# The zero dispersion limit for the Benjamin–Ono equation on the line

Patrick Gérard  Université Paris–Saclay, Laboratoire de Mathématiques d’Orsay,
CNRS, UMR 8628, 91405 Orsay, France<EMAIL_ADDRESS>
(Date: July 24, 2023)

###### Abstract.

We identify the zero dispersion limit of a solution of the Benjamin–Ono
equation on the line corresponding to every initial datum in
$L^{2}({\mathbb{R}})\cap L^{\infty}({\mathbb{R}})$. We infer a maximum
principle and a local smoothing property for this limit. The proof is based on
an explicit formula for the Benjamin–Ono equation and on the combination of
calculations in the special case of rational initial data, with approximation
arguments. We also investigate the special case of an initial datum equal to
the characteristic function of a finite interval, and prove the lack of
semigroup property for this zero dispersion limit.

###### Key words and phrases:

Benjamin–Ono equation, explicit formula, zero dispersion limit, Burgers
turbulence, Lax pair, Toeplitz operators

###### 2010 Mathematics Subject Classification:

37K15 primary, 47B35 secondary

It is a pleasure to thank R. Badreddine, D. Bambusi, Y. Brenier, M. Gallone,
L. Gassot, H. Holden, E. Lenzmann, P. Miller and N. Tzvetkov for fruitful
discussions about this problem.

##  1\. Introduction

We consider the Benjamin–Ono equation on the line with a small dispersion
$\varepsilon>0$,

(1)
$\partial_{t}u^{\varepsilon}+\partial_{x}[(u^{\varepsilon})^{2}]=\varepsilon\partial_{x}|D|u^{\varepsilon}\
,\ u^{\varepsilon}(0,x)=u_{0}(x)\ ,$

where $u_{0}\in L^{2}({\mathbb{R}})$ and is real valued. It is well known [18,
21] that equation (1) has a global solution $u^{\varepsilon}\in
C({\mathbb{R}},L^{2}({\mathbb{R}}))$, which is characterized as the strong
limit of smooth solutions with any initial data approximating the initial
datum $u_{0}$ strongly in $L^{2}({\mathbb{R}})$, and that the $L^{2}$ norm
$\|u^{\varepsilon}(t)\|_{L^{2}({\mathbb{R}})}$ is independent of the time
variable $t$.
Our purpose is the description of the weak limit of $u^{\varepsilon}(t)$ in
$L^{2}({\mathbb{R}})$ as $\varepsilon\to 0$. Such a problem is delicate even
for smooth data $u_{0}$, because the formal limit of equation (1) as
$\varepsilon$ tends to $0$ is the inviscid Burgers–Hopf equation [5, 17]

$\partial_{t}u+\partial_{x}(u^{2})=0\ ,$

which is well known to display singularities in finite time. In fact,
numerical simulations — see e.g. [8] — show that, near such singularities, the
solution $u^{\varepsilon}$ display very strong oscillations as $\varepsilon$
tends to $0$, so that the weak limit of $u^{\varepsilon}$, if it exists, is
very difficult to identify.
This zero–dispersion limit problem was first addressed by Lax and Levermore
[20] in the case of the Korteweg–de Vries equation,

(2)
$\partial_{t}v^{\varepsilon}+\partial_{x}[(v^{\varepsilon})^{2}]=\varepsilon\partial_{x}^{3}v^{\varepsilon}\
,\ v^{\varepsilon}(0,x)=v_{0}(x)\ ,$

using the integrability of this equation and the inverse scattering transform
for the Lax operator. Lax–Levermore’s pioneering work initiated a series of
remarkable results describing $v^{\varepsilon}$ for special initial data
$v_{0}$, under various assumptions on the spectral theory of the corresponding
Lax operator, see in particular [26, 7, 16, 6] and references therein. All
these references have in common the use of the inverse scattering theory and
the study of a related Riemann–Hilbert problem, for describing the solution
$v^{\varepsilon}$, and consequently depend on strong assumptions on the
initial data through the spectral theory of its Lax operator — here, a
Schrödinger operator.
Let us also mention recent approaches to similar problems for the
Fermi–Pasta–Ulam chain [9] on a wide range of time and energy scales. We refer
to [10] for a connection to the zero–dispersion limit for the KdV equation
(2).
In the case of the Benjamin–Ono equation (1) on the line, much less is known,
with the notable exceptions of [23, 22]. Again, these references use the
inverse scattering theory for the Lax operator inherited from the
integrability of the Benjamin–Ono equation. The case of the Benjamin–Ono
equation with periodic boundary conditions was studied in [11, 12] where the
inverse spectral theory of [14] was used to establish an explicit description
of the zero dispersion limit in the case of bell–shaped data. Furthermore, in
[12], an explicit formula proved in [13] was used to prove the existence of a
zero–dispersion limit for every initial datum in $L^{\infty}({\mathbb{T}})$.
This suggests that the results of [13] could also be used for the
zero–dispersion limit on the line. This is the purpose of this paper, where we
prove general results on the zero–dispersion limit of (1) without using the
inverse spectral theory of the Lax operator.

###  1.1. Statement of the results

Our first main result is the following.

######  Theorem 1.

As $\varepsilon\to 0$, the solution $u^{\varepsilon}(t)$ of (1) converges
weakly in $L^{2}({\mathbb{R}})$ to some function $ZD[u_{0}](t)$ for every
$t\in{\mathbb{R}}$. The mapping

$u_{0}\in L^{\infty}({\mathbb{R}})\cap L^{2}({\mathbb{R}})\mapsto ZD[u_{0}]$

has the following properties.

  1. (1)

For every $t\in{\mathbb{R}}$, $\|ZD[u_{0}](t)\|_{L^{2}}\leq\|u_{0}\|_{L^{2}}$,
and $ZD[u_{0}]$ is continuous on ${\mathbb{R}}$ with values in
$L^{2}({\mathbb{R}})$ endowed with the weak topology.

  2. (2)

If $u_{0}^{\delta}$ converges strongly to $u_{0}$ in $L^{2}({\mathbb{R}})$ as
$\delta\to 0$, with a uniform bound on $\|u_{0}^{\delta}\|_{L^{\infty}}$,
then, for every $t\in{\mathbb{R}}$, $ZD[u_{0}^{\delta}](t)$ converges to
$ZD[u_{0}](t)$ weakly in $L^{2}({\mathbb{R}})$.

The above theorem is already quite surprising compared to what is known for
the KdV equation. Indeed, as far as we know, the existence of a weak limit for
every time and for every initial datum $v_{0}\in L^{\infty}({\mathbb{R}})\cap
L^{2}({\mathbb{R}})$ is an open problem for equation (2). Coming back to the
Benjamin–Ono equation, recall that a similar result was proved on the torus in
[12]. In fact, on the line, one can go much further in the description of the
zero dispersion limit. Under a slight additional assumption on the regularity
of $u_{0}$ , we can relate the zero dispersion limit $ZD[u_{0}]$ to the
multivalued solution of the inviscid Burgers–Hopf equation with initial datum
$u_{0}$.

######  Theorem 2.

In the conditions of Theorem 1, assume moreover that $u_{0}$ is a $C^{1}$
function tending to $0$ at infinity as well as its first derivative. For every
$t\in{\mathbb{R}}$, the set $K_{t}(u_{0})$ of critical values of the function

$y\in{\mathbb{R}}\mapsto y+2tu_{0}(y)$

is a compact subset of measure $0$. For every connected component $\Omega$ of
$K_{t}(u_{0})^{c}$, there exists a nonnegative integer $\ell$ such that, for
every $x\in\Omega$, the equation

$y+2tu_{0}(y)=x$

has $2\ell+1$ simple real solutions

$y_{0}(t,x)<y_{1}(t,x)<\cdots<y_{2\ell}(t,x)\ ,$

and the zero dispersion limit is given by

(3) $ZD[u_{0}](t,x)=\sum_{k=0}^{2\ell}(-1)^{k}u_{0}(y_{k}(t,x))\ .$

$x$$t$$(t,x)$$\bullet$$y_{0}(t,x)$$y_{1}(t,x)$$y_{2}(t,x)$ Figure 1.1. The
crossing of characteristics

Remarks.

  1. (1)

Formula (3) was proved by Miller–Wetzel [22] – see also Miller–Xu [23] — in
the special case of a rational Klaus–Shaw initial potential, and by Gassot
[11, 12] in the special case of a general bell shaped initial potential with
periodic boundary conditions, using inverse spectral theory for the Lax
operator. Here we use a different approach and we have essentially no
constraint on the initial datum $u_{0}$.

  2. (2)

As soon as several characteristics are crossing at a point $(t,x)$, the
function $u=ZD[u_{0}]$ given by (3) is not a weak solution of the inviscid
Burgers–Hopf equation

$\partial_{t}u+\partial_{x}(u^{2})=0$

in a neighborhood of $(t,x)$, hence the convergence of $u^{\varepsilon}$ to
$u$ cannot be strong in $L^{2}_{\rm loc}$, confirming the strong oscillations
detected by numerical simulations. We refer to the end of section 3 for more
detail about this lack of compactness.

  3. (3)

Formula (3) was introduced by Y. Brenier in [1, 2, 3] to approximate the
entropic solution [19, 25] of the inviscid Burgers–Hopf equation. This formula
has the following weak version. For every test function $\varphi$ on
${\mathbb{R}}$,

(4)
$\int_{\mathbb{R}}ZD[u_{0}](t,x)\,\varphi(x)\,dx=\int_{\mathbb{R}}\varphi(y+2tu_{0}(y))u_{0}(y)(1+2tu_{0}^{\prime}(y))\,dy\
.$

A striking consequence of Theorem 2 is the following maximum principle.

######  Corollary 1.

For every $u_{0}\in L^{\infty}({\mathbb{R}})\cap L^{2}({\mathbb{R}})$, for
every $t\in{\mathbb{R}}$,

${\rm ess\,inf}u_{0}\leq{\rm ess\,inf}ZD[u_{0}](t)\leq{\rm
ess\,sup}ZD[u_{0}](t)\leq{\rm ess\,sup}u_{0}\ .$

Contrary to what is known about the inviscid limit of the Burgers–Hopf
equation [19, 25], where the maximum principle is inherited from the parabolic
structure, the above maximum principle is surprising, because similar
estimates do not hold for the solution $u^{\varepsilon}$ of the Benjamin–Ono
equation for a given $\varepsilon>0$. In fact, for periodic boundary
conditions, one can even prove that the Benjamin–Ono flow map may lose up to
$1/2$ derivatives in the $C^{\alpha}$ regularity (see section 4 of [15]).
Another consequence of Theorem 2 is the following local smoothing property.

######  Corollary 2.

For every $u_{0}\in L^{\infty}({\mathbb{R}})\cap L^{2}({\mathbb{R}})$, for
every $t\in[-T,T]$, the distributional derivative $2t\partial_{x}ZD[u_{0}](t)$
is a measure, with a bounded total variation on every finite interval of
${\mathbb{R}}$. In particular, for every $t\neq 0$, for every $s<\frac{1}{2}$,
$ZD[u_{0}](t)\in H^{s}_{\rm loc}({\mathbb{R}})$.

Finally, the combination of Theorem 1 is and Theorem 2 allows us to calculate
$ZD[u_{0}]$ for discontinuous functions $u_{0}$, for instance if $u_{0}$ is
the characteristic function of a finite interval. See section 5 below.

###  1.2. Structure of the paper.

This paper is organized as follows. In section 2, we prove Theorem 1 by using
an explicit formula recently derived for the initial value problem of the
Benjamin–Ono on the line [13]. Assuming the initial datum $u_{0}$ to be
$L^{\infty}$, it is possible to pass to the limit in this formula as
$\varepsilon$ tends to $0$ and to obtain a formula for the zero–dispersion
limit $ZD[u_{0}](t)$, using the Toeplitz operator of symbol $u_{0}$ acting on
the Hardy space $L^{2}_{+}({\mathbb{R}})$, see formula (7) below. In section
3, we transform formula (7) in the case of a rational initial datum $u_{0}$
into a linear system on ${\mathbb{C}}^{N+1}$, where $2N$ is the degree of the
denominator of $u_{0}$, and we obtain the formula (3) in this case. We also
prove that the weak limit to $ZD[u_{0}](t)$ is not strong in the case of three
crossing characteristics. In section 4, we prove Theorem 2 in its whole
generality, and we infer Corollary 1 and Corollary 2. Finally, in section 5,
we describe the zero–dispersion limit $ZD[u_{0}]$ when $u_{0}$ is the
characteristic function of the interval $]-1,1[$, and we prove that the map
$t\mapsto ZD[\ .\ ](t)$ is not a semigroup of transformations of
$L^{2}({\mathbb{R}})\cap L^{\infty}({\mathbb{R}})$.

##  2\. Proof of Theorem 1

We first recall the notation from [13]. Here $D$ denotes $-i\partial_{x}$,
$\Pi$ denotes the orthogonal projector ${\bf 1}_{D\geq 0}$ from
$L^{2}({\mathbb{R}})$ onto the Hardy space

$L^{2}_{+}({\mathbb{R}}):=\\{f\in
L^{2}({\mathbb{R}}):\forall\xi<0,\hat{f}(\xi)=0\\}\ ,$

where $\hat{f}$ denotes the Fourier transform. We recall that
$L^{2}_{+}({\mathbb{R}})$ identifies to holomorphic fonctions $f$ in the upper
half plane with

$\sup_{y>0}\int_{\mathbb{R}}|f(x+iy)|^{2}\,dx\ <+\infty\ .$

We denote by $G$ the adjoint operator, on $L^{2}_{+}({\mathbb{R}})$, of the
multiplication by $x$. The domain of $G$ consists of those functions $f$ in
$L^{2}_{+}({\mathbb{R}})$ such that the restriction of $\hat{f}$ to
$]0,+\infty[$ belongs to the Sobolev space $H^{1}$. Furthermore, for any
function $f\in L^{2}_{+}({\mathbb{R}})$ such that the restriction of $\hat{f}$
to $]0,1[$ belongs to $H^{1}(]0,1[)$, we set

$I_{+}(f)=\hat{f}(0^{+})\ .$

Finally, for $b\in L^{\infty}({\mathbb{R}})$ $T_{b}$ denotes the Toeplitz
operator defined on $L^{2}_{+}({\mathbb{R}})$ by $T_{b}(f)=\Pi(bf)$. This
operator is bounded, with $\|T_{b}\|=\|b\|_{L^{\infty}}$, and is selfadjoint
if $b$ is real valued. From [13], we know that, if $w_{0}\in
L^{\infty}({\mathbb{R}})\cap L^{2}({\mathbb{R}})$ and is real valued, the
solution of the initial value problem for the Benjamin–Ono equation,

(5) $\partial_{s}w+\partial_{x}(w^{2})=\partial_{x}|D|w\ ,\ w(0,x)=w_{0}(x)$

is given by

$w(s,x)=\Pi w(s,x)+\overline{\Pi w(s,x)},x\in{\mathbb{R}}\ ,$

where, for every complex number $x$ with ${\rm Im}(x)>0$,

(6) $\Pi w(s,x)=\frac{1}{2i\pi}I_{+}[(G-2s(D-T_{w_{0}})-x{\rm Id})^{-1}(\Pi
w_{0})]\ .$

We recall [13] that the resolvent in the right hand side of formula (6) is
well defined. Indeed, on the other hand, the operator $G-2sD$ has the
following dissipation property,

$\forall f\in{\rm Dom}(G-2sD)\ ,\ {\rm Im}\langle(G-2sD)f|f\rangle\leq 0\ ,$

and is maximal, in the sense that $G-2sD-x{\rm Id}:{\rm Dom}(G-2sD)\to
L^{2}_{+}({\mathbb{R}})$ is invertible for every complex number $x$ with ${\rm
Im}(x)>0$. On the other hand, the operator $2sT_{w_{0}}$ is bounded and
self–adjoint. The claim therefore follows from standard perturbation
theory.From a simple scaling argument, the solution $u^{\varepsilon}$ of (1)
is given by

$u^{\varepsilon}(t,x)=\varepsilon w(\varepsilon t,x)\ ,$

where $w(s,x)$ is the solution of (5) with
$w_{0}(x)=\varepsilon^{-1}u_{0}(x)$. Therefore we can write

$u^{\varepsilon}(t,x)=\Pi u^{\varepsilon}(t,x)+\overline{\Pi
u^{\varepsilon}(t,x)}\ ,\ x\in{\mathbb{R}}\ ,$

where, for every complex number $x$ with ${\rm Im}(x)>0$,

$\Pi u^{\varepsilon}(t,x)=\frac{1}{2i\pi}I_{+}[(G-2\varepsilon
tD+2tT_{u_{0}}-x{\rm Id})^{-1}(\Pi u_{0})]\ .$

On the other hand, recall that the $L^{2}$ norm of $u^{\varepsilon}(t)$ is
independent of $t$, equal to the $L^{2}$ norm of $u_{0}$, so there exists a
subsequence $\varepsilon_{j}$ tending to $0$ such that
$u^{\varepsilon_{j}}(t)$ has a weak limit in $L^{2}({\mathbb{R}})$. We claim
that all these weak limits coincide.

######  Lemma 1.

For every $t\in{\mathbb{R}}$, $u^{\varepsilon}(t)$ converges weakly in
$L^{2}({\mathbb{R}})$ to $u(t)$, with

$u(t,x)=\Pi u(t,x)+\overline{\Pi u(t,x)}\ ,\ x\in{\mathbb{R}}\ ,$

where, for every complex number $x$ with ${\rm Im}(x)>0$,

(7) $\Pi u(t,x)=\frac{1}{2i\pi}I_{+}[(G+2tT_{u_{0}}-x{\rm Id})^{-1}(\Pi
u_{0})]\ .$

###### Proof.

Since $u\in L^{2}({\mathbb{R}})\mapsto\Pi u(x)\in{\mathbb{C}}$ is a continuous
linear form for every complex number $x$ with ${\rm Im}(x)>0$, it is enough to
characterize $\Pi u(t,x)$ for ${\rm Im}(x)>0$, and even for ${\rm Im}(x)$ big
enough, by analytic continuation. We observe that, for ${\rm
Im}(x)>2|t|\|u_{0}\|_{L^{\infty}}$, the resolvent of $G-2\varepsilon
tD+2tT_{u_{0}}$ converges strongly to the resolvent of $G+2tT_{u_{0}}$ as
$\varepsilon\to 0$. Indeed, by standard perturbation theory, it is enough to
check that the resolvent of $G-2\varepsilon tD$ converges to $G$, with a
uniform bound by $({\rm Im}(x))^{-1}$ if ${\rm Im}(x)>0$, and this is
immediate from the following explicit formula,

$\hat{h}_{\varepsilon}(\xi)=i\int_{\xi}^{\infty}\hat{f}(\eta){\rm
e}^{i\varepsilon t(\eta^{2}-\xi^{2})+ix(\eta-\xi)}\,d\eta\ .$

for $h_{\varepsilon}:=(G-2\varepsilon tD-x)^{-1}f$.
Furthermore, from the equation

$i\partial_{\xi}\hat{f}_{\varepsilon}(\xi)-(2\varepsilon
t\xi+x)\hat{f}_{\varepsilon}(\xi)=\hat{g}_{\varepsilon}(\xi)\ ,\
g_{\varepsilon}:=\Pi u_{0}-2tT_{u_{0}}f_{\varepsilon}\ ,\ \xi>0\ ,$

we observe that the Fourier transform of

$f_{\varepsilon}=(G-2\varepsilon tD+2tT_{u_{0}}-x{\rm Id})^{-1}(\Pi u_{0})$

is uniformly bounded in $H^{1}(]0,1[)$ as $\varepsilon$ tends to $0$.
Consequently, if ${\rm Im}(x)>2|t|\|u_{0}\|_{L^{\infty}}$, we have

$\Pi
u^{\varepsilon}(t,x)\longrightarrow\frac{1}{2i\pi}I_{+}[(G+2tT_{u_{0}}-x{\rm
Id})^{-1}(\Pi u_{0})]\ .$

Therefore, by analytic continuation, for every $x$ such ${\rm Im}(x)>0$,

$\Pi u(t,x)=\frac{1}{2i\pi}I_{+}[(G+2tT_{u_{0}}-x{\rm Id})^{-1}(\Pi u_{0})]\
.$

which is (7). From formula (7), a similar argument allows us to prove
properties (1) and (2) in Theorem 1. ∎

######  Remark 1.

  1. (1)

Notice that it is not so easy to estimate the $L^{2}$ norm of $u(t)$ from
formula (7). However, we know that it is bounded by the $L^{2}$ norm of
$u_{0}$ because of the $L^{2}$ conservation law for the Benjamin–Ono equation.

  2. (2)

The above arguments also show that, for any convergent sequence
$t_{\varepsilon}$ of real numbers, $u^{\varepsilon}(t_{\varepsilon})$ is
weakly convergent in $L^{2}$ to $u(\lim_{\varepsilon\to 0}t_{\varepsilon})$.
Consequently, for every $T>0$, $u^{\varepsilon}$ is uniformly convergent to
$u$ in $C([-T,T],L^{2}_{w}({\mathbb{R}}))$, where $L^{2}_{w}({\mathbb{R}})$ is
$L^{2}({\mathbb{R}})$ endowed with the weak topology.

  3. (3)

If the initial datum $u_{0}$ is replaced by a sequence $u_{0}^{\varepsilon}$
strongly convergent to $u_{0}$, with a uniform bound in $L^{\infty}$, the
above arguments show that $u^{\varepsilon}(t)$ is still weakly convergent to
$ZD[u_{0}](t)$.

##  3\. Proof of Theorem 2 for a rational initial datum

In this section, we prove formula (3) if $u_{0}$ is a rational function with
real coefficients, with no pole on the real line,

$u_{0}(y)=\frac{P(y)}{Q(y)}\ ,$

where, for some positive integer $N$, $Q$ is a monic real polynomial of degree
$2N$, and $P$ is a real polynomial of degree at most $2N-1$. With no loss of
generality, we may assume — by an approximation argument in formula (7)—, that
$Q$ has only simple poles in the complex domain, so that

$u_{0}(y)=\sum_{j=1}^{N}\frac{c_{j}}{y-p_{j}}+\frac{\overline{c}_{j}}{y-\overline{p}_{j}}\
,\ {\rm Im}(p_{j})>0\ ,\ \Pi
u_{0}(y)=\sum_{j=1}^{N}\frac{\overline{c}_{j}}{y-\overline{p}_{j}}\ .$

Set $u(t):=ZD[u_{0}](t)$. We calculate directly $u(t,x)=\Pi
u(t,x)+\overline{\Pi u(t,x)}$ for $x\in{\mathbb{R}}$. Recall that, according
to formula (7), for ${\rm Im}(x)>0$,

$\Pi u(t,x)=\frac{1}{2i\pi}I_{+}[(G+2tT_{u_{0}}-x{\rm Id})^{-1}(\Pi u_{0})]\
.$

Notice that the algebraic equation $y+2tu_{0}(y)=x$ is equivalent to

$yQ(y)-xQ(y)+2tP(y)=0\ ,$

which is an equation of degree $2N+1$. Hence, if $x\in{\mathbb{R}}$, it has an
odd number $2\ell(t,x)+1$ of real zeroes. Discarding only a finite set of
points $x$ for a given $t$, we may consider the case where
$\ell(t,x)=\ell\in{\mathbb{Z}}_{\geq 0}$ and where these zeroes are simple,
ordered as

$y_{0}(t,x)<y_{1}(t,x)<\dots<y_{2\ell}(t,x)\ ,$

so that the derivative of $y+2tu_{0}(y)$ at $y=y_{k}(t,x)$ has the sign of
$(-1)^{k}$. Denote the other zeroes of this equation in the complex domain by

$y_{m}(t,x),m=2\ell+1,\dots,2N\ ,$

with ${\rm Im}(y_{2p}(t,x))>0$ and $y_{2p-1}(t,x)=\overline{y_{2p}(t,x)}$ if
$p=\ell+1,\dots,N$. The Cauchy–Riemann equations and the implicit function
theorem show that

$\frac{\partial{\rm Im}(y_{k}(t,x))}{\partial{\rm Im}(x)}=\frac{\partial{\rm
Re}(y_{k}(t,x))}{\partial{\rm
Re}(x)}=\frac{1}{1+2tu_{0}^{\prime}(y_{k}(t,x))}$

which has the sign of $(-1)^{k}$. If $x$ is shifted into the upper half plane
with a small imaginary part, we infer

${\rm Im}(y_{2k}(t,x))>0\ ,\ {\rm Im}(y_{2p-1}(t,x))<0\ ,\ k=0,\dots,N,\
p=1,\dots,N\ .$

For such a complex number $x$, let us calculate

$f_{t,x}:=(G+2tT_{u_{0}}-x{\rm Id})^{-1}(\Pi u_{0})\ .$

It turns out that $f_{t,x}$ is a rational function. Indeed, for any function
$f$ in $L^{2}_{+}({\mathbb{R}})$, we have

$Gf(y)=yf(y)+\frac{1}{2i\pi}I_{+}(f)\ ,\
T_{u_{0}}f(y)=u_{0}(y)f(y)-\sum_{j=1}^{N}c_{j}\frac{f(p_{j})}{y-p_{j}}\ .$

The first identity is a consequence of the Fourier representation of $G$ and
of the jump formula at $\xi=0$ for distributional derivatives. The second
identity comes from the following standard property of Toeplitz operators with
rational symbols,

$T_{(x-p)^{-1}}(f)(x)=\frac{f(x)-f(p)}{x-p}\ ,\ {\rm Im}(p)>0\ .$

Hence the equation on $f_{t,x}$ can be reformulated as

$(y-x+2tu_{0}(y))f_{t,x}(y)=u_{0}(y)+\lambda(t,x)+\sum_{j=1}^{N}\frac{\mu_{j}(t,x)}{y-p_{j}}\
,$

where $\lambda(t,x)$, $\mu_{j}(t,x),j=1,\dots,N$, are complex numbers such
that the rational function

$f_{t,x}(y)=\frac{u_{0}(y)+\lambda(t,x)+\sum_{j=1}^{N}\frac{\mu_{j}(t,x)}{y-p_{j}}}{y-x+2tu_{0}(y)}\
,$

belongs to the Hardy space. This imposes the cancellation of the numerator if
$y$ is any of the zeroes of the denominator in the upper half plane, and
therefore leads to the linear system

$u_{0}(y_{2k}(t,x))+\lambda(t,x)+\sum_{j=1}^{N}\frac{\mu_{j}(t,x)}{y_{2k}(t,x)-p_{j}}=0\
,\ k=0,\dots,N\ ,$

or, if $t\neq 0$,

(8)
$\lambda(t,x)+\sum_{j=1}^{N}\frac{\mu_{j}(t,x)}{y_{2k}(t,x)-p_{j}}=\frac{y_{2k}(t,x)-x}{2t}\
,\ k=0,\dots,N\ ,$

Then we have

$\Pi u(t,x)=\frac{1}{2i\pi}I_{+}(f_{t,x})=-\lim_{y\to\infty}yf_{t,x}(y)\ ,$

since $f_{t,x}(y)$ is a rational function of $y$. In view of the expression of
$f_{t,x}$, we conclude

$\Pi u(t,x)=-\lambda(t,x)\ .$

From Cramer’s formulae for the system (8),

$\displaystyle\lambda(t,x)$ $\displaystyle=$
$\displaystyle\frac{N(t,x)}{D(t,x)}\ ,$ $\displaystyle D$ $\displaystyle:=$
$\displaystyle\left|\begin{array}[]{ccccc}1&\frac{1}{y_{0}-p_{1}}&.&.&\frac{1}{y_{0}-p_{N}}\\\
1&\frac{1}{y_{2}-p_{1}}&.&.&\frac{1}{y_{2}-p_{N}}\\\ .&.&.&.&.\\\
1&\frac{1}{y_{2N}-p_{1}}&.&.&\frac{1}{y_{2N}-p_{N}}\end{array}\right|\ ,$
$\displaystyle N$ $\displaystyle:=:$
$\displaystyle\frac{1}{2t}\left|\begin{array}[]{ccccc}y_{0}-x&\frac{1}{y_{0}-p_{1}}&.&.&\frac{1}{y_{0}-p_{N}}\\\
y_{2}-x&\frac{1}{y_{2}-p_{1}}&.&.&\frac{1}{y_{2}-p_{N}}\\\ .&.&.&.&.\\\
y_{2N}-x&\frac{1}{y_{2N}-p_{1}}&.&.&\frac{1}{y_{2N}-p_{N}}\end{array}\right|\
.$

The following lemma follows from elementary manipulations on Cauchy and
Vandermonde determinants.

######  Lemma 2.

Given complex numbers $z_{0},\dots,z_{N},p_{1},\dots,p_{N}$ pairwise distinct,
we have

$\displaystyle\frac{\left|\begin{array}[]{ccccc}z_{0}&\frac{1}{z_{0}-p_{1}}&.&.&\frac{1}{z_{0}-p_{N}}\\\
z_{1}&\frac{1}{z_{1}-p_{1}}&.&.&\frac{1}{z_{1}-p_{N}}\\\ .&.&.&.&.\\\
z_{N}&\frac{1}{z_{N}-p_{1}}&.&.&\frac{1}{z_{N}-p_{N}}\end{array}\right|}{\left|\begin{array}[]{ccccc}1&\frac{1}{z_{0}-p_{1}}&.&.&\frac{1}{z_{0}-p_{N}}\\\
1&\frac{1}{z_{1}-p_{1}}&.&.&\frac{1}{z_{1}-p_{N}}\\\ .&.&.&.&.\\\
1&\frac{1}{z_{N}-p_{1}}&.&.&\frac{1}{z_{N}-p_{N}}\end{array}\right|}\ =\
\sum_{\alpha=0}^{N}z_{\alpha}-\sum_{j=1}^{N}p_{j}\ .$

###### Proof.

Using the formula for the Cauchy determinants, we have

$\displaystyle
A:=\left|\begin{array}[]{ccccc}z_{0}&\frac{1}{z_{0}-p_{1}}&.&.&\frac{1}{z_{0}-p_{N}}\\\
z_{1}&\frac{1}{z_{1}-p_{1}}&.&.&\frac{1}{z_{1}-p_{N}}\\\ .&.&.&.&.\\\
z_{N}&\frac{1}{z_{N}-p_{1}}&.&.&\frac{1}{z_{N}-p_{N}}\end{array}\right|$
$\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{N}(-1)^{\alpha}z_{\alpha}D_{\alpha}$
$\displaystyle
B:=\left|\begin{array}[]{ccccc}1&\frac{1}{z_{0}-p_{1}}&.&.&\frac{1}{z_{0}-p_{N}}\\\
1&\frac{1}{z_{1}-p_{1}}&.&.&\frac{1}{z_{1}-p_{N}}\\\ .&.&.&.&.\\\
1&\frac{1}{z_{N}-p_{1}}&.&.&\frac{1}{z_{N}-p_{N}}\end{array}\right|$
$\displaystyle=$ $\displaystyle\sum_{\alpha=0}^{N}(-1)^{\alpha}D_{\alpha}$

with

$\displaystyle D_{\alpha}$ $\displaystyle:=$
$\displaystyle\prod_{\begin{subarray}{c}\beta\neq\alpha,\gamma\neq\alpha\\\
\beta<\gamma\end{subarray}}(z_{\beta}-z_{\gamma})\prod_{j}(z_{\alpha}-p_{j})\Delta\
,$ $\displaystyle\Delta$ $\displaystyle:=$
$\displaystyle\frac{\prod_{j<k}(p_{k}-p_{j})}{\prod_{\beta,j}(z_{\beta}-p_{j})}\
.$

Consequently, we are led to evaluate the following quotient of Vandermonde
determinants,

$\displaystyle\frac{A}{B}$ $\displaystyle=$ $\displaystyle\frac{V(R)}{V(Q)}\
,\ R(z):=zQ(z)\ ,\ Q(z)=\prod_{j=1}^{N}(z-p_{j})\ ,$ $\displaystyle V(P)$
$\displaystyle:=$
$\displaystyle\left|\begin{array}[]{ccccc}P(z_{0})&1&z_{0}&.&z_{0}^{N-1}\\\
P(z_{1})&1&z_{1}&.&z_{1}^{N-1}\\\ .&.&.&.&.\\\
P(z_{N})&1&z_{N}&.&z_{N}^{N-1}\end{array}\right|\ .$

We notice that the linear form $V$ cancels on polynomials of degree at most
$N-1$, and on the polynomial $\tilde{P}$ defined as

$\tilde{P}(z):=\prod_{\alpha=0}^{N}(z-z_{\alpha})\ .$

Since

$\displaystyle R(z)-\tilde{P}(z)$ $\displaystyle=$ $\displaystyle
zQ(z)-\tilde{P}(z)=\left(\sum_{\alpha=0}^{N}z_{\alpha}-\sum_{j=1}^{N}p_{j}\right)z^{N}+{\mathbb{C}}_{\leq
N-1}[z]$ $\displaystyle=$
$\displaystyle\left(\sum_{\alpha=0}^{N}z_{\alpha}-\sum_{j=1}^{N}p_{j}\right)Q(z)+{\mathbb{C}}_{\leq
N-1}[z]\ ,$

we infer that

$\frac{V(R)}{V(Q)}=\frac{V(R-\tilde{P})}{V(Q)}=\sum_{\alpha=0}^{N}z_{\alpha}-\sum_{j=1}^{N}p_{j}\
.$

∎

Using lemma 2, we have

(15)
$\lambda(t,x)=\frac{1}{2t}\left(\sum_{\alpha=0}^{N}y_{2\alpha}(t,x)-\sum_{j=1}^{N}p_{j}-x\right)\
.$

In view of the algebraic equation

$yQ(y)-xQ(y)+2tP(y)=0\ ,\ Q(y)=\prod_{j=1}^{N}(y-p_{j})(y-\overline{p}_{j})\
,$

we have

(16) $x+\sum_{j=1}^{N}(p_{j}+\overline{p}_{j})=\sum_{\alpha=0}^{2N}y_{\alpha}\
.$

Now we make $x$ tend to the real line, so that $y_{k}(t,x)$ is real for
$k=0,1,\dots,2\ell$, and $y_{2p-1}(t,x)=\overline{y_{2p}(t,x)}$ if
$p=\ell+1,\dots,N\ .$ Consequently,

$\displaystyle u(t,x)$ $\displaystyle=$
$\displaystyle-(\lambda(t,x)+\overline{\lambda(t,x)})$ $\displaystyle=$
$\displaystyle\frac{1}{2t}\left(2x+\sum_{j=1}^{N}(p_{j}+\overline{p}_{j})-2\sum_{\alpha=0}^{\ell}y_{2\alpha}(t,x)-\sum_{\beta=2\ell+1}^{2N}y_{\beta}(t,x)\right)\
,$ $\displaystyle=$
$\displaystyle\frac{1}{2t}\left(x+\sum_{\gamma=1}^{\ell}y_{2\gamma-1}(t,x)-\sum_{\alpha=0}^{\ell}y_{2\alpha}(t,x)\right)\
,$ $\displaystyle=$ $\displaystyle\sum_{k=0}^{2\ell}(-1)^{k}u_{0}(y_{k}(t,x))\
.$

The proof of formula (3) when $u_{0}$ is rational is complete. Let us end this
section by an elementary observation about formula (3). We claim that, if
$\ell=1$, this alternate sum of solutions of the inviscid Burgers equation is
not a solution of the inviscid Burgers–Hopf equation. Indeed, let us argue by
contradiction. Assume $\ell=1$ and that $u$ satisfies the inviscid
Burgers–Hopf equation in some neighborhood $V$ of $(t,x)$ in
${\mathbb{R}}^{2}$. Writing $v_{k}(t,x):=u_{0}(y_{k}(t,x))$ for $k=0,1,2$, we
observe that $\partial_{t}v_{k}+\partial_{x}(v_{k}^{2})=0$, hence

$\partial_{x}[(v_{0}-v_{1}+v_{2})^{2}-v_{0}^{2}+v_{1}^{2}-v_{2}^{2}]=0\ $

in $V$. We can reformulate the above identity as

$\partial_{x}[(v_{1}-v_{0})(v_{1}-v_{2})]=0$

on $V$. Since we are dealing with real analytic functions, this identity
extends to the whole connected component of $V$ in the domain $W$
characterized by $\ell=1$. In other words, the function
$(v_{1}-v_{0})(v_{1}-v_{2})$ is not depending on $x$ in this domain. Since
this function tends to $0$ as $(t,x)$ tends to the boundary of $W$, we
conclude that $(v_{1}-v_{0})(v_{1}-v_{2})$ is identically $0$, which is a
contradiction since $y_{k}(t,x)+2tv_{k}(t,x)=x$, while
$y_{0}(t,x)<y_{1}(t,x)<y_{2}(t,x)$.

##  4\. Proof of Theorem 2, of the maximum principle and of the local
smoothing property

We now consider the case where $u_{0}\in L^{2}({\mathbb{R}})$ is continuously
differentiable, with $|u(x)|+|u^{\prime}(x)|\to 0$ as $x\to\infty$. Using a
standard mollifier, we may approximate $u_{0}$ in $L^{2}({\mathbb{R}})\cap
C^{1}_{b}({\mathbb{R}})$ by a sequence $u_{0}^{\delta}$ of functions in
$H^{s}({\mathbb{R}})$ for any $s\in{\mathbb{R}}$. Since $L^{2}$ rational
functions are dense in $H^{2}({\mathbb{R}})$, we may assume that
$u_{0}^{\delta}$ is rational. Let $t\in{\mathbb{R}}$, and let $K_{t}(u_{0})$
be the set of critical values of the function

$f_{t}:y\in{\mathbb{R}}\mapsto y+2tu_{0}(y)\in{\mathbb{R}}\ .$

By the Sard theorem, $K_{t}(u_{0})$ has zero Lebesgue measure. Furthermore,
since $1+2tu_{0}^{\prime}(y)\to 1$ as $y\to\infty$, the set of critical points
of $f_{t}$ is compact, so that its image $K_{t}(u_{0})$ by $f_{t}$ is compact
too. Let $\Omega$ be any connected component of $K_{t}(u_{0})^{c}$. Let $\ell$
be the nonnegative integer such that, for every $x\in\Omega$, the equation
$y+2tu_{0}(y)=x$ has $2\ell+1$ solutions, denoted by

$y_{0}(t,x)<\dots<y_{2\ell}(t,x)\ .$

Let $\omega$ be any open subinterval of $\Omega$ such that $\overline{\omega}$
is compact. For $\delta$ small enough, $\omega$ does not meet
$K_{t}(u_{0}^{\delta})^{c}$, and, for every $x\in\omega$, the equation

$y+2tu_{0}^{\delta}(y)=x$

has exacly $2\ell+1$ solutions,

$y_{0}^{\delta}(t,x)<\dots<y_{2\ell}^{\delta}(t,x)\ .$

Furthermore, for every $k\in\\{0,1,\dots,2\ell\\}$, $y_{k}^{\delta}(t,x)\to
y_{k}(t,x)$ as $\delta\to 0$. From section 3, we know that, for $x\in\omega$,

$ZD[u_{0}^{\delta}](t,x)=\sum_{k=0}^{2\ell}(-1)^{k}u_{0}^{\delta}(y_{k}^{\delta}(t,x))\
.$

Passing to the weak limit in $L^{2}(\omega)$ as $\delta\to 0$, we infer, for
almost every $x\in\omega$,

$ZD[u_{0}](t,x)=\sum_{k=0}^{2\ell}(-1)^{k}u_{0}(y_{k}(t,x))\ .$

Since $\omega$ is an arbitrary relatively compact subinterval of $\Omega$,
this proves formula (3). We now come to the proof of Corollary 1. First
consider the case where $u_{0}$ is continuously differentiable with
$|u_{0}(x)|+|u^{\prime}_{0}(x)|\to 0$ as $x\to\infty$, as in section 4. Let us
come back to formula (3),

$u(t,x)=\sum_{k=0}^{2\ell}(-1)^{k}u_{0}(y_{k}(t,x))$

where $y_{0}(t,x)<\dots<y_{2\ell}(t,x)$ are the solutions of $y+2tu_{0}(y)=x$.
In view of the monotonicity of the sequence $k\mapsto y_{k}(t,x)$, we infer
the monotonicity of the sequence $k\mapsto u_{0}(y_{k}(t,x))$. Consequently,
we have

$\min_{0\leq k\leq
2\ell}u_{0}(y_{k}(t,x))\leq\sum_{k=0}^{2\ell}(-1)^{k}u_{0}(y_{k}(t,x))\leq\max_{0\leq
2\ell}u_{0}(y_{k}(t,x))$

and finally

$\min_{y\in{\mathbb{R}}}u_{0}(y)\leq
u(t,x)\leq\max_{y\in{\mathbb{R}}}u_{0}(y)\ ,$

which is the claimed maximum principle.
The general case of $u_{0}\in L^{\infty}({\mathbb{R}})\cap
L^{2}({\mathbb{R}})$ follows by applying a standard mollifier to $u_{0}$ and
using property (2) in Theorem 1. Finally, let us prove Corollary 2. Let us
first assume that $u_{0}$ is continuously differentiable with
$|u_{0}(x)|+|u^{\prime}_{0}(x)|\to 0$ as $x\to\infty$. Using the weak version
(4) of formula (3), we have, for every $\varphi\in C^{1}({\mathbb{R}})$ with
compact support,

$\displaystyle\langle 2t\partial_{x}ZD[u_{0}](t),\varphi\rangle$
$\displaystyle=$
$\displaystyle-2t\int_{\mathbb{R}}u_{0}(y)\varphi^{\prime}(y+2tu_{0}(y))(1+2tu_{0}^{\prime}(y))\,dy$
$\displaystyle=$ $\displaystyle
2t\int_{\mathbb{R}}u_{0}^{\prime}(y)\varphi(y+2tu_{0}(y))\,dy$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\left[\frac{d}{dy}\left(\int_{-\infty}^{y+2tu_{0}(y)}\varphi(s)\,ds\right)-\varphi(y+2tu_{0}(y))\right]\,dy$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\varphi(s)\,ds-\int_{\mathbb{R}}\varphi(y+2tu_{0}(y))\,dy\
.$

By mollifying $u_{0}$, this statement still holds if $u_{0}\in
L^{\infty}({\mathbb{R}})\cap L^{2}({\mathbb{R}})$. We infer that, for every
$u_{0}\in L^{\infty}({\mathbb{R}})\cap L^{2}({\mathbb{R}})$,

(17) $2t\partial_{x}ZD[u_{0}](t)=1-\mu_{2tu_{0}}\ ,\
\int_{\mathbb{R}}\varphi(y)\,d\mu_{f}(y):=\int_{\mathbb{R}}\varphi(y+f(y))\,dy\
.$

Notice that $\mu_{f}([-R,R])\leq 2(R+\|f\|_{\infty})$. This proves the first
claim of Corollary 2. The second claim is a consequence of the Sobolev
embedding

$BV_{\rm loc}({\mathbb{R}})\subset H^{s}_{\rm loc}({\mathbb{R}})$

for every $s<\frac{1}{2}$.

######  Remark 2.

. Equation (17) implies a one–sided Lipschitz condition for $ZD[u_{0}](t)$.
This property is well-known for entropic solutions of the inviscid
Burgers–Hopf equation [24, 4].

##  5\. A special case and the lack of semigroup property

In this section, we consider the case where $u_{0}$ is the characteristic
function of a finite interval. We are going to calculate $ZD[u_{0}]$ by
approximating $u_{0}$ by a sequence of compactly supported $C^{1}$ functions.
An alternative method would be to use formula (17) above.

######  Theorem 3.

Let $u_{0}$ be the characteristic function of $]-1,1[$. Then $ZD[u_{0}]$ can
be described as follows. If $t\in]0,1]$,

$\displaystyle ZD[u_{0}](t,x)=\begin{cases}0&{\rm if}\
x\in]-\infty,-1]\cup]2t+1,+\infty[\\\ \frac{x+1}{2t}&{\rm if}\
x\in]-1,2t-1]\\\ 1&{\rm if}\ x\in]2t-1,1]\\\ 1-\frac{x-1}{2t}&{\rm if}\
x\in]1,2t+1]\ .\end{cases}$

If $t\in]1,+\infty[$,

$\displaystyle ZD[u_{0}](t,x)=\begin{cases}0&{\rm if}\
x\in]-\infty,-1]\cup]2t+1,+\infty[\\\ \frac{x+1}{2t}&{\rm if}\ x\in]-1,1]\\\
\frac{1}{t}&{\rm if}\ x\in]1,2t-1]\\\ 1-\frac{x-1}{2t}&{\rm if}\
x\in]2t-1,2t+1]\ .\end{cases}$

Furthermore, $ZD[u_{0}](-t,-x)=ZD[u_{0}](t,x)$.

$x$$u_{0}(x)$$-1$$1$$1$ Figure 5.1. The datum at $t=0$
$x$$u(t,x)$$-1$$2t-1$$\bullet$$\bullet$$1$$2t+1$$1$ Figure 5.2. The weak limit
at $t\in]0,1[$ $x$$1$$u(1,x)$$-1$$1$3$\bullet$$\bullet$ Figure 5.3. The weak
limit at $t=1$ $x$$u(t,x)$$-1$$1$$2t-1$$2t+1$$\frac{1}{t}$$\bullet$$\bullet$
Figure 5.4. The weak limit at $t\in]1,+\infty[$

###### Proof.

We use property (2) of Theorem 1 and Theorem 2. We approximate $u_{0}$ by the
following smooth function $u_{0}^{\delta}$ as $\delta\to 0^{+}$.

$\displaystyle u_{0}^{\delta}(x)=\begin{cases}0&{\rm if}\
x\in]-\infty,-1-\delta]\cup[1+\delta,+\infty[\\\ 1&{\rm if}\
x\in[-1,1]\end{cases}$

and $u_{0}^{\delta}$ is strictly increasing from $0$ to $1$ on the interval
$[-1-\delta,-1]$, strictly decreasing from $1$ to $0$ on the interval
$[1,1+\delta]$. Let assume $t\in]0,1]$. Given $x\in{\mathbb{R}}$, the
solutions of the equation $y+2tu_{0}^{\delta}(y)=x$ are as follows.
If $x<-1-\delta$, the only solution is $y_{0}^{\delta}(t,x)=x$ and therefore

$ZD[u_{0}^{\delta}](t,x)=u_{0}^{\delta}(y_{0}^{\delta}(t,x))=0\ .$

If $-1<x<2t-1$, there exists only one solution $y_{0}^{\delta}$, it is simple
and it belongs to $]-1-\delta,-1[$. Then

$ZD[u_{0}^{\delta}](t,x)=u_{0}^{\delta}(y_{0}^{\delta}(t,x))=\frac{x-y_{0}^{\delta}}{2t}\
.$

If $2t-1<x<1$, there exists only one solution $y_{0}^{\delta}(t,x)=x-2t$, it
is simple and it belongs to $]-1,1[$ . Then

$ZD[u_{0}^{\delta}](t,x)=u_{0}^{\delta}(y_{0}^{\delta}(t,x))=1\ .$

If $1+\delta<x<1+2t$ is outside a set of measure $0$, there exist three simple
solutions $y_{0}^{\delta}(t,x)=x-2t$, $y_{1}^{\delta}(t,x)\in]1,1+\delta[$ and
$y_{2}^{\delta}(t,x)=x$. Then

$ZD[u_{0}^{\delta}](t,x)=\sum_{k=0}^{2}(-1)^{k}u_{0}^{\delta}(y_{k}^{\delta}(t,x))=1-\frac{x-y_{1}^{\delta}(t,x)}{2t}\
.$

If $1+2t<x$, then there exists only one solution $y_{0}\delta(t,x)=x$ and

$ZD[u_{0}^{\delta}](t,x)=u_{0}^{\delta}(y_{0}^{\delta}(t,x))=0\ .$

Finally, passing to the limit as $\delta\to 0$ by keeping in mind that

$0\leq ZD[u_{0}^{\delta}]\leq 1$

in view of the maximum principle, we obtain the result.
The case $t>1$ can be handled similarly. Finally, the formula

$ZD[u_{0}](-t,-x)=ZD[u_{0}](t,x)$

follows from the fact that $u_{0}$ is even and therefore that the solution
$u^{\varepsilon}$ of (1) satisfies
$u^{\varepsilon}(t,x)=u^{\varepsilon}(-t,-x)$. ∎

We conclude this section by observing that the map $ZD$ does not satisfy the
semigroup property. More precisely, we show on the above example that, in
general,

$ZD[ZD[u_{0}](t)](s)\neq ZD[u_{0}](t+s)\ .$

Let us choose $t=1$ and $s>0$ in the above example. Then, according to Theorem
3,

$\displaystyle ZD[u_{0}](1,x)=\begin{cases}0&{\rm if}\
x\in]-\infty,-1]\cup]3,+\infty[\\\ \frac{x+1}{2}&{\rm if}\ x\in]-1,1]\\\
1-\frac{x-1}{2}&{\rm if}\ x\in]1,3]\ .\end{cases}$

Set $u_{1}(x):=ZD[u_{0}](1,x)$. This is a continuous, piecewise linear
function, and we can easily approximate it by $C^{1}$ functions by smoothing
it in neighbourhoods of $x=-1,1,3$. Given $s\in]0,1[$, the function $y\mapsto
y+2su_{1}(y)$ is continuous and strictly increasing, sending $]-\infty,-1[$
onto itself, $]-1,1[$ onto $]-1,2s+1[$, $]1,3[$ onto $]2s+1,3[$ and
$]3,+\infty[$ onto itself. Consequently, we obtain, after passing to the
limit,

$\displaystyle ZD[u_{1}](s,x)=\begin{cases}0&{\rm if}\
x\in]-\infty,-1[\cup]3,+\infty[\\\ \frac{x+1}{2(s+1)}&{\rm if}\
x\in]-1,2s+1[\\\ \frac{3-x}{2(1-s)}&{\rm if}\ x\in]2s+1,3]\\\ 0&{\rm if}\
x\in]3,+\infty[\ .\end{cases}$

$x$$1$$ZD[u_{1}](s,x)$$-1$$2s+1$3$\bullet$$\bullet$ Figure 5.5. The iteration
$ZD[ZD[u_{0}](1)](s)$ for $s\in]0,1[$.

Comparing with the expression of $ZD[u_{0}](1+s,x)$ from Theorem 3, we
conclude that $ZD[ZD[u_{0}](1)](s)\neq ZD[u_{0}](1+s)\ .$

######  Remark 3.

As pointed to us by Y. Brenier, this lack of semigroup property can also be
proved by observing that the semigroup associated to the entropic solution
[19, 25] of the inviscid Burgers–Hopf equation is deduced in [1, 2, 3] from
the map $ZD[\,.\,]$ through a Trotter formula. If the latter was a semigroup,
it would coincide with the former, which is obviously wrong.

## References

  * [1] Y. Brenier, Une application de la symétrisation de Steiner aux équations hyperboliques : la méthode de transport et écroulement, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 563–566.
  * [2] Y. Brenier, Résolution d’équations d’évolution quasilinéaires en dimension $N$ d’espace à l’aide d’équations linéaires en dimension $N+1$, J. Differential Equations 50 (1983), 375–390.
  * [3] Y. Brenier, Averaged multivalued solutions for scalar conservation laws, SIAM J. Numer. Anal. 21 (1984), 1013–1037.
  * [4] Y. Brenier, S. Osher, The discrete one–sided Lipschitz condition for convex scalar conservation laws, SIAM J. Num. Anal. 25(1988), 8–23.
  * [5] J. M. Burgers, A mathematical model illustrating the theory of turbulence, Advances in applied mechanics 1 (1948), 171–179.
  * [6] T. Claeys, T. Grava, Universality of the break-up profile for the KdV equation in the small Dispersion limit using the Riemann–Hilbert approach, Commun. Math. Phys 247 (2009), 979–1009.
  * [7] P. Deift, S. Venakides, X. Zhou, New results in small dispersion KdV by an extension of the steepest descent method for Riemann–Hilbert problems, Internat. Math. Res. Notices 6 (1997), 286–299.
  * [8] G. M. Fiksdal, Numerical methods for the Benjamin–Ono equation , Master Thesis under the supervision of H. Holden, Norwegian University of Science and Technology, 2013.
  * [9] M. Gallone, M. Marian, A. Ponno, S. Ruffo, Burgers turbulence in the Fermi–Pasta–Ulam–Tsingou chain, Phys. Rev. Letters 129(2022), 114101.
  * [10] M. Gallone, A. Ponno, Hamiltonian field theory close to the wave equation: from Fermi–Pasta–Ulam to water waves, arXiv: 2202.13454.
  * [11] L. Gassot, Zero-dispersion limit for the Benjamin–Ono equation on the torus with single well initial data, arXiv: 2111.06800, Comm. Mathematical Physics (2023), https://doi.org/10.1007/s00220-023-04701-0.
  * [12] L. Gassot, Lax eigenvalues in the zero-dispersion limit for the Benjamin-Ono equation on the torus, arXiv: 2301.03919, To appear in SIAM J. Math. Analysis.
  * [13] P. Gérard, An explicit formula for the Benjamin–Ono equation, arXiv: 2212.03139, to appear in Tunisian Journal of Mathematics.
  * [14] P. Gérard, T. Kappeler, On the integrability of the Benjamin–Ono equation on the torus, Comm. Pure Appl. Math. 74 (2021), 1685–1747.
  * [15] P. Gérard, T. Kappeler, P. Topalov, On smoothing properties and Tao’s gauge transform for the Benjamin–Ono equation on the torus, arXiv: 2109.00610, to appear in Annales Scient. Ecole Norm. Sup.
  * [16] T. Grava, C. Klein, Numerical solution of the small dispersion limit of Korteweg–de Vries and Whitham equations, Comm. Pure Appl. Math. 60 (2007), 1623–1664.
  * [17] E. Hopf, The partial differential equation $u_{t}+uu_{x}=\mu u_{xx}$ , Comm. Pure and Appl. Math. 3(1950), 201–230.
  * [18] A. Ionescu, C. Kenig, Global well-posedness of the Benjamin–Ono equation in low regularity spaces, J. Amer. Math. Soc. 20(2007), 753–798.
  * [19] S. N. Kruzhkov, Generalized solutions of the Cauchy problem in the large for first order nonlinear equations, Dokl. Akad. Nauk. SSSR 187(1969), 29–32.
  * [20] P. Lax, C. D. Levermore, The small dispersion limit of the Korteweg–de Vries equation, I, II, III, Comm. Pure Appl. Math. 36 (1983), 253–290, 571–593, 809–830.
  * [21] L. Molinet, D. Pilod, The Cauchy problem for the Benjamin-Ono equation in $L^{2}$ revisited, Anal. and PDE 5(2012), 365–395
  * [22] P. Miller, A. Wetzel, The scattering transform for the Benjamin–Ono equation in the small dispersion limit, Physica D 333 (2016), 185–199.
  * [23] P. Miller, Z. Xu, On the zero–dispersion limit of the Benjamin–Ono Cauchy problem for positive initial data, Comm. Pure Appl. Math. 64 (2011), 205–270.
  * [24] O. Oleinik, Discontinuous solutions of nonlinear differential equations, Uspekhi Mat. Nauk 12(1957), 3–73.
  * [25] D. Serre, Systems of conservation laws I, Cambridge University Press, 1999.
  * [26] S. Venakides, The zero dispersion limit of the Korteweg–de Vries equation for initial potentials with non–trivial reflection coefficient, Comm. Pure Appl. Math. 38 (1985), 125–155.


# Geometricities of driven transport in presence of reservoir squeezing

Javed Akhtar  Jimli Goswami  Himangshu Prabal Goswami<EMAIL_ADDRESS>aDepartment of Chemistry, Gauhati University, Jalukbari, Guwahati-781014,
Assam, India

###### Abstract

In a bare site coupled to two reservoirs, we explore the statistics of boson
exchange in the presence of two simultaneous processes: squeezing the two
reservoirs and driving the two reservoirs. The squeezing parameters compete
with the geometric phaselike effect or geometricity to alter the nature of the
steadystate flux and noise. The even (odd) geometric cumulants and the total
minimum entropy are found to be symmetric (antisymmetric) with respect to
exchanging the left and right squeezing parameters. Upon increasing the
strength of the squeezing parameters, loss of geometricity is observed. Under
maximum squeezing, one can recover a standard steadystate fluctuation theorem
even in the presence of phase different driving protocol. A recently proposed
modified geometric thermodynamic uncertainty principle is found to be robust.

##  I Introduction

Phase-different multiparametric temporal driving allows an additional leverage
over a system’s dynamics pancharatnam1956generalized ; berry1984quantal . This
leverage is due to gauge-invariant geometric observables during the system’s
time evolution which affect the driven transport and time-dependent energy
conversion processes. Additional phases during the time evolution of a system
that arise during cyclic variations of two parameter adiabatic driving are
usually referred to as Pancharatnam-Berry phases which introduce nontriviality
into a well understood system mukunda1993quantum . As a first application, the
holonomy of the parametric space was engineered to observe bias-independent
electronic pumping under slow periodic variations thouless1983quantization .
Subsequently, this paradigm was extended to hem in upon nonequilibrium systems
that exchange matter/energy with macroscopic reservoirs carollo2003geometric ;
wang2022geometric . Usually, geometric contributions in nonequilibrium quantum
systems are introduced by either driving the reservoirs’ temperatures or
chemical potentials or even the system-reservoir couplings wang2022geometric .
In such systems, the geometric effects not only actuate the steadystate
dynamics but also lead to violations of the well established fluctuation
theorems (FT) and thermodynamic uncertainity relationships (TUR)
PhysRevLett.104.170601 ; PhysRevE.106.024131 ; hino2020fluctuation , which are
otherwise robust even in the presence quantum coherences, entanglement and
quantum squeezing. These geometric effects are almost entirely quantified by
identifying its contribution to the generating function describing any
exchange processes in a nonequilibrium quantum system wang2022geometric ;
takahashi2020full ; hino2020fluctuation . The resulting generating functions,
derived from a full counting statistical (FCS) method, have an additive term
apart from the inherent dynamic term which is driving dependent and possesses
a geometric curvature in the parameter space PhysRevLett.104.170601 ;
sinitsyn2007berry . The geometric contribution in such nonequilibrium quantum
systems can also be observed during the evolution of the system’s density
matrix goswami2016geometric . Although observable, it is no longer a phase
factor and hence, is referred to as geometric phaselike effect or simply
geometricity.

Such effects have also been explored in quantum heat engines, thermoelectric
devices and molecular junctions hino2021geometrical ; PhysRevE.96.052129 ;
giri2019nonequilibrium ; eglinton2022geometric ; goswami2016geometric ;
yuge2012geometrical . Enhancement of engine’s constancy, affecting the
coherent contribution to flux, observation of giant Fano factors and
fractional quantization of the flux etc have been reported lu2022geometric ;
PhysRevE.96.052129 ; PhysRevLett.104.170601 . On a separate note, in the
absence of geometric effects, general observables like flux, higher order
fluctuations, constancy, thermodynamic affinities are also affected when
parameters describing the reservoirs are altered, eg. by introducing quantum
mechanical squeezing walls1983squeezed ; schnabel2017squeezed ;
dodonov1991physical ; puri1997coherent ; dupays2021shortcuts ;
abebe2021interaction . Squeezed reservoirs are also known to introduce
additional quantum control which have been exploited to observe nontrivial
quantum thermodynamics like additional corrective parameters on the classical
fluctuation theorem of the Crooks type holmes2019coherent  or not leading to
Jarzynski-Wojcik type of fluctuation theorem yadalam2022counting . Squeezed
states of the thermal reservoirs have also been exploited to overcome Carnot
limit in heat enginesmanzano2016entropy ; niedenzu2016operation ;
agarwalla2017quantum ; klaers2017squeezed ; newman2017performance  , violate
universal maximum power theories giri2019nonequilibrium ; PhysRevE.96.052129 ;
PhysRevE.106.024131  and introduce higher order correlated photon pairs from
MgO:LiNbO3 crystals ourjoumtsev2011observation ; mehmet2010observation . To
corral a universal understanding of the role of squeezed initial states in FTs
and TURs, several possibilities are currently under conceptualization
huang2012effects ; hsiang2021fluctuation ; talkner2013statistics . For
example, higher order fluctuations during photon transport can be maximized
due to mixing between a qubit and squeezed resonators wang2021nonequilibrium .
When treated separately, both geometricity (introduced via tuning the
reservoirs) and squeezing the reservoirs inherently affect the quantum
thermodynamics of nonequilibrium systems separately. Hence, it is a natural
question to ask about the quantum thermodynamics of nonequilibrium systems
where squeezed reservoirs are subjected to periodic modulations. This paper is
the first to address this question. Since, presence of geometricity make even
a simple model, eg. a resonant level coupled to two thermal or electronic
reservoirs non-trivial yuge2012geometrical ; goswami2016geometric , we focus
on such a system, where the reservoirs are squeezed.

In this work, we study the effect of squeezing the reservoirs on the
statistics of particle exchange when the temperature of the reservoirs are
periodically modulated. The geometricity that manifests itself into the
quantum statistics is explored in a toy model which is a bare site coupled to
two squeezed reservoirs. Such a model is passably standard and well-studied in
quantum transport giraldi2014coherence ; PhysRevLett.104.170601 ;
pekola2021colloquium . Our work focuses on identifying the competition between
squeezing and driving on the fluctuations of boson exchange within a quantum
statistical framework. We implement the acknowledged methodology of full
counting statistics (FCS) within a quantum master equation framework
esposito2009nonequilibrium . Firstly, in Sec. (II) we present our model and
the general formalism used. In Sec. (III), we show our results and analysis
after which we conclude in Sec. (V).

##  II Model and Formalism

Figure 1:  (a) Schematic diagram of two squeezed harmonic baths interacting
with a bosonic site with two Fock states ($|0\rangle$ and $|1\rangle$). The
temperatures of the two squeezed baths are time-dependent via an amplitude-
modulated phase different driving protocol as per Eq.(7)) and (b) represents a
circle in the parameter space of $T_{\ell}$ and $T_{r}$ with
$T_{\ell}^{0}=300K,T_{r}^{0}=250K$. Squeezing dependent (c) dynamic (d)
geometric cumulant generating function with squeezing parameters
($x_{\ell},x_{r}$) = (0,0),(0.7,0),(0,0.7) and $(\pi,\pi)$ for (c) outermost
to innermost curves and (d) in order of decreasing magnetude. The other
parameters are fixed throughout the manuscript at $\omega_{o}=7.4\pi THz$,
$\gamma_{\ell}=\gamma_{r}=1000THz$, $\Omega=100THz$, $A_{o}=100,\phi=\pi/4$.

A bare site coupled to two reservoirs have been thoroughly studied both in
presence and absence of squeezing. PhysRevE.73.026109 ; carpio2021quantum ;
abebe2021interaction ; kowalewska2001generalized . The site can be effectively
described by two Fock states that correspond to a boson-occupied ($|1\rangle$)
and an unoccupied state, ($|0\rangle$), separated by an energy
$\hbar\omega_{o}$ (see the appendix for the Hamiltonian). On the experimental
front, such a model can be a flexural mode of a GaAs-based nanobeam structure
piezoelectrically coupled to squeezed electronic noise (squeezed thermal
reservoirs) klaers2017squeezed  or a qubit system realizable in an NMR setup
saryal2019thermodynamic  as well as in a transmon qubit around a SQUID setup
lu2021nonequilibrium . The schematic representation of the model is shown in
Fig.(1 a). In such a nonequilibrium system, the time evolution of reduced
density matrix, $\hat{\rho}$, (within standard Born-Markov approximation
techniques) is a Pauli rate equation (decoupled from coherences) with two Fock
states, $|1\rangle$ and $|0\rangle$ acting as the boson exchanger between the
two squeezed reservoirs. When the reservoirs are driven, the rates become
time-dependent (see the appendix). Within the standard theory of full counting
statistical (FCS) formalism esposito2009nonequilibrium ; harbola2007statistics
, one can keep track of the net number of bosons exchanged, $q$, through a
moment generating vector for the reduced system in terms of the auxiliary
counting field, $\lambda$. In the Liouville space , the reduced moment
generating density vector, $|\breve{\rho}(\lambda,t)\rangle\rangle$, can be
written as,

$|\dot{\breve{\rho}}(\lambda,t)\rangle\rangle={\breve{\cal
L}}(\lambda)|\breve{\rho}(0,0)\rangle\rangle,$ (1)

where the elements of the density vector contain the populations of the
occupied and unoccupied states, $\\{\rho_{11}$, $\rho_{00}\\}$ (appendix) with
the time-dependent evolution superoperator, ${\breve{\cal L}}(\lambda,t)$,
given by

$\displaystyle{\breve{\cal L}}(\lambda,t)$
$\displaystyle=\left[\begin{array}[]{cc}-\alpha_{L}(t)-\alpha_{R}(t)&\beta_{L}(t)e^{\lambda}+\beta_{R}(t)\\\
\alpha_{L}(t)e^{-\lambda}+\alpha_{R}(t)&-\beta_{L}(t)-\beta_{R}(t)\end{array}\right].$
(4)

It is a standard practice to ignore the Lamb shifts terms so that the quantum
mechanical rates of boson exchange between the system and reservoirs can be
recast as:

$\displaystyle\alpha_{\nu}(t)$
$\displaystyle=\gamma_{\nu}\\{\cosh{(2x_{\nu})(n_{\nu}(t)+\frac{1}{2})+\frac{1}{2}}\\},$
(5) $\displaystyle\beta_{\nu}(t)$
$\displaystyle=\gamma_{\nu}\\{\cosh{(2x_{\nu})(n_{\nu}(t)+\frac{1}{2})-\frac{1}{2}}\\}.$
(6)

$\gamma_{\nu},\nu=\ell,r$ represents the coupling of the bare site and the
$\nu$-th reservoir with $n_{\nu}=(e^{\hbar\omega_{o}/T_{\nu}(t)}-1)^{-1}$
being the Bose-Einstein distribution of the $\nu$-th bath. $x_{\nu}>0$ is the
renormalized parameter responsible for squeezing the $\nu$-th harmonic bath
within the Markovian regime li2017production  (see the appendix). Within this
approximation, the squeezing properties get symmetrically distributed about
the concerned left or right squeezed bath’s frequency tanas2002squeezing . The
parametric modulation is present in the reservoirs’ temperatures,
$T_{\nu}(t)$, which we take to be of the following form,

$\displaystyle T_{\ell}(t)$ $\displaystyle:=T_{\ell}^{o}+A_{o}\cos(\Omega
t+\phi),$ (7) $\displaystyle T_{r}(t)$
$\displaystyle:=T_{r}^{o}+A_{o}\sin(\Omega t+\phi),$ (8)

$A_{o}$, $\Omega$ and $\phi$ are the amplitude, frequency and phase difference
between the driving protocols, respectively. Note that, this theory is valid
under the adiabatic evolution assumption, where the individual decay
timescales of the system and reservoirs are well separated.

In the steady state, when $\lambda=0$, a zero eigenvalue $\zeta_{o}(t)$ is
obtained from the RHS of Eq.(4). From this zero-eigenvalue, a cumulant
generating function, $S(\lambda)$ within the domain
$\lambda\in\\{-\infty,\infty\\}$, can be constructed which allows evaluation
of the $n$-th order cumulants,
$j^{(n)}=\partial_{\lambda}S(\lambda)|_{\lambda=0}$ esposito2009nonequilibrium
. In the presence of phase different driving protocol, $S(\lambda)$ is known
to be additively separable into two components, one dynamic
($S_{d}(\lambda,t)$) and a geometric term $S_{g}(\lambda,t)$. The geometric
term $S_{g}(\lambda,t)$ essentially is the source of geometricity in such
driven dynamics and is obtainable from the left eigenvector ($\langle
L_{o}(\lambda,t)|$) and the right eigenvector ($|R_{o}(\lambda,t)\rangle$) of
the r.h.s of Eq.(1) for the eigenvalue $\zeta_{o}(\lambda,t)$. It is
nonexistent if the two parameters (Eq. (7) are driven without any phase
difference,i.e $\phi=0$ yuge2012geometrical . Both the dynamic and geometric
cumulants can be evaluated as sinitsyn2007berry ; PhysRevLett.104.170601 ;
goswami2016geometric ; PhysRevLett.104.170601 ; hino2020fluctuation ;
PhysRevE.106.024131 ; takahashi2020full

$\displaystyle j^{(n)}_{d}=\partial_{\lambda}^{n}S_{d}(\lambda)_{\lambda=0}$
$\displaystyle=\displaystyle\frac{1}{t_{p}}\int_{0}^{t_{p}}\partial_{\lambda}^{n}\zeta_{o}(\lambda,t)|_{\lambda=0}(t)dt$
(9) $\displaystyle
j^{(n)}_{g}=\partial_{\lambda}^{n}S_{g}(\lambda)_{\lambda=0}$
$\displaystyle=\displaystyle\frac{1}{t_{p}}\int_{t_{p}}^{0}\partial_{\lambda}^{n}\langle
L_{o}(\lambda,t)|\dot{R}_{o}(\lambda,t)\rangle dt|_{\lambda=0}$ (10)
$\displaystyle=-\displaystyle\partial_{\lambda}^{n}\oiint_{S}{\cal
F}_{T_{\ell}T_{r}}(\lambda)dT_{\ell}dT_{r}|_{\lambda=0}$ (11)

with $t_{p}=2\pi/\Omega$ being the time period of the chosen external driving
(Eq.(7)). In Eq. (11), the integrand, $F_{T_{\ell}T_{r}}(\lambda)$, is the
known as the geometric curvature and is analogous to the Pancharatnam-Berry
curvature sinitsyn2007berry ; PhysRevLett.104.170601 ; goswami2016geometric ;
PhysRevLett.104.170601 ; hino2020fluctuation ; PhysRevE.106.024131 ;
takahashi2020full  in the $T_{\ell},T_{r}$ surface, $S$. Here, $n=1$ and $2$
correspond to the flux and noise respectively. Both the quantities depend on
the squeezing parameters, $x_{\ell}$, $x_{r}$ through the modified rates in
Eq.(5) and Eq.(6). The dynamic and geometric cumulant generating functions are
shown in Fig.(1c,d) for different squeezing parameters.

##  III Results and Discussion

By evaluating the eigensystem of Eq.(4), we can identify the smallest
eigenvalue $\zeta_{o}(\lambda,t)$ (see the appendix, Eq.(35)), from which we
numerically obtain the dynamic flux and noise using Eq.(9). The behavior of
the two dynamic cumulants ($n=1,2$) are shown in Fig. (2a and b) for equal
initial temperatures. The quantitative behavior is not that different from the
undriven case sarmah2023nonequilibrium  apart from change in magnitude. These
also retain the symmetry (antisymmetry) of the even (odd) cumulants with
respect to the exchange of the left and right squeezing parameters under equal
initial temperature ($T_{\ell}^{0}=T_{r}^{0}$) setting as well as the
saturating behavior as observed earlier for undriven case
sarmah2023nonequilibrium . The solid lines in Fig.(2) are evaluated by keeping
$x_{\ell}$ fixed while $x_{r}$ is varied. The dotted lines represent the case
when $x_{\ell}\to x_{r}$ while $x_{\ell}$ is varied, denoted by the symbol
$x_{\ell}\leftrightarrow x_{r}$ in the abscissa. This is simply because the
rates that affect the dynamic cumulants are just scaled by the hyperbolic
cosine functions (Eq.5 and Eq.(6)) and doesn’t alter the overall mathematical
structure of the eigenvalue $\zeta_{o}(\lambda,t)$ (Eq.(35)). In the figures,
we also have denoted the cumulants in absence of squeezing ($x_{\nu}=0$) and
driving as $j^{(n)}_{o}$ by defining a dimensionless ratio
$C^{(n)}_{d/g}:=j^{(n)}_{d/g}/j^{(n)}_{o}$. When$|C^{(n)}_{d/g}|>(<)1$, the
squeezing increases (decreases) the value of the cumulant in comparison to the
unsqueezed and undriven case.

Figure 2:  (a) Behavior of the absolute dynamic flux, $j^{(1)}_{d}$ as a
function of the two reservoirs’ squeezing parameters (the $\leftrightarrow$
indicates exchanging the values of the two parameters $x_{\ell}$ and $x_{r}$)
evaluated at equal initial temperatures $T_{\ell}^{0}=T_{r}^{0}=300K$. The
solid (dotted) lines are when $x_{\ell}$ is fixed ($x_{\ell}$ is changed to
$x_{r}$). Note the antisymmetry due to the exchange $x_{\ell}\leftrightarrow
x_{r}$. (b) Behavior of the scaled dynamic noise (second cumulant). Note the
equality upon exchanging the $x_{\ell}$ and $x_{r}$ values. Plot of geometric
scaled flux ((c)) and noise ((d)) highlighting the equality and antisymmetry
upon exchanging the squeezing parameters.

On substituting the left and right eigenvectors of $\breve{\cal L}(\lambda,t)$
in the geometricity term $\langle L(\lambda,t)|\dot{R}(\lambda,t)\rangle$, of
Eq.(10) we can identify the geometric flux and geometric noise. The geometric
flux is given by,

$\displaystyle j_{g}^{(1)}$
$\displaystyle=-\frac{\Omega}{2\pi}\displaystyle\int_{0}^{t_{p}}\frac{2\Gamma\cosh(2x_{\ell})\cosh(2x_{r})}{(\gamma_{\ell}X_{\ell}^{+}+\gamma_{r}X_{r}^{+})^{3}}dt$
(12)

and shown in Fig. (2c). Note that the geometric flux decays to zero at higher
values of the squeezing parameter. The geometric noise is given by

$\displaystyle j_{g}^{(2)}$
$\displaystyle=-\displaystyle\frac{\Omega}{2\pi}\int_{0}^{t_{p}}\frac{12\Gamma^{2}\cosh(2x_{\ell})\cosh(2x_{r})(X_{r}^{+}-X_{l}^{+})dt}{(\gamma_{\ell}+\gamma_{r})(\gamma_{\ell}X_{\ell}^{+}+\gamma_{r}X_{r}^{+})^{5}}$
(13)

with $\Gamma=\gamma_{\ell}\gamma_{r}(\gamma_{\ell}+\gamma_{r})$ and
$X_{\nu}^{\pm}:=\cosh(2x_{\nu})(2n_{\nu}(t)\pm 1)$. The r.h.s of Eq. (12) and
Eq.(13) are evaluated as a function of the squeezing parameters and the scaled
function is shown in Fig. (2c) and Fig. (2 d) respectively. Both the geometric
cumulants are observed decaying to zero as the squeezing parameters are
increased. Further, it is also observed that the geometric flux (odd cumulant)
is symmetric with respect to exchanging the squeezing parameters while the
second cumulant is antisymmetric, contrary to the behavior of the dynamic
cumulants. The symmetric behavior upon exchanging $x_{\ell}$ and $x_{r}$ in
the geometric flux is because the denominator in the integrand inside the
r.h.s of Eq.(12) is symmetric with respect to exchange. The noise is
antisymmetric with respect to exchange because the numerator imparts a
negative sign upon exchanging the squeezing parameters. It is interesting to
note that both the exchange symmetry and the antisymmetry doesn’t hold when
the initial temperatures are different. This is shown graphically in Fig.(3c
and d). Note that, in Eq.(13), when $X_{r}^{+}=X_{\ell}^{+}$, we obtain
$j^{(2)}_{g}=0$. This condition can be triggered by controlling the squeezing
parameters $x_{\ell}$ and $x_{r}$ and can be seen as the zero line along the
diagonal ($x_{\ell}=x_{r}$) of the contour plot in Fig.(3d). Under this same
condition, the integral in Eq.(12) is however nonzero and one observes
geometric flux without geometric fluctuations.

Figure 3:  Plot highlighting absence of symmetry and antisymmetry in the
geometric flux (a) and noise (b) under unequal initial temperatures upon
exchanging the squeezing parameters. Contour plots showing the vanishing
geometric flux (c) and noise (d) at higher values of squeezing parameters.
Note the zero values along the diagonal.

We now move on to explain why the geometric effects in the flux and
fluctuations vanish at higher squeezing values as seen in Fig.(2c,d) and
Fig.(3). This is because $S_{g}(\lambda)$ vanishes at higher values of
$x_{\ell},x_{r}$, as seen in Fig. (1d). The geometric curvature, in the
present model is of the form,

$\displaystyle F_{T_{\ell}T_{r}}(\lambda)=-\displaystyle\frac{2\Gamma
C_{\ell}C_{r}\sin(\lambda)}{\\{K+4f(\lambda)\\}^{3/2}}$ (14)

with

$\displaystyle C_{\nu}$ $\displaystyle=\frac{1}{k_{B}T_{\nu}^{2}}\hbar\omega
e^{\hbar\omega/k_{B}T_{\nu}}((n_{\nu}+1/2)\cosh(2x_{\nu})-\frac{1}{2})$ (15)
$\displaystyle K$
$\displaystyle=\gamma_{\nu}\displaystyle\sum_{\nu=\ell,r}2\cosh(2x_{\nu})(n_{\nu}+\frac{1}{2})$
(16) $\displaystyle f(\lambda)$
$\displaystyle=\displaystyle\prod_{\nu=\ell,r}\gamma_{\nu}(\cosh(2x_{\nu})(n_{\nu}+1/2)-1/2)$
$\displaystyle\times
e^{\hbar\omega/k_{B}T_{\ell}}(e^{\lambda}-1)+e^{\hbar\omega/k_{B}T_{\ell}}(e^{-\lambda}-1)$
(17)

and is analogous to the known expression for the unsqueezed case ($x_{\nu}=0$)
sarmah2023nonequilibrium . $F_{T_{\ell}T_{r}}(\lambda)$ is finite for the
unsqueezed case around $\lambda=0$. At low values of $\lambda$, the
$\sin(\lambda)$ dominates over the denominator’s $f(\lambda)$ term resulting
in the typical modified sinusoidal shape as already reported. In the present
case too, at lower values of squeezing ($x_{\ell},x_{r}\approx 0$) around
$\lambda=0$, such a behavior is shown for $S_{g}(\lambda)$ as shown in Fig.
(1d). As $x_{\nu}$ is increased, the hyperbolic terms from the squeezing
parameters start contributing more to Eq. (14) around $\lambda=0$ and changes
the overall geometricity. These squeezed parameters can now be used to gain
control or steer the underlying geometric statistics.

Note that, in general, the mathematical structure of $F_{T_{l}T_{r}}$ in
Eq.(14) is such that the numerator (denominator) has an overall squared (cube-
halved) dependence on the cosine hyperbolic terms. This structure dictates
that, as one keeps squeezing the reservoirs the denominator keeps increasing
and hence the amplitude (quantified by the co-efficients) of the
$\sin(\lambda)$ term keeps reducing, which results in lower slope around
$\lambda=0$. This causes the geometric flux and subsequent cumulants to keep
reducing and finally vanishes as shown in Fig.(1d). We can safely conclude
that squeezing the reservoirs reduces the geometricity of the driven system.
In this high-squeezing limit, even upon increasing the frequency of phase-
different driving, $\Omega\gg 1$, the statistics of exchange is solely
governed by the dynamicity of the system, i.e $S_{d}(\lambda)$. We can prove
this analytically by considering the following limiting case. Under the
assumption that $n_{\nu}\ll 1/2$ (low temperature regime), we have,

$\displaystyle C_{\nu}|_{n_{\nu}\ll 1/2}$
$\displaystyle\propto\cosh(2x_{\nu})-1$ (18) $\displaystyle K|_{n_{\nu}\ll
1/2}$ $\displaystyle\propto\displaystyle\sum_{\nu}\cosh(2x_{\nu})$ (19)
$\displaystyle f(\lambda)|_{n_{\nu}\ll 1/2}$
$\displaystyle\propto\displaystyle\prod_{\nu}(\cosh(2x_{\nu})-1)$ (20)

which results in

$\displaystyle F_{T_{\ell}T_{r}}|_{n_{\nu}\ll 1/2}$
$\displaystyle\propto\frac{\sin(\lambda)}{\displaystyle\sqrt{\sum_{\nu}\cosh^{3}(2x_{\nu})}\sqrt{\displaystyle\prod_{\nu}(\cosh(2x_{\nu})-1)}}$
(21)

In the above expression, taking either of the two limits, $x_{\ell}\to\infty$
or $x_{r}\to\infty$ results in the r.h.s being zero. Thus, squeezing the
reservoirs to its extremum kills the geometric curvature or the geometricity
resulting in $S_{g}(\lambda)=0$. The complete contour plots of the two
geometric cumulants $C^{(1)}_{g}$ and $C^{(2)}_{g}$ as a function of
$x_{\ell}$ and $x_{r}$ are shown in Fig. (3c and d). In both the plots, the
geometric effects vanish at higher values of squeezing.

##  IV Thermodynamic uncertainity relationship

For an undriven case, $j^{(n)}_{g}=0$ (when $\Omega=0$ or $\phi=0$), a
standard thermodynamic uncertainity relationship (TUR), reminiscent of a
steadystate fluctuation theorem holds, given by $F{\cal A}\geq 2k_{B}$
pietzonka2017finite ; saryal2019thermodynamic  with $F=j^{(2)}/j^{(1)}$ being
the Fano factor while ${\cal A}$ is the thermodynamic affinity of the system.
This TUR has been shown not to hold in the presence of geometric effects
PhysRevE.96.052129 .

In the present case, one can recover the standard TUR in the high squeezing
limit of either reservoir. Under maximum squeezing,
$F_{T_{l}T_{r}}(\lambda)=0$ kills the geometric contributions to the system
statistics. We can hence recover a Gallavoti-Cohen symmetry,

$\displaystyle\lim_{x_{\nu}\to\infty}\displaystyle\frac{1}{t_{p}}$
$\displaystyle\int_{0}^{t_{p}}\zeta_{o}(\lambda,t)dt$
$\displaystyle=\lim_{x_{\nu}\to\infty}\displaystyle\frac{1}{t_{p}}\int_{0}^{t_{p}}\zeta_{o}(-\lambda-\displaystyle\lim_{x_{\nu}\to\infty}{\cal
A},t)dt,$ (22)

with

$\displaystyle{\cal
A}=\log\left(\frac{\int_{0}^{t_{p}}X_{\ell}^{-}X_{r}^{+}dt}{\int_{0}^{t_{p}}X_{\ell}^{+}X_{r}^{-}dt}\right)$
(23)

where the time and squeezing dependent quantities $X_{\nu}^{\pm}$ are defined
in the text below Eq.(13). Eq.(23) reduces to the known expression
$1/T_{\ell}-1/T_{r}$ in absence of driving PhysRevLett.104.170601  that leads
to a steadystate fluctuation theorem. The recovery of the symmetry hence
allows us to recover the standard TUR,

$\displaystyle\displaystyle\lim_{x_{\nu}\to\infty}{\cal
A}\frac{\displaystyle\lim_{x_{\nu}\to\infty}j^{(2)}_{d}}{\displaystyle\lim_{x_{\nu}\to\infty}j^{(1)}_{d}}\geq
2k_{B}$ (24)

In the case of finite (but not maximal) squeezing, the geometricities are
still present. TUR in such a case has been shown to get modified by including
a geometric correction factor lu2022geometric ,

$\displaystyle\frac{j^{(2)}\Sigma}{(j^{(1)})^{2}g(\Omega)}\geq 2k_{B}$ (25)

where, $g(\Omega)$ is the driving dependent geometric correction factor and is
of the form,

$\displaystyle g(\Omega)$
$\displaystyle=\frac{1}{(1+j^{(1)}_{g}/j^{(1)}_{d})^{2}}$ (26)

Figure 4:  (a) Behavior of the geometric correction factor, $g(\Omega)$ to the
TUR as a function of $x_{\ell}.$ Note the singularity at $x_{\ell}=0.7$. At
this value of $x_{\ell}$, the thermodynamic force, ${\cal A}=0$ (dotted line).
(b) Behavior of the minimum entropy produced. It is also zero at
$x_{\ell}=0.7$. (c) Contour of $\Sigma_{min}$ as a function of squeezing
parameters. The region where it is zero is where ${\cal A}=0$. (d) Symmetry in
the minimum entropy upon exchanging the values of the squeezing parameters.

We numerically evaluate Eq.(26) and plot $g(\Omega)$ as a function of
$x_{\ell}$ in Fig.(4a) where a discontinuity is observed at $x_{\ell}=0.7$.
This discontinuity is at that point of $x_{\ell}$, where ${\cal A}=0$
($e^{\cal A}$ is shown as a vertically slanted line) that results in
$j^{(1)}_{d}=0$ in Eq.(26). Further, for any fixed value of $j^{(1)}_{g}$ in
Eq.(25), the r.h.s is greater (less) than unity when $j^{(1)}_{d}<(>)0$ and
vice versa. We have earlier shown that, for an undriven case, the direction of
the dynamic flux $j^{(1)}_{d}$ is controllable through the squeezing
parameters due to the modification of the thermodynamic affinity, ${\cal A}$
sarmah2023nonequilibrium . Thus by controlling $x_{\ell}$, we observe regions
where $g(\Omega)>1$ and $g(\Omega)<1$ characterized by a shift between these
two regions at that value of $x_{\ell}$ where ${\cal A}=0$ as seen in
Fig.(4a). The curve below unity is evaluated by maintaining positive geometric
flux and ${\cal A}>0$ ($T_{\ell}=T_{r},x_{\ell}=0.7,x_{r}=0$) so that
$g(\omega)<1$. In a standard context without squeezing, as in the work of Lu
et al lu2022geometric  where $g(\Omega)$ was introduced, the dynamic flux is
solely dependent on the temperature gradient which controls the thermodynamic
affinity and the $g(\Omega)$ is a continuous function either below unity or
above unity depending on the signs of the dynamic or geometric fluxes.

We state that this observed discontinuity doesn’t lead to violation of the
modified TUR, Eq. (25). Although, not highlighted in the earlier work
lu2022geometric , the continuity of $g(\Omega)$ in Eq. (25) as a function of a
system parameter is rather limited to positive dynamic flux, characterized by
${\cal A}>0$. As long as we maintain ${\cal A}>0$, by properly choosing
$x_{\ell}$ and $x_{r}$ values, the modified TUR given by Eq.(25) always holds
within these two separate regions. By maintaining, ${\cal A}>0$, we directly
estimate the minimum entropy production, $\Sigma_{min}$ in the presence of
geometricities,

$\displaystyle\Sigma_{min}=2k_{B}\displaystyle\frac{\left(j^{(1)}_{d}+j^{(1)}_{g}\right)^{2}}{j^{(2)}_{d}+j^{(2)}_{g}}g(\Omega)$
(27)

In the above equation, it is not possible to separate the entropy rates into
dynamic and geometric contributions. Although when $\Omega\gg 1$,
$j^{(1)}_{g}\gg j^{(1)}_{d}$, the same cannot be said for the second cumulant
which makes the denominator in Eq.(27) to have combined dynamic and geometric
contributions. Nonetheless, the modified TUR allows an easy way to evaluate
the total minimum entropy production rate. Note that, in the presence of
geometricities, evaluation of entropies with contribution from both dynamic
and geometric components is not at all straightforward yuge2013geometrical
due to production of excess entropies. We evaluate Eq.(27) and plot it as a
function of the left reservoir’s squeezing parameters in Fig.(3)b,c and d).
The dependence of $\Sigma_{min}$ on $x_{\ell}$ is nonlinear and saturates at
higher values. In Fig.(3)c), we show a contour map of $\Sigma_{min}$ for a
wide range of $x_{\ell}$ and $x_{r}$ values. There exists a wide region of
$\Sigma_{min}$ around the diagonal of the contour is where ${\cal A}\approx 0$
that results in $\Sigma_{min}$=0. This region is actually not allowed since
$g(\Omega)$ is strictly not defined. One shouldn’t substitute this zero value
in Eq. (25) and claim it as a violation of the TUR. In Fig.(3)d), we show the
existence of the exchange symmetry ($x_{\ell}\leftrightarrow x_{r}$) in the
entropy under equal temperature setting.

##  V Conclusion

We employ a full counting statistical method to derive a tilted driven quantum
master equation for a simple bosonic site coupled to two squeezed harmonic
reservoirs. The temperatures of the two squeezed reservoirs are assumed to be
adiabatically driven with a phase-different driving protocol. This allowed us
to explore the combined effect of squeezing parameters and the geometricities
or geometric phaselike contributions to the steadystate observables, the flux
(first cumulant) and the noise (second cumulant). The dynamic cumulants
exhibit similar qualitative behavior as a function of squeezing parameters to
that of what is already known for an undriven scenario, albeit with modified
magnitudes. The geometric cumulants are however affected by the squeezing
parameters. The odd (even) geometric cumulants are found to be antisymmetric
(symmetric) with respect to exchanging the left and right squeezing parameters
when the initial thermal gradient is maintained at zero. These also decay to
zero as we keep increasing the strength of the reservoirs’ squeezing
parameters. This is because an increased squeezing prohibits the generation of
geometricity in the cumulant generating function. Hence, under maximum
squeezing, one can recover a standard steadystate fluctuation theorem which
also leads to a standard thermodynamic uncertainty relation even in the
presence of phase different driving protocol. Using a recently proposed
modified geometric thermodynamic uncertainty principle, which is robust in the
presence of squeezing, we estimate the minimum entropy production rate at
finite values of dynamic flux. This minimum entropy production rate cannot be
separated into dynamic and geometric contributions. It exhibits a saturating
behavior and is also symmetric with respect to exchange of the left and right
squeezing parameters under a zero initial thermal gradient scenario.

## Acknowledgments

HPG acknowledge the support from the University Grants Commission, New Delhi
for the startup research grant, UGC(BSR), Grant No. F.30-585/2021(BSR) and the
Science and Engineering Research Board for the startup grant with file number
SERB/SRG/2021/001088.

##  Appendix A Appendix

The Hamiltonian of the bare site interacting with two bosonic reservoirs can
be written as,

$\displaystyle\hat{H}$
$\displaystyle=\displaystyle\hbar\omega_{o}\hat{b}^{\dagger}\hat{b}+\sum_{i=\nu,\nu\in
L,R}\hbar\omega_{i}\hat{a}_{i}^{\dagger}\hat{a}_{i}+\hat{V},$ (28)

with

$\displaystyle\hat{V}$ $\displaystyle=\sum_{i,\nu\in
L,R}k_{i}^{\nu}(\hat{a}_{i\nu}^{\dagger}\hat{b}+\hat{a}_{i\nu}\hat{b}^{\dagger})$
(29)

Here, $\hbar\omega_{o}\hat{b}^{\dagger}\hat{b}$ is the on-site Hamiltonian
with bare frequency $\omega_{o}$, while $\hat{b}^{\dagger}(\hat{b})$ is the
bosonic creation (annihilation operator) on the site. The second term is the
reservoir Hamiltonian with squeezed harmonic states and is a sum of two terms
that represent the left (L) and right (R) squeezed reservoirs. The single
particle operators $\hat{a}_{i\nu}^{\dagger}(\hat{a}_{i\nu})$ represent the
creation (annihilation) of a boson in the i-th mode from (of) the $\nu$-th
bath. $\hat{V}$ is the system bath coupling Hamiltonian with $k_{i}^{\nu}$
being the coupling constant for the i-th squeezed mode of the $\nu$th bath to
the bare site mode. The squeezed density matrix for the $\nu$-th reservoir
($\hat{H}_{\nu}$ being the $\nu$th reservoir Hamiltonian) is given by

$\displaystyle\hat{\rho}_{\nu}$
$\displaystyle=\frac{1}{Z}\exp\\{-\beta_{\nu}(t)\hat{S}_{\nu}\hat{H}_{\nu}\hat{S}^{\dagger}_{\nu}\\},$
(30) $\displaystyle\hat{S}_{\nu}$
$\displaystyle=\displaystyle\prod_{k}e^{\frac{1}{2}(x_{\nu}^{*}\hat{a}_{k\nu}^{{\dagger}2}-h.c)}.$
(31)

$\beta_{\nu}(t)=(k_{B}T_{\nu}(t))^{-1}$ being the inverse temperature and
$\hat{S}_{\nu}$ is the squeezing operator on the $k-$th mode of the $\nu-$th
bath, with $x_{\nu}$ being the squeezing the $\nu$-th reservoir’s squeezing
parameter dodonov2002nonclassical ; li2017production ; yadalam2022counting ;
sarmah2023nonequilibrium . Assuming that the initial density matrix is
factorisable and there is well separation between the system and reservoir
timescales (adiabaticity) li2017production , we can write down two adiabatic
Pauli-type master equations, with time-dependent squeezed rates,

$\displaystyle\dot{\rho}_{11}$
$\displaystyle=-(\gamma_{L}(1+N_{L}(t))+\gamma_{R}(1+N_{R}(t)))\rho_{11}$ (32)
$\displaystyle+(\gamma_{L}N_{L}(t)+\gamma_{R}N_{R}(t))\rho_{00}$
$\displaystyle\dot{\rho}_{00}$
$\displaystyle=(\gamma_{L}(1+N_{L}(t))+\gamma_{R}(1+N_{R}(t)))\rho_{11}$ (33)
$\displaystyle-(\gamma_{L}N_{L}(t)+\gamma_{R}N_{R}(t))\rho_{00}$

where $\langle m|\rho|m\rangle=\rho_{mm}$ represents the probability of
occupation of the occupied and unoccupied Fock states. Note that the
populations and coherences are decoupled and the equations are effectively
classical albeit with quantum mechanical rates bagrets2003full . The driving
dependent, squeezed occupation factors are given by li2017production ,

$\displaystyle N_{\nu}(t)$
$\displaystyle=\big{(}\cosh(2x_{i\nu})(n_{\nu}(t)+\frac{1}{2})-\frac{1}{2}\big{)}$
(34)

where $n_{\nu}(t)$ is the driven Bose function for the $\nu$-th squeezed bath.
Now we can recast the above two equations in the Liouville space and following
the standard procedure of FCS by introducing the auxiliary counting field,
$\lambda$ to keep track of the net number of bosons exchanged, $q$
esposito2009nonequilibrium ; harbola2007statistics , we can arrive at Eq.(4),
where the quantum mechanical rates have been redefined to
$\alpha_{\nu}(t)=\gamma_{L}(1+N_{L}(t))$ and
$\beta_{\nu}=\gamma_{\nu}N_{\nu}(t)$. The $\lambda$-dependent zero eigen value
of Eq.(4) is given by,

$\displaystyle\zeta_{o}(\lambda,t)$
$\displaystyle=-(\gamma_{\ell}X_{\ell}^{+}+\gamma_{r}X_{r}^{+})$ (35)
$\displaystyle+\sqrt{(\gamma_{\ell}+\gamma_{r})^{2}+(\gamma_{\ell}X_{\ell}^{-}+\gamma_{r}X_{r}^{-})f(\lambda)}$
$\displaystyle X_{\nu}^{\pm}$ $\displaystyle=\cosh(2x_{\nu})(2n_{\nu}(t)\pm
1),\nu=l,r$ (36) $\displaystyle f(\lambda)$
$\displaystyle=(\gamma_{\ell}e^{-\lambda}(1+X_{\ell}^{+})+\gamma_{r}e^{2\lambda}(X_{r}^{+}))$
(37)

from which the dynamic flux and noise can be numerically evaluated using Eq.
(9).

## References

  * (1) S. Pancharatnam, Generalized theory of interference, and its applications: Part i. coherent pencils, in: Proceedings of the Indian Academy of Sciences-Section A, Vol. 44, Springer India New Delhi, 1956, pp. 247–262.
  * (2) M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 392 (1802) (1984) 45–57.
  * (3) N. Mukunda, R. Simon, Quantum kinematic approach to the geometric phase. i. general formalism, Annals of Physics 228 (2) (1993) 205–268.
  * (4) D. Thouless, Quantization of particle transport, Physical Review B 27 (10) (1983) 6083.
  * (5) A. Carollo, I. Fuentes-Guridi, M. F. Santos, V. Vedral, Geometric phase in open systems, Physical review letters 90 (16) (2003) 160402.
  * (6) Z. Wang, L. Wang, J. Chen, C. Wang, J. Ren, Geometric heat pump: Controlling thermal transport with time-dependent modulations, Frontiers of Physics 17 (2022) 1–14.
  * (7) J. Ren, P. Hänggi, B. Li, Berry-phase-induced heat pumping and its impact on the fluctuation theorem, Phys. Rev. Lett. 104 (2010) 170601.  doi:10.1103/PhysRevLett.104.170601.
URL https://link.aps.org/doi/10.1103/PhysRevLett.104.170601

  * (8) S. K. Giri, H. P. Goswami, Controlling thermodynamics of a quantum heat engine with modulated amplitude drivings, Phys. Rev. E 106 (2022) 024131.  doi:10.1103/PhysRevE.106.024131.
URL https://link.aps.org/doi/10.1103/PhysRevE.106.024131

  * (9) Y. Hino, H. Hayakawa, Fluctuation relations for adiabatic pumping, Physical Review E 102 (1) (2020) 012115.
  * (10) K. Takahashi, Y. Hino, K. Fujii, H. Hayakawa, Full counting statistics and fluctuation–dissipation relation for periodically driven two-state systems, Journal of Statistical Physics 181 (6) (2020) 2206–2224.
  * (11) N. Sinitsyn, I. Nemenman, The berry phase and the pump flux in stochastic chemical kinetics, Europhysics Letters 77 (5) (2007) 58001.
  * (12) H. P. Goswami, B. K. Agarwalla, U. Harbola, Geometric effects in nonequilibrium electron transfer statistics in adiabatically driven quantum junctions, Physical Review B 93 (19) (2016) 195441.
  * (13) Y. Hino, H. Hayakawa, Geometrical formulation of adiabatic pumping as a heat engine, Physical Review Research 3 (1) (2021) 013187.
  * (14) S. K. Giri, H. P. Goswami, Geometric phaselike effects in a quantum heat engine, Phys. Rev. E 96 (2017) 052129.  doi:10.1103/PhysRevE.96.052129.
URL https://link.aps.org/doi/10.1103/PhysRevE.96.052129

  * (15) S. K. Giri, H. P. Goswami, Nonequilibrium fluctuations of a driven quantum heat engine via machine learning, Physical Review E 99 (2) (2019) 022104.
  * (16) J. Eglinton, K. Brandner, Geometric bounds on the power of adiabatic thermal machines, Physical Review E 105 (5) (2022) L052102.
  * (17) T. Yuge, T. Sagawa, A. Sugita, H. Hayakawa, Geometrical pumping in quantum transport: Quantum master equation approach, Physical Review B 86 (23) (2012) 235308\.
  * (18) J. Lu, Z. Wang, J. Peng, C. Wang, J.-H. Jiang, J. Ren, Geometric thermodynamic uncertainty relation in a periodically driven thermoelectric heat engine, Physical Review B 105 (11) (2022) 115428.
  * (19) D. F. Walls, Squeezed states of light, nature 306 (5939) (1983) 141–146.
  * (20) R. Schnabel, Squeezed states of light and their applications in laser interferometers, Physics Reports 684 (2017) 1–51.
  * (21) V. Dodonov, A. Klimov, V. Man’ko, Physical significance of correlated and squeezed states, in: Group Theoretical Methods in Physics, Springer, 1991, pp. 450–456.
  * (22) R. Puri, Coherent and squeezed states on physical basis, pramana 48 (3) (1997) 787–797.
  * (23) L. Dupays, A. Chenu, Shortcuts to squeezed thermal states, Quantum 5 (2021) 449\.
  * (24) T. Abebe, D. Jobir, C. Gashu, E. Mosisa, Interaction of two-level atom with squeezed vacuum reservoir, Advances in Mathematical Physics 2021 (2021).
  * (25) Z. Holmes, S. Weidt, D. Jennings, J. Anders, F. Mintert, Coherent fluctuation relations: from the abstract to the concrete, Quantum 3 (2019) 124.
  * (26) H. K. Yadalam, B. K. Agarwalla, U. Harbola, Counting statistics of energy transport across squeezed thermal reservoirs, arXiv preprint arXiv:2202.04011 (2022).
  * (27) G. Manzano, F. Galve, R. Zambrini, J. M. Parrondo, Entropy production and thermodynamic power of the squeezed thermal reservoir, Physical Review E 93 (5) (2016) 052120.
  * (28) W. Niedenzu, D. Gelbwaser-Klimovsky, A. G. Kofman, G. Kurizki, On the operation of machines powered by quantum non-thermal baths, New Journal of Physics 18 (8) (2016) 083012.
  * (29) B. K. Agarwalla, J.-H. Jiang, D. Segal, Quantum efficiency bound for continuous heat engines coupled to noncanonical reservoirs, Physical Review B 96 (10) (2017) 104304.
  * (30) J. Klaers, S. Faelt, A. Imamoglu, E. Togan, Squeezed thermal reservoirs as a resource for a nanomechanical engine beyond the carnot limit, Physical Review X 7 (3) (2017) 031044.
  * (31) D. Newman, F. Mintert, A. Nazir, Performance of a quantum heat engine at strong reservoir coupling, Physical Review E 95 (3) (2017) 032139.
  * (32) A. Ourjoumtsev, A. Kubanek, M. Koch, C. Sames, P. W. Pinkse, G. Rempe, K. Murr, Observation of squeezed light from one atom excited with two photons, Nature 474 (7353) (2011) 623–626.
  * (33) M. Mehmet, H. Vahlbruch, N. Lastzka, K. Danzmann, R. Schnabel, Observation of squeezed states with strong photon-number oscillations, Physical Review A 81 (1) (2010) 013814.
  * (34) X. Huang, T. Wang, X. Yi, et al., Effects of reservoir squeezing on quantum systems and work extraction, Physical Review E 86 (5) (2012) 051105.
  * (35) J.-T. Hsiang, B.-L. Hu, Fluctuation–dissipation relation for a quantum brownian oscillator in a parametrically squeezed thermal field, Annals of Physics 433 (2021) 168594.
  * (36) P. Talkner, M. Morillo, J. Yi, P. Hänggi, Statistics of work and fluctuation theorems for microcanonical initial states, New Journal of Physics 15 (9) (2013) 095001.
  * (37) C. Wang, H. Chen, J.-Q. Liao, Nonequilibrium thermal transport and photon squeezing in a quadratic qubit-resonator system, Physical Review A 104 (3) (2021) 033701.
  * (38) F. Giraldi, F. Petruccione, Coherence in a dissipative two-level system, The European Physical Journal D 68 (6) (2014) 1–7.
  * (39) J. P. Pekola, B. Karimi, Colloquium: Quantum heat transport in condensed matter systems, Reviews of Modern Physics 93 (4) (2021) 041001.
  * (40) M. Esposito, U. Harbola, S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Reviews of modern physics 81 (4) (2009) 1665.
  * (41) D. Segal, A. Nitzan, Molecular heat pump, Phys. Rev. E 73 (2006) 026109.  doi:10.1103/PhysRevE.73.026109.
URL https://link.aps.org/doi/10.1103/PhysRevE.73.026109

  * (42) P. Carpio-Martinez, G. Hanna, Quantum bath effects on nonequilibrium heat transport in model molecular junctions, The Journal of Chemical Physics 154 (9) (2021).
  * (43) A. Kowalewska-Kudlaszyk, R. Tanaś, Generalized master equation for a two-level atom in a strong field and tailored reservoirs, journal of modern optics 48 (2) (2001) 347–370.
  * (44) S. Saryal, H. M. Friedman, D. Segal, B. K. Agarwalla, Thermodynamic uncertainty relation in thermal transport, Physical Review E 100 (4) (2019) 042101.
  * (45) Y. Lu, N. Lambert, A. F. Kockum, K. Funo, A. Bengtsson, S. Gasparinetti, F. Nori, P. Delsing, Steady-state heat transport and work with a single artificial atom coupled to a waveguide: Emission without external driving, PRX Quantum 3 (2) (2022) 020305.
  * (46) U. Harbola, M. Esposito, S. Mukamel, Statistics and fluctuation theorem for boson and fermion transport through mesoscopic junctions, Physical Review B 76 (8) (2007) 085408.
  * (47) S.-W. Li, et al., Production rate of the system-bath mutual information, Physical Review E 96 (1) (2017) 012139.
  * (48) R. Tanas, Squeezing and squeezing-like terms in the master equation for a two-level atom in strong fields, Journal of Optics B: Quantum and Semiclassical Optics 4 (3) (2002) S142.
  * (49) M. J. Sarmah, A. Bansal, H. P. Goswami, Nonequilibrium fluctuations in boson transport through squeezed reservoirs, Physica A: Statistical Mechanics and its Applications 615 (2023) 128620.
  * (50) P. Pietzonka, F. Ritort, U. Seifert, Finite-time generalization of the thermodynamic uncertainty relation, Physical Review E 96 (1) (2017) 012101.
  * (51) T. Yuge, T. Sagawa, A. Sugita, H. Hayakawa, Geometrical excess entropy production in nonequilibrium quantum systems, Journal of Statistical Physics 153 (2013) 412–441.
  * (52) V. Dodonov, Nonclassical’states in quantum optics: asqueezed’review of the first 75 years, Journal of Optics B: Quantum and Semiclassical Optics 4 (1) (2002) R1.
  * (53) D. Bagrets, Y. V. Nazarov, Full counting statistics of charge transfer in coulomb blockade systems, Physical Review B 67 (8) (2003) 085316.