Hiroyuki Yabu- Topics on Boson-Fermion mixture in ultra-cold atomic gas

Topics on Boson-Fermion mixturein ultra-cold atomic gas

Hiroyuki YABU(Tokyo Metropolitan Univ.)

・Short Introduction of Japanese activities experiments theory・Boson-Fermion mixed condensates researches by our group

power, and the loading durations of these two isotopes to

make equal both numbers. Then the number of each isotope

reached about 1!106. In the cases of boson-boson mixtures,the numbers of both isotopes were 8!105 for 172Yb-174Yb,and were 3!105 for 174Yb-176Yb, respectively. For all mix-tures, the temperature was about 100 !K, which was mea-sured by the time-of-flight method. The lifetimes of the

trapped atoms were about 400 ms, which was mainly limited

by the background vapor pressure of the order of 10"9 Torr.

From the measured temperatures, atom numbers, and trap

oscillation frequencies, we estimated the atom densities to be

more than 1014/cm3 at the crossing region.

With the fermion-boson mixture (171Yb-174Yb) trapped by

the crossed FORT, we could observe the first direct evidence

of cross collisions between bosons and fermions trapped in

any optical trap. First, we tried evaporative cooling in which

the potential depth was decreased gradually by reducing the

FORT laser light intensity with an AOM inserted in the

FORT laser beam path. Initially, the FORT potential depth

was 1090 !K, which was kept for 30 ms. In the next 10 ms,the potential depth was linearly decreased from 1090 !K to700 !K, then from 700 !K to 61 !K in 50 ms, and finallyfrom 61 !K to 25 !K in 80 ms. It is noted that we have

recently demonstrated that this scheme really worked for

bosons 174Yb and resulted in the successful cooling from

100 !K to 4 !K, although the phase-space density "PSD#did not increase so much in our configuration $20%. However,in the case of fermions 171Yb, starting with 8!105 171Yb

atoms in the crossed FORT, we have found that all the atoms

escaped from the crossed FORT at the end of this sequence.

The failure of the evaporative cooling for the fermions 171Yb

should be attributed to slow thermalization due to the small

elastic collision rate between cold fermions. Then we tried

sympathetic cooling in the mixture of fermions 171Yb and

bosons 174Yb. Initially, the mixture was composed of 1

!106 171Yb atoms and 7!105 174Yb atoms. After the

scheme mentioned above, we observed that 2!104 171Yb

atoms were still remaining in the trap. Although we could not

measure the atom temperature because of the small atom

number, it was certainly below the trap potential of 25 !K.It should be noted that our observation did not show a suc-

cessful evaporative cooling of fermions 171Yb even in the

case of mixture, i.e., PSD did not increase with ramping

down the potential. However, the PSD really increased in the

case of the mixture compared with that of pure fermions,

which is direct evidence of a sympathetic cooling effect. In

this experiment, we took special care to avoid instrumental

effects. Even in the measurement for the pure fermion 171Yb

gas, we performed exactly the same experimental procedures

as those for the mixture except for closing the atomic beam

shutter during the loading of the bosons 174Yb. Conse-

quently, the observed difference between the mixture and the

pure fermion gas truly comes from collision between bosons174Yb and fermions 171Yb.

To observe clearly the effect of the cross collisions be-

tween 171Yb and 174Yb, we used a scheme of ‘‘delayed

crossed FORT’’ $14%. In this scheme, the atoms were initiallytrapped in a ‘‘single FORT,’’ by blocking the second FORT

beam with an mechanical shutter. By opening the shutter

after a certain delay time, we suddenly produced a deep and

narrow well potential in the single FORT potential. This de-

layed crossed FORT experiment is similar to the adiabatic

deformation experiment, in which atoms can be compressed

by producing adiabatically a small and steep potential in a

large and shallow trap $22%. When two atoms trapped insingle FORT collide in the crossing region, one atom is ex-

cited to a higher vibrational level, and the other atom decays

to the lower level in the crossed FORT. This process results

in the increase of the atoms in the crossing region. It is noted

that, in this scheme, the loading into the crossing region was

purely ascribed to the atom-atom elastic collisions since the

MOT beams were completely turned off before the crossed

FORT potential was produced. Therefore, the concentration

in the crossed region represents a direct probe of elastic col-

lisions between trapped atoms. The time evolution of the171Yb atom distribution in the crossed FORT is shown in

Fig. 4 for the mixture of fermions 171Yb and bosons 174Yb

and for the pure fermion gas of 171Yb atoms. Here, we have

introduced a ‘‘concentration ratio’’ defined as a ratio of the

number of atoms trapped only in the crossing region to the

total number of trapped atoms. After 100 ms from opening

the shutter, a concentration ratio for the mixture reached

about 0.08, which is considerably large compared with the

value of about 0.03 for the pure fermion gas.

FIG. 3. "Color#Absorption images of 171Yb atoms "a# and 174Yb

atoms "b# in the fermion-boson mixture trapped in the crossedFORT. Optical density increases from red to blue. Blue areas near

the center of the image are crossed FORT regions. Yellow long

areas from side to side are single FORT regions. The dimension of

each picture is 180 !m!1710 !m.

FIG. 4. Time evolution of the 171Yb concentration ratio with

(!) and without (") 174Yb, after switching on the crossed FORT.

The concentration ratio was defined as the atom number in the

crossed FORT region relative to the atom number in the whole

FORT region. The solid lines show exponential fits to the measured

data, and & indicates the time constant. The observed time scales forloadings are close to the period of the axial oscillation of the single

FORT.

RAPID COMMUNICATIONS

OPTICAL DIPOLE FORCE TRAPPING OF A FERMION- . . . PHYSICAL REVIEW A 66, 021401"R# "2002#

021401-3

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BEC experiments in JapanKuga-group (U.Tokyo) Dec. 12 ‘98

Takahashi-group (Kyoto U.) Dec. 27 ‘98

Hirano-group (Gakushuin) July. 19 ‘00

Nakagawa-group (UEC) Dec. 19 ‘01

Mukai-group (NTT) July 8 ‘02

図４は rf周波数の掃引曲線を示している。25MHzから1.5MHzまで指数関数的に掃引することで周波数を急激に下げていき、1.5MHzから910kHzまでは直線的に掃引することでゆっくりと下げていきBEC

の生成を目指した。25MHzでは原子集団の温度は 1.16mKに相当し、910kHzでは 96nKに相当する。下の図５はTOF9msで最終周波数を変えていったときの吸収像とその断面図である。TOF とはTime of

flightの略で原子集団を磁気トラップから解放して吸収像を撮影するまでの時間のことである。TOFの時間が長くなると、もしBECが生成されていればBECになっている原子集団とサーマルな原子集団とで温度が異なるために、時間による広がりが異なってくる。断面図の赤線はガウスフィティングした時の曲線である。断面図を見ると930kHzでは原子の温度が高いためサーマルな原子集団のみなのでガウス分布でフィティングできるが、920kHz になると温度の高い原子と BEC になった原子が両方存在するためガウス分布ではフィティングできなくなる。そして、910kHz ではほとんどの原子が BEC になっていると思われるためガウス分布をしていない。910kHz の吸収像から原子数 3.7×105個、温度 131nK、横方向の原子の大きさ 90μm とわかった。

930kHz

920kHz

光学密度レベル

MAX(赤)･･･４

MIN(黒)･･･０

910kHz

図５最終周波数を変化させた時の吸収像と断面図（TOF９ｍｓ）

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930kHz

920kHz




910kHz


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RbOptics & Atom laser

YbEDM of nucleusFermion, Boson-Fermion mixture

RbMulti-component BEC

RbAtomic laser, BEC on microchip

RbTechnological applications

Boson-Fermion mixed condensates　boson-fermion (B and F) mixture (trapped system)

nF

Boson : BEC Fermion : Fermi degeneracy

Same trap

Systems in study ・H-D system (Silvera, Oliva) ・4He-3He (very popular system) ・π-,K-cond. in HD nuclear matter?

Interaction between B and F leads to many interesting phenomena!

・Mean-square radius〈r2〉B/NB =

3

2ξ2

B 〈r2〉F /NF =3

4(nF + 2)ξ2

F

ξ2

B,F =

√h

mB,F ωB,F

NF = (nF + 1)(nF + 2)(nF + 3)/6 ∼ n3

F√〈r2〉F〈r2〉B

∼ N1/6 N ≡ NB = NF , ξB = ξF

: HO lengths

Experiments of B-F mixed cond. of the trapped atomic gas

VOLUME 87, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 20 AUGUST 2001

FIG. 3. Mixture of Bose and Fermi gases. Top: In situ spa-tial distributions after sympathetic cooling with NB ! 3.5 3104 and NF ! 2.5 3 104 . The Bose condensed peak (8.5 3103 atoms) is surrounded by the thermal cloud which allowsthe determination of the common temperature: T ! 1.6 mK !0.87TC ! 0.57TF . The Fermi distribution is wider becauseof the smaller magnetic moment and Fermi pressure. Bot-tom: profiles with a quasipure condensate, with NB ! 104,NF ! 4 3 103. The barely detectable thermal cloud indicates atemperature of !0.28 mK ! 0.2"1#TC ! 0.2"1#TF.

this loss rate. In the following we concentrate on this sec-ond, and simpler, evaporation scheme, sympathetic coolingof 7Li by evaporative cooling of 6Li.

In Fig. 3 in situ absorption images of bosons and fermi-ons at the end of the evaporation are shown. The bosonicdistribution shows the typical double structure: a strongand narrow peak forms the condensate at the center, sur-rounded by a much broader distribution, the thermal cloud.As the Fermi distribution is very insensitive to temperature,this thermal cloud is a very useful tool for the determina-tion of the common temperature. Note that, as cooling wasperformed only on 6Li atoms, the temperature measuredon 7Li cannot be lower than the temperature of the fermi-ons. Measuring NB, NF, the condensate fraction N0$NB,and v, we determine the quantum degeneracy of the Boseand Fermi gases. In Fig. 3 (top), the temperature is justbelow TC, T ! 1.6 mK ! 0.87TC ! 0.57TF. In Fig. 3(bottom) on the contrary, the condensate is quasipure;N0$NB ! 0.77; the thermal fraction is near our detectivitylimit, indicating a temperature of !0.28 mK # 0.2TC !

0.2"1#TF with NB ! 104 bosons and 4 3 103 fermions.The condensate fraction N0$NB as a function of T$TC isshown in Fig. 4(a), while the size of the fermi gas as afunction of T$TF is shown in Fig. 4(b). With the stronganisotropy (vrad$vax ! 59) of our trap, the theory in-cluding anisotropy and finite number effects differs sig-nificantly from the thermodynamic limit [2], in agreementwith our measurements even though there is a 20% sys-tematic uncertainty on our determination of TC and TF.We have also obtained samples colder than those presentedin Fig. 4, for which the 7Li thermal fraction is below ourdetectivity floor, indicating T , 0.2TC ! 0.2TF. Clearlya more sensitive thermal probe is required now to inves-tigate this temperature domain. An elegant method relieson the measurement of thermalization rates with impurityatoms including Pauli blocking [18,19].

Because of the small scattering length, this 7Li conden-sate has interesting properties. Time of flight images, per-formed after expansion times of 0–10 ms with N0 ! 104

condensed atoms, reveal that the condensate is one dimen-sional (1D). In contrast to condensates in the Thomas-Fermi (TF) regime, where the release of interaction energy

FIG. 4. Temperature dependence of mixtures of quantumgases: (a) normalized BEC fraction as a function of T$TC.Dashed line: theory in the thermodynamic limit. Solid line:theory including finite size and trap anisotropy [2]; (b) fermioncloud size: variance of Gaussian fit divided by the square ofFermi radius R2

F ! 2kBTF$Mv2ax as a function of T$TF. Solid

line: theory. Dashed line: Boltzmann gas.

080403-3 080403-3

7Li and 6Li (Rice U.) (2001) 7Li and 6Li (ENS, Paris) (2001)

Classical

Quantum

These exp. have been done for Fermion cooling(Sympathetic cooling)

VOLUME 88, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 22 APRIL 2002

the good and bad interspecies collisions in this mixture.The losses due to inelastic spin-exchange collisions tookplace only on a time scale of several seconds, comparableto the total evaporation time of 15 s.

In order to produce a collisionally stable Bose-Fermimixture, it is necessary to transfer the lithium atoms to thelower hyperfine state. To minimize the initial losses dueto spin-exchange collisions, this transfer should take placeas early in the cooling process as possible. On the otherhand, before lithium atoms can be efficiently trapped inthe lower hyperfine state, they must be cooled significantlybelow 330 mK. Therefore, we implemented sympatheticcooling in two stages (see Fig. 2). We optimized the ini-tial evaporation stage to reach a temperature of !50 mKin 5 s, while losing less than half of the lithium atoms, andmaintaining the conditions for efficient sodium evapora-tion. At this point, we found that a substantial fraction oflithium atoms was in the j3"2, 1"2# state. They could thusbe transferred to the j1"2, 21"2# state on a single-photonrf transition at 228 MHz, which is, to first order, inde-pendent of the magnetic field. This simplification over asimilar hyperfine transfer previously employed in [4] wasnot expected. After the rf pulse was applied, the remainingF ! 3"2 atoms were optically pumped into untrapped hy-perfine states, and expelled from the trap. If this “cleanup”light pulse was omitted, spin-exchange collisions betweenlithium atoms in different hyperfine states led to a rapidloss of atoms from the trap [18]. The overall efficiencyof our hyperfine transfer was !50%. The evaporation wasthen resumed for another 10 s. We observed efficient sym-pathetic cooling of the j1"2, 21"2# atoms, and cooled bothgases into quantum degeneracy without observable lossesin the lithium atom number.

Figure 3(a) displays the effect of sympathetic coolingon the 6Li cloud. Absorption images of the trapped 6Ligas were taken after the 23Na evaporation was terminatedat different trap depths, and the sample was allowed toequilibrate for 1 s. Cooling (from top to bottom) is seenin the shrinking of the density distribution and an increasein the peak optical density. In contrast to standard evapo-rative cooling, and the mutual cooling between two Fermispecies, the total number of atoms remains constant.

Quantitative analysis of the 6Li clouds is depicted inFig. 3(b). We performed two-dimensional fits to therecorded column densities using both a simple Gaussianmodel and a semiclassical (Thomas-Fermi) distribution fortrapped noninteracting fermions. In the latter approach,the probability that an atom has position "r and momentum"p is obtained from the Fermi-Dirac distribution for thetotal (kinetic and potential) energy of the particle. Thefitting function for the spatial distribution of atoms in thetrap is then obtained by integrating over the momentumdegrees of freedom. While the Gaussian model gives avalid description of the gas only in the classical, hightemperature limit, the Thomas-Fermi approach is validat all temperatures, as long as the number of particles in

0 1.5

Optical Density

-500 0 500

Axial distance [µm]

2 TF

1 TF

0.5 TF DF

a) b)

FIG. 3. Onset of Fermi degeneracy. Three pairs of images (topto bottom) correspond to T"TF ! 2, 1, and 0.5. (a) Columndensities of the 6Li cloud were recorded by absorption imaging.(b) Axial line density profiles and the Thomas-Fermi fits to thedata are plotted. The arrow indicates the size of the Fermidiameter, DF .

the system is large [19]. Indeed, at higher temperatures,the two fits performed equally well, and yielded the sametemperature. However, at a temperature of about 400 nK,the classical fits started to fail. This was indicated by therelative growth of the reduced x2 values, by up to 20%above the corresponding values for the fermionic fits.For the coldest samples, Gaussian fits also overestimatedthe temperature by !15%. From the fitted number ofatoms in the system (NLi ! 1.4 3 105, TF $ 670 nK),we found that the noticeable inadequacy of the classicalfits occurred at !0.6 TF , which is a clear signature ofthe Fermi degeneracy. Figure 3(b) shows projected linedensities along the axial direction of the 6Li cloud and theThomas-Fermi fits to the data. The arrow indicates thesize of the Fermi diameter, DF ! 2

p2kBTF"%mv2

z &, forthe fitted atom number. The spatial extent of the coldestcloud %T $ 330 nK $ 0.5 TF& is already comparableto the minimum size the system would assume at zerotemperature. Typical densities of the coldest lithiumsamples were !1012 cm23.

In Fig. 4, the temperature of the 6Li cloud is plotted asa function of the final 23Na trap depth. The coldest lithiumsamples were produced in coexistence with almost puresodium condensates with !2 3 106 atoms. The lifetimeof this degenerate mixture was limited to about 10 s by thethree-body decay of the BEC, while the lithium cloud hada lifetime longer than 100 s. The lifetime of either specieswas not detectably affected by the presence of the otherone. Even after sodium was lost from the trap, the energyof the lithium cloud did not increase noticeably %,5 nK"s&during its slow decay. We also compared the 6Li tempera-tures with the temperatures of the 23Na cloud, extractedfrom the thermal wings of the bosonic density distribution.In hotter samples, the two agreed to within 10%. However,for the coldest samples we observed a discrepancy betweenthe two temperatures. The lowest measurable temperature

160401-3 160401-3

collisions needed for thermalization. Even though Katoms cannot thermalize between themselves, sinces-wave collisions are forbidden for identical fermions,the thermalization with Rb happens on a time scale longerthan the mean period of oscillation in the trap, and thisensures thermalization of the K sample. Actually, wehave observed that both the density and momentum dis-tributions have Gaussian profiles that lead to the sametemperature. From the measured ! we derive a quite largemagnitude for the 40K-87Rb scattering length: jaj !330"160#100a0. Here the uncertainty is dominated by that on! and by a 40% uncertainty on the atom number. Thedirect measurement with the fermionic isotope is inagreement with the value a ! #261"170

#159a0 that we pre-viously inferred by mass scaling from collisional mea-surements on the bosonic 41K-87Rb mixture [12]. Themeasurement of a large value for jaj also confirms theattractive character of the interaction, since a positivescattering length would have been compatible only with amuch smaller magnitude [13].

By further cooling the mixture, we have evidence ofthe formation of a degenerate Fermi gas coexisting with aBose-Einstein condensate. In Fig. 1 we show a series ofabsorption images of the mixture for three different finalenergies of the evaporation ramp, which reveals the dif-ferent nature of the two degenerate gases. Both samplesare imaged in the same experimental run, by using twoshort, delayed light pulses. The images are taken after aballistic expansion appropriate to measure the momentumdistribution of the samples; in particular, the expansionlasts 4.5 ms for K and 17.5 ms for Rb. Sections of suchimages are also shown: they are taken along the verticaldirection for K and along the horizontal direction for Rb.With our experimental parameters, we have a Fermitemperature TF ! 250 nK and a critical temperature for

BEC Tc ! 110 nK for a sample composed of 104 and 2$104 atoms, respectively.

Thermometry of the system is provided by the bosoniccomponent, assuming thermal equilibrium between thetwo components. As the temperature is decreased by al-most a factor of 2 (from top to bottom in Fig. 1), Rbundergoes the phase transition to BEC, while the width ofthe fermionic component remains almost constant. A fitof the coldest K cloud with a Thomas-Fermi profile [17]gives a radius R ! 52 "m, which is consistent to within10% with the minimum radius allowed by Fermi statis-tics: R ! RF

!!!!!!!!!!!!!!!!!!!!1"!2

r!2p

, where RF ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2kBTF=%M!2

r&p

isthe Fermi radius and ! is the expansion time. For thequasipure BEC shown in Fig. 1, a fit to both the thermaland condensate components with Gaussian profiles indi-cate a condensate fraction of 60%. This implies a tem-perature T ! 80 nK of both the BEC and the Fermi gas,which corresponds to 0:3TF.

In these conditions, we see a small decrease of thenumber of K atoms on the time scale of 1 s, which isthe lifetime of the BEC in our system. This indicates thepresence of some kind of inelastic collisional process,which will be the object of future investigation.

Further evidence for the achievement of quantum de-generacy is obtained by studying the Gaussian 1=e widthof the fermionic sample as a function of temperature. Asshown in Fig. 2 the square of the width, normalized to RF,scales linearly for T > TF, indicating thermal equili-brium between K and Rb. Below TF, the data deviatefrom the behavior expected for a classical gas, and indeedthey are better reproduced by the prediction of the modelfor an ideal Fermi gas [17].

Since we have also observed degenerate mixtures inwhich the thermal fraction of the condensate was belowour detection limit of nearly 30%, the attainment of

FIG. 1 (color). Simultaneous onset of Fermi degeneracy for 40K (left) and of Bose-Einstein condensation for 87Rb (right). Theabsorption images are taken for three decreasing temperatures, after 4.5 ms of expansion for K and 17.5 ms for Rb, and the sectionsshow the profile of the momentum distributions. The bosons provide a thermometry of the system: in the coldest sample, the RbBEC has a 40% thermal component, and the temperature is T ! 0:74Tc ! 80 nK, which corresponds to T ! 0:3TF for K. TheThomas-Fermi radius of the K profile is R ! 52 "m, consistent with the radius expected for a degenerate Fermi gas.

VOLUME 89, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 7 OCTOBER 2002

150403-2 150403-2

40K and 87Rb (LENS, Firenze) (2002)

39K-40K-41K (JILA)

84,86,88-87Sr (Tokyo)

Experiments going on now!

Experiments of B-F mixed cond. of the trapped atomic gas

7Li and 23Na (MIT) (2002)

7Li cloudpower, and the loading durations of these two isotopes to

make equal both numbers. Then the number of each isotope

reached about 1!106. In the cases of boson-boson mixtures,the numbers of both isotopes were 8!105 for 172Yb-174Yb,and were 3!105 for 174Yb-176Yb, respectively. For all mix-tures, the temperature was about 100 !K, which was mea-sured by the time-of-flight method. The lifetimes of the

trapped atoms were about 400 ms, which was mainly limited

by the background vapor pressure of the order of 10"9 Torr.

From the measured temperatures, atom numbers, and trap

oscillation frequencies, we estimated the atom densities to be

more than 1014/cm3 at the crossing region.

With the fermion-boson mixture (171Yb-174Yb) trapped by

the crossed FORT, we could observe the first direct evidence

of cross collisions between bosons and fermions trapped in

any optical trap. First, we tried evaporative cooling in which

the potential depth was decreased gradually by reducing the

FORT laser light intensity with an AOM inserted in the

FORT laser beam path. Initially, the FORT potential depth

was 1090 !K, which was kept for 30 ms. In the next 10 ms,the potential depth was linearly decreased from 1090 !K to700 !K, then from 700 !K to 61 !K in 50 ms, and finallyfrom 61 !K to 25 !K in 80 ms. It is noted that we have

recently demonstrated that this scheme really worked for

bosons 174Yb and resulted in the successful cooling from

100 !K to 4 !K, although the phase-space density "PSD#did not increase so much in our configuration $20%. However,in the case of fermions 171Yb, starting with 8!105 171Yb

atoms in the crossed FORT, we have found that all the atoms

escaped from the crossed FORT at the end of this sequence.

The failure of the evaporative cooling for the fermions 171Yb

should be attributed to slow thermalization due to the small

elastic collision rate between cold fermions. Then we tried

sympathetic cooling in the mixture of fermions 171Yb and

bosons 174Yb. Initially, the mixture was composed of 1

!106 171Yb atoms and 7!105 174Yb atoms. After the

scheme mentioned above, we observed that 2!104 171Yb

atoms were still remaining in the trap. Although we could not

measure the atom temperature because of the small atom

number, it was certainly below the trap potential of 25 !K.It should be noted that our observation did not show a suc-

cessful evaporative cooling of fermions 171Yb even in the

case of mixture, i.e., PSD did not increase with ramping

down the potential. However, the PSD really increased in the

case of the mixture compared with that of pure fermions,

which is direct evidence of a sympathetic cooling effect. In

this experiment, we took special care to avoid instrumental

effects. Even in the measurement for the pure fermion 171Yb

gas, we performed exactly the same experimental procedures

as those for the mixture except for closing the atomic beam

shutter during the loading of the bosons 174Yb. Conse-

quently, the observed difference between the mixture and the

pure fermion gas truly comes from collision between bosons174Yb and fermions 171Yb.

To observe clearly the effect of the cross collisions be-

tween 171Yb and 174Yb, we used a scheme of ‘‘delayed

crossed FORT’’ $14%. In this scheme, the atoms were initiallytrapped in a ‘‘single FORT,’’ by blocking the second FORT

beam with an mechanical shutter. By opening the shutter

after a certain delay time, we suddenly produced a deep and

narrow well potential in the single FORT potential. This de-

layed crossed FORT experiment is similar to the adiabatic

deformation experiment, in which atoms can be compressed

by producing adiabatically a small and steep potential in a

large and shallow trap $22%. When two atoms trapped insingle FORT collide in the crossing region, one atom is ex-

cited to a higher vibrational level, and the other atom decays

to the lower level in the crossed FORT. This process results

in the increase of the atoms in the crossing region. It is noted

that, in this scheme, the loading into the crossing region was

purely ascribed to the atom-atom elastic collisions since the

MOT beams were completely turned off before the crossed

FORT potential was produced. Therefore, the concentration

in the crossed region represents a direct probe of elastic col-

lisions between trapped atoms. The time evolution of the171Yb atom distribution in the crossed FORT is shown in

Fig. 4 for the mixture of fermions 171Yb and bosons 174Yb

and for the pure fermion gas of 171Yb atoms. Here, we have

introduced a ‘‘concentration ratio’’ defined as a ratio of the

number of atoms trapped only in the crossing region to the

total number of trapped atoms. After 100 ms from opening

the shutter, a concentration ratio for the mixture reached

about 0.08, which is considerably large compared with the

value of about 0.03 for the pure fermion gas.

FIG. 3. "Color#Absorption images of 171Yb atoms "a# and 174Yb

atoms "b# in the fermion-boson mixture trapped in the crossedFORT. Optical density increases from red to blue. Blue areas near

the center of the image are crossed FORT regions. Yellow long

areas from side to side are single FORT regions. The dimension of

each picture is 180 !m!1710 !m.

FIG. 4. Time evolution of the 171Yb concentration ratio with

(!) and without (") 174Yb, after switching on the crossed FORT.

The concentration ratio was defined as the atom number in the

crossed FORT region relative to the atom number in the whole

FORT region. The solid lines show exponential fits to the measured

data, and & indicates the time constant. The observed time scales forloadings are close to the period of the axial oscillation of the single

FORT.

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OPTICAL DIPOLE FORCE TRAPPING OF A FERMION- . . . PHYSICAL REVIEW A 66, 021401"R# "2002#

021401-3

171Yb and 174Yb (Kyoto) (2003)

T. Miyakawa, K. Oda, T. Suzuki and HY, J. Phys. Soc. Jap. 69 (2000) 2779 (Static Properties)T. Miyakawa, T. Suzuki and HY, Phys. Rev. A62 (2000) 063613 (Collective Excitations, Sum rule)T. Miyakawa, T. Suzuki and HY, Phys. Rev. A64 (2001) 033611 (Induced Instability & Super Bose Nova)K. Oda, T. Miyakawa, HY and T. Suzuki, J. Phys. Soc. Jap. 70 (2001) 1549 (Ps-Ps Int. & Ps-BEC)T. Sogo, T. Suzuki and HY, Phys. Rev. A66 (2002) 013618 (Collective Excitations, RPA)T. Sogo and HY, Phys. Rev. A66 (2002) 043611 (Collective Ferromagnetism)T. Sogo, T. Suzuki and HY, Phys. Rev. A68 (2003) 063607 (Phase separation & Shell structure)HY, Y. Takayama and T. Suzuki, Physica B329-333P1 (2003) 25 (BF-bound state & BF-Phase Diagram)T. Miyakawa, HY and T. Suzuki, cond-mat/0401107 (1D-BF-cond. & Peierls instability)

Theoretical Studies by Our Group

We discuss briefly: ・Repulsive BF Int. Shell-structure (Phase separation)・Attractive BF Int. Induced instabilities, Super Bose Nova・Composite Fermion and Quantum Mass Action・Quantum Vortex

Fermion Shell Structure (Phase Separation)in case of strong BF repulsive interaction

h : BF Coupling Const.g：BB Coupling Const.

Critical Coupl. Const.h/g ∼ (1 − 1.3)

BF

Fermion ShellStructure

We first calculate the ground state of the system in the

mean-field approximation. At T!0 this is given by the

coupled Gross-Pitaevskii and Hartree-Fock equations !12"

!"#2

2m$2#

1

2m%ho

2 r2#g&b'r! (#h& f'r! (")0'r! (!*b)0'r! (

'5(

and

!"#2

2m$2#

1

2m%ho

2 r2#h&b'r! ("+,'r! (!-,f +,'r! (, '6(

where *b is the bosonic chemical potential, -,f is the fermion

single-particle energy in the occupied state , , the highestoccupied state giving the Fermi energy -F . The values of *b

and -F are determined by the boson number Nb and the

fermion number Nf . The boson and fermion density distri-

butions are given by

&b'r! (!Nb#)0'r! (#2, & f'r! (! .,

-,/-F

#+,'r! (#2, '7(

in terms of the bosonic and fermionic single-particle wave

functions )0 and +, which are normalized to unity. The

above set of equations determines the density distribution

and occupied single-particle wave functions, which are then

used to calculate the unoccupied single-particle energies and

wave functions by a similar set of equations as Eqs. '5( and'6(.We calculate collective oscillation modes in RPA, i.e., by

solving the linearized equation of motion for the excitation

operator

!H ,0†"$#10†, '8(

where 0† is given by a linear combination of particle-hole'p-h( creation and annihilation operators of both bosons andfermions, and 1 gives the corresponding excitation fre-

quency. Here, particles 'holes( refer to the single-particlestates unoccupied 'occupied( in the ground state in the mean-field approximation. Details of the formula and calculation

procedures are given in Ref. !12".In the actual calculation we took the number of bosons Nb

much larger than that of fermions Nf . We consider a41K'boson(- 40K'fermion( system with Nb!10

6 and Nf

$455, the precise number of the latter being dependent onthe subshell closure of fermion single-particle states for a

given value of h. The conditions for the phase separation

have been discussed in Refs. !7,8,10", and according to thesethe present choice produces a shell-like distribution of fermi-

ons surrounding a bosonic core as the boson-fermion repul-

sion becomes sufficiently strong. The boson-boson interac-

tion is fixed at abb!15.13 nm, while the boson-fermioninteraction is varied. The mass m!0.649$10"25 kg and the

oscillator frequency %ho!103 Hz give the oscillator length

2!!#/m%ho!1.27 *m.Figure 1 shows the density distribution of bosons for

h/g!1, while Fig. 2 shows that of fermions for various

values of h/g . The boson density distribution is almost inde-

pendent of the h/g value, which implies that the effect of

fermions on the bulk properties of bosons is negligibly small.

From Fig. 2 we find that the fermion density distribution

rapidly changes around h/g31 from central peaked to

surface-peaked 'shell-like( one, and the latter corresponds tothe phase separation of bosons and fermions as discussed in

Ref. !8". Below we study mostly the fermionic properties ofthe system which undergo a drastic change in the course of

the phase separation.

In Fig. 3 we show fermion single-particle levels measured

from the Fermi level -F against orbital angular momentum !

FIG. 1. Density distribution of bosons in units of 23 againstradial distance r/2 for h/g!1, where 2!(#/m%ho)

1/2.

FIG. 2. Density distribution of fermions in units of 23 for vari-ous values of boson-fermion interaction strength h/g .

SOGO, SUZUKI, AND YABU PHYSICAL REVIEW A 68, 063607 '2003(

063607-2

We first calculate the ground state of the system in the

mean-field approximation. At T!0 this is given by the

coupled Gross-Pitaevskii and Hartree-Fock equations !12"

!"#2

2m$2#

1

2m%ho

2 r2#g&b'r! (#h& f'r! (")0'r! (!*b)0'r! (

'5(

and

!"#2

2m$2#

1

2m%ho

2 r2#h&b'r! ("+,'r! (!-,f +,'r! (, '6(

where *b is the bosonic chemical potential, -,f is the fermion

single-particle energy in the occupied state , , the highestoccupied state giving the Fermi energy -F . The values of *b

and -F are determined by the boson number Nb and the

fermion number Nf . The boson and fermion density distri-

butions are given by

&b'r! (!Nb#)0'r! (#2, & f'r! (! .,

-,/-F

#+,'r! (#2, '7(

in terms of the bosonic and fermionic single-particle wave

functions )0 and +, which are normalized to unity. The

above set of equations determines the density distribution

and occupied single-particle wave functions, which are then

used to calculate the unoccupied single-particle energies and

wave functions by a similar set of equations as Eqs. '5( and'6(.We calculate collective oscillation modes in RPA, i.e., by

solving the linearized equation of motion for the excitation

operator

!H ,0†"$#10†, '8(

where 0† is given by a linear combination of particle-hole'p-h( creation and annihilation operators of both bosons andfermions, and 1 gives the corresponding excitation fre-

quency. Here, particles 'holes( refer to the single-particlestates unoccupied 'occupied( in the ground state in the mean-field approximation. Details of the formula and calculation

procedures are given in Ref. !12".In the actual calculation we took the number of bosons Nb

much larger than that of fermions Nf . We consider a41K'boson(- 40K'fermion( system with Nb!10

6 and Nf

$455, the precise number of the latter being dependent onthe subshell closure of fermion single-particle states for a

given value of h. The conditions for the phase separation

have been discussed in Refs. !7,8,10", and according to thesethe present choice produces a shell-like distribution of fermi-

ons surrounding a bosonic core as the boson-fermion repul-

sion becomes sufficiently strong. The boson-boson interac-

tion is fixed at abb!15.13 nm, while the boson-fermioninteraction is varied. The mass m!0.649$10"25 kg and the

oscillator frequency %ho!103 Hz give the oscillator length

2!!#/m%ho!1.27 *m.Figure 1 shows the density distribution of bosons for

h/g!1, while Fig. 2 shows that of fermions for various

values of h/g . The boson density distribution is almost inde-

pendent of the h/g value, which implies that the effect of

fermions on the bulk properties of bosons is negligibly small.

From Fig. 2 we find that the fermion density distribution

rapidly changes around h/g31 from central peaked to

surface-peaked 'shell-like( one, and the latter corresponds tothe phase separation of bosons and fermions as discussed in

Ref. !8". Below we study mostly the fermionic properties ofthe system which undergo a drastic change in the course of

the phase separation.

In Fig. 3 we show fermion single-particle levels measured

from the Fermi level -F against orbital angular momentum !

FIG. 1. Density distribution of bosons in units of 23 againstradial distance r/2 for h/g!1, where 2!(#/m%ho)

1/2.

FIG. 2. Density distribution of fermions in units of 23 for vari-ous values of boson-fermion interaction strength h/g .

SOGO, SUZUKI, AND YABU PHYSICAL REVIEW A 68, 063607 '2003(

063607-2

Nb = 106

Nf = 455

Boson density Fermion density

Mean field calculation(GP+HF)

Analytical estimateswith GP+TF

for various values of h/g . Occupied !hole" levels correspondto those below the horizontal line at E!0. The structure ofthe levels changes dramatically around h/g#1 in accordancewith the fermion density distribution. For h/g below(h/g)crit!0.95, single-particle levels show an oscillatorlikestructure with an effective oscillator constant, say, $eff whichdecreases towards h/g!1, while for h/g%1 the levels arearranged in a parabolic manner #!2 forming a bandlikestructure. We shall see later that the latter is typical of the

single-particle levels with surface-peaked density distribu-

tion.

Based on the single-particle states of fermions and

bosons, we construct collective oscillation modes in RPA as

coherent superposition of particle-hole excitations. We here

study monopole oscillations which are obtained by vertical

excitation of particles from the states below the E!0 line tothose above in Fig. 3, i.e., without change in the ! value ofthe particle and hole quantum numbers. We denote RPA ei-

genvalues by &n and eigenstates by "n'. Figure 4 shows thefermion monopole strength distribution as a function of ex-

citation frequency

S!&"! (n"0

")n"F0f "0'"2*!&#&n", !9"

where "0' stands for the RPA ground state, and

F0f!

1

!4+(i

fermions

ri2 !10"

is the monopole operator for fermions. Strong concentration

of the strength around &#2$eff for h/g,(h/g)crit is, at leastsuperficially, in accordance with the oscillatorlike single-

particle spectra in Fig. 3, although the coherence of various

p-h components becomes weaker towards this critical

strength: The energy-weighted sum-rule fraction of the most

collective state is more than 90% for h/g!0.8 and is 40%for 0.9 !there are a few states not resolved in the figure". Wenote that the coherence of the fermion p-h components is

brought about by the effective fermion-fermion interaction

which is induced by the boson-fermion interaction and the

propagation of bosonic excitation. This induced interaction is

of a long-range character -14. specified by the coherencelength of the Bose-Einstein condensate!BEC" -15.. Thestrength of this interaction is, however, rather weak for the

values of parameters under study. In fact, the bosonic mono-

pole strength !for the operator F0b similarly defined as F0

f ) is

FIG. 3. Fermion single-particle level spectra measured from the

Fermi level shown by the horizontal line at E!0. The levels areplotted vertically for each orbital angular momentum ! . The circlesshow the values (2n$l)/$eff$00 for h/g%1, where 00 is thefermion lowest single-particle energy, and those given by the for-

mula !26" for h/g"1.

FIG. 4. Monopole strength distribution for fermions ")n"F0f "0'"2

plotted against excitation frequency & . The strengths are given inunits of 14.

TRANSITION OF FERMIONIC SPECTRA AND . . . PHYSICAL REVIEW A 68, 063607 !2003"

063607-3

Fermion single-particle level spectra

−h

2

2m∆ → −

h2

2m∆S2

=h

2

2m〈r2〉l(l + 1)

Fermion Shell Structure (Phase Separation)

εnl = ε0 + hωeff(2n + l)

ωeff =

√1 −

h

gωho

・h/g < 1 (Harmonic-Oscillator-type)

・h/g > 1 (Rotator-type)

: Eff. HO frequency for Fermion (inc. Boson effects)

Oscillation-Rotation transition

Fermion kinetic term :

(because of Fermion shell structure)

(T. Sogo et.al., Phys. Rev. A68 (2003) 063607 )

εnl = εn +1

2Il(l + 1)

Fermions on 2D-sphere!

Induced Instability of BF-mixed Cond.Let’s consider the case with attractive BF-int.

(and repulsive BB-int.)

Two possibilities:1) Collapse of the system Induced instability and Super Bose Nova (Collapse of Attractive BEC: Bose Nova, Colorado, 2001)

2) Creation of the BF quasi-bound state(molecule) Composite-fermion state and Quantum Mass Action (Cold-molecule experiments)

Induced Instability of BF-mixed Cond. Energy Curve in Variational Method

Φ(x;R) ≡ ξ−3/2φ(x;R) =

√(32π

)3/2 Nb

(ξR)3exp

(−3x2

4R2

),

nf (x;R) = (6π2)−1[µf (R) − x2 − h|φ(x;R)|2

]3/2θ(Λ − x),

１）Variational Ansatz (originally by Beym et.al. for BEC)

：Boson Gaussian ansatz

：Fermion Thomas-Fermi Solution

R ：Radius of Boson Density Distribution（Variational Parameter）

２）Energy Curve

E(R) = E[Φ(x;R), nf (x;R)]Nb = Nf = 106

g = 0.2

α = |h|/g

F!"#$% f!"2!C

R3e!(3"2/2R2)"0, !4#

where C"(3/2&)3/2hNb . We take n f(x;R)$0 when x'" .The scaled chemical potential % f"2% f /(()) in Eq. !3# isdetermined by the normalization condition *d3x+ f(x)"Nf .

As a result, the two parameters " and % f become functions

of R. Using Eqs. !2# and !3#, the total energy E(R)

"E,-(x;R),n f(x;R). becomes

E!R #/!()#"Nb! 98 1

R2#1

2R2#2!5/2

3

&gNb

1

R3"

#Nf/Ike!R ##Iho!R ##CR!3Ib f!R #0, !5#

where g"2g/(()13) and integrals Ike ,ho ,b f(R) are definedby

Ike!R #"1

5&Nf#0

"

dxx2F!x #5/2,

Iho!R #"1

3&Nf#0

"

dxx4F!x #3/2, !6#

Ib f!R #"1

3&Nf#0

"

dxx2e!(3x2/2R2)F!x #3/2.

One of the important candidates for mixed condensate is

the 39,41K– 40K !boson-fermion# system. Their scatteringlengths are not well fixed at present, and different values

have been reported experimentally ,20–22.. In order to bequantitative and give realistic estimates of the region of sta-

bility, we take for the scattering length, abb , the41K value of

Ref. ,20.. This gives g"0.2 at a trapping frequency ) of 450

Hz. For the attractive boson-fermion interaction h , we take

several values for the ratio: 2$! h/ g"033 for the fixedvalue of g . It should be noted that the interaction strength

can be shifted using the Feshbach resonance phenomena

,23., whose applications for K-atom have been discussed in

,21,24..In Fig. 1, numerical results for E(R) in Eq. !5# are plotted

for 2"$h$/ g"0.1,1.0,2.0,2.5,3 !lines a–e# with N"Nb

"Nf"106, where three kinds of patterns can be read off:

stable !a#, metastable !b,c#, and unstable cases !d,e#. Forweak boson-fermion interaction, the system is stable against

the variation of R, and the total energy has an absolute mini-

mum. At the intermediate strength it becomes metastable

with one local minimum !lines b,c#, and, for strong attractiveinteractions, the minimum disappears and the system be-

comes unstable !lines d,e#. It should be noted that similarmetastable states appear in the BEC with attractive interac-

tions ( 7Li) ,16., but no stable states can exist in that case.As seen in Fig. 1, the stability of mixed condensate can be

judged from the small R behavior of E(R): positive diver-

gence in small R suggests a stable state. To obtain the stabil-

ity condition, we consider analytically the asymptotic expan-

sion of E(R) in R$1. For that purpose, the leading order

terms of % f(R) and "(R) in small R should be determinedfrom Eq. !4# and the normalization condition: *d3x+ f(x;R)

"Nf . To evaluate them, we assume "(R)/R$1 when R

$1, and expand the Gaussian function to the order of

(x/R)2: exp(!3x2/2R2)31!(3x2/2R2). The consistency ofthese assumptions will be shown later, and can be also

checked numerically. This assumption allows an analytical

evaluation of the integral in the normalization condition, and

we obtain

% f3!% 32& & 3/2$h$Nb

1

R3

#!3

2% 32& & 3/4!48Nf #

1/3! hNb#1/2

1

R5/2, !7#

"3% 23& 1/4% 2&

3& 3/8 !48Nf #

1/6

! hNb#1/4R5/4. !8#

It should be noted that Eq. !8# leads to "/R4R1/4$1 for R$1, which shows the consistency of our assumption. UsingEqs. !7# and !8#, in the leading order in small R, E(R) be-comes

FIG. 1. The total energy variation E(R)/(N()) of the boson-fermion mixed condensate with the Gaussian boson distribution

-(x;R) of radius R: E(R) and -(x;R) are defined in Eqs. !5# and!2#. The Thomas-Fermi density function !3# is applied for the fer-

mion distribution. N"Nb"Nf"106, g"0.2, and 2" h/ g"0.1,

1.0,2,2.5,3 for lines !a#–!e#.

T. MIYAKAWA, T. SUZUKI, AND H. YABU PHYSICAL REVIEW A 64 033611

033611-2

E!R "/!#$"%! 32& " 3/2Nb

2 # gNb

22/5!$h$Nf% 1

R3

"1

4

9

61/6! 32& " 3/4! $h$Nb"

1/2Nf4/3

1

R5/2

"O!R!2", !9"

so that, in the small R region, E(R) is dominated by the R!3

term, and a positive coefficient of the R!3 term gives the

condition for stability

'($h$

g#2!5/2

Nb

N f

. !10"

For the cases of Fig. 1 it becomes '#0.18, which shows thatonly line !a" corresponds to the stable state. This condition isalso plotted as the dashed line in Fig. 2.

From the above stability condition, lines !b" and !c" inFig. 1 are found to be metastable, so that, in principle, the

equilibrium states !at local minima R$Req) should collapse

by the quantum tunneling effects into those of R$0 throughthe potential barrier between R$0 and Req . To study the

collapses of metastable states we estimate their collective

tunneling lifetime )ct applying the Gamow theory of the

nuclear '-decay to these states *25+; we use WKB approxi-mation introducing the kinetic term of the collective motion

T$!#2/2Md2/dR2 (M is inertia mass for the collective

variable R). E(R) is approximated by the simplified linear-

plus-harmonic oscillator potential V(R) *26+ !see Fig. 3 forschematic illustration":

V!R "$# V1!R "$FR!G !0,R#Rt",

V2!R "$1

2M-2!R!Req"

2"V0 !Rt,R ",

!11"

where - is a harmonic oscillator frequency around the local

minimum. The parameters F, G, Req , and V0 are obtained

from the fit to the calculated E(R). As shown in Appendix A,

the tunneling lifetime )ct for V(R) becomes )ct!1

$D exp(!W), where D is a staying probability on barrier

surface per unit time and exp(!W) is a transmission coeffi-

cient for the potential barrier. The explicit formulas for D

and W are

D$4!2-

!&!.V

#-!1

2, !12"

W$4

3!2M

#2! .V!

#-

2" 1/2!Rt!RE""

!Req!Rt"2

aHO2

,

!13"

where RE is a WKB turning point for the energy E and .V$V1(Rt)!V0 is a barrier height. aHO$!#/M- is the har-

monic oscillator length around the equilibrium point. The

derivation of Eqs. !12" and !13" is given in Appendix A.To determine M and - of collective motion, we numeri-

cally calculated the second order coefficient in the expansion

of E(R) around Req . By adopting for - the monopole fre-

quency value calculated in the sum-rule method *14+, weextracted the value for M from this coefficient. For param-

eters (Req ,V0), *Rt ,V1(Rt)+ and RE , we use the values that

can be obtained from the numerically calculated E(R).

For the present case of N$106 and '$2.0 we obtain Rt

!RE%1, Req!Rt%3 in the unit of / , and .V%40 in theunit of N#$ *see line !c" in Fig. 1+. The inertia mass and

FIG. 2. Stability phase diagram of the boson-fermion mixed

condensate for N(Nb$Nf and '$ h/ g when g$0.2. Three re-gions exist in it: stable !S", metastable !MS", and unstable !US".The dashed line at '$0.18 corresponds to the border of the stableregion, and the solid line shows the Mo”lmer scaling law between

metastable and unstable regions. The open circles correspond to the

parameters of the states in Fig. 1, and the closed circles show nu-

merically confirmed critical states between metastable and unstable

regions.

FIG. 3. Schematic view of potential for metastable condensates.

V1(R) and V2(R) are the linear and harmonic oscillator parts of the

potential V(R). Req , Rt , and RE are the equilibrium, maximum,

and the WKB turning points, respectively. V0 is the equilibrium

energy of the potential.

INDUCED INSTABILITY FOR BOSON-FERMION MIXED . . . PHYSICAL REVIEW A 64 033611

033611-3

Meta-Stable

Unstable

Stability Phase Diagram

Stability Phase Diagram of BF-mixture

N = Nf = Bb

α = |h|/g

E!R "/!#$"%! 32& " 3/2Nb

2 # gNb

22/5!$h$Nf% 1

R3

"1

4

9

61/6! 32& " 3/4! $h$Nb"

1/2Nf4/3

1

R5/2

"O!R!2", !9"




'($h$

g#2!5/2

Nb

N f

. !10"











V!R "$# V1!R "$FR!G !0,R#Rt",

V2!R "$1

2M-2!R!Req"

2"V0 !Rt,R ",

!11"








and W are

D$4!2-

!&!.V

#-!1

2, !12"

W$4

3!2M

#2! .V!

#-

2" 1/2!Rt!RE""

!Req!Rt"2

aHO2

,

!13"
















regions.







033611-3

・Border between S and MS

E!R "/!#$"%! 32& " 3/2Nb

2 # gNb

22/5!$h$Nf% 1

R3

"1

4

9

61/6! 32& " 3/4! $h$Nb"

1/2Nf4/3

1

R5/2

"O!R!2", !9"




'($h$

g#2!5/2

Nb

N f

. !10"











V!R "$# V1!R "$FR!G !0,R#Rt",

V2!R "$1

2M-2!R!Req"

2"V0 !Rt,R ",

!11"








and W are

D$4!2-

!&!.V

#-!1

2, !12"

W$4

3!2M

#2! .V!

#-

2" 1/2!Rt!RE""

!Req!Rt"2

aHO2

,

!13"
















regions.







033611-3

・Asymptotic Exp. of Vriational Energy Curve around R=0

(T. Miyakawa, et.al, Phys. Rev. A64 (01) 033611)

|h|2

g=

4π2

62/3N

−1/6

f ∼ 12.0N−1/6

f

・Border between MS and US

(with WKB method)

Collective Excitations of BF mixed cond. Microscopic RPA calculations

１）BF in-phase oscillation２）BF out-phase oscillation

３）Fermionic oscillation４）Bosonic oscillation

単極振動

240 ph-config.

Monopole 双極子振動

464 ph-config.

Dipole ４重極振動

672 ph-config.

Quadrapole

Collective Excitations by Sum Rule Method

lation. This is because the fermionic mode costs a larger

kinetic energy and favors !!"/4 as seen in the term (Ekinb

"Ekinf )sin ! cos ! in Eq. #5$.

#c$ Dipole. General arguments %1& show that for a

harmonic-oscillator external potential a uniform shift of the

ground-state density generates an eigenstate of the system,

corresponding to the boson-fermion in-phase dipole oscilla-

tion with frequency '0. This is evident in Fig. 2#c$ and alsoin Eq. #6$ at !!0. In the regions I and III the out-of-phaseoscillation is unfavorable due to the same reason as for the

monopole mode: It loses the energy gained in the ground-

state boson-fermion configuration. For a weakly repulsive h

an interesting possibility arises: In the region II, the out-of-

phase mode of the boson-fermion oscillation lies lower than

the in-phase mode. Let us first note that at h!0 the out-of-phase oscillation frequency becomes degenerate as the in-

phase one because the bosonic and the fermionic dipole

modes are independent. One may note that at h/g!1 againthe degeneracy occurs. Here the potential term for the fer-

mion becomes almost linear to the fermion density itself

%see, Eq. #10$&. This suggests that the fermion density is de-termined almost entirely by the chemical potentials and be-

comes nearly constant as far as the boson density is finite. A

uniform dipole shift of fermions thus produces almost noeffect on bosons, and results in the degeneracy of the fre-quency. Between h/g!0 and 1, the boson-fermion repulsionis weaker for the out-of-phase oscillation than the in-phaseone #and hence the ground state$ as reflected in the sign of( .In the present paper, we studied collective oscillations of

trapped boson-fermion mixed condensates using the sum-rule approach. We introduced a mixing angle of bosonic andfermionic multipole operators so as to study if the in- orout-of-phase motion of those particles is favored as a func-tion of the boson-fermion interaction strength. For the mono-pole and quadrupole cases, the coupling of the Bose- andFermi-type oscillation is not large for moderate values of thecoupling strength h. At large values of "h/g", the low-lyingmodes become an in-phase oscillation of bosons and fermi-

ons. This is especially so for the monopole oscillation at

attractive boson-fermion interaction: The excitation energy

of this mode almost vanishes at the instability point of the

ground state. In the case of the dipole mode, in contrast, the

in-phase oscillation remains an exact eigenmode with a fixed

energy for Harmonic-oscillator potentials, while the average

energy of the out-of-phase oscillation is strongly dependent

on the boson-fermion interaction. We found that at weak

repulsive values of the interaction the out-of-phase motion

stays lower than the in-phase oscillation. In this paper we

calculated also the frequencies of the high-lying modes for

each multipole, by adopting the operators orthogonal to the

low-lying modes. These modes, too, are collective in charac-

ter and, in the present framework, their average frequencies

showed rather strong dependencies on the boson-fermion

coupling strength. Deeper insight into the collective modes

studied in this paper will require a detailed investigation of

the solutions of, e.g., the self-consistent random-phase ap-

proximation type equations that allow an arbitrary radial de-

pendence of the excitation operators. Studies in this direction

are now in progress.

FIG. 2. Excitation frequencies of collective oscillations #a$

monopole, #b$ quadrupole, #c$ dipole as functions of h/g! h/ g ,

calculated based on the Eqs. #4$–#6$. The mixing angle ! has beendetermined so as to minimize ' for each multipole operator. The

ordinates are given in the unit of '0 and are dimensionless. The

solid #dashed$ lines are the lower #higher$ energy mode for eachoscillation.

FIG. 3. Mixing angles of in/out-of-phase excitation modes,

which are determined by minimizing excitation energies in Fig. 3.

The solid line shows the monopole, the dashed line the quadrupole,

and the dotted line the dipole oscillations. Shaded areas show

mainly in-phase #hatches$ and mainly out-of-phase #cross hatches$regions in ! . Angles for pure bosonic or fermionic modes are alsoindicated.


063613-4

lation. This is because the fermionic mode costs a larger




















































063613-4lation. This is because the fermionic mode costs a larger




















































063613-4

h : BF Coupling Const.g：BB Coupling Const.

・Dipole Mode h<0 In-phase (COM motion) 0<h<1 Out-phase 1<h In-phase again! (phase sep.)・Monopole Mode h/g ～-2 Soft -> Instability

Lowest Energy Mode & Vertical Mode

Out-phase Normal

Out-phase Phase sep.

Nb = Nf = 106

Tunneling Collapse of Metastable State１）Estimated by calculations similar with that of α-decay２）For Case c),

τ−1ct /ω ∼ 108 exp(−107)

practically stable!

１）Adjust the parameters２）Parametric Resonance, et.al.

Super Bose Nova & Quantum Tunneling Effect Collapse of Unstable/Metastable State

１）Similar Collapse of BEC - Bose Nova２）High-T, High-ρ, after Collapse?

Tunneling Life-Time：

Several Possibilities for Quantum Tunneling

Collapse Exp. of BF mixture by LENS group(LENS, SCIENCE 297 (02) 2240, PR A68 (03) 043626)

Collapse of 40K-87Rb

Rb K

abf = −21+4.3−4.8 nm

abb = 5.5 nm

ωa = 2π × 24 sec−1

ωr = 2π × 317 sec−1

(for K. For Rb, /1.5)

Exp. Parameters

measures in the experiment is a collisional rate, which at

ultralow temperature has a dependence

!"Na2 #4$

on the atom number N and on the scattering length a. The

absolute atom number, in the case of a mixture of two spe-

cies, cannot be determined with an accuracy typically better

than 40%, giving rise to a 20% uncertainty on the scattering

length.

By solving Eqs. #2$ and #3$, we can investigate the groundstate and the stability of the system against the collapse. The

equations are solved with an imaginary-time evolution, em-

bedded in an iterative scheme, as in Ref. %6&. The stability ischecked by requiring that energies, chemical potentials, and

central densities converge to the final value with a precision

at least of 10!6. In Fig. 1 we show a typical ground state

configuration, calculated with the nominal values aBF"!410a0 , NB"7.5#104, NF"2#104. As already discussedin Refs. %5,6&, the mutual attraction results in an enhance-ment of the density of both species in the volume of overlap.

We note that the gravitational sag, that shifts the centers of

the two clouds, reduces the effect of the mutual attraction

with respect to the concentric case studied in Ref. %6&.The deformed ground state of the system is predicted to

lead to a series of effects that can be described within a

mean-field approach, including a modification of the fre-

quencies of collective excitations %9,11& and of the expansionof the two clouds from the trap %10&. We actually observedthe latter effect already from the first experiments %4&, wherewe had an evidence of a modified expansion of the conden-

sate in presence of the Fermi gas.

If the atom numbers are increased above some critical

values, an instability occurs. In our model, according to the

study reported in Refs. %6,16&, the signature of the instabilityis the failure of the convergence procedure during the itera-

tive evolution toward the ground state of the system. In par-

ticular, the onset of instability is characterized by an indefi-

nite growth of central densities, which triggers the

simultaneous collapse of the two species. We note that the

investigation of the actual dynamics after the system has

been driven into the unstable region would require a descrip-

tion that goes beyond Eqs. #2$ and #3$, in a similar fashion towhat happens during the collapse of a single Bose-Einstein

condensate with attractive interaction %17,18&. Here we willnot discuss these aspects, since we are concerned with the

determination of the critical values for the onset of instabili-

ties only.

In the experiment we observe the instability of the Fermi-

Bose system as the number of condensed bosons is increased

by evaporation at a fixed number of fermions %12&. The sig-nature of the instability is a sudden loss of most of the fer-

mions from the magnetic trap, which we attribute to a largely

increased three-body recombination in the collapsing

samples. Here we are interested just in the critical atom num-

bers, but we note that further experiments, for example, us-

ing a Feshbach resonance to tune aBF %13&, will be necessaryto study the collapse dynamics of the system.

To compare the predictions of the mean-field model to the

experimental findings on the instability, we have built a sta-

bility diagram, shown in Fig. 2. Here, we report in the plane

NB!NF the condensate and fermion atom numbers that we

were able to measure in the experiment for stable samples,

and compare them with the calculated critical line for insta-

bility. We note that all the data points refer to samples at

temperatures of the order of the (0.2–0.5)TF for the Fermi

gas, and with almost no detectable thermal fraction for the

BEC. In the model, the occurrence of instability depends on

both NB and NF , and also on the value of the scattering

length aBF . In Fig. 2 we have therefore plotted a family of

critical lines by varying aBF around the nominal value. Note

that the position of the critical line depends quite strongly on

the value of aBF , and it shifts toward larger atom numbers

for decreasing magnitudes of aBF . Although the critical line

calculated for the nominal value aBF"!410a0 is in rathergood agreement with the experimental observation, the one

calculated for aBF"!395a0 better fits the data. In the ex-periment we observed the collapse of the Fermi gas for the

number pairs close to the two marked data points %19&, whichappear to be actually the region of achievable number pairs

closest to the instability.

FIG. 1. Density profiles along the vertical y direction #a$ andcontour levels in the (y!z) plane #b$ of the ground-state of theFermi-Bose mixture for aBF"!410a0 , NB"7.5#104, and NF

"2#104.

MODUGNO et al. PHYSICAL REVIEW A 68, 043626 #2003$

043626-2

Density profiles before collapse (GP+TF calculation)

BF quasi-bound states and B-F mixed condensates Let’s consider the case of attractive B-F interaction.　 The B-F pairs can make quasi-bound states (BF)/Composite Fermion in B-F mixed quantum gas:

B + F ↔ BF

It may produce new phases like BF degenerate states.BF (fermion) degenerate ststes vs the BEC of unpaired bosons

If the BF binding energy is enough large, the BF pairs can exhausts bosons and the BEC will not appear in the mixed condensates!

Recently, many experiments have been done for creating molecules (especially in F+F system).

+ ↔

Law of Quantum Mass Action・We study the phase structures of atomic gas under the molecular formation process A1 +A2 ↔ A12.・Special interests are on quantum statistical effects of atoms. (BEC, Fermi degeneracy et.al.)

・For ideal atomic gas (Maxwell-Boltzmann gas), it gives “Law of Mass Action”

and no phase structures.・However, quantum statistical effects give rich phase structures.

nA12

nA1nA2

= K(T )

・Simplified models with quantum statistical effects: 1) Uniform system 2) Free Bose/Fermi gas 3) Ideal mixing system

・Another interest is in controlled ultra-cold chemical reactions

Chemical Equilibrium for A1 +A2 ↔ A12 A1-A2 mixed gas

: Total A1 density: Total A2 density

: Isolated A1 density: Isolated A2 density: A12(composite) density

Equilibrium condition for the process

・Ideal mixing condition for Quantum gasses

nA1t = nA1 + nA12

nA2t = nA2 + nA12

nA2

nA1

nA12

A1 + A2 ↔ A12

µA1 + µA2 = µA12 µA = mAc2

+ µTA

nB =(mB)3/2

√2π2

∫∞

0

√εdε

e(ε−µT

B)/kBT − 1

=λ−3

T,A

Γ(3/2)

∫∞

0

x1/2dx

ex+ν − 1

nF =(mF )3/2

√2π2

∫∞

0

√εdε

e(ε−µT

F)/kBT + 1

=λ−3

T,A

Γ(3/2)

∫∞

0

x1/2dx

ex+ν + 1

λT,A =

[2πh

2

mAkBT

]1/2

µTA1 + µ

TA2 − µ

TA12 = ∆mc

2≡ (mA12 − mA1 − mA2)c

2

: Thermal de Brogli wave length

: mass defect (standard chemical affinity)

Scaled Quantities n =

n

Nt

Nt = nA1t + nA2t

m =m

mA12

E =h

2

mA12

N2/3

t

T =kBT

E

Conservation Law of Mass∆m

mA1 + mA2

! 1 =⇒ mA12 ∼ mA1 + mA2(except mass defect in chemical equilibrium)

nA1t + nA2t = 1

Independent parameters

Three Cases 1) F1 + F2 ↔ FF12 (Composite Boson)2) B1 + F2 ↔ BF12 (Composite Fermion)3) B1 + B2 ↔ BB12 (Composite Boson)

0 ≤ mA1 ≤ 1,

0 ≤ nA1t ≤ 1,

0 ≤ T < ∞. mA1 + mA2 = 1

: total density

∆m =

+ (resonance)− (bound state)

m1 = m2 = 1/2

n1t = n2t = 1/2

T-dep. of Atom densities : B1 + F2 ⇔ BF12

: Fermi T

TF1t =(6π2n1t)2/3

2m1

B1 density (boson) & BF12 density (composite boson)

!0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

log10(T)

n12

critical

dE= 10.

dE= 4.

dE= 0.

dE= -4.

dE= -10.

µF12 > 0 µF12 < 0

・At high-T, all molecules desociate into B1+F2.・Condition for BEC of B1 (Boson) at low-T :

・When , molecule number increases around (precursor of quantum gas?)

∆m ! 0

TF1t < ∆m (100% BEC at T = 0)−(3π2)/2 < ∆m < TF1t (Partial BEC at T = 0)

∆m < −(3π2)/2 (No BEC)

µF12 ∼ 0

∆m

!0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

log10(T)

n1

critical:n1

critical:n2

dE= 10.

dE= 4.

dE= 0.

dE= -4.

dE= -10.

BEC No-BEC

nB1 = m1ζ(3/2)(

T2π

)3/2

nB

1

nB

F12

BEC-TC for B1 (boson) : B1 + F2 ⇔ BF12

Tc for B1-BEC m1 = m2 = 1/2

!20 0 20

!0.4

!0.2

0.0

0.2

0.4

0.6

dE

log

10(T

c1)

max

n1t= 0.5

max

n1t= 0.333333333

min

max

n1t= 0.666666667

−

(3π2)2/3

2

T =2πm1

(n1t

ζ(3/2)

)2/3

T =2πm1

[n1t−n2t

ζ(3/2)

]2/3

←→

BF (fermion) degenerate ststes vs the BEC of unpaired bosons

∆m

T=0 Phase Diagram of B1 + F2 ⇔ BF12

: Fermi T TF1t =(6π2n1t)2/3

2m1

n1t + n2t = 1ex. (F,BF) : F2 (fermion)-BF12(fermion molecule) coexistent phase

m1 = m2 = 1/2

(6π2)2/3

2m1

TF1t = ∆m

A

B

AB

C

BC

AC

(0, 0)

−

(3π2)2/3

2

(6π2n2t)2/3

2= −∆m

[6π2(n1t−n2t)]

2/3

2 = ∆m

!20 0 200.0

0.2

0.4

0.6

0.8

1.0

"E/h2n12-1 (n1tot+n2tot)

2/3

n2(F

)tot/(

n1to

t+n

2to

t)

(BF)

(B,F) (B,F,BF)

(F,BF)

(B,BF)

A

B

C∆m

nF

2t

m1 = m2 = 1/2 n1t = n2t = 1/2

T-dep. of Atom densities : F1 + F2 ⇔ FF12

: Fermi T TF1t =(6π2n1t)2/3

2m1

F1 density (fermion) & FF12 density (composite boson)

∆m

!0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

log10(T)

n1

2

critical

dE= 21.

dE= 4.

dE= 0.

dE= -4.

dE= -21.

nB12 = ζ(3/2)

(T

2π

)3/2

BEC No-BEC

nF

F12

!0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

log10(T)

n1

critical:n1

critical:n2

dE= 21.

dE= 4.

dE= 0.

dE= -4.

dE= -21.

TF1t = TF2t = (3π2)

2

3

µF1 > 0 µF1 < 0

nF

1Application to F + F ⇔ FF

FF Molecular Formation and BEC-BCS crossover

Atomic Fermi Gas and FF-molecule condensation

・ Observation of Fermi degeneracy 1999 JILA(Jin) (40K) ・ Change of inter-atomic int. between two Fermions with Feshbach resonance method 2003 JILA (40K), ENS (6Li)・ Molecular Formation of Fermion pair 2003 Innsbruck, ENS & Rice (6Li), JILA (40K) ・ Observation of BEC of Fermion-paired molecule 2003 Innsbruck & MIT (6Li), JILA (40K)・ Fermion pair cond. & BEC-BCS cross-over? 2004 Innsbruck (6Li) & JILA(Jin) (40K) & MIT (6Li)

Fermion-pair-molecules (boson) condensate at enough low-T

・・BEC-BCS crossover (Schafroth, Legget, Noziers & Schmidt-Rink) Attractive region : BCS state of FF Cooper pairs Repulsive region : BEC state of molecules

T < TC ∼ 0.1TF

at lower initial T=TF with increasing !B, an effect pre-dicted in [4–7].

An essential aspect of these measurements is the fastmagnetic-field sweep that pairwise projects the fermionicatoms onto molecules. It is a potential concern that thecondensation might occur during this sweep rather thanat Bhold. However, in our previous work it was shown thata magnetic-field sweep with an inverse speed less than800 !s=G was too fast to produce a molecular condensate

when starting with a Fermi gas 0.68 G on the BCS side ofthe resonance [16]. Thus, the inverse sweep speed we usein this Letter of typically 50 !s=G, while sufficientlyslow to convert 60% of the sample to weakly boundmolecules [11], is much too fast to produce a molecularcondensate.

In addition, we have checked that the observation of acondensate on the BCS side of the resonance does notdepend on this sweep speed. As seen in Fig. 5(a), muchfaster sweeps result in fewer molecules. This is consistentwith our previous study of the molecule creation process[11]. However, we find that the measured condensatefraction is independent of the sweep rate [Fig. 5(b)].Even with the lower number of molecules, and thereforea lower phase space density of the molecular gas, weobserve an essentially unchanged condensate fraction.

Finally, we note that, as in our previous measurementsperformed in the BEC limit, the measured condensatefraction always remains well below one [16]. As part ofour probing procedure the magnetic field is set well belowthe Feshbach resonance where the molecule lifetime isonly on the order of milliseconds [15,40]. This results in ameasured loss of 50% of the molecules and may alsoreduce the measured condensate fraction.

In conclusion, we have introduced a method for prob-ing the momentum distribution of fermionic atom pairsand employed this technique to observe fermionic con-densates near a Feshbach resonance. By projecting thesystem onto a molecular gas, we map out condensation of

FIG. 3 (color online). Time of flight images showing thefermionic condensate. The images, taken after the projectionof the fermionic system onto a molecular gas, are shown for!B ! 0:12, 0.25, and 0.55 G (left to right) on the BCS side ofthe resonance. The original atom cloud starts at T=TF ! 0:07,and the resulting fitted condensate fractions are N0=N ! 0:10,0:05, and 0:01 (left to right). Each image corresponds to N !100 000 particles and is an average over 10 cycles of theexperiment.

00.05

0.10

0.15

0.20

-0.4-0.2

00.2

0.40.6

0.8

-0.05

0

0.05

0.10

0.15

N /N0

T/TF

∆B (gauss)

FIG. 4 (color online). Transition to condensation as a func-tion of both !B and T=TF. The data for this phase diagramwere collected with the same procedure as shown in the inset inFig. 2 with thold " 2 ms. The area between the dashed linesaround !B ! 0 reflects the uncertainty in the Feshbach reso-nance location from the width of the feature in Fig. 1. Thesurface and contour plots are obtained using a Renka-Clineinterpolation of approximately 200 distinct data points (#)[36]. One measure of when the gas becomes strongly interactingis the criterion jkFaj > 1, where "hkF is the Fermi momentum[20,21,37,38]. For these data, j!Bj< 0:6 corresponds tojkFaj > 1.

-0.5 0 0.5 1.00

0.05

0.10

0.15

-9 -6 -3 0 3 16 18198

200

202

204

B(g

auss

)

t (ms)N0/N

∆B (gauss)

FIG. 2. Measured condensate fraction as a function of detun-ing from the Feshbach resonance !B ! Bhold $ B0. Data herewere taken for thold ! 2 ms (!) and thold ! 30 ms (4) with aninitial cloud at T=TF ! 0:08 and TF ! 0:35 !K. The area be-tween the dashed lines around !B ! 0 reflects the uncertaintyin the Feshbach resonance position based on the 10%–90%width of the feature in Fig. 1. Condensation of fermionic atompairs is seen near and on either side of the Feshbach resonance.Comparison of the data taken with the different hold timesindicates that the pair condensed state has a significantly longerlifetime near the Feshbach resonance and on the BCS (!B > 0)side. The inset shows a schematic of a typical magnetic-fieldsweep used to measure the fermionic condensate fraction. Thesystem is first prepared by a slow magnetic-field sweep towardsthe resonance (dotted line) to a variable position Bhold, indi-cated by the two-sided arrow. After a time thold the optical trapis turned off and the magnetic field is quickly lowered by"10 G to project the atom gas onto a molecular gas. Afterfree expansion, the molecules are imaged on the BEC side ofthe resonance (#).

P H Y S I C A L R E V I E W L E T T E R S week ending30 JANUARY 2004VOLUME 92, NUMBER 4

040403-3 040403-3

at lower initial T=TF with increasing !B, an effect pre-dicted in [4–7].

An essential aspect of these measurements is the fastmagnetic-field sweep that pairwise projects the fermionicatoms onto molecules. It is a potential concern that thecondensation might occur during this sweep rather thanat Bhold. However, in our previous work it was shown thata magnetic-field sweep with an inverse speed less than800 !s=G was too fast to produce a molecular condensate

when starting with a Fermi gas 0.68 G on the BCS side ofthe resonance [16]. Thus, the inverse sweep speed we usein this Letter of typically 50 !s=G, while sufficientlyslow to convert 60% of the sample to weakly boundmolecules [11], is much too fast to produce a molecularcondensate.

In addition, we have checked that the observation of acondensate on the BCS side of the resonance does notdepend on this sweep speed. As seen in Fig. 5(a), muchfaster sweeps result in fewer molecules. This is consistentwith our previous study of the molecule creation process[11]. However, we find that the measured condensatefraction is independent of the sweep rate [Fig. 5(b)].Even with the lower number of molecules, and thereforea lower phase space density of the molecular gas, weobserve an essentially unchanged condensate fraction.

Finally, we note that, as in our previous measurementsperformed in the BEC limit, the measured condensatefraction always remains well below one [16]. As part ofour probing procedure the magnetic field is set well belowthe Feshbach resonance where the molecule lifetime isonly on the order of milliseconds [15,40]. This results in ameasured loss of 50% of the molecules and may alsoreduce the measured condensate fraction.

In conclusion, we have introduced a method for prob-ing the momentum distribution of fermionic atom pairsand employed this technique to observe fermionic con-densates near a Feshbach resonance. By projecting thesystem onto a molecular gas, we map out condensation of

FIG. 3 (color online). Time of flight images showing thefermionic condensate. The images, taken after the projectionof the fermionic system onto a molecular gas, are shown for!B ! 0:12, 0.25, and 0.55 G (left to right) on the BCS side ofthe resonance. The original atom cloud starts at T=TF ! 0:07,and the resulting fitted condensate fractions are N0=N ! 0:10,0:05, and 0:01 (left to right). Each image corresponds to N !100 000 particles and is an average over 10 cycles of theexperiment.

00.05

0.10

0.15

0.20

-0.4-0.2

00.2

0.40.6

0.8

-0.05

0

0.05

0.10

0.15

N /N0

T/TF

∆B (gauss)

FIG. 4 (color online). Transition to condensation as a func-tion of both !B and T=TF. The data for this phase diagramwere collected with the same procedure as shown in the inset inFig. 2 with thold " 2 ms. The area between the dashed linesaround !B ! 0 reflects the uncertainty in the Feshbach reso-nance location from the width of the feature in Fig. 1. Thesurface and contour plots are obtained using a Renka-Clineinterpolation of approximately 200 distinct data points (#)[36]. One measure of when the gas becomes strongly interactingis the criterion jkFaj > 1, where "hkF is the Fermi momentum[20,21,37,38]. For these data, j!Bj< 0:6 corresponds tojkFaj > 1.

-0.5 0 0.5 1.00

0.05

0.10

0.15

-9 -6 -3 0 3 16 18198

200

202

204

B(g

auss

)

t (ms)N0/N

∆B (gauss)

FIG. 2. Measured condensate fraction as a function of detun-ing from the Feshbach resonance !B ! Bhold $ B0. Data herewere taken for thold ! 2 ms (!) and thold ! 30 ms (4) with aninitial cloud at T=TF ! 0:08 and TF ! 0:35 !K. The area be-tween the dashed lines around !B ! 0 reflects the uncertaintyin the Feshbach resonance position based on the 10%–90%width of the feature in Fig. 1. Condensation of fermionic atompairs is seen near and on either side of the Feshbach resonance.Comparison of the data taken with the different hold timesindicates that the pair condensed state has a significantly longerlifetime near the Feshbach resonance and on the BCS (!B > 0)side. The inset shows a schematic of a typical magnetic-fieldsweep used to measure the fermionic condensate fraction. Thesystem is first prepared by a slow magnetic-field sweep towardsthe resonance (dotted line) to a variable position Bhold, indi-cated by the two-sided arrow. After a time thold the optical trapis turned off and the magnetic field is quickly lowered by"10 G to project the atom gas onto a molecular gas. Afterfree expansion, the molecules are imaged on the BEC side ofthe resonance (#).

P H Y S I C A L R E V I E W L E T T E R S week ending30 JANUARY 2004VOLUME 92, NUMBER 4

040403-3 040403-3

∆B = 0.12, 0.25, 0.55 G

PRL 92 (04) 040403

Colorado-Jin40K

BEC-TC for FF12 (boson) : F1 + F2 ⇔ FF12

: Fermi T

TF1t =(6π2n1t)2/3

2m1

except that now the effective attractive interaction Ueff!q"depends on energy as well as momentum. Thus this thirdterm 2NC may be viewed as the effect of Cooper-pairfluctuations on N. In the strongly coupled fermion-bosonproblem we are dealing with, it should be emphasizedthat our interpretation of the separate terms in (6) as 2NBand 2NC is only meant as a way of understanding theessential physics.

In understanding the character of the renormalized bbosons described by NB in (6), it is useful to transformthe Matsubara frequency summation into a frequencyintegration, NB # $ 1

!

PqR1$1 dznB!z"Im~DD!q; i"n ! z%

i#". One sees that if the b molecule component decays

into two Fermi atoms in the presence of the Feshbachresonance, it has a finite lifetime as given by the inverseof the imaginary part of the self-energy $g2r ~!! in~DD!q; z% i#". However, since Im!!q; z% i#" in (3) isproportional to the step function "!z% 2$$ E0

q", therenormalized b molecules do not decay if their energiesare smaller than E0

q $ 2$. The energy (!q) of a stablemolecule corresponding to the pole of ~DD is given by (forE0q $ 2$ & 0)

!q # !E0q $ 2$" % '2"$ g2r ~!!!q; !q"(: (7)

Such stable long-lived molecules appear when the ‘‘ef-fective’’ threshold 2~"" ) 2"$ g2r ~!!!q; z" becomes nega-tive, since decay into two Fermi atoms is then forbidden.Around z # q # 0, we may use the static approximationfor this renormalized threshold, 2~"" ’ 2"$ g2r ~!!!0; 0".Using (3), one finds ~!!!0; 0" > 0, and hence 2~"" decreases.NB in (6) is thus found to consist of two kinds of b

bosons, NB # N%#0B % N%>0

B , i.e., stable molecules N%#0B

with infinite lifetime and quasimolecules N%>0B which

can decay into two Fermi atoms (strictly speaking,N%>0

B also includes a contribution from scattering states).Extracting the contribution of these stable molecules, weobtain N%#0

B # Ppolesq Z!q"nB!!q", where Z!q"$1 )

1% g2r@ ~!!!q;!q"

@z .A similar discussion can be used to understand the

third term 2NC defined in (6), describing the renormal-ized Cooper-pair fluctuations. When 2$ approaches 2",the interaction Ueff!q" mediated by the b molecules be-comes very strong, so that stable preformed Cooper pairsmay appear [6]. Because of the resonance coupling gr, onefinds that the energy of these strongly renormalizedCooper-pair molecules is also given by (7). We can divideNC into contributions from (stable) preformed Cooperpairs ()N%#0

C ) and from scattering states ()NscC ),

with N%#0C # g2r

2

Ppolesq

@ ~!!!q;!q"@$ Z!q"nB!!q".

These stable bosons give a very clear physical pictureof BCS-BEC crossover present in the model Hamitonianin (1) [4,5].When 2" * "F, the Fermi states are dominantand a BCS-like phase transition is found. Indeed, since$+ "F > 0 in this regime, stable bosons are absent be-cause Im ~!!!q; z" ! 0 for z > 0. The only exception is forz # 0 [since Im ~!!!q; 0" # 0]. However, the condition thata stable (undamped) boson with q # 0 appears at !q # 0given by (7) is 2$ # 2~"". This is easily shown to reduce tothe equation for Tc in (4), i.e., one finds 2$ # 2~"" ) 2"$g2r ~!!!0; 0" is equivalent to 1 # Ueff!0"!!0; 0". Thus, as iswell known in weak-coupling BCS theory, the formationof stable Cooper-pair bosons and the phase transitionoccur at the same temperature [6,7]. No long-lived bosonswith q ! 0 exist above Tc.

On the other hand, when "< 0, the Fermi states arealmost empty, so that the phase transition is expected tobe BEC-like. Indeed, since 2~"" can be negative in thisregime, stable bosons with a spectrum given by (7) can

0

0.05

0.1

0.15

0.2

0.25

0.3

Tc/

ε F

(a) U=0U=0.3

BCSBEC

-1

-0.5

0

0.5

µ(T

c)

(b)U=0.3

µ=ν

0

0.1

0.2

0.3

0.4

0.5

-1 -0.5 0 0.5 1 1.5 2 2.5

Num

ber o

f Bos

ons

(Tc)

ν/εF

(c)NB

γ=0

NBγ>0

2NCsc

NCγ=0

NF0

-0.2

0

0.2

0.4

0.6

0.8

1.0

0

0.1

0.2

0.3

0 1 2 3

Tc/

ε F

Ueff/εF

U=0.3BCS

FIG. 3. Self-consistent solution for (a) Tc and (b) $!Tc" as afunction of " for U # 0:3"F and gr # 0:6"F. In panel (a), forcomparison, we show the BEC case (gr # U # 0), and the BCSsolution [4,5] of (4) with $!T" being the chemical potential of anoninteracting Fermi gas. Panel (c) shows the number offermions N0

F, stable b molecules N%#0B , quasimolecules (finite

lifetime) N%>0B , preformed Cooper pairs N%#0

C , and the scatter-ing part Nsc

C , all at Tc, as a function of the threshold parameter2". In the case shown, stable long-lived bosons exist only for" & 0:22"F [where $ # ~""!0; 0" is negative].

VOLUME 89, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 23 SEPTEMBER 2002

130402-3 130402-3

Tc in BEC-BCS crossoverby Y. Ohashi (Tsukuba) et.al.,

(PRL89 (02) 130402)Tc for FF12-BEC

!20 0 20

!0.4

!0.2

0.0

0.2

tdE

log

10(T

c12)

max

n1t= 0.5

max

n1t= 0.333333333

max

n1t= 0.666666667

TC = 2π

[min(n1t, n2t)

ζ(3/2)

]2/3

∆mC = TF1t + TF2t

Effect of Int. fot Tc ・BCS-region : strongly dependent ・BEC-region : weakly dependent？　　１）Mean-field approx. : ～4% (in atom-gas BEC)　　２）Correlation eff.: not strong ex. Superfluid He4 : ( )TC = 3.1 K Tλ = 2.17 K

m1 = m2 = 1/2

∆m

T=0 Phase Diagram of F1 + F2 ⇔ FF12

!20 0 20 400.0

0.2

0.4

0.6

0.8

1.0

"E/h2m12-1 (n1tot+n2tot)

2/3

n1to

t/(n

1to

t+n

2to

t)

(FF)

(F1,F2,FF) (F1,F2)

(F1,FF)

(F2,FF)

C

B

A

(6π2)2/3

2m1

(6π2)2/3

2m2

TF1t = ∆m

TF2t = ∆m

TF1t + TF2t = ∆m

A

B

AB

C

BC

AC

(0, 0)

: Fermi T

TF1t =(6π2n1t)2/3

2m1

n1t + n2t = 1

ex. (F1,FF) : F1 (fermion)-FF(boson molecule) coexistent phase

m1 = m2 = 1/2

∆m

Quantum Vortex in BEC

※ Studies on Vortex & Vortex lattice of BEC are recently very hot. Much progress has been done especially for the dynamics.

!

"!

!"#$"%&&'()'*%+"',--.'

./' 01&%"2+3(#)'#4'5#"3%6'7+33(8%&'()'9#&%:;()&3%()'<#)=%)&+3%&'

#$%&'()*+! ,-.'(/*0! 12%3! %! 4*3! .-2*! (&! 0$1*.52$(+('3! %&+! 0$1*./-&+$/'(,('36! ! 7&!

0$1*./-&+$/'-.08!9%:&*'(/!52$;!2(&*0!%..%&:*!'<*90*2,*0!(&!.*:$2%.!2%''(/*0!'<%'!<%,*!=**&!

+(.*/'23!(9%:*+6!!7&!0$1*.52$(+08!+(.*/'!-=0*.,%'(-&!-5!,-.'(/*0!<%+!=**&!2(9('*+!'-!09%22!

%..%30! >$1! '-!""!,-.'(/*0?8!=-'<! (&! 2(@$(+!AB*! C"D!%&+!9-.*! .*/*&'23! (&! .-'%'(&:!:%0*-$0!

E-0*FG(&0'*(&!/-&+*&0%'*0!>EGH?!CI8!JD6!

K*!<%,*!-=0*.,*+!'<*!5-.9%'(-&!-5!<(:<23F-.+*.*+!,-.'*;!2%''(/*0!(&!%!.-'%'(&:!E-0*F

/-&+*&0*+!:%0!CAD6!L<*3!M*.*!1.-+$/*+!=3!.-'%'(&:!'<*!/-&+*&0%'*!%.-$&+!('0! 2-&:!%;(0!

$0(&:!'<*!-1'(/%2!+(1-2*!5-./*!*;*.'*+!=3!%!=2$*F+*'$&*+!2%0*.6! !N!0'.(4(&:!5*%'$.*!-5!'<*!

-=0*.,*+! 2%''(/*0! (0! '<*! *;'.*9*! .*:$2%.('38! 5.**! -5! %&3!9%O-.!+(0'-.'(-&08! *,*&!&*%.! '<*!

=-$&+%.36! ! P$/<! QN=.(4-0-,R! 2%''(/*0!M*.*! 5(.0'! 1.*+(/'*+! 5-.! @$%&'()*+!9%:&*'(/! 52$;!

2(&*0! (&! '31*F77! 0$1*./-&+$/'-.06! ! L<*! -=0*.,*+! '.(%&:$2%.! 2%''(/*0! /-&'%(&*+! -,*.! "SS!

,-.'(/*0!M('<!2(5*'(9*0!-5!0*,*.%2!0*/-&+06!!7&+(,(+$%2!,-.'(/*0!1*.0(0'*+!$1!'-!AS!06!!L<*!

2%''(/*0!/-$2+!=*!:*&*.%'*+!-,*.!%!M(+*!.%&:*!-5!.-'%'(-&!5.*@$*&/(*0!%&+!'.%1!:*-9*'.(*08!

0<*++(&:!2(:<'!-&!'<*!5-.9%'(-&!1.-/*006!!T$.!-=0*.,%'(-&!-5!2%''(/*!+(02-/%'(-&08!(..*:$2%.!

0'.$/'$.*!%&+!+3&%9(/0!(&+(/%'*!'<%'!:%0*-$0!E-0*FG(&0'*(&!/-&+*&0%'*0!9%3!=*!%!9-+*2!

030'*9!5-.!'<*!0'$+3!-5!,-.'*;!9%''*.6!

A B C D

!

T=0*.,%'(-&!-5!,-.'*;!2%''(/*06! !L<*!*;%912*0!0<-M&!/-&'%(&!>N?!"U!>E?!JI!>H?!VS!%&+!>W?!"JS!,-.'(/*06!!

L<*!,-.'(/*0!<%,*!X/.30'%22()*+X!(&!%!'.(%&:$2%.!1%''*.&6!L<*!+(%9*'*.!-5!'<*!/2-$+!(&!>W?!M%0!"!99!%5'*.!

=%22(0'(/!*;1%&0(-&!M<(/<!.*1.*0*&'0!%!9%:&(5(/%'(-&!-5!IS6!

(Vortices in Rb-BEC, ENS, 2000)

(Vortices in Na-BEC, MIT, 2001)

VOLUME 83, NUMBER 13 P HY S I CA L R EV I EW LE T T ER S 27 SEPTEMBER 1999

FIG. 2 (color). Condensate images (100 mm on a side) in which the imaging laser is tuned such that only j2! is visible. (a) Theinitial image of the vortex, (b) after p"2 of the interconversion pulse, and (c) after completion of the p pulse. The vortex is nowinvisible (in j1!) and the interior fluid is imaged (in j2!). (d) The normalized difference in densities between the local average ringand interior densities [(a) and (c), respectively] and the phase interferogram density (b) for each corresponding point in the images.This is approximately the cosine of the local phase difference f between the vortex state and the interior state [14]. The valuesare shown only for those regions where the densities of each state were high enough to give adequate signal to noise for the phasereconstruction. (e) The radial average at each angle u around the ring is shown in (d). (The data are repeated after the azimuthalangle 2p to better show the continuity around the ring.)

different from the j2! state vortices (Fig. 3c) due to thedifferent scattering lengths. The j2! vortex ring sinks intoward the trap center (Fig. 3c), and then rebounds and

apparently breaks up. This pattern is repeatedly observed

in measurements of the j2! vortices. Conversely, the equi-librium position of a j1! state fluid is obtained by “float-ing” outside the j2! state fluid. The inner radius of thej1! state ring shrinks slowly as the interior j2! fluid decaysaway with a #1 s lifetime. A variety of additional be-

haviors has been seen for j1! vortices. Initial asymmetryis very sensitive to beam position and condensate slosh.

For small differences in the initial vortex state density

distribution, asymmetric density distributions sometimes

develop and/or “heal” between 0.5 and 1 s.

The nonrotating j2! fluid within a j1! vortex is analo-gous to the defects that pin vortices in superconductors.

This j2! “defect” can be removed quickly, by a prop-erly tuned laser pulse, or allowed to decay slowly (as in

Figs. 3a and 3b). Thus this system can be varied between

two relevant physical limits. In the limit of a large repul-

sive central potential (produced by a large amount of j2!fluid) the system most closely resembles quantized flow in

a fixed, 3D toroidal potential. The vortex core is pinned

and its size is determined by the central potential. Fig-

ures 3a and 3b are near this limit for t ! 0 and 200 ms.In this case the density distribution of the vortex state is

vulnerable to instabilities due to the fact that the relative

densities of j1! and j2! may evolve with relatively littleenergy cost so long as the total density remains constant

[10]. In the opposite limit (a small amount of j2! fluid),the central potential is negligible and the size of the vor-

tex core (in equilibrium) is determined entirely by its own

centrifugal barrier. Figures 3a and 3b at t ! 600 ms areevolving towards this limit. In Fig. 3a, we see that the in-

terior j2! fluid is no longer pinning the vortex at 600 ms,but rather is being dragged around by the precessing vor-

tex. The j2! vortex provides an interesting mixture of theabove limits. The j1! fluid “floats” to the outside so thereis no pinning of the vortex core, but the tendency towards

density instabilities remains.

Expanded studies of stability issues are under way. We

also expect to be able to observe interesting transitional

behavior between these limiting cases. For example, it

is a straightforward extension of our method to create

an l ! 2 vortex. In the presence of a strong pinning

potential, l ! 2 vortices should be stable, but in the

weak potential limit, l ! 2 vortices are predicted to

spontaneously bifurcate [4].

2500

(Vortex in Rb-BEC, Colorado, 1999)

※Numerical Simulation of Vortex creation by Tsubota-group (Osaka City U.) in solving the Gross-Pitaevski equation.

http://matter.sci.osaka-cu.ac.jp/bsr/tsubotag/tsubotasim-j.html

(K. Kasamatsu et.al, PR. A 66 (2002) 053606)

Quantum Vortex and Vortex Lattice in BF mixed Cond.!10 0 10

0.00

0.02

0.04

0.06

r

"z (

y=

0)

a) Ω = 1.5 , Ef = 52.7 , Lz = 34.4

!10 0 100.00

0.01

0.02

0.03

r

"z (

y=

0)

b) Ω = 2.0 , Ef = 37.0 , Lz = 66.4

!10 0 100.000

0.005

0.010

0.015

r

"z (

y=

0)

c) Ω = 2.5 , Ef = −4.87 , Lz = 141

Fig. 14: Fermi気体の光学密度 (gBBNB = 1.E4 , NF = 969 , gBF /gBB = 0.5、右図は y = 0)

43

!10 0 100.000

0.005

0.010

0.015

x

"z(y=0)

a) Ω = 1.5 , µ = 26.3 , Lz = 9.52 , a = 2.24

!10 0 100.000

0.002

0.004

0.006

0.008

x

"z(y=0)

b) Ω = 2.0 , µ = 16.0 , Lz = 25.9 , a = 1.92

!10 0 100.000

0.001

0.002

0.003

0.004

0.005

x

"z(y=0)

c) Ω = 2.5 , µ = −19.7 , Lz = 138 , a = 1.71

Fig. 12: Bose凝縮体の光学密度 (gBBNB = 1.E4、右図は y = 0)

40

Boson (Vortex Lattice) Fermion (trapped in vortex cores)

Giant Vortex & Quantum Giddy 　at Large Angular Velocity

(by T.Nishimura & HY)

spontaneously scattered photons blasts atoms out of the

condensate.

Using this new technique we are able to substantially

increase the vorticity in the BEC and can now routinely

obtain condensates containing more than 250 vortices

[Fig. 1(b)]. Since these vortex cores are too small to be

imaged when the BEC is held in trap, the images in

Figs. 1–3 and 6 are taken after having let the BEC expand.

For this, a rapid adiabatic passage technique is employed

to change the hyperfine state of the atoms from the

original trapped jF ! 1; mF ! "1i state to the anti-

trapped jF ! 2; mF ! "1i state. Subsequently the mini-mum of the magnetic trapping field is rapidly moved

below the position of the atoms so that the atoms are

supported against gravity while at the same time they are

radially expelled [17]. After a chosen expansion time the

magnetic fields are switched off and the atoms are imaged

along the original axis of rotation. This technique allows

us to expand the condensate to an adjustable diameter that

can exceed 1.4 mm [18].

If the removal of central atoms is driven strongly

enough, a giant vortex appears. The core is surrounded

by a ring-shaped superflow. The circulation of this super-

flow is given by summing up all the ‘‘missing’’ phase

singularities in the core and can assume very large values.

Using the fact that for large amounts of circulation the

rotation of the velocity field can be obtained classically,

we determine that in some cases the BEC supports a ring-

shaped superflow with a circulation of up to 60 quanta

around the core.We can easily control the amount of this

circulation by changing the intensity or the duration of

the laser beam that removes atoms along the axis of

rotation.

In our experiment the formation of the giant vortex

comes about in a sequence of very distinct stages as

shown in Fig. 1. For this expansion image sequence a

rapidly rotating BEC is first formed by our evaporative

spin-up technique. Then the atom-removal laser is ap-

plied with a fixed power of 8 fW for a variable amount of

time, followed by a 10 ms in-trap evolution time and a

45 ms expansion in the antitrapping configuration de-

scribed above. During the atom removal an rf shield is

left on to constantly remove uncondensed atoms that tend

to decelerate the condensate rotation. Figure 1(a) shows

the result of only the evaporative spin-up. This particular

condensate contains 180 vortices and has a Thomas-

Fermi radius of 63:5 !m when held in the trap. When

the atom-removal laser is applied for 14 s as in Fig. 1(b),

the number of vortices is increased to 250 and the

Thomas-Fermi radius to 71 !m. The rotation rate, deter-mined from side view aspect-ratio images, has increased

from 0:94!" to 0:97!". After atom-removal times of 15

to 20 s, the vortex lattice becomes disordered [Figs. 1(c)

and 1(d)], and the giant vortex core starts to develop in the

center [Figs. 1(e) and 1(f)]. An enlarged view of the core

region in this early stage of giant vortex core formation is

seen in Fig. 1(i), which nicely demonstrates how the

individual vortices in the center consolidate. For the

FIG. 2. Lattice reforming after giant vortex formation.

(a) BEC after evaporative spin-up. (b) Effect of a 60 fW, 2.5 s

laser pulse. (c),(d) Same as (b), but additional 10 s (c) and 20 s

(d) in-trap evolution time after the end of the laser pulse.

Images taken after a sixfold expansion of the BEC.

FIG. 3. Core developing after a 5 ms short, 2.5 pW laser

pulse. In-trap evolution time after the end of the pulse is

(a) 0.5 ms, (b) 10.5 ms, (c) 20.5 ms, and (d) 30.5 ms. Images

taken after sixfold expansion of BEC. (e) Zoomed-in core

region of (c).

FIG. 1. Different stages of the giant vortex formation process.

(a) Starting point: BEC after evaporative spin-up. (b)–(h)

Laser shone onto BEC for (b) 14 s, (c) 15 s, (d) 20 s, (e) 22 s,

(f) 23 s, (g) 40 s, and (h) 70 s. Pictures are taken after 5.7-fold

expansion of the BEC. Laser power is 8 fW. (i) Zoomed-in core

region of (f).

P H Y S I CA L R EV I EW L E T T ER S week ending2 MAY 2003VOLUME 90, NUMBER 17

170405-2 170405-2

cf. Giant vortex in Rb BEC

(Colorado, PRL90 (03) 170405)

・Quantum Hall effects・Yrast line

Ω ≡ 2ω/ωHO = 2.5

Summary and PerspectiveSummary and Perspective・Some aspects of boson-fermion mixtures of ultra-cold atomic gas 　has been discussed.

・Other topics: 1) Peierls instability, periodic BEC and (spin) density waves in quasi-1D boson-fermion mixture (Miyakawa, Yabu, Suzuki, & Nakano, Yabu) 2) Boson-mediated superconducting state (Sogo, Yabu, Suzuki, in study) 3) Boson-Fermion Cooper pairing (Schuck, Suzuki, Yabu)

＊ Recently, BF-Feshbach resonances have been observed: 6Li - 23Na : MIT-group, cond-mat/0406129 87Rb - 40K : Colorado(Jin)-group, cond-mat/0406208 We can expect much progress in BF studies in near future.

・Applications to the relativistic systems, including subnuclear phenomena (Meson cond., Hyperon matter, Diquark cond.)

Documents

Hiroyuki Yabu- Topics on Boson-Fermion mixture in ultra-cold atomic gas