Supersymmetric Response of Bose-Fermi Mixture to Photoassociation

0.43 MB·English

by  T. Shi

Supersymmetric Response of Bose-Fermi Mixture to Photoassociation
T. Shi, Yue Yu, and C. P. Sun
Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China
(Dated: January 26, 2010)
We study supersymmetric (SUSY) responses to a photoassociation(PA) process in a mixture
of Bose molecules b and Fermi atoms f which turn to mutual superpartners for a set of proper
parameters. We consider the molecule b to be a bound state of the atom f and another Fermi
atom F with different species. The b-f mixture and a free F atom gas are loaded in an optical
lattice. The SUSY nature of the mixture can be signaled in the response to a photon-induced
0 atom-molecule transition: While two new types of fermionic excitations, an individual b particle-f
1 holepaircontinuumandtheNambu-Goldstone-fermion-like(orgoldstino-like)collectivemode,are
0 concomitantforagenericb-f mixture,theformeriscompletelysuppressedintheSUSYb-f mixture
2
andthezero-momentummodeofthelatterapproachestoanexacteigenstate. ThisSUSYresponse

n can be detected by means of the spectroscopy method, e.g., the PA spectrum which displays the
a molecular formation rate of Ff →b.
J
6 PACSnumbers: 67.85.Pq,37.10.Jk,11.30.Pb
2

] Introduction — Recently, studies in the supersymmetry (a) photoassociation process (b) Feynman diagram
r
(SUSY)foramixtureofcoldBoseandFermiatomshave
e
h made spectacular progress [1–3]. In such a cold atomic F
t system, however, a Bose atom never transits to a Fermi
o
P
t. atom, its superpartner or vice verse. In addition to the Bose- Fermi mixture R
a nonrelativity, this is another essential difference of this
m low-energySUSYfromtheSUSYinhigh-energyphysics.
- Forthelatter,suchSUSYdecayprocessesarealwaysan-
d ticipated, e.g., a quark (lepton) may emit or absorb a 1
n P
gauginoanddecaystoasquark(slepton), the superpart- D 0
o 0
c ner of the quark (lepton) [4]. w
[ w 2 1
ToexposetheinterestingSUSYnatureofthemixture,
P P
3 the effective ”decay” process must be introduced. For - d Eb 0 U R
v 0 bf
a cold atomic SUSY mixture with Bose-Einstein con-
6
densation, there is an effective decay of SUSY genera-
9
9 tors since they behave as the fermion annihilation and
FIG.1: (Color online) (a)Up-panel: Theopticallattice with
3 creation operators [3]. Therefore, the SUSY excitations
cold particles. The gray (green), black (red), and white dots
9. can be simulated by a boson-enhanced fermionic exci- denote Fermi atoms f, F, and the molecule b, respectively.
0 tation. As a result, a Nambu-Goldstone-fermion-like Low panel: The PA processes of two atoms to one molecule
9 (or ”goldstino-like”) mode in the condensation phase of withthebindingenergyE . (b)TheFeynmandiagramforthe
b
0
bosons couldbe observedby means of the single-particle linearresponsetheory: TheGreenfunctionofQiscalculated
:
v spectroscopy [5, 6]. byRPA.Up-panel: Thewavy(red)linesanddotted(red)line
i denotethefreeGreenfunctionsofphotonandF,respectively.
X ToachieveanexactSUSYmixture,thesystemparam- Lowpanel: Thesolid(green)curvesanddotted(black)curves
r eters have to be fine-tuned, which requires elaborate ex- denotethe free Green functions of f and b, respectively.
a
perimental setups and then loses the generality. In this
article, we explore how to observe the SUSY response
by means of a spectroscopy measurement, even if the SUSY in relativistic theory.
mixture deviates slightly from the SUSY and the bosons We consider a mixture of Bose molecules b and Fermi
do not condense to form a whole ordered phase. This atomsf with on-siteinteractionina d-dimensionalopti-
can resolve the fine-tuning restraints in measuring the callattice (d=2,3)(see Fig.1(a)). With properly tuned
SUSY response. On the other hand, the explicit break- interactions and hopping amplitudes, this b-f mixture
ingofthe SUSYmaycreatenew excitations,the bosonic may become SUSY [3]. We are interested in a special
particle-fermionicholeindividualcontinuousexcitations, kindofmoleculeb,aboundstateoff andanotherspecies
other than the collective goldtino-like mode. Although ofFermiatomF withbindingenergyE ,andrestrictour
b
ourtheoryisnonrelativistic,thecreationofthesenewex- analysis to the normal phase of the b-f mixture [7]. To
citations due to SUSY explicit breaking should be quite probe the SUSY behaviors,we loada free Fermi atomF
general . This may be a helpful point to the study of gas, which does not interact both b and f directly. In a
2
photoassociation(PA)process[8],thetransitionsbetween The PA processes are realized by simultaneously shin-
two atoms and one molecule, i.e., Ff b, are induced ing two laser beams with frequencies ω1 and ω2 into the
↔
by two laser beams with frequencies ω1 and ω2. For the lattice (shown in Fig. 1(a)). The ω1-beam may turn two
SUSY b-f mixture, this resembles a high energy physics free atoms f and F into a higher energy bound state 1
| i
process: a ‘quark’ or a ‘lepton’ (f) absorbs a ‘fermionic which then may transit to the molecule b by emitting
gaugino’ (‘absorbs’ a F and emits a photon) and decays a photon with frequency ω2. Meanwhile, the molecule b
toa‘squark’ora‘slepton’(b)orviceverse. (Onecanalso mayalsobeexcitedto 1 bytheω2-beamandthenisun-
| i
consider f to be a Fermi molecule formed by the bound bound with some probability by emitting a photon with
stateofaBoseatombandaFermiatomF,i.e.,processes frequency ω1. For large detuning ∆0, the state 1 can
| i
Fb f. We will study these processes separately.) be eliminated adiabatically, so that the PA is modeled
↔
For a negative detuning δ0 = ω2 ω1 Eb, we show by the tight-binding Hamiltonian
− −
that the molecule dissociation process b Ff is forbid-
den. In the formationprocess Ff b, tw→o types of new Hex = (gib†ifiFieiδ0t+H.c.), (2)
→ i
fermionic excitations, an individual (bosonic) particle- X
(fermionic) hole pair continuum and a collective mode, wherethedetuningδ0 =ωc Eb withtheeffectivedriven
−
emergewhentheSUSYintheb-f mixtureisslightlybro- frequency ωc =ω2 ω1, and gj =g0exp( ik0 rj) with
ken. For a SUSY b-f mixture, the former is completely g0 ∝ ddrexp(−ik−0·r)wb∗(r)wf(r)wF(r)−being·the cou-
pling intensity independent of the site.
suppressed while the latter in zero-momentum becomes
R
In the k-space, the Hamiltonian H is rewritten as
anexacteigenstate,theGoldstinomode[3]. Inthissense, ex
PwAe rsepgeacrtdrutmhesisedeixrceicttaltyiornelsaatesdthteo SthUeStYheremspoolencsuelsa.rTfohre- Hex =g0√ρ( Q†k−k0Fkeiδ0t+H.c.), (3)
k
X
mation rate varying as the detuning and faithfully de-
where Q† = b† f /√N and aα = aαexp( ik
scribes these two types of excitations. The position of k p p+k p k j j − ·
peak in the PA spectrum determines the frequency of rj)/√V (whePreV standsforthevolumeaPndaαk standfor
thecollectivezero-momentummode. Thismolecularfor- bk, fk, and Fk). ρ=N/ is the total density of b and f
V
mation rate is measured by the number variation of the with the particle number N = (nb+nf).
i i i
F atoms in time. Experimentally, the number counting
Molecular formation rate — TPhe formation rate of the
of atoms is much simpler than detecting the single atom
molecules b can be counted by the PA variation of F-
spectrum.
fermion number R = ∂ ψ(t) N ψ(t) for ψ(t) being
t F
h | | i | i
Model setup — The system illustrated in Fig. 1(a) is thetimeevolutionfromthegroundstate G = g F of
| i | i| i
described by a Hamiltonian H = H0 + Hex, where H0. It follows from the linear response theory that
FHe0sh=baHchbfr+esHoFnawnicteh[H9],bfth=esHcab+ttHerfin+gVle.nBgtyhmsbeaentwseoefnthFe R=2g02ρ ImDR(k,−δ0), (4)
k
and the b-f mixture can be adjusted to negligibly small. X
where the retarded Green function is given by
In the tight-binding approximation, one has
∞ n (x) n (εF)
Hα = − tαaαi†aαj −µα aαi†aαi, (1) DR(k,ω)=Z−∞dxA(k−k0,x)x−f εFk −−ωf−ik0+, (5)
Xhiji Xi in terms of one loop calculations (Fig. 1(b)). The sin-
U
V = bb nb(nb 1)+U nbnf, gle particle dispersions are εα = 2t d cosk
2 i i − bf i i k − α s=1 s −
Xi Xi µα where the lattice spacings are set toPbe the unit.
n (x) is the Fermi distribution at temperature T and
f
where aαi = bi, fi, and Fi (α = b, f, and F) are the the spectral function A(k,ω) = ImΠR(k,ω)/π is
annihilation operators of b, f and F at site i; µα and defined by the retarded Green fu−nctionΠ (k,ω) =
nαi =aαi†aαi are chemical potentials and the number op- i ∞dt g Q (t),Q†(0) g eiωt. R
erators at site i. The definitions of the hopping ampli- − 0 h |{ k k }| i
At sufficiently low temperature, the pole and branch
tudes t and the interaction strengths U by the Wan- R
α αβ cutinEq.(5)arenotqualitativelyaffectedbyT andnor
nier function w (r) can be found in the literature [10].
α is the molecular formation rate. For simplicity, we take
Thespatialinhomogeneityofopticallatticestrappingthe
a zero temperature approximation in our calculation. It
atoms and molecules has been omitted. For the subsys-
follows from Eq. (5) that the rate R= R R
b→Ff Ff→b
temb-f mixture,theHamiltonianH isSUSYinvariant −
bf contains two parts :
if t = t , U = U and µ = µ [1, 3]. In order to
b f bb bf b f
prevent the phase separation, the parameters obey, e.g., Rb→Ff = 2πg02ρA(k−k0,εFk −δ0)θ(δ0−εFk),
4πtfρfUbb >Ub2f in2-dimensions,whereρf isthedensity Xk
oobfefyaintgomthsi[s1c1o].nWditeiocnh.oosethe parametersofthe system RFf→b = 2πg02ρA(k−k0,εFk −δ0)θ(−εFk). (6)
k
X
3
which respectively are the dissociation rate for b Ff
and the formation rate for Ff b. → 3.93 (a) Goldstino mode (b)
→
individual
Collective and individual fermionic modes — In order E (k)
c modes
to obtain R for the weak interactions, we perturbatively
calculate Π (k,ω)=ρ−1[Π−1(k,ω)+U ]−1, which for-
R 0 bf 3.9
mally results from the equation of motion of Qk. It then 0 k 0.04 kF kc kQ k
followsfromthe randomphaseapproximation(RPA) il-
lustrated by the “bubble”in (Fig. 1(b)) that
3.99 50
Π0(k,ω)=Z (d2dπp)dnfω(ε−fp)E+kpn+b(εibk0++p), (7) w 0 w =3.93 (c) A(w ) 3.996 i n mdoivdiedsu(adl)
0
whereE =εb εf +2ρ δU+U ρwith δU =U
U andkρp =Nk+/p−; np(x) isbthe Bobsfe distribution. Tbbh−e 3.76 0
bf b b b
isolated pole andVbranch cut of Π (k,ω) describe the - 0.05 0 0.05 3.9 4.5
R
collective and individual SUSY excitations of Q† g . dt w
k| i
Next we consider the elementary excitations in the
two-dimensional lattice with f atoms at half filling, i.e., FIG.2: (Coloronline)Thedispersions,spectrumandspectral
ρf = 0.5. For the SUSY b-f mixture, i.e., δU = 0 and function of the excitations for ∆µ = 3.93 and Ubb = Ubf =
0.1; t is taken as the unit. (a) The dispersion of collective
δt = t t = 0, the dispersion of the collective modes, b
b f
− 2 modefortheSUSYmixture. (b)ThespectrumfortheSUSY
E (k) ∆µ α k for the small k [3], is read out
c system: thedashed(red)anddotted(blue)curvesdenotethe
≃ − | | | |
from the poles of the retarded Green function ΠR(k,ω) dispersionsofcollectivemodesforθ=0andπ/5,respectively.
(seeFig.2(a)),where∆µ=µf µb. Forlarge k,theen- Thesolid(blue)curvedenotesthedispersionofatomsF fora
− | |
ergyEc(k)=Ec(k ,θ) depends not only on k but also small Fermimomentum kF. (c) The frequencies of Q†0|gifor
| | | |
on the angle θ = arctan(k /k ). The energy E (k ,θ) differentδt. (d)Thespectralfunctionofthezero-momentum
y x c
of Q† g decreases as k increases for a fixed θ. T|he| re- for δt = −0.1: The sharp peak is the shifted goldstino-like
tardekd| Gireen’s functio|n|Π (0,ω) = (ω ∆µ+i0+)−1 mode.
R
−
possesses a pole ω = ∆µ, which corresponds to the
goldstino-like excitation Q† g . This recovers the re-
0| i amine the dependence ofthe spectralfunction on the in-
sultinRef.[3]. Theexcitationspectrumis schematically
teractingstrengthandfindthatthehumpheightmaybe
shown in Fig. 2(b).
depressed as the interaction becomes stronger, e.g., the
For the b-f mixture deviating slightly from SUSY,
height is lower than 0.5 for U = U = 0.5 comparing
bf bb
the retarded Green function Π (0,ω) has an isolated
R with 10 in Fig. 2(d) for U =U =0.1.
bf bb
pole and a branch cut, which correspond to a collec- ∼
In order to study the PA spectrum of the molecular
tive fermionic mode and individual (bosonic) particle-
formationrate,wediscusstheexcitationspectrumshown
(fermionic) hole pair continuum modes, respectively.
in Fig. 2(b). For some momenta k, there is a collective
The pole in ω0 < E0 −4δt for δt > 0 (or ω0 < E0 for mode(dashedredcurve)belowtheindividualcontinuum.
δt < 0) describes the shifted goldstino-like mode. The
For other momenta k , the dispersion of the collective
c
frequenciesω0 ofcollectivezero-momentummodefordif- mode merges into the continuum. However, for small
ferent δt are shown in Fig. 2(c). For k = 0, the pole of
ΠR(k,ω) has the form Ec′(k) ≃ ω0 − α6 ′|k|2 for small mthoeminedntivuimduka,ltchoenrteinaulwuamy.seFxoirstcsoancvoelnleiecnticvee,mwoeddeebfienleowa
k. Remarkably, the branch cut l0 of ΠR(0,ω) emerges, critical momentum k (θ), so that for a fixed θ, when
| | Q
whichdescribesindividualzero-momentummodes. Here,
k > k (θ) the negative frequencies of the mode Q
Q k
l0 =[E0 4δt,E0]forδt>0(or[E0,E0 4δt]forδt<0), e|m|erge, i.e., A(k,ω < 0) = 0 when k > k (θ), and
− − Q
andE0 =∆µ+2ρbδU+Ubfρ. Noticethatfortheweakin- A(k,ω <0)=0 when k <6 k (θ). | |
Q
teractionsU andU the SUSY breakingfromδU does | |
bb bf
not develop a branch cut but only shifts the positions PA spectrum — The rate R varies as detuning δ0 or the
of the pole and branch cut. The pole and the branch light frequency ωc. Measurement of the b boson forma-
cut can be seenin the spectral function A(k,ω), i.e., the tion rate varying as δ0 is called the PA spectrum S(δ0).
peak and the hump in Fig. 2(d) for k=0. Note that For a long wave photon, the coupling gj varies slowly in
for the SUSY b-f, the branch cut length l0 of ΠR(0,ω) space and the rates in Eq. (6) are approximately inde-
shrinks to zero so that the individual continuum modes pendent of k0.
of zero momentum are completely suppressed. On the AccordingtoEq.(6),thedissociationrateR does
b→Ff
other hand, as the b-f mixture deviates from the SUSY, notvanishonlyifδ0 >εFk andA(k,εFk−δ0)6=0. Because
the goldstino-like mode is gradually suppressed. We ex- thespectralfunctionA(k,x<0)=0isdefinedbythere-
6
4
8 1 decreases as δt increases. As a result, the peak height
(a) (b) is lowered wh|ile|its position is shifted to δ0 = ω0. The
−
S ratio RFf→b(ω0)/R0 is shown in Fig. 3(b) for different
RFffi b(w 0)/R0 δt and a small µeff, where RFf→b(ω0) is the value of
0 0.48 RFf→b at δ0 = −ω0. Remarkably, a minor hump de-
3.7 4.5 - 0.05 0 0.05 velops in the region δ0 l0 due to the emergence of
d dt individualmodes(See Fig∈.3(a)). Asthesystemdeviates
0
further from SUSY, the individual modes are enhanced
due to the sum rules dωA(0,ω)=1. The temperature
FIG. 3: (Color online) The PA spectra with the same pa-
may suppress and broaden the peak and hump. These
rameters as those in Fig. 2. (a) The detuning dependence R
characters of the PA spectrum in the b-f mixture are
for δt = 0.05 and −0.05, the solid (red) and dashed (blue)
lines, respectively. The unit of S is 2πg02ρNF and NF/N is experimentally measurable SUSY responses to the light
taken to be 0.01. (b) The major peak values for different δt. field.
Thisshowsthatthepeakvalueissuppressedwhenthesystem Conclusions — We studied how to observe the SUSY
deviates from SUSY. nature of the b-f mixture in optical lattices through PA
spectra. For the Bose molecules formedwith twospecies
of Fermi atoms, we showed that the photon induced
tarded Green function, it does not vanish only when the
atom-moleculetransitiondisplaysthesignalofSUSY.As
energies for collective modes or individual modes of Q
k
the response to the PA processes, a fermionic individual
are negative for the large k >k (θ). We consider a di-
Q
| | continuum and the goldstino-like mode were found. The
luteFermigasF withthechemicalpotentialµ 4t ,
F F
the dispersion relation turns out εFk = tF |k|2∼−−µeff, PtiAonspraetcetruofmFcfanexpb.licBitleycawuistenetshsethgeolmdsotliencou-llaikrefomrmodae-
where µeff = µF +4tF. In this case, the fermion F pos- →
in zero momentum turns to be the exact eigen state for
sessesa smallFermimomentum kF = µeff/tF which is
the SUSY mixture, the major peak in the PA spectrum
much smaller than k (θ) for small deviating δt. There-
Q
fore, εF is alwayspositive when k >kp (see Fig. 2(b)). reflects the SUSY response to the light field, even if the
Fortheknegativedetuningδ0,the|co|nditiQonδ0 >εFk isnot mixture is not fine-tuned to a SUSY one.
asnanetgdiasθfit(ievδde0−idneεttFkhu)encirnaegnginamnoetd|bksem|n>aollnkF-Qze(erθrm)o.isTimmhouamltteaisnn,teuAomu(kslk,yεFkf.o−rTtδhh0ies) uinsgeTfhuhilesdaiiduscetuhasoisrnisotnthhse.anYekaYrJliiiesnrbgsirntaatLgeeifuaolnftdtohPKiseunwngoYrZkah.nagTnghfoifrsosrwhotahrrke-
F
finishes our proof of R =0. is supported in part by National Natural Science Foun-
b→Ff
The vanishing of Rb→Ff for the negative δ0 and small dation of China, the national programfor basic research
Fermi momentum k can be understood in a more of MOST of China and a fund from CAS.
F
straightforwardway. The transitionb Ff is described
→
by the Hermite conjugate term (H.c.) in the Hamilto-
nianH ,whichisahigh-frequencyoscillationtermwhen
ex
δ0 < 0. Hence, the Fermi golden rule results in that
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| i
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