Anomalous fermion bunching in density-density correlation

0.16 MB·English

by  Hongwei Xiong

Anomalous fermion bunching in density-density correlation
Hongwei Xiong∗1,2 and Shujuan Liu1,3,2
1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, P. R. China
2Graduate School of the Chinese Academy of Sciences, P. R. China
3Center for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, P. R. China
7 (Dated: February 6, 2008)
0
0 We consider theoretically density-density correlation of identical Fermi system by including the
2 finiteresolution of a detector and delta-function term omitted in the ordinary method. Wefindan
anomalous fermion bunching effect, which is a quantum effect having no classical analogue. This
n
anomalous fermion bunchingis studied for ultracold Fermi gases released from a three-dimensional
a
opticallattices. Itisfoundthatthisanomalousfermionbunchingissupportedbyarecentexperiment
J
(T. Rom et al Nature444, 733 (2006)).
9
1

Demonstrated firstly by Hanbury Brown and Twiss tribution is considered, there would be an anomalous
]
r [1], quantum noise correlation is a fundamental prob- fermion bunching effect under appropriate conditions.
e
h lem in identical Bose and Fermi system, and has im- This anomalousfermion bunching effect originatesphys-
t portant application in astronomy, quantum optics, con- ically from the delta function in the anticommutation
o
densed matter physics and subatomic physics [2, 3, 4, 5] relation of fermion field operators, omitted in the ordi-
.
at etc. In the last ten years,quantum noise correlationwas nary method in calculating the density-density correla-
m studied experimentally for cold Bose and Fermi gases tion. Theconsiderationoffiniteresolutionofthedetector
[6, 7, 8, 9, 10, 11, 12, 13, 14] with the remarkable de- will avoid the notorious divergence in the delta-function
-
d velopment of cold atom physics. Together with these term, and lead to anomalous bunching effect. The ob-
n
experimentaladvances,intensive theoreticalstudies con- servation of obvious anomalous fermion bunching effect
o
tribute largely to our understanding of correlation effect requestsspecialconditions andparameters. Fortunately,
c
[ for cold atomic system. The theoretical studies about we notice that there is already an anomalous fermion

high-order correlation have extended from a single har- bunching in the experimental data of Ref. [13].
1
monicallytrappedBosegas[15] to differentmatter state Because we will give a comparison between our the-
v
1 such as pair correlations of fermionic superfluid [16, 17] ory and experimental data to support the anomalous
7 anddifferenttrappingpotentialsuchashigh-ordercorre- fermion bunching, we first introduce briefly the elegant
4 lationof coldatoms in an opticallattice [18]. The finite- experimentinRef. [13]. In this experiment,anultracold
1 temperature effect [19] and low-dimensional effect [20] Fermigasof40Kwasfirstlypreparedinthecombinedpo-
0
about high-order correlation were also studied theoreti- tential of an optical trap and a 3D (three-dimensional)
7
0 cally. optical lattice. After free expansion of 10 ms by sud-
/ Thetheoreticalstudiesforfermionicsuperfluid[16,17] denly switching off the combined potential, 2D (two-
t
a andone-dimensionalultracoldFermigas[21]haveshown dimensional) density images were recorded with a CCD
m
that quantum noise correlation provides important in- (charge-coupled device) camera by illuminating a reso-
- formation about the fundamental quantum features of nantlaser along the vertical(z) direction. See Fig. 1(a).
d
n ultracold Fermi gases. Most recently, there is a clear ob- The2Ddensity-densitycorrelationwasobtainedbydeal-
o servation of fermion antibunching in a degenerate Fermi ing with appropriately a set of 2D density images.
c gasreleasedfromathree-dimensionalopticallattice[13]. The starting point of our theory is the 2D density-
v: The fermion antibunching physically arises from the an- density correlation function given by
arXi tncilocuoscmiloamnssupictraainlticaoinnpalerloeilgnauteiho.ingIhto-foisrfidaeelrdmcaoonprierfreeaslattatoitroison,n.anoBdfePstihaduueslsihtehaxes- C2(d,t)= hRhnh(hrnb(−r−d/d2,/t2),iti)dnbh(hrn(+rd+/d2,/t2),iti)didi2drd2r.
R (1)
observation of the antibunching effect in density-density b b
Here n is a 2D density operator. r ≡ {x,y} and
correlation for fermionic atoms released from an optical
d ≡ {d ,d }. To deal with appropriately the layered
x y
lattice [13], fermion antibunching was also observed for b
distribution of fermionic atoms in 3D optical lattice and
electrons [22, 23, 24] and neutrons [25].
absorptionimagingalongz direction, the 2Ddensity op-
In the present Letter, we find that when the resolu- erator takes the form n(r,t) = gΨ†(r,t)Ψ(r,t). The op-
tion of a detector for the measurement of density dis- erator g has the propberty thatbbf(g)a†ibjaij = f(gij).
D E
g = f with f being the occupation number
ij bkz ijkz ijkz b b b
inthePlatticesiteindexedby {i,j,k }. Thesummationis
z
∗[email protected] aboutthelatticesitealongz direction. gij representsthe
2
overall particle number in a string of lattice sites along withzerovalueoutsidethisregion. Whenboththedelta-
z direction. a is an annihilation operator for the atom functiontermandspatialresolutionareconsidered,more
ij
in an equivalent 2D lattice site indexed by {i,j}. The accurate results are obtained by calculating
b
2D density-density correlation function is dependent on
d. C2(d,t) < 1 corresponds to the fermion antibunch- hhn(r−d/2,t)n(r+d/2,t)iid =
ing effect, while C (d,t) > 1 corresponds an anomalous
2 b b
fermion bunching effect.
d2s d2s hn(r−d/2+s ,t)n(r+d/2+s ,t)i.
We give here a brief introduction about the reasonfor Z 1Z 2 1 2
Ξ Ξ
the aboverules in the transformationfrom3Dsystemto b b (5)
2D density-density correlation measured by a CCD. Be- Here Ξ denotes the region −∆ /2 < s < ∆ /2 and
d x d
cause the density images were recordedin Ref. [13] with −∆ /2 < s < ∆ /2. In the above formula, hi repre-
d y d d
a CCD camera by illuminating a resonant laser along z sents the average due to the finite resolution of the de-
direction,specialconsiderationshouldbegivenaboutthe tector. This average has been considered in the starting
2D density-density correlation for a 3D system. The 2D point (1).
density distribution is From equation (1), we get the following approximate
result
n(r,t) = dz Ψ†(r,z,t)Ψ(r,z,t)
= ZΣijgijD|φbij(r,t)|2b. E (2) C2(d,t)≈1− (cid:12)(cid:12)Σjpgjpe−i2Nπ(2dxj+dyp)/l(cid:12)(cid:12)2 + 2Sπl2∆lp24dΣσi2jNgi22j.
(6)
Here φ (r,t) is the wave function of the atom initially Here l = 2π~t/ml with m being the atomic mass and
ij p
in the lattice site {i,j}. Because in 2D density-density l being the spatial period of the optical lattice. S is
p
correlation, the coordinate z has been integrated, it is the area of the overlapping region between Ξ (deter-
1
convenient to define the 2D density operator as mined by −∆ /2 < x < ∆ /2 and −∆ /2 < y < ∆ /2)
d d d d
and Ξ (determined by −∆ /2 < x−d < ∆ /2 and
2 d x d
n(r,t)=gΨ†(r,t)Ψ(r,t). (3) −∆ /2 < y−d < ∆ /2). N is the total particle num-
d y d
ber. In equation (6), the second term represents the
b bb b
Withtheaboverulesaboutg,hn(r,t)iisthesameasthe antibunching behavior. This term is obtained by omit-
result given by equation (2). With this definition of 2D ting the effect of spatial resolution and under the con-
b b
density operator and g, it is straightforward to get the dition l/σ > L /l and l ≫ ∆ l /σ. Here L and σ
0 p d p 0
finalresultofthe2Ddensity-densitycorrelationfunction. represent respectively the overallwidth of the Fermi gas
b
From Eq. (1), it is easy to get and wavepacketwidth of an atom in a lattice site before
switching off the combined potential. The last term in
hn(r−d/2,t)n(r+d/2,t)i equation(6)physicallyoriginatesfromthedelta-function
= bg2Ψ†(r−d/b2,t)Ψ(r+d/2,t) δ(d) term in the anticommutation relation of field operators.
D E This term is obtained by considering the spatial resolu-
+bhnb(r−d/2,t)ihnb(r+d/2,t)i tion and under the condition l/σ>L0/lp.
2 Using the experimental parameters in Ref. [13] and
− bgΨ†(r−d/2,tb)Ψ(r+d/2,t) . (4)
equation (6), Fig.lb gives C (d,t) for flight time of 10
(cid:12)D E(cid:12) 2
(cid:12)(cid:12) bb b (cid:12)(cid:12) ms. Besides eight dark dots representing antibunching,
The delta-function term in the above formula is or-
atthecenterlocationwenoticeabrightdotrepresenting
dinarily omitted [13, 26] because the divergent prop-
bunching. Fig.1cgivesfurther one ofthe darkdot, while
erty and δ(d) = 0 for d 6= 0. This term originates
Fig.1d gives further the bright dot. Eight dark dots are
from the delta function in the anticommutation rela-
due to the second term in equation (6). This term is −1
tion Ψ(r+d/2,t),Ψ†(r−d/2,t) =δ(d),andthusac-
ford /l=i andd /l=j with i andj being integers. At
n o x y
countsbfor a pure quabntum effect. the locations of these dark dots, the last term in equa-
Bunching (antibunching) corresponds to a peak (dip) tion (6) is zero. The bright dot at the center is due to
in density-density correlation. For d → 0, the delta- thelastterminequation(6),whichislargerthan1close
function term in equation (4) means a divergent bunch- to d = 0 and d = 0. If the last term in equation (6)
x y
ingbehavior. Whenthefinitewidth∆ ofspatialresolu- is not included, there should be a dark dot at the center
d
tion is considered, it is understandable that there would (see Fig.1e). In Ref. [13], the theoretical model with-
be an effect of increasing the width and decreasing the out the last term is used to interpret their experimental
height (depth) in the peak (dip) of the bunching (an- data. It is quite interesting to note that eight dark dots
tibunching) behavior. Roughly speaking, δ(d) may be (rather than nine dark dots without the considerationof
replaced by a function with height 1/∆2 in the region the last term in equation (6)) were observed in Ref. [13]
d
−∆ /2 < d < ∆ /2 and −∆ /2 < d < ∆ /2, and (see Fig.2c and its figure caption in this reference)! In
d x d d y d
3
field theory. The analyses about δ2n(x,t) show clearly
thattheδ-functiontermcannotbeomittedsimply. Sim-
ilarlytoourstudiesofthedensity-densitycorrelation,the
divergenceintheδ-functiontermcanbeavoidedbecause
the resolution of a detector always has a width. This is
equivalent to the presentation that creation or annihila-
tion of a particle at an infinitesimal point is impossible
because this would mean an infinite energy exchange.
Notethattheanomalousfermionbunchingdoesnotvi-
olateinanysensethePauliexclusionprincipleforidenti-
calfermions. Theanomalousfermionbunchingoriginates
fromthe delta-function term in the anticommutationre-
lation of field operators. Of course, the Pauli exclusion
FIG. 1: Fermion antibunching and bunching in density- principle will try to destroy the anomalous bunching ef-
densitycorrelation function. (a)Themeasurementofdensity fect. The second term in (6) reflects the Pauli exclusion
distributionforaFermigasreleasedfromathree-dimensional
principle, and leads to the antibunching effect, and even
optical lattice. (b) Two-dimensional density-density correla-
cancompletelydestroythe bunching effectatthe center.
tionobtainedfromequation(6)andexperimentalparameters
in Ref. [13]. The theoretical results agree with the exper- In summary, our studies give an anomalous fermion
imental data. In particular, an anomalous fermion bunch- bunching, and we have given a strong evidence by ana-
ing is shown clearly by the bright dot at the center, which lyzing a recent experimental data in Ref. [13]. The con-
has been supported strongly in Ref. [13]. (c) Illustration of dition to observe the anomalous fermion bunching is in
a dark dot (antibunching effect) in Fig.1b. (d) Illustration
fact quite rigorous. It is not surprising that this unique
of the bright dot (bunching effect). (e) The density-density
quantum effect is found accidentally in Ref. [13] with
correlation near the center without the consideration of the
highly developed experimental technique. The experi-
delta-function term in the ordinary theory. In these figures,
thespatial coordinate is in unit of µm. mental technique in Ref. [13] gives us chance to test
further our theory. (i) The last term in equation (6)
increaseswith the increasingofflighttime. Thus, we ex-
Fig.2d of Ref. [13], the obvious and “mysterious” peak pectatransitionprocessfromtheantibunchingtobunch-
atthe centerofdensity-densitycorrelationshowsfurther ing at the center of C (d,t), with the increasing of the
2
the anomalous fermion bunching behavior. This com- flighttime. (ii)Thedistributionoffermionicatomsinthe
parison gives strong experimental evidence for anoma- latticesites{i,j,k }isdeterminedbyi2+j2+α2k2 ≤R2
z z z
lous fermion bunching. It is clear that the condition for at zero temperature. Here α is determined by the har-
z
observing fermion bunching is l2lp2Σijgi2j >2π∆2dσ2N2. monic potential due to the optical trap. Simple calcu-
The measurement of density-density correlation ex- lations give Σ g2/N2 ≃ 0.93(α N)−2/3. We see that
ij ij z
tracts the correlationinformation in quantum noise. We decreasing α has an effect of enhancing the anomalous
z
discuss further the physical mechanism of the bunching fermion bunching effect for identical particle number.
effectforFermisystemthroughtheconsiderationofden- This dependence on α would give us further chance to
z
sity fluctuations. The density fluctuations are defined test our theory. We believe further experimental stud-
as ies would deep largely our understanding of the indis-
tinguishability of identical particles, wave packet local-
δ2n(x,t)= lim hn(x,t)n(y,t)i−hn(x,t)i2. (7)
y→x ization in quantum measurement process and high-order
b b b correlation. High resolution of a detector is required to
Simple calculations give
revealanomalousfermionbunching effect. The astonish-
ing resolution of STM (scanning tunneling microscope)
δ2n(x,t)= lim Ψ†(x,t)Ψ(y,t) δ(x−y)−hn(x,t)i2.
y→xD E and AFM (atomic force microscope) etc suggests that
b b b (8) the experimental and theoretical studies about electrons
We see that there is a divergent δ-function term in in an atom or molecule may lead to a new regime about
δ2n(x,t). Itisobviousthattheomissionoftheδ-function the studies of high-order correlation,quantum noise and
termwillleadtoabsurdresultofnegativedensityfluctu- quantum measurement etc.
ations. Thedivergentδ-functiontermisapurequantum The δ-function term in the anticommutation relation
effect by noting that it comes fromthe anticommutation is the direct reason for the anomalous fermion bunching
relation between field operators. There is another origin effect. For Bose system, the inclusion of the δ-function
for this δ-function term that the field operators Ψ(x,t) term in the commutation relation of field operators may
and Ψ†(x,t) comprise infinite modes, rather than only alsoplayimportantroleindensity-densitycorrelation. It
b
the modes occupied by particles before a measurement. iseasytounderstandthattheinclusionoftheδ-function
b
Infact,thispropertyisanessentialpropertyofquantum term will lead to an enhanced boson bunching effect at
4
d=0. Infact,inarecentexperimentaboutthedensity- WestbrookandI.Bouchoule,Phys.Rev.Lett.96,130403
density correlation of ultracold bosonic atoms released (2006).
from an optical lattice, there is a clear enhanced bo- [12] I. B. Spielman,W. D.Phillips and J. V.Porto, Preprint
cond-mat/0606216 (2006).
son bunching effect at d = 0 in C (d,t) (see Fig. 2(c)
2
[13] T. Rom, Th. Best, D. van Oosten, U. Schneider, S.
in [8], where the brightness in the center dot is much
F˝olling,B.ParedesandI.Bloch,Nature444,733(2006).
higher than other eight bright dots, and this can not
[14] T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen,
be explained without the considerationof the δ-function V.Krachmalnicoff, M.Schellekens,A.Perrin,H.Chang,
term.). This sort of enhanced boson bunching effect is D. Boiron, A. Aspect and C. I. Westbrook Preprint
also implied strongly in [12]. These analyses show that cond-mat/0612278 (2006).
anomalous fermion bunching effect and enhanced boson [15] M. Naraschewski and R. J. Glauber, Phys. Rev. A 59,
4595 (1999).
bunching effect have common physicalorigin—the indis-
[16] Ehud Altman, Eugene Demler and Mikhail D. Lukin,
tinguishability of identical particles and the δ-function
Phys. Rev.A 70, 013603 (2004).
term in the anticommutation (commutation) relation of
[17] C. Lobo, I. Carusotto, S. Giorgini, A. Recati and S.
field operators. Stringari, Phys. Rev.Lett. 97, 100405 (2006).
This work is supported by NSFC under Grant Nos. [18] Radka Bach and Kazimierz Rzaz˙ewski, Phys. Rev. A
֒
10634060, 10474117 and NBRPC under Grant Nos. 70, 063622 (2004); C.Ates, Ch.Moseley andK.Ziegler,
2006CB921406,2005CB724508and also funds fromChi- Phys. Rev.A 71, 061601(R) (2005).
nese Academy of Sciences. [19] N. M. Bogoliubov, C. Malyshev, R. K. Bullough and J.
Timonen, Phys. Rev.A 69, 023619 (2004).
[20] D.S.Petrov, G.V.Shlyapnikovand J. T.M. Walraven,
Phys. Rev.Lett. 85, 3745 (2000); J. O. Andersen,U.Al
KhawajaandH.T.C.Stoof,Phys.Rev.Lett.88,070407
(2002);ChristopheMoraandYvanCastin,Phys.Rev.A
[1] R.H. Brown and R. Q.Twiss, Nature 177, 27 (1956). 67,053615(2003);D.L.LuxatandA.GriffinPhys.Rev.
[2] L. Mandel and E. Wolf, Optical coherence and quantum A67,043603(2003); Jean-S´ebastienCauxandPasquale
Optics (Cambridge Univ.Press, Cambridge, 1995). Calabrese, Phys. Rev. A 74, 031605(R) (2006); N. P.
[3] M. O. Scully and M. S. Zubairy, Quantum optics (Cam- Proukakis, Phys.Rev.A 74, 053617 (2006).
bridgeUniv. Press, Cambridge, 1997).
[21] L. Mathey, E. Altman and A. Vishwanath, Preprint
[4] G. Baym, Acta Phys.Pol. B 29, 1839 (1998).
cond-mat/0507108 (2005).
[5] D.H.Boal,C.K.GelbkeandB.K.Jennings,Rev.Mod.
[22] W. D.Oliver, J. Kim, R.C. Liu and Y.Yamamoto, Sci-
Phys.62, 553 (1990). ence 284, 299 (1999).
[6] MasamiYasudaandFujioShimizu,Phys.Rev.Lett.77,
[23] M. Henny,S. Oberholzer, C. Strunk, T. Heinzel, K. En-
3090 (1996). sslin,M.HollandandC.Sch˝onenberger,Science284,296
[7] M.Schellekens,R.Hoppeler,A.Perrin,J.VianaGomes,
(1999).
D.Boiron, A.Aspectand C. I.Westbrook, Science 310, [24] H. Kiesel, A. Renz and F. Hasselbach, Nature 418, 392
648 (2005).
(2002).
[8] S.F˝olling,F.Gerbier,A.Widera,O.Mandel,T.Gericke
[25] M.Iannuzzi,A.Orecchini,F.Sacchetti,P.FacchiandS.
and I. Bloch, Nature 434, 481 (2005).
Pascazio, Phys.Rev.Lett. 96, 080402 (2006).
[9] Anton O˝ttl, Stephan Ritter, Michael K˝ohl and Tilman [26] L. E. Ballentine, Quantum mechanics: a modern devel-
Esslinger, Phys. Rev.Lett. 95, 090404 (2005). opment (World Scientific Publishing, Singapore, 1998).
[10] M. Greiner, C. A. Regal, J. T. Stewart and D. S. Jin,
Phys.Rev.Lett. 94, 110401 (2005).
[11] J. Esteve, J.-B. Trebbia, T. Schumm, A. Aspect, C. I.