5. Application 1: Soliton dynamics

In this section, we will use the EFT that was developed in Section 4 to study the properties of dark solitons in Fermi superfluids.

#### 5.1. What is a dark soliton?

Solitons are nonlinear solitary waves that maintain their shape while propagating through a medium at a constant velocity. They are found as the solution of nonlinear wave equations and emerge in a wide variety of physical systems, including optical fibers, classical fluids, and plasmas. More recently, they have also become a subject of interest in superfluid quantum gases [19–23]. In these systems, solitons appear most often in the form of dark solitons, which are characterized by a localized density dip in the uniform background and a jump in the phase profile of the order parameter. The magnitude of this density dip and phase jump are intrinsically connected to the velocity vs with which the soliton propagates through the superfluid, as illustrated in Figure 4. The higher the soliton velocity, the smaller the phase jump and

Figure 4. Example of the density profile (upper row) and phase profile (lower row) of a dark soliton for different soliton velocities vs relative to the critical velocity vc.

soliton depth become. Above a certain critical velocity vc, the phase jump and density dip will disappear completely and a dark soliton solution no longer exists.

#### 5.2. Solution for a one-dimensional dark soliton

For the case of a dark soliton in a one-dimensional (1D) Fermi superfluid with a uniform background, the EFT provides an exact analytical solution for the pair field [24]. To describe the dynamics of the system, it is necessary to move from the imaginary-time action functional (30) to the real-time one, using the formal replacements.

$$
\pi \to \text{it} \tag{39}
$$

H ¼ Ω<sup>s</sup> þ

value at the position of the soliton:

with

x<sup>0</sup> ¼ x � vst and t

be written as

C 2m

convenient to write the pair field Ψð Þ x; t as

<sup>L</sup> ¼ � <sup>κ</sup>ð Þ<sup>a</sup> <sup>a</sup><sup>2</sup> <sup>∂</sup><sup>θ</sup>

∂t

<sup>þ</sup> <sup>Q</sup> � <sup>4</sup>Rj j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> <sup>a</sup><sup>2</sup> � �j j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> <sup>∂</sup><sup>a</sup>

ρqpð Þ¼ a

ρsfð Þ¼ a

along with the soliton and has its origin at the soliton center. It follows that

f xð Þ¼ � vst f x<sup>0</sup> ð Þ, <sup>∂</sup>

C <sup>m</sup> j j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> <sup>a</sup><sup>2</sup>

<sup>∇</sup>x<sup>Ψ</sup> � <sup>∇</sup>x<sup>Ψ</sup> � � � <sup>E</sup>

2m

<sup>∇</sup>xj j <sup>Ψ</sup> <sup>2</sup> � �<sup>2</sup>

As mentioned above, a dark soliton in a superfluid is mainly characterized by a jump in the phase profile and a dip in the amplitude profile of the order parameter. Therefore, it is

Moreover, since a soliton is a localized perturbation, we write the modulus as a product of the constant background value ∣Ψ∞∣ and a relative amplitude að Þ x; t that modifies the background

2

∂t � �<sup>2</sup>

<sup>κ</sup>ð Þ¼ <sup>a</sup> D að Þj j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup>

<sup>C</sup> � <sup>4</sup>Ej j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> <sup>a</sup><sup>2</sup>

Here, we added Ωsð Þ a<sup>∞</sup> to the original Lagrangian to obtain a regularized Lagrangian density in which energy values are considered with respect to the energy of the uniform system. The superfluid density ρsf determines how much the pair condensate resists gradients in its phase field, while the quantum pressure ρqp is a consequence of the fact that the condensate also resists gradients in the pair density. We will further limit ourselves to a 1D problem in which the soliton propagates with constant speed vs in the x-direction on a uniform background. This assumption can be implemented through the condition that the space-time dependence of the pair field satisfies the relation f xð Þ¼ ; t f xð Þ � vst . We then perform a change of variables

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>∂</sup>

If we further drop the primes, the Lagrangian density (46) in the soliton frame of reference can

<sup>m</sup> j j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup>

<sup>0</sup> ¼ t, corresponding to a transformation to the frame of reference that moves

∂t ¼ ∂ ∂t <sup>0</sup> � vs

<sup>∂</sup>x<sup>0</sup> , <sup>∂</sup>

ρqpð Þa ð Þ ∇xa

<sup>þ</sup> <sup>Q</sup>j j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> <sup>a</sup><sup>2</sup> <sup>∂</sup><sup>θ</sup>

Ψð Þ¼ x; t ∣Ψð Þ x; t ∣e

Substituting this form for the pair field in the field Lagrangian (42), we find

� <sup>½</sup>Ωsð Þ� <sup>a</sup> <sup>Ω</sup>sð Þ <sup>a</sup><sup>∞</sup> � � <sup>1</sup>

<sup>þ</sup> <sup>Q</sup> <sup>∂</sup><sup>Ψ</sup> ∂t ∂Ψ

<sup>∂</sup><sup>t</sup> � <sup>R</sup> <sup>∂</sup>j j <sup>Ψ</sup> <sup>2</sup> ∂t !<sup>2</sup>

An Effective Field Description for Fermionic Superfluids http://dx.doi.org/10.5772/intechopen.73058

<sup>i</sup>θð Þ <sup>x</sup>;<sup>t</sup> : (44)

<sup>ρ</sup>sfð Þ<sup>a</sup> ð Þ <sup>∇</sup>x<sup>θ</sup> <sup>2</sup>

, (47)

: (49)

∂

<sup>∂</sup>x<sup>0</sup> : (50)

, (48)

∣Ψð Þ x; t ∣ ¼ ∣Ψ∞∣að Þ x; t : (45)

∂t � �<sup>2</sup> , : (43)

19

(46)

$$\mathcal{S}\_{\text{FFT}}(\boldsymbol{\beta}) \to -i\,\mathcal{S}\_{\text{FFT}}(t\_b, t\_a). \tag{40}$$

From the relation between the real-time action functional and the Lagrangian density L,

$$S\_{\rm EFT}(t\_b, t\_a) = \int\_{t\_a}^{t\_b} \mathbf{d}t \int \mathbf{dx} \mathcal{L}\_\prime \tag{41}$$

we subsequently find the following expression for L:

$$\begin{split} \mathcal{L} = i \frac{D}{2} \left( \overline{\Psi} \frac{\partial \Psi}{\partial t} - \frac{\partial \overline{\Psi}}{\partial t} \Psi \right) - \Omega\_s - \frac{\mathbb{C}}{2m} \left( \nabla\_\mathbf{x} \overline{\Psi} \cdot \nabla\_\mathbf{x} \Psi \right) + \frac{E}{2m} \left( \nabla\_\mathbf{x} |\Psi|^2 \right)^2 \\ + Q \frac{\partial \overline{\Psi}}{\partial t} \frac{\partial \Psi}{\partial t} - R \left( \frac{\partial |\Psi|^2}{\partial t} \right)^2, \end{split} \tag{42}$$

where the Hamiltonian density H is defined as

An Effective Field Description for Fermionic Superfluids http://dx.doi.org/10.5772/intechopen.73058 19

$$\mathcal{H} = \Omega\_\text{s} + \frac{\mathbb{C}}{2m} \left( \nabla\_\mathbf{x} \overline{\Psi} \cdot \nabla\_\mathbf{x} \Psi \right) - \frac{E}{2m} \left( \nabla\_\mathbf{x} |\Psi|^2 \right)^2 + Q \frac{\partial \overline{\Psi}}{\partial t} \frac{\partial \Psi}{\partial t} - R \left( \frac{\partial |\Psi|^2}{\partial t} \right)^2 \,. \tag{43}$$

As mentioned above, a dark soliton in a superfluid is mainly characterized by a jump in the phase profile and a dip in the amplitude profile of the order parameter. Therefore, it is convenient to write the pair field Ψð Þ x; t as

$$
\Psi(\mathbf{x},t) = |\Psi(\mathbf{x},t)|e^{i\theta(\mathbf{x},t)}.\tag{44}
$$

Moreover, since a soliton is a localized perturbation, we write the modulus as a product of the constant background value ∣Ψ∞∣ and a relative amplitude að Þ x; t that modifies the background value at the position of the soliton:

$$|\Psi(\mathbf{x},t)| = |\Psi\_{\approx}|a(\mathbf{x},t). \tag{45}$$

Substituting this form for the pair field in the field Lagrangian (42), we find

$$\begin{split} \mathcal{L} &= -\kappa(a)a^2 \frac{\partial \theta}{\partial t} - \left[\Omega\_s(a) - \Omega\_s(a\_\circ)\right] - \frac{1}{2}\rho\_{\rm qp}(a)(\nabla\_\mathbf{x}a)^2 - \frac{1}{2}\rho\_{\rm st}(a)(\nabla\_\mathbf{x}\theta)^2 \\ &+ \left(Q - 4R|\Psi\_\circ|^2a^2\right)|\Psi\_\circ|^2\left(\frac{\partial a}{\partial t}\right)^2 + Q|\Psi\_\circ|^2a^2\left(\frac{\partial \theta}{\partial t}\right)^2, \end{split} \tag{46}$$

with

soliton depth become. Above a certain critical velocity vc, the phase jump and density dip will

Figure 4. Example of the density profile (upper row) and phase profile (lower row) of a dark soliton for different soliton

For the case of a dark soliton in a one-dimensional (1D) Fermi superfluid with a uniform background, the EFT provides an exact analytical solution for the pair field [24]. To describe the dynamics of the system, it is necessary to move from the imaginary-time action functional

τ ! it (39)

dxL, (41)

<sup>∇</sup>xj j <sup>Ψ</sup> <sup>2</sup> � �<sup>2</sup>

(42)

� � ! �iSEFTð Þ tb; ta : (40)

E 2m

disappear completely and a dark soliton solution no longer exists.

SEFT β

From the relation between the real-time action functional and the Lagrangian density L,

� <sup>Ω</sup><sup>s</sup> � <sup>C</sup> 2m

ðtb ta dt ð

<sup>∂</sup><sup>t</sup> � <sup>R</sup> <sup>∂</sup>j j <sup>Ψ</sup> <sup>2</sup> ∂t !<sup>2</sup>

<sup>∇</sup>x<sup>Ψ</sup> � <sup>∇</sup>x<sup>Ψ</sup> � � <sup>þ</sup>

,

SEFTð Þ¼ tb; ta

<sup>þ</sup><sup>Q</sup> <sup>∂</sup><sup>Ψ</sup> ∂t ∂Ψ

5.2. Solution for a one-dimensional dark soliton

velocities vs relative to the critical velocity vc.

18 Superfluids and Superconductors

(30) to the real-time one, using the formal replacements.

we subsequently find the following expression for L:

<sup>Ψ</sup> <sup>∂</sup><sup>Ψ</sup> <sup>∂</sup><sup>t</sup> � <sup>∂</sup><sup>Ψ</sup> ∂t Ψ

where the Hamiltonian density H is defined as

� �

L ¼ i D 2

$$\kappa(a) = D(a) \left| \Psi\_{\simeq} \right|^2,\tag{47}$$

$$\rho\_{\rm qp}(a) = \frac{\mathbb{C} - 4E|\Psi\_{\simeq}|^2 a^2}{m} \left| \Psi\_{\simeq} \right|^2 \,\,\,\,\,\tag{48}$$

$$\rho\_{\rm sf}(a) = \frac{\mathbb{C}}{m} |\Psi\_{\rm co}|^2 a^2. \tag{49}$$

Here, we added Ωsð Þ a<sup>∞</sup> to the original Lagrangian to obtain a regularized Lagrangian density in which energy values are considered with respect to the energy of the uniform system. The superfluid density ρsf determines how much the pair condensate resists gradients in its phase field, while the quantum pressure ρqp is a consequence of the fact that the condensate also resists gradients in the pair density. We will further limit ourselves to a 1D problem in which the soliton propagates with constant speed vs in the x-direction on a uniform background. This assumption can be implemented through the condition that the space-time dependence of the pair field satisfies the relation f xð Þ¼ ; t f xð Þ � vst . We then perform a change of variables x<sup>0</sup> ¼ x � vst and t <sup>0</sup> ¼ t, corresponding to a transformation to the frame of reference that moves along with the soliton and has its origin at the soliton center. It follows that

$$f(\mathbf{x} - \boldsymbol{\upsilon}\_s t) = f(\mathbf{x}'), \qquad \frac{\partial}{\partial \mathbf{x}} = \frac{\partial}{\partial \mathbf{x}'}, \qquad \frac{\partial}{\partial t} = \frac{\partial}{\partial t'} - \boldsymbol{\upsilon}\_s \frac{\partial}{\partial \mathbf{x}'}.\tag{50}$$

If we further drop the primes, the Lagrangian density (46) in the soliton frame of reference can be written as

$$\mathcal{L} = \kappa(a)a^2 \upsilon\_s \frac{\partial \theta}{\partial \mathbf{x}} - \left[\Omega\_s(a) - \Omega\_s(a\_\circ)\right] - \frac{1}{2}\tilde{\rho}\_{\rm op}(a) \left(\frac{\partial a}{\partial \mathbf{x}}\right)^2 - \frac{1}{2}\tilde{\rho}\_{st}(a) \left(\frac{\partial \theta}{\partial \mathbf{x}}\right)^2. \tag{51}$$

with the modified superfluid density and quantum pressure

$$\tilde{\rho}\_{\rm qp}(a) = \frac{\mathbb{C} - 4E|\Psi\_{\circ\circ}|^2 a^2}{m} |\Psi\_{\circ\circ}|^2 - 2\left(\mathbb{Q} - 4\mathbb{R}|\Psi\_{\circ\circ}|^2 a^2\right) |\Psi\_{\circ\circ}|^2 v\_{s'}^2 \tag{52}$$

$$
\tilde{\rho}\_{s\xi}(a) = \frac{\mathbb{C}}{m} |\Psi\_{\circ\circ}|^2 a^2 - 2Q |\Psi\_{\circ\circ}|^2 a^2 v\_s^2. \tag{53}
$$

From the above expression for Lð Þ a; θ , we can now find the equations of motion for the relative amplitude field a xð Þ and the phase field θð Þx :

$$\frac{\partial}{\partial t} \left( \frac{\partial \mathcal{L}}{\partial (\partial\_l a)} \right) + \frac{\partial}{\partial x} \left( \frac{\partial \mathcal{L}}{\partial (\partial\_x a)} \right) = \frac{\partial \mathcal{L}}{\partial a},\tag{54}$$

X að Þ¼ Ωsð Þ� a Ωsð Þ a<sup>∞</sup> , (62)

<sup>¼</sup> 0 and a xð Þj<sup>x</sup>!�<sup>∞</sup> <sup>¼</sup> <sup>1</sup>, (65)

X að Þ� <sup>v</sup><sup>2</sup>

<sup>2</sup>ρ~sfð Þ<sup>a</sup> , (63)

An Effective Field Description for Fermionic Superfluids http://dx.doi.org/10.5772/intechopen.73058 21

<sup>s</sup>Y að Þ � �: (64)

<sup>s</sup>Y að Þ, (66)

sY að Þ, (67)

<sup>s</sup>Y að Þ¼ <sup>0</sup> 0: (69)

, the temperature T=TF, the imbalance

: (68)

Y að Þ¼ <sup>κ</sup>ð Þ<sup>a</sup> <sup>a</sup><sup>2</sup> � <sup>κ</sup><sup>∞</sup>

∂2 a <sup>∂</sup>x<sup>2</sup> <sup>¼</sup> <sup>∂</sup> ∂a

While the above equation does not allow for a straightforward solution for a as a function of the position x, it can be solved for x as a function of a instead. Using the boundary conditions

<sup>¼</sup> X að Þ� <sup>v</sup><sup>2</sup>

ρ~qpð Þa X að Þ� v<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>ρ</sup>~qp <sup>a</sup><sup>0</sup> ð Þ <sup>q</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X a<sup>0</sup> ð Þ� v<sup>2</sup> <sup>s</sup>Y a<sup>0</sup> ð Þ <sup>p</sup> da<sup>0</sup>

þ ρ~qpð Þa

∂a ∂x � �<sup>2</sup>

1 ffiffiffi 2 p ða a0

X að Þ� <sup>0</sup> <sup>v</sup><sup>2</sup>

¼ 1 2

Here, a<sup>0</sup> ¼ a xð Þ ¼ 0 is the relative amplitude at the center of the soliton, which is found as the

chemical potential ζ, and the soliton velocity vs, formulae (60) and (68) allow us to calculate the complete pair field profile of the dark soliton. For example, the soliton density and phase

The dark soliton solution derived in the previous section has been employed in the description of various soliton phenomena in superfluid Fermi gases. For instance, adding a small twodimensional perturbation to the exact 1D solution allows for a description of the snake instability mechanism [25], which makes the soliton decay into vortices if the radial width of the system is too large [23, 26]. We have also studied collisions between dark solitons by numerically evolving two counter-propagating 1D solitons in time [27]. As an example of an

we find

for a dark soliton

solution of

1 2 ∂ρ~qp ∂a

we find that (64) can be integrated, yielding:

∂a ∂x � �<sup>2</sup>

∂a ∂x � � � � x!�∞

> 1 2 ρ~qpð Þa

> > ⇔ ∂x ∂a � �<sup>2</sup>

⇔ x ¼ �

For given values of the interaction parameter ð Þ kFas �<sup>1</sup>

5.3. Dark solitons in imbalanced Fermi gases

profiles in Figure 4 were calculated using the above expressions.

� �<sup>2</sup>

$$\frac{\partial}{\partial t} \left( \frac{\partial \mathcal{L}}{\partial (\partial\_l \theta)} \right) + \frac{\partial}{\partial x} \left( \frac{\partial \mathcal{L}}{\partial (\partial\_x \theta)} \right) = \frac{\partial \mathcal{L}}{\partial \theta}. \tag{55}$$

Starting with the equation for the phase field, we easily find:

$$\frac{\partial}{\partial \mathbf{x}} \left( \kappa(a) a^2 v\_s - \tilde{\rho}\_{st}(a) \frac{\partial \theta}{\partial \mathbf{x}} \right) = 0 \tag{56}$$

$$
\Leftrightarrow \quad \frac{\partial \theta}{\partial \mathbf{x}} = \frac{\kappa(a)a^2 v\_s + \alpha}{\tilde{\rho}\_{\text{sf}}(a)}. \tag{57}
$$

The integration constant α can be determined through the boundary condition for a dark soliton:

$$\frac{\partial \theta}{\partial \mathbf{x}} \to 0 \quad \text{for} \quad \mathbf{x} \to \pm \infty. \tag{58}$$

which yields α ¼ �vsκ<sup>∞</sup> with κ<sup>∞</sup> ¼ κð Þ a<sup>∞</sup> and thus

$$\frac{\partial \Theta}{\partial \mathbf{x}} = \frac{\upsilon\_{\rm s}}{\check{\rho}\_{\rm sf}(a)} \left( \kappa(a) a^2 - \kappa\_{\rm ss} \right). \tag{59}$$

If we set θð Þ¼ �∞ 0, the phase profile of the superfluid is given by

$$\Theta(\mathbf{x}) = \upsilon\_s \int\_{-\infty}^{\mathbf{x}} \frac{\kappa(a(\mathbf{x}')) \, a^2(\mathbf{x}') - \kappa\_{\ast\ast}}{\tilde{\rho}\_{\ast\mathbf{f}}(a(\mathbf{x}'))} \, d\mathbf{x}'. \tag{60}$$

Next, we derive the equation of motion for a xð Þ:

$$\frac{\partial}{\partial \mathbf{x}} \left( -\tilde{\rho}\_{\mathbf{q}\mathbf{p}}(a) \frac{\partial a}{\partial \mathbf{x}} \right) = \frac{\partial}{\partial a} \left( \kappa(a) a^2 \right) \upsilon\_s \frac{\partial \theta}{\partial \mathbf{x}} - \frac{\partial \Omega\_s}{\partial a} - \frac{1}{2} \frac{\partial \tilde{\rho}\_{\mathbf{q}\mathbf{p}}}{\partial a} \left( \frac{\partial a}{\partial \mathbf{x}} \right)^2 - \frac{1}{2} \frac{\partial \tilde{\rho}\_{\mathbf{st}}}{\partial a} \left( \frac{\partial \theta}{\partial \mathbf{x}} \right)^2. \tag{61}$$

Inserting the solution for the derivative of the phase field (59) and defining

An Effective Field Description for Fermionic Superfluids http://dx.doi.org/10.5772/intechopen.73058 21

$$X(a) = \Omega\_{\sf s}(a) - \Omega\_{\sf s}(a\_{\sf s}),\tag{62}$$

$$Y(a) = \frac{\left(\kappa(a)a^2 - \kappa\_{\ast\ast}\right)^2}{2\tilde{\rho}\_{st}(a)},\tag{63}$$

we find

<sup>L</sup> <sup>¼</sup> <sup>κ</sup>ð Þ<sup>a</sup> <sup>a</sup><sup>2</sup>

20 Superfluids and Superconductors

vs ∂θ

ρ~qpð Þ¼ a

ρ~sfð Þ¼ a

amplitude field a xð Þ and the phase field θð Þx :

with the modified superfluid density and quantum pressure

C

<sup>C</sup> � <sup>4</sup>Ej j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> <sup>a</sup><sup>2</sup>

∂ ∂t

∂ ∂t

Starting with the equation for the phase field, we easily find:

which yields α ¼ �vsκ<sup>∞</sup> with κ<sup>∞</sup> ¼ κð Þ a<sup>∞</sup> and thus

Next, we derive the equation of motion for a xð Þ:

∂a ∂x ¼ ∂ ∂a

∂

<sup>∂</sup><sup>x</sup> �ρ~qpð Þ<sup>a</sup>

� �

∂ ∂x

⇔

∂θ

∂θ <sup>∂</sup><sup>x</sup> <sup>¼</sup> vs ρ~sfð Þa

If we set θð Þ¼ �∞ 0, the phase profile of the superfluid is given by

θð Þ¼ x vs

ðx �∞

<sup>κ</sup>ð Þ<sup>a</sup> <sup>a</sup><sup>2</sup> � �vs

Inserting the solution for the derivative of the phase field (59) and defining

∂θ <sup>∂</sup><sup>x</sup> � <sup>∂</sup>Ω<sup>s</sup>

<sup>∂</sup><sup>x</sup> � <sup>½</sup>Ωsð Þ� <sup>a</sup> <sup>Ω</sup>sð Þ <sup>a</sup><sup>∞</sup> � � <sup>1</sup>

<sup>m</sup> j j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> <sup>a</sup><sup>2</sup> � <sup>2</sup>Qj j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> <sup>a</sup><sup>2</sup>

∂L ∂ ∂ð Þ ta � �

∂L ∂ ∂ð Þ <sup>t</sup>θ � �

<sup>κ</sup>ð Þ<sup>a</sup> <sup>a</sup><sup>2</sup>

∂θ

2 ρ~qpð Þa

<sup>m</sup> j j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> � <sup>2</sup> <sup>Q</sup> � <sup>4</sup>Rj j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> <sup>a</sup><sup>2</sup> � �

v2

∂L ∂ ∂ð Þ xa � �

∂L ∂ ∂ð Þ <sup>x</sup>θ � �

> ∂θ ∂x

<sup>¼</sup> <sup>∂</sup><sup>L</sup>

<sup>¼</sup> <sup>∂</sup><sup>L</sup>

From the above expression for Lð Þ a; θ , we can now find the equations of motion for the relative

þ ∂ ∂x

þ ∂ ∂x

vs � ρ~sfð Þa

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>κ</sup>ð Þ<sup>a</sup> <sup>a</sup><sup>2</sup>vs <sup>þ</sup> <sup>α</sup>

<sup>κ</sup>ð Þ<sup>a</sup> <sup>a</sup><sup>2</sup> � <sup>κ</sup><sup>∞</sup>

<sup>κ</sup> a x<sup>0</sup> ð Þ ð Þ <sup>a</sup><sup>2</sup> <sup>x</sup><sup>0</sup> ð Þ� <sup>κ</sup><sup>∞</sup>

<sup>∂</sup><sup>a</sup> � <sup>1</sup> 2 ∂ρ~qp ∂a

<sup>ρ</sup>~sf a x<sup>0</sup> ð Þ ð Þ dx<sup>0</sup>

The integration constant α can be determined through the boundary condition for a dark soliton:

� �

∂a ∂x � �<sup>2</sup>

� 1 2 ρ~sfð Þa

> j j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> v2

<sup>s</sup> : (53)

<sup>∂</sup><sup>a</sup> , (54)

<sup>∂</sup><sup>θ</sup> : (55)

¼ 0 (56)

<sup>ρ</sup>~sfð Þ<sup>a</sup> : (57)

� �: (59)

� 1 2 ∂ρ~sf ∂a

: (60)

∂θ ∂x � �<sup>2</sup>

: (61)

<sup>∂</sup><sup>x</sup> ! 0 for <sup>x</sup> ! �∞: (58)

∂a ∂x � �<sup>2</sup>

∂θ ∂x � �<sup>2</sup>

: (51)

<sup>s</sup> , (52)

$$\frac{1}{2}\frac{\partial\tilde{\rho}\_{\rm qp}}{\partial a}\left(\frac{\partial a}{\partial x}\right)^2 + \tilde{\rho}\_{\rm qp}(a)\frac{\partial^2 a}{\partial x^2} = \frac{\partial}{\partial a}\left(\mathcal{X}(a) - v\_s^2 \mathcal{Y}(a)\right). \tag{64}$$

While the above equation does not allow for a straightforward solution for a as a function of the position x, it can be solved for x as a function of a instead. Using the boundary conditions for a dark soliton

$$\left. \frac{\partial a}{\partial x} \right|\_{x \to \pm \infty} = 0 \quad \text{and} \quad a(\mathbf{x})|\_{x \to \pm \infty} = 1,\tag{65}$$

we find that (64) can be integrated, yielding:

$$\frac{1}{2}\tilde{\rho}\_{\text{GP}}(a)\left(\frac{\partial a}{\partial \mathbf{x}}\right)^2 = X(a) - \upsilon\_s^2 Y(a),\tag{66}$$

$$\Leftrightarrow \left(\frac{\partial x}{\partial a}\right)^2 = \frac{1}{2} \frac{\tilde{\rho}\_{\text{qp}}(a)}{X(a) - v\_s^2 Y(a)},\tag{67}$$

$$\Leftrightarrow \mathbf{x} = \pm \frac{1}{\sqrt{2}} \int\_{a\_0}^{a} \frac{\sqrt{\tilde{\rho}\_{\rm qp}(\mathbf{a}')}}{\sqrt{X(\mathbf{a}') - \upsilon\_s^2 Y(\mathbf{a}')}} d\mathbf{a}'. \tag{68}$$

Here, a<sup>0</sup> ¼ a xð Þ ¼ 0 is the relative amplitude at the center of the soliton, which is found as the solution of

$$X(a\_0) - \upsilon\_s^2 Y(a\_0) = 0.\tag{69}$$

For given values of the interaction parameter ð Þ kFas �<sup>1</sup> , the temperature T=TF, the imbalance chemical potential ζ, and the soliton velocity vs, formulae (60) and (68) allow us to calculate the complete pair field profile of the dark soliton. For example, the soliton density and phase profiles in Figure 4 were calculated using the above expressions.

#### 5.3. Dark solitons in imbalanced Fermi gases

The dark soliton solution derived in the previous section has been employed in the description of various soliton phenomena in superfluid Fermi gases. For instance, adding a small twodimensional perturbation to the exact 1D solution allows for a description of the snake instability mechanism [25], which makes the soliton decay into vortices if the radial width of the system is too large [23, 26]. We have also studied collisions between dark solitons by numerically evolving two counter-propagating 1D solitons in time [27]. As an example of an application, we will give a short description of the influence of spin-imbalance on dark solitons, a topic that was studied in detail in [18].

providing the system with more space to store the excess component. The fact that a dark soliton in an imbalanced superfluid Fermi gas has to drag along additional particles changes its effective mass, which in turn influences its general dynamical properties [18]. Moreover, since a soliton plane provides more space to accommodate the excess component than a vortex core, the presence of spin imbalance has been found to stabilize dark solitons with respect to

As a second application, the time-independent version of the theory is considered in order to derive the stable vortex structure. For the description of the vortex, the quantum velocity field

where θ is the phase field from the hydrodynamical description (44). In the time-independent case, the action (30) reduces to the free energy (times the inverse temperature), which is given by:

The free energy was written in a more compact<sup>3</sup> form using the hydrodynamical description (48), (49), (62) and (70). As an application of the effective field theory, the general structure of a superfluid vortex will be numerically determined and compared with the commonly used variational hyperbolic tangent. A more detailed description on vortices in superfluids and

Both in the classical and the quantum sense, a vortex is defined as a line in the fluid around which there is a circulating flow. In order to quantify this rotation around an axis, the circula-

where γ is a closed contour and v the superfluid velocity field (70). A distinct feature of superfluids4 is that the circulation κ is only allowed to take on values which are integer multiples of the circulation quantum h=m. In superfluids, circulation is always carried by quantized vortices.

This quantization of the circulation can be derived using the definition of the velocity field (70).

κ ¼ ∮ <sup>γ</sup>

Upon substitution, the circulation (72) can be written as:

Where again the free energy at infinity was subtracted to obtain a well behaved free energy.

In the case of a superconductor, the quantized value is given by the magnetic flux.

8

<sup>ρ</sup>sfð Þ<sup>a</sup> <sup>v</sup><sup>2</sup>

ð Þþ x 1 2

v rð Þ� ds, (72)

∇xθ, (70)

An Effective Field Description for Fermionic Superfluids http://dx.doi.org/10.5772/intechopen.73058

<sup>ρ</sup>qpð Þ<sup>a</sup> j j <sup>∇</sup>x<sup>a</sup> <sup>2</sup>

: (71)

23

<sup>v</sup> <sup>¼</sup> <sup>ℏ</sup> m

<sup>d</sup>rFð Þ <sup>a</sup>; <sup>∇</sup>xa; <sup>x</sup> with <sup>F</sup>ð Þ¼ <sup>a</sup>; <sup>∇</sup>xa; <sup>x</sup> X að Þþ <sup>1</sup>

the snake instability [25].

v will be used, defined as:

their behavior can be found in [30].

6.1. What is a vortex?

tion κ is defined as:

3

4

F ¼ ð

6. Application 2: the vortex structure

In ultracold Fermi gases, the amount of atoms in each spin population can be tuned experimentally, allowing for the possibility of having unequal amounts of spin-up and spin-down particles [28, 29]. In that case, when particles of different spin type pair up and form a superfluid state, an excess of unpaired particles will remain in the normal state, which in turn can have interesting effects on other phenomena in the system, including dark solitons. In the context of the EFT, we control the population imbalance by setting the value of the imbalance chemical potential ζ, defined in (18). Figure 5a and b shows respectively the fermion particle density n xð Þ and spinpopulation density difference δn xð Þ (both with respect to the bulk density n∞) along a stationary dark soliton for ð Þ kFas �<sup>1</sup> <sup>¼</sup> 0 (unitarity), <sup>T</sup> <sup>¼</sup> <sup>0</sup>:1TF, and for different values of <sup>ζ</sup>. The density and density difference profiles are calculated using formulas (22) and (23) in a mean-field local density approximation. From the left figure, we observe that as we raise the imbalance chemical potential, the fermion density at the soliton center increases and the soliton broadens. However, we also know that, for a stationary dark soliton, the pair density at the center is always zero (as shown in the upper left panel of Figure 4), which means that the particles filling up the soliton are unpaired particles. This is confirmed by the right figure, which shows that the density difference between spin-up and spin-down particles in the soliton center increases with ζ. The same effects are observed across the whole BEC-BCS crossover.

As the imbalance between the spin components in the Fermi gas increases, so does the amount of unpaired particles that cannot participate in the superfluid state of pairs. While some of these normal state particles can coexist with the pair condensate as a thermal gas, it is energetically favorable for the remaining excess to be spatially separated from the superfluid. In this context, the soliton dip is a very suitable location to accommodate the excess particles and consequently fills up with an increasing amount of unpaired particles as the imbalance gets higher. Also, the broadening of the soliton with increasing imbalance might be a way of

Figure 5. Fermion density (left figure) and density difference (right figure) profiles of a dark soliton for ð Þ kFaS �<sup>1</sup> <sup>¼</sup> 0 at temperature T=TF ¼ 0:1, for different values of the imbalance chemical potential ζ. The densities are given with respect to the bulk density n∞.

providing the system with more space to store the excess component. The fact that a dark soliton in an imbalanced superfluid Fermi gas has to drag along additional particles changes its effective mass, which in turn influences its general dynamical properties [18]. Moreover, since a soliton plane provides more space to accommodate the excess component than a vortex core, the presence of spin imbalance has been found to stabilize dark solitons with respect to the snake instability [25].
