Download Quantum Nonlinear Resonances in Atom Optics

Low dimensional ion crystals
Giovanna Morigi
Universitat Autonoma de Barcelona
Low dimensional ion crystals
In collaboration with
Shmuel Fishman
Technion, Haifa
Gabriele De Chiara
Universitat Autonoma de Barcelona
Tommaso Calarco
ITAMP, Cambridge, and University of Trento
One-component plasma
Coupling parameter:
Wigner-Seitz radius
This parameter measures the strength of correlations
In an infinite homogeneous gas it determines the phase
Strong correlations:
Crystallization (transition to spatial order):
1D: Crossover. Long range order at T=0
Coulomb crystal of ions Properties
Gas of ionized atoms:
Singly-ionized alkali-earth metal / Radiate in the visible.
Imaging: Fluorescence by laser light
Confinement by external potentials:
The ions are trapped by Paul (radiofrequency) or Penning traps
Unscreened Coulomb repulsion
Crystallization:
Low thermal energies are achieved through laser-cooling.
Interparticle distance of the order of 10 micrometers
Trap geometry shapes the crystal
The ion chain
(M. Drewsen and coworkers, Aahrus)
p j2


2
N
1
1
Q
2 2
2
2
2
H 
 m  x j  t  y j  z j   
2
2 j 1 i  j ri , j
j 1 2m
N
One dimensional structure (ion chain)
Charges distribution at equilibrium
Continuum limit: develop a suitable mean field for 1D
Linear density:
Length of the chain:
at leading order in 1/log N
D. Dubin, PRE 1997.
Linear fluctuations
of the classical ground state
1) Evaluation of the density of states and spectrum of the excitations
2) Quantization of the eigenmodes in the regime of stability:
Thermodynamic properties
3) The stability of the chain:
a) thermal instabilities
b) quantum instabilities
c) Structural instabilities:
Phase transition to a zig-zag configuration
Linear fluctuations of
the classical ground state
(R. Blatt and coworkers, Innsbruck)
Harmonic vibrations
around the equilibrium positions
with
and
Fourier modes
Eigenvalue problem
Some properties
No uniform distribution of charges along the trap axis
It implies that
Bloch theorem does not apply: The excitation are NOT phononic waves
It is a dimensional effect:
In one dimension the correlation energy is crucial
Long-range interaction + one-dimension:
The dynamics are NOT the one-dimensional limit of a three dimensional
mean field description.
Long wavelength modes
Continuum approximation (away from ends):
Long wavelength modes
Rescaled variables / Continuum approximation / Perturbative expansion
Leading order in
Jacobi Polynomials differential equation!
Eigenmodes:
Axial Eigenfrequencies:
Transverse Eigenfrequencies:
Spectra of excitations
Long wave-length modes: Jacobi polynomials type of excitations
Short wave-length modes: Phononic waves type of excitations
(solved using Dyson's theory for oscillators chains with random springs)
G. M. and Sh. Fishman, PRL 2004; PRE 2004.
Statistical mechanics
of the chain at equilibrium
Statistical Mechanics
Quantization of the vibrations
Canonical ensemble
One-dimensional behaviour:
Thermodynamic limit:
Density in the center n  0  
3N
4 L
fixed
Specific Heat
low temperature estimate
Non extensive behaviour at low temperatures in the
thermodynamic limit: c  1/ ln N
a
•Due to long-range correlations
•It is a quantum effect (at high-T Dulong-Petit holds)
Coefficient of thermal expansion

1 L
T 
T
L
T 

P, N
3

T Ca
2L
1
 ln N 
3/ 2
For a usual harmonic uniform crystal:
Ca
heat capacity
T
compressibilty
1
T 
ln N
T  0
Equivalence of ensembles
Relative energy fluctuation
 ln N
1

 
Ca  N
1/ 2



Thermal Stability:
Thermal energy much smaller than equilibrium energy;
Displacement much smaller than spacing between atoms
Stability condition in thermodynamic limit
Q2
ln N
a
>>
k BT
Structural instability:
phase transition
to a zigzag configuration
J.P. Schiffer, PRL 1993.
Chain to Zigzag: Previous works
Molecular dynamics
J.P. Schiffer, PRL 1993.
Numerical simulations, Piacente et al, PRB 2004:
Study the derivative of the free energy for a finite crystal at
the critical point:
behave like a second order phase transition.
Structural stability of the chain
Zigzag mode
Our theory gives
Critical aspect ratio
G. M. and Sh. Fishman, PRL 2004; PRE 2004.
Chain to Zig-Zag:
second-order phase transition
Zigzag mode
Zigzag mode is the soft mode
Symmetry breaking: line to plane
Order parameter: Equilibrium
distance from the axis
Control field: Transverse confinement
Landau-Ginzburg theory of the
Chain to Zig-Zag transition
Sh. Fishman, G. de Chiara, T. Calarco, G.M.
Questions
Definition of a temperature for this system?
Measurement of the thermodynamic function?
Quantum mechanical effects?
Outlook
Coupling to the internal degrees of freedom
Coupling to the photonic mode of an optical resonator
Quantum Stability:
Size of one-particle wave packet « Interparticle spacing
Typical parameters: One particle wave packet: 30 nm
Interparticle distance: 10 mm
Phonon-like approximation
i  kja t 
q j  Aj e
slowly varying
Jacobi
Phonon-like
Aj
Short wavelength modes
Nearest neighbor approximation
Assume slow variation of
Apply method developped in
Short wavelength: Density of states
Density of states vanishes at
=0