Esteban Calzetta- Analog cosmology with spinor Bose-Einstein condensates

8/3/2019 Esteban Calzetta- Analog cosmology with spinor Bose-Einstein condensates

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Analog cosmology with spinor

Bose-Einstein condensatesEsteban Calzetta

CONICET and Universidad de Buenos AiresArgentina

Analo cosmolo with s inor Bose-Einstein condensates . 1/


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Summary

It is well known that both a particle with spin and aquantum field in a Bianchi type IX Universe arerelated to the quantum mechanical top. We will

combine these insights to show that the dynamics of acold gas of atoms with total momentum F = 0 maybe used to explore the behavior of quantum matter

near a cosmological singularity.



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BECs

Because of the propensity of bosons to bunchtogether, at low enough temperature the lowest lyingone-particle mode for a system of bosonic particles

may acquire a macroscopic occupation number, thatis, of the order of the total particle number. This is thephenomenon of Bose-Einstein condensation. The

particles in the condensate are coherent, whichleads to a number of observable phenomena such assuperfluidity.Bose-Einstein condensates are being activelyinvestigated and have a number of applications, notleast their role in a possible implementation of aquantum computer



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To condense or not to condense

For a system of free bosons all particles fall to thecondensate at zero temperature. If there is a repulsiveinteraction then a compromise is reached, and there is

a nonzero number of noncondensate particles even atabsolute zero.For attractive interactions, in principle the condensed

state is unstable. However, if the increase of densitybrings an increase in kinetic energy which outweighsthe loss of potential energy, then a stable state may befound. This happens when the total particle number isless than a critical value.



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Field theory of BEC

For a field-theoretic description of BEC we begin witha second - quantized field operator (x, t) whichremoves an atom at the location x at times t. It obeys

the canonical commutation relations

[ (x, t) , (y, t)] = 0 (1)

(x, t) , (y, t)

= (x y) (2)



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Field theory of BEC

The dynamics of this field is given by the Heisenbergequations of motion

i t

= [H, ] (3)

H =

ddx

H + Vint

,

(4)

H = 2

2M2

+ Vtrap (x) (5)



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Classical action

The Heisenberg equation of motion

i

t = H +

Vint

(6)

is also the classical equation of motion derived fromthe action

S =

dd+1x i

t

dt H (7)



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Particle number conservation

The theory is invariant under a global phase change ofthe field operator

ei

, ei

(8)The constant of motion associated with this U(1)invariance through Noether theorem is the totalparticle number.



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Condensate wave function

There is a special one-particle state, with wavefunction 0, which, upon condensation, acquires amacroscopic occupation number N0, comparable to

the total number of particles N. We call this state thecondensate. We decompose into its componentproportional to 0 and an orthogonal component

= + , where = a00. The operator a0 is thedestruction operator for the condensate.

Analo cosmolo with s inor Bose-Einstein condensates . /

art c e m er onser ng


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art c e um er onserv ngFormalism

The PNC approach is a systematic expansion ininverse powers of the total particle number. To lowestorder 0 obeys the classical equation derived from the

action, with the addition of a chemical potential termto enforce the normalization. This is theGross-Pitaievskii equation. Higher quantum effects

may be described by a suitable effective action



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Analog Gravity with BECs

Up to now there were essentially two ways of doinganalog gravity with BECs

focus on metrics: if the condensate flow isinhomogeneous, long wavelength perturbationsof the condensate behave as a relativistic scalarfield in an effective metric



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Analog Gravity with BECs

Up to now there were essentially two ways of doinganalog gravity with BECs

focus on metrics: if the condensate flow isinhomogeneous, long wavelength perturbationsof the condensate behave as a relativistic scalarfield in an effective metric

focus on mechanisms: one could also mimic aspecific mechanism without duplicating thewhole metric. The best known example is using

the Bose-Nova condensate collapse experiment tostudy cosmological particle creation.



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Analog Cosmology

Our work is similar to the metric approach in that weshall use Bose-Einstein condensates to emulate aquantum field in a Bianchi type IX spacetime, but it is

also close to the Bose-Nova work in that the necessarycondensate configuration can be obtained in realisticexperimental situations - indeed, several relevant

experiments have already been performed: L. E.Sadler et al., Nature 443, 312 (2006); T. Lahaye et al.,Nature 448, 672 (2007); M. Vengalattore et al. ,quant-ph/0712.4182, etc.



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The Bianchi type IX Universe

The Bianchi type IX (Mixmaster) Universe is the mostgeneral homogeneous but anisotropic model. Themetric of the spatial sections can then be written in

terms of Misner parameters as

ds

2

B = e

2e

2+

2

1 + e

++3

2

2 + e

+

3

2

3(9)

where the nonintegrable 1-forms a satisfy

da = 12

abcb c (10)

where abc is the totally antisymmetric symbol.



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Why Bianchi IX

It has been conjectured by Belinsky, Lifshitz andKhalatnikov that the Universe behaved locally like avacuum Bianchi type IX Universe near the

cosmological singularity. Moreover, the Mixmastermodel is known to be chaotic



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Why Quantum Bianchi IX

There are several interesting cosmological questionsthat require a semiclassical Bianchi IX model

Can a Bianchi IX Universe dissipate itsanisotropy?



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Can it avoid recollapse?



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Can it avoid recollapse?

Can there be a smooth transition from a BianchiIX to an inflationary evolution?



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Spinor BECs

In a spinor BEC, the sum F of the electron spin S andthe nuclear spin I is not zero in the ground state.Examples are rubidium (F = 1 ), sodium (F = 2 )

and chromium (F = 3 ).Spinor condensates show local point interactions,spin-exchange interactions and dipole-dipole

interactions. The strenght of each of these can becontrolled independently by tayloring the confiningpotential and/or applying external electromagneticfields



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From spinor BECs to Bianchi IX

The space of states of any spinning particle is asubspace of the Hilbert space of a quantum top (N.Rosen, Phys. Rev. 82, 621 (1951). Introduce Euler

angles

J

K

N



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From spinor BECs to Bianchi IX

The spin operators are identified with differentialoperators in the configuration space of the top

Fx =

i

cos[]

sin[]cos[]

sin[]

+

sin[]

sin[]

Fy = i

sin[]

+ cos[]cos[]

sin[]

cos[]

sin[]

Fz =

i

(

L. S. Schulman used this to develop a path integral for

spin (Phys. Rev. 176, 1558 (1968))



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The free spinor condensate

We are almost there. Because of the hyperfinesplitting, the Hamiltonian H for a spinor condensatecontains a term (A/2)F2 (for rubidium,

A = 2 106eV). After second quantization, this termbecomes (a = (,,))

H0 = A2

d3 g0 gab0 a b (12)

where g0ab is the metric of the sphere, given by

ds20 = d2 + d2 + d2 + 2 cos dd (13)

Analo cosmolo with s inor Bose-Einstein condensates . 1 /


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Turn on B

Under the influence of an external magnetic field B,there appear linear and quadratic Zeeman shifts. Theformer can be elliminated by going to a rotating

frame, and the latter adds to the second quantizedHamiltonian a term

HZ =

2

BB

2

8A

d3 g0 (14)



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Bianchi IX at last

To see that this is indeed a Bianchi type IX metric,identify

1 = d + cos d

2 = cos d + sin sin d

3 = sin d + cos sin d (17)



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Bianchi IX at last

The metric can then be written in terms of Misnerparameters as

ds2B = e2

e2+21 + e++322 + e

+323

(18)

where = 0, + = and

=1

3

ln 1 + B2 (19)



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Final remarks

We have shown that a spinor BEC has a naturalrepresentation as a field on a Bianchi type IXUniverse whose scale factor and shape depend

upon the external magnetic field. This opens upopportunities for both cosmology and BECphysics



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Final remarks



For cosmology: build a bounce in the laboratory!



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Final remarks



For cosmology: build a bounce in the laboratory! For BEC: bring the sustantial literature on

quantum fields on homogeneous spaces to bear

on the computation of quantum corrections to theGross-Pitaievskii equation.



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Final remarks



For cosmology: build a bounce in the laboratory! For BEC: bring the sustantial literature on

quantum fields on homogeneous spaces to bear

on the computation of quantum corrections to theGross-Pitaievskii equation.

More next time!


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Esteban Calzetta- Analog cosmology with spinor Bose-Einstein condensates