, is a challenging problem in 
scientific, civilian, and military arenas. The problem 
becomes more acute for short exposure times, or 
equivalently, low average photon occupation number. 
The principles of image reconstruction were first 
suggested by Michelson1 and later codified in the van 
Cittert-Zernike theorem2. The theorem states that the 
source intensity distribution, I(x,y), is the Fourier 
transform of the equal time complex degree of 
coherence j(u,v). 
           dvduvujdvduevujyxI
vuvu
vyuxki


,,
,/,,
(1) 
In a long-baseline interferometer (LBI) the thermal 
photons from a distant source are collected by two 
apertures, A1 and A2. A1 is at a fixed position P(0,0) 
in the u-v plane, and A2 is at P(u,v). As A2 scans the 
u-v plane, the difference of the accumulated phase , 
along the paths from the source to A1 and A2, is 
modulated.  
When the thermal photons from A1 and A2 are 
interfered on a beam splitter, the modulation of  
generates the fringe pattern j(u,v). The SNR of the 
fringe pattern j(u,v), SNRj , is related to SNRTh of the 
average occupation number, n  . 
SNRTh = n / n (1+ n) , and for 1n ,
SNRTh Þ n .  At low n  SNRj decreases, and to 
obtain high resolution images an increasing exposure 
time is required. In this paper we address the problem 
of increasing SNRj above the thermal limit of 
SNR j @ n and thus reducing the exposure times  
for given image resolution. 
 The problem of increasing SNRj above its 
thermal limit is closely related to minimization of the 
standard deviation
  of the phase difference  
between the two arms of a Mach-Zehnder 
interferometer. A good summary of recent work on 
this subject is found in Ref. [3].  
We studied the effects of photon addition or 
subtraction technology (PAST) on one of the output 
ports of the LBI beam splitter where the fringes are 
generated, as shown in Figure 1. The PAST idea was 
first suggested, in a different context, in Ref. [4].  To 
implement the PAST concept, the state density 
matrix ˆ is formally modified by the addition or 
subtraction process as5 
 
       
        0],ˆˆˆ[Tr,ˆˆˆˆ
0,]ˆˆˆ[Tr,ˆˆˆˆ
††
††




maaNNaa
maaNNaa
mmmm
mmmm

  
              (2) 
We determined theoretically that PAST significantly 
improves SNRj and consequently the resolution of the 
image. We refer to the formulation of PAST, given in 
Eq. (2), as the mathematical protocol (MP). 
In the theoretical approach, we considered 
only the single-mode case. As we discuss below, this 
is not a restrictive assumption for 1n . The single-
mode thermal probability distribution of photons, a 
Bose-Einstein (B-E) distribution, is     11/  nn nnnp  
For a rectangular spectral distribution, all the modes 
share the same average occupation number n , the 
distribution is [2], 
   
 
nnn
n
n
nnp  


 )/1()/1(
!!1
!1
,, 


 
  
2 
 
It can be shown that  nnp ,,  approaches the B-E 
distribution for 1n  and arbitrary . 
This behavior justifies using the single mode case in 
our study where 1n .  
The LBI simulation geometry is schematized in 
Fig. 1. It is equivalent to a Mach-Zehnder 
Interferometer (MZI).  
A single-mode thermal beam of infinite time 
duration, with an average occupation number of n2 , 
enters one port of a 50-50 beam splitter 
 
FIG. 1 Schematic used in the computation to emulate 
the LBI 
 and the other port is open to vacuum. The phase 
shift  in the upper arm of the MZI, represents the 
phase accumulation difference between the paths from 
the object to A1 and A2. The entities within the yellow 
box in Figure 1 emulate the thermal object whose 
output is collected by two separated apertures in the 
LBI. The input density matrix 
VacThIn  ˆˆ 

 has 
been evaluated in terms of the MZI output states 5n  
and 6n  using standard methods, and with the help of 
a quantum simulation package in Mathematica6. The 
resulting density matrix at the output of the second 
balanced beam splitter BS2 is  
 
 
 
 
       
6ˆ56ˆ5ˆ
*
66
*
55
0 0 0
1
0
2ˆ1ˆ2ˆ1ˆ1
,)'(,)(
!!)!()!(
!
21
2
0,0,
21
2
ˆ
kknkkncccc
kkknkn
n
n
n
nn
n
n
kkknkn
n
n
k
n
k
n
n
n
n
n
Out








  






 
with   215 iec   and   216 ieic  .  The 
subscripts refer to modes 5 and 6 in Fig. 3. Next we 
apply the MP described by Eq. (2) and set 

mˆ and

mˆ
to be the density matrices corresponding to the 
addition or subtraction of m photons respectively, 
0m  for addition, 0m  for subtraction. 
By tracing over mode 5 we obtain the photon 
distributions at detector D1, 
66 PandP for photon 
addition and subtraction respectively. 
   
 
  
  
   
 
 
  
   mn
n
n
n
mn
n
n
n
nm
mn
nnPnP
n
n
mnm
n
nnPnP


















6
6
5
6
6
5
12
2
6
6
0
655666
12
2
6
6
0
655666
2cos21
2cos2
!!
!
,,
2cos21
2cos2
!!
!
,




  
    (7) 
The average occupation numbers at D1 for photon 
addition or subtraction are 
     0,2cos212cos2 226 

mnmnN
m
  
     (3a) 
  0,2cos2)1( 26 

mnmN
m
   
      
     (3b) 
Two interesting results emerge from Eqs. (3). 
First, from Eq. (3a) for  = , all the photons are 
collected at D1. This is the standard result for a 
balanced MZI. Secondly, either for photon addition or 
subtraction, the average occupation number is 
augmented. The increase in the average occupation 
number is larger for photon addition than subtraction. 
This is the result of the modified photon distribution 
that happens by adding or subtracting photons. The 
modification of the photon distributions will be 
presented below. 
Finally the fringe SNR’s at D1 are 
  
      
 
 
0,
2cos21
2cos2)1(
SNR
0,
2cos22cos211
2cos2)1(
SNR
2
2
22
22
6
6










m
n
nm
m
nnm
nmm
N
N




     (4) 
3 
 
 In Fig. 2 the Mathematical Protocol (MP) 

6P  probability distributions and SNR are plotted for 
m = 0, 1, 5, 10 and 5.0n .  
FIG. 2 MP Probability distributions and SNR at 
detector D1, see Fig. 3, for photon addition  
The m = 0 B-E distribution is monotonically 
decreasing. For m ≥ 1 (photon addition) the 
distribution acquires a bell-shaped form.   
From the figure it is also obvious that the average 
occupation number increases when m > 0. It does so 
more rapidly with addition rather than subtraction (not 
shown) of photons. The MP photon count SNR grows 
significantly with increasing m values. The benefits 
(not shown) are however weaker in the case of photon 
subtraction. 
The MP results have been applied to the 
simulation of imaging of a galaxy. The  
FIG. 3. MP numerically reconstructed image of a 
galaxy with 5.0n and m=0, 1, 5, 10.  
image reconstruction, as seen in Fig. 3, is based on Eq. 
(1) where the SNR is given in Eq. (4).  The results 
indicate a clear improvement from m = 0 to m = 10. 
Several features of the galaxy that are unresolved at m 
= 0, become much more discernable at m = 10.  
 
The MP used here is a mathematical abstraction. We 
need to devise a physical protocol (PP) that performs 
either photon subtraction or addition. The simplest PP 
is based on photon subtraction via a beam splitter and 
post-selection. We refer to it as PP-BS. In Fig. 4 we 
show the PP-BS schematic.  
FIG. 4 Schematic for the simulation of PP-BS. BS4, 
BS5, BS6 are the beam splitters simulating the 
detector losses. 
  BS3 occasionally performs photon subtraction, and 
the D3 detector performs post-selection. We included 
losses, via beam splitters BS4, BS5, BS6 with 
transmission T1, T2 and T3 respectively, to accurately 
emulate the experimental results. We analyzed two 
cases: a) post-selection measurement in which m 
photons are subtracted; b) “click” detection, where any 
number, larger than zero, of photons are subtracted. 
We use the same MP methodology to study the PS-BS 
case, and briefly summarize the results below. The 
probability of success for m subtracted photons and 
the photon probability distribution are, respectively
 
    
    
0,
2cos121
2cos12
)(
12
3
2
3 





m
nTT
nTT
Tp
m
m
m

   
     
  )!()2/cos())((21
)2/cos()1(21)2/cos(
!)!(
2
12
331
12
3
2
114
nmTTTTn
TTnTTn
nm
nP
nm
mn
n








 
In the PP-BS with m = –1 case, the distribution shift is 
mitigated by the presence of losses. In the absence of 
losses PP-BS ⇨ MP (for m=-1), hence the relevance 
of mitigating the losses. Finally the click detection 
distribution and SNR (not shown) are quite similar to 
the PP-BS for lown . 
The corresponding PP-BS SNR are 
   
 
     
    
0,
2cos121
2cos21
SNR
2cos21
2cos2
SNR
2
31
2
1
2
1
2
1
0
14
14






m
TTTTn
TTnm
TTn
TTn
mN
N




 
               (4) 
4 
 
There is a tradeoff between SNR and the probability 
of success, where the larger the T the larger the SNR 
at the expense of a very low probability of success. In 
the experiments we choose T ≈ 0.8.  This rules out the 
use of PP-BS for practical applications; it is, however, 
of great relevance to experimentally verify that, if 
photon subtraction could be performed in a 
deterministic process, it does improve the SNR of a 
thermal beam. 
 A corroborating experimental effort was 
performed at the University of Rochester and a 
succinct summary is given here. We use a CW laser 
whose light is passed through a rotating ground glass 
to create thermal statistics, coupled into a single-mode 
fiber (SMF) and then into an MZI.  One of the output 
ports of the MZI is coupled to a multimode fiber 
(MMF) and onto an avalanche photo diode (APD) 
detector.  The other output port leads to a half-wave 
plate (HWP) and polarizing beam splitter (PBS) to 
implement photon subtraction before terminating in 
two APDs.  The full detailed description and analysis 
of the experiment can be found in7.  
We determine the photon statistics from the 
APD measurements. The thermal source has a 
coherence time of roughly one microsecond (adjusted 
to match that of the laser) whereas the APD dead time 
is significantly shorter at about 50 ns. Thus, for a 
relatively small (< 4) number of photons in one source 
coherence time, the number of APD pulses 
approximates well the true number of photons in a 
coherence time of the laser. As described above, the 
detector quantum efficiency can be taken account of in 
the modeling by adding another beam splitter to the 
path.  
We generated the MZI fringes and evaluated 
the SNR.  The results are shown in Fig. 5 which shows 
1A. A.  Michelson, Phil. Mag. 30, 1 (1890) and Astrophys. J. 51, 
27 (1920) 
2  L Mandel, E Wolfe, “ Optical Coherence and Quantum Optics”, 
Cambridge University Press, 1995 
3 Y Ben-Aryeh, J. Opt. Soc. Am. B Vol. 29, No 10 (2012) 2754 
and  J. Dunningham & T. Kim (2006): Journal of Modern Optics, 
53:4, 557-571 
4 G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991). 
SNR as a function of MZI phase . These results agree 
with the predictions of Eq. (5).  The agreement  
FIG. 5 Comparison of theoretical predictions (lines) 
and experimental results (dots) for SNR as function of 
in degrees. Red, no photon subtraction, blue and 
greed one and two photons subtracted. With 
permission from [7] 
 
between theory and experiment is good and fully 
validates our conjecture that photon subtraction and, 
by induction, photon addition, lead to significantly 
improved SNR for single mode thermal beams.  
 We are working on a deterministic PP based 
on the injection of excited atoms in the thermal 
photons beam. A similar approach for photon 
subtraction and possibly photon addition is discussed 
in8. Here a photon is deterministically subtracted from 
a photon beam by a nearby atom due to destructive 
interference resulting in a Raman transfer of the atom 
from the ground state.  
  
Acknowledgements: This research was developed with 
funding from the Defense Advanced Research Project 
Agency (DARPA) and by Boeing IR&D. The views, 
opinions, and/or findings expressed are those of the 
authors and should not be interpreted as representing 
the official views or policies of the Department of 
Defense or the U.S. Government. Approved for Public 
Release, Distribution Unlimited 
5 D Braun, P Jian, O Pinel, N Treps “ Precision measurements with 
photon-subtracted or photon-added Gaussian States” 
arXiv:1403.1397v1 [quant-ph] 6 March 2014 
6 www.wolfram.com/ 
7 S. M. Hashemi Rafsanjani, et al. ”Enhanc ed 
Phase Estimation by Photon Subtraction”, to be 
published. 
8 S Rosenblum et al. Nature Photonics Vol. 10, January 2016 
                                                             


Enhanced Thermal Object Imaging by Photon Addition or Subtraction

Enhanced Thermal Object Imaging by Photon Addition or Subtraction

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