Parametric Resonance in Bose-Einstein...

Parametric resonance Variational approach Equations of motion Isotropic stability Anisotropic stability Conclusions

Parametric Resonancein Bose-Einstein Condensates

William Cairncross1,2 and Axel Pelster3,4

1Institut fur Theoretische Physik, Freie Universitat Berlin, Germany2Faculty of Physics, Engineering Physics & Astronomy, Queens University, Kingston, Canada

3Hanse-Wissenschaftskolleg, Delmenhorst, Germany4Fachbereich Physik und Forschungszentrum OPTIMAS, Technische Universitat

Kaiserslautern, Germany

arXiv:1209.3148


Outline

1 Parametric resonancePendulum physicsMathieu equationBEC

2 Variational approach

3 Equations of motionEquilibrium position

4 Isotropic stabilityNon-homogeneous Mathieu equationResults

5 Anisotropic stabilityCoupled Mathieu equationsResults

6 Conclusions


Parametric resonance

Parametric oscillator: harmonic oscillator with time-dependentparameters

Parametric resonance: resonant behaviour of a parametricoscillator

Destabilization of Stabilization ofstable equilibrium unstable equilibrium

Swing Paul trap (Nobel Prize 1989)

Kapitza pendulum


Inverted pendulum with a vertically oscillated pivot

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Inverted-pendulum-with-a-vertically-oscillated-pivot.mp4Media File (video/mp4)


Pendulum physics

Driving amplitude A, frequency

Equation of motion

(t) +

g

l+A2

lcos t

sin(t) = 0

Linearize:sin(t) ' (t)

With definitions

c = 4gl2

q = 2Al

2t = t x(t) = (t)

Mathieu equation

x(t) +c 2q cos 2t

x(t) = 0


Mathieu equation

x(t) +[c 2q cos 2t

]x(t) = 0

Floquet theory: on stability borders, x(t) is - or 2-periodic.One method: Fourier series ansatz

x(t) =n=0

An cos(n t) +n=1

Bn sin(n t)

Obtain decoupled systems

n=0

An

[(cn2) cos(n t)q cos

((n1) t

)q cos

((n+1) t

)]= 0

n=1

Bn

[(cn2) sin(n t) q sin

((n1) t

) q sin

((n+ 1) t

)]= 0


Mathieu equationContinued

Infinite matrix equations truncate for approx. solution

Vanishing determinants for nontrivial An, Bn

266666666664

c q 0

2 q c 4 q

0 q c 16

.

.

.

...

377777777775

266666666664

A0

A2

A4

.

.

.

377777777775= 0,

266666666664

c 4 q 0

q c 16 q

0 q c 36

.

.

.

...

377777777775

266666666664

B0

B2

B4

.

.

.

377777777775= 0,

266666666664

c q 0

2 q c 1 q

0 q c 9

.

.

.

...

377777777775

266666666664

A1

A3

A5

.

.

.

377777777775= 0,

266666666664

c 1 q 0

q c 9 q

0 q c 25

.

.

.

...

377777777775

266666666664

B1

B3

B5

.

.

.

377777777775= 0

(q, c) for vanishing determinant gives stability borders


Mathieu equationStability diagram

c

q

0 5 10 155

0

5

10


Bose-Einstein Condensate

Extreme Tunability of Interactions in a 7Li Bose-Einstein CondensateS. E. Pollack et al., PRL 102, 090402 (2009)

Tuning of scattering length by Feshbach resonance

a(B) = aBG

(1

B B

)Collective excitation of a Bose-Einstein condensate by modulation ofthe atomic scattering lengthK. M. F. Magalhaes et al., PRA 81, 053627 (2010)

B(t) = Bav + B cos t, a = aav + a cos t

where

aav = a(Bav), a =aBGB

(Bav B)2


Bose-Einstein Condensate

Analogous stability behaviour for BEC?NONLINEAR BOSE-EINSTEIN-CONDENSATE DYNAMICS . . . PHYSICAL REVIEW A 84, 013618 (2011)

15

20

25

30

35

0 200 400 600 800 1000 1200

Axi

al c

onde

nsat

e w

idth

t

variationalGP numerics

1

1.1

1.2

1.3

1.4

1.5

1.6

0 50 100 150 200 250

Rad

ial c

onde

nsat

e w

idth

t


1

1.1

1.2

220 225 230 235 240

FIG. 1. (Color online) Time-dependent axial and radial conden-sate widths calculated as root-mean-square averages. Comparisonof the numerical solution of time-dependent GP equation with asolution obtained using the Gaussian approximation for the actualexperimental parameters in Eq. (8) and ! = 0.05.

In Fig. 1, we plot the resulting time-dependent axial and radialcondensate widths rms(t) and zrms(t) calculated as root-mean-square values

rms(t) =

2

dz

0 d |(,z,t)|2 2 , (9)

zrms(t) =

2

dz

0 d |(,z,t)|2 z2 , (10)

and compare them with numerical solutions of Eqs. (5) and (6).We assume that initially the condensate is in the ground state. Inthe variational description, this translates into initial conditionsu(0) = u0, u(0) = 0, uz(0) = uz0, and uz(0) = 0, whereu0 and uz0 are time-independent solutions of Eqs. (5) and(6), while in GP simulations we reach the ground state byperforming an imaginary-time propagation [36]. For solvingthe GP equation (2), we use the split-step Crank-Nicolsonmethod [36]. It is evident that we have a good qualitativeagreement between the two approaches.

The main result obtained previously by using the Gaussianapproximation is an analytical estimate for the frequencies ofthe low-lying collective modes [5,6]. In this paper, we considerexcitations induced by a modulation of the interaction strengthand focus on the properties of the quadrupole and breathingmode. We assume that the external trap is stationary, thus pre-venting excitations of the dipole (Kohn) mode, correspondingto the center-of-mass motion. By linearizing Eqs. (5) and (6)around the equilibrium widths u0 and uz0, frequencies of

both the quadrupole Q0 and the breathing mode B0 wereobtained:

B0,Q0 =

2[(

1 + 2 p4u20u

3z0

)

(

1 2 + p4u20u

3z0

)2+ 8

(p

4u30u2z0

)2] 12.

(11)

For the repulsive interaction, the quadrupole mode has a lowerfrequency and is characterized by out-of-phase radial andaxial oscillations, while in-phase oscillations correspond to thebreathing mode. In the case of the experiment [19], Eq. (11)yields

Q0 = 0.035375, B0 = 2.00002. (12)

We emphasize that, although based on the Gaussian ansatz, thevariational approximation reproduces exactly the frequenciesof collective modes not only for the weakly interacting BECbut also for the strongly interacting BEC in the Thomas-Fermi regime [4,5]. Therefore, it represents a solid analyticaldescription of BEC dynamics.

However, due to the nonlinear form of the underlyingGP equation, we expect nonlinearity-induced shifts in thefrequencies of low-lying modes with respect to the valuesin Eq. (11) calculated using the linear stability analysis. Inparticular, our goal is to describe collective modes induced bythe harmonic modulation of the interaction strength in Eq. (7).In the case of a close matching of the driving frequency ! andone of the BEC eigenmodes, we expect resonances (i.e., largeamplitude oscillations). Here, the role of the nonlinear termsbecomes crucial and nonlinear phenomena become visible, aswe discuss in the next section.

III. SPHERICALLY SYMMETRIC BEC

Using a simple symmetry-based reasoning, we concludethat a harmonic modulation of interaction strength in the caseof a spherically symmetric BEC (i.e., = 1) leads to theexcitation of the breathing mode only, so that u(t) = uz(t) u(t). This fact simplifies numerical and analytical calculations,and this is why we first consider this case before we embarkto the study of a more complex axially symmetric BEC.

Thus, the system of ordinary differential Eqs. (5) and (6)reduces to a single equation:

u(t) + u(t) 1u(t)3

p(t)u(t)4

= 0 . (13)

The equilibrium condensate width u0 satisfies

u0 1u30

pu40

= 0 , (14)

and a linear stability analysis yields the breathing modefrequency

0 =

1 + 3u40

+ 4pu50

. (15)

013618-3

NONLINEAR BOSE-EINSTEIN-CONDENSATE DYNAMICS . . . PHYSICAL REVIEW A 84, 013618 (2011)

15

20

25

30

35

0 200 400 600 800 1000 1200

Axi

al c

onde

nsat

e w

idth

t


1

1.1

1.2

1.3

1.4

1.5

1.6

0 50 100 150 200 250

Rad

ial c

onde

nsat

e w

idth

t


1

1.1

1.2

220 225 230 235 240

FIG. 1. (Color online) Time-dependent axial and radial conden-sate widths calculated as root-mean-square averages. Comparisonof the numerical solution of time-dependent GP equation with asolution obtained using the Gaussian approximation for the actualexperimental parameters in Eq. (8) and ! = 0.05.

In Fig. 1, we plot the resulting time-dependent axial and radialcondensate widths rms(t) and zrms(t) calculated as root-mean-square values

rms(t) =

2

dz

0 d |(,z,t)|2 2 , (9)

zrms(t) =

2

dz

0 d |(,z,t)|2 z2 , (10)

and compare them with numerical solutions of Eqs. (5) and (6).We assume that initially the condensate is in the ground state. Inthe variational description, this translates into initial conditionsu(0) = u0, u(0) = 0, uz(0) = uz0, and uz(0) = 0, whereu0 and uz0 are time-independent solutions of Eqs. (5) and(6), while in GP simulations we reach the ground state byperforming an imaginary-time propagation [36]. For solvingthe GP equation (2), we use the split-step Crank-Nicolsonmethod [36]. It is evident that we have a good qualitativeagreement between the two approaches.

The main result obtained previously by using the Gaussianapproximation is an analytical estimate for the frequencies ofthe low-lying collective modes [5,6]. In this paper, we considerexcitations induced by a modulation of the interaction strengthand focus on the properties of the quadrupole and breathingmode. We assume that the external trap is stationary, thus pre-venting excitations of the dipole (Kohn) mode, correspondingto the center-of-mass motion. By linearizing Eqs. (5) and (6)around the equilibrium widths u0 and uz0, frequencies of

both the quadrupole Q0 and the breathing mode B0 wereobtained:

B0,Q0 =

2[(

1 + 2 p4u20u

3z0

)

(

1 2 + p4u20u

3z0

)2+ 8

(p

4u30u2z0

)2] 12.

(11)

For the repulsive interaction, the quadrupole mode has a lowerfrequency and is characterized by out-of-phase radial andaxial oscillations, while in-phase oscillations correspond to thebreathing mode. In the case of the experiment [19], Eq. (11)yields

Q0 = 0.035375, B0 = 2.00002. (12)

We emphasize that, although based on the Gaussian ansatz, thevariational approximation reproduces exactly the frequenciesof collective modes not only for the weakly interacting BECbut also for the strongly interacting BEC in the Thomas-Fermi regime [4,5]. Therefore, it represents a solid analyticaldescription of BEC dynamics.

However, due to the nonlinear form of the underlyingGP equation, we expect nonlinearity-induced shifts in thefrequencies of low-lying modes with respect to the valuesin Eq. (11) calculated using the linear stability analysis. Inparticular, our goal is to describe collective modes induced bythe harmonic modulation of the interaction strength in Eq. (7).In the case of a close matching of the driving frequency ! andone of the BEC eigenmodes, we expect resonances (i.e., largeamplitude oscillations). Here, the role of the nonlinear termsbecomes crucial and nonlinear phenomena become visible, aswe discuss in the next section.

III. SPHERICALLY SYMMETRIC BEC

Using a simple symmetry-based reasoning, we concludethat a harmonic modulation of interaction strength in the caseof a spherically symmetric BEC (i.e., = 1) leads to theexcitation of the breathing mode only, so that u(t) = uz(t) u(t). This fact simplifies numerical and analytical calculations,and this is why we first consider this case before we embarkto the study of a more complex axially symmetric BEC.

Thus, the system of ordinary differential Eqs. (5) and (6)reduces to a single equation:

u(t) + u(t) 1u(t)3

p(t)u(t)4

= 0 . (13)

The equilibrium condensate width u0 satisfies

u0 1u30

pu40

= 0 , (14)

and a linear stability analysis yields the breathing modefrequency

0 =

1 + 3u40

+ 4pu50

. (15)

013618-3

Excitation of Bose-Einstein Condensates (BECs) by harmonicmodulation of the scattering lengthI. Vidanovic, A. Balaz, H. Al-Jibbouri, and A. Pelster, PRA84, 013618 (2011).

Geometric Resonances in Bose-Einstein Condensates with Two- andThree-Body InteractionsH. Al-Jibbouri, I. Vidanovic, A. Balaz, and A. Pelster,arXiv:1208.0991.

Excellent agreement with Gross-Pitaevskii Equation


Variational approach

Lagrangian

L(t) =L(r, t) dr ,

Lagrange density

L(r, t) = i~2

(

t

t

) ~

2

2m||2 V (r)||2 g

2||4

Gaussian variational ansatzPhys. Rev. Lett. 77, 5320 (1996)Phys. Rev. A 56, 1424 (1997)

G(, z, t) = N (t) exp1

2

2

u(t)2+

z2

uz(t)2

+ i

`2(t) + z

2z(t)

Time-dependent normalization

N (t) = 1

32 u2(t)uz(t)


Variational approachcontinued

Euler-Lagrange equations

d

dt

L

q Lq

= 0, q {ui, i

}Phases

(t) =m u2~u

, z(t) =m uz2~uz

Dimensionless parameters:

= t, ui() =ui(t)aho

, aho =

~

m

Dimensionless driving

p() = p0 + p1 cos(

), p0 =

2

Naavaho

, p1 =

2

Naaho


Equations of motion

Equations of motion

u + u =1u3

+p()u3 uz

, uz + 2uz =1u3z

+p()u2 u

2z

Isotropic condensate: u = uz = u and = 1Reduction to one ODE:

u+ u =1u3

+p()u4

Stationary solutions:

u0 =1u30

+p0

u30 uz0, 2uz0 =

1u3z0

+p0

u20 u2z0

Isotropic case:

u0 =1u30

+p0u40


Equations of motionEquilibrium position continued

Equilibrium condition: u50 u0 = p0 (isotropic condensate)

u50 u0-1

0

1

1.50.5

p0

u0

u0 u0+

p0

pcrit

Figure: Equilibrium widths u0 of a Bose-Einstein Condensate subject toattractive interactions.


Equations of motionMathieu equation

Linearize about equilibrium position u0

u() = u0 + u()

Taylor expand nonlinear terms to first order in u

1(u0 + u)3

=1u30 3 u

u40+ . . . ,

1(u0 + u)4

=1u40 4 u

u50+ . . .

With definitions

q = 8 p1u50

(

)22t =

c = 4(

)2(5 1

u40

)x(t) = u()

Obtain an inhomogeneous Mathieu equation

x(t) +[c 2 q cos(2 t)

]x(t) = u0

2q cos(2 t)


Isotropic stabilityNon-homogeneous term

Stability unaffected by non-homogeneous term

x(t) +[c 2q cos 2t

]x(t) = u0

2q cos 2t

Infinite determinant method:c q 02 q c 4 q

0 q c 16...

. . .

M

A0

A2

A4

...

=

0

u02 q0

...

Coefficients: An (detM)1Stability borders coefficients divergeTransform diagram for relevant parameters


Isotropic stabilityResults

case 2: u0+

p1/p0

(b)

case 1: u0

/

p1/p0

(a)

0 0.5 10 0.5 10

2

4

0

5

10

15

p1/p0

/

p1/p0

0 0.5 10 50

5

10

0

20

40

60


Anisotropic stabilityCoupled Mathieu equations

Equations of motion

u + u =1u3

+p()u3 uz

, uz + 2uz =1u3z

+p()u2 u

2z

Linearize: ui = ui0 + uiDefinitions:

2t = , q = p1,

x(t) =(

u()

uz()

), A = 4

(

)2( 4 p0u30u2z02p0

u30u2z0

32 + 1u4z0

),

f = 4(

)2 p1u30uz0p1

u20u2z0

, Q = 2 ( )2( 3

u40uz0

1u30u

2z0

2u30u

2z0

2u20u

3z0

)

Coupled, inhomogeneous Mathieu equations:

x(t) + [A 2qQ cos(2t)] x(t) = f cos(2t)


Anisotropic stabilityCoupled Mathieu equations continued

Non-homogeneity does not affect stability J. Slane et al., J.Nonlinear Dynamics and Systems Theory, 11 (2) (2011).

Floquet ansatz:

x(t) =

n=u2ne(+2in)t

Recursion relation[A + ( + 2in)2 I

]u2n qQ (u2n+2 + u2n2) = 0

Ladder operators

S2n ={A + [ + 2i (n+ 1)]2 I qQS2n2

}1qQ

Continued matrix inversionA+

2Iq2Q

hA + ( + 2i)

2 . . .i1

+hA + ( 2i)2 . . .

i1 ffQ

u0 = 0

Vanishing determinant for stability borders


Anisotropic stabilityResults, case 1: u0


Anisotropic stabilityResults, case 2: u0+


Conclusions and Outlook

Analogous physics: BEC and pendulum

Stabilize unstable equilibrium

Experimental possibilities

Dipolar BEC Dipolar Fermi Gas4.4. Low-Lying Excitations

0 5 10 15 20 25 300

1

2

3

4

5

6R

Rz

!bdd

a)

0 1 2 3 4 5 60

2

4

6

8

10

!bdd

Unstable

Metastable

Stable

b)

Figure 4.3.: a) Aspect ratio = R/Rz as a function of the dipolar interaction strength #bdd fordifferent values of = (#bdd = 0). For #

bdd < 1, the transcendental equation (4.15)

gives a single condensate aspect ratio , which corresponds to a stable configuration andis depicted in solid curves. For #bdd > 1, a second, unstable branch, depicted in brokencurves, shows up for the aspect ratio, which meets the stable one at #bdd = #

b,critdd . For

#bdd > #b,critdd , no solution is available. b) Stability diagram of a cylinder-symmetric Bose-

Einstein condensate. The blue line depicts the critical value of the interaction strength#b,critdd , above which no solution is available for Eqs. (4.42) and (4.43). The colored arearepresents stable configurations, in which an additional unstable solution is present (lightgreen) or not (gray).

constant. Indeed, the symmetry of the problem allows to understand by analogy the properties ofthe system in the inverse case, i.e., when y is hold fixed and x changes. We show in Fig. 4.4a) theaspect ratio x as a function of the dipolar interaction strength #bdd for x = 4 and y = 2, 3, 4, 5, 6.The solid lines represent the stable solutions of the stationary versions of Eqs. (4.41) while the brokenones mark the unstable solutions. The orange curve depicts the cylinder-symmetric case for x =y = 4. As y increases to y = 5 (purple) and y = 6 (gray), the maximum value of the interactionstrength supporting a stable solution increases, the contrary being true for decreasing y. This can beunderstood as a consequence of the fact that a larger trap aspect ratio implies a more pancake-shapedcloud, which favors the repulsive part of the dipolar interaction.

The absence of the cylinder symmetry can be displayed in a more dramatic way by considering thestability diagram of a dipolar condensate. Fig. 4.4b) shows the value of the critical interaction strengthas a function of the trap aspect ratio x. In addition to the cylinder-symmetric curve y = x (black),the stability diagram is also calculated and shown for y = 2x (red) and y = x/2 (blue). We omitthe classification of the corresponding regions in order to highlight the importance of the asymmetryfor the stability diagram.

4.4. Low-Lying Excitations

The study of the low-lying excitations is a very important diagnostic tool for the physics of coldatoms. In this section, we will discuss these excitations in a dipolar condensate by giving a generaldescription of the eigenvectors as well as semi-analytic expressions for the frequencies of oscillation inthe cylinder-symmetric dipolar condensate.

67

7.4. Static Properties

0 2 4 6 8 10 120

1

2

3

4

5

6

7

!dd

RxxRz

f

Figure 7.1.: Aspect ratio in real space Rxx/Rz for a cylinder-symmetric trap with x = y =1, 2, 3, 4, 5, 6, 7 (bottom to top). The upper branch (continuous) corresponds to a localminimum of the total energy, while the lower branch (dotted) represents an extremum butnot a minimum.

of the energy (7.31) shows that the system cannot have a global minimum for any non-vanishing "fdd.This can be seen by noticing that the stabilization comes from the factor K2 R2 whereas thedipolar interaction goes with R3, rendering the energy unbounded from below. Nonetheless, for weakenough dipolar interactions a local minimum might exist, to which the system would return after asmall perturbation. The regions satisfying this property will be called stable, while inflection pointsand local maxima will be denoted unstable equilibrium points. The mathematical criterion behindthis classification scheme is given by the eigenvalues of the Hessian matrix associated with the foureffectively independent variables of the problem.

One of the consequences of the unboundedness of the internal energy is that, for each value of theinteraction strength "fdd, where the system has a stable configuration, there is also another unstableone. This can be seen by considering the aspect ratio of the cloud, which is depicted in Fig. 7.1 fordifferent values of the trap aspect ratio x = y, as a function of "fdd. Here, we recognize that thestable branch (continuous) of the real space aspect ratio starts at "fdd = 0 with Rx = Rz = 1 andextends itself until the value "f,critdd , where it meets the unstable branch (dotted). For "

fdd > "

f,critdd , no

stationary solution for the equations (7.34) exists. The unstable branch, on the other hand, possessesa vanishing aspect ratio for "fdd = 0. This is due to the fact that the dipole-dipole interaction tendsto stretch the sample along the polarization direction. For a small value of "fdd, the unbounded energysolution is obtained with Rx 0 and, consequently, Rx/Rz 0, although the Thomas-Fermi radiusin the axial direction Rz remains finite. We remark that the upper branch corresponds to a localminimum of the energy such that the Hessian matrix has only positive eigenvalues, while the lowerone is an extremum but not a minimum, corresponding to a Hessian matrix with at least one negativeeigenvalue. The corresponding graph for a dipolar Bose-Einstein condensate shown in Fig 4.3 in theThomas-Fermi regime bears a crucial difference: unstable solutions only become available for "bdd > 1[97]. The physical reason for this effect is that in dipolar condensates the stabilization comes from thecontact interaction Eq. (4.30), which scales with R3, just like the dipole-dipole interaction.

In order to study the effect of a triaxial trap on the static properties of a dipolar Fermi gas, we explorefurther the symmetry f(x, y) = f(y, x) of the anisotropy function as defined by Eq. (4.34). Due tothis symmetry, we only need to discuss the aspect ratio Rxx/Rz since the properties of Ryy/Rz can

117

bdd =Cdd3g

, fdd =Cdd4

(M33

~5

)1/2N1/6

A.R.P. Lima and A. Pelster, PRA 81, 021606(R)/1-4 (2010)and PRA 84, 041604(R)/1-4 (2011)


Thank you for your attention

Generously sponsored by the DAAD RISE program

Parametric resonancePendulum physicsMathieu equationBEC

Variational approachEquations of motionEquilibrium position

Isotropic stabilityNon-homogeneous Mathieu equationResults

Anisotropic stabilityCoupled Mathieu equationsResults

Conclusions

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