Esteban Calzetta- Analog cosmology with spinor Bose-Einstein condensates
Transcript of 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
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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.
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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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Why Quantum Bianchi IX
There are several interesting cosmological questionsthat require a semiclassical Bianchi IX model
Can a Bianchi IX Universe dissipate itsanisotropy?
Can it avoid recollapse?
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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?
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)
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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
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
For cosmology: build a bounce in the laboratory!
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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
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
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
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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