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Fermion ensembles that show statistical bunching
Volume 124, number 1,2
PHYSICS LETTERS A
14 September 1987
FERMION ENSEMBLES THAT S H O W STATISTICAL BUNCHING M.P. SILVERMAN Department of Physics, Trinity College, Hartford, CT06106, USA Received 3 June 1987; accepted for publication 8 July 1987 Communicated by J.P. Vigier
Contrary to the general expectation that Fermi-Dirac statistics limit fermion clustering behaviour to that of antibunching, it is shown that particular fermion ensembles can give rise to fermion bunching. These ensembles comprise correlated multifermion states entering both input ports of an SU(2) fermion interferometer. The beams leaving the two output ports exhibit a positive cross-correlation under appropriate conditions.
It is well known theoretically and demonstrated experimentally that photons manifest a variety of clustering behaviours depending upon the coherence properties of the source. For example, chaotic light gives rise to photon bunching, as demonstrated by Brown and Twiss [ 1,2] and interpreted specifically in a quantum statistical context by Purcell [ 3] and Kahn [4]. Coherent light (e.g. laser light above threshold) manifests no bunching; photon counts are distributed randomly in time as demonstrated by Arecchi et al. [ 5]. Photon antibunching occurs for quantum states of light with fixed photon number (see discussion in ref. [6]) which is a system very difficult to produce experimentally and for resonance fluorescence from single two-level atoms, as predicted by Carmichael and Walls [7] and demonstrated by Kimble et al. [8]. The fundamentally significant problem of elucidating the clustering behaviour of fermion beams has so far been largely unexplored. In the absence of a thorough general study, it is widely suspected that fermions always exhibit clustering of the antibunching type as a result of the Fermi-Dirac limitation of one fermion per cell of phase space. Purcell [3], in his analysis of the Brown-Twiss experiment, pointed out that a split beam of thermal electrons should exhibit a negative cross-correlation indicative of fermion antibunching. Subsequent detailed studies have treated the observability of antibunching in beams of thermal neutrons [ 9,10] and field-emission elec-
trons [ 11 ]. No example of clustering behaviour other than antibunching has so far been reported for fermions. It is the objective of this paper to show that fermion bunching is theoretically possible. It will be demonstrated that, as is the case with light, the nature of fermion fluctuations and correlations depends not only on the type of quantum statistics (which is universal for all fermions), but also on the specific linear superpositions of quantum states encompassing the particle ensemble. I have shown previously [ 12 ] that fermion ensembles for which the density operator factorises in a momentum-spin projection representation into a product of single-particle momentum-spin projection distribution functions give rise to fermion antibunching. Density operators with this property characterise diverse fermion sources among which are thermal electrons, thermal neutrons, and field-emission electrons. In this paper I consider particular ensembles of highly correlated multi-fermion states for which the density operator does not so factorise; I show that these ensembles manifest a positive cross-correlation, and hence statistical bunching, in a split-beam experiment of the Brown-Twiss kind. (Preliminary reports of positive fermion cross-correlations were given in ref. [ 13 ].) In previous studies of the quantum effects of a vector potential and gravitational potential on the clustering behaviour of fermions [ 14,15 ], experimental configurations employing a single fermion
0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
27
Volume 124, n u m b e r 1,2
PHYSICS LETTERS A
14 S e p t e m b e r 1987
relations. The input and output annihilation operators for a passive '~ interferometer are related to within a global phase factor as follows:
R
d , ( k , s ) = ( l r ~ r ~ l e '°'e+ltLt2te ioJ2)bj(k,s)
!o
+ i ( l & t e le i°/e-[t, relel°/2)b~_(k,s),
(2a)
d2(k, s) = - i ( It, r~_le io,e_ Ir, le le i°'2) b,(k,s) + ( Ir~r_~ le i°~2+lt~tzlei°"~-)b2(k,s),
(2b)
BS 2
$2
L ~
Fig. 1. F e r m i o n M a c h - Z e h n d e r interferometer. F e r m i o n s from sources S j, Se, enter through input ports 1, 2, respectively; b e a m s exiting the o u t p u t ports to detectors D~, De are cross-correlated by c o u n t e r / c o r r e l a t o r C.
beam at the input port of the fermion interferometer were analysed. Fig. 1. illustrates a Mach-Zehnder fermion interferometer wherein both input ports are utilised. The two entering beams from sources Sl and $2 are split at beam splitter BSI, pass via mirrors M~ and M2 to beam splitter BS2 and thence to detectors D. and D2 whose instantaneous outputs are crosscorrelated by counter/correlator C and displayed by recording device R. The fermion field operators for both input and output ports are expressible (up to an irrelevant normalisation factor) as
~P,( x, t) = ~, b,( k, s) exp[ i( k x - o J t ) ] , k,s
/ = i n p u t ports 1,2,
(la)
where (r~,t~) and (r>t~) are the reflectance and transmittance amplitudes of BS~ and BS2, respectively, and 0 is the phase difference incurred between upper and lower beam components. It is assumed that the beam is quasimonochromatic, so that the momentum spread is much less than the main particle momentum, Ak<< ko, for which case 0 is, to a good approximation, independent of the momentum distribution. (There are also cases, such as in the Aharonov-Bohm effect [ 14] where 0 depends only on an external parameter (magnetic flux) and is rigorously independent of path length and momentum distribution.) In what follows BSI and BS2 are taken to be 50-50 beam splitters, i.e. Ir,12= It,12= 1/2. The operators corresponding to number of particles received at D~ and D2 during an arbitrary sampiing time are N(D,)=~d~(k,s)~d,(k,s),
i=1,2,
(3)
k,s
the mean number of counts received by each detector is correspondingly N(D,) =Tr{fiN(D,) ~ ,
(4)
where fi is the density operator of the input ensemble. The cross-correlation of particle fluctuations [ 15 ] is given by C(DI,
D2)
= Tr{/J[N(DI ) - N ( D , )] [N(D2) - N(De) ] }
cI)~(x, t) = ~ d,(k, s) e x p [ i ( k x - o ) t ) ] , k,s
=Tr{I~]V(D,)N(D2)}-N(D,)N(D2). i = output ports 1,2,
( 1b)
where the fermion input (output) annihilation and creation operators, b,(k, s) and bt,(k, s) (d,(k, s) and d~(k, s)) satisfy standard fermion anticommutation 28
(5)
~ By a passive interferometer is m e a n t one that preserves the particle d i s t r i b u t i o n over m o m e n t u m states. This requires that r,t, be pure imaginary, and IrA 2 + I t,I 2 = 1. The phases have been chosen so that r, = I r, I exp ( in/2 ), t, = I t, I-
Volume 124,number 1,2
PHYSICSLETTERSA
The density operator characterising a two-port fermion input may be expanded in eigenstates ]nl(ks);nE(k's')) of the input particle number operators
N t = ~ bi(k,s)*b,(k,s),
i=1,2,
(6)
k.s
is the operator for total number of particles passing through the interferometer during a counting time, and
I~=-
with eigenvalues nl, r/2 for ports 1, 2, respectively. The designation n~(ks) in the ket label is an abbreviation for the momentum and spin projection distribution [(kl, Sl )(k2, $2)...(kni, Sn,)] over n~particles. The total number of particles passing through the interferometer is conserved, i.e. Nl +N2 = At(D,) +N(D2). For a single-port fermion input of mean particle number n~- and variance (Anl)2, characterised by a density operator of the "diagonal" form
~= ~ p(n~(ks))lnl(ks);O)(nl(ks);OI,
(7a)
14 September 1987
~. [bl(k,s)*b2(k',s')*b2(k,s) bl(k',s') k,s,k' ,s'
+bl(k,s)tb2(k',s')*b2(k',s ') bl(k,s)] .
(8f)
The operators Ji (i= x, y, z) satisfy the commutation relations of angular momentum operators; in addition [Z, ff/] =0
(9a)
and [J,, N] = 0 .
(9b)
the cross-correlation has been shown to be [ 15 ]
The operators needed for determination of the cross-correlation may be expressed in terms of N and the SU(2) generators by means of the relations
C(DI, D2) = [(An~ )2 - h~-~](sin 0)2/4,
~r(Dl ) =N/2 + Jz cos 0 + J , sin 0,
(10a)
J(D2) = ~ / 2 - ~
(10b)
nl,k,s
(7b)
where the fermion ensemble is subpoissonian. The variance is thus less than the mean, and the resulting negative cross-correlation is indicative of antibunching [ 3 ]. For values of 0 for which sin 0 vanishes the output beams are uncorrelated. Considerations of phase sensitivity in an interferometer have led recently to the introduction of a set of correlation fermion states [ 16] that are basis states of an SU(2) algebra constructed from the fermion annihilation and creation operators as follows J,- = ½Z [bl(k,s)*b2(k,s) k,s
+b2(k, s)*bl (k, s)] ,
(8a)
Jv - 1 ~ [bt(k,s)*b2(k,s) • -
2i
~,s
-b2(k,s)*bl(k,s)] , L = 1 (]Wl--N2),
(8b)
(8c)
the Casimir operator is of the form `72 = (]V/Z) [(~/72) + 1] + fig, where
cos 0 - J x sin 0,
]VD=/~r(Dl )-N(D2) =2(Jz cos 0 +Jx sin 0) , N-(D, ) fi(D2) = [/~(Dl )2 +~(D2)2 - N ~ ] / 2 .
(10d)
SU(2) basis states of the form In/2; m ) , which are eigenstates of j2 and J: with eigenvalues (n/2)[(n/2) + 1] and m, respectively, (i.e. eigenstates of ~r and /~ with eigenvalues n and 0, respectively) can be constructed from the Inx; n2) basis through use of "raising" and "lowering" operators ,7+=,Tx+,Tr The SU(2) correlated state In/2; n/2) is equivalent to the particle number state In;0) corresponding to all particles entering port 1. Repeated application of J generates the remaining states [j;m) of the (j=n/2; W=0) irreducible representation according to In/2; m) = [n!(n/2-m)!/(n/2+ra)!]-1/2
xJn_/2-m In~2;n12) . (8d)
(10c)
(1 la)
The state In/2; m) resulting from n / 2 - m applications of J_ therefore has n / 2 - m fermions entering 29
Volume 124, number 1,2
PHYSICS LETTERS A
port 2 and n / 2 + m fermions entering port 1. For example, in a two-particle system comprising quantum states a = (kl, s l) and b = (k2, s2 ) the S U ( 2 ) correlated states may be expressed as 1;1 ) = I ( a , b ) ; O ) , 1;0) = [ I b ; a ) - l a ; b )
(1 lb) ]/2 ,
1 . - 1 ) = - I 0;(a,b) >
(llc) (1 ld)
(the phase convention for application of annihilation and creation operators to fermion states is given in ref. [17]). The SU (2) correlated states form a complete basis with respect to which one can expand the density operator of the two-port fermion input. Consider first an ensemble of fermion input states characterised by a density operator diagonal in a basis o f SU (2) correlated states. From eqs. (2), (3), (5), (10) it may be shown that the cross-correlation of outputs at D, and D2 is
14 September 1987
of sharp n and m, then the variances (An) 2 and ( A m ) 2 vanish, and the eigenvalues n, = n / 2 + m and n 2 = n / 2 - m are also sharp. The cross-correlation. which is expressible as C ( D I , De) =-
[n~n2 + (n, +n~)/2](sin 0)2/2
(13a)
is again intrinsically negative except at selected phase angles for which it vanishes. (C) In the more general case of an input beam comprising an appropriate mixture of S U ( 2 ) correlated states there is a range of angles, given by 4Ci2/[2nl n2 + n-- (An) e ] > (tan 0) -~
(13bl
for which the cross-correlation can be positive. For example, at 0 = 0 the condition (An) 2 > 4(Arn) 2
(13c)
is the variance about the mean total particle number r~ entering the interferometer and
results in a positive cross-correlation. This condition is most directly met by an ensemble of variable total particle number ( ( A n ) 2 > 0 ) , but fixed particle difference ( ( A m ) 2 = 0 ) at the input ports. An input comprising the states In/2; m ) and 1n/2+2: rn) with Iml <~n/2 leads to C(D~, D 2 ) = C l 2 = 1/4. Consider, next, an ensemble comprising a mixture of linear superpositions of SU (2) correlated states. The density operator in the S U ( 2 ) basis is not diagonal. Of particular interest is an ensemble of states of the form
(Am) 2 =m 2 --m
IS(n) ) = [ In/2; 1/2) + In~2: - 1/2)
C ( D I , D2) = (An)2/4 - (Am)2(cos 0) z 2 )(sin 0)2/2,
-n2/4+~/2-m
(12a)
where (An) 2 = n 2 - n 2
(12b)
2
(12c)
is the variance about r~ = (n~ - n 2 )/2, one half of the difference in particle number o f the entering beams. Eq. (12a) may also be expressed in the form C ( D I , D 2 ) = C j 2 ( c o s 0)2 + [(An) 2 -- rT-- 2nl n2 ](sin 0)2/4 ,
(12d)
where C~2 =nl/'/2 - n , rt 2 = ( z ~ n ) 2 / 4 - ( A m ) 2 .
(12e)
Consider the following special cases: (A) For an input beam all particles of which enter port 1 (the case previously treated in ref. [ 15 ]), one has ( A n ) 2 = 4 ( A m ) 2 = ( A n l ) 2, and the cross-correlation reduces to eq. (7b) as expected. (B) If the input is in a pure S U ( 2 ) correlated state 30
]/x,,"~2
(n odd integer).
( 14a )
with (An)2> 1. It has been shown [16] that such states lead to a phase sensitivity of 1/n (rather than 1/v/n for single-port states) in a fermion interferometer. It will be shown here that statistical bunching can result from an ensemble of such states. The crosscorrelation is given by C ( D ~ , D 2 ) = [(An) 2 - (cos 0)~]/4 +[(An)2--(~+3)(t~--l)/2](sinO)2/8.
(14b)
For 0 = 0 a positive cross-correlation results from the condition (An)2> 1, which is characteristic of the proposed ensemble. In conclusion, it has been demonstrated that par-
Volume 124, number 1,2
PHYSICS LETTERSA
ticular ensembles of multiparticle fermion states, wherein the fermions enter both i n p u t ports of a ferm i o n interferometer, can give rise to positive crosscorrelation of beams at the output ports a n d hence to statistical bunching. In particular, fermion bunching will be manifested (for zero phase difference between b e a m c o m p o n e n t s ) by an ensemble of those S U ( 2 ) correlated states (the I S ( n ) ) states) that enhance the fermion interferometer phase sensitivity. It should be noted that the results obtained are i n d e p e n d e n t of the m o m e n t u m spin-projection distribution (providing that the condition of quasim o n o c h r o m a t i c i t y is m a i n t a i n e d ) . Although there is at present no k n o w n source of S U ( 2 ) m u l t i f e r m i o n correlated states, there is no theoretical barrier to their existence. The striking statistical properties of these states and their linear superpositions (which resemble in some ways "squeezed" optical states [ 18 ]) should provide incentive for their experimental realisation. This work was partially supported by a Faculty Research G r a n t of T r i n i t y College.
14 September 1987
References [1] R. Hanbury Brown and R.Q. Twiss, Phil. Mag. 45 (1956) 663. [2] R. Hanbury Brownand R.Q. Twiss, Nature 177 (1956) 27. [3] E.M. Purcell, Nature 178 (1956) 1449. [4] F.D. Kahn, Opt. Acta 5 (1958) 93. [5] F.T. Arecchi, E. Gatti and A. Sona, Phys. Lett. 20 (1966) 27. [6] R. Loudon, The quantum theory of light (Clarendon, Oxford, 1983) p. 220. [7] H.J. Carmichael and D.F. Walls, J. Phys. B 9 (1976) L43, 1199. [8] H.J. Kimble, M. Dagenais and L. Mandel, Phys. Rev. Lett. 39 (1977) 691. [9] S. Boffi and G. Caglioti, Nuovo Cimento B 41 (1966) 651. [ 10] S. Boffi and G. Caglioti, Nuovo Cimento B 3 (1971) 262. [ 11 ] M.P. Silverman, Phys. Lett. A 120 (1987) 442. [ 12] M.P. Silverman, Nuovo Cimento D 97 (1987) 200. [13] M.P. Silverman, Effects of potentials on fermion antibunching, Schr6dinger Centenary Conference, Imperial College, London (31 March-3 April 1987); Theoretical study of electron antibunching in a field-emission beam, Meeting of the Americal Physical Society, Arlington, Virginia (20-23 April 1987). [14] M.P. Silverman, Phys. Lett. A 118 (1986) 155. [ 15] M.P. Silverman, Phys. Lett. A 122 ( 1987 ) 226. [ 16] B. Yurke, Phys. Rev. Lett. ( 1986 ) 1515. [17] L.I. Schiff, Quantum mechanics, 3rd Ed. (McGraw-Hill, New York, 1968) p. 508. [ 18] J.R. Klauder, Bull. Amer. Phys. Soc. 32 (1987) 1087.
31
PHYSICS LETTERS A
14 September 1987
FERMION ENSEMBLES THAT S H O W STATISTICAL BUNCHING M.P. SILVERMAN Department of Physics, Trinity College, Hartford, CT06106, USA Received 3 June 1987; accepted for publication 8 July 1987 Communicated by J.P. Vigier
Contrary to the general expectation that Fermi-Dirac statistics limit fermion clustering behaviour to that of antibunching, it is shown that particular fermion ensembles can give rise to fermion bunching. These ensembles comprise correlated multifermion states entering both input ports of an SU(2) fermion interferometer. The beams leaving the two output ports exhibit a positive cross-correlation under appropriate conditions.
It is well known theoretically and demonstrated experimentally that photons manifest a variety of clustering behaviours depending upon the coherence properties of the source. For example, chaotic light gives rise to photon bunching, as demonstrated by Brown and Twiss [ 1,2] and interpreted specifically in a quantum statistical context by Purcell [ 3] and Kahn [4]. Coherent light (e.g. laser light above threshold) manifests no bunching; photon counts are distributed randomly in time as demonstrated by Arecchi et al. [ 5]. Photon antibunching occurs for quantum states of light with fixed photon number (see discussion in ref. [6]) which is a system very difficult to produce experimentally and for resonance fluorescence from single two-level atoms, as predicted by Carmichael and Walls [7] and demonstrated by Kimble et al. [8]. The fundamentally significant problem of elucidating the clustering behaviour of fermion beams has so far been largely unexplored. In the absence of a thorough general study, it is widely suspected that fermions always exhibit clustering of the antibunching type as a result of the Fermi-Dirac limitation of one fermion per cell of phase space. Purcell [3], in his analysis of the Brown-Twiss experiment, pointed out that a split beam of thermal electrons should exhibit a negative cross-correlation indicative of fermion antibunching. Subsequent detailed studies have treated the observability of antibunching in beams of thermal neutrons [ 9,10] and field-emission elec-
trons [ 11 ]. No example of clustering behaviour other than antibunching has so far been reported for fermions. It is the objective of this paper to show that fermion bunching is theoretically possible. It will be demonstrated that, as is the case with light, the nature of fermion fluctuations and correlations depends not only on the type of quantum statistics (which is universal for all fermions), but also on the specific linear superpositions of quantum states encompassing the particle ensemble. I have shown previously [ 12 ] that fermion ensembles for which the density operator factorises in a momentum-spin projection representation into a product of single-particle momentum-spin projection distribution functions give rise to fermion antibunching. Density operators with this property characterise diverse fermion sources among which are thermal electrons, thermal neutrons, and field-emission electrons. In this paper I consider particular ensembles of highly correlated multi-fermion states for which the density operator does not so factorise; I show that these ensembles manifest a positive cross-correlation, and hence statistical bunching, in a split-beam experiment of the Brown-Twiss kind. (Preliminary reports of positive fermion cross-correlations were given in ref. [ 13 ].) In previous studies of the quantum effects of a vector potential and gravitational potential on the clustering behaviour of fermions [ 14,15 ], experimental configurations employing a single fermion
0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
27
Volume 124, n u m b e r 1,2
PHYSICS LETTERS A
14 S e p t e m b e r 1987
relations. The input and output annihilation operators for a passive '~ interferometer are related to within a global phase factor as follows:
R
d , ( k , s ) = ( l r ~ r ~ l e '°'e+ltLt2te ioJ2)bj(k,s)
!o
+ i ( l & t e le i°/e-[t, relel°/2)b~_(k,s),
(2a)
d2(k, s) = - i ( It, r~_le io,e_ Ir, le le i°'2) b,(k,s) + ( Ir~r_~ le i°~2+lt~tzlei°"~-)b2(k,s),
(2b)
BS 2
$2
L ~
Fig. 1. F e r m i o n M a c h - Z e h n d e r interferometer. F e r m i o n s from sources S j, Se, enter through input ports 1, 2, respectively; b e a m s exiting the o u t p u t ports to detectors D~, De are cross-correlated by c o u n t e r / c o r r e l a t o r C.
beam at the input port of the fermion interferometer were analysed. Fig. 1. illustrates a Mach-Zehnder fermion interferometer wherein both input ports are utilised. The two entering beams from sources Sl and $2 are split at beam splitter BSI, pass via mirrors M~ and M2 to beam splitter BS2 and thence to detectors D. and D2 whose instantaneous outputs are crosscorrelated by counter/correlator C and displayed by recording device R. The fermion field operators for both input and output ports are expressible (up to an irrelevant normalisation factor) as
~P,( x, t) = ~, b,( k, s) exp[ i( k x - o J t ) ] , k,s
/ = i n p u t ports 1,2,
(la)
where (r~,t~) and (r>t~) are the reflectance and transmittance amplitudes of BS~ and BS2, respectively, and 0 is the phase difference incurred between upper and lower beam components. It is assumed that the beam is quasimonochromatic, so that the momentum spread is much less than the main particle momentum, Ak<< ko, for which case 0 is, to a good approximation, independent of the momentum distribution. (There are also cases, such as in the Aharonov-Bohm effect [ 14] where 0 depends only on an external parameter (magnetic flux) and is rigorously independent of path length and momentum distribution.) In what follows BSI and BS2 are taken to be 50-50 beam splitters, i.e. Ir,12= It,12= 1/2. The operators corresponding to number of particles received at D~ and D2 during an arbitrary sampiing time are N(D,)=~d~(k,s)~d,(k,s),
i=1,2,
(3)
k,s
the mean number of counts received by each detector is correspondingly N(D,) =Tr{fiN(D,) ~ ,
(4)
where fi is the density operator of the input ensemble. The cross-correlation of particle fluctuations [ 15 ] is given by C(DI,
D2)
= Tr{/J[N(DI ) - N ( D , )] [N(D2) - N(De) ] }
cI)~(x, t) = ~ d,(k, s) e x p [ i ( k x - o ) t ) ] , k,s
=Tr{I~]V(D,)N(D2)}-N(D,)N(D2). i = output ports 1,2,
( 1b)
where the fermion input (output) annihilation and creation operators, b,(k, s) and bt,(k, s) (d,(k, s) and d~(k, s)) satisfy standard fermion anticommutation 28
(5)
~ By a passive interferometer is m e a n t one that preserves the particle d i s t r i b u t i o n over m o m e n t u m states. This requires that r,t, be pure imaginary, and IrA 2 + I t,I 2 = 1. The phases have been chosen so that r, = I r, I exp ( in/2 ), t, = I t, I-
Volume 124,number 1,2
PHYSICSLETTERSA
The density operator characterising a two-port fermion input may be expanded in eigenstates ]nl(ks);nE(k's')) of the input particle number operators
N t = ~ bi(k,s)*b,(k,s),
i=1,2,
(6)
k.s
is the operator for total number of particles passing through the interferometer during a counting time, and
I~=-
with eigenvalues nl, r/2 for ports 1, 2, respectively. The designation n~(ks) in the ket label is an abbreviation for the momentum and spin projection distribution [(kl, Sl )(k2, $2)...(kni, Sn,)] over n~particles. The total number of particles passing through the interferometer is conserved, i.e. Nl +N2 = At(D,) +N(D2). For a single-port fermion input of mean particle number n~- and variance (Anl)2, characterised by a density operator of the "diagonal" form
~= ~ p(n~(ks))lnl(ks);O)(nl(ks);OI,
(7a)
14 September 1987
~. [bl(k,s)*b2(k',s')*b2(k,s) bl(k',s') k,s,k' ,s'
+bl(k,s)tb2(k',s')*b2(k',s ') bl(k,s)] .
(8f)
The operators Ji (i= x, y, z) satisfy the commutation relations of angular momentum operators; in addition [Z, ff/] =0
(9a)
and [J,, N] = 0 .
(9b)
the cross-correlation has been shown to be [ 15 ]
The operators needed for determination of the cross-correlation may be expressed in terms of N and the SU(2) generators by means of the relations
C(DI, D2) = [(An~ )2 - h~-~](sin 0)2/4,
~r(Dl ) =N/2 + Jz cos 0 + J , sin 0,
(10a)
J(D2) = ~ / 2 - ~
(10b)
nl,k,s
(7b)
where the fermion ensemble is subpoissonian. The variance is thus less than the mean, and the resulting negative cross-correlation is indicative of antibunching [ 3 ]. For values of 0 for which sin 0 vanishes the output beams are uncorrelated. Considerations of phase sensitivity in an interferometer have led recently to the introduction of a set of correlation fermion states [ 16] that are basis states of an SU(2) algebra constructed from the fermion annihilation and creation operators as follows J,- = ½Z [bl(k,s)*b2(k,s) k,s
+b2(k, s)*bl (k, s)] ,
(8a)
Jv - 1 ~ [bt(k,s)*b2(k,s) • -
2i
~,s
-b2(k,s)*bl(k,s)] , L = 1 (]Wl--N2),
(8b)
(8c)
the Casimir operator is of the form `72 = (]V/Z) [(~/72) + 1] + fig, where
cos 0 - J x sin 0,
]VD=/~r(Dl )-N(D2) =2(Jz cos 0 +Jx sin 0) , N-(D, ) fi(D2) = [/~(Dl )2 +~(D2)2 - N ~ ] / 2 .
(10d)
SU(2) basis states of the form In/2; m ) , which are eigenstates of j2 and J: with eigenvalues (n/2)[(n/2) + 1] and m, respectively, (i.e. eigenstates of ~r and /~ with eigenvalues n and 0, respectively) can be constructed from the Inx; n2) basis through use of "raising" and "lowering" operators ,7+=,Tx+,Tr The SU(2) correlated state In/2; n/2) is equivalent to the particle number state In;0) corresponding to all particles entering port 1. Repeated application of J generates the remaining states [j;m) of the (j=n/2; W=0) irreducible representation according to In/2; m) = [n!(n/2-m)!/(n/2+ra)!]-1/2
xJn_/2-m In~2;n12) . (8d)
(10c)
(1 la)
The state In/2; m) resulting from n / 2 - m applications of J_ therefore has n / 2 - m fermions entering 29
Volume 124, number 1,2
PHYSICS LETTERS A
port 2 and n / 2 + m fermions entering port 1. For example, in a two-particle system comprising quantum states a = (kl, s l) and b = (k2, s2 ) the S U ( 2 ) correlated states may be expressed as 1;1 ) = I ( a , b ) ; O ) , 1;0) = [ I b ; a ) - l a ; b )
(1 lb) ]/2 ,
1 . - 1 ) = - I 0;(a,b) >
(llc) (1 ld)
(the phase convention for application of annihilation and creation operators to fermion states is given in ref. [17]). The SU (2) correlated states form a complete basis with respect to which one can expand the density operator of the two-port fermion input. Consider first an ensemble of fermion input states characterised by a density operator diagonal in a basis o f SU (2) correlated states. From eqs. (2), (3), (5), (10) it may be shown that the cross-correlation of outputs at D, and D2 is
14 September 1987
of sharp n and m, then the variances (An) 2 and ( A m ) 2 vanish, and the eigenvalues n, = n / 2 + m and n 2 = n / 2 - m are also sharp. The cross-correlation. which is expressible as C ( D I , De) =-
[n~n2 + (n, +n~)/2](sin 0)2/2
(13a)
is again intrinsically negative except at selected phase angles for which it vanishes. (C) In the more general case of an input beam comprising an appropriate mixture of S U ( 2 ) correlated states there is a range of angles, given by 4Ci2/[2nl n2 + n-- (An) e ] > (tan 0) -~
(13bl
for which the cross-correlation can be positive. For example, at 0 = 0 the condition (An) 2 > 4(Arn) 2
(13c)
is the variance about the mean total particle number r~ entering the interferometer and
results in a positive cross-correlation. This condition is most directly met by an ensemble of variable total particle number ( ( A n ) 2 > 0 ) , but fixed particle difference ( ( A m ) 2 = 0 ) at the input ports. An input comprising the states In/2; m ) and 1n/2+2: rn) with Iml <~n/2 leads to C(D~, D 2 ) = C l 2 = 1/4. Consider, next, an ensemble comprising a mixture of linear superpositions of SU (2) correlated states. The density operator in the S U ( 2 ) basis is not diagonal. Of particular interest is an ensemble of states of the form
(Am) 2 =m 2 --m
IS(n) ) = [ In/2; 1/2) + In~2: - 1/2)
C ( D I , D2) = (An)2/4 - (Am)2(cos 0) z 2 )(sin 0)2/2,
-n2/4+~/2-m
(12a)
where (An) 2 = n 2 - n 2
(12b)
2
(12c)
is the variance about r~ = (n~ - n 2 )/2, one half of the difference in particle number o f the entering beams. Eq. (12a) may also be expressed in the form C ( D I , D 2 ) = C j 2 ( c o s 0)2 + [(An) 2 -- rT-- 2nl n2 ](sin 0)2/4 ,
(12d)
where C~2 =nl/'/2 - n , rt 2 = ( z ~ n ) 2 / 4 - ( A m ) 2 .
(12e)
Consider the following special cases: (A) For an input beam all particles of which enter port 1 (the case previously treated in ref. [ 15 ]), one has ( A n ) 2 = 4 ( A m ) 2 = ( A n l ) 2, and the cross-correlation reduces to eq. (7b) as expected. (B) If the input is in a pure S U ( 2 ) correlated state 30
]/x,,"~2
(n odd integer).
( 14a )
with (An)2> 1. It has been shown [16] that such states lead to a phase sensitivity of 1/n (rather than 1/v/n for single-port states) in a fermion interferometer. It will be shown here that statistical bunching can result from an ensemble of such states. The crosscorrelation is given by C ( D ~ , D 2 ) = [(An) 2 - (cos 0)~]/4 +[(An)2--(~+3)(t~--l)/2](sinO)2/8.
(14b)
For 0 = 0 a positive cross-correlation results from the condition (An)2> 1, which is characteristic of the proposed ensemble. In conclusion, it has been demonstrated that par-
Volume 124, number 1,2
PHYSICS LETTERSA
ticular ensembles of multiparticle fermion states, wherein the fermions enter both i n p u t ports of a ferm i o n interferometer, can give rise to positive crosscorrelation of beams at the output ports a n d hence to statistical bunching. In particular, fermion bunching will be manifested (for zero phase difference between b e a m c o m p o n e n t s ) by an ensemble of those S U ( 2 ) correlated states (the I S ( n ) ) states) that enhance the fermion interferometer phase sensitivity. It should be noted that the results obtained are i n d e p e n d e n t of the m o m e n t u m spin-projection distribution (providing that the condition of quasim o n o c h r o m a t i c i t y is m a i n t a i n e d ) . Although there is at present no k n o w n source of S U ( 2 ) m u l t i f e r m i o n correlated states, there is no theoretical barrier to their existence. The striking statistical properties of these states and their linear superpositions (which resemble in some ways "squeezed" optical states [ 18 ]) should provide incentive for their experimental realisation. This work was partially supported by a Faculty Research G r a n t of T r i n i t y College.
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References [1] R. Hanbury Brown and R.Q. Twiss, Phil. Mag. 45 (1956) 663. [2] R. Hanbury Brownand R.Q. Twiss, Nature 177 (1956) 27. [3] E.M. Purcell, Nature 178 (1956) 1449. [4] F.D. Kahn, Opt. Acta 5 (1958) 93. [5] F.T. Arecchi, E. Gatti and A. Sona, Phys. Lett. 20 (1966) 27. [6] R. Loudon, The quantum theory of light (Clarendon, Oxford, 1983) p. 220. [7] H.J. Carmichael and D.F. Walls, J. Phys. B 9 (1976) L43, 1199. [8] H.J. Kimble, M. Dagenais and L. Mandel, Phys. Rev. Lett. 39 (1977) 691. [9] S. Boffi and G. Caglioti, Nuovo Cimento B 41 (1966) 651. [ 10] S. Boffi and G. Caglioti, Nuovo Cimento B 3 (1971) 262. [ 11 ] M.P. Silverman, Phys. Lett. A 120 (1987) 442. [ 12] M.P. Silverman, Nuovo Cimento D 97 (1987) 200. [13] M.P. Silverman, Effects of potentials on fermion antibunching, Schr6dinger Centenary Conference, Imperial College, London (31 March-3 April 1987); Theoretical study of electron antibunching in a field-emission beam, Meeting of the Americal Physical Society, Arlington, Virginia (20-23 April 1987). [14] M.P. Silverman, Phys. Lett. A 118 (1986) 155. [ 15] M.P. Silverman, Phys. Lett. A 122 ( 1987 ) 226. [ 16] B. Yurke, Phys. Rev. Lett. ( 1986 ) 1515. [17] L.I. Schiff, Quantum mechanics, 3rd Ed. (McGraw-Hill, New York, 1968) p. 508. [ 18] J.R. Klauder, Bull. Amer. Phys. Soc. 32 (1987) 1087.
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