Physic Labs

Condensed matter physics

Bose–Einstein condensation: theory

Cool a uniform ideal Bose gas below its critical temperature Tc=2πℏ2mkB(nζ(3/2))2/3T_c=\frac{2\pi\hbar^2}{mk_B}\left(\frac{n}{\zeta(3/2)}\right)^{2/3} and verify that the condensate fraction grows as N0/N=1−(T/Tc)3/2N_0/N=1-(T/T_c)^{3/2}.

Research

Equipment

  • 3D canvas: condensate-fraction curve and occupation of quantum states
  • Slider “Nhiệt độ” (temperature T)
  • Slider “Mật độ / tần số” (density / trap frequency)
  • Slider “Số trạng thái” (number of states) and “Tạm dừng chuyển động” button

Procedure

  1. Cool toward T_c

    Start above TcT_c and lower the temperature slider. Above TcT_c the ground-state fraction N0/NN_0/N stays near zero; right at TcT_c the ground state begins to fill macroscopically. Read where the curve lifts off and note the displayed TcT_c.

  2. Check the power law

    Continue cooling and read N0/NN_0/N at two temperatures below TcT_c, e.g. T/Tc=0.5T/T_c=0.5 and 0.80.8. Check the values against N0/N=1−(T/Tc)3/2N_0/N=1-(T/T_c)^{3/2} (about 0.65 and 0.28) and observe the ground-state bar in the state-occupation figure grow accordingly.

  3. Vary density and trap

    Adjust the density/frequency slider and see TcT_c shift: a denser gas condenses warmer, Tc∝n2/3T_c\propto n^{2/3}. Then change the number of states and describe which idealizations — uniformity, no interactions, thermodynamic limit — the model keeps and which it drops.

Simulation

Experiment history

In 1924 Satyendra Nath Bose sent Einstein a derivation of Planck's radiation law based on counting indistinguishable photons. Einstein translated the paper, extended the statistics to massive particles, and predicted in 1925 that an ideal gas of bosons would accumulate in its ground state below a critical temperature — the Bose–Einstein condensate. The prediction waited seventy years for refrigeration: in 1995 Eric Cornell and Carl Wieman produced a condensate in rubidium-87 vapor cooled below 170 nK, and Wolfgang Ketterle followed with sodium, work honored by the 2001 Nobel Prize. Real condensates interact, so their exact thresholds and fractions differ from the ideal-gas curve shown here.

Related physicists

Albert EinsteinLouis de BroglieEric Allin CornellCarl Edwin WiemanWolfgang Ketterle

Related library topics

Bose–Einstein condensation: theory