PHYSICAL REVIEW A 91, 033628 (2015)
Quantum quench phase diagrams of an s-wave BCS-BEC condensate
E. A. Yuzbashyan,
1
M. Dzero,
2
V. G u r a r i e ,
3
and M. S. Foster
1,4
1
Center for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA
2
Department of Physics, Kent State University, Kent, Ohio 44240, USA
3
Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
4
Department of Physics and Astronomy, Rice University, Houston, Texas 7700, USA
(Received 9 January 2015; published 23 March 2015)
We study the dynamic response of an s-wave BCS-BEC (atomic-molecular) condensate to detuning quenches
within the two-channel model beyond the weak-coupling BCS limit. At long times after the quench, the condensate
ends up in one of three main asymptotic states (nonequilibrium phases), which are qualitatively similar to those
in other fermionic condensates defined by a global complex order parameter. In phase I the amplitude of the
order parameter vanishes as a power law, in phase II it goes to a nonzero constant, and in phase III it oscillates
persistently. We construct exact quench phase diagrams that predict the asymptotic state (including the many-body
wave function) depending on the initial and final detunings and on the Feshbach resonance width. Outside of
the weak-coupling regime, both the mechanism and the time dependence of the relaxation of the amplitude of
the order parameter in phases I and II are modified. Also, quenches from arbitrarily weak initial to sufficiently
strong final coupling do not produce persistent oscillations in contrast to the behavior in the BCS regime. The
most remarkable feature of coherent condensate dynamics in various fermion superfluids is an effective reduction
in the number of dynamic degrees of freedom as the evolution time goes to infinity. As a result, the long-time
dynamics can be fully described in terms of just a few new collective dynamical variables governed by the
same Hamiltonian only with “renormalized” parameters. Combining this feature with the integrability of the
underlying (e.g., the two-channel) model, we develop and consistently present a general method that explicitly
obtains the exact asymptotic state of the system.
DOI: 10.1103/PhysRevA.91.033628 PACS number(s): 67.85.De, 34.90.+q, 74.40.Gh
I. INTRODUCTION
The problem of a superconductor driven out of equilibrium
by a sudden perturbation goes back many decades. Early
studies [1–6] addressed small deviations from equilibrium
using linearized equations of motion. An important result was
obtained by Volkov and Kogan [3], who discovered a power
law oscillatory attenuation of the Bardeen-Cooper-Schriffer
(BCS) order parameter for nonequilibrium initial conditions
close to the superconducting ground state.
In the past decade it was realized that even large deviations
from equilibrium are within the reach of appropriate theo-
retical methods. Recent studies, motivated by experiments
in cold atomic fermions, focused on quantum quenches,
nonequilibrium conditions created by a sudden change in
the superconducting coupling strength. Barankov et al. [7],
in a paper that set off a surge of modern research in this
long-standing problem [8–24] in the context of quantum gases,
found that for initial conditions close to the unstable normal
state, the order parameter exhibits large anharmonic periodic
oscillations.
Subsequently, Yuzbashyan et al. [16] developed an ana-
lytical method to predict the s tate of the system at large
times based on the integrability of the underlying BCS model.
This work extended Volkov and Kogan’s result to l arge
deviations from equilibrium and showed that the oscillation
frequency is twice the nonequilibrium asymptotic value of the
order parameter, a conclusion confirmed by recent terahertz
pump pulse experiments in Nb
1-x
Ti
x
N films [25,26]. Later
studies [17,18] mapped out the full quantum quench “phase
diagram” for weakly coupled s-wave BCS superconductors
finding that three distinct regimes occur depending on the
strength of the quench: Volkov-and-Kogan-like behavior,
persistent oscillations, and exponential vanishing of the order
parameter. Most recent research [27–30] fueled by exper-
imental breakthroughs [25,31,32] investigates nonadiabatic
dynamics of s-wave BCS superconductors in response to fast
electromagnetic perturbations. Closely related subjects devel-
oping in parallel are exciton dynamics [33], collective neutrino
oscillations [34,35], quenched p-wave superfluids [36,37], etc.
Most existing work addressed the dynamics in the BCS
regime and, in particular, quenches such that the interaction
strength is weak both before and after the quench. This was
so that the system always remains in the BCS regime, since
the physics of the condensate beyond this regime was not
sufficiently well understood. However, a superfluid made up
of cold atoms can be as well quenched from the BCS to
the Bose-Einstein condensation (BEC) regime or within the
BEC regime. With few exceptions [23,36,37], these types of
quenches are not adequately studied in the existing literature.
Our paper aims to close this gap and analyze all possible
interaction quenches throughout the BCS-BEC crossover in a
paired superfluid, including BCS-to-BEC, BEC-to-BCS, and
BEC-to-BEC quenches. We fully determine the steady state of
the system at large times after the quench: the asymptote of the
order parameter, as well as the approach to the asymptote; the
many-body wave function; and certain observables, such as
the radio-frequency absorption spectrum and the momentum
distribution. In the BCS limit, we recover previous results.
Beyond this limit the dynamics is quantitatively and some-
times qualitatively different. For example, the power law in
the Volkov-and-Kogan-like attenuation changes in the BEC
regime, exponential vanishing is replaced with a power law,
and persistent oscillations first change their form and then
1050-2947/2015/91(3)/033628(43) 033628-1 ©2015 American Physical Society
YUZBASHYAN, DZERO, GURARIE, AND FOSTER PHYSICAL REVIEW A 91, 033628 (2015)
disappear altogether after a certain threshold for quenches
from any initial (e.g., arbitrarily weak) to sufficiently strong
final coupling. We believe an experimental verification of
the predictions of this work is within a reach of current
experiments in cold atomic systems.
The long-time dynamics can be determined explicitly due
to a remarkable reduction mechanism at work, so that at large
times the system is governed by an effective interacting Hamil-
tonian with just a few classical collective spin or oscillator
degrees of freedom. In a sense, the system “flows in time” to a
much simpler Hamiltonian. This observation, combined with
the integrability of the original Hamiltonian (see below), lead
to a method originally proposed in Ref. [16] for obtaining the
long-time asymptote (steady state) of integrable Hamiltonian
dynamics in the continuum (thermodynamic) limit. Here we
improve this method as well as provide its comprehensive and
self-contained review including many previously unpublished
results and steps. We do so in the context of the s-wave
BCS (one channel) and inhomogeneous Dicke (two-channel)
models, but with some modifications the same method also
applies to all known integrable pairing models [38–44], such
as p + ip superfluids [36,37], integrable fermion or boson
pairing models with nonuniform interactions [45,46], Gaudin
magnets (central spin models), and potentially can be extended
to a much broader class of integrable nonlinear equations.
The purpose of this paper is therefore twofold. First, it
serves as an encyclopedia of quantitatively exact predictions,
new and old, for the quench dynamics of real s-wave BCS-BEC
condensates in two and three spatial dimensions. Readers
primarily interested in this aspect of our work will find most
of the relevant information in the Introduction, Sec. VII,
and Conclusion. In particular, Sec. IDconcisely summarizes
our main results and provides a guide to other sections that
contain further results and details. Our second goal is to
develop and thoroughly review a method for determining the
far-from-equilibrium dynamics in a certain class of integrable
models. We refer readers interested in learning about the
method to Sec. II. Also, from this viewpoint, Secs. III and IV
should be considered as applications of our approach and
Sec. V as a related development.
A major experimental breakthrough with ultracold atoms
was achieved in 2004, when they were used to emulate s-wave
superconductors with an interaction strength that can be varied
at will [47,48]. The experimental control parameter is the
detuning ω, the binding energy of a two-fermion bound state
(molecule). This parameter determines the strength of the
effective interaction between fermions and can be varied both
slowly and abruptly with the help of a Feshbach resonance.
Moreover, it is straightforward to make time-resolved mea-
surements of t he subsequent evolution of the system. Thus,
cold atoms provide a natural platform to study quenches in
superfluids and in a variety of other setups [49,50].
At large ω we have fermionic atoms with weak effective
attraction that form a paired superfluid, an analog of the
superconducting state of electrons in a metal. As ω is
decreased, the atoms pair up into bosonic molecules which
then Bose condense. It was argued for a long time that both
the paired superfluid and the Bose-condensed molecules are in
the same phase of the fermionic gas, named the BCS-BEC
condensate [51,52]. As ω decreases, the strength of the
effective interaction (coupling) between fermions increases
from weak to strong and the system undergoes a BCS-BEC
crossover. At ω 2ε
F
, where ε
F
is the Fermi energy, the
system is deep in the BCS regime, while at large negative ω
it is deep in the BEC regime. It is not known how to r ecreate
such a crossover in a conventional solid-state superconductor
since the interaction strength cannot be easily adjusted.
In a quantum quench setup the system is prepared in the
ground state at a detuning ω
i
.Att = 0 the detuning is suddenly
changed, ω
i
→ ω
f
.Att>0 the system evolves with a new
Hamiltonian H (ω
f
). The main goal is to determine the state
of the system at large times, t →∞.
A. Models and approximations
We consider two closely related models in this paper in both
two and three dimensions. The first one is the well-known
two-channel model that describes two species of fermionic
atoms interacting via an s-wave Feshbach resonance
ˆ
H
2ch
=
p,σ =↑,↓
p
ˆ
a
†
pσ
ˆ
a
pσ
+
q
ω +
q
2
4m
ˆ
b
†
q
ˆ
b
q
+g
pq
ˆ
b
†
q
ˆ
a
q
2
+p,↑
ˆ
a
q
2
−p,↓
+
ˆ
b
q
ˆ
a
†
q
2
−p,↓
ˆ
a
†
q
2
+p,↑
. (1.1)
It is convenient to think of the two types of fermions of mass
m and energy
p
= p
2
/2m as spin-up and spin-down, created
and annihilated by operators
ˆ
a
†
pσ
and
ˆ
a
pσ
. The interaction term
converts two fermions into a bosonic molecule and vice versa
at a rate controlled by the parameter g. Molecules are created
and annihilated by
ˆ
b
†
q
and
ˆ
b
q
and have a binding energy ω.
The parameter g is set by the type of atoms and the specifics
of a particular Feshbach resonance and cannot be changed
in a single experiment; ω can be varied at will by varying the
magnitude of the magnetic field applied during the experiment.
This model describes atoms in the BCS regime when ω is
large, which undergo a crossover to the BEC regime as ω is
decreased.
A parameter with dimensions of energy important for our
analysis of this model is g
2
ν
F
, where ν
F
is the bulk density of
states (proportional to the total volume) at the Fermi energy
F
. A well-known parameter,
γ =
g
2
ν
F
F
, (1.2)
controls whether the resonance is narrow γ 1 or broad
γ 1. This parameter is the dimensionless atom-molecule
interaction strength or, equivalently, the resonance width.
A very convenient feature of the narrow resonance is that,
regardless of the regime of the system, controlled by ω,the
system is adequately described with mean-field theory [53].
This is already clear from the form of the Hamiltonian: Small
γ implies that interaction g is small.
Broad resonances, on the other hand, correspond to large
g. Under those conditions it is possible to integrate out the
molecules
ˆ
b
q
to arrive at a simpler Hamiltonian [53] describing
fermions interacting via a short-range attractive interaction
033628-2
QUANTUM QUENCH PHASE DIAGRAMS OF AN s-WAVE . . . PHYSICAL REVIEW A 91, 033628 (2015)
with variable strength,
ˆ
H
1ch
=
p,σ =↑,↓
p
ˆ
a
†
pσ
ˆ
a
pσ
−
λ
ν
F
pp
q
ˆ
a
†
q
2
−p,↓
ˆ
a
†
q
2
+p,↑
ˆ
a
q
2
+p
,↑
ˆ
a
q
2
−p
,↓
, (1.3)
where
λ =
g
2
ν
F
ω
=
γε
F
ω
. (1.4)
This is the single (one)-channel, or BCS, model, which is
the second model we analyze in this paper. It also describes
the BCS-BEC crossover as ω is decreased (λ is increased).
However, while in the BCS and (to some extent) in the BEC
regimes corresponding to large and small λ, respectively,
mean-field theory holds in equilibrium, for the intermediate
values of λ (neither large nor small) the mean-field theory
is known to break down. A special value of λ in the middle
of the regime unaccessible to the mean-field theory already
in equilibrium is called the unitary point. It corresponds to
the interaction strength where molecules are about to be
formed. Noncondensed molecules play an important role in
the description of the unitary point and its special properties
are a s ubject of many studies in the literature [52,54].
Just as in earlier work on the far-from-equilibrium su-
perconductivity, we analyze the quench dynamics in the
mean-field approximation where no molecules are transferred
into or out of the BCS-BEC condensate after the quench;
i.e., the dynamics of the condensate is decoupled from
the noncondensed modes. We analyze the validity of this
approximation for nonequilibrium steady states produced by
quenches in the two-channel model in Appendix A. We find
that the situation is similar to that in equilibrium [53]. In the
case of a broad Feshbach resonance, mean field is expected
to hold for quenches where both initial and final detunings
are far from the unitary point. A quench into the unitary point
is a very interesting problem addressed by some publications
before [55], but the method we employ here is not applicable
to this case.
Nevertheless, a variety of quenches are still accessible to
our description even when the resonance is broad, includ-
ing BCS → BCS, BCS → BEC, BEC → BCS, and BEC →
BEC, where BCS and BEC stand for the value of the interaction
strength far weaker or far stronger than that at the unitary point.
In the case of BCS-BEC superfluids formed with interactions
generated by narrow Feshbach resonances, the mean-field
theory treatment is valid even at the threshold of the formation
of the bound state and throughout the BCS-BEC crossover.
Here we consider quenches of the detuning ω for both narrow
and broad resonances within the mean field. Note that in the
case of the one-channel model we expect the mean field on the
BEC side to be valid only in the far BEC limit where the ground
state essentially consists of noninteracting Bose-condensed
molecules [56].
In the mean-field treatment the condensate is described by
the q = 0 part of the Hamiltonian (1.1), which is decoupled
from q = 0 terms in this approximation. The Hamiltonian
therefore becomes
ˆ
H
2ch
=
p
2
p
ˆ
s
z
p
+ ω
ˆ
b
†
ˆ
b + g
p
(
ˆ
b
†
ˆ
s
−
p
+
ˆ
s
+
p
ˆ
b), (1.5)
where
ˆ
s
−
p
=
ˆ
a
p↑
ˆ
a
−p↓
,
ˆ
s
z
p
=
1
2
ˆ
a
†
p↑
ˆ
a
p↑
+
ˆ
a
†
−p↓
ˆ
a
−p↓
− 1
(1.6)
are Anderson pseudospin-
1
2
operators [1] and
ˆ
b =
ˆ
b
q=0
.
Hamiltonian (1.5) is also known as inhomogeneous Dicke
or Tavis-Cummings model. In a quantum quench problem
we need to solve Heisenberg equations of motion for this
Hamiltonian for given initial conditions
d
ˆ
s
p
dt
=
ˆ
B
p
×
ˆ
s
p
,
d
ˆ
b
dt
=−iω
ˆ
b − ig
ˆ
J
−
,
(1.7)
ˆ
J =
p
ˆ
s
p
,
ˆ
B
p
= 2g
ˆ
b + 2
p
ˆ
z,
where
ˆ
b =
ˆ
b
x
ˆ
x +
ˆ
b
y
ˆ
y,
ˆ
b
x
, and −
ˆ
b
y
are Hermitian and anti-
Hermitian parts of the operator
ˆ
b =
ˆ
b
x
− i
ˆ
b
y
, and
ˆ
x,
ˆ
y,
ˆ
z are
coordinate unit vectors.
The second step in the mean-field treatment of the two-
channel model is to replace Heisenberg operator
ˆ
b(t)inthe
first equation of motion in Eq. (1.7) with its time-dependent
quantum-mechanical average,
ˆ
b(t) →
ˆ
b(t)≡b(t), which is
expected to be exact in thermodynamic limit as long as the
q = 0 state is macroscopically occupied at all times. This
replacement can be shown to be exact in equilibrium using
the exact solution for the spectrum of the inhomogeneous
Dicke model [38,57] and numerically for the time-dependent
problem [58]. Upon this replacement equations of motion be-
come linear in operators and taking their quantum-mechanical
average, we obtain
˙
s
p
=
B
p
×s
p
,
˙
b =−iωb − igJ
−
,
(1.8)
J =
p
s
p
,
B
p
= 2g
b + 2
p
ˆ
z,
where s
p
=
ˆ
s
p
. These are Hamiltonian equations of motion
for a classical Hamiltonian,
H
2ch
=
p
2
p
s
z
p
+ ω
¯
bb + g
p
(
¯
bs
−
p
+ bs
+
p
), (1.9)
which describes a set of angular momenta (classical spins or
vectors) coupled to a harmonic oscillator. Here,
¯
b denotes the
complex conjugate of b. These dynamical variables obey the
Poisson brackets
s
a
p
,s
b
k
=−ε
abc
δ
pk
s
c
p
, {b,
¯
b}=i, (1.10)
where a, b, and c stand for spatial indicies x, y, and z.
Similar steps in the case of the single-channel model (1.3)
lead to a classical spin Hamiltonian,
H
1ch
=
p
2
p
s
z
p
−
λ
ν
F
p,p
s
−
p
s
+
p
, (1.11)
together with the corresponding equations of motion.
033628-3
YUZBASHYAN, DZERO, GURARIE, AND FOSTER PHYSICAL REVIEW A 91, 033628 (2015)
An important characteristic of the system both in and out
of equilibrium is the superfluid order parameter or the gap
function defined in the two-channel model as
(t) =−g
ˆ
b(t)=−gb(t) ≡
x
(t) − i
y
(t). (1.12)
In the one-channel limit, this expression turns into
1ch
(t) =
λ
ν
F
p
ˆ
a
p↑
(t)
ˆ
a
−p↓
(t)=
λ
ν
F
p
s
−
p
. (1.13)
The magnitude |(t)| of the order parameter is known as
the Higgs or amplitude mode for its similarity with the
Higgs boson [20,59] and its time-dependent phase represents
a Goldstone mode. Note, however, that out of equilibrium
the gap function does not entirely determine the state of the
system. It specifies the effective magnetic field acting on each
spin according to Eq. (1.8), but there is still a certain freedom
in how the spin moves in this field. For example, even for
a constant field the spin can precess around it, making an
arbitrary constant angle with its direction.
In the above models we took a free single-particle spectrum,
ε
p
= p
2
/2m, and labeled states with momenta p. This choice
is not essential for our analysis. We can as well consider an ar-
bitrary spectrum ε
i
. The pairing is then between pairs of time-
reversed states [60,61]; see also the first two pages in Ref. [13]
for more details. For example, in Hamiltonian (1.5) this results
in relabeling
ˆ
s
p
→
ˆ
s
i
,
ˆ
a
p↑
ˆ
a
−p↓
→
ˆ
a
i↑
ˆ
a
i↓
,
ˆ
a
†
p↑
ˆ
a
p↑
→
ˆ
a
†
i↑
ˆ
a
i↑
,
etc., where the state |i ↓ is the time-reversed counterpart of
|i ↑. Our results below depend only on the density of the
single-particle states ν(ε) in the continuum limit regardless of
whether these states are characterized by momenta p or any
other set of quantum numbers i.
B. Ground state
In the ground state
(t) =
0
e
−2iμt
, (1.14)
where the magnitude
0
is time independent. Apart from
an overall rotation about the z axis with frequency 2μ,the
ground state is a static solution of the equations of motion that
minimizes H
2ch
. The minimum is achieved when each spin is
directed against its effective magnetic field, i.e.,
s
−
p
=
0
e
−2iμt
2E(ε
p
;
0
,μ)
,s
z
p
=−
ε
p
− μ
2E(ε
p
;
0
,μ)
, (1.15)
where
E(ε; ,μ) ≡
(ε − μ)
2
+
2
. (1.16)
Note that the length of the spin s
p
= 1/2. This is because the
ground state is a tensor product of single spin-
1
2
wave functions
and s
p
=
ˆ
s
p
.
The equation of motion (1.8)forb yields
|J
−
|=
(ω − 2μ)
0
g
2
, (1.17)
which implies a self-consistency equation for
0
(ω − 2μ)
g
2
=
p
1
2E(ε
p
;
0
,μ)
. (1.18)
Further, the Hamiltonian (1.9) conserves
n =
bb +
p
s
z
p
+
1
2
, (1.19)
which is the average total number of bosons and fermion pairs.
This number is related to
0
and the chemical potential μ as
2n =
2
2
0
g
2
+
p
1 −
ε
p
− μ
E(ε
p
;
0
,μ)
. (1.20)
The Fermi energy ε
F
is the chemical potential of the
fermionic atoms at zero temperature in the absence of any
interaction, when only fermions are present. It provides an
overall energy scale and it is convenient to measure all energies
in units of the Fermi energy. Thus, from now on, we set
everywhere below
ε
F
= 1. (1.21)
Below we often switch from discrete to continuum (ther-
modynamic limit) formulations. In the former version, there
are N discrete single-particle energy levels ε
p
with certain
degeneracy each. Any quantity A
p
we consider in this paper
depends on p only through ε
p
, A
p
= A(ε
p
). For example, all
spins s
p
on a degenerate level ε
p
are parallel at all times
and effectively merge into a single vector. There are N such
vectors, so we count N distinct classical spins.
In thermodynamic limit, energies ε
p
form a continuum
on the positive real axis, i.e., are described by a continuous
variable ε with a density of states ν(ε) that depends on the
dimensionality of the problem
ν(ε) = ν
F
f (ε), (1.22)
where ν
F
is the bulk density of states (proportional to
the system volume) at the Fermi energy, f (ε) = 1intwo
dimensions (2D), and f (ε) =
√
ε in 3D. Summations over
p turn into integrations,
p
A
p
→ ν
F
∞
0
A(ε)f (ε)dε. (1.23)
With only fermions present, the total particle number is
2n =
1
0
2ν(ε)dε =
4
d
ν
F
, (1.24)
where d = 2,3 is the number of spatial dimensions. Interaction
redistributes this number between fermions and bosons as in
Eq. (1.20). Combining Eqs. (1.20) and (1.24) and taking the
continuum limit, we obtain
4
d
=
2
2
0
γ
+
∞
0
⎡
⎣
1 −
ε − μ
(ε − μ)
2
+
2
0
⎤
⎦
f (ε)dε, (1.25)
where γ is the dimensionless resonance width defined in
Eq. (1.2).
Similarly, Eq. (1.18) becomes in the thermodynamic limit
2ω − 4μ
γ
=
ε
0
f (ε)dε
(ε − μ)
2
+
2
0
, (1.26)
033628-4
QUANTUM QUENCH PHASE DIAGRAMS OF AN s-WAVE . . . PHYSICAL REVIEW A 91, 033628 (2015)
0 0.2 0.4 0.6 0.8
1
max
-12
-8
-4
0
F
FIG. 1. (Color online) Ground-state chemical potential μ for the
two-channel model in 3D in units of the Fermi energy ε
F
as a function
of the ground-state gap
0
for various resonance width γ . μ(
0
)is
calculated from Eqs. (1.25)and(1.26). Note that in the two-channel
model
0
is bounded from above by
max
.
where ε
is the high-energy cutoff. In 3D it can be eliminated
by an additive renormalization of the detuning ω; see, e.g.,
Ref. [53]. This, however, does not affect our results for the
quench dynamics as they depend on the difference between
the initial and final values of the detuning.
Equations (1.25) and (1.26) contain two independent
parameters not counting the cutoff. For example, we can
choose γ and ω and determine μ and
0
from these equations,
or choose γ and
0
and determine μ and ω etc.; see Fig. 1 for a
plot of μ(
0
) for various γ in 3D. Note also that
2
0
= g
2
¯
bb is
proportional to the number of bosons and is therefore limited
by the total number of particles. Equation (1.25) implies
0
2γ
d
=
max
. (1.27)
C. Quench setup and initial conditions
In a quantum quench setup we prepare the system in a
ground state at a certain detuning ω
i
; i.e., the initial state is
s
−
p
(t = 0) =
0i
2E(ε
p
;
0i
,μ
i
)
,
(1.28)
s
z
p
(t = 0) =−
ε
p
− μ
i
2E(ε
p
;
0i
,μ
i
)
,
where
0i
,μ
i
are the ground-state values determined by
Eqs. (1.25) and (1.26) with ω = ω
i
. We t hen quench the
detuning ω
i
→ ω
f
and evolve the system with the two-channel
Hamiltonian (1.9) starting from the initial state (1.28)att = 0.
The state of the system is fully determined by the many-
body wave function, which in the mean-field treatment is at all
times a product state of the form
|(t)=|ψ(t)⊗(
ˆ
b
†
)
n(t)
|0, (1.29)
where n(t) =|b(t )|
2
and |ψ(t ) is the fermionic part of the
wave function:
|ψ(t )=
p
[u
p
(t) + v
p
(t)
ˆ
a
†
p↑
ˆ
a
†
−p↓
]|0. (1.30)
Bogoliubov amplitudes u
p
(t),v
p
(t) obey the Bogoliubov de
Gennes (BdG) equations
i
∂
∂t
u
p
(t)
v
p
(t)
=
p
(t)
¯
(t) −
p
u
p
(t)
v
p
(t)
, (1.31)
with the normalization condition |u
p
|
2
+|v
p
|
2
= 1. Apart
from an overall time-dependent phase (which is important
for certain observables), these equations are equivalent to the
classical spin equations of motion (1.8) and spins are related
to the amplitudes as
s
−
p
s
p
= 2u
p
v
p
,
s
z
p
s
p
=|v
p
|
2
−|u
p
|
2
, (1.32)
where s
p
is the length of the spin. For quench initial conditions
s
p
= 1/2, as explained below Eq. (1.16).
Each quench is uniquely characterized by three parameters:
the resonance width γ = g
2
ν
F
and the initial ω
i
and final ω
f
values of the detuning in units of the Fermi energy. Indeed,
ω
i
and γ determine
0i
and μ
i
and thus the initial condition,
while the equations of motion (1.7) in the thermodynamic limit
depend only on ω
f
and γ . To see the latter, note that model
parameters enter the equation of motion for spin s
p
≡s(ε
p
)
only through =−gb, while the equation of motion for the
bosonic field b can be equivalently written as
˙
=−iω
f
+ iγ
∞
0
s
−
(ε)f (ε)dε. (1.33)
Instead of ω
i
,ω
f
we find it more convenient to characterize
the quench by
0i
,
0f
, the ground-state gaps corresponding
to these values of the detuning. As discussed below Eq. (1.26),
for a given γ , the detuning ω uniquely determines
0
and
vice versa. Note that
0f
has nothing to do with the time-
dependent gap function (t) and in particular with the large-
time asymptote (t →∞). Whenever (t) goes to a constant
at large times, we denote this constant
∞
.
D. Main results
Our main result is a complete description of the long-time
dynamics of two- and one-channel models (1.9) and (1.11)
in two and three spatial dimensions following a quench of
the detuning ω
i
→ ω
f
(coupling λ
i
→ λ
f
in the one-channel
model) in the thermodynamic limit. A key effect that makes
such a description possible is a drastic reduction in the number
of effective degrees of freedom as t →∞. It turns out that the
large-time dynamics can be expressed in t erms of just a few
new collective spins plus the oscillator in the two-channel case
that are governed by the same Hamiltonians (1.9) and (1.11)
only with new effective parameters replacing ε
p
and ω.The
number of collective spins is m = 0, 1, or 2 and m =−1, 0,
or 2 for one- and two-channel models, respectively, depending
on the quench. The difference is due to the presence of the
oscillator degree of freedom in the latter case. For example,
m =−1 means that the effective large-time Hamiltonian H
red
not only has no spins, but also the oscillator b is absent; i.e.,
H
red
= 0. This reduction effect combined with integrability of
classical Hamiltonians (1.9) and (1.11) allows us to determine
the state of the system (its many-body wave function) at t →
∞. We explain this method in detail in Sec. II. This section
033628-5
![FIG. 12. (Color online) Order parameter (t) vanishes whenever the square of the Lax vector L2(u) has no isolated roots. Panel (a) shows real, Re[c], and imaginary, Im[cm], parts of the roots cm, and panel (b) shows the corresponding | (t)| for a detuning quench in a 3D two-channel model with γ = 0.1 and N = 1024 spins. There are N + 1 pairs of complex conjugate continual roots whose imaginary parts scale as 1/N so that in the N → ∞ limit they form a continuum on the real axis. Here 0i = 0.34 max, 0f = 8.1 × 10−3 max, μi = 0.91εF , and δω = 3.45γ .](/figures/fig-12-color-online-order-parameter-t-vanishes-whenever-the-3vichdyy.webp)
