Macroscopic population of excited states for Bose-Einstein condensation?

I am currently learning about Bose-Einstein condensation (BEC). I understand that the ground state is rapidly populated when the temperature goes below the critical temperature. This macroscopic occupation of the ground state is the BEC.

However, most texts I have read immediately assert that all excess particles occupy the ground state. Is there a reason why the BEC cannot occupy the excited states?

• What do you mean by "excess particles"? – GiorgioP Mar 4 at 19:11

The expected boson occupation for a state with energy $$\epsilon_j$$ is given by

$$\langle n_j \rangle = \frac{1}{e^{\beta (\epsilon_j - \mu)}-1} = \frac{1}{z^{-1}e^{\beta\epsilon_j}-1} = \frac{z}{e^{\beta \epsilon_j}-z}$$

Here $$\beta = \frac{1}{kT}$$ and I have defined the fugacity $$z=e^{\beta\mu}$$ where $$\mu$$ is the chemical potential. Without changing the physics we can add an arbitrary offset to the energies so that the ground state energy, $$\epsilon_0 = 0$$. We see that for $$\langle n_j \rangle$$ to be positive it is necessary that $$0.

To understand BEC we must understand the behaviour of $$z$$ as a function of temperature. For concreteness I assume a fixed atom number $$N$$ and a 3D isotropic harmonic oscillator with frequency $$\omega_0$$. The energy levels are then space by $$\hbar \omega$$. It is then reasonable to define a dimensionless temperature $$\tilde{T} = \frac{kT}{\hbar \omega_0}$$. Below I plot $$z$$ as a function of $$\tilde{T}$$ for a non-interacting bosonic gas in a 3D harmonic oscillator potential for various atom numbers $$N$$. This plot shows fugacity $$z$$ versus temperature $$\tilde{T}$$ for atom number $$N = \{10^1, 10^2, 10^3, 10^4\}$$ from left to right. We see that as $$T$$ decreases $$z$$ increases linearly towards $$1$$ until $$T_c$$ at which point $$z$$ saturates to 1. As $$T$$ is lowered below $$T_c$$ $$z$$ still increases but now more slowly since it has saturated. The transition from linear growth to saturate becomes sharper and more "phase transition-y" as atom number $$N$$ is increased.

Let us now consider the ground state population.

$$\langle n_0 \rangle = \frac{z}{1-z}$$

We see that as $$z \rightarrow 1$$ that $$\langle n_0 \rangle$$ will become very large.

Now consider the first excited state population.

$$\langle n_1 \rangle = \frac{z}{e^{\frac{\epsilon_1}{kT}}-z}$$

I note that for experimental BECs the quantity $$\frac{\epsilon}{kT} \ll 1$$*. I will now consider two limits of this function, the $$T>T_c$$ and $$T limits.

For $$T>T_c$$ we have that $$z<1$$ so we can approximate $$e^{\beta \epsilon_1} \approx 1$$ and write

\begin{align} \langle n_1 \rangle \approx \frac{z}{1-z} \end{align}

As temperature decreases towards $$T_c$$ this function increases since $$z$$ increases towards $$1$$.

For $$T we have $$z\approx 1$$ and we can no longer approximate $$e^{\beta \epsilon_1} \approx 1$$ so we have

$$\langle n_1 \rangle \approx \frac{1}{e^{\beta \epsilon_1}-1} \approx \frac{1}{1+\beta \epsilon_1-1} \approx \frac{kT}{\epsilon_1}$$

We see that this function decreases as $$T$$ is decreased.

Thus we see that the excited state population $$\langle n_1 \rangle$$ decreases as $$T$$ is either increased or decreased away from $$T_c$$. Thus, the excited state population has a maximum at $$T=T_c$$. The question then of whether the excited state can be macroscopically occupied is a question of how large the excited state population is at the critical temperature. For a 3D Harmonic oscillator we have

$$kT_c \approx \hbar \omega N^{\frac{1}{3}}$$

So the fraction of atoms in the first excited states at the transition is

$$\frac{n_1}{N} \approx \frac{N^{\frac{1}{3}}}{N} = N^{-\frac{2}{3}}$$

So we see that the excited state fraction decreases as the total atom number $$N$$ increases and we move deeper and deeper into the thermodynamic limit. Below I plot the excited state fraction as a function of $$\frac{T}{T_c}$$ for $$N = \{10^2, 10^3, 10^4, 10^5\}$$ atoms. See W. Ketterle and N.J. van Druten, Phys. Rev. A 54, 656 at W. Ketterle and N.J. van Druten, Phys. Rev. A 54, 656 for a more thorough discussion of finite atom number effects in Bose-Einstein condensation.

*This is a critically important point about Bose-Einstein condensation. The energy corresponding to the critical temperature is MUCH larger than the energy corresponding to the first excited state. There is a trivial single particle effect which is that if you decrease the temperature so much that $$kT \ll \epsilon_i$$ then of course you expect to find most particles in the ground state. This would be true even of a classical gas of distinguishable particles. I emphatically point out that this is not the physics of BEC.