# Steady state thermodynamics of ideal gas in shear flow

Karol Makuch<EMAIL_ADDRESS>Institute of Physical Chemistry, Polish
Academy of Sciences Kasprzaka 44/52, 01-224 Warszawa  Konrad Giżyński
Institute of Physical Chemistry, Polish Academy of Sciences Kasprzaka 44/52,
01-224 Warszawa  Anna Maciołek  Institute of Physical Chemistry, Polish
Academy of Sciences Kasprzaka 44/52, 01-224 Warszawa  Max-Planck-Institut für
Intelligente Systeme Stuttgart, Heisenbergstr. 3, D-70569 Stuttgart, Germany
Paweł J. Żuk  Institute of Physical Chemistry, Polish Academy of Sciences
Kasprzaka 44/52, 01-224 Warszawa  Department of Physics, Lancaster University,
Lancaster LA1 4YB, United Kingdom  Robert Hołyst<EMAIL_ADDRESS>Institute of Physical Chemistry, Polish Academy of Sciences Kasprzaka 44/52,
01-224 Warszawa

###### Abstract

Equilibrium thermodynamics describes the energy exchange of a body with its
environment. Here, we describe the global energy exchange of an ideal gas in
the Coutte flow in a thermodynamic-like manner. We derive a fundamental
relation between internal energy as a function of parameters of state. We
analyze a non-equilibrium transition in the system and postulate the extremum
principle, which determines stable stationary states in the system. The
steady-state thermodynamic framework resembles equilibrium thermodynamics.

##  I Introduction

There is mounting evidence that the exchange of energy of a macroscopic steady
state system with its environment can be described in an equilibrium
thermodynamic-like fashion. It has been considered in the 90s of the twentieth
century for systems with chemical reactions that may constantly occur in time
(ross1988thermodynamics, ; hunt1990thermodynamic, ; ross1992thermodynamic, ).
Such a steady state situation goes beyond equilibrium thermodynamics which
excludes any macroscopic flows and currents
(Thermodynamics_and_an_Introduction_to_Thermostatistics_2ed_H_Callen, ).
Around a similar time, the thermodynamic-like approach was introduced for
systems with shear-flow (evans1980thermodynamics, ; hanley1982thermodynamics,
; evans1983computer, ; evans1986rheology, ) and is still under development
(daivis2003steady, ; taniguchi2004steady, ; daivis2008thermodynamic, ;
baranyai2018thermodynamic, ). Another class of macroscopic systems for which
the attempt to introduce thermodynamic description has been undertaken is
systems with heat flow (nakagawa2019global, ; nakagawa2022unique, ;
Zhang2021continuous, ; holyst2022thermodynamics, ; Holyst2023Steady, ).

The above approaches to formulate steady state thermodynamics focus on
chemical reactions and shear flow, both in homogeneous temperature profiles or
systems with heat flow without chemical reactions and macroscopic flows. If
steady state thermodynamics is ever formulated on general grounds, it is now
at its inception. About two decades ago, Oono and Paniconi (oono1998steady, )
and Sasa and Hatano (sasa2006steady, ) postulated a thermodynamic-like
description based on a general footing. However, because the descriptions were
postulated, they require validation and further investigation on the eventual
limitation. In particular, it is not clear whether the nonequilibrium entropy
defined by the integral of the local entropy density over the volume of the
system (nakagawa2022unique, ) or rather the excess heat-based entropy
(oono1998steady, ) should be used to construct the principles of steady-state
thermodynamics. These two entropies are not equivalent
(holyst2022thermodynamics, ). It shows that the steady state thermodynamics is
far from complete, and the core fundamental questions still remain open
(Holyst2023Steady, ; nakagawa2022unique, ). It is worth mentioning stochastic
thermodynamics, which focuses on thermodynamic notions for the system on the
level of individual trajectories (seifert2012stochastic, ; ding2022unified, ).
Here we focus on a reduced, macroscopic description of a steady state system
(speck2021modeling, ).

Allowing various macroscopic constant fluxes drives the system from
equilibrium to a steady state and opens up phenomena that eludes equilibrium
thermodynamics. Take for example a quiescent liquid in a uniform gravitational
field in equilibrium. This situation assumes a uniform temperature and no
macroscopic flows, which, together with the equations of state, determine the
thermodynamic parameters at each point of the system. Allowing a vertical or
horizontal heat flow radically changes the situation. The flow of heat
combined with the gravitational force can cause regular mass movement, either
because the denser liquid is at the top and the thinner is at the bottom or
because gravity unevenly acts on areas with different horizontal densities.
This unstable situation causes a mass movement in the atmosphere, oceans,
planets, and stars (bodenschatz2000recent, ). The core feature of this
phenomenon is the coupling between the heat flow and the mass flow. However,
the question arises whether some reduced description for non-equilibrium
steady states also emerges from hydrodynamics. Can it take the shape of
equilibrium thermodynamics, in which the system’s behavior is described by
only a few parameters and the rules of extremum containing only these
parameters?

Here we present a thermodynamic-like description of a system with heat and
mass flow coupling. We consider ideal gas in shear flow with a dissipative
temperature profile shown schematically in Fig. 1. We introduce the first law
for this system that describes different ways of exchanging the system’s
energy with its environment. We also consider a movable wall as a
thermodynamic constraint in the system. We introduce the extremum principle
that determines the position of the wall. We show that there is a critical
shear in the system above which the equilibrium position of the internal wall
becomes unstable, and the system undergoes a nonequilibrium second order phase
transition. We give a complete thermodynamic-like description of this steady
state system in which the position of the movable wall is a thermodynamic
constraint. For the vanishing shear flow, the steady state extremum principle
directly reduces to the minimum principle in the corresponding problem in
thermodynamics.

##  II System

We consider ideal monoatomic gas between two two parallel walls at walls’
temperature $T_{0}$, as shown in Fig. 1. The upper wall moves with speed
$V_{w}$ moving in $x$-direction. The system is described by irreversible
thermodynamic equations: the continuity equation, Navier-Stokes equation,
energy balance and two equations of state (Groot_Mazur_Non-
equilibrium_thermodynamics, ). We assume that the system is translationally
invariant in $x$ and $y$ directions, therefore the fields depend only on $z$
coordinate. In particular, equations of state are given by,

$p\left(z\right)=n\left(z\right)k_{B}T\left(z\right),$ (1)
$u\left(z\right)=\frac{3}{2}n\left(z\right)k_{B}T\left(z\right),$ (2)

with pressure $p\left(z\right),$ volumetric particle number density
$n\left(z\right)$, temperature $T\left(z\right)$, Boltzmann constant $k_{B}$,
and the volumetric energy density, $u\left(z\right)$. The velocity field also
depends on $z$ coordinate only and is oriented in the direction determined by
the moving wall,

$\mathbf{v}\left(x,y,z\right)={\bf e}_{x}v\left(z\right).$ (3)

Figure 1: Ideal gas in shear flow. The gas is confined by the immobile bottom
wall and by the upper wall moving with velocity $V_{w}$. There is also a
thermally insulating and moving internal wall that divides the gas into two
subsystems. Shearing motion causes energy dissipation in the system, which
changes the temperature profile. The heat flows out from the system through
the confining walls at temperature $T$. Colors schematically show the
temperature profile (red - hotter, blue - colder).

##  III Steady state

We describe the system within linear response theory (Groot_Mazur_Non-
equilibrium_thermodynamics, ). Therefore, the system is locally described by
equation of states (1) and (2) supplemented with mass, momentum and energy
conservation. We neglect bulk viscosity. We assume that the shear viscosity
coefficient $\eta$ does not depend on the local state of the gas, so the
viscosity does not depend on local density and temperature. The translational
symmetry for the pressure, $p\left({\bf r}\right)=p\left(z\right)$, and for
the velocity given by Eq. (3) reduces the $z$-component of the Navier-Stokes
equation,

$\rho\left(\partial_{t}{\bf v}+\left({\bf v}\cdot\nabla\right){\bf
v}\right)=-\nabla p+\eta\Delta{\bf
v}+\frac{1}{3}\eta\nabla\left(\nabla\cdot{\bf v}\right),$ (4)

to the formula, $dp\left(z\right)/dz=0$, while the $x$-component of the
Navier-Stokes equation gives $0=\eta d^{2}v\left(z\right)/dz^{2}$. Therefore
the pressure in the system is homogeneous,

$p\left(z\right)=p,$ (5)

and, using the boundary conditions $v\left(0\right)=0$ and
$v\left(L\right)=V_{w}$, there is a shear flow in the system,

$\displaystyle v\left(z\right)$ $\displaystyle=\frac{z}{L}V_{w}.$ (6)

The continuity equation,

$\partial_{t}\rho+{\bf v}\cdot\nabla\rho=-\rho\nabla\cdot{\bf v},$ (7)

with mass density $\rho\left(z\right)$ and the above velocity field is
satisfied automatically for any density profile. The energy balance equation
is given by,

$\displaystyle\rho\left(\partial_{t}u_{m}+{\bf v}\cdot\nabla u_{m}\right)$
$\displaystyle=\kappa\Delta T-p\nabla\cdot{\bf v}$
$\displaystyle+2\eta\mathring{\left[\text{$\nabla$}{\bf
v}\right]}^{s}:\mathring{\left[\text{$\nabla$}{\bf v}\right]}^{s},$ (8)

where $\mathring{\left[\text{$\nabla$}{\bf
v}\right]}_{ij}^{s}=\left(\nabla_{i}\mathbf{v}_{j}+\nabla_{j}\mathbf{v}_{i}-\delta_{ij}\frac{2}{3}\nabla\cdot{\bf
v}\right)/2$ denotes the symmetrized and traceless velocity gradient matrix,
the colon is the double contraction, $\delta_{ij}$ denotes the Kronecker
delta, and $u_{m}=u/\rho$ is the mass density of the internal energy.
Utilizing the shear velocity profile (6) and the translational invariance of
the internal energy density, $u_{m}\left(z\right)$, the energy balance
equation reduces to the equation for the temperature profile

$0=\kappa\frac{d^{2}}{dz^{2}}T\left(z\right)+\eta\left(\frac{V_{w}}{L}\right)^{2}.$
(9)

The above equation has the following form,
$0=\kappa\frac{d^{2}}{dz^{2}}T\left(z\right)+\lambda$, where $\lambda$ is the
volumetric heating rate,

$\lambda=\eta\left(\frac{V_{w}}{L}\right)^{2}.$ (10)

We also introduce the dimensionless parameter

$D\equiv\frac{\eta V_{w}^{2}}{\kappa T_{0}}.$

##  IV Shear flow without the internal wall

We first consider the shear flowing system shown in Fig. 1, but without the
internal wall. In this situation, the boundary conditions are that on the
upper and lower surface and the temperature profile is given by,

$T\left(z\right)=T_{0}\left[-\frac{1}{2}\frac{\lambda L^{2}}{\kappa
T_{0}}\left(\frac{z}{L}\right)^{2}+\frac{1}{2}\frac{\lambda L^{2}}{\kappa
T_{0}}\frac{z}{L}+1\right].$ (11)

Number of particles in the surface area $A$ of the translationally invariant
system is determined by the particle density,
$N=A\int_{0}^{L}dz\,n\left(z\right)$, which with the mechanical equation of
state (1) and the above temperature profile gives

$p=\frac{Nk_{B}T_{0}}{AL}\psi\left(\frac{\eta V_{w}^{2}}{\kappa
T_{0}}\right),$ (12)

where

$\psi(D)\equiv\frac{D}{8\sqrt{\frac{D}{8+D}}\tanh^{-1}\sqrt{\frac{D}{8+D}}}.$
(13)

Combination of equations of state (1) and (2) gives the following internal
energy density $u\left(z\right)=3p\left(z\right)/2.$ Total internal energy in
the system is given by

$U=A\int_{0}^{L}dz\,u\left(z\right)=\frac{3}{2}ALp,$ (14)

where we used the fact, that the pressure in the system is constant. Utilizing
Eq. (12) in the above equation we get,

$U=\frac{3}{2}Nk_{B}T_{0}\psi\left(\frac{\eta V_{w}^{2}}{\kappa
T_{0}}\right).$ (15)

We determine the kinetic energy due to macroscopic motion,
$E_{k}=A\int_{0}^{L}dz\,mn\left(z\right)v^{2}\left(z\right)/2,$ where $m$ is a
mass of a single ideal gas molecule. We obtain

$E_{k}=mN\frac{V_{w}^{2}}{2}\varepsilon\left(\frac{\eta V_{w}^{2}}{\kappa
T_{0}}\right),$ (16)

where

$\varepsilon\left(D\right)=\frac{2}{D}\left(\frac{4+D}{4}\frac{1}{\psi\left(D\right)}-1\right).$
(17)

Utilizing Eqs. (15), (16) and (17) we obtain the relation between the kinetic
and internal energy,

$E_{k}\frac{\eta}{mN\kappa T_{0}}=\frac{4+\frac{\eta V_{w}^{2}}{\kappa
T_{0}}}{4}\frac{\frac{3}{2}Nk_{B}T_{0}}{U}-1.$ (18)

##  V Transition between steady states and energy balance

We consider transitions between steady states in the system. The system is in
one steady state at time $t_{i}$ , after which we slightly change one or more
control parameters that appear in (or are related to) the governing equations.
For example, we slowly modify the velocity of the wall, the position of the
upper wall (volume $V$) or the external temperature $T_{0}$. In particular, we
increase the velocity of the wall by
$V_{w}\left(t\right)=V_{w}+\left(t-t_{i}\right)dV_{w}/\left(t_{f}-t_{i}\right)$
for $t_{i}<t<t_{f}$, which modifies velocity of the wall from $V_{w}$ to
$V_{w}+dV_{w}$. This induces a time dependent hydrodynamic field. After time
$t_{f}$ the system reaches another steady state close to the previous one due
to a small change in the control parameters. We focus on steady,
nonequilibrium states with a constant number of particles in the system.

Previously we have analyzed the energy balance for an ideal gas in a heat flow
(holyst2022thermodynamics, ) during the transition between steady states. The
considerations necessary for Couette flow go along the same line. We do not
repeat them but give a sketch. We monitor the change of the energy between
times $t_{i}$ and $t_{f}$,

$dU=U\left(t_{f}\right)-U\left(t_{i}\right)=\int_{t_{i}}^{t_{f}}dt\,\frac{dU\left(t\right)}{dt},$

where

$U\left(t\right)=\int_{V(t)}d^{3}r\,\rho\left({\bf r},t\right)u\left({\bf
r},t\right),$ (19)

and $\rho\left({\bf r},t\right)$ is the density and $u\left({\bf r},t\right)$
is the internal energy density per unit mass at position ${\bf r}$ and time
$t$. We then represent the time derivative of the above integral utilizing the
internal energy balance equation (8). Applying analysis from
(holyst2022thermodynamics, ) leads to the conclusion that the system exchanges
the energy with the environment according to the following formula,

$dU=\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudQ+\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudW+\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudD,$ (20)

with the excess heat defined by

$\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudQ=-\int_{t_{i}}^{t_{f}}dt\int_{\partial
V\left(t\right)}d^{2}r\,\hat{n}\cdot\left(J_{q}\left(t\right)-J_{q,st}\left(t\right)\right),$
(21)

where $\hat{n}$ is the vector normal to the surface of volume $V$ and
$J_{q}\left(t\right)$ is the heat flux in the system, with the work related to
the expansion of the gas defined by

$\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudW=-\int_{t_{i}}^{t_{f}}dt\int_{V\left(t\right)}d^{3}r\,p\nabla\cdot{\bf
v},$ (22)

and the excess dissipation defined by,

$\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudD=\int_{t_{i}}^{t_{f}}dt\int_{V\left(t\right)}d^{3}r\,\left(\Pi:\left[\nabla{\bf
v}\right]-\Pi_{st}\left(t\right):\left[\nabla{\bf
v}_{st}\left(t\right)\right]\right),$ (23)

where $\left[\nabla{\bf v}\right]_{ij}=\nabla_{i}{\bf v}_{j}$ is the gradient
velocity matrix. The subscript $st$ in the above equations means that the
quantity should be taken at a stationary state that corresponds to the control
parameters at time $t$. The first term in the internal energy balance (20),
$\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudQ$, is the heat on top of the
steady state heat exchanged during the transition between steady states
(oono1998steady, ). The second term in (20) is the volumetric work performed
by that wall (more specifically, by the perpendicular component of the force
of the wall) when the system is compressed. The third term $\mkern
3.0mu\mathchar 22\relax\mkern-12.0mudD$ is the energy dissipation in the
system on top of the steady state dissipation.

Because the pressure in a steady state is spatially uniform, the work from Eq.
(22) for slow transitions between steady states reduces as follows, $\mkern
3.0mu\mathchar
22\relax\mkern-12.0mudW\approx-p\int_{t_{i}}^{t_{f}}dt\int_{V\left(t\right)}d^{3}r\,\nabla\cdot{\bf
v}$. The integrals of the divergence with the Gauss theorem reduce to the
change of the volume, and the work reduces to the volumetric work that also
appears in equilibrium thermodynamics,

$\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudW=-pdV.$

To determine the exchange of the kinetic energy, we proceed along the same
line as for the internal energy above. Thus we use the kinetic energy balance
equation (Groot_Mazur_Non-equilibrium_thermodynamics, ),

$\partial_{t}\frac{1}{2}\rho{\bf v}^{2}=-\nabla\cdot\left(\frac{1}{2}\rho{\bf
v}^{2}{\bf v}+P\cdot{\bf v}\right)+P:\left[\nabla{\bf v}\right],$ (24)

where the pressure tensor $P$ in the system is given by

$\displaystyle P_{ij}$ $\displaystyle=p\delta_{ij}+\Pi_{ij},$
$\displaystyle\Pi$ $\displaystyle=-2\eta\mathring{\left[\text{$\nabla$}{\bf
v}\right]}^{s}.$

Analysis of the change of the kinetic energy
$dE_{k}=E_{k}\left(t_{f}\right)-E_{k}\left(t_{i}\right)=\int_{t_{i}}^{t_{f}}dt\,\frac{dE_{k}\left(t\right)}{dt}$
leads to the conclusion that during the transition between steady states, the
kinetic energy changes according to the following equation,

$dE_{k}=\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudW_{w}-\mkern
3.0mu\mathchar 22\relax\mkern-12.0mudD,$ (25)

where

$\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudW_{w}=\int_{t_{i}}^{t_{f}}dt\int_{\partial
V\left(t\right)}d^{2}r\,\hat{n}\cdot\left(\Pi\cdot{\bf
v}-\Pi_{st}\left(t\right)\cdot{\bf v}_{st}\left(t\right)\right)$

According to the above equations, the kinetic energy may change in two ways.
$\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudW_{w}$ is the work performed by
the upper wall on top of the steady work performed to keep the shearing steady
state. It may be called the excess shear work of the wall. This effect has
been recently discussed by Baranyai (baranyai2018thermodynamic, ).

The differential of the total energy, $E=U+E_{k}$ follows from Eqs. (20) and
(25) and is given by,

$dE=\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudQ-pdV+\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudW_{w}.$ (26)

The total energy changes as a result of the excess heat $\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudQ$, volumetric work $-pdV$, and the excess shear work
$\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudW_{w}$. Contrary to the energy
balance equations (20) and (25), there is no excess dissipation $\mkern
3.0mu\mathchar 22\relax\mkern-12.0mudD$ defined by Eq. (23) in the above total
energy balance equation. It follows that the excess dissipation $\mkern
3.0mu\mathchar 22\relax\mkern-12.0mudD$ describes an internal transfer of
internal and kinetic energy. During the steady state transition, part of the
kinetic energy changes to internal energy through the kinetic energy
dissipation that occurs on top of the steady state dissipation.

##  VI Nonequilibrium entropy

By introducing

$\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudP\equiv\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudQ+\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudD$ (27)

we obtain the internal energy balance (20) in the following form,

$dU=\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudP-pdV.$ (28)

The above formula holds in the space of the control parameters, which are
temperature $T$, the volume of the system $V$, and the speed of the moving
wall $V_{w}$. We do not change the number of particles $N$. The above formula
has the same form as the internal energy of ideal gas without the macroscopic
kinetic energy but with volumetric heating (holyst2022thermodynamics, ;
Zhang2021continuous, ). Because the pressure in a shear flow system is
constant, Eq. (5), and due to the ideal gas mechanical equation of state, Eq.
(14), we obtain,

$p=\frac{2}{3}\frac{U}{V}.$ (29)

We notice the following two facts. First, the internal energy balance in the
transition between steady states given by Eq. (28) is very similar to the
equilibrium thermodynamics expression $dU_{eq}=\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudQ_{eq}-pdV$. We see that the differential $\mkern
3.0mu\mathchar 22\relax\mkern-12.0mudP$ plays the role of the heat
differential in equilibrium thermodynamics. Second, the expression for
pressure (29) is also the equilibrium ideal gas pressure mechanical equation,
$p=2U_{eq}/3V.$ The above two similarities between steady state and the
equilibrium thermodynamic differentials suggest that there exists the
integrating factor for $\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudP$
differential,

$\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudP=T^{*}dS^{*},$ (30)

where nonequilibrium (effective) entropy is given by the following fundamental
equation for the internal energy,

$S^{*}\left(U,V\right)=Nk_{B}\log\left[\frac{V}{N}\left(\frac{U}{N}\right)^{3/2}\right]+Ns_{0},$
(31)

while the nonequilibrium temperature

$T^{*}=\frac{\partial U\left(S^{*},V\right)}{\partial S^{*}}.$ (32)

The pressure is determined by $p=-\partial U\left(S^{*},V\right)/\partial V$.
Whether the differential $\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudP$ has
integrating factor (is exact) can also be shown directly by proving that the
differential $\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudP=dU/T^{*}\left(U,V,V_{w}\right)+p\left(U,V,V_{w}\right)/T^{*}\left(U,V,V_{w}\right)dV+0\cdot
dV_{w}$ has equal derivatives, $\partial\left(1/T^{*}\right)/\partial
V=\partial\left(p/T^{*}\right)/\partial U$ and that the differential does not
explicitly depend on $V_{w}$ (Spivak_M_Calculus_on_manifolds, ).

##  VII System with internal wall

Now we focus on the system with the internal wall shown in Fig. 1. The wall
splits the system into two parts, upper 1 and lower 2. In a steady state, the
velocity field in the system is the shear flow given by Eq. (6). Therefore the
velocity of the internal wall is $V_{w}^{s}=V_{w}z_{w}/L$, where $z_{w}$ is
the position of this wall. The wall is adiabatically insulating, which changes
one of the boundary conditions. There is no heat flow through the wall.
Without the wall the temperature profile would be given by the symmetric
parabola, Eq. (11), but the insertion of the adiabatic wall in the middle of
the system leads to the situation with half of the parabolic profile. Using
this observation, we conclude that from the perspective of the heat flow
equation, we can use the solution without the internal wall to determine the
energy of subsystem $2$ as follows,
$U_{2}\left(T,A,L_{2},V_{w,2}^{u}\right)=U\left(T,A,2L_{2},2V_{w,2}^{u}\right)/2$,
where $V_{w,2}^{u}$ is the speed of the upper wall in subsystem 2 and $L_{2}$
is the vertical length of the subsystem. Because the change to the moving
reference frame does not change the internal energy, we can deduce the
internal energy of the upper system in a similar manner. The internal energy
in both subsystems is given by,

$\displaystyle U_{i}\left(T,A,L_{i},V_{w,i}^{u}-V_{w,i}^{l}\right)$
$\displaystyle=\frac{1}{2}U\left(T,A,2L_{i},2\left(V_{w,i}^{u}-V_{w,i}^{l}\right)\right)\text{
for }i=1,2,$

where $V_{w,i}^{u}$ is the speed of the upper wall in subsystem $i$, and
$V_{w,i}^{l}$ is the speed of the lower wall in subsystem $i$. We have
$V_{w,2}^{l}=0$ and $V_{w,2}^{u}=V_{w,1}^{l}$.

Using the properties of the equations of state for ideal gas we can show that
the differentials

$\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudP_{i}\equiv
dU_{i}^{int}+p_{i}dV_{i}\text{ for }i=1,2,$ (33)

have integrating factors in agreement with Eq. (30),

$\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudP_{i}=T_{i}^{*}dS_{i}^{*}\text{
for }i=1,2.$ (34)

On the other hand, the analysis of the energy balance similar to the one
presented in the previous section leads to the conclusion that $\mkern
3.0mu\mathchar 22\relax\mkern-12.0mudP_{1/2}=\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudQ_{1/2}+\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudD_{1/2}$ is the sum of excess heat and excess
dissipation in each subsystem. The differentials $\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudQ_{1/2}$ and $\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudD_{1/2}$ are determined by a direct application of
formulas (21) and (23) with the replacement of the volume of integration with
the volume for a particular subsystem.

Until now, we have discussed the energy balance independently for each
subsystem. But it is particularly interesting to consider a perpendicular
motion of the internal wall. As follows from the dynamics, the normal
component of the pressure tensor on the wall in a steady state comes solely
from hydrostatic pressure. The wall may move vertically only due to the
pressure difference, $p_{1}\neq p_{2}.$ It’s natural position is where these
pressures are balanced. Therefore the knowledge of the effective entropy
$S^{*}$ and volume for each subsystem is sufficient to determine the force
acting on the wall. The wall’s horizontal velocity enters the problem
indirectly. The shear velocity plays the role of energy dissipation. The speed
of the wall appears in the energy balance equation (9) in the source term. It
thus plays the role of volumetric heating,
$\lambda\equiv\eta\left(V_{w}/L\right)^{2}$.

The ideal gas with volumetric heating and an internal wall has already been
considered previously (Zhang2021continuous, ). Such a system exhibits
continuous nonequilibrium phase transition. Similarly, the system with
macroscopic kinetic energy shown in Fig. 1 will exhibit the phase transition
as well for the following reason. Let’s focus on the position of the internal
wall $z_{w}$ and upper wall velocity, $V_{w,2}^{u}$, keeping the external
walls temperature $T$, the total volume of the system, $L_{1}+L_{2}=L$, and
other parameters constant. Number of particles in both subsystems is equal,
$N_{1}=N_{2}$. Because the hydrostatic pressure gives the sole force normal to
the walls, the motion of the internal wall can be determined by the
fundamental relations (31) for each subsystem. The pressure follows from
$p_{i}=p_{i}\left(U_{i},V_{i},N_{i}\right)$. Moreover, the internal energy in
steady state is determined by Eq. (15). The equivalence is explicit after
comparing Eq. (15) to the expression (5) from the reference
(Zhang2021continuous, ). This reasoning leads to the conclusion that from the
perspective of the position of the internal wall, the steady state behavior of
both subsystems is the same. The speed of the upper wall for the system with
kinetic energy must be related to the volumetric heating $\lambda$ from
(Zhang2021continuous, ) by $\lambda\equiv\eta\left(V_{w}/L\right)^{2}$.

Because there is one-to-one correspondence, we conclude that once the upper
wall’s speed increases, the shear flow volumetrically heats the system. For
the value, $\lambda L^{2}/4\kappa T\approx 4.55344$, equivalently $\eta
V_{w}^{2}/4\kappa T_{0}\approx 4.55344$, the central position of the wall
ceases to be stable. Above the critical speed, the wall spontaneously chooses
one of the two stable positions and moves away from the center.

As we show below, it is possible to introduce an extremum principle that
determines the spontaneous position of the internal wall. Let’s consider an
additional external force $F_{w}$ perpendicular to the wall. In a steady state
it is determined by the pressure difference,
$F_{ext}=-A\left(p_{2}-p_{1}\right)$. The force performs work on the system,
$\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudW_{ext}=-A\left(p_{2}-p_{1}\right)\,dz_{w}.$ Because the
spontaneous direction of the wall is toward the equilibration of pressures,
the system reaches “equilibrium” for the position of the wall $x$, for which
the work done by the external force is negative,

$\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudW_{ext}<0.$ (35)

The above phenomenological fact we use to construct the thermodynamic-like
extremum principle. When the upper wall does not move vertically, the work of
the external force is related to the volumetric work in both subsystems,

$\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudW_{ext}=-p_{1}dV_{1}-p_{2}dV_{2}.$ (36)

This equation, by utilizing the internal energy balance given by Eqs. (33) and
the nonequilibrium entropy, is expressed as follows,

$\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudW_{ext}=dU_{1}-T_{1}^{*}dS_{1}^{*}+dU_{2}-T_{2}^{*}dS_{2}^{*}.$
(37)

Equivalently, it is given by

$\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudW_{ext}=dU_{12}-\mkern
3.0mu\mathchar 22\relax\mkern-12.0mudP_{12},$ (38)

where

$dU_{12}=dU_{1}+dU_{2}$ (39)

and

$\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudP_{12}=\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudP_{1}+\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudP_{2}.$ (40)

Eq. (38) puts us in the same position as in equilibrium thermodynamics: if
$\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudP_{12}$ has an integrating factor
and the corresponding potential, $\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudP_{12}=T_{12}^{*}dS_{12}^{*}$, then the condition of
minimal work (35) by means of expression (38) follows the condition of the
minimum of the energy for ‘adiabatically insulated’ system defined by
$S_{12}^{*}=const$ surface. This surface is considered in the space of
$T_{0},V_{1},V_{2},V_{w}$, because we keep the number of particles ,
$N_{1},N_{2}$, and other parameters constant. Let’s notice that expressions
(35-40) for the case of no shear flow (no heating), $V_{w}=0$, reduces to the
problem of finding the position of the wall that separates two homogeneous
ideal gases. Thus, in equilibrium thermodynamics, $\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudP$, becomes the role the heat differential. The
nonequilibrium situation has been discussed earlier (holyst2022thermodynamics,
) and it suggests the minimum principle as follows.

The position of the wall is determined by the minimum of total internal
energy, $U_{12},$ as a function of parameters of states
$S_{1}^{*},V_{1},S_{2}^{*},V_{2},$ for the constraint

$S_{12}^{*}=S_{1}^{*}+rS_{2}^{*}.$ (41)

The latter condition denotes steady ‘adiabatic insulation’. In this principle
there appears parameter $r$. Let’s apply the above principle to verify whether
it gives a proper position of the wall and to find the meaning of $r$
parameter. We apply the above principle with the constraint of total volume of
the system fixed,

$V=V_{1}+V_{2}.$ (42)

The total energy is given by

$U\left(S_{1}^{*},V_{1},S_{2}^{*},V_{2}\right)=U\left(S_{1}^{*},V_{1}\right)+U\left(S_{2}^{*},V_{2}\right),$
(43)

where $U\left(S_{1}^{*},V_{1}\right)$ follows from the fundamental relation
(31). Minimization of the above energy with constraints (41) and (42) allows
us to use $S_{1}^{*}$ and $V_{1}$ as two independent parameters and minimize
the function
$U\left(S_{1}^{*},V_{1}\right)+U\left(\left(S_{12}^{*}-S_{1}^{*}\right)/r,V-V_{1}\right)$.
Its first derivatives lead to the following necessary conditions for the
minimum of the function,

$\displaystyle T_{1}^{*}-\frac{1}{r}T_{2}^{*}$ $\displaystyle=0,$ (44)
$\displaystyle p_{1}-p_{2}$ $\displaystyle=0.$

It proves that the necessary condition for the minimum in the presented
extremum principle leads to equality of pressures. The first of the above
equations gives the interpretation of $r$ parameter. It is the nonequilibrium
temperature ratio. Notice that when $r=1$, the entropy becomes additive (Eq.
(41)) and the minimum principle reduces directly to the equilibrium minimum
principle.

In the above reasoning the minimum principle follows from the fact, that the
internal energy balance is determined by $dU_{i}=\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudP_{i}-p_{i}dV_{i}$ with homogeneous pressure that is
solely determined by the internal energy and volume,
$p_{i}=p_{i}\left(U_{i},V_{i}\right)$. Both assumptions also hold when
transport coefficients depend on parameters of states (Holyst2023Steady, ). In
particular, if the viscosity or heat conductivity depends on density or
temperature. For this reason, the derived minimum energy principle is also
valid beyond the regime of Onsager’s linear response theory (Groot_Mazur_Non-
equilibrium_thermodynamics, ).

It is also worth noting that the inequality (35) which determines the
direction of spontaneous motion of the wall, with the use of Eq. (36) and Eq.
(26) applied to each subsystem is expressed by

$\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudW_{ext}=dE_{12}-\mkern
3.0mu\mathchar 22\relax\mkern-12.0mudQ_{12}-\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudW_{w12},$

with energy differential, $dE_{12}=dE_{1}+dE_{2}$, excess heat $\mkern
3.0mu\mathchar 22\relax\mkern-12.0mudQ_{12}=\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudQ_{1}+\mkern 3.0mu\mathchar 22\relax\mkern-12.0mudQ_{2}$
and excess work performed by wall, $\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudW_{w12}=\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudW_{w1}+\mkern 3.0mu\mathchar
22\relax\mkern-12.0mudW_{w2}$. The above formula suggests an approach to
identify the quantity that is minimized in the system by subtraction from
total energy all other terms (apart from work related to the change of the
constraint) that are present in the energy balance equation (here, excess work
performed by the wall and heat) which describe the ways the system exchanges
energy with the environment. Such a way could lead to a mnemotechnic rule to
identify the physical quantity to generalize the second law in other
situations than considered in this paper (e.g., with the exchange of
particles).

##  VIII Summary

In this article, we investigate whether there is a description similar to
equilibrium thermodynamics for out-of-equilibrium steady states. We consider
this problem in the context of the Couette flow, where there is a mass current
(velocity field), energy dissipation, and a continuous flow of heat. Each
feature independently excludes the theory of equilibrium thermodynamics and
its tools.

Since, in general, it is still unclear that such a simple, equilibrium
thermodynamic-like description of nonequilibrium steady states is possible, we
have chosen the most straightforward possible system, which we believe
includes the coupling of mass flow and heat current, i.e., an ideal gas in
shear flow. We show that nonequilibrium entropy exists, which describes how
the system gains energy through excess heat or dissipation. In addition, the
nonequilibrium entropy allows us to construct the principle of minimum energy,
which leads to the proper state of the system after removing the constraint,
which is the force acting on the wall in the system. It proves the existence
of a description of a system with an out-of-equilibrium heat and mass flow,
which practically takes the form of equilibrium thermodynamics and reduces to
the principle of minimum energy in equilibrium thermodynamics when the shear
flow disappears.

The thermodynamic-like description introduced in this paper leads to further
questions inspired by equilibrium thermodynamics. Particularly interesting are
the procedures for measuring non-equilibrium state parameters, effective
temperature and entropy, and measuring excess heat, dissipation, and work the
wall performs. It is worth noting that the obtained result goes along the line
of recent experiments of Yamamoto et al., who extended calorimetry for the
measurements of the excess heat of sheared fluids (yamamoto2023calorimetry, ).
Another issue is the possibility of developing the description of interacting
systems, where the fundamental relationship for van der Waals gas with heat
flow can be introduced. It is also interesting to ask about Maxwell’s
identities, which in equilibrium thermodynamics appear very practical.
Finally, it is fascinating to check whether such a description exists for
other systems with coupled heat flow and mass current in the atmosphere and
the chemical industry.

## Acknowledgement

P.J.Z. would like to acknowledge the support of a project that has received
funding from the European Union’s Horizon 2020 research and innovation program
under the Marie Skłodowska-Curie grant agreement No. 847413 and was a part of
an international co-financed project founded from the program of the Minister
of Science and Higher Education entitled "PMW" in the years 2020–2024;
agreement No. 5005/H2020-MSCA-COFUND/2019/2.

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# Nondegenerate two-photon absorption in ZnSe: Experiment and theory

L. Krauss-Kodytek  Experimentelle Physik 2, Technische Universität Dortmund,
44227 Dortmund, Germany  W.-R. Hannes  T. Meier  Department of Physics,
Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany  C. Ruppert
M. Betz  Experimentelle Physik 2, Technische Universität Dortmund, 44227
Dortmund, Germany

###### Abstract

We experimentally and theoretically investigate the nondegenerate two-photon
absorption coefficient $\beta(\omega_{1},\omega_{2})$ in the prototypical
semiconductor ZnSe. In particular, we provide a comprehensive data set on the
dependence of $\beta(\omega_{1},\omega_{2})$ on the nondegeneracy parameter
$\omega_{1}/\omega_{2}$ with the total frequency sum $\omega_{1}+\omega_{2}$
kept constant. We find a substantial increase of the two-photon absorption
strength with increasing $\omega_{1}/\omega_{2}$. In addition, different
crystallographic orientations and polarization configurations are
investigated. The nonlinear optical response is analyzed theoretically by
evaluating the multiband semiconductor Bloch equations including inter- and
intraband excitations in the length gauge. The band structure and the matrix
elements are taken as an eight-band ${\bm{k}\cdot}{\bm{p}}$ model. The
simulation results are in very good agreement with the experiment.

two-photon absorption, nonlinear optics, ultrafast spectroscopy of
semiconductors, semiconductor Bloch equations

##  I Introduction

Semiconductors are characterized by large third-order optical nonlinearities,
making them excellent materials for nonlinear photonic applications. These
nonlinearities exist in both refractive and absorptive fashions and have been
experimentally studied and theoretically modelled in many materials. Among
those phenomena, two-photon absorption (2PA) and its wavelength dependence are
particularly well investigated [1, 2]. Many previous studies are related to
degenerate configurations where, e.g., the strength of the 2PA is studied in a
$z$-scan experiment with one driving laser field. Less attention has been paid
to the case of nondegenerate configurations even though they can occur in a
variety of configurations. Nondegenerate 2PA is triggered by two laser fields
at different frequencies. Some experiments have studied driving fields close
to degeneracy [3], others $\omega$/2$\omega$ configurations [4]. Also
situations with widely nondegenerate fields have attracted attention [5, 6].
Here a massive enhancement of the two-photon absorption strength with
increasing nondegeneracy parameter $\omega_{1}/\omega_{2}$ is observed.

Theoretical studies [7, 8, 9, 10] have suggested characteristic scaling laws
for the spectral and bandgap dependence of the two-photon absorption
coefficient $\beta$. While most of these dependencies are well established,
the nondegenerate enhancement, which is best characterized by the dependence
on the ratio $\omega_{1}/\omega_{2}$ for a fixed sum frequency
$\omega_{1}+\omega_{2}$, has been theoretically investigated mainly by rather
simple two-band models so far [11, 12, 13]. A validation by a detailed
comparison to experiment, meaning data sets for $\beta(\omega_{1},\omega_{2})$
with $\omega_{1}+\omega_{2}=\text{const.}$, would be highly desirable.

With this paper we fill this gap and provide several results for the
nondegenerate two-photon absorption coefficient $\beta(\omega_{1},\omega_{2})$
for the prototypical semiconductor zinc selenide (ZnSe). We investigate
configurations where the polarization of the two driving fields is either
aligned parallel or perpendicular and additionally distinguish between
different crystallographic orientations. We compare these results to numerical
simulations of the two-photon absorption strength for this material, using the
8-band Kane model in combination with the semiconductor Bloch equations (SBE).
Very good agreement with the experiment is achieved, without the use of any
fitting parameter, for the magnitude of $\beta(\omega_{1},\omega_{2})$ and its
dependence on $\omega_{1}/\omega_{2}$, the crystallographic direction, and the
polarization direction.

##  II Experimental details

The experimental setup is based on a Ti:sapphire regenerative amplifier laser
system (Coherent RegA 9000). It provides optical pulses with a central photon
energy of 1.55 eV, a pulse energy of $8\,\upmu$J and a pulse duration of
$\approx 60\,$fs at a repetition rate of 250 kHz. Using a beam splitter a
major fraction of these pulses is fed into an optical parametric amplifier
(Coherent OPA 9400/9450) pumped by the second harmonic of these pulses. It
emits signal and idler pulses with a fixed sum energy of
$\hbar$$\omega_{\mathrm{sig}}+\hbar$$\omega_{\mathrm{idl}}=3.1\,$eV. The
corresponding individual tuning ranges are
$\lambda_{\mathrm{sig}}=(510-800)\,$nm and
$\lambda_{\mathrm{idl}}=(1855-800)\,$nm. Note that the configuration of
$\lambda_{\mathrm{sig}}=\lambda_{\mathrm{idl}}=800\,$nm is directly provided
by the RegA system, but is included in the above tuning ranges. Thus, the OPA
allows to systematically investigate frequency configurations
$\omega_{\mathrm{sig}}/\omega_{\mathrm{idl}}$ from unity to 3.6. The pulse
lengths and energies depend on the wavelength configuration and range from
$\tau_{\mathrm{sig}}$ = $\tau_{\mathrm{idl}}=(70-460)\,$fs (full width at half
maximum), $\varepsilon_{\mathrm{sig}}\leq$ 140 nJ and
$\varepsilon_{\mathrm{idl}}\leq$ 40 nJ. Both beams are linearly polarized.

The experimental setup is schematically shown in Fig. 1. In essence, we
measure the strength of the 2PA in a pump-probe fashion with temporally
overlapping signal and idler pulses. To this end, the idler pulse passes a
motorized linear translation stage to adjust the time delay
$\tau_{\mathrm{d}}$ between the two pulses. Afterwards, the signal and idler
beam are superimposed on the sample. An angle of 20∘ between the two beams
allows us to easily separate them afterwards. Note that the angle within the
sample (9∘) is even smaller due to refraction and, thus, signal and idler
directions can be assumed to be parallel inside the sample. A broadband
$\lambda$/2-waveplate allows the polarization of the idler pulse to be rotated
by 90∘. In this way, we can compare 2PA measurements where the signal and
idler pulses are polarized either parallel or perpendicular to each other. In
the following, these two configurations will be referred to as co-polarized
and cross-polarized. The idler pulse is blocked after the sample. The
detection of the signal transmittance is performed with the lock-in technique
referenced to a modulation of the idler beam with a mechanical chopper.

Figure 1: Experimental setup for the measurement of nondegenerate 2PA.

For the quantitative analysis of the two-photon absorption coefficient
$\beta$, a precise knowledge of the spot diameters in the sample plane is
important. To this end, we use the knife-edge-method with a titanium plate
mounted in the sample plane acting as a knife. The peak irradiances required
for the calculation of $\beta$ also depend on the pulse durations. To measure
these pulse durations of the idler, we use a commercial autocorrelator (APE
Mini TPA) that can be equipped with different detector units depending on the
actual wavelength of the pulse. The signal pulses could also be temporally
characterized here. However, the duration of the signal is easily determined
by the data analysis described below.

We now turn towards the characterization of the samples. Zinc selenide (ZnSe)
belongs to the group of II-VI-semiconductors with zincblende structure. The
material is a direct semiconductor with a bandgap of
$\varepsilon_{\mathrm{g}}$ = 2.7 eV at room temperature [14]. As for the 2PA
the sum energy of the involved photons needs to exceed the bandgap, the laser
system in combination with the OPA, delivering pulses with
$\hbar$$\omega_{\mathrm{sig}}+\hbar$$\omega_{\mathrm{idl}}=3.1\,$eV, is
perfectly suited for the investigation of ZnSe. The excess energy above the
bandgap is $\Delta\varepsilon=400\,$meV. We investigate ZnSe specimens with in
both (100) and (110) orientation. The sample thicknesses are $42\,\upmu$m for
the (100) oriented and $37\,\upmu$m for the (110) oriented sample,
respectively. These thicknesses are achieved by mechanical grinding of
commercial wafers. The thicknesses are chosen to ensure homogeneous excitation
conditions across the sample depth. The orientation of the light polarizations
relative to the crystallographic axes is shown in Fig. 2 for both samples.

Figure 2: Orientation of the signal and idler polarizations with respect to
the crystallographic axes for the (100) and (110) ZnSe sample.

##  III Theoretical model

One of the simplest band-structure models suitable for investigating 2PA in
zincblende semiconductors for different relative polarization directions
between the two pulses is the 8-band Kane model with nonparabolicity [15].
This model includes the lowest conduction band and the light-hole, heavy-hole
and split-off valence bands. It was used in Refs. [16, 3] and provided an
improved frequency-dependence of the (nondegenerate) $\beta$ over the two-band
models used in earlier studies [7, 8, 9, 10]. However, the model is still
isotropic and thus does not capture the dependence on the crystal orientation.
From measurements, such an anisotropy has been known to exist for a long time.
In Refs. [17, 18] it was shown that the anisotropy of the (degenerate) $\beta$
is mainly due to the influence of higher conduction bands, not by serving as
intermediate states (at least for the considered frequencies), but by inducing
warping of the valence bands. For the numerical approach described here we
also use an 8-band Kane model, but include the interaction with higher
conduction bands in second-order perturbation theory [19] which leads to
higher-order terms in the band energies $\varepsilon_{k}$. Where available, we
compare our findings with the predictions made by two-band models in
transition rate [20, 21, 12] or susceptibility [11, 22, 23] approaches.

A major difficulty in the combination of dynamical equations in the length
gauge with rather realistic band-structure models is the random gauge of the
numerically obtained eigenfunctions/matrix elements at each ${\bm{k}}$-point.
We build on earlier works [24, 25, 26, 27], to reconstruct a globally smooth
gauge in the modeled region of ${\bm{k}}$-space. This is achieved by applying
a parallel transport gauge (PTG), where additionally the lines in
${\bm{k}}$-space are connected via a common starting point. With this
procedure further gauge transformations during the numerical time evolution
are not required for any polarization setting. Since we transform the gauge
such that connected bands are mixed [26] (not only where they are exactly or
nearly degenerate), the same transformation needs to be applied to all other
matrices (denoted by the hat symbol; note that the matrices other than
$\bm{\xi_{k}}$ transform as usual operators). The dynamical equation for the
single particle density matrix $\rho_{\bm{k}}$ can be expressed (independently
of the gauge) as

$\displaystyle\frac{\partial}{\partial t}\hat{\rho}_{\bm{k}}$
$\displaystyle=\frac{i}{\hbar}\big{[}\hat{h}_{\bm{k}}-e\bm{E}(t)\cdot\hat{\bm{\xi}}_{\bm{k}},\hat{\rho}_{\bm{k}}\big{]}+\frac{1}{\hbar}{e\bm{E}}(t)\cdot{\nabla}_{\bm{k}}\hat{\rho}_{\bm{k}}.$
(1)

The optical response is determined by the local spatial average of the mean
microscopic current density,

$\displaystyle\bm{J}$
$\displaystyle=-e\frac{1}{L^{3}}\sum_{\bm{k}}\mathrm{Tr}(\hat{\rho}_{\bm{k}}\hat{\bm{v}}_{\bm{k}}).$
(2)

The input matrices ($h_{\bm{k}},\bm{\xi_{k}},\bm{v_{k}}$) to Eqs. (1) and (2)
are ground state properties defined as follows. In the original gauge,
$h_{\bm{k}}$ is the diagonal matrix of the single-particle band energies
$\varepsilon_{k}$. The velocity matrix $\bm{v_{k}}$ is defined as

$\displaystyle\hbar\hat{\bm{v}}_{\bm{k}}={\nabla}_{\bm{k}}\hat{h}_{\bm{k}}+i\big{[}\hat{h}_{\bm{k}},\hat{\bm{\xi}}_{\bm{k}}\big{]},$
(3)

where in the original gauge the second term corresponds to the interband
contribution. Note that Eqs. (2) and (3) hold for both the original gauge and
our PTG, which leaves $h_{\bm{k}}$ smooth. The transition dipoles
$e\bm{\xi}_{\bm{k}}^{\lambda\lambda^{\prime}}$ are obtained as matrix elements
of the position operator from the periodic parts of the Bloch states,

$\displaystyle\bm{\xi}_{\bm{k}}^{\lambda\lambda^{\prime}}$
$\displaystyle\equiv i\langle
u^{\lambda}_{\bm{k}}|\nabla_{\bm{k}}|u^{\lambda^{\prime}}_{\bm{k}}\rangle.$
(4)

This also holds for the intraband ($\lambda=\lambda^{\prime}$) elements which,
however, requires a smooth gauge. The gradient terms in Eqs. (1), (3), and (4)
are evaluated numerically as symmetrized difference quotient in the
discretized ${\bm{k}}$-space.

Our approach is fully consistent in the sense that different gauges lead to
the same result for the optical response. Eq. (1) corresponds to the SBE in
the length gauge. In this study, the commonly used phenomenological dephasing
and relaxation terms are not included, since the corresponding time scales are
expected to be longer than the pulse length in the experiment and because the
effect of dephasing is essentially limited to a broadening of spectral
features. Likewise, many-body effects are not expected to dominate as the sum
frequencies are sufficiently high above the band gap and are thus omitted as
in previous studies on 2PA. We are interested in 2PA resulting from the
simultaneous absorption of one signal and one idler photon. This process is
isolated from other responses via a directional order expansion in the two
electric fields (where $\bm{E}$(t) in Eq. (1) is the sum of both, i.e.,
$\bm{E}$(t)$=$$\bm{E}_{\mathrm{sig}}$(t) $+$$\bm{E}_{\mathrm{idl}}$(t)) [13].
The absorbed two-photon energy is given by the time-integrated product of the
signal-directed third-order current $\tilde{{\bm{J}}}\equiv\bm{J}^{(3)(1|0)}$
with the signal electric field. Normalization with respect to the incident
intensities leads to the 2PA coefficient [13, 28]

$\displaystyle\beta(\omega_{1},\omega_{2})$
$\displaystyle=\frac{1}{\epsilon_{0}^{2}c^{2}}\frac{1}{n_{\mathrm{sig}}n_{\mathrm{idl}}}\frac{\displaystyle\int\mathrm{d}t\,\langle\tilde{{\bm{J}}}(t)\cdot\tilde{{\bm{E}}}_{\mathrm{sig}}^{*}(t)\rangle}{\displaystyle\int\mathrm{d}t\hat{E}_{\mathrm{sig}}^{2}(t)\hat{E}_{\mathrm{idl}}^{2}(t)},$
(5)

where $n_{\mathrm{pulse}}\equiv n(\omega_{\mathrm{pulse}})$ is the linear
refractive index at the frequency of the (signal or idler) pulse. Since the
variation of $n_{\mathrm{sig}}n_{\mathrm{idl}}$ in the considered parameter
range (with fixed sum frequency) is small, we approximate the frequency-
dependent refractive index $n(\omega)$ by [29]

$\displaystyle n^{2}=A+\frac{B\lambda^{2}}{\lambda^{2}-C^{2}},$ (6)

where $\lambda$ is the wavelength in $\upmu\text{m}$, and the parameters for
ZnSe have been experimentally determined as: $A=4.00$, $B=1.90$, and
$C^{2}=0.113$. [29] Thus, in our case the factor
$n_{\mathrm{sig}}n_{\mathrm{idl}}$ increases by about 6% when changing the
nondegeneracy parameter $\omega_{1}/\omega_{2}$ from unity to 4 (corresponding
to the range considered in Sec. IV), with the sum frequency fixed at 3.1 eV.

In the numerical simulations, signal and idler pulse are assumed to be co-
propagating along the sample orientations given in Fig. 2. We use pulses with
a Gaussian envelope of duration
$\tau_{\mathrm{sig}}=\tau_{\mathrm{idl}}\approx 29.44\,$fs (full width at half
maximum of the pulse intensity). Since the computational expense increases
higher than linearly with the pulse durations (due to demanding finer
${\bm{k}}$\- and $t$-resolutions and a longer duration of the integration), we
have refrained from simulating longer pulses to keep the numerical effort
within resaonable limits. However, in our case of fixed sum frequency, the
higher spectral resolution provided by longer pulses is not required since it
does not significantly alter the shown spectra (as confirmed by simulations
using dimensionally reduced $k$-grids). As mentioned in Sec. IV, the
evaluation of experiments with different pulse durations also provided
practically identical 2PA coefficients.

The band-structure input is provided by the 8-band Kane model with remote band
contributions, using the parameters tabulated in Ref.
[[][.Theexplicitmatrixformofthe8-bandKanemodelisgiveninTableC.8,
whereweneglectalltermsproportionaltoKane'sparameters($B_8v^±$and$B_7v$).TableD.1listsbulkbandstructureparameters.TheeffectivemassandLuttingerparametershavetobereducedaccordingtoTableC.9.]Winkler2003].
In order to investigate different crystallographic orientation, a rotation of
the crystal is applied for the (110) sample, such that the quantization axis
of the total angular momentum (the $z$-axis in the ${\bm{k}\cdot}{\bm{p}}$
theory) corresponds to the propagation direction. The simulated 3D
${\bm{k}}$-space is resolved by a Cartesian grid of sufficiently large side
length (here 0.2 $\pi/a_{0}$, where $a_{0}$ is the lattice constant), with the
$\Gamma$-point in the center. For the chosen resonance frequency of
$\hbar$$\omega_{\mathrm{sig}}$ \+ $\hbar$$\omega_{\mathrm{idl}}$ about 400 meV
above the band gap, the accuracy of the 8-band ${\bm{k}\cdot}{\bm{p}}$ model
may be limited. We can, however, estimate that this together with the limited
resolution and the short pulse duration causes an uncertainty of the obtained
values for $\beta$ of roughly 15%.

The perturbation expansion of the SBE (1) can be solved analytically in the
cw-limit, which has been demonstrated for the $\chi^{(3)}$-response within a
simple two-band model with ${\bm{k}}$-independent dipole matrix element in
Ref. [13]. That approach results in a scaling $\beta\sim
n_{\mathrm{sig}}^{-1}n_{\mathrm{idl}}^{-1}\omega_{\mathrm{sig}}^{-3}\omega_{\mathrm{idl}}^{-4}(\omega_{\mathrm{sig}}^{2}+\omega_{\mathrm{idl}}^{2})^{2}$
(not shown), which corresponds to a slightly stronger increase of the 2PA with
the nondegeneracy parameter than our present numerical results with
${\bm{k}}$-dependent dipole matrix elements and than the experimental results.
If we alter the model of Ref. [13] by the following ${\bm{k}}$-dependence of
the transition dipole, $\xi_{\bm{k}}=\xi_{0}(\omega_{0}/\omega_{\bm{k}})^{2}$,
we find $\beta\sim
n_{\mathrm{sig}}^{-1}n_{\mathrm{idl}}^{-1}\omega_{\mathrm{sig}}^{-3}\omega_{\mathrm{idl}}^{-4}$,
which for fixed sum frequency is the same frequency dependence as obtained
from the approaches of Refs. [21] and [11] and also agrees well with the
results shown below (see curves labeled ‘2-band’ in Fig. 4). Identifying a
similar scaling rule for the cross-polarized case is less trivial since it
would require analyzing the magnitude and directions of the complex transition
dipoles in a more realistic band-structure model including spin.

##  IV Results and discussion

To experimentally investigate the nondegenerate 2PA coefficient
$\beta(\omega_{\mathrm{sig}},\omega_{\mathrm{idl}})$, depending on the
frequency ratio $\omega_{\mathrm{sig}}/\omega_{\mathrm{idl}}$ of the two
involved photons, we measure various data points within the range of possible
frequency combinations of signal and idler photons. Our experiment and the
detection scheme is designed to measure 2PA whereby one photon from each beam
triggers 2PA. In order to suppress degenerate 2PA with two photons from the
same beam, the energies of the signal and idler photons are chosen to be
$\varepsilon_{g}/2<\hbar\omega_{\mathrm{sig}}<\varepsilon_{g}$ and
$\hbar\omega_{\mathrm{idl}}<\varepsilon_{g}/2$. With this choice the idler
alone cannot trigger 2PA. The idler is chosen as the pump beam in our pump-
probe approach. The signal acts as the probe beam. Its intensity is much
weaker than the idler such that also the signal alone does not induce
significant 2PA. The spot sizes of the pulses are adjusted that the idler spot
is at least 2.5 times larger than the signal spot. As a result, the intensity
of the idler across the signal spot is practically identical to its peak
intensity such that spatial averaging is not necessary for the extraction of
$\beta$.

Figure 3: Exemplary pump-probe transients for a co- and cross-polarized 2PA
measurement at $\lambda_{\mathrm{sig}}=570\,$nm and
$\lambda_{\mathrm{idl}}=1341\,$nm for the (100) oriented ZnSe sample. The blue
(red) data correspond to the co- (cross-) polarized configuration. Solid lines
represent a fit according to the model by Negres et al. [30].

Fig. 3 shows an exemplary pump-probe transient for a co- and cross-polarized
measurement at a wavelength combination of $\lambda_{\mathrm{sig}}=570\,$nm
and $\lambda_{\mathrm{idl}}=1341\,$nm for the (100) oriented ZnSe sample. The
signal and idler pulses have Gaussian temporal pulse shapes and the measured
idler pulse length is $\tau_{\mathrm{idl}}=98.2\,$fs (full width at half
maximum). While our pulse durations are somewhat longer than the ones in the
simulation, we do not expect an impact for the resulting 2PA coefficients. In
fact, by using different alignments of the OPA, we have repeated the
experiment with different pulse durations for one wavelength combination.
After incorporating the actual pulse durations into the data analysis, we
found practically identical 2PA coefficients

To evaluate the 2PA coefficient $\beta$ we use the theoretical approach for
the normalized signal transmittance of Negres et al. [30]. In short, the
results of Negres et al. establish an analytical formula for the shape of the
pump-probe transients for our experimental configuration. It assumes pulses
with a Gaussian temporal envelope. It takes into account the well- known group
velocity mismatch of signal and idler pulses. While the approach of Negres et
al. in principle takes linear absorption into account as well, we set this
linear absorption to zero. This assumption is valid because the linear
absorption across the bandgap is supressed by the chosen photon energies
($<\epsilon_{\text{g}}$). Free carrier absorption can be excluded as the pump-
probe traces do not show transient absorption beyond the temporal overlap of
pump and probe pulse. A fit of this model to the experimental transients
reveals the co- and cross-polarized 2PA absorption coefficients
$\beta_{\parallel}$, $\beta_{\perp}$, and, in addition, the duration of the
signal pulse. Two examples for such a fit are displayed in Fig. 3 and show
excellent agreement between the model and the measurement.

For the specific wavelength combination presented in Fig. 3, the fit
represents curves with $\beta_{\parallel}=12.93\,_{-1.4}^{+1.6}\,$cm/GW with
$\tau_{\mathrm{sig}}=(120.7\pm 0.9)\,$fs and
$\beta_{\perp}=4.77\,_{-0.5}^{+0.6}\,$cm/GW with
$\tau_{\mathrm{sig}}=(120.1\pm 1.7)\,$fs. The pulse lengths are given as the
full width at half maximum. The spot sizes for this example are
$A_{\text{sig}}=2100\,\upmu\text{m}^{3}$ and
$A_{\text{idl}}=30100\,\upmu\text{m}^{3}$. We estimate the errors for $\beta$
by varying the input parameters within the measurement inaccuracies. Since all
the pump-probe data sets look rather comparable and the analysis follows
analogously to the above procedure, we now restrict the presentation to the
values for the two-photon absorption coefficient $\beta$ and its dependence on
the nondegeneracy, polarization, and crystal orientation.

Figure 4: 2PA coefficients of ZnSe as a function of the nondegeneracy
parameter. The corresponding polarization settings for co- and cross-polarized
cases and each crystallographic orientation are shown in Fig. 2. The curves
labeled ‘theory’ are obtained from simulations with $\tau=50\,$fs pulses. The
curves labeled ‘2-band’ correspond to $\beta_{\parallel}\sim
n_{\mathrm{sig}}^{-1}n_{\mathrm{idl}}^{-1}\omega_{\mathrm{sig}}^{-3}\omega_{\mathrm{idl}}^{-4}$,
where the prefactor has been chosen for best fit with the numerical results.

We now turn to the comparison of the experimentally observed 2PA coefficients
for various nondegeneracy parameters
$\omega_{\mathrm{sig}}/\omega_{\mathrm{idl}}$ and the two crystallographic
orientations. As seen in Fig. 4, the values for
$\beta_{\parallel}(\omega_{\mathrm{sig}}/\omega_{\mathrm{idl}})$ and
$\beta_{\perp}(\omega_{\mathrm{sig}}/\omega_{\mathrm{idl}})$ increase with the
nondegeneracy parameter. Specifically, we find an approximately fivefold
increase of $\beta$ when comparing configurations with
$\omega_{\mathrm{sig}}/\omega_{\mathrm{idl}}=3.6$ to the degenerate case
$\omega_{\mathrm{sig}}/\omega_{\mathrm{idl}}=1$. Note that the scatter of the
experimental data also increases with
$\omega_{\mathrm{sig}}/\omega_{\mathrm{idl}}$ because the noise of the optical
parametric amplifier increases for widely nondegenerate operation. The
increase of the two-photon absorption strength with higher degrees of
nondegeneracy has been theoretically predicted in several studies [13, 11,
12]. When comparing the two crystallographic orientations, we find very
similar magnitudes for
$\beta_{\parallel}(\omega_{\mathrm{sig}}/\omega_{\mathrm{idl}})$ at the same
frequency combinations. In contrast, the values for
$\beta_{\perp}(\omega_{\mathrm{sig}}/\omega_{\mathrm{idl}})$ for the
(110)-oriented sample are consistently higher than for the (100)-oriented
sample. We will elaborate further on this fact below.

It is also instructive to compare our results to previous studies of 2PA in
bulk ZnSe. So far, comparable results are only available for degenerate, co-
polarized configurations and 800 nm pulses. In this case, we find average
$\beta$-values of 3.9 cm/GW for the (100)-sample and 4.9 cm/GW for the
(110)-sample. Our experimental results agree well with the value of 3.5 cm/GW
independently reported in literature by Balu et al. [31] and Fishman et al.
[5]. Balu et al. used a polycrystalline ZnSe specimen. Fishman et al. do not
give particular specifications of their sample.

Fig. 4 also contains the theoretical results for the same crystallographic
orientations and polarization configurations. Given the uncertainties in both
experiment and theory, the agreement for all four settings is surprisingly
good. Without using any fitting parameter, the theory yields a very similar
behavior of $\beta$ as a function of the nondegeneracy parameter, and even its
absolute magnitudes are very close to the measured ones.

Figure 5: Ratio between cross- and co-polarized 2PA. Same data (experiment and
theory) as in Fig. 4.

We finally elaborate on the anisotropy of $\beta$ in terms of the different
polarization configurations and its dependence on the nondegeneracy. A useful
parameter to characterize the polarization anisotropy is the ratio
$\beta_{\perp}/\beta_{\parallel}$ for co- and cross-polarized signal and idler
beams. These results are summarized in Fig. 5. The first observation is that
the values of $\beta_{\perp}/\beta_{\parallel}$ for the (100) oriented sample
are clearly smaller than for the (110) sample. More surprisingly, we find a
significant increase of $\beta_{\perp}/\beta_{\parallel}$ with increasing
$\omega_{\mathrm{sig}}/\omega_{\mathrm{idl}}$. This result has not been
reported in any previous study on nondegenerate 2PA. An advantage of
inspecting this ratio is its lower noise due to cancelling out of certain
error sources in the original quantities.

The numerical results from the 8-band Kane model lie within the experimental
uncertainties. The experimental result for $\beta_{\perp}/\beta_{\parallel}$
might be slightly overestimated due to the oblique idler propagation in the
nominally co-polarized case. As discussed above, the numerical results come
with their own uncertainty - in particular, the slight decrease of
$\beta_{\perp}/\beta_{\parallel}$ in the vicinity of the degenerate case might
result from the finite resolution in ${\bm{k}}$-space. Otherwise, the
properties of $\beta_{\perp}/\beta_{\parallel}$ found in the experiment as
described above are well captured by our simulations.

To explain the microscopic origin of this behavior is complicated in the
considered model, since the 8$\times$8-Kane Hamiltonian with remote-band
contributions cannot be diagonalized analytically. However, we have evaluated
a simpler model, in particular an anisotropic two-band model with complex
${\bm{k}}$-independent transition dipoles, obtained from zone-center wave
functions [32]. In this model, $\beta_{\perp}$ is given by $\beta_{\parallel}$
with two out of four excitation paths dropped for symmetry reasons. This
results in a different frequency dependence for both cases and in the
following dependence of the ratio on the nondegeneracy parameter,

$\displaystyle\frac{\beta_{\perp}}{\beta_{\parallel}}$
$\displaystyle=\frac{1+(\omega_{1}/\omega_{2})^{4}}{[1+(\omega_{1}/\omega_{2})^{2}]^{2}},$
(7)

which equals $1/2$ for the degenerate case and increases towards unity for the
strongly nondegenerate case. In comparison with the realistic model, this
increase is too strong as a consequence of the neglected ${\bm{k}}$-dependence
of transiton dipoles.

##  V Conclusions

In summary, we have carefully investigated the nondegenerate two-photon
absorption coefficient $\beta(\omega_{1},\omega_{2})$ in the prototypical
semiconductor ZnSe for a fixed sum frequency $\omega_{1}+\omega_{2}$. We find
a substantial increase of the two-photon absorption strength with increasing
$\omega_{1}/\omega_{2}$ as predicted by different theoretical approaches.
Specifically, we find an about fivefold increase of
$\beta(\omega_{1},\omega_{2})$ for $\omega_{1}/\omega_{2}=3.5$ when compared
to the degenerate case. A numerical model based on eight-band
${\bm{k}\cdot}{\bm{p}}$ calculations and semiconductor Bloch equations
including inter- and intraband excitations agrees quantitatively with the
experiment in terms of the dispersion of $\beta(\omega_{1},\omega_{2})$ as
well as the crystalline and polarization anisotropy. The results are important
for a more detailed understanding of nonlinear two-color interactions in
semiconductors and their applications in nonlinear photonics.

###### Acknowledgements.

We acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG) in the
framework of the CRC – TRR142 (grant No. 231447078, project A07). We also
thank the PC2 (Paderborn Center for Parallel Computing) for providing
computing time.

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