7. Ideal Bose Systems Thermodynamic Behavior of an Ideal Bose Gas Bose-Einstein Condensation in Ultracold Atomic Gases Thermodynamics of the Blackbody Radiation The Field of Sound Waves Inertial Density of the Sound Field Elementary Excitations in Liquid Helium II

7.1. Thermodynamic Behavior of an Ideal Bose Gas From § 6.1-2, Bose gas : Grand partition function Grand potential  =   BE condensation DOS

Correction for a(0) = 0 : ( to handle ) ( See App.F for rigorous justification ) E.g. : # of particles in ground state : is negligible for N0 , V   

Bose-Einstein functions

= # of particles in ground state 

U, z ( n 3 ) << 1  for z < 1.  Calculated using Mathematica

Virial Coefficients ( z << 1 )  al = Virial Coefficients = volume per particle Calculated using Mathematica

CV ( z << 1 )   CV has max. Known : Calculated using Mathematica   CV has max. Known :

z ≤ 1 = density of excited particles    = # of particles in the ground state N0 1 10 100 z 0.5 0.91 0.99   BEC ( Bose-Einstein Condensation )

Bose-Einstein Condensation ( BEC ) Superconductor : Condensatiion in momentum space Superfluid : Condensatiion in coordinate space  Condition for BEC is or with  Condensate = mixture of 2 phases : Normal phase (excited particles) Condensed phase (ground state p’cles)  for T <<TC

For is obtained by solving For is obtained by solving  Calculated using Mathematica For is obtained by solving For is obtained by solving 

P ( T ) for all z as V    for T < TC   T < TC    ½ PMB (TC)

( Determines z for given n & T. ) for all z for T  TC For T > TC , N0 ~ O(1)  ( Determines z for given n & T. ) Calculated using Mathematica Mixture ( z = 1 )  Virial expansions for T >> TC Inaccessible ( z > 1 ) Bose gas Classical Transition line ( P  T 5/2  T ) normal phase ( z < 1)

CV   For T < TC For T = TC

For T < TC For T > TC    

with = CV / T discontinuous at TC : Prob.7.6 classical value  

 Transition London : He I – He II transition is a BE condensation. Calculated using Mathematica m = 6.65  1024 g. V = 27.6 cm3 / mole v = V / NA = 4.58  1023 TC = 3.13 K Exp: TC = 2.19 K He4 He II He I

Isotherms For isotherms, N, T = const. & z is a function of v = V / N determined by Setting  & z is determined by for &

Transition line : P( v = vC ) , i.e., For v < vC indep of v Transition line : P( v = vC ) , i.e., T > T Calculated using Mathematica

Adiabats Fundamental thermodynamic equation : see Reichl §2.E 

Since z = 1 for T > TC , z = const  T for an adiabatic process.  const z  const n 3 Hence, for an adiabatic process i.e.   Same as the ideal classical gas.

 Prob 7.4-5  5/3 for T >> TC > 5/3 otherwise   for T = TC Mixed phase region (T < TC ) :  ( No contribution from N0 )

7.2. Bose-Einstein Condensation in Ultracold Atomic Gases Magneto-optical traps (MOTs) to cool 104 neutral atoms / molecules at T ~ nK : Step 1 : T ~  K 3 orthogonal pairs of opposing laser beams with  Stationary atoms not affected. Moving atoms Doppler shifted to absorb photon & recoil. Re-emit photons are isotropic.  Atoms slowed. Recoil limit :  

Step 2 : T ~ 100 n K Laser off. Anisotropic, harmonic potential at trap center created by B(r) . m = magnetic moment of atom Evaporative cooling :  adjusted to resonance to remove highest energy atoms.  Degeneracy of the level is Prob 3.26

DOS a ( )   

Grand Potential ( F =   ) Grand partition function 

N   V = const for a trap Onset of BEC : z = 1, T = TC, N = Ne = # of trapped atoms.   For a given T , z is given by

T > TC : Obs. ~ 170 nK T < TC : is finite in the TD limit (N ,V   ) . Occupancy of 1st excited state :  0 in the TD limit.

7.2.A. Detection of the BEC Harmonic oscillator : Linear size of ground state along x is Linear size of thermal distribution of excited atoms is ( equipartition theorem ) :  For  = 2 ( 100 Hz), T = 100 nK, Time of flight measurement of momentum distribution f ( p ) : B turned off  atomic cloud expands for 100 ms according to f ( p ). ( v ~ 1 mm/s  x ~ 100 m. ) Cloud illuminated with laser at resonant  shadow on CCD. ( size & shape of shadow n( r , t ) gives f ( p ) at t = 0 ) 3. For long times, n0 ( r , t ) is anisotropic, while ne ( r , t ) is isotropic.

n0 For a 1-D harmonic oscillator in its ground state In the plane wave basis ( p-representation ) :  Mathematica At t = 0, B is turned off so that for t > 0, H = p2 / 2m :

n anisotropyic for large t ( BEC signature )  Mathematica t = 0 t > t t~10ms n anisotropyic for large t ( BEC signature )

nexcited Semi-classical treatment : ensemble average done in phase space : BE statistics is used for f : with For t > 0

n loses anisotropy for large t . t~10ms Mathematica

87Rb Anisotropy is BEC signature.

7.2.B. Thermodynamic Properties of the BEC  Alternatively : = same result  for T > TC Setting z = 1 : for T < TC ( U = 0 for condensate )

Calculated using Mathematica

V is const in trap    T > TC : T < TC :

Calculated using Mathematica