Non-Gaussian continuous-variable entanglement and steering

0.42 MB·English

by  M. K. Olsen

Non-Gaussian continuous-variable entanglement and steering
M. K. Olsen and J. F. Corney
School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia.
(Dated: February 1, 2013)
Two Kerr-squeezed optical beams can be combined in a beamsplitter to produce non-Gaussian
continuous-variable entangled states. We characterize the non-Gaussian nature of the output by
calculating the third-order cumulant of quadrature variables, and predict the level of entanglement
that could be generated by evaluating the Duan-Simon and Reid Einstein-Podolsky-Rosen criteria.
These states have the advantage over Gaussian states and non-Gaussian measurement schemes in
that the well known, efficient and proven technology of homodyne detection may be used for their
characterisation. A physical demonstration maintaining the important features of the model could
be realised using two optical fibres, beamsplitters, and homodyne detection.
3 PACSnumbers: 42.50.Dv,42.50.Lc,03.67.Bg,03.67.Mn
1
0
2 INTRODUCTION strated.
Inthispaper,weuseasingle-modeanharmonicoscilla-
n
a Continuousvariable(CV)systemsprovideflexibleand tor[11]todeterminethenon-Gaussianentanglementthat
J can in principle be achieved with Kerr-squeezed states.
powerful means for implementing quantum information

0 We characterise the non-Gaussian statistics through
schemes [1], in large part because there are mature and
3 higher-order cumulants and gauge the level of entan-
precisetechniquesformeasuringthequadraturesoflight,

] mostofwhicharefamiliarfromclassicalcommunications glement by calculating the Duan-Simon and Einstein-
h technologies. Despite the need to deal with transmission Podolsky-Rosen correlations.
p
losses, recent work has demonstrated useful distances
-
t comparable to those achieved with discrete-variable sys-
n TESTING FOR NON-GAUSSIAN STATISTICS
tems [2]. Furthermore, research and development has
a
u progressed to the stage where CV quantum key distri-
AGaussianstatecanbemostsimplydefinedasastate
q butions systems have advantages over discrete-variable
[ with a Gaussian Wigner function, i.e. a state whose
methods [3–5].

marginal distributions are Gaussian. For a CV state,
1 TheremainingstumblingblocktothewideruseofCV
thedeparturesfromnon-Gaussianbehaviourcanthusbe
v systems is that the most readily available CV systems
5 characterised by the skewness of the distributions of its
and the the most developed detection techniques pro-
7 quadraturemoments,asrevealedinnonzerohigher-order
duce only Gaussian statistics. This limitation rules out
4 cumulants [12].
7 tasks such as entanglement distillation [6], quantum er- We define the generalised quadrature Xˆ(θ) at quadra-
. rorcorrection[7],andquantumcomputation. Onewayof
1 ture at angle θ as
introducing non-Gaussian statistics is through nonlinear
0
3 measurements [8], but this approach negates one of the Xˆ(θ)=aˆe−iθ+aˆ†eiθ, (1)
1 main advantages of CV systems, namely, the highly de-
v: velopedtechnologythatisavailableforperformingGaus- so that the canonical Xˆ quadrature is found at θ = 0,
i sian homodyne measurements. with conjugate Yˆ =Xˆ(π).
X 2
In this paper, we proceed along an alternative ap- ForaGaussiandistribution,allcumulantshigherthan
ar proach, namely to use CV sources that produce non- second order vanish, and therefore we can test for non-
Gaussian outputs. The importance of this area of re- Gaussian statistics by a the presence of a nonzero third-
search was shown recently by Ohliger et al. [9], who order cumulant:
demonstrated there are serious limitations to the use
κ (θ)=(cid:104)Xˆ3(θ)(cid:105)+2(cid:104)Xˆ(θ)(cid:105)3−3(cid:104)Xˆ(θ)(cid:105)(cid:104)Xˆ2(θ)(cid:105). (2)
of Gaussian states for quantum information tasks which 3
may be avoided by developing useful and relatively sim-
While κ (cid:54)= 0 is a sufficient condition for non-Gaussian
3
ple non-Gaussian sources.
statistics, it is not a necessary one. In particular, κ
3
Non-Gaussian light can be produced by means of a
will vanish for a symmetric distribution in phase space.
χ(3) nonlinear medium, such as a single-mode optical fi-
In the presence of such symmetry, the fourth-order mo-
bre. PairsofsuchKerrsqueezedbeamscanbecombined
ment κ provides the lowest-order test for non-Gaussian
4
onabeamsplittertoproduceCVentangledstates,ashas
behaviour:
been experimentally produced using polarisation squeez-
ing [10]. However, the non-Gaussian character of these κ (θ)=(cid:104)Xˆ4(θ)(cid:105)+2(cid:104)Xˆ(θ)(cid:105)4−3(cid:104)Xˆ2(θ)(cid:105)2−(cid:104)Xˆ(θ)(cid:105)κ (θ).
4 3
states has not to our knowledge been explicitly demon- (3)
2
To analytically calculate these moments, we use the fol-
The fourth-order cumulant can be used to infer the lowing expectation values and their complex conjugates:
negativity of the Wigner function [13], which is con-
sidered to be a direct measure of the nonclassicality of (cid:104)aˆ2(t)(cid:105) = α2e−2iθe−4iχte|α|2(cos4χt−isin4χt−1),
θ
a state. It also allows comparison to the nonclassical (cid:104)aˆ3(t)(cid:105) = α3e−3iθe−9iχte|α|2(cos6χt−isin6χt−1),
states that have been experimentally demonstrated to θ
be non-Gaussian, such as the number state [8] and the (cid:104)aˆ†θaˆ2θ(t)(cid:105) = α∗α2e−iθe−3iχte|α|2(cos2χt−isin2χt−1),
photon-subtracted squeezed vacuum [14, 15]. For both (cid:104)a4(t)(cid:105) = α4e−4iθe−i16χte|α|2(cos8χt−isin8χt−1),
θ
of these states, κ scales quadratically with number. For
4 (cid:104)a†a3(t)(cid:105) = α∗α3e−2iθe−i8χte|α|2(cos4χt−isin4χt−1).
the number-state for example, θ θ
(12)
κ =−6n(n+1). (4)
4
These equations reveal several kinds of contributions
In the analysis below, we will determine the regimes in
to the dynamics, with different time scales. The sin and
whichtheKerr-squeezedstateisskewedtoasimilarlevel.
cos terms in the exponents can each be expanded, and
forsufficientlysmallinteractiontimeχt,wecankeepthe
NON-GAUSSIAN STATISTICS IN THE first two terms in each, i.e. up to fourth order in time.
KERR-SQUEEZED STATE We are left with a number of different contributions to
the exponent.
The Hamiltonian for the single-mode model, ignoring First, there is the nonlinear phase factor proportional
any effects due to loss and excess noise, is toNχt, whereN =|α|2. Thismean-fieldfrequencyshift
can be removed by a switch to a rotating frame, i.e. set-
H=(cid:126)χ(cid:0)a†a(cid:1)2, (5) ting θ =θ +2Nχt.
0
Second, the real exponent proportional to Nχ2t2 is
where χ represents the third-order nonlinearity of the
responsible for squeezing, although in order to obtain
medium and aˆ is the bosonic annihilation operator for
quantum squeezing, i.e. below the coherent-state level,
the electromagnetic field mode.
we also require the zero-point phase factors.
For an input Glauber-Sudarshan coherent state,
Finally, there are the third and fourth order terms
|α(cid:105)=e−|α|2/2(cid:88)∞ √αn |n(cid:105), (6) Nχ3t3 and Nχ4t4, which for large N give the leading
n! order contribution to the third and fourth order cumu-
n=0
lants and hence are responsible for most of the skewness
where|n(cid:105)representsaFockstateoffixednumber,wemay we see in the quadrature statistics.
find analytical expressions for all the operator moments
In typical Kerr squeezing experiments, the number of
necessary to calculate the first four cumulants.
photons is large N (cid:29) 1 in order to compensate for a
The Heisenberg equation of motion for aˆ can formally
weak nonlinearity χ (cid:28) 1. In this limit, we can derive a
be solved to give
simple expression for the third-order cumulant of the Y
aˆ(t)=e−iχt(2aˆ†aˆ+1)aˆ(0), (7) quadraturerotatingatthemean-fieldfrequency,whichis
where skewness is most evident:
whose expectation value in a coherent state is
(cid:104)aˆ(t)(cid:105) = αe−iχte|α|2(cos2χt−isin2χt−1). (8) κ (cid:16)π(cid:17)(cid:39)−256√1 (χNt)3. (13)
3 2 N
Defining aˆ ≡ aˆe−iθ, we can write the mean of a
θ
quadrature moment as
The validity of this expression is demonstrated in
(cid:104)Xˆ(θ,t)(cid:105)=(cid:104)aˆ (t)+aˆ†(t)(cid:105), (9) Fig. 1, which plots the third-order cumulant for various
θ θ photon numbers. The exact results for N > 106 are in-
forwhichwealreadyhaveasolution. Thesecondmoment distinguishable on this time scale from the simple cubic
is growth described by Eq. (13). Time is here scaled by
(cid:104)Xˆ2(θ,t)(cid:105)=(cid:104)1+2aˆ†aˆ +aˆ†2+aˆ2(cid:105), (10) Nχ in order compare results that give the same Kerr ef-
θ θ θ θ fect. Onthisscale,thethird-ordercumulantdecreasesin
where we have dropped the time argument on the RHS proportion to the square root of the number of photons.
for simplicity. The third- and fourth-order moments are Note however, that the absolute size of the third-order
cumulantincreaseswithparticlenumber,ataratefaster
(cid:104)Xˆ3(θ)(cid:105) = (cid:104)aˆ†θ3+3aˆ†θ2aˆθ+3aˆ†θaˆ2θ+aˆ3θ+3aˆ†θ+3aˆθ(cid:105), than (cid:104)Xˆ(cid:105)3:
(cid:104)Xˆ4(θ)(cid:105) = (cid:104)a4+4a†a3+6a†2a2+4a†3a +a†4
θ θ θ θ θ θ θ θ κ
+6a2θ+12a†θaθ+6a†θ2+3(cid:105). (11) (cid:104)Xˆ3(cid:105)3 ∼N. (14)
3
0x 106 100 AgTahine, fofourrthla-rogrederphcoutmonulannutmκb4eriss tphleottceudmiunlaFntig.ap2-.
1000
proaches a limiting scaling behaviour:
−0.5
1
−1 κ ∝ (χNt)4, (15)
4 N
−1.5 0x 104 10000
N which gives the same relative growth of
* #3 −2 −2 κ4 ∼N, (16)
" −2.5 * N"#3 −−64 100000 (cid:104)Xˆ(cid:105)4
−3 althoughifthetimeisadjustedasafunctionofN tokeep
−8 the Kerr squeezing constant, the fourth-order cumulant
−3.5 decreases in proportion to the particle number.
−10 1000000
0 5 10 15 20
N! t Figures 3 and 4 show the cumulants as a function of
−4
0 5 10 15 20 25 N, for a fixed value of χNt = 25. One can clearly see
N! t
two different regimes of behaviour, with the crossover
between the two occurring just above N ∼ 104. For κ
4
the number-state result is also plotted for comparison.
FIG. 1: (colour online) The third order cumulant, κ , of One can see that κ /(Nt)4 scales as described above for
3 4
Yˆ = Xˆ(π/2) in a rotating frame as a function of time, for large N, but for small N is limited to values of the order
various photon numbers ranging from 100 to 106, as labeled. ofcorrespondingnumber-stateresults(increasingwithN
In this and subsequent plots, time is scaled by the inverse of
quadratically). ThisresultsuggeststhataKerr-squeezed
themean-fieldinteractionstrengt√handhenceisadimension- statecanbeasnon-Gaussianbythismeasureasthenum-
less quantity; κ is scaled by 1/ N. The dashed lines give
3
ber state, for sufficiently long interaction time.
the exact results (Eqs. (12)) and the solid line gives the ap-
proximate result (Eq. (13)), which is accurate for large N or
small mean-field interaction time Nχt. The inset shows κ
3
for N =1000 in more detail.
11
x 10
0 100
1000
3
−2 κ
10000
x 109
0
−104
−4 −1
N
* −2 100000
4
" −6 * N"4−−43
103 104 105 106
−5 N
−8
−6
1000000
−70 5 10 N! t 15 20 25 FIG. 3: (colour online) Third-order cumulant κ3(π2) of the
Kerr-squeezedstateatfixedχNt=25asafunctionofN. The
0 5 10 15 20 25 behaviouratN (cid:38)105 revealsthescalingbehaviourdescribed
N! t
byEq.(13);thebehaviouratN (cid:46)104isduetothesaturation
effect seen in the inset to Fig. 1.
FIG.2: (colouronline)Thefourth-ordercumulant,κ ofYˆ =
4
Xˆ(π/2) in a rotating frame as a function of time, for various
photonnumbersrangingfrom100to106,aslabeled. Timeis QUADRATURE VARIANCES, ENTANGLEMENT
scaledbythemean-fieldinteractionstrength,andκ3 isscaled AND EINSTEIN-PODOLSKY-ROSEN
by 1/N. For large N or small mean-field interaction time CORRELATIONS
Nχt, the results approach the same limiting curve ∝t4. The
insetshowsκ forN =1000inmoredetail,bothfortheKerr-
4 Althoughentangledstateshavealreadybeenpredicted
squeezedstate(dashedline)andforthenumberstate(dotted
to be produced by the intracavity nonlinear coupler [16,
line).
17], the linearisation process used to obtain the spectra
4
110000000000
4
3.5
−106 3
100
2.5
e
−108 nc
a 2
κ4 vari 10
1.5
−1010
1
10 100 1000
1000
0.5
−1012 100
1000
10000 10000
0
102 103 104 105 106 0 0.5 1 1.5 2 2.5
N Nχ t
FIG. 4: (colour online) Fourth-order cumulant κ (π) of the
Kerr-squeezed state at fixed χNt = 25 as a fun4cti2on of N FIG. 5: (colour online) The variances in the Xˆ (solid line)
(crosses). The squares give the fourth-order cumulant of a and Yˆ (dashed line) quadratures as a function of mean-field
numberstatewiththesamenumberofphotons. ForN (cid:46)103 interactiontimeNχt. Onthisscale,thesqueezingresultsfor
theKerr-squeezedresultsaturatestothatofthenumberstate different numbers are indistinguishable for N >104.
(see inset to Fig. 2). For N (cid:38) 104, the results follow the
scaling as given in Eq. (15).
For notational convenience, we will now make the sim-
plification Xˆ → Xˆ , and similarly for Yˆ . This allows
in those cases forces Gaussian statistics on the outputs. ustodefinetbhjevariajncesofthebeamsplitbtjeroutputsas,
Here, we wish to produce entangled states that maintain
non-Gaussianstatistics,sowewillproceedbymixingthe V(Xˆ ) = ηV(Xˆ )+(1−η)V(Yˆ ),
1 a1 a2
outputs of two Kerr oscillators on a beamsplitter [18], as V(Xˆ ) = (1−η)V(Yˆ )+ηV(Xˆ ),
experimentally demonstrated by [10]. We will now show 2 a1 a2
V(Yˆ ) = ηV(Yˆ )+(1−η)V(Xˆ ),
the quadrature variances, as we need squeezed states in 1 a1 a2
ordertoobtainentangledmodesintheoutputs. Wenote V(Yˆ2) = (1−η)V(Xˆa1)+ηV(Yˆa2). (19)
that, while this could be done by mixing one squeezed
Along with the covariances,
mode with vacuum, better results in terms of the degree
of violation of the relevant inequalities are obtained by V(Xˆ ,Xˆ ) = −(cid:112)η(1−η)(cid:104)V(Xˆ ,Yˆ )+V(Xˆ ,Yˆ )(cid:105),
mixing two squeezed states. The Heisenberg uncertainty 1 2 a1 a1 a2 a2
principle demands that V(Yˆ ,Yˆ ) = (cid:112)η(1−η)(cid:104)V(Xˆ ,Yˆ )+V(Xˆ ,Yˆ )(cid:105),
1 2 a1 a1 a2 a2
(cid:16) (cid:17) (cid:16) π (cid:17)
V Xˆ(θ) V Xˆ(θ+ ) ≥1, (17) (20)
2
we now have all the expressions needed to calculate
so that any quadrature with variance below one is
the quantities necessary to check for violation of the
squeezed. Fig. 5 shows the variances of the canoni-
cal Xˆ = Xˆ(0) and Yˆ = Xˆ(π) quadratures with time continuous-variable Duan-Simon inequality [19, 20]. For
2 the purposes of this article, we define this as
again scaled by Nχt so that, apart from small-number
effects, the same level of squeezing is obtained for differ- V(Xˆ ±Xˆ )+V(Yˆ ∓Yˆ )≥4, (21)
1 2 1 2
entphotonnumbers. Asforthecumulants,theresultsfor
N > 104 cannot be distinguished on this time scale. In with any violation of this inequality being sufficient to
fact, above N =1000, the small-number effects only ap- demonstrate the presence of entanglement for a non-
pear as small differences amount of squeezing, so for the Gaussian state. The result for this correlation, with
remainder of the paper, we quote results for N =1000. η = 0.5, is shown in Fig. 6. The (red) dotted line gives
Considering a beamsplitter with reflectivity η and la- themaximumviolation,optimisedoverquadratureangle
belling the inputs by aˆ1 and aˆ2 and the outputs by ˆb1 θ. Clearly the outputs from two Kerr oscillators mixed
andˆb2, we find on a beamsplitter can give a continuous-variable non-
ˆb = √ηaˆ +i(cid:112)1−ηaˆ , Gaussian entangled bipartite resource.
1 1 2 As shown by Wiseman et al. [21] and Cavalcanti et
ˆb = i(cid:112)1−ηaˆ +√ηaˆ . (18) al. [22], the inseparability of the system density matrix
2 1 2
5
5 2
ns 4
o 1.5
ati
el
r 3
r
o
c R
n P 1
o E θ = 0
m 2
Si
−
uan 1 0.5 θopt(χ t)
D
0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
χ t x 10−3 χ t x 10−3
FIG. 6: (colour online) Duan-Simon correlations (LHS of FIG. 7: (colour online) Reid EPR correlation (LHS of
Eq.(21))aftermixingona50:50beamsplitter,asafunction Eq. (22)) after mixing on a 50 : 50 beamsplitter as a func-
of interaction time χt, for N=1000. A value below 4 signifies tion of interaction time χt, for N = 1000. A value below 1
entanglement. The(blue)solidlineusesthecanonicalXˆ −Xˆ signifies a demonstration of the EPR paradox. The dashed,
1 2
andYˆ +Yˆ quadraturesandthegreen(dash-dotted)lineuses lowerlineisoptimisedforquadratureangle,whiletheupper,
1 2
thecanonicalXˆ +Xˆ andYˆ −Yˆ quadratures,whilethe(red) solid line is for the canonical quadratures.
1 2 1 2
dotted line is optimised for quadrature angle at each time.
can be shown to be:
describes a set of states which includes within it subsets 1
κ (X ) = √ (κ (X )−κ (Y ))
which are more deeply non-classical than evidenced by 3 1 8 3 a1 3 a2
entanglementalone,suchasthosewhichdemonstratethe 1
κ (X ) = (κ (X )+κ (Y ))+4(cid:104)X (cid:105)κ (X ),
Einstein-Podolsky-Rosen (EPR) paradox [23]. For our 4 1 4 4 a1 4 a2 1 3 1
purposes here, we will use the inequality developed by (24)
Reid [24], written as
whichconfirmsthatskewedinputstoabeamsplitterlead
V (Xˆ )V (Yˆ )≥1, (22) toskewedoutputs, withcumulantsgenerallyofthesame
inf bj inf bj order of magnitude.
where j =1,2 and
CONCLUSIONS
(cid:104) (cid:105)2
V(Xˆ ,Xˆ )
V (Xˆ ) = V(Xˆ )− bj bk , In summary, we have employed a simple model of χ(3)
inf bj bj V(Xˆ ) nonlinear process to demonstrate that violations of the
bk
(cid:104) (cid:105)2 Duan-SimonandReidentanglementcriteriaoccuratthe
V(Yˆ ,Yˆ )
V (Yˆ ) = V(Yˆ )− bj bk . (23) same time as significant departures from Gaussian be-
inf bj bj V(Yˆ ) haviour. For sufficiently long interaction time Nχt, the
bk nonlinear interaction will skew the distribution of the
quadrature variables, leading to large third and fourth
From the expressions given above for the Duan-Simon
order cumulants.
criterion, it can be seen that all the moments necessary
Moreover, such non-Gaussian entanglement occurs in
to calculate these expressions are available analytically.
regimesaccessibletoopticalfibreexperiments[25]. How-
AsshowninFig.7,thetwomodesafterthebeamsplitter
ever, for accurate quantitative predictions, one would
exhibit a strong violation of the Reid inequality.
need to go beyond the single-mode model, to include the
Finally,weconsidertheskewnessofthefinalentangled effects of pulse dynamics and extra noise sources, using
state. For a 50:50 beam splitter, the third- and fourth- the simulations methods, for example, that were used in
order cumulants of the X-quadrature in output port 1 [26].
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