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)

H



j N

  1

p

j

2 2

m

 1 2

m

  2

x

2

j

 

t

2 

y

2

j



z

2

j

   1 2

j N

  1

Q

2

r

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

 3 4

N L

fixed

Specific Heat low temperature estimate Non extensive behaviour at low temperatures in the thermodynamic limit:

c a

 1/ ln

N

•Due to long-range correlations •It is a quantum effect (at high-T Dulong-Petit holds)

Coefficient of thermal expansion 

T

 1

L

 

L



T

  3 2

L



T C a C a



T

heat capacity compressibilty 

T

  ln 1

N

 3/ 2 

T

 1 ln

N

For a usual harmonic uniform crystal: 

T

 0

Equivalence of ensembles

Relative energy fluctuation

 1

C a

     ln

N N

    1/ 2

Thermal Stability:

Thermal energy much smaller than equilibrium energy; Displacement much smaller than spacing between atoms Stability condition in thermodynamic limit

Q

2 ln

N a

>>

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 m m

Phonon-like approximation

q j



A e j

 

t

 slowly varying

A j

Jacobi Phonon-like

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