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Bose-Einstein condensation in ultracold atomic gases

1. What is being studied?

We're looking at Bose–Einstein condensation (BEC) of a dilute gas of atoms (like \(^{87}\text{Rb}\) or \(^{23}\text{Na}\)) that are:

  • cooled to nanokelvin temperatures
  • held in a magnetic trap that looks, near its center, like a 3D harmonic oscillator.

So: think of a bunch of bosonic atoms sitting in a 3D "bowl-shaped" potential.

2. How do they cool and trap the atoms?

Laser cooling (optical molasses)

  • Three pairs of counter-propagating laser beams nearly resonant with an atomic transition shine on the atoms.
  • Atoms moving toward a beam are Doppler-shifted into resonance and preferentially absorb photons from that direction.
  • Each absorption + random re-emission gives a small momentum kick against the atom's motion → atoms slow down.
  • There is a recoil limit: you can't cool below the temperature set by the photon momentum.

Magnetic trap

After laser cooling, the lasers are turned off. A spatially varying magnetic field creates an effective potential

\[ V(r) = \frac{1}{2} m \big(\omega_x^2 x^2 + \omega_y^2 y^2 + \omega_z^2 z^2\big) \]

(eq. (1))

This is just a 3D harmonic oscillator with frequencies \(\omega_x,\omega_y,\omega_z\). The atoms sit in this "magnetic bowl."

Evaporative cooling

The trap depth is lowered so that the highest-energy atoms escape. The remaining atoms collide and re-thermalize at a lower temperature. Repeat → you cool the gas further, into the BEC regime.

3. Single-particle energy levels in the trap

If we ignore interactions, each atom is just a 3D harmonic oscillator. Its energy levels are

\[ \varepsilon_{l_1,l_2,l_3} = \hbar\omega_1 l_1 + \hbar\omega_2 l_2 + \hbar\omega_3 l_3 + \frac{1}{2}\hbar(\omega_1+\omega_2+\omega_3), \]

(eq. (2))

where \(l_1,l_2,l_3 = 0,1,2,\dots\) are quantum numbers. So: energy is a sum of three independent oscillator energies.

Often they focus on the excitation energy (ignoring the constant zero-point term).

4. Density of states and thermodynamic potential

For many atoms and many levels, it's convenient to replace the discrete levels by a smooth density of states \(a(\varepsilon)\). For an anisotropic trap they get

\[ a(\varepsilon) = \frac{\varepsilon^2}{2(\hbar\omega_0)^3}, \]

(eq. (3))

where \(\omega_0 = (\omega_1\omega_2\omega_3)^{1/3}\) is the geometric mean frequency.

Using this, they write the grand potential (thermodynamic potential)

\[ \Pi(\mu,T) = -\frac{(kT)^4}{2(\hbar\omega_0)^3} g_4(z), \]

(eq. (4))

where

  • \(\mu\) is the chemical potential,
  • \(z = e^{\beta\mu}\) is the fugacity,
  • \(g_n(z)\) are standard Bose functions.

You don't need the details; the point is: all thermodynamics is encoded in \(\Pi\), which depends on \(T\), \(\mu\) and the trap frequency.

5. Number of atoms and the critical temperature \(T_c\)

The total number of excited atoms (not in the ground state) is

\[ N(\mu,T) = \left( \frac{kT}{\hbar\omega_0} \right)^3 g_3(z) \]

(eq. (5)).

  • As you cool the system, \(\mu\) increases towards 0, so \(z \to 1\).
  • BEC occurs when \(\mu = 0\) (i.e. \(z=1\)), because then the excited states can no longer hold all the atoms and the extra atoms pile into the ground state.

Set \(z=1\) and \(N = N_{\rm excited}\) to get the critical temperature

\[ \frac{kT_c}{\hbar\omega_0} = \left(\frac{N}{\zeta(3)}\right)^{1/3} \]

(eq. (6)), where \(\zeta(3) = g_3(1)\approx 1.202\).

So:

  • Larger \(N\) → higher \(T_c\).
  • Tighter trap (larger \(\omega_0\)) → higher level spacing → lower \(T_c\).

For \(T < T_c\), the number of excited atoms is

\[ N_{\text{excited}} = \zeta(3)\left(\frac{kT}{\hbar\omega_0}\right)^3 = N\left(\frac{T}{T_c}\right)^3, \]

(eq. (7))

so the condensate fraction is

\[ \frac{N_0}{N} = 1 - \left(\frac{T}{T_c}\right)^3, \]

(eq. (8)).

Compare with the uniform Bose gas, where the exponent is \(3/2\); here the trap changes that to \(3\).

Figure 7.7 shows that this simple formula fits the experimental condensate fraction pretty well.

6. Detection of the condensate: sizes and time-of-flight

Sizes in the trap

  • Ground-state size in direction \(\alpha\) (\(x,y,z\)):
\[ a_\alpha = \sqrt{\frac{\hbar}{m\omega_\alpha}} \]

(eq. (9)).

This is basically the spatial width of the harmonic oscillator ground state.

  • Size of the thermal cloud in the same direction:
\[ a_{\text{thermal}} = a_\alpha \sqrt{\frac{kT}{\hbar\omega_\alpha}} \]

(eq. (10)).

Since \(kT \gg \hbar\omega_\alpha\), the thermal cloud is much bigger than the condensed cloud.

So at low \(T\) you have a tiny, dense condensate in the center plus a big, fuzzier thermal halo.

Time-of-flight experiment

Instead of looking inside the trap, experimentalists:

  1. Suddenly switch off the trap at time \(t=0\).
  2. Let the cloud expand freely for ~100 ms.
  3. Shine a resonant laser and take a shadow image on a CCD camera.

Key points:

  • During free expansion, each atom just keeps its momentum: \(r(t) = r(0) + \frac{p}{m}t\).
  • So the spatial distribution after a long time is basically the momentum distribution the atoms had at \(t=0\).

Atoms in:

  • the condensate have low momentum → they expand slowly → form a sharp, narrow peak in the image.
  • the thermal cloud have higher, Maxwell-like momenta → they expand more → form a broad background.

This is why BEC images show a tall, narrow central spike on top of a wide hill (Figures 7.8 and 7.9).

7. Distributions of condensed and non-condensed atoms

They then write more detailed formulas:

Condensed part

The ground-state wavefunction \(\psi_0(r,t)\) spreads in time (like a Gaussian wavepacket). Its density is

\[ n_0(r,t) = N_0 |\psi_0(r,t)|^2 \]

(eq. (11)), which gives a narrow anisotropic peak.

Excited part / thermal cloud

Treated semiclassically. At \(t=0\) the phase-space distribution is the usual Bose–Einstein form

\[ f(r,p,0) = \frac{1}{\exp\left[\frac{p^2}{2m} + V(r) - \mu\right]/kT - 1} \]

(eq. (12)).

After release, each point in phase space just shifts according to free motion, so \(f(r,p,t) = f\big(r + \tfrac{pt}{m}, p, 0\big)\) (eq. (13)).

Integrating over momenta gives the spatial density of excited atoms at time \(t\), \(n_{\text{excited}}(r,t)\) (eqs. (14)–(15)).

Important qualitative statement:

  • At early times (\(\omega_0 t \ll 1\)), both the condensed and thermal distributions reflect the anisotropy of the original trap.
  • At late times (\(\omega_0 t \gg 1\)), the thermal cloud becomes spherically symmetric (because its momentum distribution was isotropic), but the condensate remains anisotropic, reflecting its original squeezed shape. That's another clear signature of BEC.

Figure 7.8: theoretical time-of-flight density (condensed + excited). Figure 7.9: real experimental images showing the same evolution.

8. Thermodynamic properties of the condensate

Now they use the grand potential to compute energy and heat capacity.

Internal energy

They write the internal energy as

\[ U(\mu,T) = 3 \frac{(kT)^4}{(\hbar\omega_0)^3} g_4(z) \]

(eq. (17)).

Using the relation between \(z\), \(T\), and \(N\), we can express \(U\) in terms of \(T/T_c\).

Result:

\[ \frac{U}{NkT_c} = \begin{cases} 3\left(\dfrac{T}{T_c}\right)^4 \dfrac{\zeta(4)}{\zeta(3)} & T \le T_c,\\[6pt] 3\left(\dfrac{T}{T_c}\right)^4 \dfrac{g_4(z)}{\zeta(3)} & T \ge T_c, \end{cases} \]

(eq. (19)).

Figure 7.10(b) shows this scaled internal energy versus \(T/T_c\). Figure 7.12 compares it with experiments: the full curve and points agree quite well.

Heat capacity at constant \(N\)

Heat capacity is

\[ C_N(T) = \left(\frac{\partial U}{\partial T}\right)_N. \]

There's some algebra using the dependence of \(\mu\) (or \(z\)) on \(T\), which leads to a nice final expression

\[ \frac{C_N}{Nk} = \begin{cases} \displaystyle \frac{12\zeta(4)}{\zeta(3)}\left(\frac{T}{T_c}\right)^3 & T < T_c,\\[8pt] \displaystyle \frac{1}{\zeta(3)}\left(\frac{T}{T_c}\right)^3 \left[12 g_4(z) - 9\frac{g_3^2(z)}{g_2(z)}\right] & T > T_c, \end{cases} \]

(eq. (20)).

So:

  • For \(T<T_c\), \(C_N \propto T^3\).
  • Near \(T_c\) there is a kink / jump in \(C_N\) (eq. (21) gives the limiting values from below and above).

Figure 7.11 shows this: a pronounced peak in \(C_N\) at \(T_c\). Figure 7.12 shows experimental data for the internal energy and the change of slope around \(T_c\), consistent with the theoretical discontinuity in heat capacity.

9. Big picture

Putting it all together:

  • The harmonic trap quantizes the motion of bosonic atoms and changes their thermodynamics compared to a uniform gas.
  • Below a critical temperature \(T_c \propto \hbar\omega_0 N^{1/3}\), a macroscopic number of atoms fall into the lowest trap state: Bose–Einstein condensate.
  • The condensate fraction falls as \(1 - (T/T_c)^3\).
  • In real experiments, the BEC is seen as a narrow, anisotropic central peak in time-of-flight images, on top of a broad thermal background.
  • Thermodynamic quantities like internal energy and heat capacity show clear signatures (a kink or jump) at \(T_c\), and these match experiments quite well.