Table Of ContentNonclassical correlation properties of radiation fields
Werner Vogel∗
Arbeitsgruppe Quantenoptik, Institut fu¨r Physik, Universit¨at Rostock, D-18051 Rostock, Germany
Afullcharacterizationofnonclassicalspace-timedependentcorrelationsofradiationisformulated
intermsofnormallyandtime-orderedfieldcorrelationfunctions. Itdescribesnotonlytheproperties
of initially prepared multimode radiation fields, but also the dynamics of radiation sources. Some
of these correlation effects occur in theresonance fluorescence of a single two-level atom.
PACSnumbers: 03.70.+k,42.50.Dv,42.50.Ar
8 Nonclassical effects of radiation have not lost any ticethisnumberwilldependontheuseddetectionsetup.
0 part of their attraction since the early days of quantum Consider an operator function fˆ,
0
physics [1, 2, 3]. It took over seven decades until Ein-
2 fˆ≡fˆ[Eˆ(−)(1),...,Eˆ(−)(k),Eˆ(+)(k),...,Eˆ(+)(1)], (1)
steins postulate ofthe existenceofphotons couldbe ver-
n
ifiedbyacleardemonstrationoftheantibunchingofpho- depending on the positive and negative frequency parts
a
J tons [4]. Other demonstrations of nonclassical radiation of the electric field operators, Eˆ(±)(i), where i ≡ (ri,ti)
2 properties, such as sub-Poissonian photon statistics [5] is the space-time argument (i = 1,...,k). We expand
1 and quadrature squeezing [6] were following soon. the operator fˆas
Inthefieldofquantumopticsthestudyofnonclassical ∞
]
h effects of radiation was widely based on the Glauber- fˆ= c{ni,mi}[Eˆ(−)(1)]n1...[Eˆ(−)(k)]nk
p Sudarshan P-function [7, 8]. Whenever it fails to have {ni,Xmi}=0
- the properties of a probability density, then the state is
nt said to be nonclassical [9, 10]. This condition describes ×[Eˆ(+)(k)]mk...[Eˆ(+)(1)]m1, (2)
a prominent examples of nonclassical effects. where the notation {n ,m } is used for the dependence
u i i
q Recently some possibilities of a complete characteri- on n1,...,nk,m1,...,mk.
[ zation of nonclassicality have been developed for a sin- Classicality of normally and time-ordered correlation
gle mode of the radiation field. Observable conditions properties can be defined as
2
v couldbe derived,whicharebasedoncharacteristicfunc- ∀fˆ: h◦fˆ†fˆ◦i≥0, (3)
1 tions [11, 12] or on moments [13, 14]. Both approaches ◦ ◦
5 lead to infinite hierarchies of nonclassicality conditions which generalizes the single-mode condition [14]. The
9 in terms of the corresponding quantities. The character- ◦ ... ◦ notation (cf. [22]) denotes both normal (: ··· :
0 ◦ ◦
istic function approach has already been used in some notation)andtime-ordering,withincreasingtimesinthe
.
6 experiments [15, 16]. fields Eˆ(−) and Eˆ(+) from left to right and right to left,
0
Nonclassicality of multimode radiation has been con- respectively. By generalizing the standard definition of
7
0 sidered in some special cases only. Entanglement, a the P-function (see [7, 8, 22]) to the functional
: special nonclassical property, is considered to be use-
v k
i ful for quantum information processing. A complete P[{E(+)(i)}]=h◦ δˆ(Eˆ(+)(i)−E(+)(i))◦i, (4)
X characterization in terms of moments is known for bi- ◦ ◦
iY=1
r partite continuous-variable entangled states whose par-
a the field correlation functions can be formally written
tiallytransposeddensityoperatorexhibitssomenegativ-
as those of classical stochastic processes. They behave
ities [17], for some special cases see also [18, 19]. There
like classical ones, if P[{E(+)(i)}] has the properties
exist attempts to generalize the method for the multi-
of a classical joint probability density. In the classi-
mode case [20] and for bound entanglement [21].
callimit the operatorscorrespondto classicalquantities:
In the present Letter we provide a general characteri- Eˆ(±)(i)→E(±)(i)andfˆ→f,orderingprescriptionsare
zation of nonclassical correlation properties of radiation
no longer needed, and averages become classical ones,
fields in terms of space-time dependent (normally and
h...i . In this case the left-hand side (lhs) of Eq. (3),
cl
time-ordered) field correlation functions. Beyond the
characterization of initially prepared quantum states of h◦fˆ†fˆ◦i→h|f|2i , (5)
◦ ◦ cl
free multimode radiation, this approach also describes
is indeed nonnegative in general.
the nonclassicaleffects caused by the dynamics of radia-
Based on this results, a radiation field shows nonclas-
tion sources. The physical realization and the detection
sicalnormallyandtime-orderedcorrelationpropertiesin
of the new correlation effects are also considered.
k space-time points, iff
Letus dealwithfieldcorrelationpropertiesinanarbi-
trary but fixed number of k space-time points. In prac- ∃fˆ: h◦fˆ†fˆ◦i<0. (6)
◦ ◦
2
This completely defines the nonclassical correlation ef- serve the quantity h◦fˆ†fˆ◦i for any operator fˆ.
◦ ◦
fects inthe chosenspace-timepoints. Itis difficult, how-
ever,tohandlethenonclassicalityconditioninthisform. We reformulate the condition (6) solely in terms of
The operator fˆmust be considered for all choices of the field correlation functions. Inserting Eq. (2) into the lhs
coefficients c . Moreover, it is unclear how to ob- of Eq. (6), we obtain a quadratic form:
{ni,mj}
∞
h◦fˆ†fˆ◦i= c∗ c h◦[Eˆ(−)(1)]n1+q1...[Eˆ(−)(k)]nk+qk[Eˆ(+)(k)]mk+pk...[Eˆ(+)(1)]m1+p1 ◦i.(7)
◦ ◦ {pi,qi} {ni,mi} ◦ ◦
{pi,qi,Xni,mi}=0
The necessary and sufficient condition for any violation outline the procedure and consider some examples.
of the nonnegativity of the quadratic form requires the
negativity of at least one of the principal minors of the The nonclassicality conditions obtained from negativ-
form(7). Inviewofthelengthyexpressionswemayonly ities of the second-order minor read as
2
◦[Eˆ(−)(1)]n1+q1...[Eˆ(−)(k)]nk+qk[Eˆ(+)(k)]mk+pk...[Eˆ(+)(1)]m1+p1 ◦ >
◦ ◦
(cid:12)D E(cid:12)
(cid:12) (cid:12)
(cid:12) ◦[Iˆ(1)]n1+m1...[Iˆ(k)]nk+mk ◦ ◦[Iˆ(1)]p1+q1...[Iˆ(k)]pk+qk(cid:12)◦ , (8)
◦ ◦ ◦ ◦
D ED E
where Iˆ(i) = Eˆ(−)(i)Eˆ(+)(i) is the intensity operator. Specifying the minors is equivalent to start with a sim-
These conditions are sufficient for the existence of non- plified form of the operator fˆ, such as
classical correlations in the chosen space-time points.
The necessary and sufficient conditions require the con-
fˆ=c [Eˆ(+)(1)]m+c [Eˆ(−)(2)]n+c :[Iˆ(3)]p :, (9)
sideration of the minors of all higher orders. 1 2 3
Let us give an example for nonclassicality conditions
basedonthird-orderminors,leadingtoinequalitiescom- with redefinedcoefficients c . The resulting nonclassical-
i
posed of sums of products of three correlationfunctions. ity condition reads as
h:[Iˆ(1)]m :i h◦[Eˆ(−)(1)]m[Eˆ(−)(2)]n◦i h◦[Eˆ(−)(1)]m[Iˆ(3)]p◦i
◦ ◦ ◦ ◦
(cid:12)h◦[Eˆ(+)(1)]m[Eˆ(+)(2)]n◦i h:[Iˆ(2)]n :i h◦[Iˆ(3)]p[Eˆ(+)(2)]n◦i(cid:12)<0. (10)
(cid:12) ◦ ◦ ◦ ◦ (cid:12)
(cid:12) h◦[Eˆ(+)(1)]m[Iˆ(3)]p◦i h◦[Iˆ(3)]p[Eˆ(−)(2)]n◦i h:[Iˆ(3)]2p :i (cid:12)
(cid:12) ◦ ◦ ◦ ◦ (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
More generally, the minor contains more complex corre- Schwarz inequality, for example cf. [22]. The intensity
lation functions, such as those in the condition (8). correlation function on the lhs is nonnegative, so that
We mayfurther simplify the conditions(8). By choos- the absolute value can be omitted. In the given form
ing the only nonvanishing powers as n = m = p = theantibunchingconditionalsoappliesfornonstationary
1 1 2
q =1, we get radiation [23]. The stationary condition was used in the
2
pioneeringphoton-antibunchingexperimentinresonance
◦Iˆ(1)Iˆ(2)◦ > :[Iˆ(1)]2 : :[Iˆ(2)]2 : . (11) fluorescence [4].
◦ ◦ r
D E D ED E Itisstraightforwardtoformulatehigher-ordergeneral-
This is the photon-antibunching condition in its general izationsoftheantibunchingcondition(11). InEq.(8)we
form, which can be also derived as a violation of the choose for the nonvanishing powers n = m = N −n,
1 1
3
n = m = M −m, q = p = n, and q = p = m (N ≥n and M ≥m), which leads to
2 2 1 1 2 2
◦[Iˆ(1)]N[Iˆ(2)]M ◦ > ◦[Iˆ(1)]2(N−n)[Iˆ(2)]2(M−m)◦ ◦[Iˆ(1)]2n[Iˆ(2)]2m◦ . (12)
◦ ◦ r ◦ ◦ ◦ ◦
D E D ED E
If we further specify m=M and n=0, the condition In general it depends on the chosen radiation sources,
in particular on their intensity correlation properties,
whether this form of the condition is stronger than the
◦[Iˆ(1)]N[Iˆ(2)]M ◦ > :[Iˆ(1)]2N : :[Iˆ(2)]2M :
◦ ◦ r form (15) or vice versa.
D E D ED E
(13) It is also of interest to formulate nonclassicality con-
represents a direct higher-ordergeneralizationof the an- ditions including the full field strength. Equating the
tibunching condition (11). coefficientsofEˆ(−)(1)andEˆ(+)(1)intheoperatorfˆ,the
From the nonclassicality condition (8) one may also positive frequency part in Eq. (16) is replaced with the
obtainconditionsforthe normallyandtime-orderedcor- field strength operator, Eˆ(1)=Eˆ(−)(1)+Eˆ(+)(1). Now
relations of intensity and field strength. Such correla- the nonclassicality condition is of the form
tions have been studied in the resonance fluorescence
of a single atom [24] and their measurement has been
◦Eˆ(1)Iˆ(2)◦ > :[Eˆ(1)]2 : :[Iˆ(2)]2 : . (18)
analyzed [25, 26]. A recently proposed method of bal- ◦ ◦ r
(cid:12)D E(cid:12) D ED E
ancedhomodynecorrelationmeasurementscombinesthe (cid:12) (cid:12)
(cid:12) (cid:12)
advantages of balanced homodyning with those of corre- The field strength-intensity correlations are no longer
lation techniques [27]. By using a strong local oscillator compared with the quantum statistical properties of the
it allows one to detect correlationfunctions of higher or- intensity alone. Note that nonclassical correlations of
ders, even when the overallquantum efficiency is small. intensity and field strength of lowest order have been
Letusconsideratypicalexample. ChoosinginEq.(8) discussed under special conditions, such as for Gaussian
the non-zero powers as n = m +p with p ≥ 0, we fluctuations [26].
i i i i
obtain the condition In the following we deal with the question of whether
ornotnonclassicalcorrelationpropertiesofthe types in-
◦[Eˆ(−)(1)]p1...[Eˆ(−)(k)]pk[Iˆ(k)]mk...[Iˆ(1)]m1 ◦ troduced above may occur in the irradiation of realistic
◦ ◦
(cid:12)D E(cid:12) sources. As a simple example we consider the resonance
(cid:12) (cid:12)
(cid:12) >rD◦◦[Iˆ(1)]2m1+p1...[Iˆ(k)]2mk+pk ◦◦E. ((cid:12)14) flcfu.oere.gs.ce[2n2c]e. oIfnathsiisncgalesetwwoe-hleavveel daetotamil,edfokrntohweletdhgeeoroyf
the high-order correlationfunctions to be considered. In
Itgivessomeinsightinthenonclassicalcorrelationprop-
particular,the occurrenceofphotonantibunching iswell
erties of intensity and field strength operators, by com-
known. The general concept of nonclassicalcorrelations,
paring normally and time-ordered correlation functions
however, applies to arbitrary radiation sources. For ex-
containingthefieldstrength(herethenegativefrequency
ample,itcouldserveforamorecompletecharacterization
part) with intensity correlation functions. In the lowest
ofthe radiationinthe mentionedcavityexperiment[26],
order we may choose p = m = 1 as the only nonvan-
1 2 but also for many other radiation sources.
ishing powers, which yields
Let us start to consider nonclassical effects based on
intensity correlation measurements of higher order, see
◦Eˆ(−)(1)Iˆ(2)◦ > ◦Iˆ(1)[Iˆ(2)]2◦ . (15) the condition (12). When at leastone of the values ofN
◦ ◦ r ◦ ◦
(cid:12)D E(cid:12) D E or M becomes two or larger,both sides of the inequality
(cid:12) (cid:12)
(cid:12) (cid:12) become zero, due to the fact that a single atom cannot
We can also formulate conditions for intensity-field
simultaneously emit two photons. Higher-order nonclas-
strength correlations of other types. Let us restrict the
generality of the operator fˆ, for example by setting sicaleffectsofthistypedonotoccurinresonancefluores-
cence from a two-level atom, their observation requires
fˆ=c Eˆ(+)(1)+c Iˆ(2). (16) other types of radiation sources.
1 2
Consider now the situation for field strength-intensity
Inserting this expression into the condition (6), we get correlations. Whenever a term [Eˆ(±)(i)]n with n ≥ 2
occurs (i = 1,...,k) on the lhs of the condition (8),
the corresponding correlation function vanishes for the
◦Eˆ(−)(1)Iˆ(2)◦ > Iˆ(1) :[Iˆ(2)]2 : . (17)
◦ ◦ r single-atom resonance fluorescence. That is, such types
(cid:12)D E(cid:12) D ED E
(cid:12) (cid:12)
(cid:12) (cid:12)
4
of nonclassical correlations do not exist. Based on this timode quantum states in a nonclassical initial prepara-
knowledge, however, we may formulate a variety of non- tion, but also to nonclassical properties emerging from
classicalityconditionsinterms ofthose higher-ordercor- the dynamics of radiation sources.
relationfunctionswhichdonotcontaintermsofthemen-
The author gratefully acknowledges valuable com-
tioned type. This leads to inequalities of the form
ments by C. Di Fidio, M. Fleischhauer, Th. Richter and
E. Shchukin. This work was supported by the Deutsche
◦◦Eˆ(−)(1)...Eˆ(−)(l)Iˆ(l+1)...Iˆ(k)◦◦ > Forschungsgemeinschaft.
(cid:12)D E(cid:12)
(cid:12) (cid:12)
(cid:12) ◦Iˆ(1)...Iˆ(l)[Iˆ(l+1)]2...[Iˆ(k)]2(cid:12)◦ , (19)
r ◦ ◦
D E
for 1 < l < k. To provide a nontrivial characteriza- ∗
Electronic address: [email protected]
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