Spatial interference between pairs of disjoint optical paths with a single chaotic source

0.85 MB·

by  Michele Cassano

Spatial interference between pairs of optical paths
with a chaotic source
Michele Cassano
6
1
Dipartimento Interateneo di Fisica, Universitá degli Studi di Bari, 70100 Bari, Italy
0
2

n Milena D’Angelo
a
J Dipartimento Interateneo di Fisica, Universitá degli Studi di Bari, 70100 Bari, Italy

1 Instituto Nazionale Di Fisica Nucleare, sez. di Bari, 70100 Bari, Italy
2

] Augusto Garuccio
h
p Dipartimento Interateneo di Fisica, Universitá degli Studi di Bari, 70100 Bari, Italy
-
t Instituto Nazionale Di Fisica Nucleare, sez. di Bari, 70100 Bari, Italy
n
a
u
Tao Peng
q
[
Department of Physics, University of Maryland, Baltimore County, Baltimore, MD

2 21250, USA
v
5
4
Yanhua Shih
0
5 Department of Physics, University of Maryland, Baltimore County, Baltimore, MD
0
21250, USA
.
1
0
6 Vincenzo Tamma
1
: Institut für Quantenphysik and Center for Integrated Quantum Science and
v
Technology (IQST), Universität Ulm, D-89069 Ulm, Germany
i
X
E-mail: [email protected]
r
a
Abstract. We demonstrate a second-order spatial interference effect with a chaotic
source. This phenomenon arises from the spatial indistinguishability between pairs
of optical paths emerging from correlation measurements in the photon-number
fluctuations, providing a deeper understanding of the physics of multi-photon
interference and spatial coherence. We show how this effect can be used to simulate a
controlled-NOT gate, as well as arbitrary-order on-demand entanglement correlations
in more general interferometric networks.
PACS numbers: 42.50.-p,42.50.Ar
Spatial interference between pairs of optical paths with a chaotic source 2
1. Introduction
The fundamental physics of temporal and spatial coherence and interference with
thermal light is at the very heart of the experiments performed in the mid-1950s
by Hanbury Brown and Twiss. These experiments triggered the development of the
quantum optics field [1, 2, 3, 4]. Indeed, in the last 60 years, numerous fundamental
studies [5, 6, 7, 8, 9, 10] of the physics of multi-photon interference have been performed,
with applications in imaging [11, 12, 13, 14, 15, 16, 17], quantum information processing
[18, 19, 20, 21, 22], metrology [23, 24], N-photon tomography, entanglement generation
[9, 25] and entanglement simulation [26, 27, 28, 29, 30].
Recently, a second-order correlation phenomenon based on the interference between
pairs of optical paths has been demonstrated [10] in the temporal domain with a
chaotic source, providing a deeper fundamental understanding of the physics of second-
order coherence beyond the Hanbury Brown and Twiss effect. As shown in [10], this
phenomena can be also used to simulate a controlled-NOT (CNOT) gate operation
[31, 32, 33].
In this paper, we demonstrate theoretically a second-order interference effect in
the spatial domain analogous to the one demonstrated in the temporal domain [10].
The proposed interferometer (Fig. 1) is characterized by two identical two-pinhole
masks illuminated by chaotic light, and placed in two separate arms. The distance
between the two pinholes in both masks is larger than the transverse coherence length
of the chaotic source. Hence, no first-order interference exists behind each mask.
Nonetheless, we show that second-order interference between two pairs of spatially
indistinguishable optical paths can be observed by performing correlation measurements
at the interferometer output (sect. 2). Furthermore we demonstrate that the proposed
interferencephenomenoncanbeused(Fig. 2)tosimulateaparticularclassofcontrolled-
unitary gates, including a CNOT gate. We also propose (sect. 3) a modified setup (Fig.
3) that can be used for the implementation of higher-order interferometric networks.
2. Basic interference effect
In the proposed interferometer (Fig. 1), chaotic light is split by a balanced non-
polarizing beam splitter. Two identical two-pinhole masks are placed, in the output
arms of the beam splitter, at the same distance z from the source. The pinholes at the
positions are indicated as 1,2 for the upper mask and 1(cid:48),2(cid:48) for the lower mask. The
light transmitted by the masks reaches two point-like detectors, D and D , placed at
C T
the same distance f from the masks. A correlation measurement in the fluctuation of
the number of photons is finally performed.
Let us start with the general expression of the second-order correlation function
[34, 17]
G(2)(x ,x ) ∝ (cid:104)n(x )n(x )(cid:105) = (cid:104)n(x )(cid:105)(cid:104)n(x )(cid:105)+(cid:104)∆n(x )∆n(x )(cid:105), (1)
i j i j i j i j
with i = 1,2 and j = 1(cid:48),2(cid:48), where n represents the photon number and ∆n the
Spatial interference between pairs of optical paths with a chaotic source 3
x
C
f
x D
M C
C
2
1
z
BS
<Δn(x ) Δn(x )>
C T
z
1’
x
S
2’
S T
x
2a M
f
D
T
x
T
Figure 1. Scheme of the proposed interferometer for observing spatial second-order
interference between indistinguishable pairs of optical paths. The light, emitted from
a one-dim chaotic sources of size 2a, after being split on a balanced non-polarizing
beam splitter propagates through two identical two-pinhole masks placed at the same
distance z from the source and reaches two point-like detectors, D and D , placed
C T
at distance f from the masks. A correlation measurement between the fluctuations of
the number of photons detected by D and D is performed.
C T
photon-number fluctuation around the mean. We consider now the case of a quasi-
monochromatic chaotic source, which, for simplicity, is also assumed to be 1-dim, of
length 2a, and H-polarized. The input chaotic light state is described by [34, 35]
(cid:34) (cid:35)
(cid:90)
(cid:89) (cid:79)
ρˆ = d2α P({α }) |α (cid:105) (cid:104)α |,
H κ,H κ,H κ,H S κ,H
κ κ
with the Glauber-Sudarshan probability distribution [3, 36]
(cid:32) (cid:33)
(cid:89) 1 |α |2
κ,H
P({α }) = exp − ,
κ,H
π(cid:104)n (cid:105) (cid:104)n (cid:105)
κ κ
κ
where α are H-polarized coherent states, in the mode κ associated with the x
κ,H
component of the transverse wave vector, and (cid:104)n (cid:105) is the corresponding average photon
κ
number, which is assumed for simplicity to be constant [35]. In this case Eq. (1) reduces
Spatial interference between pairs of optical paths with a chaotic source 4
to [13]
G(2)(x ,x ) = G(1)(x )G(1)(x )+|G(1)(x ,x )|2,
i j i j i j
with G(1) the first-order correlation function. Therefore G(2)(x ,x ) depends on two
i j
contributions: the first one, G(1)(x )G(1)(x ), is a constant background; the second one,
i j
|G(1)(x ,x )|2, istheinterferenceterm. Thebackgroundcanberemovedbyperforminga
i j
correlation measurement between the fluctuations of the number of photons [17]. Due to
the position (or momentum) correlation at the heart of the ghost imaging [13, 14, 15, 16]
(or the HBT [1, 2]) effect, the outcome of this measurement [17, 34]
(cid:104)∆n(x )∆n(x )(cid:105) ∝ (cid:12)(cid:12)G(1)(x ,x )(cid:12)(cid:12)2
i j i j
isdifferentfromzeroforallthepossiblepairsofpaths(i,j) = (1,1(cid:48)),(2,2(cid:48)),(1,2(cid:48)),(2,1(cid:48)),
providedtherelativedistancebetweeneachpairofpinholesissmallerthanthetransverse
coherence length of the source (l ), which is: |x −x | (cid:28) l . Now an interesting result
coh i j coh
comes out by working in the hypothesis that
(A) the pinholes, in each mask, are separated by a distance greater than the transverse
coherence length of the source, namely
|x −x | (cid:29) l |x −x | (cid:29) l ; (2)
1 2(cid:48) coh 1(cid:48) 2 coh
(B) the corresponding pairs of pinholes at the two masks are within the transverse
coherence length, which is
|x −x | (cid:28) l |x −x | (cid:28) l . (3)
1 1(cid:48) coh 2 2(cid:48) coh
In fact, in this case, only the two pairs of paths (1,1(cid:48)) and (2,2(cid:48)) (each one associated
with two paths spatially coherent with respect to each other), contribute to the
correlation, while no contribution comes from the two pairs of paths (1,2(cid:48)) and (2,1(cid:48)),
namely
(cid:104)∆n(x )∆n(x )(cid:105) (cid:54)= 0 ⇔ (i,j) = (1,1(cid:48)),(2,2(cid:48)).
i j
Whathappensifweperformcorrelationmeasurementsafterthetwo-pinholemasks?
Intuitively one may expect that the only two possible contributions, associated with the
two alternatives (1,1(cid:48)) and (2,2(cid:48)), would add incoherently. However, as we shall show,
they give rise to a counter-intuitive interference effect. To demonstrate this result we
evaluate the correlation between the photon-number fluctuations ∆n(x ) and ∆n(x )
C T
measured at equal detection times by the detectors D and D , respectively, placed at
C T
the transverse position x and x , behind the two-pinhole masks, namely
C T
(cid:104)∆n(x )∆n(x )(cid:105) ∝ (cid:12)(cid:12)G(1)(x ,x )(cid:12)(cid:12)2. (4)
C T C T
Here,
G(1)(x ,x ) = Tr[ρˆ Eˆ−(x )Eˆ+(x )] (5)
C T H C C T T
is the first-order correlation function calculated at x ,x , where Eˆ+(x ) and Eˆ−(x )
C T d d d d
are, respectively, the positive and negative frequency part of the electric field operator
Spatial interference between pairs of optical paths with a chaotic source 5
at the position x , namely
d
(cid:114)(cid:126)ω (cid:90)
Eˆ(+)(x ) = ı dκ e−ıωtaˆ (κ), (6)
d d 2(cid:15) d
0
where ω is the detected frequency, t is the detection time and aˆ (κ) is the annihilation
d
operator associated with the transverse mode κ, at the detector D , with d = C,T. By
d
propagating the field from the detector to the source plane, we can write
aˆ (κ) = g{κ;S,x }aˆ (κ), (7)
d d s
where g{κ;S,x } is the Green’s function that describes the propagation of the mode
d
κ from the source S to the detector D , placed in x , and aˆ (κ) is the annihilation
d d S
operator at the source S associated with the mode κ.
As demonstrated in Appendix B, in the paraxial approximation and by using the
conditions given in Eqs. (2) and (3), Eq. (4) becomes
(cid:104)∆n(x )∆n(x )(cid:105) ∝ (cid:12)(cid:12)G(1) (x ,x )+G(1) (x ,x )(cid:12)(cid:12)2, (8)
C T 1,1(cid:48) C T 2,2(cid:48) C T
where G(1) and G(1) indicate the contributions to the correlation measurement coming
1,1 2,2(cid:48)
from the two pairs of paths (1,1(cid:48)) and (2,2(cid:48)), respectively; in fact, we have
G(1)(x ,x ) ∝ B∗(x )B (x )FT (cid:8)|A(x )|2(cid:9)[(x −x )/(2al )], (9)
i,j C T i C j T s i j coh
where B∗(x ) and B (x ) are two phase factors (see Eq. (A.8)), FT {|A(x )|2}[χ]
i C j T s
represents the Fourier transform, calculated in χ, of the source intensity profile, which
we have assumed to be constant and different from zero only over an extension 2a (see
Eq. (B.7)) so that l = λz/(2a). Hence, on the one hand, due to the conditions in Eq.
coh
(2), the contributions G(1) and G(1) are null; on the other hand, the conditions in Eq.
1,2(cid:48) 2,1(cid:48)
(3) guarantee that the contribution G(1) with i = 1,2, are maximum:
i,i(cid:48)
G(1)(x ,x ) ∝ aB∗(x )B (x ),
i,i(cid:48) C T i C i(cid:48) T
G(1) (x ,x ) = G(1) (x ,x ) = 0. (10)
1,2(cid:48) C T 2,1(cid:48) C T
Therefore the correlation between the fluctuations of the number of photons depends on
the interference between the first-order correlation functions associated with the pairs of
paths (1,1(cid:48)) and (2,2(cid:48)). In fact, these two pairs of paths are indistinguishable even if the
corresponding pairs of pinholes are separated much further than the coherence length
of the source. Such indistinguishability arises from the fact that each contribution G(1)
i,j
(Eq. (9)), for each of the four possible pairs of paths (i,j) = (1,1’), (2,2’), (1,2’), (2,1’),
depends only on the relative distance x −x between the point-like pinholes i and j as
i j
compared to the transverse coherence length, and the paths (1,1’) and (2,2’) are the only
ones corresponding to pinholes having a relative distance smaller than the transverse
coherence length (conditions (2) and (3)). Therefore only these two pairs of paths have
non-zero contribution (Eq. (10)) and interfere (Eq. (8)).
As shown in Appendix B, Eq. (8) can be written as
(cid:104)∆n(xC)∆n(xT)(cid:105) ∝ (cid:12)(cid:12)1+eıφ(x1,x2,x1(cid:48),x2(cid:48),xC,xT)(cid:12)(cid:12)2, (11)
Spatial interference between pairs of optical paths with a chaotic source 6
where
ω ω
φ(x ,x ,x ,x ,x ,x ) = (x2 +x2 −x2 −x2)+ (x x −x x −x x +x x ),
1 2 1(cid:48) 2(cid:48) C T 2ch 1 2(cid:48) 1(cid:48) 2 cf C 2 T 2(cid:48) C 1 T 1(cid:48)
(12)
with 1/h = 1/z +1/f. In the condition x = x , Eq. (12) reduces to
C T
ω
φ (x ,x ,x ,x ,x ) = [(x2 −x2)−(x2 −x2 ))]
xC=xT 1 2 1(cid:48) 2(cid:48) C 2ch 1 2 1(cid:48) 2(cid:48)
x ω
C
− [(x −x )−(x −x )].
1 2 1(cid:48) 2(cid:48)
cf
Interestingly, the phase φ (x ,x ,x ,x ,x ) consists of two terms: each term
xC=xT 1 2 1(cid:48) 2(cid:48) C
depends on the difference between the relative positions (the values of the positions
are squared in the first term) of the pinholes in each mask; furthermore, the second
term depends on the detector position x = x . Therefore, the measured interference
C T
provides simultaneous information about the two relative positions of the pinholes in
the two masks. For example, this information can be retrieved by varying the positions
of both detectors. This would not be possible by measuring the intensity separately at
the two interferometer output ports, since no first-order interference can be observed.
This effect is of interest for imaging and metrology applications.
3. Quantum gate simulation
In this section we demonstrate that the novel physical effect described by Eq. (8) can
be exploited for simulating a controlled-U gate, with U described by the matrix
φ φ
(cid:32) (cid:33)
0 eıφ
U ..= . (13)
φ eıφ 0
Let us start by describing a genuine controlled-U gate. Given two-qubit input
φ
states |φ (cid:105) |φ (cid:105) , where
C C T T
|φ (cid:105) ..= cosφ |H(cid:105) +sinφ |V(cid:105) ,
C C C C C C
and
|φ (cid:105) ..= cosφ |H(cid:105) +sinφ |V(cid:105) ,
T T T T T T
the controlled-U gate operates on the input states, by giving the following output
φ
entangled state [18]
|ψ(cid:105) = cosφ |H(cid:105) |φ (cid:105) +eıφsinφ |V(cid:105) |φ(F)(cid:105) ,
C C T T C C T T
where
|φ(F)(cid:105) ..= sinφ |H(cid:105) +cosφ |V(cid:105) .
T T T T T T
The polarization-dependent joint detection probability associated with the state |ψ(cid:105) is
P ..= |(cid:104)θ ,θ |ψ(cid:105)|2 = (cid:12)(cid:12)cosφ cosθ cos(φ −θ )+eıφsinφ sinθ sin(φ +θ )(cid:12)(cid:12)2.
U C T C C T T C C T T
φ
(14)
Spatial interference between pairs of optical paths with a chaotic source 7
In particular, for φ = 0 the controlled-U gate reduces to a CNOT gate [18] and the
φ
polarization-dependent joint detection probability in Eq. (14) becomes
P ..= |(cid:104)θ ,θ |ψ(cid:105)|2 = |cosφ cosθ cos(φ −θ )+sinφ sinθ sin(φ +θ )|2.
CNOT C T C C T T C C T T
(15)
In order to simulate a controlled-U gate we propose the interferometer in Fig. 2
φ
which is identical to the one in Fig. 1, with the only addition of polarization optics.
The interferometer consists of three parts: the first one prepares the initial polarization
state in the “control” input port C and in the “target” input port T ; the second one
implements polarization transformations along the control and target output channels;
the final part consists of the measurement process.
x
C
θ
C
D
C
C
2
V 1
H
BS
<Δn(x ,θ ) Δn(x , θ )>
C C T T
1’
x
F
S
2’
S T
D

T
2a
θ
T
x
T
Figure 2. Interferometer for the simulation of controlled-U gates. In the first
φ
part of the interferometer, the initial polarization state of the light is prepared. The
second part, from the ports C and T to the ports C and T, respectively, performs a
polarization-dependent transformation. Correlation measurements in the fluctuations
ofthenumberofphotonsareperformedattheinterferometeroutput. R andR are
φC φT
twohalf-waveplatesthatrotatethepolarizationoftheanglesφ andφ ,respectively;
C T
F is a half-wave plate implementing a flip from the horizontal (H) polarization to the
vertical (V) polarization and vice versa; H, V, θ and θ represent the polarization
C T
directions for the corresponding polarizers.
In the first part of the setup the H-polarized chaotic light impinges on a balanced
Spatial interference between pairs of optical paths with a chaotic source 8
non-polarizing beam splitter and then propagates through two half-wave plates R and
φ
C
R .
φ
T
The second part of the setup consists of a “control” path, connecting the ports C
and C, and a “target” path, connecting the ports T and T. Similar to the setup in
Fig. 1, both in the control and in the target paths light goes through identical two-
pinhole masks. However, in the control path, two polarizers oriented along the H and
V directions are placed just before pinholes 1 and 2, respectively, while in the target
path a half-wave plate oriented at π/4 is placed just before the pinhole 2(cid:48).
Let us now describe the detection process. A polarizer, oriented along the direction
θ ..= (cosθ sinθ )T,withd = C,T,isplacedinfrontofeachdetector. Apolarization-
d d d
dependent correlation measurement between the fluctuations of the number of photons
∆n(x ,θ ) and ∆n(x ,θ ), detected, respectively, by D and D , is then performed.
C C T T C T
As shown in Appendix C, if the conditions given in Eqs. (2) and (3) are satisfied,
in paraxial approximation the correlation between the fluctuations of the number of
photons is proportional to the joint detection probability typical of a controlled-U
φ
gate, namely
(cid:12) (cid:12)2
(cid:104)∆n(x ,θ )∆n(x ,θ )(cid:105) ∝ (cid:12)G(1) (x ,θ ,x ,θ )+G(1) (x ,θ ,x ,θ )(cid:12) ∝ P , (16)
C C T T (cid:12) 1,1(cid:48) C C T T 2,2(cid:48) C C T T (cid:12) Uφ
with φ(x ,x ,x ,x ,x ,x ) as defined in Eq. (12). Here, we find again two interfering
1 2 1(cid:48) 2(cid:48) C T
contributions G(1) and G(1) associated with the propagation through the two pairs of
1,1(cid:48) 2,2(cid:48)
pinholes (1,1(cid:48)) and (2,2(cid:48)), respectively. However, different from the setup in Fig. 1,
these interfering contributions are now polarization dependent. In particular:
(A) the control path 1, associated with the polarization mode H, is correlated with the
target path 1’, where the polarization is not modified;
(B) the control path 2, associated with the polarization mode V, is correlated with the
target path 2’, where the polarization is flipped from H to V, and vice versa.
Interestingly, the resulting second-order interference pattern is proportional to the
probability P associated with a controlled-U gate, with φ defined in Eq. (12). In
U φ
φ
particular, when
|φ(x ,x ,x ,x ,x ,x )| (cid:28) 1, (17)
1 2 1(cid:48) 2(cid:48) C T
Eq. (16) reduces to
(cid:104)∆n(x ,θ )∆n(x ,θ )(cid:105) ∝ P ,
C C T T CNOT
with P defined in Eq. (15), leading to the simulation of a CNOT gate operation.
CNOT
The condition reported in Eq. (17) can be experimentally obtained, for example, by
performing the detections at equal positons x = x in the condition
C T
2l ah
|x2 −x2| (cid:28) coh ,
i(cid:48) i πz
for i = 1,2, with l = λz/(2a).
coh
Spatial interference between pairs of optical paths with a chaotic source 9
4. Toward higher order correlations
The basic idea described in the previous section can be generalized in order to simulate
higher-order entanglement correlations. To this end, we modify the setup of Fig. 2 by
replacing the two masks with two Mach-Zehnder-like interferometer, as shown in Fig.
3. It is worth noticing that this modified setup is the spatial equivalent of the setup
proposed in [10] in the temporal domain. In fact, by employing the technique proposed
in [10], several such interferometers can be combined to simulate higher-order networks.
In this perspective, it is worth demonstrating that also the interferometer in Fig. 3 can
simulate controlled-U gates and, in particular, a CNOT gate.
φ
δ
c
H
1
PBS PBS
x
2 C
C
V
θ
C D
C
S <Δn(x ,θ ) Δn(x , θ )>
BS C C T T

2a
D
F θ T
T

2’
T x
T
BS BS
1’
δ
T
Figure 3. ModifiedversionoftheinterferometerreportedinFig. 2forthesimulation
of controlled-U gates. This interferometric configuration can be used as building
φ
blocks for the implementation of more general higher-order networks [10].
As mentioned above, the two masks of the scheme in Fig. 2 are substituted by
two interferometers: the “control” interferometer, connecting the input port C with the
output port C, and the “target” interferometer, connecting the input port T with the
output port T. The control interferometer consists of two polarizing beam splitters and
two tiltable mirrors. The mirror in one path is tilted by an angle δ with respect to
C
the alignment position (δ = 0) corresponding to the two interferometric paths 1 and 2
C
to have equal lengths. Hence the tilting angle δ determines the difference between the
C
Spatial interference between pairs of optical paths with a chaotic source 10
optical paths 1 and 2, and the PBS plays the role of the polarizers placed on the mask
in arm C of the setup reported in Fig. 2. The target interferometer is a standard Mach-
Zehnder interferometer, with a half wave plate F, analogously to the one employed in
the setup in Fig. 2 along the path 2(cid:48). Similar to the conditions (Eqs. (2) and (3))
imposed in the setups of Figs. 1 and 2, the tilting angles δ and δ satisfy, in the small
C T
angle approximation, the conditions
l l
coh coh
|δ | (cid:29) , |δ | (cid:29) , (18)
C T
2z¯ 2z¯
l
coh
|δ −δ | (cid:28) , (19)
C T
2z¯
where z¯ is the longitudinal distance from the mirrors to the detectors. As in Fig. 1, we
shall measure the correlation in the fluctuations of the number of photons detected by
D and D , which are placed at the same longitudinal distance z from the source.
C T
As demonstrated in Appendix E, in the condition
|x −x | (cid:28) l (20)
C T coh
and in the paraxial approximation, the correlation between the photon-number
fluctuations reads
(cid:12) (cid:12)2
(cid:104)∆n(x ,θ )∆n(x ,θ )(cid:105) ∝ (cid:12)G(1) (x ,θ ;x ,θ )+G(1) (x ,θ ;x ,θ )(cid:12) ∝ P , (21)
C C T T (cid:12) 1,1(cid:48) C C T T 2,2(cid:48) C C T T (cid:12) Uφ
where P is defined in Eq. (14), with φ given by the corresponding expression of Eq.
U
φ
(12), namely
2ω
(cid:2) (cid:3)
φ(x ,x ,δ ,δ ) = z¯2(δ2 −δ2)+z¯(x δ −x δ ) .
C T C T cz C T C C T T
Hence, similar to the setup in Fig. 2, the interference between the two pairs of paths
(1,1(cid:48)) and (2,2(cid:48)) leads to the simulation of a controlled-U operation, with U defined
φ φ
in Eq. (13). Also in this case the operation of a CNOT gate can be simulated if
|φ(x ,x ,δ ,δ )| (cid:28) 1,
C T C T
which can be achieved experimentally by choosing
l a
|δ2 −δ2| (cid:28) coh ,
C T 2z¯2π
and
l a
coh
|x δ −x δ | (cid:28) ,
C C T T
2z¯π
with l = λz/(2a).
coh
5. Discussions
We have theoretically demonstrated a second-order spatial interference effect emerging
fromcorrelationmeasurementsinthephoton-numberfluctuations. Wehaveshowedthat