# A shorter proof on recent iterative algorithms constructed by the relaxed
$(u,v)$-cocoercive mappings and a similar case for inverse-strongly monotone
mappings

E. Soori  Department of Mathematics, Lorestan University, P.O. Box 465,
Khoramabad, Lorestan, Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this short note, using the class of the relaxed $(u,v)$-cocoercive mappings
and $\alpha$-inverse strongly monotone mappings, we prove that if an important
condition holds then we can prove the convergence of the proposed algorithm,
more shorter than the original proof.

###### Key words and phrases:

Inverse-strongly monotone mapping, Strongly monotone mapping, Relaxed
$(u,v)$-cocoercive mapping

###### 2010 Mathematics Subject Classification:

47H09, 47H10

##  1\.  Introduction and preliminaries

Let $C$ be a nonempty closed convex subset of a real Hilbert space $H$. Recall
the following well known definitions:

  1. (1)

a mapping $A:C\rightarrow H$ is said to be inverse-strongly monotone, if there
exist $\alpha>0$ such that

$\langle Ax-Ay,x-y\rangle\geq\alpha\|Ax-Ay\|^{2},$

for all $x,y\in C$. Such a mapping $A$ is also called $\alpha$-inverse-
strongly monotone.

  2. (2)

a mapping $A:C\rightarrow H$ is said to be strongly monotone, if there exists
a constant $\alpha>0$ such that

(1) $\langle Ax-Ay,x-y\rangle\geq\alpha\|x-y\|^{2}.$

  3. (3)

let $C$ be a nonempty closed convex subset of a real Hilbert space $H$.
Suppose that $B:C\rightarrow H$ is a nonlinear map. $B$ is said to be relaxed
$(u,v)$-cocoercive, if there exist two constants $u,v>0$ such that

$\langle Bx-By,x-y\rangle\geq(-u)\|Bx-By\|^{2}+v\|x-y\|^{2},$

for all $x,y\in C$. For $u=0$, $B$ is $v$-strongly monotone. Clearly, every
$v$-strongly monotone map is a relaxed $(u,v)$-cocoercive map.

In this paper, using the relaxed $(u,v)$-cocoercive mappings, we make the
proof of some recent iterative algorithms more shorter than the original one.
Then we find a similar case for the inverse-strongly monotone mappings.

##  2\. The comments for the relaxed $(u,v)$-cocoercive mappings

Let $C$ be a nonempty closed convex subset of a real Hilbert space $H$.

G. Cai and S. Bu [1], W. Chantarangs and C. Jaiboon [2], S. Imnang [5], C.
Jaiboon and P. Kumam [11], P. Kumam and C. Jaiboon [15], X. Qin, M. Shang and
H. Zhou [19], X. Qin, M. Shang and Y. Su [22], X. Qin, M. Shang and Y. Su
[23], considered some iterative algorithm for finding a common element of the
set of the fixed points of nonexpansive mapping and the set of the solution of
a variational inequality $VI(C,A)$, where $A$ is a relaxed $(u,v)$-cocoercive
mapping of $C$ into $H$. But, there is a condition that makes the proof more
shorter than the original one.

The important condition is $\alpha$-expansiveness:

######  Condition 1.

Let $B$ be a self mapping on $C$. Suppose that there exists a positive integer
$\alpha$ such that:

$\|Bx-By\|\geq\alpha\|x-y\|.$

for all $x,y\in C$.

######  Lemma 2.1.

Let $A$ be a relaxed $(m,v)$-cocoercive mapping and $\epsilon$-Lipschitz
continuous such that $v-m\epsilon^{2}>0$ and $VI(C,A)\neq\emptyset$. Then $A$
is an $(v-m\epsilon^{2})$-expansive mapping and $VI(C,A)$ is singleton.

###### Proof.

Let $x_{1},x_{2}\in VI(C,A)$. Then

(2) $\langle Ax_{1},y-x_{1}\rangle\geq 0,$

for each $y\in C$, and

(3) $\langle Ax_{2},y-x_{2}\rangle\geq 0,$

for each $y\in C$.
Substituting $x_{1}$ in (3) and $x_{2}$ in (2), we have $\langle
Ax_{1},x_{2}-x_{1}\rangle\geq 0$ and $\langle Ax_{2},x_{1}-x_{2}\rangle\geq
0$. Adding them, we have

(4) $\langle Ax_{2}-Ax_{1},x_{2}-x_{1}\rangle\leq 0.$

But we know that $A$ is monotone. Indeed since $A$ is $\epsilon$-Lipschitz
continuous for each $x,y\in C$ we have

$\displaystyle\langle Ax-Ay,x-y\rangle$ $\displaystyle\geq(-m)\|Ax-
Ay\|+v\|x-y\|$ $\displaystyle\geq(-m\epsilon^{2})\|x-y\|+v\|x-y\|$ (5)
$\displaystyle=(v-m\epsilon^{2})\|x-y\|\geq 0,$

then we have $\langle Ax_{2}-Ax_{1},x_{2}-x_{1}\rangle\geq 0$, hence from (4),
we have

(6) $\langle Ax_{2}-Ax_{1},x_{2}-x_{1}\rangle=0.$

But if $x_{1}\neq x_{2}$, from (2) we have

$\displaystyle\langle
Ax_{2}-Ax_{1},x_{2}-x_{1}\rangle\geq(v-m\epsilon^{2})\|x_{2}-x_{1}\|>0,$

that is contradiction with (14). Hence, $VI(C,A)$ is singleton. From (2), we
have

$\|Ax-Ay\|\geq(v-m\epsilon^{2})\|x-y\|,$

then $A$ is $(v-m\epsilon^{2})$-expansive. ∎

Now, we are ready to investigate [15] for example, in the following comment
and we will bring a shorter proof for Theorem 3.1 in [15] for the relaxed
$(u,v)$-cocoercive mappings:

######  Comment 2.2.

Consider Theorem 3.1 in [15] and the $\xi$-Lipschitz continuous and relaxed
$(m,v)$-cocoercive mapping $B$ in Theorem 3.1. From Lemma 2.1 and the
condition (C7) of Theorem 3.1 in [15] and the condition $\xi$-Lipschitz
continuous on $B$ we have that $VI(E,B)$ is singleton, that is, there exists
an element $p\in E$ such that $VI(E,B)=\\{p\\}$ hence $\Gamma=\\{p\\}$ in
Theorem 3.1. From the condition (C7) of Theorem 3.1 in [15], we may assume
that $(v-m\xi^{2})>0$, hence from Lemma 2.1, $B$ is
$(v-m\epsilon^{2})$-expansive, i.e,

(7) $\|Bx-By\|\geq(v-m\xi^{2})\|x-y\|,$

then the condition 1 is valid for $B$. The authors have also proved in the
relation (3.25) in the page 521 that

(8) $\displaystyle\lim_{n}\|Bz_{n}-Bp\|=0.$

Now, putting $x=z_{n}$ and $y=p$ in (7), from (7) and (8), we have

$\displaystyle\lim_{n}\|z_{n}-p\|=0,$

hence, $z_{n}\rightarrow p$, therefore here we get one of the main claims of
Theorem 3.1. Note that since $\Gamma=VI(E,B)=\\{p\\}$ hence obviously $p$
belongs to other intersections in $\Gamma$ in Theorem 3.1. Then we can remove
all parts of the proof from the relation (3.33) in page 524 to end of the
proof of theorem 3.1 in [15] and another extra part of the proof.

In the following comment we introduce some more similar results.

######  Comment 2.3.

Refer the readers to [1, 2, 5, 11, 15, 19, 22, 23] and similar results, to see
more similar works. The authors of such these articles, using the relation

$\displaystyle\lim_{n}\|Bx_{n}-Bp\|=0,$

for relaxed $(u,v)$-cocoercive mapping $B$, have proved that $\\{x_{n}\\}$
converges to a unique element of the set $\mathfrak{F}$ which is for some
conditions and situations similar to comment 2.2 about [15], so we can make
their proofs shorter than the original proofs as in the comment 3.2.

##  3\. The comments for the inverse strongly monotone mappings

J. Chen, L. Zhang, T. Fan [3], S. Takahashia and W. Takahashi [27], H. Iiduka
and W.Takahashi [4], K. R. Kazmi, Rehan Ali and Mohd Furkan [14], X. Qin and
M. Shang [22], M. Zhang [30], S. Shan and N. Huang [25], T. Jitpeera and P.
Kumam [7], S. Peathanom and W. Phuengrattana[17], Piri [18], X. Qin, M. Shang
and Y. Su [23] and M. Lashkarizadeh Bami and E. Soori [16] considered some
iterative methods for finding a common element of the set of the fixed points
of nonexpansive mapping and the set of the solution of a variational
inequality $VI(C,A)$, where $A$ is an $\alpha$-inverse strongly monotone
mapping of $C$ into $H$. But, there is a case that is similar to
$(m,v)$-cocoercive mappings. First we see the following Lemma for
$\alpha$-inverse strongly monotone mappings.

######  Lemma 3.1.

Let $A$ be an $\alpha$-inverse strongly monotone mapping and
$VI(C,A)\neq\emptyset$. Suppose that $A$ is also an $\gamma$-expansive
mapping. Then $VI(C,A)$ is singleton.

###### Proof.

Since $A$ is an $\alpha$-inverse strongly monotone mapping, we have

(9) $\langle Ax-Ay,x-y\rangle\geq\alpha\|Ax-Ay\|^{2}\geq 0,$

then $A$ is monotone. Since $A$ is $\gamma$-expansive, we have

(10) $\|Ax-Ay\|\geq\gamma\|x-y\|.$

Therefore, we conclude that $A$ is one to one, because if $Ax=Ay$, then from
(10), $\|x-y\|=0$, hence $x=y$.

Let $x_{1},x_{2}\in VI(C,A)$. Then

(11) $\langle Ax_{1},y-x_{1}\rangle\geq 0,$

for each $y\in C$, and

(12) $\langle Ax_{2},y-x_{2}\rangle\geq 0,$

for each $y\in C$.
Substituting $x_{1}$ in (12) and $x_{2}$ in (11), we have $\langle
Ax_{1},x_{2}-x_{1}\rangle\geq 0$ and $\langle Ax_{2},x_{1}-x_{2}\rangle\geq
0$. Adding them, we have

(13) $\langle Ax_{2}-Ax_{1},x_{2}-x_{1}\rangle\leq 0.$

Since $A$ is monotone, then we have $\langle
Ax_{2}-Ax_{1},x_{2}-x_{1}\rangle\geq 0$, hence from (13), we have

(14) $\langle Ax_{2}-Ax_{1},x_{2}-x_{1}\rangle=0.$

Then from (9) we have $Ax_{2}=Ax_{1}$, since $A$ is one to one, we get
$x_{2}=x_{1}$. Then $VI(C,A)$ is singleton. ∎

Now as for $(m,v)$-cocoercive mappings, if $A$ is $\alpha$-strongly monotone
mapping and $\gamma$-expansive, i.e, the condition 1 holds for $A$, we can
make short the proof of the algorithms. For clearing our discussion, we
investigate [4] for example, in the following comment:

######  Comment 3.2.

Consider Theorem 3.1 in [4]. Since $A$ is an $\alpha$-strongly monotone
mapping, if we consider the extra condition $\gamma$-expansiveness in theorem
3.1 in [4], i.e,

(15) $\|Ax-Ay\|\geq\gamma\|x-y\|.$

for all $x,y\in C$, then from Lemma 3.1, we have $VI(C,A)$ is singleton and
hence $F(S)\cap VI(C,A)$ is singleton, i.e, for example $F(S)\cap
VI(C,A)=VI(C,A)=\\{u\\}$ for an element $u\in C$. The authors have also proved
in the line 8 from below in the page 5 that

(16) $\displaystyle\lim_{n}\|Ax_{n}-Au\|=0,$

Now, put $x=x_{n}$ and $y=u$ then from (16) and (15), we have

$\displaystyle\lim_{n}\|x_{n}-u\|=0,$

hence, $x_{n}\rightarrow u$ therefore in the line 8 from below in the page 5,
we get the main claims of Theorem 3.1. Note that since $F(S)\cap
VI(C,A)=VI(E,B)=\\{u\\}$ hence obviously $u\in F(S)$. Then we can remove all
parts of the proof from the line 8 from below in the page 345 to end of the
proof of theorem 3.1 in [4] and another extra part of the proof.

In the following comment we introduce some more similar results.

######  Comment 3.3.

The structure of main results in [3, 4, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17,
18, 22, 23, 24, 25, 26, 27, 28, 29, 30] is similar to [4]. The authors of such
these articles, using the relation

$\displaystyle\lim_{n}\|Ax_{n}-Ap\|=0,$

for $\alpha$-inverse strongly monotone mapping $A$, have proved that
$\\{x_{n}\\}$ converges to a unique element of the set $\mathfrak{F}$ which is
for some conditions and situations similar to comment 3.2 about [4].

##  4\. Acknowledgements

This author wishes to thank of the Lorestan university for their support.

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# Periodic dynamics of population-imbalanced fermionic condensates in optical
lattices: identifying FFLO structures

Avinaba Mukherjee<EMAIL_ADDRESS>Raka Dasgupta
<EMAIL_ADDRESS>Department of Physics, University of Calcutta, $92$ A. P.
C. Road, Kolkata $700009$, India

###### Abstract

We investigate the dynamics of a population-imbalanced two-species fermionic
system trapped in an optical lattice. The paired fermions here can form
bosonic molecules via Feshbach coupling in the presence of an external
magnetic field. It is shown that the natural fluctuations of the condensate
fraction is periodic beyond a threshold Feshbach detuning; and below this
threshold value the condensate shows no oscillation at all. The oscillation
frequency vs. detuning curve is linear in nature. The slope and intercept of
this line is shown to carry important information about the amount of
imbalance present in the system, and also about the nature of the pairing that
takes place in the population-imbalanced system. Most importantly, it can help
identifying the Fulde-Ferrel-Larkin-Ovchinikov (FFLO) phase as its signature
is different from the other possible phases.

###### pacs:

71.10Ca, 05.30.-d, 42.50Lc, 37.10Jk

##  I Introduction

Population-imbalanced ultracold fermionic systems remain an active field of
research since past two decades. While it is known that Feshbach-coupled two-
species fermionic gases can demonstrate a BCS-BEC crossover as the effective
interaction is varied [1, 2, 3, 4, 5], the situation changes if the system
resembles a spin-polarized state i.e., one species is more populated than the
other. In this case, instead of the weakly interacting homogeneous BCS state
(that demands both species to be equally populated), one would have novel
pairing structures that can accommodate the unpaired fermions. Several
theoretically proposed phases exist for such systems, including (i) Breached
pair (BP) [6, 7, 8, 9, 10, 11, 12] (ii) Phase separation (PS) [13, 14, 15] and
(iii) Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states [16, 17, 18].

The tunability associated with the ultracold atomic systems went a step ahead
with the advent of optical lattices [19, 20, 21, 22] as now the geometry as
well as the dimension of the lattice could be easily controlled by using the
laser beams. As Fermionic superfluidity could now be realized in the optical
lattices [5, 23, 24], it opened up the possibility of studying the non-BCS
exotic pairing phases in the presence of the lattice potentials. For example,
it was suggested that the noise correlation of the two component Fermi gas in
an optical lattices would show different signatures for BP and FFLO phases
[25]. How the phase diagram evolves with a varying filling fraction and a
varying strength of lattice parameters has been studied in the context of BP
[26, 27, 28], FFLO [27, 28, 29, 30], Phase Separation [32, 27, 31] , and also
for specifically 1D systems [33, 34]. Other important features, like, the
collective modes in FFLO [35] and the stability properties of FFLO [36, 37,
38] have also been studied in the presence of the optical lattices. However,
it is extremely difficult to find direct experimental signatures of the
theoretically predicted exotic phases (except the PS state [15]), and a few
indirect methods has been proposed in the past [39, 40, 41, 42, 43, 44, 45,
46, 12, 47].

In this paper, we study the mean-field level fluctuation dynamics of the
population-imbalanced fermionic condensate in an optical lattice for 1D, 2D
and 3D. We observe that the dynamics is periodic, and the oscillation
frequency changes almost linearly with a varying Feshbach detuning. We find
that the information about both the amount of population imbalance and the
pairing phase can be extracted from the slope and intercept of this straight
line. We propose that this periodic dynamics can be used as an indirect probe
to indicate the FFLO state, as it gives a distinct signature as compared to
the other phases both in 2D and 3D. Considering the fact that currently
several aspects about this FFLO phase is being earnestly pursued
theoretically[48, 49, 50], and yet, obtaining its experimental signature has
proven to be somewhat tricky : we believe that such an indirect method might
turn out to be extremely useful.

The paper is organized as follows. The basic theoretical model and the
dynamical equations are described in Sec. II. In Sec. III, possible pairing
phases are discussed and the relevant parameters are set up in different
dimensions. The frequency vs. detuning plots are presented in Sec. IV, and it
is shown that the oscillation dynamics is periodic. How the slope of the
frequency vs. detuning curve varies with the amount of population imbalance is
discussed in Sec. V, and the same for the intercept is done in Sec. VI. In
Sec. VII, we provide the analytical reasoning for the numerical results of the
previous two sections. The results are summarized in Sec. VIII.

##  II Model Hamiltonian and dynamics

Here we consider a system of the ultracold fermionic atoms in an optical
lattice, that can form a bosonic molecule by means of Feshbach resonances. The
system can be described by an extended version of the Hubbard Hamiltonian

$\begin{split}H=&\sum_{\langle\vec{j}.\vec{l}\rangle}(t-\mu_{\uparrow}\delta_{\vec{l},\vec{j}})a^{\dagger}_{\vec{j}\uparrow}a_{\vec{l}\uparrow}+\sum_{\langle\vec{j}.\vec{l}\rangle}(t-\mu_{\downarrow}\delta_{\vec{l},\vec{j}})a^{\dagger}_{\vec{j}\downarrow}a_{\vec{l}\downarrow}\\\
&+g_{1}\sum_{\vec{j}}{n_{\vec{j}\uparrow}n_{\vec{j}\downarrow}}+g_{2}\Big{(}\sum_{\vec{j}}b^{\dagger}_{\vec{j}}a_{\vec{j}\downarrow}a_{\vec{j}\uparrow}+\sum_{\vec{j}}a^{\dagger}_{\vec{j}\uparrow}a^{\dagger}_{\vec{j}\downarrow}b_{\vec{j}}\Big{)}\\\
&+[\epsilon_{b}-(\mu_{\uparrow}+\mu_{\downarrow})]\sum_{\vec{j}}{b^{\dagger}_{\vec{j}}b_{\vec{j}}}\end{split}$
(1)

Here, up-spin $(\uparrow)$ and down-spin $(\downarrow)$ represent either two
different atomic species, or two different hyperfine states of the same atom.
$\mu_{\uparrow}$ and $\mu_{\downarrow}$ are their respective chemical
potentials. The hopping amplitude (assumed to be same for both the species) is
$t$. Also, $a^{\dagger}_{\vec{j}}$ and $a_{\vec{j}}$ are respectively the
fermionic atom creation and annihilation operators for site
$\vec{j}=(j_{x},j_{y},j_{z})$, and their bosonic counterparts for molecule
creation and destruction are given by $b^{\dagger}_{\vec{j}}$ and
$b_{\vec{j}}$. The number operator for up-spin and down-spin fermions at
$\vec{j}$ th site are $n_{\vec{j}\uparrow}$ and $n_{\vec{j}\downarrow}$
respectively. The on-site interaction strength is $g_{1}$, while $g_{2}$ is
the additional interaction strength of the feshbach variety which couples two
fermionic atoms to form a bosonic molecule.The threshold energy of the
composite bosonic molecule energy band is $\epsilon_{b}$, and it is the
pivotal adjustable “detuning" parameter in this model.

This Hamiltonian takes the following form in momentum space :

$\begin{split}H=&\sum_{\vec{k}}\epsilon_{\vec{k}\uparrow}a^{\dagger}_{\vec{k}\uparrow}a_{\vec{k}\uparrow}+\sum_{\vec{k}}\epsilon_{-\vec{k}\downarrow}a^{\dagger}_{-\vec{k}\downarrow}a_{-\vec{k}\downarrow}\\\
&+g_{1}\sum_{\vec{k},\vec{k}^{\prime}}a^{\dagger}_{\vec{k}\uparrow}a^{\dagger}_{-\vec{k}+\vec{q}\downarrow}a_{-\vec{k}^{\prime}+\vec{q}\downarrow}a_{\vec{k}^{\prime}\uparrow}\\\
&+g_{2}(\sum_{\vec{q},\vec{k}^{\prime}}b^{\dagger}_{\vec{q}}a_{-\vec{k}^{\prime}+\vec{q}\downarrow}a_{\vec{k}^{\prime}\uparrow}+\sum_{\vec{k},\vec{q}^{\prime}}a^{\dagger}_{\vec{k}\uparrow}a^{\dagger}_{-\vec{k}+\vec{q}^{\prime}\downarrow}b_{\vec{q}^{\prime}})\\\
&+[\epsilon_{b}-(\mu_{\uparrow}+\mu_{\downarrow})]\sum_{\vec{q}}b^{\dagger}_{\vec{q}}b_{\vec{q}}\end{split}$
(2)

$\epsilon_{\vec{k}\uparrow}$ and $\epsilon_{-\vec{k}\uparrow}$ are the energy-
momentum dispersion relation for up-spin and down-spin respectively. Also,
$a^{\dagger}_{\vec{k}\uparrow}$, $a_{\vec{k}\uparrow}$ are the creation and
annihilation operators corresponding to momentum $k$ for one fermionic
species, and $a^{\dagger}_{\vec{k}\downarrow}$, $a_{\vec{k}\downarrow}$ are
the corresponding operators for the other fermionic species. The bosonic
molecule creation and annihilation operators in the momentum space is given by
$b^{\dagger}_{\vec{q}}$ and $b_{\vec{q}}$ respectively. Here $\vec{q}$ is the
momentum of the composite bosons, and has magnitude zero for BCS-type of
pairing, and a non-zero magnitude if an exotic pairing structure like FFLO is
involved. More details about these pairing phases are discussed in III.2.

We define

$O_{\vec{k},\vec{q}}=\langle
a_{-\vec{k}+\vec{q}\downarrow}a_{\vec{k}\uparrow}\rangle$ (3)

The equilibrium conditions are given by

$\frac{\partial O^{eq}_{\vec{k},\vec{q}}}{\partial t}=0\hskip
15.0pt\mbox{and}\hskip 15.0pt\frac{\partial b^{eq}_{\vec{q}}}{\partial t}=0$
(4)

Now, we consider the intrinsic and spontaneous quantum fluctuations on top of
the equilibrium base states. We write,
$O_{\vec{k},\vec{q}}=O^{eq}_{\vec{k},\vec{q}}+\tilde{O}_{\vec{k},\vec{q}}$ and
$b_{\vec{q}}=b^{eq}_{\vec{q}}+\tilde{b}_{\vec{q}}$. Here
$\tilde{O}_{\vec{k},\vec{q}}$ is the fluctuation in $O_{\vec{k},\vec{q}}$ and
$\tilde{b}_{\vec{q}}$ is the fluctuation in $b_{\vec{q}}$. The scheme
resembles the one used in [47] for studying population-imbalanced uniform
systems as well as trapped systems under local density approximation. However,
we have extended that to incorporate the effects of optical lattices that are
more relevant for present-day experiments.

By calculating the commutation relations of $a_{\vec{k}\uparrow}$,
$a_{-\vec{k}+\vec{q}\downarrow}$, $b_{\vec{q}}$ and $O_{\vec{k},\vec{q}}$
respectively with the Hamiltonian, and imposing the equilibrium conditions
which mentioned as above, we arrive at two dynamical equations :

$\begin{split}i\hbar\frac{\partial\tilde{O}_{\vec{k},\vec{q}}}{\partial
t}=&\sum_{\vec{k}}(\epsilon_{\vec{k}\uparrow}+\epsilon_{-\vec{k}+\vec{q}\downarrow})\tilde{O}_{\vec{k},\vec{q}}-(g_{1}\sum_{\vec{k}}\tilde{O}_{\vec{k},\vec{q}}+g_{2}\tilde{b}_{\vec{q}})\end{split}$
(5) $i\hbar\frac{\partial\tilde{b}_{\vec{q}}}{\partial
t}=g_{2}\sum_{\vec{k}}\tilde{O}_{\vec{k}}+(\epsilon_{b}-(\mu_{\uparrow}+\mu_{\downarrow}))\tilde{b}_{\vec{q}}$
(6)

By taking the fourier transform of Eqs. (5, 6), we obtain

$\sum_{\vec{k}}\tilde{O}_{\vec{k},\vec{q}}(\omega)=\frac{g_{2}\tilde{b}_{\vec{q}}(\omega)f_{1}(\omega)}{1-g_{1}f_{1}(\omega)}$
(7)

Here

$f_{1}(\omega)=\sum_{\vec{k}}\frac{1}{\epsilon_{\vec{k}\uparrow}+\epsilon_{-\vec{k}+\vec{q}\downarrow}+\hbar\omega}$
(8)

By putting Eq. (7) in Eq. (6) we get

$b_{\vec{q}}(\omega)\Big{(}\epsilon_{b}+\hbar\omega+\frac{g^{2}_{2}f_{1}(\omega)}{1-g_{1}f_{1}(\omega)}\Big{)}=0$
(9)

Thus, in the Fourier expansion of $b_{\vec{q}}(t)$, only those
$b_{\vec{q}}(\omega)$ would survive for which

$\epsilon_{b}+\hbar\omega+\frac{g^{2}_{2}f_{1}(\omega)}{1-g_{1}f_{1}(\omega)}=0$
(10)

Therefore, if there is a single solution of $\omega$ for which Eq. (10) is
satisfied, $b_{\vec{q}}(t)=b_{0}+b_{1}e^{i\omega t}$, and the condensate
fraction $|b_{\vec{q}}(t)|^{2}$ would show an oscillatory dynamics with
frequency $\pm\omega$. If there are two such solutions $\omega_{1}$ and
$\omega_{2}$ :
$b_{\vec{q}}(t)=b_{0}+b_{1}e^{i\omega_{1}t}+b_{2}e^{i\omega_{2}t}$, the
condensate fraction would have three periodic components : $\pm\omega_{1}$,
$\pm\omega_{2}$, $\pm(\omega_{1}-\omega_{2})$, and similarly for higher number
of allowed $\omega$ values. If no real solution is found for Eq. (10), that
the condensate fraction would have no oscillatory dynamics at all.

As evident from Eq. (8), the quantity $f_{1}(\omega)$ involves a sum over all
paired region in the momentum space and is dependent on (i) the dimension of
the system, (ii) the amount of population imbalance present in the system, and
(iii) the nature of pairing. Thus, the solution for $\omega$ obtained from Eq.
(10), that translates to the frequency of oscillation of the condensate
fraction, would contain information about these factors as well.

##  III Different variants of the population imbalanced system

###  III.1 Dimension of the system

Most of the ultracold-atomic experiments offer a tremendous flexibility when
it comes to the dimension of the system. For example, 1-dimensional (1D)
optical lattice can be formed when the potential is shallower in one direction
but it is much deeper in the other two directions [51, 52, 53, 54]. Converting
the summation in Eq. (8) to an integral,

$f_{1}(\omega)=\frac{L}{2\pi\hbar}\int\frac{dk}{\epsilon_{\vec{k}\uparrow}+\epsilon_{-\vec{k}+\vec{q}\downarrow}+\hbar\omega}$
(11)

Here $\epsilon_{\vec{k}\uparrow}=2t(\cos{ka})-\mu_{\uparrow}$,
$\epsilon_{-\vec{k}\downarrow}=2t(\cos{ka})-\mu_{\downarrow}$. $L$ and $a$ are
the quantization length and lattice constant of the optical lattice
respectively. Similarly, 2-dimensional (2D) optical lattice can be formed when
the potential is deeper in one direction but shallower in the other two
directions [55].

$f_{1}(\omega)=\frac{A}{(2\pi\hbar)^{2}}\int\frac{dk_{x}dk_{y}}{\epsilon_{\vec{k}\uparrow}+\epsilon_{-\vec{k}+\vec{q}\downarrow}+\hbar\omega}$
(12)

In this case,
$\epsilon_{\vec{k}\uparrow}=2t(\cos{k_{x}a}+\cos{k_{y}a})-\mu_{\uparrow}$,
$\epsilon_{-\vec{k}\downarrow}=2t(\cos{k_{x}a}+\cos{k_{y}a})-\mu_{\downarrow}$
and $A$ is the quantization area of the optical lattice.

In 3-dimensional (3D) optical lattices, the potential depths are comparable in
three possible directions [56]. Here,

$f_{1}(\omega)=\frac{V}{(2\pi\hbar)^{3}}\int\frac{dk_{x}dk_{y}dk_{z}}{\epsilon_{\vec{k}\uparrow}+\epsilon_{-\vec{k}+\vec{q}\downarrow}+\hbar\omega}$
(13)

In this case, $\epsilon_{\vec{k}\uparrow}=2t(\cos k_{x}a+\cos k_{y}a+\cos
k_{z}a)-\mu_{\uparrow}$, $\epsilon_{\vec{k}\downarrow}=2t(\cos k_{x}a+\cos
k_{y}a+\cos k_{z}a)-\mu_{\downarrow}$ and $V$ is the quantization volume of
the optical lattice. In Eqs. (12) and (13), it is assumed that the lattice
constant is of the same value in all the directions.

###  III.2 Possible pairing phases

Figure 1: The structure of different pairing phases in population imbalanced
fermionic systems in 1D : (a) BP1 (b) BP2 (c) PS (d) FFLO

If a system has population imbalance between up-spin and down-spin then
different exotic phases arise, because the paring cannot entirely take place
in the BCS-way. The phases that we consider here are (i) Breached-Pair-1 state
(BP1) or Sarma phase (ii) Breached pair-2 state (BP2) (iii) phase separation
(PS) , and (iv) FFLO.

Figure 2: The momentum-space structure of BP1 phase in 2D, the blue shell
contains superfluid and the red region contains free fermi gas Figure 3: The
momentum-space structure of BP2 phase in 2D, the blue regions contains
superfluid and the red region contains free fermi gas

In Fig. 1, the blue region and the red dashed regions indicate the paired and
unpaired region respectively for a one-dimensional Fermi gas, and different
types of pairing structures are depicted. Breached pair or Sarma phase appears
at $T=0$, $\Delta\neq 0$, $P\neq 0$ (here, $\Delta$ is the energy gap and
$P=\frac{n_{\uparrow}-n_{\downarrow}}{n_{\uparrow}+n_{\downarrow}}$ is the
population imbalance). In the BP1 (Fig. 1 (a)) structure for a 1-dimensional
system, we get the paired region near the ends of the momentum-space line, and
the unpaired free fermi gas remains in the middle. In BP2 (Fig. 1 (b))
structure for 1-dimension, we get the paired region at both ends of the line
and also in the middle, and the “breached" regions with a single species exist
in-between. Phase separation (Fig. 1 (c)) is defined as the coexistence of
superfluid ($\Delta\neq 0$, $P=0$) and normal states ($\Delta=0$, $P\neq 0$).
In Fig. 1 (c), $-k_{1}$ to $k_{1}$ is the range of paired region and $-k_{2}$
to $k_{2}$ is the range of the unpaired region in a momentum space. FFLO (Fig.
1 (d)) state is a BCS-like paired state but with a non-zero total momentum.
FFLO state is not translationally invariant like BCS and BP phases. Here, the
gap parameter is $\Delta(r)=\Delta e^{2i\vec{q}.\vec{r}}$ ($q>0$) where
$\vec{r}$ is the distance from the origin.

Figure 4: (a) Usual BCS type of pairing (b) FFLO type of pairing

Out of these phases, BP1, BP2 and FFLO involve specific structures in
momentum-space. Although these phases have been predicted long back, direct
experimental signatures of these phases are difficult to obtain. In contrast,
phase separation appears in the real space, and is easier to identify
experimentally [15].

Figure 5: (a) low attraction strength between red and blue atoms (b) moderate
attraction strength between red and blue atoms which generate the phase
separation state

In 2D or 3D, the BP1 (Fig. 2) is a two-shell structure of the free fermi gas
and the superfluid in momentum space. The BP2 state (Fig. 3) in 2D and 3D is a
three-shell structure of the free fermi gas and the superfluid in a momentum
space. In Fig. 4, the nature of FFLO pairing is depicted where there is the
non-zero momentum pairing between red (up-spin) and blue atoms (down-spin)
[57] in 2D. The phase-separated state, on the other hand, originates when the
up-spin (red) and down-spin (blue) atoms pair with each other and repel the
redundant atoms [58], leading to a phase separation between the paired
superfluid fermions and the excess unpaired fermions (as shown in Fig. 5). The
integration limits for solving Eqs. (11), (12) and (13) later have been chosen
in accordance with the pairing structures described here, and is elaborated
further in IV.

###  III.3 System parameters

In this subsection, we set up the parameter values for different types of
paring structures to numerically solve Eq. (10). We choose the following
values for the optical lattice depth : $V_{x}=6E^{F}_{R}$ and
$V_{y}=V_{z}=40E^{F}_{R}$ for 1D; $V_{x}=V_{y}=6E^{F}_{R}$ and
$V_{z}=40E^{F}_{R}$ for 2D; and $V_{x}=V_{y}=V_{z}=6E^{F}_{R}$ for 3D. Here
$E^{F}_{R}$ is the recoil Fermi energy. The wavelength of the laser beam that
creates the optical lattice is chosen to be $\lambda=825$nm as in [59]. In $x$
direction the hopping amplitude is given by [60, 61, 62, 63, 64, 24, 65, 66,
59]

$\frac{t}{E^{F}_{R}}=\frac{4}{\sqrt{\pi}}\Big{(}\frac{V_{x}}{E^{F}_{R}}\Big{)}^{\frac{3}{4}}e^{-2\sqrt{V_{x}/E^{F}_{R}}}$
(14)

Hopping in $y$ and $z$ directions can be calculated in the similar fashion.
The interatomic coupling $g_{1}$ is given by [65]

$g_{1}=\sqrt{\frac{8}{\pi}}ka_{bg}E^{F}_{R}\Big{(}\frac{V_{x}}{E^{F}_{R}}\Big{)}^{\frac{1}{4}}\Big{(}\frac{V_{y}}{E^{F}_{R}}\Big{)}^{\frac{1}{4}}\Big{(}\frac{V_{z}}{E^{F}_{R}}\Big{)}^{\frac{1}{4}}$
(15)

The background scattering length is $a_{bg}$ for different fermionic species
(${}^{6}\mbox{Li}$ and ${}^{40}\mbox{K}$) are given in Table 1. If the optical
lattice potential is quite deep ($\geq 5E^{F}_{R}$), only the lowest two
energy bands remain relevant. In this case, the additional interaction in the
form of Feshbach coupling is given by,

$\begin{split}g_{2}=&\sqrt{\frac{4\pi\hbar^{2}a_{bg}\Delta
B\mu_{co}}{m_{F}}}({\int dx\mathcal{W}^{B}_{x}[\mathcal{W}^{F}_{x}]^{2}}{\int
dy\mathcal{W}^{B}_{y}[\mathcal{W}^{F}_{y}]^{2}}\\\ &{\int
dz\mathcal{W}^{B}_{z}[\mathcal{W}^{F}_{z}]^{2}})\end{split}$ (16)

Here $\mathcal{W}^{B}_{x}$, $\mathcal{W}^{B}_{y}$, $\mathcal{W}^{B}_{z}$ are
Wannier functions for bosons along x, y, z direction respectively, and
$\mathcal{W}^{F}_{x}$, $\mathcal{W}^{F}_{y}$, $\mathcal{W}^{F}_{z}$ are their
fermionic counterparts [67]. Again, for sufficiently deep lattice potentials,
the Wannier functions can be approximated by harmonic oscillator
wavefunctions. For $x$-direction, we thus take the bosonic Wannier function to
be

$\mathcal{W}^{B}_{x}=\Big{(}\frac{m_{B}\omega^{B}_{x}}{\pi\hbar}\Big{)}^{\frac{1}{4}}e^{-(m_{B}\omega^{B}_{x}x^{2})/2\hbar}$
(17)

and fermionic Wannier function to be

$\mathcal{W}^{F}_{x}=\Big{(}\frac{m_{F}\omega^{F}_{x}}{\pi\hbar}\Big{)}^{\frac{1}{4}}e^{-(m_{F}\omega^{F}_{x}x^{2})/2\hbar}$
(18)

as in [68]. Here, $m_{B}$ and $m_{F}$ are the masses of the bosonic molecule
and fermionic atoms respectively. In addition, we consider $\mu_{co}=2\mu_{B}$
[69], $\omega^{B}_{x}=\frac{2}{\hbar}\sqrt{V_{x}E^{B}_{R}}$ [70],
$\omega^{F}_{x}=\frac{2}{\hbar}\sqrt{V_{x}E^{F}_{R}}$ [70]. Here,
$E^{B}_{R}(=\frac{\hbar^{2}k^{2}}{2m_{B}})$ is the bosonic recoil energy and
$\Delta B$ is the resonance width. Similarly, $\mathcal{W}^{F}_{y}$ and
$\mathcal{W}^{B}_{y}$ can be computed from $V_{y}$; while
$\mathcal{W}^{F}_{z}$ and $\mathcal{W}^{B}_{z}$ can be computed from $V_{z}$.

Combining all these, the coupling $g_{2}$ can be expressed as :

$\begin{split}g_{2}=\sqrt{\frac{4\pi a_{bg}\Delta
B\mu_{co}}{m_{F}\hbar}}\Big{(}\frac{2m_{B}}{\pi}\Big{)}^{\frac{3}{4}}\Big{(}E^{B}_{R}\Big{)}^{\frac{3}{8}}\Big{(}V^{B}_{x}V^{B}_{y}V^{B}_{z}\Big{)}^{\frac{1}{8}}\\\
\Big{(}\frac{2}{2+\sqrt{\frac{m_{B}}{m_{F}}}}\Big{)}^{\frac{3}{2}}\end{split}$
(19)

The complete list of system parameters for 1D, 2D and 3D that we compute, and
use for solving the dynamical equations are given in Tables 2, 3 and 4 in
Appendix A.

##  IV Variation of frequency with detuning

In this section, we present the numerical solutions of Eq. (10) in different
dimensions, for different types of pairing structures in a population-
imbalanced system. The result is presented in the form of oscillation
frequency ($\omega$) of the condensate fraction vs. detuning ($\epsilon_{b}$)
of the Feshbach resonance for two distinct fermionic systems :
${}^{6}\mbox{Li}$ and ${}^{40}\mbox{K}$.

###  IV.1 Dynamics in 1-dimensional systems

Here the value of $f_{1}(\omega)$ in Eq. (10) has to be evaluated by using Eq.
(11). The limit of the integration is chosen in accordance with the structure
of the pairing phases as shown in Fig. 1. As an example, for fractional
imbalance, $P=0.5$, we put the limits of the $k$ value of the paired region as
$-\frac{\pi}{2}$ to $-\frac{\pi}{3}$ and $\frac{\pi}{3}$ to $\frac{\pi}{2}$
for BP1. The paired region in BP2 is taken as $k=-\frac{\pi}{2}$ to
$k=-\frac{5\pi}{12}$, $k=-\frac{\pi}{12}$ to $k=\frac{\pi}{12}$ and
$k=\frac{5\pi}{12}$ to $k=\frac{\pi}{2}$. The paired region in PS spans from
$k=-\frac{\pi}{6}$ to $k=\frac{\pi}{6}$. In FFLO, the paired region is
$k=\frac{\pi}{6}$ to $k=\frac{\pi}{2}$.

The $\omega$ vs. $\epsilon_{b}$ plots for ${}^{6}\mbox{Li}$ near the narrow
resonance (resonance width $\Delta B=0.1$ G , resonance position
$B_{0}=543.25$ G) is shown in Fig. 6. Here we find that for all values of
$\epsilon_{b}$, there exists a single value of $\omega$. It implies that the
dynamics of $b_{\vec{q}}$ is periodic :

$b_{\vec{q}}(t)=b_{0}+b_{1}e^{i\omega t}$ (20)

Therefore, the condensate fraction goes as

$|b_{\vec{q}}(t)|^{2}=|b_{0}|^{2}+|b_{1}|^{2}+b_{0}^{\dagger}b_{1}e^{i\omega
t}+b_{1}^{\dagger}b_{0}e^{-i\omega t}$ (21)

So, essentially it would contain a $\cos{\omega t}$ component. Thus, the
dynamics of the condensate fraction would be periodic as well.

Figure 6: $\omega$ vs. $\epsilon_{b}$ plot at fractional imbalance $P=0.5$ for
various phases in 1D : (a) BP1 (b) BP2 (c) PS (d) FFLO of ${}^{6}\mbox{Li}$
near narrow resonance. Inset : the enlarged view of the no oscillation region
indicated by the gap near the resonance point.

The $\omega$ vs. $\epsilon_{b}$ curve is almost linear in nature. There is a
small region with no real solution for $\omega$ in the central region ( Inset
of Fig. 6). The plot for ${}^{40}\mbox{K}$ (resonance width $\Delta B=7.8$ G,
$B_{0}=202.10$ G) shows a similar pattern in Fig. 7, but the region with no
solution for $\omega$ is wider. This suggests that the condensate fraction
shows no oscillation if the magnitude of the detuning is lower than a
threshold value, and there is a periodic oscillation beyond it.

Figure 7: $\omega$ vs. $\epsilon_{b}$ plot at fractional imbalance $P=0.5$ for
various phases : (a) BP1 (b) BP2 (c) PS (d) FFLO of ${}^{40}\mbox{K}$ in
narrow resonance width at 1D

We would like to mention that the $\omega$ vs. $\epsilon_{b}$ curve for the
broad resonance in ${}^{6}\mbox{Li}$ ($\Delta B=300$ G, $B_{0}=834.15$ G)
shows a similar behavior. However, the range over which no real solution of
$\omega$ is obtained is much wider in this case, and one obtains the straight-
line like curve only when the magnitude of the detuning is very large. For the
relatively broader resonance of ${}^{40}\mbox{K}$ ($\Delta B=9.7$ G,
$B_{0}=224.21$ G), the $\omega$ vs. $\epsilon_{b}$ curve is almost similar to
the narrower resonance case. These sets of plots (broad resonances of
${}^{6}\mbox{Li}$ and ${}^{40}\mbox{K}$) are not included in the present
manuscript.

One point to note is that, in all these plots, no remarkable difference is
spotted among the $\omega$ vs. $\epsilon_{b}$ curves belonging to the
different pairing structures. Thus, these cannot directly be used to
differentiate among possible exotic phases.

###  IV.2 Dynamics in 2-dimensional systems

Here, the limits of the integration in Eq. (12) is chosen keeping in mind that
the momentum space phase diagram in 2D is a circle. We set the total radius of
the circle as unity, i.e., all other momenta are scaled by the Fermi momentum
$k_{F}$ of the majority species. For fractional imbalance $P=0.5$, we put the
limits of the paired region as $k=\sqrt{2/3}$ to $1$ for BP1. The paired
region in BP2 spans from $k=0$ to $k=\sqrt{1/6}$, and $k=\sqrt{5/6}$ to $1$.
The paired region in PS extends from $k=0$ to $k=\sqrt{1/3}$. The paired
region in FFLO is $k=1-\sqrt{1/3}$ to $1$.

The $\omega$ vs. $\epsilon_{b}$ plots for ${}^{6}\mbox{Li}$ near the narrow
resonance is shown in Fig. 8. Here, too, we find that for all values of
$\epsilon_{b}$, there exists a single value of $\omega$, meaning the dynamics
of the condensate fraction is periodic as in Eq. (20).

Figure 8: $\omega$ vs. $\epsilon_{b}$ plot at fractional imbalance $P=0.5$ for
various phases in 2D : (a) BP1 (b) BP2 (c) PS (d) FFLO of ${}^{6}\mbox{Li}$
near narrow resonance. Inset : the enlarged view of the no oscillation region,
indicated by the gap near the resonance point of negative detuning side.

The $\omega$ vs. $\epsilon_{b}$ curve is almost linear in nature, and again,
there is a region with no real solution near the resonance point (Inset of
Fig. 8) of negative detuning side. The plot for ${}^{40}\mbox{K}$ shows a
similar pattern in Fig. 9. This no-solution region appears mostly in the
negative detuning side in case of ${}^{6}\mbox{Li}$, and along both the
detuning sides in case of ${}^{40}\mbox{K}$. However, the region of no
solution near the $\epsilon_{b}=0$ point is less wider here than the analogous
region in the 1D case. Though the curve is nearly linear, there is a very
small curvature near the resonance.

Figure 9: $\omega$ vs. $\epsilon_{b}$ plot at fractional imbalance P=0.5 for
various phases : (a) BP1 (b) BP2 (c) PS (d) FFLO of ${}^{40}\mbox{K}$ in
narrow resonance width at 2D

In all these plots, too, the different pairing structures cannot be
distinguished solely from the $\omega$ vs. $\epsilon_{b}$ curves.

###  IV.3 Dynamics in 3-dimensional systems.

The $\omega$ vs. $\epsilon_{b}$ plots for ${}^{6}\mbox{Li}$ (narrow resonance)
and ${}^{40}\mbox{K}$ (narrow resonance) are shown in Fig. 10 and Fig. 11
respectively. The momentum space phase diagram in 3D is a sphere, and we set
the total radius of the sphere as unity. As for the limits of the integration
in Eq. (13) for fractional imbalance $P=0.5$, the paired region spans from
$k=(2/3)^{\frac{1}{3}}$ to 1 for BP1, from $k=0$ to $k=(1/6)^{\frac{1}{3}}$,
and $k=(5/6)^{\frac{1}{3}}$ to $1$ for BP2. The one for PS is $k=0$ to
$k=(1/3)^{\frac{1}{3}}$ and for FFLO is $k=1-(1/3)^{\frac{1}{3}}$ to $1$.

Here, too, we find that for all values of $\epsilon_{b}$, there exists a
single value of $\omega$, meaning the oscillation of the condensate fraction
is periodic as in Eq. (20).

Figure 10: $\omega$ vs. $\epsilon_{b}$ plot at fractional imbalance P=0.5 for
various phases in 3D : (a) BP1 (b) BP2 (c) PS (d) FFLO of ${}^{6}\mbox{Li}$ in
narrow resonance. Inset : the enlarged view of the no oscillation region,
indicated by the gap near the resonance point of negative detuning side.

The same trend (i.e., a periodic oscillation except near the resonance point)
is observed in this case as well. The span of the region of no solution is
less than the corresponding regions in 1D or 2D. The $\omega$ vs.
$\epsilon_{b}$ curve is almost linear.

Figure 11: $\omega$ vs. $\epsilon_{b}$ plot at fractional imbalance P=0.5 for
various phases : (a) BP1 (b) BP2 (c) PS (d) FFLO of ${}^{40}\mbox{k}$ in
narrow resonance width at 3D

Thus, by comparing Fig. 6 , Fig. 8 and Fig. 10, and also by comparing Fig. 7,
Fig. 9 and 11, we see that the phase plots do not differ much if the dimension
is altered. Also, no major difference vis à vis the choice of the atoms
(${}^{6}\mbox{Li}$ vs. ${}^{40}\mbox{k}$) is found.

We would like to highlight here that the frequencies of oscillation of the
condensate fraction falls well within experimentally detectable range. If the
range of detuning is $5$ G to $50$ G [71] away from the resonance point
($B_{0}=543.25$ G [69] for narrow resonance width of ${}^{6}\mbox{Li}$. ), we
get the frequency in GHz range for all possible dimensions. As an example, in
Fig. 6, 7, 8, 9, 10, 11 the $\epsilon_{b}=1000$ point in our scale implies a
detuning of $0.5$ G apart from the resonance point. The corresponding
oscillation frequency is $\omega=1000$ in our convention, that translates to a
frequency $0.01$ GHz.

Additionally, we observe that if the amount of fractional imbalance is
changed, then the qualitative nature of the curves remains the same, but some
quantitative changes are involved. To quantify these changes, we cast each
$\omega$ versus $\epsilon_{b}$ plot in the form

$\omega=a\epsilon^{2}_{b}+b\epsilon_{b}+c$ (22)

where $a$ is found to be very small ($a\rightarrow 0$). Here $b$ and $c$ are
the slope and intercept of the line respectively.

In the next section we study the effects of population imbalance on the slope
($b$) of these $\omega$ vs. $\epsilon_{b}$ plots.

##  V variation of slope with population imbalance

In this section, we study how the $\omega$ vs. $\epsilon_{b}$ plots in Sec. IV
change with a varying amount of population imbalance. For this, we extract the
slope $b$ from the linearized form of the curves as in Eq. (22). The result is
presented in the form of slope ($b$) versus fractional imbalance ($P$) plot
for ${}^{6}\mbox{Li}$. We observe that these curves are of parabolic in
nature, either concave or convex : depending on the dimensionality and the
nature of the pairing structure.

###  V.1 Characteristics of 1-dimensional systems

Fig. 12 shows that if $P$ increases then the absolute value of the slope
decreases for all four phases in ${}^{6}\mbox{Li}$ in 1D for frequency range
$16\leq\omega\leq 1000$. Moreover, all these plots look parabolic, and are
concave in nature.

Figure 12: $b$ vs. $P$ plots for various phases in positive frequency and
negative detuning domain : (a) BP1 (b) BP2 (c) PS (d) FFLO of
${}^{6}\mbox{Li}$ at narrow resonance width in 1D

###  V.2 Characteristics of 2-dimensional systems

Fig. 13 corresponds to $b$ vs $P$ values for different phases in population-
imbalanced ${}^{6}\mbox{Li}$ in 2D for frequency range $9\leq\omega\leq 1000$,
and it shows that if $P$ increases, the magnitude of the slope decreases. The
plots of BP1, BP2 and PS phases are parabolic and concave in nature. The plot
for FFLO phase, however, offers a striking difference : it is convex.

Figure 13: $b$ vs. $P$ plot for various phases in positive frequency and
negative detuning domain : (a) BP1 (b) BP2 (c) PS (d) FFLO of
${}^{6}\mbox{Li}$ at narrow resonance width in 2D

###  V.3 Characteristics of 3-dimensional systems

Fig. 14 shows that, like the 2D case, the $b$ vs. $P$ plots for 3D, too, are
concave in nature for BP1, BP2 and PS for frequency range $13\leq\omega\leq
1000$. The corresponding plot for FFLO phase is convex.

Figure 14: $b$ vs. $P$ plot for various phases in positive frequency and
negative detuning domain : (a) BP1 (b) BP2 (c) PS (d) FFLO of
${}^{6}\mbox{Li}$ at narrow resonance width in 3D

We would like to point out that Fig. 12, Fig. 13 and Fig. 14 correspond to the
positive frequency (and negative detuning) sides of Fig. 6, Fig. 8 and Fig. 10
respectively (and their equivalent plots for other values of $P$). The plots
for the negative frequency (and positive detuning) sides are qualitatively
similar, and therefore, are not presented separately. The plots for
${}^{40}\mbox{K}$ near the narrow resonance show the same tendencies, and are
not included either.

Thus, we find that the $b$ vs. $P$ plots for FFLO provide a different
signature from the other novel phases in 2D and 3D. This can be used as a tool
to detect FFLO, the only phase with non-zero momentum pairing. Moreover, the
slope of the $\omega$ versus $\epsilon_{b}$ curve in any dimension can be used
to extract the population-imbalance present in the system by means of Fig. 12,
Fig. 13 and Fig. 14.

In the next section we try to study the effect of $P$ on the intercept of the
linearized form of the $\omega$ vs. $\epsilon_{b}$ curve.

##  VI Variation of intercept with fractional imbalance

In this section, the linearized form of the $\omega$ vs. $\epsilon_{b}$ plot
is studied and the intercept is extracted. Here we investigate the effect of
the population imbalance on the value of this intercept. The result is
presented in the form of intercept ($c$) versus fractional imbalance ($P$)
plot for ${}^{6}\mbox{Li}$. We find that these plots are of parabolic in
nature, and are either concave or convex.

###  VI.1 Characteristics in 1-dimensional system

Fig. 15 and Fig. 16 show that if the fractional imbalance increases then the
absolute value of the intercept decreases for all the phases in
${}^{6}\mbox{Li}$ for frequency range $16\leq\omega\leq 1000$. The plots are
concave in nature for both positive frequency (negative detuning), and
negative frequency (positive detuning) domains.

Figure 15: $c$ vs. $P$ plot for various phases in positive frequency and
negative detuning domain : (a) BP1 (b) BP2 (c) PS (d) FFLO of
${}^{6}\mbox{Li}$ at narrow resonance width in 1D

Figure 16: $c$ vs. $P$ plot for various phases in negative frequency and
positive detuning domain : (a) BP1 (b) BP2 (c) PS (d) FFLO of
${}^{6}\mbox{Li}$ at narrow resonance width in 1D

###  VI.2 Characteristics in 2-dimensional system

Fig. 17 and Fig. 18 show that if the fractional imbalance increases then the
absolute value of the intercept decreases for different phases in
${}^{6}\mbox{Li}$ for frequency range $9\leq\omega\leq 1000$. The plots for
BP1, BP2 and PS are concave in nature for both positive frequency (negative
detuning) and negative frequency (positive detuning) domain. However, the plot
corresponding to the FFLO phase is convex

Figure 17: $c$ vs. $P$ plot for various phases in positive frequency and
negative detuning domain : (a) BP1 (b) BP2 (c) PS (d) FFLO of
${}^{6}\mbox{Li}$ at narrow resonance width in 2D

Figure 18: $c$ vs. $P$ plot for various phases in negative frequency and
positive detuning domain : (a) BP1 (b) BP2 (c) PS (d) FFLO of
${}^{6}\mbox{Li}$ at narrow resonance width in 2D

###  VI.3 Characteristics in 3-dimensional system

Here we find that (Fig. 19 and Fig. 20) the $c$ vs. $P$ plots for BP1, BP2 and
PS are concave for both positive frequency (negative detuning) and negative
frequency (positive detuning) side for frequency range $13\leq\omega\leq
1000$. The plot for FFLO is, on the other hand, convex for both positive
frequency (negative detuning) and negative frequency (positive detuning) side.

Figure 19: $c$ vs. $P$ plot for various phases in positive frequency and
negative detuning domain : (a) BP1 (b) BP2 (c) PS (d) FFLO of
${}^{6}\mbox{Li}$ at narrow resonance width in 3D

Figure 20: $c$ vs. $P$ plot for various phases in negative frequency and
positive detuning domain : (a) BP1 (b) BP2 (c) PS (d) FFLO of
${}^{6}\mbox{Li}$ at narrow resonance width in 3D

Thus, the slope ($b$ vs. $P$) and the intercept ($c$ vs. $P$) of the frequency
vs. detuning graph lead to a distinct signature for the FFLO phase in both 2D
and 3D.

##  VII Analyzing the nature of the slopes and the intercepts

In this section, we try to explain the results presented in V and VI, by means
of providing analytical arguments and physical justifications. We first
establish that there would be only one non-trivial solution for $\omega$ from
Eq. (10). Then we argue that the absolute value of the slope and the intercept
of the $\omega$ vs. $\epsilon_{b}$ curve increases with a decreasing
population imbalance. Finally, we demonstrate how (i) the
slope($b$)/intercept($c$) of the $\omega$ vs. $\epsilon_{b}$ curve changes
from one phase to another, and (ii) the $b$ vs. $P$ and $c$ vs. $P$ curves
give a distinct signature for the FFLO phase in 2D and 3D.

###  VII.1 A single branch of frequency

The function $f_{1}(\omega)$ can be approximated by a simpler function

$f_{1}(\omega)\approx\frac{\alpha}{\omega}.$ (23)

Here the value of $\alpha$ depends on (i) dimension of the system, (ii) amount
of population imbalance present, and (iii) nature of the pairing. In Fig. 21
we plot $f_{1}(\omega)$ and $\alpha$/$\omega$ for 1D to show how close the two
functions are. In this plot, the blue solid line and red dashed line represent
$f_{1}(\omega)$ and $\frac{\alpha}{\omega}$ respectively. So, Eq. (10) will
now be

$\epsilon_{b}=-\omega-\frac{g^{2}_{2}\alpha}{\omega-g_{1}\alpha}$ (24)

Simplifying above equation we get

$\begin{split}\omega_{\pm}=\frac{g_{1}\alpha-\epsilon_{b}}{2}\pm\frac{\sqrt{(\epsilon_{b}-g_{1}\alpha)^{2}-4\alpha(g^{2}_{2}-g_{1}\epsilon_{b})}}{2}\end{split}$
(25)

From the above equation we obtain two roots. One root is close to zero and the
other root is prominent. For positive $\epsilon_{b}$, $\omega_{-}>>\omega_{+}$
and $\omega_{+}\approx 0$. . Similarly, For negative $\epsilon_{b}$,
$\omega_{+}>>\omega_{-}$ and $\omega_{-}\approx 0$. Thus, only one non-trivial
solution for $\omega$ exists, and that explains the nature of the plot in Fig.
6, 7, 8, 9, 10, 11. No oscillation region of these plots implies imaginary
$\omega$. The fluctuation of the condensate fraction is, therefore, purely
periodic as in Eq. (21).

###  VII.2 Effect of the population imbalance on the slope and intercept of
frequency vs. detuning curves

The limits of the integration in Eqs. (11), (12) and (13) are controlled by
the fractional imbalance $P$. So, $f_{1}(\omega)$ depends on $P$. If
$f_{1}(\omega)$ is approximated as in Eq. (23), we find that if $P$ decreases
then $\alpha$ increases and vice versa. A set of $\alpha$ vs. $P$ values is
presented in Table [5] for the PS state, and the overall trend remains the
same for the other phases as well.

####  VII.2.1 Variation of slope with fractional imbalance

The slope of the $\omega$ vs. $\epsilon_{b}$ curve, as per Eq. (25) is

$\begin{split}b=\frac{d\omega_{\pm}}{d\epsilon_{b}}=-\frac{1}{2}\pm\frac{1}{2}\frac{\epsilon_{b}+g_{1}\alpha}{\sqrt{(\epsilon_{b}-\alpha
g_{1})^{2}-4\alpha(g^{2}_{2}-\epsilon_{b}g_{1})}}\end{split}$ (26)

We consider the non-trivial branch always, i.e., for $\epsilon_{b}>0$, $b$ is
calculated using $\omega_{-}$; while for $\epsilon_{b}<0$, $b$ is calculated
using $\omega_{+}$. Accordingly, $b$ vs. $\alpha$ is plotted in Fig. 22. It
shows that $b$ increases linearly with $\alpha$ when other parameters are held
constant. Thus, if $P$ decreases then $b$ increases. This can explain the
traits shown in Fig. 12, 13, and 14.

Figure 21:  Plot of $f_{1}(\omega)$ denoted by blue solid line and its nearest
function denoted by red dashed line vs. $\omega$ Figure 22:  $b$ vs. $\alpha$
plot for $\epsilon_{b}<0$

####  VII.2.2 Variation of intercept with fractional imbalance

Figure 23: $\omega$ vs. $\epsilon_{b}$ plot for ${}^{6}\mbox{Li}$ (narrow
resonance width) at $P=0.5$ for $\epsilon_{b}>0$ of PS state in 1D.

The $\omega_{+}$ vs. $\epsilon_{b}$ curve for ${}^{6}\mbox{Li}$ (narrow
resonance width) at $P=0.5$ , as obtained from Eq. (25) is presented in Fig.
23, corresponding to $\epsilon_{b}>0$ and PS state in 1D. It clearly indicates
a positive intercept, i.e., $c>0$. It can be argued that if $\omega_{-}\approx
0$ in Eq. (25), then

$\omega_{+}\approx-\epsilon_{b}+g_{1}\alpha$ (27)

Thus, $c=g_{1}\alpha$. Since if $P$ decreases then $\alpha$ increases, it
implies that if $P$ decreases then the intercept $c$ increases, and vice
versa.

###  VII.3 The nature of pairing as reflected in Eq. 8

If we consider different types of pairing (e.g., BP1/BP2/PS/FFLO), then the
limits of the integrations in Eqs. (11), (12), (13) differ from one phase to
another, affecting the value of $f_{1}(\omega)$. For example, if we choose a
particular fractional imbalance ($P$=0.5, say) then the order of
$f_{1}(\omega)$ is PS$>$BP2$>$BP1 for both positive and negative frequency
regions and all possible dimensions, as shown in Table 6, and 7. The magnitude
of $f_{1}(\omega)$ for FFLO lies between the $f_{1}(\omega)$ values of BP2 and
BP1 phase in 1D. On the other hand, for 2D and 3D, the magnitude of
$f_{1}(\omega)$ for FFLO phase is greater than the magnitudes of
$f_{1}(\omega)$ for all other phases for positive frequency and less than all
other phases for negative frequency respectively. The corresponding numerical
values for a few specific frequency values are presented in Tables 8, 9, 10
and 11.

###  VII.4 Concavity or convexity of slope vs. fractional imbalance plot

One striking feature of the slope ($b$) vs. population imbalance ($P$) and
intercept ($c$) vs. population imbalance ($P$) plots is that the FFLO phase
gives a completely distinct signature. In both the positive and negative
frequency domains, the curves are concave for all other phases, viz., BP1, BP2
and PS, while for FFLO it is convex.

We try to elaborate on this concavity and convexity. The slope of the $b$ vs
$P$ curve is given by

$\frac{\partial b}{\partial P}=\frac{\partial
b}{\partial\alpha}\frac{\partial\alpha}{\partial P}$ (28)

$\frac{\partial b}{\partial\alpha}$ is almost constant for all phases. So,
whether the curve will be a concave or a convex one depends solely on
$\frac{\partial\alpha}{\partial P}$. For BP1, BP2 and PS phases, the $\alpha$
vs. $P$ curve in the positive frequency domain shows a concave nature (the
curve corresponding to the PS phase is shown in Fig. 24 (a)). In contrast, the
curve for FFLO state shows a convex nature (Fig. 24 (b)).

Figure 24: $\alpha$ vs. $P$ plot for various phases in positive frequency and
negative detuning domain : (a) BP1/BP2/PS (b) FFLO of ${}^{6}\mbox{Li}$
(narrow resonance width)

Similar arguments can be put forward for the intercept vs. population
imbalance curves as well. All these explain the concavity and convexity
corresponding to the exotic phases as reported in Sec.V and Sec.VI.

##  VIII Discussion

In this work, we theoretically studied the dynamics of the population-
imbalanced ultracold fermions in an optical lattice. Considering a Feshbach
coupling that enables the fermionic atoms to form bosonic molecules, the
fluctuation of the condensate fraction is probed. We showed that beyond a
small threshold value of the Feshbach detuning, the dynamics of the condensate
fraction is always periodic in nature. This implies that the oscillation is
not just a small “fluctuation", rather, it is embedded in the mean-field
description of the system itself : the periodic dynamics is sustained
throughout the course of time. This result is independent of the amount of
imbalance present in the system, as well as of the pairing structure.

In particular, we considered different pairing structures that arise in
population-imbalanced fermionic systems, viz., Breached Pair (BP1 and BP2),
Phase Separation (PS), Fulde-Ferrel-Larkin-Ovchinikov(FFLO). Using the exact
values of the Feshbach-resonance parameters of ${}^{6}\mbox{Li}$ and
${}^{40}\mbox{K}$, we calculated the oscillation frequencies of the condensate
fraction for each of these phases, and in different dimensions. The result is
presented in the form of frequency ($\omega$) versus Feshbach detuning
($\epsilon_{b}$) plots. Again, we observed that neither the nature of the
pairing (i.e., whether it is BP1/BP2/PS/FFLO), nor the choice of the samples (
e.g., ${}^{6}\mbox{Li}$ or ${}^{40}\mbox{K}$) / dimensionality (1, 2 or 3
dimension) affects the overall qualitative description : The $\omega$ vs.
$\epsilon_{b}$ curve always shows a straight-line like behaviour. This also
implies that the dynamics of the population-imbalanced system in an optical
lattice is markedly different from that in a harmonic trap/ homogeneous
system.

However, the equation of this straight line is dependent on the dimension of
the system, the nature of the pairing, and most importantly, the amount of
imbalance present. Thus, the slope and intercept of this line can be used to
characterize the population-imbalanced system : like the amount of imbalance
can be deduced for a certain structure from the slope of this line, if the
slope/ intercept is known for the same system at some other fixed imbalance
values. Next the slope and intercept of this straight line are plotted with
respect to the fractional imbalance. In this case, the FFLO phase gives a
distinct signature. In 2D or 3D, the curve for FFLO shows a convex nature
(while the curve for all other phases are concave ones).

The FFLO phase is fundamentally different from the other phases in the sense
that it hosts a non-zero momentum pairing (and a broken translational
symmetry) and that is reflected in this pattern. Thus, if one can construct
the The $\omega$ vs. $\epsilon_{b}$ lines for a fixed population imbalance by
experimentally varying the Feshbach magnetic field, and then repeat the same
with different values of imbalance, one can have a wonderful insight in the
nature of the pairing. As the detection of exotic phases like FFLO is quite a
difficult and elusive process, such an indirect probe might turn out to be of
great use in the long run.

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##  Appendix A Calculation of parameters for Li-6 and K-40

In this section, we compute the $g_{1}$ and $g_{2}$ values for
${}^{6}\mbox{Li}$ and ${}^{40}\mbox{K}$ corresponding to different amounts of
fractional imbalance and dimension for ${}^{6}\mbox{Li}$ and ${}^{40}\mbox{K}$
sample. Building element of $g_{1}$ and $g_{2}$ for both ${}^{6}\mbox{Li}$ and
${}^{40}\mbox{K}$ has been shown in Table [1] in the page of tables. Table 2,
Table 3 and Table 4 are given in the page of table for various dimension of
system parameters.

Table 1: Building elements of $g_{1}$ and $g_{2}$ [69] Sample name | resonance width ($\Delta B$) | resonance position ($B_{0}$) | background scattering length ($a_{bg}$)
---|---|---|---
${}^{6}\mbox{Li}$ |  $0.1$ G |  $543.25$ G | $59a_{0}$
${}^{6}\mbox{Li}$ |  $-300$ G |  $834.149$ G | $-1450a_{0}$
${}^{40}\mbox{K}$ |  $7.8$ G |  $202.10$ G | $174a_{0}$
${}^{40}\mbox{K}$ |  $9.7$ G |  $224.21$ G | $174a_{0}$
Table 2: System parameters for 1D Sample name | resonance width ($\Delta B$) | J (set $0.065E^{F}_{r}=1$) |  $g_{1}$(set $0.065E^{F}_{r}=1$) |  $g_{2}$ (set $0.065E^{F}_{r}=1$)
---|---|---|---|---
${}^{6}\mbox{Li}$ |  $0.1$ G | $-1$ | $5.785$ | $17.077$
${}^{6}\mbox{Li}$ |  $-300$ G | $-1$ | $-137.785$ | $4566.969$
${}^{40}\mbox{K}$ |  $7.8$ G | $-1$ | $17.062$ | $613.215$
${}^{40}\mbox{K}$ |  $9.7$ G | $-1$ | $17.062$ | $683.862$
Table 3: System parameters for 2D Sample name | resonance width ($\Delta B$) | J (set $0.065E^{F}_{r}=1$) |  $g_{1}$(set $0.065E^{F}_{r}=1$) |  $g_{2}$ (set $0.065E^{F}_{r}=1$)
---|---|---|---|---
${}^{6}\mbox{Li}$ |  $0.1$ G | $-1$ | $3.6$ | $13.477$
${}^{6}\mbox{Li}$ |  $-300$ G | $-1$ | $-85.738$ | $3602.677$
${}^{40}\mbox{K}$ |  $7.8$ G | $-1$ | $10.615$ | $483.723$
${}^{40}\mbox{K}$ |  $9.7$ G | $-1$ | $10.615$ | $539.676$
Table 4: System parameters for 3D Sample name | resonance width ($\Delta B$) | J (set 0.065$E^{F}_{r}=1$) |  $g_{1}$(set $0.065E^{F}_{r}=1$) |  $g_{2}$ (set $0.065E^{F}_{r}=1$)
---|---|---|---|---
${}^{6}\mbox{Li}$ |  $0.1$ G | $-1$ | $2.246$ | $10.631$
${}^{6}\mbox{Li}$ |  $-300$ G | $-1$ | $-53.369$ | $2841.6$
${}^{40}\mbox{K}$ |  $7.8$ G | $-1$ | $6.6$ | $381.738$
${}^{40}\mbox{K}$ |  $9.7$ G | $-1$ | $6.6$ | $425.538$
Table 5: Variation of nearest form of $f_{1}(\omega)$ with fractional imbalance for PS fractional imbalance | paired region | actual form of $f_{1}$($\omega=16$ to $1000$) | numerator of nearest function( $\frac{\alpha}{\omega}$)
---|---|---|---
$0.1$ |  $-\frac{9\pi}{22}$ to $\frac{9\pi}{22}$ | $\frac{0.63662ArcTanh\frac{-2.9924-0.748591\omega}{15.9999931+(0.005242-\omega)\omega}}{16+(0.005242-\omega)\omega}$ | $0.38$
$0.3$ |  $-\frac{7\pi}{26}$ to $\frac{7\pi}{26}$ | $\frac{0.63662ArcTanh\frac{-1.796-0.450\omega}{15.999+(0.0157-\omega)\omega}}{15.999+(0.016-\omega)\omega}$ | $0.28$
$0.5$ |  $-\frac{\pi}{6}$ to $\frac{\pi}{6}$ | $\frac{0.63662ArcTanh\frac{-1.06829-0.2679\omega}{15.99+(0.0262-\omega)\omega}}{15.9998+(0.0262-\omega)\omega}$ | $0.18$
$0.7$ |  $-\frac{3\pi}{34}$ to $\frac{3\pi}{34}$ | $\frac{0.63662ArcTanh\frac{-0.555-0.139\omega}{15.999+(0.036-\omega)\omega}}{15.9997+(0.036-\omega)\omega}$ | $0.1$
$0.9$ |  $-\frac{\pi}{38}$ to $\frac{\pi}{38}$ | $\frac{0.63662ArcTanh\frac{-0.16449-0.0413603\omega}{15.999+(0.046-\omega)\omega}}{15.9995+(0.046-\omega)\omega}$ | $0.028$
Table 6: Order of $f_{1}(\omega)$ for positive frequency in 1D for narrow resonance width of ${}^{6}\mbox{Li}$ fractional |  |  |
---|---|---|---
imbalance | frequency | nature of pairing | $f_{1}(\omega=20)$
$0.5$ |  $16$ to $20$ | BP1 | $0.008791$
$0.5$ |  $16$ to $20$ | BP2 | $0.00947262$
$0.5$ |  $16$ to $20$ | PS | $0.01031$
$0.5$ |  $16$ to $20$ | FFLO | $0.00924235$
Table 7: Order of $f_{1}(\omega)$ for negative frequency in 1D for narrow resonance width of ${}^{6}\mbox{Li}$ fractional |  |  |
---|---|---|---
imbalance | frequency | nature of pairing | $f_{1}(\omega=-20)$
$0.5$ |  $-20$ to $-16$ | BP1 | $-0.00793369$
$0.5$ |  $-20$ to $-16$ | BP2 | $-0.00754065$
$0.5$ |  $-20$ to $-16$ | PS | $-0.00699348$
$0.5$ |  $-20$ to $-16$ | FFLO | $-0.0076237$
Table 8: Order of $f_{1}(\omega)$ for positive frequency in 2D for narrow resonance width of ${}^{6}\mbox{Li}$ fractional |  |  |
---|---|---|---
imbalance | frequency | nature of pairing | $f_{1}(\omega=20)$
$0.5$ |  $10$ to $20$ | BP1 | $0.00195351$
$0.5$ |  $10$ to $20$ | BP2 | $0.00205625$
$0.5$ |  $10$ to $20$ | PS | $0.00215206$
$0.5$ |  $10$ to $20$ | FFLO | $0.004984$
Table 9: Order of $f_{1}(\omega)$ for positive frequency in 3D for narrow resonance width of ${}^{6}\mbox{Li}$ fractional |  |  |
---|---|---|---
imbalance | frequency | nature of pairing | $f_{1}(\omega=20)$
$0.5$ |  $13$ to $20$ | BP1 | $0.000580916$
$0.5$ |  $13$ to $20$ | BP2 | $0.000624014$
$0.5$ |  $13$ to $20$ | PS | $0.00065759$
$0.5$ |  $13$ to $20$ | FFLO | $0.00178948$
Table 10: Order of $f_{1}(\omega)$ for negative frequency in 2D for narrow resonance width of ${}^{6}\mbox{Li}$ fractional |  |  |
---|---|---|---
imbalance | frequency | nature of pairing | $f_{1}(\omega=-20)$
$0.5$ |  $-20$ to $-1$ | BP1 | $-0.00100408$
$0.5$ |  $-20$ to $-1$ | BP2 | $-0.000981385$
$0.5$ |  $-20$ to $-1$ | PS | $-0.00095865$
$0.5$ |  $-20$ to $-1$ | FFLO | $-0.00243312$
Table 11: Order of $f_{1}(\omega)$ for negative frequency in 3D for narrow resonance width of ${}^{6}\mbox{Li}$ fractional |  |  |
---|---|---|---
imbalance | frequency | nature of pairing | $f_{1}(\omega=-20)$
$0.5$ |  $-20$ to $-1$ | BP1 | $-0.000185725$
$0.5$ |  $-20$ to $-1$ | BP2 | $-0.000182148$
$0.5$ |  $-20$ to $-1$ | PS | $-0.000179089$
$0.5$ |  $-20$ to $-1$ | FFLO | $-0.000532398$