Bogoliubov Transformation and Bose Gas Spectrum

Hello! Welcome to the next lesson in our exploration of collective phenomena in quantum systems.

In our previous lesson, we derived the Gross-Pitaevskii equation (GPE). We saw that it provides a powerful mean-field description of the condensate itself—the macroscopic ground state of a weakly interacting Bose gas. However, the GPE doesn't explicitly describe the low-energy excitations that can exist on top of this condensate.

Today, we will delve into the nature of these elementary excitations. Our goal is to go beyond the mean-field ground state and understand the collective modes of the entire many-body system. This will lead us to one of the most important results in the theory of quantum gases and will provide the microscopic origin for superfluidity.

By the end of this lesson, you will be able to:

• Apply the Bogoliubov transformation to diagonalize the Hamiltonian for a weakly interacting Bose gas and analyze its spectrum.

This procedure is a cornerstone of many-body physics. You'll find the approach of diagonalizing a Hamiltonian to find quasiparticle excitations is conceptually analogous to the method we used to find the Bogoliubov quasiparticles in BCS theory, but applied here to a bosonic system.

1. The Hamiltonian for a Homogeneous Bose Gas

Let's begin with the Hamiltonian for a homogeneous gas of bosons interacting via a contact potential, which we introduced in the last lesson. In second quantization and working in momentum space, the Hamiltonian is:

$$\hat{H}=\sum _{\mathbf{k}}\frac{{\hslash }^2{k}^2}{2m}{\hat{a}}_{\mathbf{k}}^{†}{\hat{a}}_{\mathbf{k}}+\frac{g}{2V}\sum _{\mathbf{k},\mathbf{p},\mathbf{q}}{\hat{a}}_{\mathbf{p}+\mathbf{q}}^{†}{\hat{a}}_{\mathbf{k}-\mathbf{q}}^{†}{\hat{a}}_{\mathbf{k}}{\hat{a}}_{\mathbf{p}}$$

where $V$ is the volume and $g=4\pi {\hslash }^{2}a/m$ is the interaction strength.

The interaction term, with its four operators, makes the problem difficult. To proceed, we use the Bogoliubov approximation. At low temperatures, most particles occupy the zero-momentum ground state, forming the condensate. We can therefore treat the number of particles in the condensate, $N_0$, as being macroscopically large. This allows us to replace the operators for the condensate mode, ${\hat{a}}_0$ and ${\hat{a}}_0^{†}$, with a classical number, $\sqrt{N_0}$.

$${\hat{a}}_0,{\hat{a}}_0^{†}⟶\sqrt{N_0}$$

We also assume that the number of particles in excited states ($\mathbf{k}\ne 0$) is small compared to $N_0$, so we can approximate $N_0\approx N$, the total number of particles.

Applying this approximation means we systematically expand the four-operator interaction term and keep only the terms that are at most quadratic in the excited state operators ${\hat{a}}_{\mathbf{k}\ne 0}$. The dominant contributions come from processes where two of the four operators are condensate operators. This simplifies the Hamiltonian dramatically.

The following resource provides a detailed and clear derivation of the approximated Hamiltonian.

After performing the approximation, the Hamiltonian takes the form (ignoring constant energy terms for a moment):

$$\hat{H}\approx \sum _{\mathbf{k}\ne 0}\left(\frac{{\hslash }^2{k}^2}{2m}+g{n}_0\right){\hat{a}}_{\mathbf{k}}^{†}{\hat{a}}_{\mathbf{k}}+\frac{1}{2}g{n}_0\sum _{\mathbf{k}\ne 0}\left({\hat{a}}_{\mathbf{k}}^{†}{\hat{a}}_{-\mathbf{k}}^{†}+{\hat{a}}_{\mathbf{k}}{\hat{a}}_{-\mathbf{k}}\right)$$

where $n_0=N_0/V$ is the condensate density.

Notice the structure of this Hamiltonian. It has a diagonal part ($\sim {\hat{a}}_{\mathbf{k}}^{†}{\hat{a}}_{\mathbf{k}}$), which looks like a number operator for free particles with a shifted energy. However, it also has crucial off-diagonal terms: ${\hat{a}}_{\mathbf{k}}^{†}{\hat{a}}_{-\mathbf{k}}^{†}$, which creates a pair of particles with opposite momenta from the condensate, and ${\hat{a}}_{\mathbf{k}}{\hat{a}}_{-\mathbf{k}}$, which annihilates such a pair. These terms mean that the original "bare" particles are not the true energy eigenstates of the system.

2. Diagonalization via the Bogoliubov Transformation

To find the true elementary excitations, we need to diagonalize this Hamiltonian. We do this with the Bogoliubov transformation, which defines a new set of bosonic operators, ${\hat{b}}_{\mathbf{k}}$, as a linear combination of the old ones:

$$\begin{gathered} {\hat{a}}_{\mathbf{k}}=u_k{\hat{b}}_{\mathbf{k}}+v_k{\hat{b}}_{-\mathbf{k}}^{†} \\ {\hat{a}}_{\mathbf{k}}^{†}=u_k{\hat{b}}_{\mathbf{k}}^{†}+v_k{\hat{b}}_{-\mathbf{k}} \end{gathered}$$

The coefficients $u_k$ and $v_k$ are real functions that we will choose to diagonalize $\hat{H}$. To ensure that the new operators ${\hat{b}}_{\mathbf{k}}$ also obey bosonic commutation relations, $[{\hat{b}}_{\mathbf{k}},{\hat{b}}_{{\mathbf{k}}^{\prime }}^{†}]={\delta }_{\mathbf{k},{\mathbf{k}}^{\prime }}$, we must impose the condition:

$$u_k^2-v_k^2=1$$

This condition is automatically satisfied by parametrizing them as $u_k=\cosh ({\theta }_k)$ and $v_k=\sinh ({\theta }_k)$.

The process is now straightforward:

1. Substitute the transformation into the quadratic Hamiltonian.
2. Collect terms proportional to ${\hat{b}}_{\mathbf{k}}^{†}{\hat{b}}_{\mathbf{k}}$, ${\hat{b}}_{\mathbf{k}}^{†}{\hat{b}}_{-\mathbf{k}}^{†}$, and ${\hat{b}}_{\mathbf{k}}{\hat{b}}_{-\mathbf{k}}$.
3. Choose $u_k$ and $v_k$ (or ${\theta }_k$) such that the coefficients of the off-diagonal terms, ${\hat{b}}_{\mathbf{k}}^{†}{\hat{b}}_{-\mathbf{k}}^{†}$ and ${\hat{b}}_{\mathbf{k}}{\hat{b}}_{-\mathbf{k}}$, are zero.

The resource you just read carries out this procedure explicitly in section 4.3 (equations 4.20 through 4.28). After this algebraic work, the Hamiltonian becomes perfectly diagonal:

$$\hat{H}=E_0+\sum _{\mathbf{k}\ne 0}{E}_k{\hat{b}}_{\mathbf{k}}^{†}{\hat{b}}_{\mathbf{k}}$$

This is a remarkable result. Our complex system of interacting bosons is now described as a simple gas of non-interacting quasiparticles (created by ${\hat{b}}_{\mathbf{k}}^{†}$), each with energy $E_k$.

3. The Bogoliubov Excitation Spectrum

The energy of these quasiparticle excitations—the Bogoliubov spectrum—is found to be:

$$E_k=\sqrt{{\left(\frac{{\hslash }^2{k}^2}{2m}\right)}^2+2g{n}_0\left(\frac{{\hslash }^2{k}^2}{2m}\right)}=\sqrt{{ϵ}_k^0({ϵ}_k^0+2g{n}_0)}$$

where ${ϵ}_k^0={\hslash }^2{k}^2/2m$ is the kinetic energy of a free particle. This dispersion relation is the key physical prediction of the theory.

To understand its meaning, let's analyze its behavior in two limits, as detailed in section 4.4.3 of the LINK handout.

Low Momentum Limit (${ϵ}_k^0\ll g{n}_0$)

In this limit, the second term under the square root dominates:

$$E_k\approx \sqrt{2g{n}_0\left(\frac{{\hslash }^2{k}^2}{2m}\right)}=\left(\sqrt{\frac{g{n}_0}{m}}\right)\hslash k=c_s\hslash k$$

This is a linear, phonon-like dispersion relation. The excitations are collective density waves—sound waves—propagating through the condensate. The slope, $c_s=\sqrt{g{n}_0/m}$, is precisely the speed of sound.

High Momentum Limit (${ϵ}_k^0\gg g{n}_0$)

In this limit, the first term under the square root dominates:

$$E_k\approx \sqrt{{\left(\frac{{\hslash }^2{k}^2}{2m}\right)}^2}=\frac{{\hslash }^2{k}^2}{2m}+g{n}_0$$

(Note: A more careful expansion gives the $+g{n}_0$ shift). This recovers the quadratic dispersion of a free particle, shifted by the mean-field interaction energy. At high energies, the excitations behave like individual particles punching through the condensate, feeling only its average potential.

The following video provides an excellent summary and connects this theoretical spectrum directly to experiments.

The crossover from linear to quadratic dispersion defines a characteristic length scale, the healing length $\xi =\hslash /\sqrt{2mg{n}_0}$, which we encountered in the last lesson. For momenta $k\ll 1/\xi$, the excitations are collective phonons; for $k\gg 1/\xi$, they are particle-like.

4. Physical Consequences

The Bogoliubov theory gives us more than just the energy spectrum; it changes our picture of the ground state and provides the microscopic origin of superfluidity.

Quantum Depletion

What is the ground state of our interacting system? It's the vacuum of the quasiparticles: ${\hat{b}}_{\mathbf{k}}|G⟩=0$ for all $\mathbf{k}\ne 0$. But how many real particles are in excited states ($\mathbf{k}\ne 0$) in this ground state? We can calculate the expectation value:

$$⟨{\hat{n}}_{\mathbf{k}}⟩=⟨G|{\hat{a}}_{\mathbf{k}}^{†}{\hat{a}}_{\mathbf{k}}|G⟩=⟨G|(u_k{\hat{b}}_{\mathbf{k}}^{†}+v_k{\hat{b}}_{-\mathbf{k}})(u_k{\hat{b}}_{\mathbf{k}}+v_k{\hat{b}}_{-\mathbf{k}}^{†})|G⟩=v_k^2$$

Since $v_k^2>0$, this means that even at absolute zero temperature, there is a finite population of real particles in excited momentum states. This effect is called quantum depletion. The interactions scatter some particles out of the condensate, "depleting" it. This is a pure quantum many-body effect, absent in a non-interacting gas. As discussed in the video segment from 07:01 to 10:06, this depletion is small (~1%) in dilute atomic gases but can be very large (~90%) in a strongly interacting system like liquid Helium-4.

Test your understanding!

Consider the expression for the Bogoliubov coefficients $u_k$ and $v_k$ from equation (4.26) in resource LINK. In the low momentum limit ($k\to 0$), how do $u_k$ and $v_k$ behave? What does this imply about the relationship between a real particle and the quasiparticles in this limit?

Show answer

In the limit $k\to 0$, the quasiparticle energy $E_k\to 0$. The expression for $v_k^2$ becomes:
$v_k^2=\frac{1}{2}\left(\frac{{ϵ}_k^0+g{n}_0}{E_k}-1\right)\approx \frac{1}{2}\left(\frac{g{n}_0}{c_s\hslash k}\right)\sim \frac{1}{k}$
Both $v_k^2$ and $u_k^2=1+v_k^2$ diverge as $k\to 0$. This means that a single low-momentum particle (${\hat{a}}_k$) is a superposition of a very large number of quasiparticles. The low-energy excitations are truly collective, involving coherent motion of many underlying atoms.

Superfluidity and Landau's Criterion

The shape of the Bogoliubov spectrum is the key to understanding superfluidity. According to Landau's criterion, an object moving through a fluid at velocity $\mathbf{v}$ can only dissipate energy by creating an excitation if it is energetically favorable. This condition is $v>v_c$, where the critical velocity $v_c$ is given by:

$$v_c=\underset{\mathbf{k}}{\min }\frac{E_k}{\hslash k}$$

For a gas of free particles, $E_k={\hslash }^2{k}^2/2m$, so $E_k/(\hslash k)=\hslash k/2m$, and the minimum is zero. Any velocity can create an excitation. However, for the Bogoliubov spectrum, the ratio $E_k/(\hslash k)$ has a non-zero minimum, which is the speed of sound, $c_s$.
Therefore, as long as an object moves slower than the speed of sound ($v<c_s$), it cannot create excitations and thus moves without friction. This is superfluidity.

Conclusion

In this lesson, we have moved beyond the static mean-field picture of a BEC to understand its dynamic properties. By applying the Bogoliubov approximation and transformation, we diagonalized the Hamiltonian for a weakly interacting Bose gas.

Key Takeaways:

• The Bogoliubov transformation allows us to reinterpret an interacting system of bosons as a non-interacting gas of quasiparticles.
• The Bogoliubov spectrum $E_k$ for these quasiparticles is linear (phonon-like) at low momenta and quadratic (particle-like) at high momenta.
• The linear part of the spectrum corresponds to sound waves, and its slope defines the speed of sound $c_s$.
• The shape of the spectrum provides a microscopic justification for superfluidity via Landau's criterion, with $v_c=c_s$.
• Even at $T=0$, interactions cause quantum depletion, populating momentum states outside the condensate.

Preview of the Next Lesson:

We have now examined two monumental examples of collective quantum phenomena: superconductivity in fermion systems and superfluidity in boson systems. We will now shift our focus to a different, but equally profound, collective effect in a two-dimensional electron gas subjected to a strong magnetic field: the Integer Quantum Hall effect. We will discover how principles of topology lead to the emergence of perfectly quantized plateaus in conductivity, and we will introduce the Chern number as the underlying topological invariant.