# Two-phonon propagation in a 4-level thermal quantum nanomachine

Y. Lai  School of Physics and CRANN Institute, Trinity College Dublin, Dublin
2, D02 PN40, Ireland  The Blackett Laboratory, Department of Physics, Imperial
College London, South Kensington Campus, SW7 2AZ, London, United Kingdom  C.
McDwyer  School of Physics and CRANN Institute, Trinity College Dublin, Dublin
2, D02 PN40, Ireland  P. Karwat  School of Physics and CRANN Institute,
Trinity College Dublin, Dublin 2, D02 PN40, Ireland  Department of Theoretical
Physics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego
27, 50-370 Wrocław, Poland  O. Hess  School of Physics and CRANN Institute,
Trinity College Dublin, Dublin 2, D02 PN40, Ireland  The Blackett Laboratory,
Department of Physics, Imperial College London, South Kensington Campus, SW7
2AZ, London, United Kingdom

###### Abstract

Acoustic waves, as science of sound, is an established field in physics. In
analogy with light, engineers adapt tools from optic fields for manipulating
sound waves to fight with cancer cells. Here we present a heat-gradient driven
nanomachine concept for simultaneous emission of two phonons, i.e. a
convertion of heat into a sonic wave output. Our theoretical work sheds light
on nanoscale components or systems that could be used for future ultrasound
devices.

##  I Introduction

Recent progress in polariton-driven phonon lasing [1] shows that there is much
more to discover [2, 3]. As lasers in optics, as sound waves in acoustics, in
analogy with light they could be used as a tool, i.e. acoustic tweezer [4, 5],
and play an important role to fight with cancer cells. Scientists are adopting
ideas from optics to manipulate sound weaves [6, 7] instead. Recent
experimental works suggest that ultrasound could not only trap, but also
compress a cancer cell [8]. Another motivation to study acoustics is fact,
that a phononic medium offers a wider range of wavelengths, usually not
accessible to conventional lasers [9, 10, 11].

In this paper, we present a thermal quantum nanomachine, i.e. a phonon lasing
concept [12] based on 4-level system with two 2-level subsystems, able to
convert heat into phononic output. As typical ultrasound devices used to fight
with cancer cells operate in the range from 20 kHz to several GHz, we focus on
energy scale with tenths of meV. In the end, we analyze some thermodynamic
aspects of our phonon medium.

##  II Model

Figure 1: (a) Sketch of the system. (b) Two scenarios, left: generation of two
phonons, i.e. transitions $|4\rangle$ $\rightarrow$ $|3\rangle$ and
$|2\rangle$ $\rightarrow$ $|1\rangle$ in the middle subsystem QS M.; right:
generation of a photon, i.e. transitions $|3\rangle$ $\rightarrow$
$|2\rangle$.

The system under study illustrated in Fig. 1 is composed of centre 4-level
quantum subsystem (QS M) interacting with two 2-level subsystems (QS L/R) at
adjacent sides, while the whole system is coupled to heat baths with different
temperatures. The heat-gradient [13, 14, 15] is necessary for excitation flow,
that allows the transitions between $|4\rangle$ $\leftrightarrow$ $|3\rangle$
and $|1\rangle$ $\leftrightarrow$ $|2\rangle$ in the middle subsystem to be
accompanied with phonon emission and absorption. These two-level subsystems
work as filters to avoid thermalization with a mean temperature in QS M. We
assume that only the middle subsystem is coupled to external lattice
displacement field.

The QSs are coupled in a way that excitations can be exchanged between
adjacent sites as denoted in Fig. 1(b), i.e. the middle system interacts with
both parts (i.e. top and bottom), while the interaction between top and bottom
system is suppressed. The corresponding Hamiltonian reads

$\displaystyle\hat{H}_{\mathrm{int}}=\lambda_{\mathrm{MT}}\bigg{(}\hat{h}^{(\mathrm{TM})}\otimes\hat{1}^{\mathrm{(B)}}\bigg{)}+\lambda_{\mathrm{MB}}\bigg{(}\hat{1}^{\mathrm{(T)}}\otimes\hat{h}^{(\mathrm{MB})}\bigg{)}$
(1)

where $\lambda_{\mathrm{MT/MB}}$ is the coupling parameter. The coupling

$\hat{h}^{(\mathrm{TM})}=\hat{P}_{21}^{\mathrm{(T)}}\otimes\bigg{(}\hat{P}_{12}^{\mathrm{(M)}}+\hat{P}_{14}^{\mathrm{(M)}}+\hat{P}_{23}^{\mathrm{(M)}}+\hat{P}_{24}^{\mathrm{(M)}}\bigg{)}+\mathrm{h.c.}$
(2)

is given by the respective projection operators and analogous for
$\hat{h}^{(\mathrm{MB})}$. The coupling is taken to be weak such that the
energy contribution of the interaction is small compared to the energy
contained in the system.

Each of the two edge QSs is coupled locally to a heat bath of different
temperature. To describe the coupling, we make use of a Quantum Master
Equation within a Lindblad form, which accounts for the non-equilibrium
situation in our system. For this, we set up the equation of motion for the
density matrix $\hat{\rho}$ via

$\frac{d\hat{\rho}}{dt}=-\frac{i}{\hbar}[\hat{H}_{\mathrm{sys}}+\hat{H}_{\mathrm{int}},\hat{\rho}]+\hat{D}_{\mathrm{H}}(\hat{\rho})+\hat{D}_{\mathrm{C}}(\hat{\rho})\,.$
(3)

$\hat{D}_{\mathrm{H}}$ and $\hat{D}_{\mathrm{C}}$ are the dissipators to the
hot (left) and cold (right) heat bath, respectively,

$\hat{D}_{\mathrm{H}}(\hat{\rho})=\sum_{k=1}^{2}\Gamma_{k}(T_{\mathrm{H}})\bigg{(}\hat{L}^{(\mathrm{T})}_{k}\hat{\rho}\hat{L}^{(\mathrm{T})\dagger}_{k}-\frac{1}{2}[\hat{L}^{(\mathrm{T})\dagger}_{k}\hat{L}^{(\mathrm{T})}_{k},\hat{\rho}]_{+}\bigg{)}$
(4)

with the Lindblad operators

$\hat{L}^{(\mathrm{T})}_{1}=\hat{P}^{(\mathrm{T})}_{21}\otimes\hat{1}^{(\mathrm{M,B})},\qquad\hat{L}^{(\mathrm{T})}_{2}=\hat{P}^{(\mathrm{T})}_{12}\otimes\hat{1}^{(\mathrm{M,B})}\,.$
$\Gamma_{k}(T_{\mathrm{H}})=\frac{\gamma}{1+\exp\big{\\{}(-1)^{k-1}\delta_{\mathrm{L}}/k_{\mathrm{B}}T_{\mathrm{H}}\big{\\}}}.$
(5)

containing the distribution function [16]. The dissipator describing the
coupling to the cold (right) heat bath $\hat{D}_{\mathrm{C}}(\hat{\rho})$ is
analogue.
Our goal is to achieve an inversion between states:
$|4\rangle^{(\mathrm{M})}\rightarrow|3\rangle^{(\mathrm{M})}$,
$|2\rangle^{(\mathrm{M})}~{}\rightarrow~{}|1\rangle^{(\mathrm{M})}$, which
later will be coupled to a phonon mode, and
$|3\rangle^{(\mathrm{M})}\rightarrow|2\rangle^{(\mathrm{M})}$ (see Fig. 1(b))
for the coupling to a photon mode.

The energy of typical acoustic phonons lie in the order of a few meV. Setting
the energy of the lower state $|1\rangle^{(\mathrm{M})}$ to zero, we
accordingly chose $\delta_{\mathrm{M_{+}}}=30$ meV and
$\delta_{\mathrm{M_{-}}}=25$ meV, thus
$\delta_{\mathrm{M_{+}}}-\delta_{\mathrm{M_{-}}}=5$ meV. The energies of the
edge state are set to $\delta_{\mathrm{T}}=\delta_{\mathrm{M_{+}}}=30$ meV and
$\delta_{\mathrm{B}}=\delta_{\mathrm{M_{-}}}=25$ meV. If not stated otherwise,
the parameters are set to
$\lambda=\lambda_{\mathrm{MT}}=\lambda_{\mathrm{MB}}=0.03$ meV,
$\gamma=\gamma_{\mathrm{H}}=\gamma_{\mathrm{C}}=3$ ps-1. As initial condition
we assume that the whole system is in its ground state
$|1\rangle^{(\mathrm{T})}\otimes|1\rangle^{(\mathrm{M})}\otimes|1\rangle^{(\mathrm{B})}$.

##  III phonon lasing

For simplicity, we assume two single acoustic-phonon modes, as realized in a
cavity formed, e.g., in semiconductor superlattices. Denoting $b,b^{\dagger}$
as the bosonic operators of the phonon mode and $\omega$ their frequency, the
phonon Hamiltonian is composed of the free phonon system and the carrier-
phonon coupling

$\hat{H}_{\mathrm{ph}}=\hbar\omega_{1}\hat{b}_{1}^{\dagger}\hat{b}_{1}+\hbar\omega_{2}\hat{b}_{2}^{\dagger}\hat{b}_{2}+\hat{H}_{\mathrm{c-ph}}\,.$
(6)

The energy of the phonon is chosen to be resonant with the energy difference
in QS M with $\hbar\omega=5$ meV. The phonon mode is coupled only to the
middle QS M via

$\hat{H}_{\mathrm{c-ph}}=\hat{1}^{(\mathrm{T})}\otimes\hat{h}^{(\mathrm{M})}\otimes\hat{1}^{(\mathrm{B})}$
(7)

with

$\hat{h}^{(\mathrm{M})}=\hbar
g\big{(}\hat{b}^{\dagger}\hat{P}^{(\mathrm{M})}_{34}+\hat{b}\hat{P}^{(\mathrm{M})}_{43}+\hat{b}^{\dagger}\hat{P}^{(\mathrm{M})}_{12}+\hat{b}\hat{P}^{(\mathrm{M})}_{21}\big{)}.$
(8)

Here $g$ is the real coupling constant between the QS and phonon mode. We
assume that the system can relax from the state $|4\rangle^{(\mathrm{M})}$ to
state $|3\rangle^{(\mathrm{M})}$ and from the state $|3\rangle^{(\mathrm{M})}$
to state $|2\rangle^{(\mathrm{M})}$ by the emission of a phonon, while the
reverse transition is possible by absorbing a phonon.

A measurable quantity for the phonons is the lattice displacement, which is
connected to the phonon operators via

$\langle\hat{u}\rangle=u_{0}\left(\langle\hat{b}^{\dagger}\rangle+\langle\hat{b}\rangle\right)=u^{(+)}+u^{(-)},$
(9)

where we defined $u^{(+)}=u_{0}\langle\hat{b}\rangle$ and
$u^{(-)}=u_{0}\langle\hat{b}^{\dagger}\rangle$ as well as the single phonon
amplitude $u_{0}$.

Heisenberg equation.

$i\hbar\frac{d}{dt}\hat{b}^{\dagger}=-\hbar\omega\hat{b}^{\dagger}-\hbar
g|3\rangle\langle 2|$ $i\hbar\frac{d}{dt}\hat{b}=+\hbar\omega\hat{b}+\hbar
g|2\rangle\langle 3|$
$\frac{d}{dt}\hat{b}^{\dagger}=i\omega\hat{b}^{\dagger}+ig|3\rangle\langle 2|$
$\frac{d}{dt}\hat{b}=-i\omega\hat{b}-ig|2\rangle\langle 3|$
$\frac{d}{dt}u=iu_{0}\omega(\langle\hat{b}^{\dagger}\rangle-\langle\hat{b}\rangle)+iu_{0}g(|3\rangle\langle
2|-|2\rangle\langle 3|)$

Introducing the phonon coupling, we extend the equations of motion to

$\frac{d\hat{\rho}}{dt}=-\frac{i}{\hbar}[\hat{H}_{\mathrm{sys}}+\hat{H}_{\mathrm{int}}+\hat{H}_{\mathrm{ph}},\hat{\rho}]+\hat{D}_{\mathrm{H}}(\hat{\rho})+\hat{D}_{\mathrm{C}}(\hat{\rho}),$
(10)

For the lattice displacement this leads to the following rate Eq. (10).

$\frac{du_{1}}{dt}=-\Gamma
u_{1}+iC\bigg{(}\rho_{34}^{(\mathrm{M})}(t)+\rho_{43}^{(\mathrm{M})}(t)\bigg{)},$
(11) $\frac{du_{2}}{dt}=-\Gamma
u_{2}+iC\bigg{(}\rho_{12}^{(\mathrm{M})}(t)+\rho_{21}^{(\mathrm{M})}(t)\bigg{)},$
(12)

where we introduced the phonon dephasing rate $\Gamma$ (assuming the same for
both modes) and defined the coupling constant $C=u_{0}g$. For our simulations
we assume $\Gamma=2$ ps-1, $g=2.25$ ps-1, $u_{0}=20$ pm. We further assume
that always a very small, but finite displacement is present.

Then the dynamics can be described by currents. We take a trace the Quantum
Master Equations over all the subsystems at first. The reduced density
operators only describe the states for corresponding subsystems. The diagonal
elements of the reduced density matrix indicate the occupation of states in
corresponding subsystems.

$\rho^{\mathrm{(L)}}=Tr_{\mathrm{MR}}(\rho),\rho^{\mathrm{(R)}}=Tr_{\mathrm{LM}}(\rho),\rho^{\mathrm{(M)}}=Tr_{\mathrm{LR}}(\rho).$
(13)

The time evolution equations of these occupations of the energy levels in
subsystems are derived from reduced master equations

$\dot{\rho}^{\mathrm{(L)}}=Tr_{\mathrm{MR}}(\dot{\rho}),\dot{\rho}^{\mathrm{(R)}}=Tr_{\mathrm{LM}}(\dot{\rho}),\dot{\rho}^{\mathrm{(M)}}=Tr_{\mathrm{LR}}(\dot{\rho}).$
(14)

The transitions between different states result in the change of states. To be
consistent with previous research [17], we define the thermal particle
currents to describe the transition induced by the thermal reservoirs

$J^{\mathrm{(L)}}_{1\rightarrow
2}=\Gamma^{\mathrm{(L)}}_{\mathrm{1}}(T_{\mathrm{H}})\rho^{\mathrm{(L)}}_{11}-\Gamma^{\mathrm{(L)}}_{\mathrm{2}}(T_{\mathrm{H}})\rho^{\mathrm{(L)}}_{22},$
(15) $J^{\mathrm{(R)}}_{1\rightarrow
2}=\Gamma^{\mathrm{(R)}}_{\mathrm{1}}(T_{\mathrm{C}})\rho^{\mathrm{(R)}}_{11}-\Gamma^{\mathrm{(R)}}_{\mathrm{2}}(T_{\mathrm{C}})\rho^{\mathrm{(R)}}_{22},$
(16) $J^{\mathrm{(M)}}_{\mathrm{D},1\rightarrow
2}=\Gamma^{\mathrm{(M12)}}_{\mathrm{1}}(T_{\mathrm{sys}})\rho^{\mathrm{(M)}}_{11}-\Gamma^{\mathrm{(M12)}}_{\mathrm{2}}(T_{\mathrm{sys}})\rho^{\mathrm{(M)}}_{22},$
(17) $J^{\mathrm{(M)}}_{\mathrm{D},3\rightarrow
4}=\Gamma^{\mathrm{(M34)}}_{\mathrm{1}}(T_{\mathrm{sys}})\rho^{\mathrm{(M)}}_{33}-\Gamma^{\mathrm{(M34)}}_{\mathrm{2}}(T_{\mathrm{sys}})\rho^{\mathrm{(M)}}_{44}.$
(18)

Then the independent equations of occupation (diagonal terms of density
matrix) can be rewritten as

$\frac{d}{dt}\rho^{\mathrm{(L/R)}}_{22}=J^{\mathrm{(L/R)}}_{1\rightarrow
2}+J^{\mathrm{(M)}}_{\mathrm{H/C}},$ (19)

$\frac{d}{dt}\rho^{\mathrm{(M)}}_{22}=J^{\mathrm{(M)}}_{3\rightarrow
2}-J^{\mathrm{(M)}}_{2\rightarrow 1},$ (20)
$\frac{d}{dt}\rho^{\mathrm{(M)}}_{33}=J^{\mathrm{(M)}}_{4\rightarrow
3}-J^{\mathrm{(M)}}_{3\rightarrow 2},$ (21)
$\frac{d}{dt}\rho^{\mathrm{(M)}}_{44}=J^{\mathrm{(M)}}_{1\rightarrow
4}-J^{\mathrm{(M)}}_{4\rightarrow 3},$ (22)

where

$J^{\mathrm{(M)}}_{\mathrm{H}}=-J^{\mathrm{(M)}}_{\mathrm{H},1\rightarrow
4}+J^{\mathrm{(M)}}_{\mathrm{H},4\rightarrow
3}+J^{\mathrm{(M)}}_{\mathrm{H},3\rightarrow
2}+J^{\mathrm{(M)}}_{\mathrm{H},2\rightarrow 1},$ (23)
$J^{\mathrm{(M)}}_{\mathrm{C}}=J^{\mathrm{(M)}}_{\mathrm{C},1\rightarrow
4}+J^{\mathrm{(M)}}_{\mathrm{C},4\rightarrow
3}-J^{\mathrm{(M)}}_{\mathrm{C},3\rightarrow
2}+J^{\mathrm{(M)}}_{\mathrm{C},2\rightarrow 1},$ (24)

$J^{\mathrm{(M)}}_{2\rightarrow 1/4\rightarrow
3}=\sum_{\mathrm{i}=\mathrm{H,C,Ph,D}}J^{\mathrm{(M)}}_{\mathrm{i},2\rightarrow
1/4\rightarrow 3},$ (25) $J^{\mathrm{(M)}}_{3\rightarrow 2/1\rightarrow
4}=J^{\mathrm{(M)}}_{\mathrm{H},3\rightarrow 2/1\rightarrow
4}+J^{\mathrm{(M)}}_{\mathrm{C},3\rightarrow 2/1\rightarrow 4},$ (26)

We assume that our phonon laser operates in steady-state
($d\rho^{\mathrm{(M)}}_{ii}/dt=0$), obtaining: $J^{\mathrm{(L)}}_{1\rightarrow
2}=-J^{\mathrm{(M)}}_{\mathrm{H}}$, $J^{\mathrm{(R)}}_{1\rightarrow
2}=-J^{\mathrm{(M)}}_{\mathrm{C}}$, $J^{\mathrm{(M)}}_{3\rightarrow
2}=J^{\mathrm{(M)}}_{2\rightarrow 1}$, $J^{\mathrm{(M)}}_{4\rightarrow
3}=J^{\mathrm{(M)}}_{3\rightarrow 2}$, $J^{\mathrm{(M)}}_{1\rightarrow
4}=J^{\mathrm{(M)}}_{4\rightarrow 3}$.

As shown in the previous research, the energy mismatch have a large negative
impact on the interaction strength [18]. We assume that non-resonant
interaction is negligible weak, therefore the non-resonant transition will be
neglected. The relations between currents become

$J^{\mathrm{(L)}}_{1\rightarrow
2}=-J^{\mathrm{(M)}}_{\mathrm{H}}=J^{\mathrm{(M)}}_{\mathrm{H},1\rightarrow
4},$ (27) $J^{\mathrm{(R)}}_{1\rightarrow
2}=-J^{\mathrm{(M)}}_{\mathrm{C}}=J^{\mathrm{(M)}}_{\mathrm{C},3\rightarrow
2},$ (28) $J^{\mathrm{(M)}}_{\mathrm{Ph},2\rightarrow
1}+J^{\mathrm{(M)}}_{\mathrm{D},2\rightarrow
1}=J^{\mathrm{(M)}}_{\mathrm{C},3\rightarrow 2},$ (29)
$J^{\mathrm{(M)}}_{\mathrm{C},3\rightarrow
2}=J^{\mathrm{(M)}}_{\mathrm{Ph},4\rightarrow
3}+J^{\mathrm{(M)}}_{\mathrm{D},4\rightarrow 3},$ (30)
$J^{\mathrm{(M)}}_{\mathrm{Ph},4\rightarrow
3}+J^{\mathrm{(M)}}_{\mathrm{D},4\rightarrow
3}=J^{\mathrm{(M)}}_{\mathrm{H},1\rightarrow 4}.$ (31)

We found the currents are identical everywhere when it operated under the
steady state. This identical current is defined as $J$.

##  IV Results

To demonstrate the performance of this thermal phonon lasing, we run the
simulation under two different cases: the phonon-free case and with lattice
displacement field. In the first case, the coupling between the middle four-
level system and phonon field is neglected ( the coupling strength $g$ is set
to be 0). In phonon coupled case, we take into account the coupling to
demonstrate the amplification of phonon field. Before that, we define an
important physical property associated with lasing action: inversion. The
total inversion is defined as
$(\rho^{\mathrm{(M)}}_{22}-\rho^{\mathrm{(M)}}_{11})+(\rho^{\mathrm{(M)}}_{44}-\rho^{\mathrm{(M)}}_{33})$.
This phonon laser is assumed to be operated under resonance condition
($\varepsilon^{\mathrm{(T)}}_{12}=\varepsilon^{\mathrm{(M)}}_{14}$ and
$\varepsilon^{\mathrm{(B)}}_{12}=\varepsilon^{\mathrm{(M)}}_{23}$), which is
shown as the optimized condition to achieve maximum inversion [18].

In the simulation, parameters are set to be:
$\varepsilon^{\mathrm{(T)}}_{12}=\varepsilon^{\mathrm{(M)}}_{14}=30$
$\mathrm{meV}$,
$\varepsilon^{\mathrm{(B)}}_{12}=\varepsilon^{\mathrm{(M)}}_{23}=25$
$\mathrm{meV}$, $\lambda_{\mathrm{ML}}=\lambda_{\mathrm{MR}}=0.03$
$\mathrm{meV}$, $\gamma_{\mathrm{H}}=\gamma_{\mathrm{C}}=1\mathrm{ps}^{-1}$,
$\gamma_{\mathrm{sys12}}=\gamma_{\mathrm{sys34}}=\gamma_{\mathrm{sys}}$. The
temperatures for the left reservoir (hot bath) and right reservoir (cold bath)
are set to be $400\mathrm{K}$ and $100\mathrm{K}$ correspondingly.

Figure 2: The envelope of lattice displacement fields,
$\langle\hat{b}_{1}\rangle$ (orange) and $\langle\hat{b}_{2}\rangle$ (blue).

Then in the simulation phonon field presented, the parameters associated with
phonons are set to be: $\Gamma=1$ $\mathrm{ps}^{-1}$, $g=2.25$
$\mathrm{ps}^{-1}$, and $u_{0}=20$ $\mathrm{pm}$. We also change the some
parameters to the following value:
$\varepsilon^{\mathrm{(T)}}_{12}=\varepsilon^{\mathrm{(M)}}_{14}=29$
$\mathrm{meV}$,
$\varepsilon^{\mathrm{(B)}}_{12}=\varepsilon^{\mathrm{(M)}}_{23}=25$
$\mathrm{meV}$, $\lambda_{\mathrm{ML}}=\lambda_{\mathrm{MR}}=0.08$
$\mathrm{meV}$, $\gamma_{\mathrm{H}}=\gamma_{\mathrm{C}}=5\mathrm{ps}^{-1}$,
$\gamma_{\mathrm{sys12}}=\gamma_{\mathrm{sys34}}=\gamma_{\mathrm{sys}}=0$. In
Fig. 2, we present the expectation value of the phonon field operators
$\langle\hat{b}_{1}\rangle$ and $\langle\hat{b}_{2}\rangle$ as a function of
time. The initial values of $\langle\hat{b}_{1}\rangle$ and
$\langle\hat{b}_{2}\rangle$ are set to be $10^{-3}$, which are caused by the
vacuum fluctuation of the phonon field. The phonon field
$\langle\hat{b}_{1}\rangle$ and $\langle\hat{b}_{2}\rangle$ increases
dramatically at beginning, therefore shows its ability to amplify the phonon
field. However, the damping is too strong, therefore the amplified field
cannot last for long. We can observe only one pulse in
$\langle\hat{b}_{1}\rangle$, associated with the initial fast increasing of
the population inversion between $|1\rangle$ and $|2\rangle$. The phonon field
$\langle b_{1}\rangle$ is amplified to the maximum simultaneous when the
coupling is turned on. The maximum amplified phonon field is much larger than
initial conditions. However, dynamics of $\langle\hat{b}_{2}\rangle$ is
totally different. There are several pulses in $\langle\hat{b}_{2}\rangle$,
associated with the several increasing of the inversion between $|3\rangle$
and $|4\rangle$. The amplitude of each pulse is twice smaller compared with
$\langle\hat{b}_{1}\rangle$.

Figure 3: Lattice displacement field amplitude - full dependence.

##  V Thermodynamic Analysis

According to the first law of thermodynamics (law of conservation of energy),
the energy of the system changes when it have heat exchanges or power inpur or
output. Therefore heat absorption from hot reservoir is related to the energy
change in left subsystem, written as

$\dot{Q}_{\mathrm{H}}=\varepsilon^{\mathrm{(T)}}_{12}J^{\mathrm{(T)}}_{1\rightarrow
2}$ (32)

This also holds for the heat releasing to the cold reservoir

$\dot{Q}_{\mathrm{C}}=\varepsilon^{\mathrm{(B)}}_{12}J^{\mathrm{(B)}}_{1\rightarrow
2}$ (33)

The dissaption released heat

$\dot{Q}_{\mathrm{D},12}=\varepsilon^{\mathrm{(M)}}_{12}J^{\mathrm{(M)}}_{\mathrm{D},1\rightarrow
2}$ (34)
$\dot{Q}_{\mathrm{D},34}=\varepsilon^{\mathrm{(M)}}_{34}J^{\mathrm{(M)}}_{\mathrm{D},4\rightarrow
3}$ (35)

The heat exchanges accompany with the change of entropy. The total entropy
production of the environment is

$\dot{S}=-\frac{\dot{Q}_{\mathrm{H}}}{T_{\mathrm{H}}}+\frac{\dot{Q}_{\mathrm{C}}}{T_{\mathrm{C}}}+\frac{\dot{Q}_{\mathrm{D},12}}{T_{\mathrm{sys}}}+\frac{\dot{Q}_{\mathrm{D},34}}{T_{\mathrm{sys}}}\geq
0$ (36)

If there is no dissapator

$\dot{Q}_{\mathrm{D},34}=\dot{Q}_{\mathrm{D},12}=0$ (37)

When the phonon lasers is operated under steady-state, we have

$\frac{\varepsilon^{\mathrm{(B)}}_{12}}{\varepsilon^{\mathrm{(T)}}_{12}}\geq\frac{T_{\mathrm{C}}}{T_{\mathrm{H}}}$
(38)

The minimum temperature difference to drive this system can be determined by
the Clausius Inequality (Eq.36)

$\Delta
T\geq(\frac{\varepsilon^{\mathrm{(T)}}_{12}}{\varepsilon^{\mathrm{(B)}}_{12}}-1)T_{\mathrm{C}}$
(39)

We define this minimum temperature as the thermal threshold to achieve lasing
action. Below this threshold, this device cannot generate any phonon output.
The thermal threshold is found to be the linear function of the temperature of
thermal reservoir and the phonon energy, which is consistent with previous
numerical result [18].

Analog to the heat engine, we can define the efficiency of our phonon laser

$\eta_{\mathrm{ideal}}=\frac{\dot{Q}_{\mathrm{H}}-\dot{Q}_{\mathrm{C}}}{\dot{Q}_{\mathrm{H}}}=1-\frac{\varepsilon^{\mathrm{(B)}}_{12}}{\varepsilon^{\mathrm{(T)}}_{12}}\leq
1-\frac{T_{\mathrm{C}}}{T_{\mathrm{H}}}=\eta_{\mathrm{Carnot}}$ (40)

The upper bound is found to be the Carnot Efficiency, which is expected when
this phonon laser is treated as heat engine. While when the dissipation is
consider and the laser is operated under irreversible condition, some
additional term is added to reduce the efficiency.

$\eta=\frac{\dot{Q}_{\mathrm{H}}-\dot{Q}_{\mathrm{C}}-\dot{Q}_{\mathrm{D},12}-\dot{Q}_{\mathrm{D},34}}{\dot{Q}_{\mathrm{H}}}$
(41)

$\eta=1-\frac{T_{\mathrm{C}}}{T_{\mathrm{H}}}-\frac{T_{\mathrm{C}}\dot{S}}{Q_{\mathrm{H}}}-(1-\frac{T_{\mathrm{C}}}{T_{\mathrm{sys}}})\frac{(\dot{Q}_{\mathrm{D},12}+\dot{Q}_{\mathrm{D},12})}{Q_{\mathrm{H}}}\leq\eta_{\mathrm{Carnot}}$
(42)

The partition of the dissipation and the total entropy production contribute
to the reduction of the efficiency. The temperature of the middle system also
have a influence on the efficiency.To increase the temperature of middle
system and reduce the dissipation is able to optimize the efficiency.

##  VI Conclusion

We proposed a four-level phonon laser medium composed of quantum subsystems.
The explicit expression of Hamiltonian of this coupled quantum system have
been presented to illustrate the coupling mechanism between energy filters and
gain medium, and between the gain medium and phonon field. The system has been
described using the density matrix formalism, of which the motion is
determined by Quantum Master Equation within Linblad form. We have shown that
this phonon lasing medium can achieve population inversion as well as
amplification of lattice displacement field. Moreover, the diagonal terms of
Quantum Master Equations have been rewritten by currents and simplified by the
resonance assumption. The verification of this resonance assumption has been
shown as the result of numerical methods we used. Based on the modified
Quantum Master Equation, we conduct a thermodynamics analysis by defining the
heat flow. We found that there is an upper bound of efficiency of this phonon
laser concept, which is known as Carnot efficiency.

## References

  * [1] D. L. Chafatinos, A. S. Kuznetsov, S. Anguiano, A. E. Bruchhausen, A. A. Reynoso, K. Biermann, P. V. Santos, and A. Fainstein, Nature Communications 11, 4552 (2020).
  * [2] R. Wang, P. Täschler, Z. Wang, E. Gini, M. Beck, and J. Faist, ACS Photonics 9, 426 (2022).
  * [3] R. C. Keitel, M. Aellen, B. l. Feber, A. A. Rossinelli, S. A. Meyer, J. Cui, and D. J. Norris, Nano Letters 21, 8952 (2021).
  * [4] L. Jiang, H. Chen, Y. Zeng, Z. Tan, J. Wu, J. Xing, and J. Zhu, ACS Applied Materials & Interfaces (2022).
  * [5] Y. Gu, C. Chen, J. Rufo, C. Shen, Z. Wang, P.-H. Huang, H. Fu, P. Zhang, S. A. Cummer, Z. Tian, and T. J. Huang, ACS Nano 14, 14635 (2020).
  * [6] V. M. Jooss, J. S. Bolten, J. Huwyler, and D. Ahmed, Science Advances 8, eabm2785 (2022).
  * [7] S. Yves, M. I. N. Rosa, Y. Guo, M. Gupta, M. Ruzzene, and A. Alù, Sci. Adv. 9, e2200181 (2022).
  * [8] Y. Zeng, J. Hao, J. Zhang, L. Jiang, S. Youn, G. Lu, D. Yan, H. Kang, Y. Sun, K. K. Shung, K. Shen, and Q. Zhou, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 69, 1889 (2022).
  * [9] B. Gu and I. Franco, Phys. Rev. A 98, 063412 (2018).
  * [10] O. S. Ojambati, K. B. Arnardottir, B. W. Lovett, J. Keeling, and J. J. Baumberg, Few-emitter lasing in single ultra-small nanocavities, 2021.
  * [11] R. McKenna, S. Corbett, S. T. Naimi, D. Mickus, D. McCloskey, and J. F. Donegan, IEEE Journal of Selected Topics in Quantum Electronics 28, 1 (2022).
  * [12] P. Karwat, G. Pasławski, P. Damery, Y. Yang, F. Bello, and O. Hess, Near-field plasmonics for generation of phonon lasing in a thermal nanomachine, 2021.
  * [13] C. Chamon, E. R. Mucciolo, L. Arrachea, and R. B. Capaz, Phys. Rev. Lett. 106, 135504 (2011).
  * [14] M. Hasegawa and T. Kato, Journal of the Physical Society of Japan 87, 044709 (2018).
  * [15] M. Gluza, J. a. Sabino, N. H. Ng, G. Vitagliano, M. Pezzutto, Y. Omar, I. Mazets, M. Huber, J. Schmiedmayer, and J. Eisert, PRX Quantum 2, 030310 (2021).
  * [16] H. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
  * [17] J. Wang, Y. Lai, Z. Ye, J. He, Y. Ma, and Q. Liao, Phys. Rev. E 91, 050102 (2015).
  * [18] P. Karwat, D. E. Reiter, T. Kuhn, and O. Hess, Phys. Rev. A 98, 053855 (2018).