It is shown that a linear superposition of two macroscopically
distinguishable optical coherent states can be generated using a single photon
source and simple all-optical operations. Weak squeezing on a single photon,
beam mixing with an auxiliary coherent state, and photon detecting with
imperfect threshold detectors are enough to generate a coherent state
superposition in a free propagating optical field with a large coherent
amplitude ($\alpha>2$) and high fidelity ($F>0.99$). In contrast to all
previous schemes to generate such a state, our scheme does not need photon
number resolving measurements nor Kerr-type nonlinear interactions.
Furthermore, it is robust to detection inefficiency and exhibits some
resilience to photon production inefficiency.Comment: Some important new results added, to appear in Phys.Rev.A (Rapid
  Communication

Jeong, H.

Kim, M. S.

Lund, A. P.

Ralph, T. C.

English

arXiv

A. P. Lund

H. Jeong

T. C. Ralph

M. S. Kim

Crossref

Conditional production of superpositions of coherent states with inefficient photon detection

It is shown that a linear superposition of two macroscopically distinguishable optical coherent states can be generated using a single photon source and simple all-optical operations. Weak squeezing on a single photon, beam mixing with an auxiliary coherent state, and photon detecting with imperfect threshold detectors are enough to generate a coherent state superposition in a free propagating optical field with a large coherent amplitude (alpha>2) and high fidelity (F>0.99). In contrast to all previous schemes to generate such a state, our scheme does not need photon number resolving measurements nor Kerr-type nonlinear interactions. Furthermore, it is robust to detection inefficiency and exhibits some resilience to photon production inefficiency

University of Queensland eSpace

Conditional production of superpositions of coherent states with inefficient photon detection
A. P. Lund,1 H. Jeong,1 T. C. Ralph,1 and M. S. Kim2
1Center for Quantum Computer Technology, Department of Physics, University of Queensland, St Lucia, Qld 4072, Australia
2School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom
(Received 3 January 2004; revised manuscript received 26 March 2004; published 17 August 2004)
It is shown that a linear superposition of two macroscopically distinguishable optical coherent states can be
generated using a single photon source and simple all-optical operations. Weak squeezing on a single photon,
beam mixing with an auxiliary coherent state, and photon detecting with imperfect threshold detectors are
enough to generate a coherent state superposition in a free propagating optical field with a large coherent
amplitude sa.2d and high fidelity sF.0.99d. In contrast to all previous schemes to generate such a state, our
scheme does not need photon number resolving measurements nor Kerr-type nonlinear interactions. Further-
more, it is robust to detection inefficiency and exhibits some resilience to photon production inefficiency.
DOI: 10.1103/PhysRevA.70.020101 PACS number(s): 03.65.Ta, 03.65.Ud, 42.50.Xa
Since Schrödinger suggested his famous cat paradox [1],
there has been great interest in generating and observing a
quantum superposition of a macroscopic system. The com-
ponent states composing such a superposition state should be
macroscopically distinguishable, i.e., they should give mac-
roscopically distinct measurement outcomes [2]. Two coher-
ent states can be discriminated by a homodyne measurement,
which can be considered a macroscopic measurement, when
they are well separated in the phase space. Therefore, a su-
perposition of two optical coherent states with sufficiently
large amplitudes and with a p phase difference between
these amplitudes is considered a realization of such a mac-
roscopic superposition.
Recently, the coherent state superposition (CSS) in a free
propagating optical field has been studied for application to
quantum information processing including quantum telepor-
tation [3,4], quantum computation [5–7], entanglement puri-
fication [8] and error correction [9]. In particular, it was
found that quantum computation can be realized using only
linear optics and photon counting, given pre-arranged CSSs
[6,7]. In this framework, initial CSSs of amplitude aø2 are
required for efficient quantum computation with simple op-
tical networks [7].
Unfortunately, it is extremely demanding to generate a
free propagating CSS using current technology. It is well
known that the CSS can be generated from a coherent state
by a nonlinear interaction in a Kerr medium [10]. However,
Kerr nonlinearity of currently available nonlinear media is
extremely small and attenuation is not negligible compared
with the required level to generate a CSS [11].
Some alternative methods have been studied to generate a
superposition of macroscopically distinguishable states based
upon conditional measurements [12,13]. A crucial drawback
of these schemes is that a highly efficient detector which can
discriminate photon numbers is necessary. Cavity quantum
electrodynamics has been studied to enhance nonlinearity
[14]. Some success has been reported in creating such super-
position states within high Q cavities in the microwave [15]
and optical [16] domains. However, all the suggested
schemes for quantum information processing with coherent
states [3–9] require a free propagating CSS.
In this letter, we show that a free propagating optical CSS
can be generated with a single photon source and simple
optical operations. A CSS with a small coherent amplitude
sał1.2d and high fidelity sF.0.99d can be deterministically
generated by squeezing a single photon. A large CSS sa.2d
with high fidelity sF.0.99d can be obtained in a non-
deterministic way from small CSSs. Weak squeezing, beam
mixing with an auxiliary coherent field and photon detecting
with threshold detectors are enough to generate such large
CSSs given a single photon source. Remarkably, neither dis-
crimination of photon numbers nor xs3d nonlinear interac-
tions are required in our scheme. Furthermore, our scheme is
robust to detection inefficiency and somewhat resilient to
photon production inefficiency. In a more general sense, our
examples reveal some previously unrealized relations be-
tween the quantum states of harmonic oscillators: we learn
that the first excited energy eigenstates can be converted to
superpositions of large coherent states by linear operations
and projections.
A CSS can be defined as uCSSwsadl=Nwsadsual
+eiwu−ald, where Nwsad is a normalization factor, ual is a
coherent state of amplitude a, and w is a real local phase
factor. The amplitude a is assumed to be real for simplicity
without loss of generality. In this paper we refer to the mag-
nitude of a as the size of the CSS. Note that CSSs such as
uCSS±sadl=N±sadsual± u−ald are called even and odd CSSs,
respectively, because the even (odd) CSS, uCSS+sadl
suCSS
−
sadld, always contains an even (odd) number of pho-
tons.
An arbitrarily large CSS can be produced out of arbi-
trarily small CSSs using the simple experimental setup de-
picted in Fig. 1. Let us first illustrate this procedure with a
simple example. Suppose that one has a collection of identi-
cal small odd CSSs with known amplitude ai. Two of the
small CSSs are selected and incident onto a 50:50 beam
splitter BS1 as
uCSS
−
saidlauCSS−saidlb→
BS1
u0l fsu˛2ailg + u− ˛2ailgd
− su˛2ail f + u− ˛2ail fdu0lg,
s1d
where the normalization factor is omitted on the right-hand
side. One can then say that if one could condition on detect-
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ing u0lg, a larger CSS with amplitude ˛2ai would be ob-
tained at mode f . An additional step is therefore needed to
unambiguously discriminate between the vacuum and coher-
ent states u±˛2ailg with inefficient detectors. Another 50:50
beam splitter, BS2, mixes the field at mode g and an auxil-
iary coherent state u˛2ailc as
uBS1l f ,gu˛2ailc→
BS2
u0l fsu2ailt1u0lt2 + u0lt1u2ailt2d − su˛2ail f
+ u− ˛2ail fduailt1u− ailt2, s2d
where uBS1l f ,g represents the right-hand side of Eq. (1) and
the normalization factor is omitted. Finally, photodetectors A
and B are set to detect photons at modes t1 and t2. The
remaining state at mode f is selected only when both the
detectors detect any photon(s) at the same time. In this case,
it is obvious that the right-hand side of Eq. (2) is reduced to
larger CSSs. If either of the detectors fails to click, the re-
sulting state is discarded. This process can be recursively
applied until a sufficiently large CSS is obtained. Suppose
that an even CSS with amplitude aø2 is required while the
initial amplitude of small odd CSSs is ai=1/˛2. After a suf-
ficient number of CSSs of the amplitude ˛2ai are obtained,
the second step will be taken with the same experimental
setup and another auxiliary coherent state u2ail. In this sec-
ond stage, larger even CSSs of amplitude 2ai will be gained
from pairs of even CSSs of ˛2ai. Eventually, the amplitude
will reach the required value by four recursive applications
of the process, i.e., a=4ai<2.83.
The process described above can be generalized for arbi-
trarily small CSSs with known amplitudes as shown in Fig.
1. Suppose two small CSSs, uCSSwsadl and uCSSfsbdl, with
amplitudes a and b. The reflectivity r and transmitivity t of
BS1 are set to r=b /˛a2+b2 and t=a /˛a2+b2, where the
action of the beam splitter is represented by
Bˆ a,bsr , tdualaublb= uta+rbl fu−ra+ tblg. The other beam split-
ter BS2 is a 50:50 beam splitter sr= t=1/˛2d regardless of
the conditions and the amplitude g of the auxiliary coherent
field is determined as g=2ab /˛a2+b2. The resulting state
for mode f then becomes uCSSw+fsAdl~ uAl+eisw+fdu−Al,
whose coherent amplitude A=˛a2+b2 is larger than both a
and b. The relative phase of the resulting CSS is the sum of
the relative phases of the input CSSs. The success probability
Pw,fsa ,bd for a single iteration is
Pw,fsa,bd =
e−4a
2b2/sa2+b2dse2a
2b2/sa2+b2d
− 1d2f1 + cossw + fde−2sa
2+b2dg
2s1 + cos we−2a
2
ds1 + cos fe−2b
2
d
,
which is plotted for a number of different combinations in
Fig. 2. The success probability approaches 1/2 as the ampli-
tudes of initial CSSs become large. Note that the probabili-
ties depend on the type of CSSs (odd or even) used. The
effect of detector inefficiency is just to decrease this success
probability.
We now show that a small odd CSS with ał1.2 is sur-
prisingly well approximated by a squeezed single photon.
The single mode squeezing operator is Sˆ srd=e−sr/2dsaˆ2−aˆ†2d,
where r is the squeezing parameter and aˆ is the annihilation
operator. This operator reduces quantum noise of a vacuum
state in the phase quadrature by a factor of e−2r. When the
squeezing operator is applied to a single photon the resultant
state can be expanded in terms of photon number states as
Sˆ srdu1l = o
n=0
‘
stanh rdn
scosh rd3/2
˛s2n + 1d!
2nn!
u2n + 1l . s3d
The state contains only odd photon numbers and has coeffi-
cients decaying exponentially as n increases in a similar
fashion to an odd CSS. The fidelity of this state with an odd
CSS is
FIG. 2. The success probabilities of the CSS-amplifying process
in Fig. 1 for the input fields of two identical odd CSSs (solid line),
two identical even CSSs (dashed line), and even and odd CSSs
(dotted line).
FIG. 1. A schematic of the non-deterministic CSS-amplification
process. See the text for details.
LUND et al. PHYSICAL REVIEW A 70, 020101(R) (2004)
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Fsr,ad = ukCSS
−
saduSsrdu1lu2 =
2a2 expfa2stanh r − 1dg
scosh rd3s1 − expf− 2a2gd
.
Figure 3 shows the maximized fidelity on the y axis plotted
against a range of possible values for a for the desired odd
CSS. Some example values are: F=0.999 99 for amplitude
a=1/2, F=0.9998 for a=1/˛2, and F=0.997 for a=1,
where the maximizing squeezing parameters are r=0.083, r
=0.164, and r=0.313, respectively. First note that for a very
close to zero the fidelity approaches unity. When a→0, r
→0 and hence the squeezing operator Sˆ srd approaches the
identity transformation. An odd CSS with a very close to
zero has a significant contribution from a single photon and
very little from higher odd photon numbers. This is the rea-
son for the high fidelity as a tends to zero. The fidelity re-
mains high for a near zero as one can match the three photon
contribution to the CSS by the squeezing operator whilst still
being able to neglect higher order photon number terms.
Eventually as a increases, higher photon numbers cannot be
matched and so as a tends to infinity, the fidelity tends to
zero.
As the fidelity between a squeezed single photon and an
ideal small CSS is extremely high, it can be conjectured that
a large CSS distilled from squeezed single photons by our
scheme will also be very close to an ideal large CSS. In what
follows, we will show that this conjecture is true for a
ł2.5.
In order to calculate multiple iterations we need to use
numerical techniques. We are using coherent states of some
bounded coherent amplitude and superpositions there of.
Provided the coherent amplitudes are not too large, the most
significant contributions to these states are Fock states of low
number. For computations here the lowest thirty Fock states
were used. This provides a very good approximation for co-
herent states with ał2.5. All 29 possible “click” events are
included for all detectors.
If one wished to create a CSS with a particular a with n
CSS-amplification steps, then initial CSSs with ai=a /˛2n
are required. As the number of steps increases the required ai
decreases. When generating a large CSS out of the squeezed
single photon states the fidelity maximizes for a particular
number of iterations. Figure 4 shows the maximum possible
fidelity using this process in (a) and the number of steps in
(b) against the desired a in the CSSs. For example, four
iterations starting from the initial amplitude ai=1/2 are re-
quired to gain the maximum fidelity F=0.995 for a=2. It is
evident from Fig. 4 that high fidelity, F.0.99, can be ob-
tained up to a=2.5. The error rate for discrimination be-
tween coherent states with a= ±2.5 via a classical measure-
ment (homodyne detection) is only 3310−7.
Current technology does not produce pure single photon
states; the single photon is always in a mixture with the
vacuum as pu0lk0u+ s1− pdu1lk1u, where p is the inefficiency
of the photon production. Hence the squeezed single photon
state will also be a mixture with a squeezed vacuum. How-
ever, an interesting aspect of our scheme is that it may be
somewhat resilient to the photon production inefficiency be-
cause its first iteration purifies the mixed CSSs while ampli-
fying them. The initial input states for the CSS amplification
process from the imperfect single photon source are
ra,b,c = fs1 − pd2uS1lkS1u ^ uS1lkS1u + p2uS0lkS0u ^ uS0lkS0u
+ ps1 − pdsuS0lkS0u ^ uS1lkS1u + uS1lkS1u ^ uS0lkS0udga,b
^ suglkgudc, s4d
where uS0l=Sˆ srdu0l and uS1l=Sˆ srdu1l. Here, the terms with p2
and ps1− pd are undesired error terms where either (or both)
of the single photons is missing. Note that the initial ampli-
tude is required to be small to produce a large CSS with high
fidelity. Provided such a small amplitude, input states inci-
dent onto the beam splitters in our experimental setup con-
tain approximately only two (or slightly more than two) pho-
tons. In such cases the probability of simultaneous clicks at
detectors A and B in Fig. 1 will significantly decrease when
any of the single photons is missing. In other words, the
undesired cases will rarely be selected for the next iteration
of the amplification process. We have obtained numerical
results for the initial amplitude ai=1/2 as follows by the
FIG. 4. (a) The maximum fidelity obtained in our scheme vs the
coherent amplitude. (b) The number of iterations which gives the
maximum fidelity vs the coherent amplitude.
FIG. 3. The fidelity between an odd CSS and squeezed single
photon. The odd CSS is extremely well approximated by the
squeezed single photon for a small coherent amplitude, ał1.2.
CONDITIONAL PRODUCTION OF SUPERPOSITIONS OF PHYSICAL REVIEW A 70, 020101(R) (2004)
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methods that we have already explained. If p=0.4, the fidel-
ity of the initial CSS, which is a mixture with a squeezed
vacuum, is F=0.60 but it will become F=0.89 by the first
iteration. Thus a larger CSS of significantly high fidelity is
produced. If p=0.25 sp=0.05d, the fidelity of the initial CSS
is F=0.750 sF=0.950d but becomes F=0.941 sF=0.990d by
the first iteration. Such purification effects for multiple itera-
tions are beyond the scope of this letter but deserve further
investigations.
In the CSS amplification process, the zero amplitude co-
herent states that occur in the detection modes in Eq. (2) may
be slightly different from zero because of imperfect mode
matching at beam splitters. This will lead to a small prob-
ability of accepting the wrong state. Good mode matching is
a requirement in any linear optical network where one
wishes to measure manifestly quantum mechanical effects.
Highly efficient mode matching of a single photon from
parametric down conversion and a weak coherent state from
an attenuated laser beam at a beam splitter has been experi-
mentally demonstrated using optical fibers [19]. Such tech-
niques could be employed for the implementation of our
scheme.
The dark count rate of photodetectors will affect the fidel-
ity of the CSSs. Currently, highly efficient detectors have
relatively high dark count rates while less efficient detectors
have very low dark count rates [18]. We emphasize again
that our scheme does not require highly efficient detectors
because the inefficiency of the detectors does not affect the
quality of CSSs even though it decreases the success prob-
ability. Silicon avalanche photodiodes operating at the visible
wavelength have relatively high efficiency and a small dark
count rate, which is preferred in our proposal.
The single photons required for our scheme could be gen-
erated conditionally from a down-converter [17]. This is a
xs2d process (like squeezing) and does not require photon
number resolving detection. Once free propagating CSSs are
generated, they can be detected by homodyne measurements,
which can be highly efficient in quantum optics experiments.
Our scheme non-deterministically generates large CSSs.
However, a non-deterministic CSS source is useful enough
for quantum information processing [6,7]. Efficient gate op-
erations for coherent-state quantum computation [7] are
based on teleportation via an entangled coherent state [3,4].
Entangled coherent states can be simply generated from
CSSs using a beam splitter and can be used as off-line re-
sources for quantum computation. We note that such en-
tanglement of macroscopically distinguishable states is per-
haps more closely aligned with Schrödinger’s original
concept [1].
In conclusion, we have proposed a simple all-optical
scheme to generate a linear superposition of macroscopically
distinguishable coherent states in a propagating optical field.
We have found a previously unrealized connection between
squeezed number states and superpositions of coherent states
as well as the interesting additive properties of the latter. In
stark contrast to all previous schemes our scheme requires
neither xs3d nonlinearity nor photon number resolving detec-
tion to generate a macroscopic superposition state.
Note added. The first two authors contributed equally to
this work.
We would like to thank A. Gilchrist for useful comments.
This work was supported by the Australian Research Council
and the University of Queensland Excellence Foundation.
M.S.K. acknowledges the UK Engineering and Physical Sci-
ence Research Councils and the Korea Research Foundation
(2003-070-C00024) for financial support.
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LUND et al. PHYSICAL REVIEW A 70, 020101(R) (2004)
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It is shown that a linear superposition of two macroscopically distinguishable optical coherent states can be generated using a single photon source and simple all-optical operations. Weak squeezing on a single photon, beam mixing with an auxiliary coherent state, and photon detecting with imperfect threshold detectors are enough to generate a coherent state superposition in a free propagating optical field with a large coherent amplitude (a>2) and high fidelity (F>0.99) . In contrast to all previous schemes to generate such a state, our scheme does not need photon number resolving measurements nor Kerr-type nonlinear interactions. Furthermore, it is robust to detection inefficiency and exhibits some resilience to photon production inefficiency.Full Tex

Lund, A.

Ralph, T.

Kim, M.

Griffith Research Online

Conditional Production of Superpositions of Coherent States with Inefficient PhotonDetectionA.P. Lund,1,∗ H. Jeong,1,∗ T.C. Ralph,1 and M.S. Kim21Center for Quantum Computer Technology, Department of Physics,University of Queensland, St Lucia, Qld 4072, Australia2School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom(Dated: February 1, 2008)It is shown that a linear superposition of two macroscopically distinguishable optical coherentstates can be generated using a single photon source and simple all-optical operations. Weak squeez-ing on a single photon, beam mixing with an auxiliary coherent state, and photon detecting withimperfect threshold detectors are enough to generate a coherent state superposition in a free prop-agating optical field with a large coherent amplitude (α > 2) and high fidelity (F > 0.99). Incontrast to all previous schemes to generate such a state, our scheme does not need photon numberresolving measurements nor Kerr-type nonlinear interactions. Furthermore, it is robust to detectioninefficiency and exhibits some resilience to photon production inefficiency.Since Schro¨dinger suggested his famous cat paradox[1], there has been great interest in generating and ob-serving a quantum superposition of a macroscopic sys-tem. The component states composing such a superposi-tion state should be macroscopically distinguishable, i.e.,they should give macroscopically distinct measurementoutcomes [2]. Two coherent states can be discriminatedby a homodyne measurement, which can be considered amacroscopic measurement, when they are well separatedin the phase space. Therefore, a superposition of twooptical coherent states with sufficiently large amplitudesand with a pi phase difference between these amplitudesis considered a realization of such a macroscopic super-position.Recently, the coherent state superposition (CSS) ina free propagating optical field has been studied forapplication to quantum information processing includ-ing quantum teleportation [3, 4], quantum computation[5, 6, 7], entanglement purification [8] and error correc-tion [9]. In particular, it was found that quantum compu-tation can be realized using only linear optics and photoncounting, given pre-arranged CSS’s [6, 7]. In this frame-work, initial CSS’s of amplitude α ≥ 2 are required forefficient quantum computation with simple optical net-works [7].Unfortunately, it is extremely demanding to generate afree propagating CSS using current technology. It is wellknown that the CSS can be generated from a coherentstate by a nonlinear interaction in a Kerr medium [10].However, Kerr nonlinearity of currently available non-linear media is extremely small and attenuation is notnegligible compared with the required level to generate aCSS [11].Some alternative methods have been studied to gen-erate a superposition of macroscopically distinguishable∗The first two authors contributed equally to this work.states based upon conditional measurements [12, 13].A crucial drawback of these schemes is that a highlyefficient detector which can discriminate photon num-bers is necessary. Cavity quantum electrodynamics hasbeen studied to enhance nonlinearity [14]. Some successhas been reported in creating such superposition stateswithin high Q cavities in the microwave [15] and opti-cal [16] domains. However, all the suggested schemesfor quantum information processing with coherent states[3, 4, 5, 6, 7, 8, 9] require a free propagating CSS.In this Letter, we show that a free propagating opticalCSS can be generated with a single photon source andsimple optical operations. A CSS with a small coherentamplitude (α ≤ 1.2) and high fidelity (F > 0.99) canbe deterministically generated by squeezing a single pho-ton. A large CSS (α > 2) with high fidelity (F > 0.99)can be obtained in a non-deterministic way from smallCSS’s. Weak squeezing, beam mixing with an auxiliarycoherent field and photon detecting with threshold de-tectors are enough to generate such large CSS’s given asingle photon source. Remarkably, neither discrimina-tion of photon numbers nor χ(3) nonlinear interactionsare required in our scheme. Furthermore, our scheme isrobust to detection inefficiency and somewhat resilient tophoton production inefficiency. In a more general sense,our examples reveal some previously unrealized relationsbetween the quantum states of harmonic oscillators: welearn that the first excited energy eigenstates can be con-verted to superpositions of large coherent states by linearoperations and projections.A CSS can be defined as |CSSϕ(α)〉 = Nϕ(α)(|α〉 +eiϕ| − α〉), where Nϕ(α) is a normalization factor, |α〉is a coherent state of amplitude α, and ϕ is a real localphase factor. The amplitude α is assumed to be real forsimplicity without loss of generality. In this paper werefer to the magnitude of α as the size of the CSS. Notethat CSS’s such as |CSS±(α)〉 = N±(α)(|α〉 ± | −α〉) arecalled even and odd CSS’s respectively because the even(odd) CSS, |CSS+(α)〉 (|CSS−(α)〉), always contains an2even (odd) number of photons.An arbitrarily large CSS can be produced out of arbi-trarily small CSS’s using the simple experimental set-updepicted in Fig. 1. Let us first illustrate this procedurewith a simple example. Suppose that one has a collectionof identical small odd CSS’s with known amplitude αi.Two of the small CSS’s are selected and incident onto a50:50 beam splitter BS1 as|CSS−(αi)〉a|CSS−(αi)〉b BS1−→ |0〉f(|√2αi〉g + | −√2αi〉g)−(|√2αi〉f + | −√2αi〉f)|0〉g (1)where the normalization factor is omitted on the righthand side. One can then say that if one could condi-tion on detecting |0〉g, a larger CSS with amplitude√2αiwould be obtained at mode f . An additional step istherefore needed to unambiguously discriminate betweenthe vacuum and coherent states |±√2αi〉g with inefficientdetectors. Another 50:50 beam splitter, BS2, mixes thefield at mode g and an auxiliary coherent state |√2αi〉cas|BS1〉f,g|√2αi〉c BS2−→ |0〉f(|2αi〉t1|0〉t2 + |0〉t1|2αi〉t2)−(|√2αi〉f + | −√2αi〉f)|αi〉t1| − αi〉t2 (2)where |BS1〉f,g represents the right hand side of Eq. (1)and the normalization factor is omitted. Finally, pho-todetectors A and B are set to detect photons at modest1 and t2. The remaining state at mode f is selectedonly when both the detectors detect any photon(s) atthe same time. In this case, it is obvious that the righthand side of Eq. (2) is reduced to a larger CSS’s. If ei-ther of the detectors fails to click, the resulting state isdiscarded. This process can be recursively applied un-til a sufficiently large CSS is obtained. Suppose that aneven CSS with amplitude α ≥ 2 is required while theinitial amplitude of small odd CSS’s is αi = 1/√2. Aftera sufficient number of CSS’s of the amplitude√2αi areobtained, the second step will be taken with the sameexperimental set-up and another auxiliary coherent state|2αi〉. In this second stage, larger even CSS’s of am-plitude 2αi will be gained from pairs of even CSS’s of√2αi. Eventually, the amplitude will reach the requiredvalue by four recursive applications of the process, i.e.,α = 4αi ≈ 2.83.The process described above can be generalized for ar-bitrarily small CSS’s with known amplitudes as shownin Fig. 1. Suppose two small CSS’s, |CSSϕ(α)〉 and|CSSφ(β)〉, with amplitudes α and β. The reflectivityr and transmitivity t of BS1 are set to r = β/√α2 + β2and t = α/√α2 + β2, where the action of the beam split-ter is represented by Bˆa,b(r, t)|α〉a|β〉b = |tα+rβ〉f |−rα+tβ〉g. The other beam splitter BS2 is a 50:50 beam split-ter (r = t = 1/√2) regardless of the conditions and theamplitude γ of the auxiliary coherent field is determined2 2(  α + β )a bf gBS2t1 t2BS1γcDetector BDetector A(β)|small CSS      >|large CSS              >(α)|small CSS      >FIG. 1: A schematic of the non-deterministic CSS-amplification process. See text for details.as γ = 2αβ/√α2 + β2. The resulting state for mode fthen becomes |CSSϕ+φ(A)〉 ∝ |A〉+ ei(ϕ+φ)|−A〉, whosecoherent amplitude A =√α2 + β2 is larger than bothα and β. The relative phase of the resulting CSS is thesum of the relative phases of the input CSS’s. The suc-cess probability Pϕ,φ(α, β) for a single iteration isPϕ,φ(α, β) =(1− e−2α2β2α2+β2 )2[1 + cos(ϕ+ φ)e−2(α2+β2)]2(1 + cosϕe−2α2)(1 + cosφe−2β2),which is plotted for a number of different combinationsin Fig. 2. The success probability approaches 1/2 as theamplitudes of initial CSS’s becomes large. Note that theprobabilities depend on the type of CSS’s (odd or even)30.5 1 1.5 20.10.20.30.4αPFIG. 2: The success probabilities of the CSS-amplifying pro-cess in Fig. 1 for the input fields of two identical odd CSS’s(solid line), two identical even CSS’s (dashed line), and evenand odd CSS’s (dotted line).used. The effect of detector inefficiency is just to decreasethis success probability.We now show that a small odd CSS with α ≤ 1.2is surprisingly well approximated by a squeezed singlephoton. The single mode squeezing operator is Sˆ(r) =e−r2(aˆ2−aˆ†2), where r is the squeezing parameter and aˆ isthe annihilation operator. This operator reduces quan-tum noise of a vacuum state in the phase quadrature bya factor of e−2r. When the squeezing operator is appliedto a single photon the resultant state can be expanded interms of photon number states asSˆ(r) |1〉 =∞∑n=0(tanh r)n(cosh r)32√(2n+ 1)!2nn!|2n+ 1〉 . (3)The state contains only odd photon numbers and has co-efficients decaying exponentially as n increases in a simi-lar fashion to an odd CSS. The fidelity of this state withan odd CSS isF (r, α) = |〈CSS−(α)|S(r)|1〉|2 = 2α2 exp[α2(tanh r − 1)](cosh r)3(1− exp[−2α2]) .Fig. 3 shows the maximized fidelity on the y-axis plot-ted against a range of possible values for α for the de-sired odd CSS. Some example values are: F = 0.99999for amplitude α = 1/2, F = 0.9998 for α = 1/√2, andF = 0.997 for α = 1, where the maximizing squeezingparameters are r = 0.083, r = 0.164, and r = 0.313respectively. Firstly note that for α very close to zerothe fidelity approaches unity. When α → 0, r → 0 andhence the squeezing operator Sˆ(r) approaches the iden-tity transformation. An odd CSS with α very close tozero has a significant contribution from a single photonand very little from higher odd photon numbers. This isthe reason for the high fidelity as α tends to zero. Thefidelity remains high for α near zero as one can match thethree photon contribution to the CSS by the squeezingoperator whilst still being able to neglect higher orderphoton number terms. Eventually as α increases, higherphoton numbers cannot be matched and so as α tends toinfinity, the fidelity tends to zero.0.5 1 1.5 2Α0.880.90.920.940.960.98FidelityFIG. 3: The fidelity between an odd CSS and squeezed singlephoton. The odd CSS is extremely well approximated bythe squeezed single photon for a small coherent amplitude,α ≤ 1.2.As the fidelity between a squeezed single photon and anideal small CSS is extremely high, it can be conjecturedthat a large CSS distilled from squeezed single photonsby our scheme will also be very close to an ideal largeCSS. In what follows, we will show that this conjectureis true for α ≤ 2.5.In order to calculate multiple iterations we need touse numerical techniques. We are using coherent statesof some bounded coherent amplitude and superpositionsthere of. Provided the coherent amplitudes are not toolarge, the most significant contributions to these statesare Fock states of low number. For computations here thelowest thirty Fock states were used. This provides a verygood approximation for coherent states with α ≤ 2.5. All29 possible “click” events are included for all detectors.If one wished to create a CSS with a particular αwith n CSS-amplification steps, then initial CSS’s withαi = α/√2nare required. As the number of steps in-creases the required αi decreases. When generating alarge CSS out of the squeezed single photon states thefidelity maximizes for a particular number of iterations.Fig. 4 shows the maximum possible fidelity using thisprocess in (a) and the number of steps in (b) againstthe desired α in the CSS’s. For example, four iterationsstarting from the initial amplitude αi = 1/2 is requiredto gain the maximum fidelity F = 0.995 for α = 2. It isevident from Fig. 4 that high fidelity, F > 0.99, can beobtained up to α = 2.5. The error rate for discrimina-tion between coherent states with α = ±2.5 via a classicalmeasurement (homodyne detection) is only 3× 10−7.Current technology does not produce pure single pho-ton states; the single photon is always in a mixture withthe vacuum as p |0〉 〈0|+ (1− p) |1〉 〈1|, where p is the in-efficiency of the photon production. Hence the squeezedsingle photon state will also be a mixture with a squeezedvacuum. However, an interesting aspect of our scheme isthat it may be somewhat resilient to the photon pro-duction inefficiency because its first iteration purifies themixed CSS’s while amplifying them. The initial inputstates for the CSS amplification process from the imper-fect single photon source are4ρa,b,c =[(1− p)2 |S1〉 〈S1| ⊗ |S1〉 〈S1|+ p2 |S0〉 〈S0| ⊗ |S0〉 〈S0|+ p(1 − p)(|S0〉 〈S0| ⊗ |S1〉 〈S1|+ |S1〉 〈S1| ⊗ |S0〉 〈S0|)]a,b⊗ (|γ〉〈γ|)c, (4)0.5 1 1.5 2 2.50.990.9920.9940.9960.9981αMax Fidelity0.5 1 1.5 2 2.5012345αIterations(a)(b)FIG. 4: (a) The maximum fidelity obtained in our scheme vsthe coherent amplitude. (b) The number of iterations whichgives the maximum fidelity vs the coherent amplitude.where |S0〉 = Sˆ(r) |0〉 and |S1〉 = Sˆ(r) |1〉. Here, theterms with p2 and p(1 − p) are undesired error termswhere either (or both) of the single photons is missing.Note that the initial amplitude is required to be smallto produce a large CSS with high fidelity. Provided sucha small amplitude, input states incident onto the beamsplitters in our experimental setup contain approximatelyonly two (or slightly more than two) photons. In suchcases the probability of simultaneous clicks at detectorsA and B in Fig. 1 will significantly decrease when anyof the single photons is missing. In other words, theundesired cases will rarely be selected for the next it-eration of the amplification process. We have obtainednumerical results for the initial amplitude αi = 1/2 asfollows by the methods that we have already explained.If p = 0.4, the fidelity of the initial CSS, which is a mix-ture with a squeezed vacuum, is F = 0.60 but it willbecome F = 0.89 by the first iteration. Thus a largerCSS of significantly high fidelity is produced. If p = 0.25(p = 0.05), the fidelity of the initial CSS is F = 0.750(F = 0.950) but becomes F = 0.941 (F = 0.990) by thefirst iteration. Such purification effects for multiple it-erations are beyond the scope of this Letter but deservefurther investigations.In the CSS amplification process, the zero amplitudecoherent states that occur in the detection modes inEq. (2) may be slightly different from zero because of im-perfect mode matching at beam splitters. This will leadto a small probability of accepting the wrong state. Goodmode matching is a requirement in any linear optical net-work where one wishes to measure manifestly quantummechanical effects. Highly efficient mode matching of asingle photon from parametric down conversion and aweak coherent state from an attenuated laser beam at abeam splitter has been experimentally demonstrated us-ing optical fibers [19]. Such techniques could be employedfor the implementation of our scheme.The dark count rate of photodetectors will affect thefidelity of the CSS’s. Currently, highly efficient detectorshave relatively high dark count rates while less efficientdetectors have very low dark count rates [18]. We em-phasize again that our scheme does not require highlyefficient detectors because the inefficiency of the detec-tors does not affect the quality of CSS’s even though itdecreases the success probability. Silicon avalanche pho-todiodes operating at the visible wavelength have rela-tively high efficiency and a small dark count rate, whichis preferred in our proposal.The single photons required for our scheme could begenerated conditionally from a down-converter [17]. Thisis a χ(2) process (like squeezing) and does not requirephoton number resolving detection. Once free propa-gaing CSS’s are generated, they can be detected by ho-modyne measurements, which can be highly efficient inquantum optics experiments.Our scheme non-deterministically generates largeCSS’s. However, a non-deterministic CSS source is usefulenough for quantum information processing [6, 7]. Effi-cient gate operations for coherent-state quantum compu-tation [7] are based on teleportation via an entangled co-herent state [3, 4]. Entangled coherent states can be sim-ply generated from CSS’s using a beam splitter and canbe used as off-line resources for quantum computation.We note that such entanglement of macroscopically dis-tinguishable states is perhaps more closely aligned withSchro¨dinger’s original concept [1].In conclusion, we have proposed a simple all-opticalscheme to generate a linear superposition of macroscopi-cally distinguishable coherent states in a propagating op-tical field. We have found a previously unrealized con-nection between squeezed number states and superposi-tions of coherent states as well as the interesting additive5properties of the latter. In stark contrast to all previousschemes our scheme requires neither χ(3) nonlinearity norphoton number resolving detection to generate a macro-scopic superposition state.We would like to thank A. Gilchrist for useful com-ments. This work was supported by the Australian Re-search Council and the University of Queensland Excel-lence Foundation. M.S.K. acknowledges the UK Engi-neering and Physical Science Research Councils and theKorea Research Foundation (2003-070-C00024) for finan-cial support.[1] E. Schro¨dinger, Naturwissenschaften. 23, 807-812; 823-828; 844-849 (1935).[2] M.D. Reid, quant-ph/0101062 and references therein.[3] S.J. van Enk and O. Hirota, Phys. Rev. A 64, 022313(2001).[4] H. Jeong, M.S. Kim, and J. Lee, Phys. Rev. A 64, 052308(2001).[5] H. Jeong and M.S. Kim, Phys. Rev. A 65, 042305 (2002).[6] T.C. Ralph, W.J. Munro, and G. J. Milburn, Proceedingsof SPIE 4917, 1 (2002); quant-ph/0110115.[7] T.C. Ralph et al., Phys. Rev. A 68, 042319 (2003).[8] H. Jeong and M.S. Kim, Quantum Information and Com-putation 2, 208 (2002); J. Clausen, L. Kno¨ll, and D.-G.Welsch, Phys. Rev. A 66, 062303 (2002).[9] P. T. Cochrane, G. J. Milburn, and W. J. Munro, Phys.Rev. A 59, 2631 (1999); S. Glancy, H. Vasconcelos, andT.C. Ralph, quant-ph/0311093.[10] B. Yurke and D. Stoler, Phys. Rev. Lett. 57, 13 (1986).[11] R.W. Boyd, J. Mod. Opt 46, 367 (1999).[12] S. Song, C.M. Caves and B. Yurke, Phys. Rev. A. 41,5261 (1990).[13] M. Dakna et al., Phys. Rev. A. 55, 3184 (1997).[14] Q.A. Turchette et al., Phys. Rev. Lett. 75, 4710 (1995).[15] M. Brune et al., Phys. Rev. Lett. 77, 4887 (1996).[16] C. Monroe et al., Science 272, 1131 (1996).[17] A.I. Lvovsky et al., Phys. Rev. Lett. 87, 050402 (2001).[18] S. Takeuchi et al., Appl. Phys. Lett. 74, 1063 (1999).[19] T.B. Pittman and J.D. Franson, Phys. Rev. Lett. 90,240401 (2003).

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Conditional Production of Superpositions of Coherent States with Inefficient Photon Detection

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