Universita degli Studi di Milano

Facolta di Scienze Matematiche, Fisiche e Naturali

Laurea Triennale in Fisica

Variational dynamics of ultracoldfermion-boson mixtures

Relatore: Dott. Nicola Manini

Correlatore: Dott. Luca Salasnich

Nicola Ferri

Matricola n◦ 675270

A.A. 2007/2008

Codice PACS: 03.75.-b

Variational dynamics of ultracold

fermion-boson mixtures

Nicola Ferri

Dipartimento di Fisica, Universita degli studi di Milano,

Via Celoria 16, 20133 Milan, Italy

26/02/2008

Abstract

We study the mean-field dynamic of a interacting boson-fermion mix-

ture in an harmonic trap, by means of a Gaussian variational ansatz.

We find concentric stationary solutions in the repulsive regime and in the

regime of weak boson-fermion attraction. Above a threshold value of the

repulsion parameter, a stable symmetry-broken stationary solution bifur-

cates, characterized by the displacement of the center of mass of the bosonic

and fermionic clouds in opposite directions away from the trap center. We

observe linear and nonlinear oscillations of the width and center of mass

of the boson and fermion distributions whenever we let the system evolve

starting from a non stationary initial condition.

Advisor: Dr. Nicola Manini

Co-Advisor: Dr. Luca Salasnich

3

Contents

1 Introduction 5

2 The formalism 5

2.1 Dimensionless units . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 The Gaussian ansatz 8

4 Euler-Lagrange equation 10

4.1 Stationary points . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Discussion 19

Bibliography 23

4

1 Introduction

Ultracold atoms in electromagnetic traps provide a fashionable playground for ex-

ploring many-body dynamics in simple idealized configurations or in complicated

interesting situations [1, 2, 3]. In particular rarefied boson-fermion mixtures [4, 5]

present a rich bulk phase diagram [6], and nontrivial dynamical properties where

experimental investigation is only starting [3, 5, 7, 8, 9, 10]. In the present thesis,

we consider the rather idealized case of a zero-temperature dilute mixture com-

posed of Nb atomic bosons forming an interesting Bose-Einstein condensate and

Nf attractive atomic fermions in the superfluid state. The mixture is trapped by

an external optical field which induces a spherically-symmetric harmonic confine-

ment.

At extremely low temperatures dilute bosonic atoms form a pure Bose-

Einstein condensate, which is a bosonic superfluid. Also attractive fermions

equally distributed into two hyperfine states form a superfluid of Cooper pairs:

a fermionic superfluid. Both bosonic and fermionic superfluids are characterized

by an irrotational velocity field. We use the quantum irrotational hydrodynamics

(QIH) to study our superfluid Bose-Fermi mixture.

We assume that the motion of the boson and fermion densities is driven by

two coupled fields: the macroscopic wave function ψb(r, z, t) of the Bose-Einstein

condensate and the Ginzburg-Landau order parameter ψf (r, z, t) of the Cooper

pairs of fermionic atoms.

We use for the fields ψj(r, z, t) (j = b, f) a cylindrically-symmetric Gaussian

ansatz. This ansatz depends on variational parameters which characterize the

position and width of each atomic clouds. Within this simplifying hypothesis we

integrate the action over spatial variables and obtain an effective action which

provides the equations of motion for the variational parameters. We study the

stationary solutions and the numerical dynamics of the parameters.

2 The formalism

The QIH action functional A of the Bose-Fermi mixture is given by

A =

∫

(Lb + Lf + Lbf) d3r dt , (1)

5

where Lb is the bosonic Lagrangian density, Lf is the fermionic one, and Lbf is

the Lagrangian density of the Bose-Fermi interaction.

We consider the bosonic Lagrangian density Lb defined by

Lb =i

2~(ψ∗

b

∂

∂tψb − ψb

∂

∂tψ∗b ) +

~2

2mbψ∗b∇2ψb −

1

2gb|ψb|4 − Vb|ψb|2 , (2)

where Vb(r) is the external potential acting on the atomic bosons. gb is the

boson-boson interaction strength associated to the s-wave interatomic scattering

length ab by the formula gb = 4π~2ab

mbwith mb the mass of the single bosonic atom.

ψb(r, t) is the macroscopic wave function of the Bose-Einstein condensate, such

that nb(r, t) = |ψb(r, t)|2 is the local number density of bosonic atoms. The total

number of bosonic atoms reads

Nb =

∫

|ψb(r, t)|2d3r ,

where integration extends through all three dimensional space.

The fermionic Lagrangian density Lf is defined as

Lf =i

2~(ψ∗

f

∂

∂tψf − ψf

∂

∂tψ∗f ) +

~2

4mfψ∗f∇2ψf − 2ε(2|ψf |2)|ψf |2 − 2Vf |ψf |2 (3)

where 2Vf(r) is the external potential acting on a Cooper pair of mass 2mf , where

mf is the mass of a single fermionic atom. ε is the bulk energy per particle of the

fermions, which is a function of the number density nf(r, t) = 2|ψf(r, t)|2 where

ψf (r, t) is the Ginzburg-Landau order parameter of the superfluid Fermi gas, i.e.

the macroscopic wave function of Cooper pairs. The total number of fermionic

atoms is given by the normalization

Nf = 2

∫

|ψf (r, t)|2 d3r .

The bulk energy per particle of a two-spin component attractive Fermi gas can

be expressed by the following expression

ε(nf) =3

5

~2k2f

2mf

G

(

1

ahkF

)

, (4)

where G(y) depends on the local inverse interaction parameter y = 1/(ahkf(nf))

is a function of position r and time t, because the Fermi momentum is kf(r, t) =

(3π2nf (r, t))1/3 the local Fermi wave vector. af is the fermion-fermion scattering

length.

6

In the experiments on the BCS-BEC crossover the scattering length af of

superfluid fermions is changed by using an external magnetic field: the Feshbach

resonance technique. af is varied from large negative values, so that y ≪ −1

and the Fermi superfluid is a BCS state of weakly bound Cooper pairs, to large

positive values, so that y ≫ 1 and the fermions form a weakly repulsive Bose gas

of molecules. In the first case we have G(y) = 1 + 10/(9πy) + O(1/y2), in the

second case G(y) = 5aM/(18πahy) + O(y5/2), giving the asymptotic behavior of

G(y). Another special limit is y = 0, the so called unitarity limit, in which one

expects that the energy per particle is proportional to a non-interacting Fermi

gas with coefficient G(0) = 0.42.

An analytical formula of the universal function G(y) which fits the numer-

ical results from Monte Carlo simulations [11] and obeys the above asymptotic

expressions is G(y) = α1 − α2 arctan(

α3yβ1+|y|β2+|y|

)

, with the specific value for αi

and βi given in Ref.[12].

The Lagrangian density Lbf of the Bose-Fermi interaction is given by the

simple form

Lbf = −2gbf |ψb|2|ψf |2 , (5)

where gbf =2π~2abf

mrwhere is the Bose-Fermi interaction strength, with mr =

mbmf/(mb + mf) as the reduced mass and abf the Boson-Fermion scattering

length.

Finally the external trapping potential is taken of the simple harmonic form,

Vj(x, y, z) =1

2mjω

2j (x

2 + y2 + z2), j = b, f. (6)

We assume generally different trapping frequencies for bosons and fermions.

In the present preliminary investigation we assume spherical symmetry of the

trap. The trapping potential for bosons defines a natural harmonic-oscillator

length scale ah = ~1/2m

−1/2b ω

−1/2b .

2.1 Dimensionless units

The Lagrangian is conveniently expressed in dimensionless form by using natural

units for each physical quantity, as listed in Table 1. Tildes identify the quantities

in reduced dimensionless units, e.g. t = ωbt, r = r/ah, Lj = Lj/~ωb, ψj = a3/2h ψj .

7

Physical quantity units used

Mass mb

Time ω−1b

Length ah =√

~

mbωb

Energy ~ωbDensity a−3

h

Table 1: Natural units used in the calculations

The bosonic part is

Lb =a3h

~ωbLb =

a3h

~ωb

[

ωbi~a−3

h

2(ψ∗

b

∂

∂tψb − ψb

∂

∂tψ∗b ) +

~2

2mba5h

ψb∗∇2ψb+

− 1

2

1

~ωba3h

4π~2ab

mb|ψb|4 − a−1

h mbω2b

1

2r2|ψb|2

]

=

=i

2(ψ∗

b

∂

∂tψb − ψb

∂

∂tψ∗b ) +

1

2ψ∗b ∇2ψb − gb

1

2|ψb|4 −

1

2r2|ψb|2. (7)

Likewise the fermionic part is

Lf =a3h

~ωbLf =

i

2(ψ∗

f

∂

∂tψf − ψf

∂

∂tψ∗f ) + λ

1

4ψ∗f∇2ψf − 2ε(2|ψf |2)|ψf |2 −

1

λ′r2|ψf |2 (8)

where ε(2|ψf |2) = 1~ωbε(2

|ψf |2

a3h

).

Finally, the interaction Lagrangian term is

Lbf =a3h

~ωbLbf = −2gbf |ψb|2|ψf |2 . (9)

The several dimensionless parameters introduced above are:

λ =mb

mf, λ

′

=mbωbmfωf

, gb =4πabah

, gbf =2πabfmb

ahmr. (10)

Hence, we shall use the rescaled units and suppress all tildes from now on, to

keep the notation simple.

3 The Gaussian ansatz

Even though the external potential is assumed to be spherically-symmetric, we

choose a cylindrical symmetric Gaussian ansatz allowing for independent axial

8

displacements of the Fermion and Boson clouds

ψj(x, y, z, t) = ψj(r, z, t) = Aje−

r2+(z−z0j(t))2

2σj (t)2 ei[αj(t)(r2+z2)+pj(t)(z−z0j (t))] . (11)

The 4+4 parameters defining the Gaussian fields carry a time dependence which

is often left implicit σj = σj(t), αj = αj(t), pj = pj(t), z0j = z0j(t), with

j = b, f . The field normalization is

Ab =

√

Nb

σ3bπ

3/2, Af =

√

Nf

2σ3fπ

3/2. (12)

We substitute the ansatz into Eqs.(7), (8) and (9) and integrate the total

Lagrangian density over space:

LTOT =

∫

(Lb + Lf + Lbf)d3r = Lb + Lb + Lbf . (13)

The integration is carried out in cylindrical coordinates.

The resulting bosonic part of Lagrangian is given by:

LbNb

= −3

2αbσ

2b + pbz0b − αbz

20b (14)

−[

3

4

1

σ2b

+ 3α2bσ

2b + 2α2

bz20b +

1

2p2b + 2pbαbz0b

]

−[

gb2

Nb

σ3b (2π)

32

+3

4σ2b +

1

2z20b

]

.

The terms in the first line are generated by time derivatives in Eq.(7). The

first term is the same that one finds in the spherical case, the second and the

third term arise from the motion of the Gaussian center of mass. In the second

line we have the terms generated from the Laplacian. The first and the second

term are ordinary terms of the spherical case, the three other terms again relate

to center of cent-re-mass motion. The third line contains the self-interaction term

and the terms derived from the external harmonic potential.

The fermionic part of the Lagrangian is given by

LfNf

= −3

4αfσ

2f +

1

2pf z0f −

1

2αfz

20f (15)

−[

3

16

1

σ2f

+3

4α2fσ

2f +

1

8p2f +

1

2α2fz

20f +

1

2pfαfz0f

]

λ

−[

4√π

Φ(Nf

σ3fπ

32

) + (3

4σ2f +

1

2z20f )

1

λ′

]

,

9

with

Φ(x) =

∫ +∞

0

ε(xe−ξ2

)e−ξ2

ξ2dξ. (16)

The Bose-Fermi interaction takes the following form:

Lbf = −gbfNfNb

[

1

π

1

(σ2b + σ2

f)

]32

e−

(z0f−z0b)2

(σ2f+σ2

b) . (17)

It is convenient to organize the terms in LTOT into two groups, LTOT = T−E,

as follows

T = − 3

2Nb[σ

2b αb + 2α2

bσ2b ] −Nb[αbz

20b − pbz0b] − 2Nb[α

2bz

20b + pbαbz0b] (18)

− 3

4Nf [σ

2f αf + λα2

fσ2f ] −

Nf

2[αfz

20f − pf z0f ]

− Nf

2[α2fz

20fλ+ pfαfz0fλ] ,

E =3

4Nb[

1

σ2b

+ σ2b ] +

3

4Nf [

λ

4σ2f

+σ2f

λ′] +

gb2

N2b

(2π)32

1

σ3b

+ (19)

+Nf

2[z20f

λ′+

1

4p2fλ] +

Nb

2[p2b + z2

0b] +

+4Nf√π

Φ(Nf

σ3fπ

32

) +gbfNfNb

π32 (σ2

f + σ2b )

32

e−

(z0f−z0b)2

σ2f+σ2

b .

4 Euler-Lagrange equation

We write now the Euler-Lagrange equation for the boson parameters. To simplify

the notation we use the shorthand S2 = σ2f + σ2

b , Z = (z0f − z0b).

From the αb-Euler-Lagrange equation ∂∂t

∂L∂αb

− ∂L∂αb

= 0 we obtain:

σb = −2

3

z0bz0bσb

+ 2αbσb +4

3

αbz20b

σb+

2

3

pbz0bσb

. (20)

From the equation ∂∂t

∂L∂pb

− ∂L∂pb

= 0 we obtain:

z0b = pb + 2αbz0b . (21)

10

We use both equations, by inserting Eqs.(21) into (20) and obtain

αb =σb2σb

. (22)

We use now ∂∂t

∂L∂z0b

− ∂L∂z0b

= 0, i.e.

pb = −4α2bz0b − 2pbαb − z0b − 2αbz0b − gbfNfπ

− 32 e−(Z

S)2 2Z

S5. (23)

∂∂t

∂L∂σb

− ∂L∂σb

= 0 yields the last equation of motion for the boson parameters

αb =1

2σ4b

− 2α2b −

1

2+

1

2gb

Nb

(2π)32

1

σ5b

+ (24)

−gbfNfπ− 3

2

[

−e−(ZS

)2

S5+

2Z2e−(ZS

)2

3S7

]

.

We plug now equations (24), (21) and (22) into equation (23) to obtain:

pb = −z0bσ4b

+σ2b

σ2b

z0b − ˙z0bσbσb

− gbNb

(2π)32

z0bσ5b

− gbfNf (1

π)

32S−52Ze−(Z

S)2 + (25)

−2z0bgbfNfe−(Z

S)2(

1

π)

32S−5 + 2z0bgbfNfe

−(ZS

)2(1

π)

32S−7Z2 =

= −z0bσ4b

+σ2b

σ2b

z0b − ˙z0bσbσb

− gbNb

(2π)32

z0bσ5b

+

+2Nf gbfπ− 3

2e−(ZS

)2[

− Z

S5− z0bS5

+2z0bZ

2

3S7

]

.

We substitute αb from Eq. (24) and pb from Eq. (21) into Eq. (23) and then

we use the relation for αb in (22) to obtain an equation for pb involving only σb e

zb for bosons and fermions at the right-hand side.

The Euler-Lagrange equations for fermion parameters are:

σf =1

2αfσfλ , (26)

z0f =1

2pfλ+ αfz0fλ , (27)

αf = −λα2f −

1

λ′+

λ

4σ4f

+8Nf

π2σ5f

Φ′(Nf

σ3fπ

32

) + (28)

+2gbfNbe−(Z

S)2π− 3

2

[

S−5 − 2Z2S−7

3

]

.

11

Like for bosons we obtain an expression for pf :

pf = −2α2fz0fλ− pfαfλ− 2z0f

λ′− 2αfz0f + 2gbfNbπ

− 32 e−(Z

S)2 2Z

S5= (29)

=8σ2

fz0f

λσ2f

− z0fλ

2σ4f

− 4z0f σfσfλ

− 16Nfz0fπ2σ5

f

Φ′(Nf

σ3fπ

32

) +

+2gbfNbe−(Z

S)22π− 3

2

[

Z

S5− z0fS5

+2z0fZ

2

3S7

]

.

Differences between fermion and boson include the λ factor, the self-interaction

term, and the sign of one term in pj . Several factors of 2 come from ψf describing

the motion of Cooper pairs, as opposed to individual bosons.

The terms involving Φ′ in Eqs.(28) and (29) derive from: ∂∂σf

Φ = ∂x∂σf

∂∂x

Φ(x) =

− 3Nf

σ4fπ

32Φ′(x), where x =

Nf

σ3fπ

32. The integral definition of Φ yields

Φ′(x) =∂

∂x

∫ +∞

0

ξ2ε(xe−ξ2

)dξ =

∫ +∞

0

ε(xe−ξ2

)ξ2e−2ξ2. (30)

We substitute the relation ∂∂ξ

[

ε(xe−ξ2)]

= ε(xe−ξ2)xe−ξ

2(−2ξ) into the integral

(30) and integrate by parts, to obtain:

∫ +∞

0

ε(xe−ξ2

)[x(−2ξ)e−ξ2

]

[ −1

2xξ

]

e−ξ2

ξ2dξ = (31)

= ε(xe−ξ2

)(− ξ

2xe−ξ

2

)

∣

∣

∣

∣

+∞

0

−∫ +∞

0

ε(xe−ξ2

)

[

−e−ξ2

2x+ξ2e−ξ

2

x

]

dξ =

=1

x

∫ +∞

0

ε(xe−ξ2

)e−ξ2

[

1

2− ξ2

]

dξ = Φ′(x)

The last equation is the expression we use to evaluate Φ′ numerically.

4.1 Stationary points

We first study the eight equations of motion for the eight parameters derived

above, to find the stationary points. These equilibrium positions are characterized

by all parameters being independent of t, so that all that terms vanish. σj = z0j =

12

αj = pj = 0. (j = b, f) we obtain a set of algebraic equations:

z0b − gbfNfπ− 3

2e−(ZS

)2 2Z

S5= 0, (32)

1

2σ4b

− 1

2+

1

2gb

Nb

(2π)32

1

σ5b

− gbfNfπ− 3

2

[

−e−(ZS

)2

S5+

2

3

Z2e−(ZS

)2

S7

]

= 0, (33)

z0fλ′

− gbfNbπ− 3

2e−(ZS

)2 2Z

S5= 0, (34)

λ

4σ4f

− 1

λ′+

8Nf

π2σ5f

Φ′(Nf

σ3fπ

32

) + 2gbfNbe−(Z

S)2π− 3

2

[

S−5 − 2

3Z2S−7

]

= 0, (35)

where we have used the simpler relations αj = pj = 0, with j = b, f , derived from

Eqs. (21), (22), (26), (27).

As an example, we consider a mixture of 6Li and 23Na with Nb = 2 · 109,

Nf = 6 · 105, with the parameters indicated in Fig.1, and explore the role of

the parameter gbf gauging the boson-fermion interaction. Through a numerical

calculation we obtain that in the attractive regime gbf < 0 the boson and fermion

clouds reach the equilibrium at z0j = 0, as one expects. Instead, with gbf > 0, we

observe that at first the equilibrium positions remain around zero, but when gbfincreases above a critical value, here gbf = 0.0048, the center of both fermion and

boson clouds start to leave the origin. As one expects, the bosons displacement

z0b assumes much smaller value than that of fermions at the same gbf , due to

the much larger number of bosons that keeps their position much more fixed in

interaction with the external harmonic potential. The typical behavior is shown

in Fig.1.

The same calculation provides the stationary value of the two σj . The results

are plotted in Fig.2. We observe that for gbf < 0.0048 the decreasing attraction

and then the increasing repulsive regime causes an increasing spread of the widths,

thus an increase of both σb and σj . This increasing regime continues along the

unstable solution, characterized by z0f = z0b = 0 (dotted lines in Fig.2), but at

gbf ≥ 0.00485 a new displaced stable solution arises with z0j 6= 0, see Fig.1. The

two clouds separate, and for further increase of gbf the values of σj decrease. Note

that the variations of σf are much larger than those of σb, again due to Nf ≪ Nb.

4.2 Time evolution

We investigate now the time evolution of the boson-fermion mixture through the

numerical solution of the Euler-Lagrange equations (21), (22), (24), (25), (26),

13

0 0.05 0.1 0.15g

bf

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

z 0b

0

10

20

30

40

z 0f

0.0048

Figure 1: The stationary values of z0f (solid line) and z0b (dot dashed

line), as a function of the dimensionless boson-fermion interaction

strength gbf for a mixture of Nb = 2 · 109 23Na and Nf = 6 · 105

6Li. We consider ab = 10 nm, af = 10 nm, ωb = 200π Hz, and

ωf = 321.81π Hz, so that λ = 3.833, λ′ = 2.393, gb = 0.0473636, and

gf = 0.00376908. The arrow indicates the critical value of gbf beyond

which nonzero equilibrium positions arise.

14

-0.1 -0.05 0 0.05 0.1 0.15g

bf

22.67

22.68

22.69

σ b

0

2

4

6

8

σ f

Figure 2: The stationary values of σf (solid line) and σb (dot dashed

line), as a function of gbf for a mixture of Nb = 2 · 109 23Na and

Nf = 6 ·105 6Li. The unstable solution corresponding to z0f = z0b = 0

is depicted by dotted lines. This calculation is based on the same

parameters as in Fig.1. The peak of boson and fermion widths coincide

with the critical value gbf = 0.048, where a nonzero displacement of

the boson and fermion Gaussian starts off.

15

-4e-06

-2e-06

0

2e-06

4e-06

z 0b

-0.008

0

0.008

z 0f

22.69085

22.6909

σ b

0 10 20 30 40t

6.89196

6.89198

σ f

Figure 3: Time calculation in the weakly repulsive regime. Nb = 2·109

23Na, Nf = 6 · 105 6Li atoms. We assume ab = 10 nm, af = 10 nm,

abf = 10 nm, ωb = 200π Hz, ωf = 321.81π Hz, λ = mb

mf= 3.833,

λ′ = mbωb

mfωf= 2.393, ah = ( ~

mbωb)

12 = 2.65317 · 10−6 m, mr =

mbmf

mb+mf=

7.90188 · 10−27 Kg, gb = 4πab

ah= 0.0473636, gf =

af

ah= 0.00376908

and gbf =2πabfmb

ahmr= 0.004. The initial conditions are: z0b(0) = 0

(equilibrium), z0f (0) = 0.01, σb(0) = 22.6909 (equilibrium), σf (0) =

6.89197 (equilibrium), and αj(0) = pj(0) = 0.

(27), (28), (29).

The solution of these eight coupled differential equations of motion gives

us the detailed time dependence of the two Gaussians. We have checked energy

conservation in the numerical integration: the value of H =∑

i piqi − L, where

pi = ∂L∂qi

and qi are the eight dynamical variables, remains constant along the

trajectories. We consider only a few example trajectories for a few choices of

initial conditions z0j(0), σj(0), and typical values of the interaction parameter

gbf .

We first investigate a weakly repulsive regime to evaluate the relations be-

tween the two z0j and the two σj . We start off the two σj at their stationary

16

-2-1012

z 0b

50

52

54

56

z 0f

22.68

22.69

22.7

σ b

0 10 20 30 40t

2.5

3

3.5

4

σ f

Figure 4: Time evolution in the strongly-repulsive regime. For the

parameters see Fig.3, except gbf = 0.8. The initial conditions are:

z0b(0) = −2, z0f (0) = 50, σb(0) = 22.70, σf (0) = 3.3. The equilibrium

position: z0b(eq)= −0.00666513, z0f (eq)= 53.1683, σb(eq)= 22.6889,

σf (eq)= 3.26125. Note that the initial z0b is fixed a value very far

from the equilibrium point.

value not to disturb the other variables too strongly. In Fig.3 we see that with

the only displacement of z0f from the stationary point we have the excitation

also of a visible oscillation of of z0b. This oscillatory motion is disturbed by the

exchange of energy with the two widths σj: even if their oscillations are small this

collective interaction between the variables causes the modulation of the peaks

in z0b. An antiphase correlation between z0b and z0f is recognizable.

The second regime we study is the strong repulsion regime. The interaction

term is much above the critical point, so we expect that the oscillations of the

variables being modulated by this large repulsion value. We set the two σj in a

value near the equilibrium value to observe their weak interactions with the other

variables. The strong interaction between the two symmetry-broken z0j tends to

make the clouds push away each other, see Fig.4. Instead, contrary to what we

observed for weak repulsion, the fermions oscillate at the same basic frequency

17

0 5 10 15 20t

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004z

z0b

z0f

Figure 5: Attractive regime: the parameters are the same in Fig.3, but

gbf = −0.1. The initial conditions: z0b(0) = −0.001, z0f (0) = 0.001,

σb(0) = 21, σf (0) = 2. The equilibrium position are: z0b(eq)= 0,

z0f (eq)= 0, σb(eq)= 22.6744, σf (eq)= 2.157383. Initially, both z0jdiffer from the stationary point by a small value.

and with the same phase as the bosons. We can deduce that in this regime the

oscillations of z0b decides the dynamic of the system even though the fermions

have an initial z0f different from the equilibrium point. The influence of the initial

value of z0f remain confined in the higher-frequency oscillations of z0f . σb shows

a fairly harmonic oscillation. σf shows an evident exchange of energy with z0f .

The large oscillations of the fermionic parameters are due to the relatively small

number of fermions compared to bosons.

Finally we investigate the attractive regime, z0j reach equilibrium at zero.

We study the dynamics with z0j(0) 6= 0 which shows the attractive interaction

that one clouds exerts on the other. In Fig.5 we see clearly that the z0b attract in

a strong manner z0f that follows the oscillating dynamics of z0b and also makes

faster oscillations on its own. The dynamics of the two σj are reported in Fig.6.

We note that the σb dynamic is an oscillation around its stationary point. With

the small value of z0j and the attractive regime the variables σj oscillates without

large interaction with the other variables, and so we do not observe secondary

peaks or any sort of intermode interference.

18

20

22

24

26σ b

0 5 10 15 20t

2

2.2

2.4

σ f

Figure 6: The parameters, stationary points and the initial conditions

are the same in Fig.5. The two σj start near their stationary value.

In Fig.7 the initial conditions differs from Fig.5 only for the initial values

of the two z0j(0), which are much larger. In this plot we see that the dynamics

for the two variables is very similar to that in Fig.5 except for the oscillation

amplitude that are larger now. We can conclude that despite these strongly dis-

torted initial conditions the system undergoes a strong attraction that regulates

its dynamics and produces a qualitatively similar movement for a large amplitude

of z0j oscillations. In Fig.8 we observe that with initial condition for the two z0jfar from zero (the stationary value) we have a sizable energy exchange involving

the variables σj . In particular σf shows secondary higher frequency peaks due to

nonlinear energy transfer to the corresponding mode.

5 Discussion

The simple variational ansatz proposed here, with all its obvious limits, provides

a nice description of the collective dynamics of a mixture of boson and fermions.

we find that when fermions and bosons attract each other, a model where the two

clouds are both centered at the trap minimum is perfectly appropriate. In the

19

0 5 10 15 20t

-10

-5

0

5

10

15

z

z0b

z0f

Figure 7: The parameters and the stationary points are the same

in Fig.5. The initial conditions: z0b(0) = −5, z0f (0) = 2, σb(0) =

21, σf (0) = 2: both the z0j are set in a value very different from

equilibrium.

20

20

22

24

σ b

0 5 10 15 20t

2

2.4

σ f

Figure 8: The parameters are in Fig.3. The initial conditions and the

stationary points are the same as in Fig.7. The σj start from initial

value near the stationary point.

21

repulsive regime instead a nice symmetry-broken solution arises when we allow

for the Gaussian clouds to displace.

An important extension of the present theory shall involves anisotropic Gaus-

sian in anisotropic traps, as often done in experiment.

22

References

[1] S. Giorgini, L. P. Pitaevskii, and S. Stringari, arXiv:0706.3360 (2007).

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[3] G. Modugno, G. Roati, F. Riboli, F. Ferlaino, R. J. Brecha, and M. Inguscio,

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[6] L. Salasnich and F. Toigo, Phys. Rev. A 75, 013623 (2007).

[7] A.G. Truscott, K.E. Strecker, W.I. McAlexander, G.B. Partridge, and R.G.

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[8] F. Schreck, L. Khaykovich, K.L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles,

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23

Ringraziamenti

In primis un ringraziamento al relatore ed al correlatore per la grande pazienza

dimostrata nel seguirmi praticamente tutti i giorni per parecchio tempo e per

l’attenzione che hanno dato a questo lavoro.

Ringrazio gli aspiranti fisici che hanno condiviso insieme a me le gioie, ma soprat-

tutto i dolori durante questi tre anni. Tra tutti, in particolare, Castelli, Carla,

Santo, Colli, Poz, Carmen, Macocchi, 50, i wackener cioe Nardo, Il Giaguaro e

Brio, tutta l’aula Laser e tutta l’aula ∼.

Un ringraziamento ai miei coinquilini Nicolo e Gianluca ed al padrone di casa

Alberto per i momenti belli che hanno reso piu divertente e leggera la routine

quotidiana.

Ringrazio i miei amici che ormai da tempo mi sopportano e spero che lo facciano

ancora per molto: Duda, Kioz, Andre, Cv, Raffa, Faustozzi, Gianno, Leo, Ste e

tutti gli altri che non ho citato.

Per ultimi, ma non per importanza ringrazio i miei genitori senza l’aiuto dei

quali non avrei potuto realizzare questo lavoro, mia sorella e tutti i parenti per il

supporto e l’interesse che mi e stato dato.

24