# Lasing in a ZnO waveguide: clear evidence of polaritonic gain obtained by
monitoring the continuous exciton screening.

Geoffrey Kreyder  Institut Pascal, Université Clermont Auvergne, CNRS,
Clermont INP, 63000 Clermont-Ferrand, France  Léa Hermet  Institut Pascal,
Université Clermont Auvergne, CNRS, Clermont INP, 63000 Clermont-Ferrand,
France  Pierre Disseix  Institut Pascal, Université Clermont Auvergne, CNRS,
Clermont INP, 63000 Clermont-Ferrand, France  François Médard
<EMAIL_ADDRESS>Institut Pascal, Université Clermont Auvergne, CNRS,
Clermont INP, 63000 Clermont-Ferrand, France  Martine Mihailovic  Institut
Pascal, Université Clermont Auvergne, CNRS, Clermont INP, 63000 Clermont-
Ferrand, France  François Réveret  Institut Pascal, Université Clermont
Auvergne, CNRS, Clermont INP, 63000 Clermont-Ferrand, France  Sophie Bouchoule
Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay,
91120 Palaiseau, France  Christiane Deparis  Université Côte d’Azur, CNRS,
CRHEA, rue Bernard Gregory, Sophia Antipolis, 06560 Valbonne, France  Jesús
Zuñiga-Pérez  Université Côte d’Azur, CNRS, CRHEA, rue Bernard Gregory, Sophia
Antipolis, 06560 Valbonne, France  Majulab, International Research Laboratory
IRL 3654, CNRS, Université Côte d’Azur, Sorbonne Université, National
University of Singapore, Nanyang Technological University, Singapore,
Singapore  Joël Leymarie  Institut Pascal, Université Clermont Auvergne, CNRS,
Clermont INP, 63000 Clermont-Ferrand, France

###### Abstract

The coherent emission of exciton-polaritons was proposed as a means of
lowering the lasing threshold because it does not require the dissociation of
excitons to obtain an electron-hole plasma, as in a classical semiconductor
laser based on population inversion. In this work we propose a method to prove
clearly the polaritonic nature of lasing by combining experimental
measurements with a model accounting for the permittivity change as a function
of the carrier density. To do so we use angle resolved photoluminescence to
observe the lasing at cryogenic temperature from a polariton mode in a zinc
oxide waveguide structure, and to monitor the continuous shift of the
polaritonic dispersion towards a photonic dispersion as the optical intensity
of the pump is increased (up to 20 times the one at threshold). This shift is
reproduced thanks to a model taking into account the reduction of the
oscillator strength and the renormalization of the bandgap due to the
screening of the electrostatic interaction between electrons and holes.
Furthermore, the measurement of the carriers lifetime at optical intensities
in the order of those at which the polariton lasing occurs enables us to
estimate the carrier density, confirming that it is lower than the
corresponding Mott density for zinc oxide reported in the literature.

##  I Introduction

First introduced by Hopfield [1] and Agranovich [2] to describe the light
propagation in bulk semiconductors, exciton-polaritons have been extensively
investigated since 1992 in vertical microcavities [3]. Exciton-polaritons,
hereafter referred to as polaritons, result from the strong coupling of
photons and excitons, which are electron-hole pairs in Coulombic interaction.
Contrary to the particles involved in an electron-hole plasma, polaritons are
bosons and when their density exceeds a critical value, they condensate into a
single state which emits coherent light, resulting in polariton lasing [4].
The working mechanism is not the stimulated emission of photons as in
conventional lasers, but the stimulated polariton scattering induced by the
final state occupation [5]. This process is predicted to occur at a much lower
particle density [6] with a smaller gain length than the half cavity length
required in a conventional laser [7]. This has promoted polariton lasers as
promising candidates for low-power coherent sources compatible with large
packing densities and, thus, they are particularly adapted to the field of
optical interconnections where a drastic reduction of the energy-consumption
is required [8]. However, the maximum operation temperature of a polariton
laser depends on the exciton stability, which is material dependent. Zinc
oxide (ZnO) is a wide-bandgap semiconductor with stable and robust excitons
for which polariton lasing has been demonstrated up to room temperature [9].
It is thus an attractive material for realising polariton lasers, provided
that the carrier density is kept below the Mott density, which corresponds to
the dissociation of all excitons into unbound electrons and holes [10].

Waveguides, which confine photons through total internal reflection, represent
a geometry for polariton lasers alternative to the most standard vertical one
[11]. Polaritonic waveguides combine numerous advantages: (i) inherent very
low radiative losses, given that they operate at wave vectors outside the
light cone, (ii) a significant spatial overlap between the excitonic medium
and the electromagnetic field, and (iii) an eased fabrication, as they consist
only of a semiconductor slab embedded between two optical cladding layers.
Strong coupling in waveguides has been first demonstrated in structures based
on J-aggregates [12] and gallium arsenide [11]. However, these materials are
only suitable for low power or low temperature applications. Recently, other
groups have reported promising results for future devices based on guided
polaritons working at room temperature: polariton lasing and amplification in
ZnO [13], long-range coherent polariton condensate flow in a halide perovskite
waveguide [14], ultrafast polariton modulation with GaN quantum wells [15] and
also polariton lasing in GaN-based circular guides [16].

In this work we propose a method to clearly evidence the polaritonic nature of
lasing. To do so we have measured accurately the continuous shift of the
polariton dispersion in a ZnO-based waveguide at cryogenic temperature for
different pumping intensities, from below lasing threshold to far above lasing
threshold. The experimental results are confronted with a modelling of the
permittivity accounting for the progressive screening of bound electron-hole
pairs. As the optical intensity of the pump is increased, the laser energy
moves from the polaritonic dispersion, whose shape is mostly controlled by
excitonic contributions to the dielectric function, towards the dispersion of
the photonic waveguide mode (for intensities 20 times larger than the
polariton lasing threshold). This observation is explained because of the
screening of the electrostatic Coulomb interaction, leading to a reduction of
the oscillator strength and a bandgap renormalisation, thus modifying the
complex permittivity of the guiding medium. The observed shift of the
dispersion to smaller wave vectors, as predicted by our model, enables us to
determine a reduction by 22% of the oscillator strength at the polariton
lasing threshold. We also perform time-resolved photoluminescence to determine
the decay time of carriers allowing us to calculate a carrier density at
threshold of $\mathrm{3\times 10^{17}\,cm^{-3}}$. A careful comparison with
the Mott density value obtained from the literature confirms the existence of
polaritons at such a density.

##  II Sample structure and optical setup

The structure investigated in the present study consists of a ZnO based
optical waveguide (50 nm thick) grown by molecular beam epitaxy on a
$\mathrm{Zn_{0.72}Mg_{0.28}O}$ buffer layer (1000 nm) deposited onto an
m-plane ZnO substrate [17]. A $\mathrm{Zn_{0.72}Mg_{0.28}O}$ upper cladding
(100 nm) was added to prevent surface recombinations and to obtain a better
confinement of the electromagnetic field. A sketch of the structure is
displayed in Fig.1a. During the epitaxial growth, cracks are naturally induced
by the mismatch of both the lattice parameters and the thermal expansion
coefficients of ZnO and $\mathrm{Zn_{0.72}Mg_{0.28}O}$. Cracks form
perpendicular to the in-plane c-axis of the crystal and are irregularly
spaced: in the current sample the typical distance between two cracks lies
between 5 to 40 µm typically (see the optical microscopy image of the sample
in Fig.1b. Due to the refractive index contrast between ZnO and air, cracks
can act as mirrors and thus two parallel cracks form a Fabry-Perot resonator.
As displayed on Fig.1b, a $\mathrm{SiO_{2}}$ grating ($\mathrm{100\times
100\,\mu{}m^{2}}$) with a $\Lambda$=190 nm period was elaborated by electron
beam lithography on a hydrogen silsesquioxane (HSQ) resin on top of the
structure to enable the outcoupling of the guided waves. Due to the
difficulties to perfectly align the mask along the crystallographic axes,
there is a small twist angle between the grating and the c-axis of about
$4^{\circ}$. Consequently, strictly speaking we measure the projection of the
wave vector along z, but this results in a negligible variation.

Figure 1: (a) Sketch of the sample structure with the definition of axes used
in the calculation of guided modes, the emission angle $\theta$ at the surface
and the period $\Lambda$ of the grating. The ZnO guiding layer is 50 nm thick
with the propagation constant $\beta$ depicted in yellow. The square modulus
of the electric field for the TE0 mode is plotted in orange. (b) Image of the
sample surface obtained using an optical microscope showing a
$\mathrm{100\times 100\,\mu{}m^{2}}$ diffraction grating (dark square) and
cracks perpendicular to the c-axis (purple arrow) as darker lines. The twist
angle between the grating and the in-plane c-axis is smaller than
$\mathrm{4^{\circ}}$ leading to a negligible error on the wave vector
measurement.

The Bragg’s diffraction law relates the emission angle $\theta$ outside the
sample to the propagation constant $\beta$ of the mode:

$\beta=\frac{2\pi}{\lambda_{0}}\,sin(\theta)-m\,\frac{2\pi}{\Lambda}\quad\mathrm{with}\quad
m\in\mathbb{Z}$ (1)

The dispersion of the guided polaritons is thus measured by Fourier-space
spectroscopy combined with a real-space filtering. The emission of the sample
was collected through a 100x NUV microscope objective with a 0.5 numerical
aperture and its Fourier plane (focal plane) was imaged onto the spectrometer
slit by two spherical lenses. The spectrometer has a 1 m focal length and a
1200 grooves per mm grating to obtain a resolution better than 0.05 nm. For
spatial filtering a pinhole was placed in the focal plane of the first lens;
the pinhole diameter of 0.8 mm corresponds to an investigated surface of 200
$\mathrm{\mu{}m^{2}}$ on the sample. This spatial filtering enabled us to
select the light emitted only from a small area of the grating coupler
deposited on the sample but also to avoid, as much as possible, collecting
light from adjacent cavities. A Q-Switched laser emitting at 266 nm with a 400
ps pulse duration and 20.6 kHz repetition rate was used for the excitation. We
have used a side excitation, focusing the laser through an aspherical lens
with a spot surface of 1220 $\mathrm{\mu{}m^{2}}$.

For the time resolved photoluminescence (TRPL), the waveguide is excited using
the third harmonic of a titanium-sapphire laser with a pulse duration of 150
fs. After being spectrally dispersed by a 600 groove per mm grating, the
luminescence is temporally analysed using a streak-camera.

##  III Numerical calculation of guided modes dispersion and permittivity
model to account for varying carrier density

In order to analyse the experimental results, guided modes in the waveguide
are calculated by solving Maxwell’s equations with proper boundary conditions.
In the region between two cracks, the structure is invariant by translation
along y and z axes (see Fig.1a) and the propagation equation for transverse
electric (TE) polarisation modes writes:

$\begin{split}\frac{d^{2}E_{y}(x)}{dx^{2}}+(k_{0}^{2}\epsilon_{r}-\beta^{2})\,E_{y}(x)=0\\\
\mathrm{with}\quad\vec{E}=E_{y}(x)\,e^{(i(\omega{}t-\beta{}z))}\,\vec{u}_{y}\end{split}$
(2)

where $k_{0}$ is the vacuum wave vector and $\beta$ the propagation constant
in the waveguide which becomes complex when losses are taken into account
through the permittivity $\epsilon_{r}$. The continuity of transverse
components of the electric and magnetic fields at each interface leads to an
implicit equation whose resolution is carried out in the complex plane to
determine the propagation constant $\beta$ as a function of the energy E. We
limit our study to the TE0 mode, for which the electric field confinement is
the strongest.

The complex permittivity of ZnO is based on the value measured by
spectrometric ellipsometry at 300 K [18, 19]. We have modelled the excitonic
transitions with harmonic oscillators and removed them to obtain a background
value $\epsilon_{b}$ that retains only the contribution of both band-to-band
and higher energies transitions. At low temperature, the background
permittivity is blueshifted by 70 meV corresponding to the bandgap change from
300 K to 5 K [20]. Then, the permittivity can be expressed using the
transition energy $E_{j}$, the broadening $\gamma_{j}$ and the oscillator
strength $f_{j}$ of ZnO excitons as follows:

$\epsilon(E)=\epsilon_{b}(E)+\sum_{j=A,B}\frac{f_{j}}{E_{j}^{2}-E^{2}+i\gamma_{j}E}$
(3)

At low temperature and low density, we used excitonic parameters from the work
by Mallet et al [21]. On Fig.2, the light cone of
$\mathrm{Zn_{0.72}Mg_{0.28}O}$ is also represented in light blue using a
constant optical index of 1.9 [22] and the photonic TE0 mode (purple dashed
line) is determined when the excitonic contribution is removed ($f_{j}$ = 0)
from the permittivity. An example of the calculated dispersion curve for the
waveguide at a temperature of 5 K is displayed as a purple line in Fig.2,
evidencing clearly the polaritonic nature of the guided mode with lower and
upper polariton branches being separated by a Rabi splitting $\Omega$ of 134
meV. This value is smaller than the previously published one [13] as the model
used in our first work coupled a linear dispersion of the TE0 mode (with a
constant effective refractive index) with a harmonic oscillator. It has been
demonstrated in reference [23] that a coupled oscillator model strongly
overestimates the deviation of the polaritonic dispersion from the photonic
one. Moreover, this is a crude model as it does not account for the dispersive
nature of the refractive index and its variation as a function of the carrier
density. Thus a clear demonstration of the polaritonic nature of the mode, in
the absence of the upper polariton branch, can only be obtained by monitoring
the shift of its dispersion from a polaritonic one to a photonic one,
following thereby the Coulomb potential screening of excitons up to the Mott
transition.

Figure 2: Numerical calculation of the guided polariton dispersion in purple
showing a Rabi splitting of 134 meV between the lower and upper branches. The
TE0 photonic mode (light purple dashed line) is obtained using only the
background permittivity of ZnO without excitonic contributions. The
$\mathrm{Zn_{0.72}Mg_{0.28}O}$ cone (light blue) corresponds to a linear
dispersion with a constant refractive index of 1.9. The red dotted line
indicates the mean energy of the excitonic transitions.

As the carrier density increases, the Coulomb interaction between electrons
and holes is progressively screened leading to the vanishing of excitons at
the Mott density. Both experimental and theoretical results in bulk
semiconductors have shown that the exciton energy does not depend on the
carrier density [24, 25]. This is explained by a compensation between the
reduction of the exciton binding energy $\mathrm{E_{b}}$ and the shrinkage of
the bandgap. We propose to model the effect of the carrier density using only
one parameter $\eta$, which represents the percentage of oscillator strength
remaining for a given carrier density n:

$\eta=\frac{f(n)}{f(n=0)}$ (4)

As both the oscillator strength and the binding energy are linked to the
exciton Bohr radius, we can express the permittivity as a function of $\eta$:

$\begin{split}\epsilon(E)=\epsilon_{b}\left(E-E_{b}\times\left[1-\eta^{1\textfractionsolidus
3}\right]\right)\\\
+\sum_{j=A,B}\frac{\eta\,f_{j}}{E_{j}^{2}-E^{2}+i\gamma_{j}E}\end{split}$ (5)

The first term of the equation accounts for the shrinkage of the bandgap and,
thus, the spectral of the background contribution, while the second term
concerns the reduction of the amplitude of the excitonic transition due to the
screening of excitons by carriers. The term $[1-\eta^{1\textfractionsolidus
3}]$ comes from the fact that the excitonic binding energy is proportional to
$a_{B}^{-1}$ where $a_{B}$ is the excitonic Bohr radius, while the oscillator
strength is proportional to $a_{B}^{-3}$ [26].

##  IV Experimental measurement of the polariton dispersion as a function of
optical pumping intensity

Figure 3: Photoluminescence spectra recorded at low temperature
($\mathrm{T_{sample}\,=\,5K}$) for optical intensities increasing from 16.4
$\mathrm{kW.cm^{-2}}$ to 96.4 $\mathrm{kW.cm^{-2}}$ (left panel); the
integrated intensity is plotted as a function of the optical intensity of the
pump power (right panel) for the polaritonic emission (pink, 3.31 to 3.345 eV)
and for excitonic transitions (green, 3.345 to 3.40 eV) ; results are shown on
a log-log scale to evidence the superlinear increase for polaritons (p=2.1)
and the supralinear one for excitons (p=0.4) above the threshold at
$\mathrm{D_{t}h}$ = 30.6 $\mathrm{kW.cm^{-2}}$.

The photoluminescence of the ZnO waveguide has been recorded at low
temperature ($\mathrm{T_{sample}\,=\,5\,K}$). The spectra are shown on the
left panel of Fig.3 for various optical intensities. For the lowest intensity
($\mathrm{D\,=\,16.4\,kW.cm^{-2}}$), the spectrum is dominated by a broad
emission centered at 3.36 eV corresponding to the emission of both donor-bound
and free excitons. The broadening is due to strain fluctuation in the
structure as further discussed below. As the optical intensity is increased,
another peak appears at lower energy, which progressively resolves into a comb
of sharp lines: these lines are successive modes of the Fabry-Pérot cavity
formed by two consecutive cracks. Their intensity increases clearly faster
than the excitonic emission. To prove unambiguously the nonlinearity of the
emission, we have plotted on the right side of Fig.3 the integrated intensity
which is proportional to the light emitted, versus the optical pumping
intensity D. Above a threshold at $\mathrm{D_{th}\,=\,30.6\,kW.cm^{-2}}$, the
area of the lower energy peak (integrated between 3.31 and 3.345 eV, pink)
increases as a power law with an exponent of 2.1, whereas the excitonic
emission (integrated between 3.345 and 3.340 eV, green) increases only as
$\mathrm{D^{0.4}}$. Indeed, the emission attributed to polaritons is
superlinear while the excitonic one is supralinear, associated to an excitonic
reservoir depletion that will be considered in the discussion section. Note
that an extensive study of lasing in the same structure, with more
experimental points, has already been published in reference [13].

The dispersion relation of the ZnO waveguide is also investigated
experimentally at the same cryogenic temperature ($\mathrm{T_{sample}=5K}$)
for optical intensities ranging from 2.9 $\mathrm{kW.cm^{-2}}$ to 636
$\mathrm{kW.cm^{-2}}$. The emission is recorded as a function of the
wavelength in ordinates and, thanks to the dispersion grating, with an angular
resolution in abscissa. The results are shown in a false colours logarithmic
scale on Fig.4 for four selected intensities. On each panel, the calculated
dispersions are also plotted using various percentage of oscillator strength:
$\eta$=1 corresponds to the low-density regime (white dashed line), $\eta$=0
accounts for the permittivity at the Mott density (white dashed dot line) and
the best fit to the experimental dispersion using a variable $\eta$ is
depicted in light purple.

Figure 4: Experimental dispersion curves of the polaritonic guided mode at
$\mathrm{T_{sample}=5K}$ as a function of the optical intensity of the
excitation laser. Calculated dispersions are also plotted using a variable
parameter $\eta$ (percentage of oscillator strength) accounting for the
exciton screening. The white dashed curve correspond to $\eta$=1, the white
dotted one to $\eta$=0 and the magenta lines are best fits to the experimental
data. Each panel corresponds to a different optical intensity: (a) 2.9
$\mathrm{kW.cm^{-2}}$, (b) 31.8 $\mathrm{kW.cm^{-2}}$ corresponding to lasing
threshold, (c) 95.4 $\mathrm{kW.cm^{-2}}$ and (d) 636 $\mathrm{kW.cm^{-2}}$.

For the lowest optical intensity (Fig.4a – 2.9 $\mathrm{kW.cm^{-2}}$), a
strong dispersionless emission is observed at energies between 3.35 eV and
3.38 eV. It corresponds to the photoluminescence of uncoupled excitons whose
emission cannot be guided in the waveguide and are thus emitted vertically.
This luminescence is mainly dominated by two broad peaks at 3.3765 eV and
3.3605 eV: they are attributed respectively to free excitons and neutral donor
bound excitons, in agreement with reference [20] for unstrained ZnO. The
significant broadening originates from large strain variations in the area
between the cracks, as we spatially integrate the emission from excitons
emitting close and far from the cracks. The presence of excitons confirms that
the carrier density is well below the Mott density. At lower energy, a
dispersive feature corresponds to the emission of the guided polaritons. The
white dashed line is the best fit of the experimental data: it allows us to
fix the physical parameters of the structure assuming the oscillator strength
is not affected at the lowest excitation ($\eta$=1). The dispersion relation
of the waveguide is also reported at the Mott density ($\eta$=0, dotted curve)
for comparison.

As the excitation is increased (Fig.4b – 31.8 $\mathrm{kW.cm^{-2}}$, 4c – 95.4
$\mathrm{kW.cm^{-2}}$ and 4d – 636 $\mathrm{kW.cm^{-2}}$), the dispersion
curve shifts continuously towards the low emission angle, corresponding to
lower propagation constants for a given energy. This is due to screening
effects as the charge carrier density is increased: bandgap renormalization
and oscillator strength reduction both lead to a lower permittivity value,
which blueshifts the guided mode dispersion (at constant wavevector). This
behaviour is well accounted for by our model if we decrease the percentage of
oscillator strength $\eta$ from 1 to 0.78 (Fig.4b), 0.66 (Fig.4c) and 0.48
(Fig.4d) as it can be seen from the agreement between the fit (purple lines)
and the experimental data.

Another interesting phenomenon appears on Fig.4b, where discrete peaks appear
superimposed on top of the polaritonic dispersion: they correspond to the
resonances of a Fabry-Pérot cavity formed by two cracks that become visible as
the gain overcomes the losses. This is a clear evidence of lasing which it is
corroborated by the nonlinear increase of the integrated intensity as a
function of the excitation shown on Fig.3. The peaks are aligned on the
calculated dispersion corresponding to $\eta$=0.78: this constitutes a
conclusive proof of the polaritonic nature of the lasing mechanism. The
nonlinear emission is still visible at higher power densities (Fig.4c and 4d)
but the peaks become broader as other adjacent cavities with different sizes
contribute to the collected emission. Simultaneously, the energy of the lasing
modes decreases, as predicted by theoretical calculations based on semi-
classical Boltzmann equations by Solnyshkov et al [27]. This behaviour can be
explained by an enhanced polariton relaxation along the lower polariton branch
due to increased polariton-polariton interactions. At 636
$\mathrm{kW.cm^{-2}}$, an optical pumping intensity corresponding to 20 times
the one at threshold, the dispersion curve comes even closer to the $\eta$=0
curve (dotted white line). The uncertainty on the permittivity used in the
calculations as well as the smaller energy difference between the photonic and
the polaritonic dispersions at this energy, do not allow us to conclude any
longer whether the lasing is based on polaritons or on an electron-hole
plasma. We have to mention that on the last two figures, a blurred emission is
detected at high angles and away from any polaritonic dispersion (at angles
between $\mathrm{20^{\circ}}$ and $\mathrm{30^{\circ}}$): this is due to the
diffraction of the undispersive emission from the cracks on the pinhole used
to spatially select the emission.

##  V Discussion on the lasing regime

The polariton laser is fed from a reservoir of uncoupled excitons with large
wave vectors. We propose to determine now the density of carriers n in this
reservoir. To do so, we have measured the decay time of the excitonic
transitions that do not couple to the waveguide and that can be recorded in
the luminescence spectrum emitted from the surface. Time resolved measurements
are shown on Fig.5 for two different optical intensities corresponding to n in
the range of $\mathrm{10^{15}}$ to $\mathrm{5\times 10^{16}\,cm^{-3}}$, as
detailed below. To obtain the decay time of carriers in the reservoir (both
free excitons and eventually ionized electron hole pairs, as explained later),
we have measured the photoluminescence decay of the waveguide with a temporal
resolution smaller than 5 ps and integrated over an energy range from 3.3765
eV (free A exciton energy from Fig.4a) up to 3.41 eV (3 times
$\mathrm{k_{B}T_{e}}$ with $\mathrm{T_{e}\sim 130K}$ the excitonic temperature
at lasing threshold). We have plotted the natural logarithm of the
luminescence intensity I subtracted from the background $\mathrm{I_{0}}$ to
unambiguously prove the monoexponential decay. The lifetime of the carriers
appears not to depend on the carrier density and we will consequently use a
mean value $\tau$=40 ps for calculations.

Figure 5: Time resolved photoluminescence of the free excitonic transition
(3.3765 $\pm$ 0.0025 eV), and eventually ionized electron-hole pairs (energy
up to 3.41 eV), for a carrier density n=$\mathrm{2.3\times 10^{15}\,cm^{-3}}$
(green squares) and n=$\mathrm{4.7\times 10^{16}\,cm^{-3}}$ (blue circles).
The ordinate axis corresponds to the natural logarithm of the intensity to
evidence the mono-exponential decay. The green and blue dotted lines are
guides for the eyes.

The carrier density can be easily determined for these measurements using the
following formula, where the laser pulse is assumed to be a Dirac delta
function:

$n=\frac{E_{pulse}}{E_{laser}}\,\times\,\frac{\alpha_{ZnO}}{S}$ (6)

We define $E_{pulse}$ the energy per pulse transmitted to the ZnO layer,
$\mathrm{S\,=\,2000\,\mu{}m^{2}}$ the surface of the spot and $E_{laser}$ =
4.66 eV the energy of the excitation laser. The part of the photons absorbed
by the $\mathrm{Zn_{0.72}Mg_{0.28}O}$ cladding is taken into account using
$\mathrm{\alpha_{ZnMgO}\,=\,1.1\,\times\,10^{5}\,cm^{-1}}$ [22] as well as the
transmission coefficients of the interfaces between each layer. The generated
carriers are assumed to be not transferred from the
$\mathrm{Zn_{0.72}Mg_{0.28}O}$ cladding to the ZnO guide as they are mostly
trapped into alloy fluctuations at low temperature. As the ZnO thickness is
less than the optical absorption length [28], the relevant thickness to
consider for the evaluation of n is 1/$\mathrm{\alpha_{ZnO}}$ with
$\mathrm{\alpha_{ZnO}=1.9\,\times\,10^{5}\,cm^{-1}}$ [29]. Using the above
formula, we can estimate that the carrier density is varying from
$\mathrm{2.3\times 10^{15}\,cm^{-3}}$ for the green curve in Fig.5 to
$\mathrm{4.7\times 10^{16}\,cm^{-3}}$ for the blue one.

Coming back to the dispersion measurement of Fig.4, the pulse duration of the
laser (400 ps) is large enough compared to the carrier lifetime (40 ps) and we
can use the quasi-continuous approximation. The carrier density is now written
as:

$n=\frac{D}{E_{laser}}\,\times\,\tau\,\times\,\alpha_{ZnO}$ (7)

We can thus deduce a carrier density
$\mathrm{n_{th}=3\,\times\,10^{17}\,cm^{-3}}$ for an optical pumping intensity
$\mathrm{D\,=\,30.6\,kW.cm^{-2}}$ corresponding to the polariton lasing
threshold. This value has to be compared to the Mott density
$\mathrm{n_{Mott}}$ from the literature. Using the static screening model
developed by Versteegh et al in reference [30], we were able to calculate
$\mathrm{n_{Mott}=7\,\times\,10^{17}\,cm^{-3}}$ for an excitonic temperature
of 130K, which is determined under the high excitation conditions. This value,
compared to $\mathrm{n_{t}h}$, further confirms the polaritonic nature of the
laser.

As the carrier density is increased (see Fig.4c and 4d), we expect to reach
the Mott density; however, this phenomenon is not clearly observed, even at
636 $\mathrm{kW.cm^{-2}}$. One possible process explaining this, at least
partially, is the depletion of the excitonic reservoir by the stimulated
scattering of polaritons to the final state, its population being clamped
above the polariton lasing threshold [31]. Indeed, we have demonstrated on
Fig.3 that the excitonic emission increases only with an exponent 0.4 above
threshold. Thus, a pumping intensity 20 times larger than the one at threshold
should lead to an increase of the carrier density by only a factor of 3.3. The
optical intensity of 636 $\mathrm{kW.cm^{-2}}$ would thus correspond to a
carrier density of $\mathrm{10\,\times\,10^{17}\,cm^{-3}}$, close to the Mott
transition in zinc oxide. This estimation is consistent with our inability to
discriminate experimentally, at this large pumping intensity, the nature of
the lasing gain (Fig.4d).

##  VI Conclusion

We have studied at cryogenic temperature a ZnO-based waveguide that supported
guided polariton modes. Thanks to outcoupling gratings deposited on the
surface of the structure, the dispersion curve of the TE0 mode has been
experimentally measured using Fourier space spectroscopy to prove its
polaritonic nature. The numerical resolution of Maxwell equations allows to
compare the measured dispersions with a purely photonic one, where the
excitonic contributions to the permittivity have been precisely removed,
showing a significant difference. As the power density of the optical
excitation is increased, a nonlinear emission is observed at a threshold of
30.6 $\mathrm{kW.cm^{-2}}$, with a small shift of the dispersion. A model has
been developed to account for carrier screening effects in the excitonic
reservoir through the reduction of the oscillator strength and bandgap
renormalization. It demonstrates that the lasing is of polaritonic nature as
the oscillator strength is only reduced by 22% at threshold. The lifetime of
carriers ($\tau$=40 ps) in the reservoir has been determined through time
resolved photoluminescence and it is used to compute the carrier density at
threshold. The value of $\mathrm{n_{th}=3\,\times\,10^{17}\,cm^{-3}}$ is found
to be lower than the Mott density reported in the literature, corroborating
that the emission originates from polaritons.

The complex permittivity model accounting for excitonic screening used in this
work, combined to the measurement of the polaritonic dispersion as a function
of the optical intensity, can be used for other polaritonic systems containing
different materials, even at high temperatures. Our description of the
permittivity may be improved with a more formal description of the interacting
electron-hole gas as described in the literature. Doing so, we could also
access to the whole dielectric function below and above the Mott transition to
simulate the gain from the electron-hole plasma and to predict the transition
to a classical laser.

###### Acknowledgements.

The authors acknowledge funding from the French National Research Agency
(ANR-16-CE24-0021 PLUG-AND-BOSE)

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††institutetext: Center for Gravitation and Cosmology, College of Physical
Science and Technology, Yangzhou University, 180 Siwangting Road, Yangzhou
City, Jiangsu Province 225002, China

# Distinguishing pure and thermal states by Landauer’s principle in open
systems

Hao Xu<EMAIL_ADDRESS>
###### Abstract

Starting from Polchinski’s thought experiment on how to distinguish between
pure and thermal states, we construct a specific system to study the
interaction between qubit and cavity quantum field theory (QFT) in order to
provide a more operational point of view. Without imposing any restrictions on
the initial states of qubit and cavity QFT, we compute the evolution of the
system order by order by the perturbation method. We choose Landauer’s
principle, an important bound in quantum computation and quantum measurement,
as the basis for the determination of the thermal state. By backtracking the
initial state form, we obtain the conditions that must be satisfied by the
cavity QFT: the expectation value of the annihilation operator should be zero,
and the expectation value of the particle number operator should satisfy the
Bose-Einstein distribution. We also discuss the difference between the thermal
state and a possible alternative to the thermal state: the canonical thermal
pure quantum (CTPQ) state.

##  1 Introduction

Prepare two sealed containers, each containing an equal amount of ice at
absolute zero. We bring one of the containers into thermal equilibrium with a
heat source large enough to reach a final temperature of 400K. The other
container is heated with a laser to reach the same energy as the first
container. The ice in both containers is now vaporized. The ice in the first
container in contact with the large heat source ends up in a thermal
equilibrium state, which we define as the canonical thermal state, while the
ice in the second container is in a pure state. We have both containers in
front of you, can you tell which is which?

This is a thought experiment proposed by Polchinski in his review of the black
hole information loss problem Polchinski:2016hrw . Polchinski believes that
the thought experiment may shed some light to the black hole information loss
paradox, because the steam will collapse into the black hole if we heat both
containers even more. In AdS/CFT we can correspond bulk physics, such as black
hole formation and evaporation, to the boundary conformal field theory. For an
AdS black hole, its event horizon changes the structure of spacetime. For
example, when we compute the entanglement entropy of a region $A$ on the
boundary field theory using the Ryu-Takayanagi formula Ryu:2006bv ; Ryu:2006ef
, we find that the minimal surfaces corresponding to $A$ and the its
complement $\bar{A}$ are different, and they are separated by the event
horizon of the black hole. This also shows that the boundary field theory
should be a mixed state. However, if the black hole has not yet formed or has
evaporated, there is no event horizon. The minimal surfaces corresponds to $A$
and $\bar{A}$ coincide, and the field theory is still in a pure state. See
e.g. Figure 1. Understanding the black hole information loss paradox requires
a better knowledge of the relationship between pure and mixed states, as well
as the problem of unitarity during black hole evolution Danielsson:1999fa ;
Takayanagi:2010wp ; Roy:2015pga .

Figure 1: The minimal surfaces for the holographic calculation of entanglement
entropy in the AdS spacetime. For the AdS black hole we have
$\gamma_{A}\neq\gamma_{\bar{A}}$ (left), while in the case of black hole
formation, we can find that $\gamma_{A}=\gamma_{\bar{A}}$ (right). This is
because the collapsing shell (dashed circle) has not yet formed the event
horizon.

In the review, Polchinski argues that it is impossible to distinguish between
pure and mixed states. Polchinski’s point is that if we know the exact form of
a pure state, we can act on it directly with the projection operator to get
identity, whereas acting on a mixed state gives us essentially zero. However,
we do not know how the pure state is prepared, so the only thing we can do to
improve accuracy is to prepare many copies and thus take multiple
measurements.

From this point of view, Polchinski’s conclusion is undoubtedly correct.
However, when we return to an operational way of thinking about the
distinction between pure and thermal states, we encounter new confusions.
First, the definition of a thermal state is precise. In principle, the density
matrix of the thermal state is uniquely determined whenever we are given the
Hamiltonian and the temperature of the system. However, there seems to be a
wider range of choices in the pure states, which is the heart of Polchinski’s
argument that we do not know how the pure state is prepared. This leads to
confusion as to why it is so difficult for us to distinguish a rigorously
well-defined system. Second, we need to define the notion of
“distinguishable”. From an operational point of view, to distinguish means to
make measurements. Therefore, the theory of open quantum systems is the
relevant theoretical framework for our research Zeh1970 ; RevModPhys.76.1267 ;
Breuer . The fact that two systems are indistinguishable implies that the same
result is obtained for all possible physical measurements.

We now ask the question: _what results do we get when we use a detector to
probe separately a thermal state and a pure state that is supposedly
indistinguishable from the thermal state(if such pure state exists)?_
Fortunately, we do already have a partial result, namely Landauer’s principle
landauer1961 ; landauer1996 . Landauer’s principle relates the entropy change
of a system to the heat dissipated into a reservoir during any logically
irreversible computation, providing a theoretical limit of energy consumption
throughout the process. It gives a direct link between information theory and
thermodynamics, and establishes that _information is physical_. In 2013, after
a series of controversial discussions, Reeb and Wolf proposed a general and
minimal setup to tighten Landauer’s principle by a quantum statistical physics
approach that need to be satified at the same time Reeb2013 : (i) both the
“system” $S$ and “reservoir” $R$ are described by Hilbert spaces, (ii) $R$ is
initially in a thermal state, (iii) $S$ and $R$ are initially uncorrelated,
and (iv) the process proceeds by unitary evolution. If all the four
assumptions are satisfied, the Landauer’s principle can be expressed as

$\Delta Q\geqslant T_{R}\Delta S.$ (1)

The quantity $\Delta
Q:=\text{tr}\left[{H}_{R}(\rho^{\prime}_{R}-\rho_{R})\right]$ is the heat
transferred to the reservoir $R$, where ${H}_{R}$ is the Hamiltonian of $R$,
while $\rho^{\prime}_{R}$ and $\rho_{R}$ denote the final and initial state of
$R$ respectively. The $T_{R}$ is the temperature of $R$, and $\Delta
S:=S(\rho_{S})-S(\rho^{\prime}_{S})$ is the von Neumann entropy change between
the initial state $\rho_{S}$ and the finial state $\rho^{\prime}_{S}$ of the
system $S$.

Let us analyze the four assumptions one by one. First, both $S$ and $R$ can be
described by Hilbert spaces means that both $S$ and $R$ are quantum systems.
This is the easiest of the four assumptions to fulfill. Second, the initial
state of $R$ is a thermal state implies that we have a strict restriction on
the density matrix of the initial state of $R$, i.e.,
$\frac{e^{-H_{R}/T_{R}}}{\text{Tr}(e^{-H_{R}/T_{R}})}$, although there is no
restriction on the form of the Hamiltonian $H_{R}$ and temperature $T_{R}$.
Third, $S$ and $R$ are initially uncorrelated means that the density matrix of
the total system in the initial state is the direct product of the density
matrices of $S$ and $R$. The connection between $S$ and $R$ is established
only after we introduce interactions. Fourth, the unitarity implies that the
entropy of the total remains constant at all times. This is also one of the
motivations for the study of Landauer’s principle, that since the von Neumann
entropy remains constant in unitary evolution, it can no longer be used as a
measure of irreversibility. We can use the difference between the two sides of
Eq.(1), called the entropy production, to measure irreversibility
Landi:2020bsq . If all four assumptions are satisfied, the entropy production
must be non-negative.

For the four assumptions above, we find that the second one puts a strict
restriction on the form of the initial state of $R$, i.e., the thermal state.
If we claim that a pure state is indistinguishable from the thermal state, it
must be able to play the role of thermal state in Landauer’s principle, so we
can in principle just start verifying whether Landauer’s principle holds when
this pure state is used as a reservior. Of course, Reeb and Wolf’s proof is
rigorous and requires the density matrix of the thermal state. However, to
better understand the relationship between thermal and pure states, as well as
other mixed states, it is helpful to consider an interaction process without
specifying the initial state of the reservoirs, so that we can backtrack the
form of the initial state that Landauer’s principle can be achieved and then
examine if there may be certain pure states that can satisfy the corresponding
conditions. In the present work we will revisit the relationship between pure
and mixed states in terms of system and reservoir interactions. We will choose
as simple a model as possible, i.e., a qubit as the system and a cavity QFT as
a reservoir, with the interaction type being linear Xu:2021buk ; Xu:2022juv .
Except that the initial state must be a direct product of system and
reservior, we will not specifically require the initial state form for the
total system, allowing us to draw more general conclusions.

The remainder of the paper is organized as follows. In section 2 we present
our model and calculate the time evolution of the total system by perturbation
method. We will present two theorems, which shows that in order for Landauer’s
principle to hold, the expectation value of the annihilation operator must be
zero and the average number of particles in the cavity QFT must satisfy the
Bose-Einstein distribution. In section 3 we discuss a particular state that is
thought to be an alternative to the thermal state: the canonical thermal pure
quantum (CTPQ) state. We examine how it relates to and differs from the
thermal state. In section 4 we give a brief summary of our main results and
close with conclusions.

##  2 The model

The total Hamiltonian $\hat{H}_{\text{total}}$ of the qubit-cavity QFT system
can be written as

$\hat{H}_{\text{total}}=\hat{H}_{0}+\hat{H}_{\text{int}},$ (2)

and the $\hat{H}_{0}$ is the sum of the free Hamiltonian of qubit and cavity
QFT

$\hat{H}_{0}=\hat{H}+\hat{H}_{\text{field}},$ (3)

where $\hat{H}=\frac{\Omega}{2}{{\hat{\sigma}}_{z}}$ ($\hat{\sigma}_{z}$ is
the Pauli matrix and $\Omega$ is the energy gap between ground state
$|g\rangle$ and excited state $|e\rangle$ of the qubit) is the qubit
Hamiltonian and
$\hat{H}_{\text{field}}=\sum_{j=1}^{\infty}\omega_{j}\hat{a}^{{\dagger}}_{j}\hat{a}_{j}$
is the Hamiltonian associated to the cavity QFT. The interaction Hamiltonian
is given by

$\hat{H}_{\text{int}}=\lambda\chi(\tau)\hat{\mu}\hat{\phi}[x(\tau)],$ (4)

in which $\lambda$ is a weak coupling constant so that we can apply
perturbation method, $\chi(\tau)$ is the switching function that controls the
interaction, $\tau$ denotes proper time, $\hat{\mu}$ is monopole operator
which allows population to be exchanged between energy levels, and
$\hat{\phi}[x(\tau)]$ is the field operator at the position of the qubit in
the cavity.

In order to solve the time evolution of the system, we move to the interaction
picture. In principle the $\hat{\mu}$ should be a linear combination of Pauli
matrices. Since the $\hat{H}_{0}$ already contains $\hat{\sigma}_{z}$, the
$\hat{\sigma}_{z}$ in $\hat{\mu}$ is commutive with $\hat{H}_{0}$, so it will
only produce Poincaré recurrences and does not contribute to the energy change
of the qubit Hornberger2009 . Therefore, in this work we will discuss
$\hat{\mu}=\hat{\sigma}_{x}$ as an example, and the results for
$\hat{\sigma}_{y}$ can be obtained in a similar way. Defining
$\hat{\sigma}_{x}:=\hat{\sigma}_{+}+\hat{\sigma}_{-}$, which satisfies
$\hat{\sigma}_{+}|g\rangle=|e\rangle$ and
$\hat{\sigma}_{-}|e\rangle=|g\rangle$, in the interaction picture we have

$\hat{\mu}(\tau)=\hat{\sigma}^{+}e^{i\Omega\tau}+\hat{\sigma}^{-}e^{-i\Omega\tau},$
(5)

and

$\displaystyle\hat{\phi}[x(\tau)]=\sum_{j=1}^{\infty}\left(\hat{a}_{j}e^{-i\omega_{j}t(\tau)}u_{j}\left[x(\tau)\right]+\hat{a}^{\dagger}_{j}e^{i\omega_{j}t(\tau)}u_{j}^{*}\left[x(\tau)\right]\right),$
(6)

where the expression of $u_{j}\left[x(\tau)\right]$ depends on the boundary
condition.

The time evolution operator of the system from time $\tau=0$ to
$\tau=T$111This $T$ should not be confused with temperature. is given by the
Dyson series:

$\displaystyle\hat{U}(T,0)=$
$\displaystyle\mathbb{1}\underbrace{-i\int^{T}_{0}d\tau\hat{H}_{\text{int}}(\tau)}_{\hat{U}^{(1)}}$
$\displaystyle\underbrace{+(-i)^{2}\int^{T}_{0}d\tau\int^{\tau}_{0}d\tau^{\prime}\hat{H}_{\text{int}}(\tau)\hat{H}_{\text{int}}(\tau^{\prime})}_{\hat{U}^{(2)}}+...$
$\displaystyle\underbrace{+(-i)^{n}\int^{T}_{0}d\tau...\int^{\tau^{(n-1)}}_{0}d\tau^{(n)}\hat{H}_{\text{int}}(\tau)...\hat{H}_{\text{int}}(\tau^{(n)})}_{\hat{U}^{(n)}},$
(7)

so the density matrix of the total system at a time $\tau=T$ will be

$\rho_{T}\\!=\\!\big{[}\mathbb{1}+\hat{U}^{(1)}+\hat{U}^{(2)}+\mathcal{O}(\lambda^{3})\big{]}\rho_{0}\big{[}\mathbb{1}+\hat{U}^{(1)}+\hat{U}^{(2)}+\mathcal{O}(\lambda^{3})\big{]}^{\dagger},$
(8)

and we can write $\rho_{T}$ order by order as

$\rho_{T}=\rho^{(0)}_{T}+\rho^{(1)}_{T}+\rho^{(2)}_{T}+\mathcal{O}(\lambda^{3}),$
(9)

where

$\displaystyle\rho^{(0)}_{T}$ $\displaystyle=\rho_{0},$ (10)
$\displaystyle\rho^{(1)}_{T}$
$\displaystyle=\hat{U}^{(1)}\rho_{0}+\rho_{0}\hat{U}^{(1)\dagger},$ (11)
$\displaystyle\rho^{(2)}_{T}$
$\displaystyle=\hat{U}^{(1)}\rho_{0}\hat{U}^{(1)\dagger}+\hat{U}^{(2)}\rho_{0}+\rho_{0}\hat{U}^{(2)\dagger}.$
(12)

Tracing out the field(qubit) part we can obtain the reduced density matrix of
the qubit(field). With the evolution equation in place, we also need to know
the initial state of the system. For the qubit, we choose it to be

$\rho_{\text{0}}^{S}=\begin{pmatrix}p&x\\\ x&1-p\end{pmatrix},$ (13)

and without loss of generality, we assume that $0<p<1/2$ and $x$ is a real
number satisfying $-\sqrt{p(1-p)}\leq x\leq\sqrt{p(1-p)}$. Here the $x$
represents the coherence of the qubit. When $x=0$, the qubit is completely
decoherent, and when $x=\pm\sqrt{p(1-p)}$, the qubit is in a pure state.

We choose the initial state for each mode of the cavity QFT to be some
arbitrary state with density matrix expressed as

$\rho_{\text{0}}^{f}=\bigotimes_{j=1}^{\infty}\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|,$
(14)

and for each integer $j$ we have a set of the positive coefficients $c_{i}$
satisfy the normalization condition $\sum_{i}c_{i}=1$. If for some $i$ there
is $c_{i}=1$ and all the rest of the coefficients are zero, then the
corresponding mode is in the pure state $|\psi_{i}\rangle$, otherwise the mode
is in a mixed state.

The inital state of the total system is then

$\rho_{0}=\begin{pmatrix}p&x\\\
x&1-p\end{pmatrix}\otimes\bigotimes_{j=1}^{\infty}\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|,$
(15)

and we can compute the evolution of the total system order by order.

###  2.1 The order of $\lambda$

In this subsection we will prove the following theorem:

Theorem 1: _In the order of $\lambda$, the von Neumann entropy of the qubit
remains unchanged, and its energy changes just the opposite of the cavity QFT.
For Landauer’s principle to hold under all possible conditions, we must have
$\langle\hat{a}_{j}\rangle:=\sum_{i}c_{i}\langle\psi_{i}|\hat{a}_{j}|\psi_{i}\rangle=0$
for the mode $\omega_{j}=\Omega$, and as a result, $\Delta Q=0$._

Proof: Setting $\chi(\tau)=1$ for $0\leqslant\tau\leqslant T$, the operator
$\hat{U}^{(1)}$ can be written as

$\hat{U}^{(1)}=\frac{\lambda}{i}\sum_{j=1}^{\infty}(\hat{\sigma}^{+}\hat{a}_{j}^{\dagger}I_{+,j}+\hat{\sigma}^{+}\hat{a}_{j}I_{-,j}^{*}+\hat{\sigma}^{-}\hat{a}_{j}^{\dagger}I_{-,j}+\hat{\sigma}^{-}\hat{a}_{j}I_{+,j}^{*}),$
(16)

where

$I_{\pm,j}:=\int^{T}_{0}d\tau~{}e^{i\left[\pm\Omega\tau+\omega_{j}t(\tau)\right]}u_{j}\left[x(\tau)\right].$
(17)

For any boundary condition, if the qubit is located at the same position, then
$u_{j}\left[x(\tau)\right]$ gives a constant value. In the mode
$\omega_{j}=\Omega$, the $I_{-,j}$ will just be proportional to the time.
However, in other cases, the integration in $I_{\pm,j}$ provides only the
oscillatory terms and decays rapidly as the deviation increases222If we
consider quantum field theory in full space, the summation over $j$ becomes an
integral of momentum, and the integral of
$e^{i\left[\pm\Omega\tau+\omega_{j}\tau\right]}$ corresponds to the Dirac
Delta function $\delta(\pm\Omega+\omega_{j})$, so only the mode
$\omega_{j}=\Omega$ contributes.. _For the remainder of this work, unless
otherwise noted, we will all be discussing only the mode $\omega_{j}=\Omega$._

We have

$\displaystyle\hat{U}^{(1)}\rho_{\text{in}}$
$\displaystyle=\hat{U}^{(1)}\left[\begin{pmatrix}p&x\\\
x&1-p\end{pmatrix}\otimes\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|\right]$
$\displaystyle=\frac{\lambda}{i}\left[\begin{pmatrix}x&1-p\\\
0&0\end{pmatrix}\hat{a}_{j}\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|I_{-,j}^{*}+\begin{pmatrix}0&0\\\
p&x\end{pmatrix}\hat{a}_{j}^{\dagger}\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|I_{-,j}\right].$
(18)

Similarly

$\displaystyle\rho_{\text{in}}\hat{U}^{(1)\dagger}=-\frac{\lambda}{i}\left[\begin{pmatrix}x&0\\\
1-p&0\end{pmatrix}\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|\hat{a}_{j}^{\dagger}I_{-,j}+\begin{pmatrix}0&p\\\
0&x\end{pmatrix}\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|\hat{a}_{j}I_{-,j}^{*}\right].$
(19)

Tracing out the field part, we have the reduced density matrix of the qubit

$\displaystyle\lambda\begin{pmatrix}2x\text{Im}(\langle\hat{a}_{j}\rangle
I_{-,j}^{*})&-i(1-2p)\langle\hat{a}_{j}\rangle I_{-,j}^{*}\\\
i(1-2p)\langle\hat{a}_{j}^{\dagger}\rangle
I_{-,j}&-2x\text{Im}(\langle\hat{a}_{j}\rangle I_{-,j}^{*})\end{pmatrix},$
(20)

where the expectation value $\langle\hat{a}_{j}^{(\dagger)}\rangle$ is defined
as
$\langle\hat{a}_{j}^{(\dagger)}\rangle:=\sum_{i}c_{i}\langle\psi_{i}|\hat{a}_{j}^{(\dagger)}|\psi_{i}\rangle$.
Other expectation values in this work are defined in the same way as
$\langle\hat{A}\rangle:=\sum_{i}c_{i}\langle\psi_{i}|\hat{A}|\psi_{i}\rangle$
below.

Diagonalizing the initial density matrix of the qubit (13) we can have the
initial von Neumann entropy of the qubit written as

$\displaystyle S=-p_{+}\ln{p_{+}}-p_{-}\ln{p_{-}},$ (21)

where

$\displaystyle p_{\pm}=\frac{1}{2}\pm\frac{1}{2}\sqrt{4p^{2}+4x^{2}-4p+1}.$
(22)

Suppose after some time the density matrix becomes

$\begin{pmatrix}p&x\\\ x&1-p\end{pmatrix}+\begin{pmatrix}\delta p&-\delta d\\\
-\delta d^{*}&-\delta p\end{pmatrix}.$ (23)

Diagonalizing this density matrix we will have

$\displaystyle
p_{\pm}=\frac{1}{2}\pm\frac{1}{2}\sqrt{4p^{2}+4x^{2}-4p+1-8x\text{Re}(\delta
d)+(8p-4)\delta p+4\delta p^{2}+4|\delta d|^{2}}.$ (24)

In the order of $\lambda$, from (20) we know

$\delta p=2\lambda x\text{Im}(\langle\hat{a}_{j}\rangle
I_{-,j}^{*}),\qquad\text{Re}(\delta
d)=-\lambda(1-2p)\text{Im}(\langle\hat{a}_{j}\rangle I_{-,j}^{*}),$ (25)

so we have

$-8x\text{Re}(\delta d)+(8p-4)\delta p=0,$ (26)

thus the von Neumann entropy of the qubit remains unchanged in the
$\lambda$-order.

Tracing out the qubit part, we have the reduced density matrix of the cavity
QFT written as

$\frac{\lambda}{i}x\left[\hat{a}_{j}\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|I_{-,j}^{*}+\hat{a}_{j}^{\dagger}\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|I_{-,j}-\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|\hat{a}_{j}^{\dagger}I_{-j}-\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|\hat{a}_{j}I_{-,j}^{*}\right],$
(27)

and the energy change of the field will be the trace after applying the
Hamiltonian of the field to the above matrix. We have

$\displaystyle\Delta Q$
$\displaystyle=\frac{\lambda}{i}x\omega_{j}[\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\hat{a}_{j}\rangle
I_{-,j}^{*}+\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\hat{a}_{j}^{\dagger}\rangle
I_{-,j}-\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle
I_{-,j}-\langle\hat{a}_{j}\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle
I_{-,j}^{*}]$ $\displaystyle=-2\lambda
x\omega_{j}\text{Im}(\langle\hat{a}_{j}\rangle I_{-,j}^{*}),$ (28)

where we have used the algebraic relations of the operators. Since
$\omega_{j}=\Omega$, we can easily find the energy changes in the qubit

$\text{Tr}\left[\frac{\Omega}{2}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\begin{pmatrix}\delta p&-\delta d\\\ -\delta d^{*}&-\delta
p\end{pmatrix}\right]=\Omega\delta p=2\lambda
x\Omega\text{Im}(\langle\hat{a}_{j}\rangle I_{-,j}^{*})$ (29)

is just the opposite of the part of the cavity QFT.

Since the von Neumann entropy of the qubit in a $\lambda$-order perturbation
remains unchanged, the right-hand side of (1) is zero regardless of the
temperature $T$ in the field theory, so we only need to have $\Delta Q\geq 0$,
which implies $x\text{Im}(\langle\hat{a}_{j}\rangle I_{-,j}^{*})\leq 0$. Since
$x$ can be positive or negative, we must have
$\text{Im}(\langle\hat{a}_{j}\rangle)=0$.

The above analysis can also be generalized to the case
$\hat{\mu}(\tau)=\hat{\sigma}_{y}(\tau)$. We can still define
$\hat{\mu}(\tau):=\hat{\sigma}^{+}e^{i\Omega\tau}+\hat{\sigma}^{-}e^{-i\Omega\tau}$,
but now the $\hat{\sigma}^{\pm}$ satisfies
$\hat{\sigma}_{+}|g\rangle=-i|e\rangle$ and
$\hat{\sigma}_{-}|e\rangle=i|g\rangle$. The reduced density matrix of the
qubit reads

$\displaystyle\lambda\begin{pmatrix}-2x\text{Re}(\langle\hat{a}_{j}\rangle
I_{-,j}^{*})&(1-2p)\langle\hat{a}_{j}\rangle I_{-,j}^{*}\\\
(1-2p)\langle\hat{a}_{j}^{\dagger}\rangle
I_{-,j}&2x\text{Re}(\langle\hat{a}_{j}\rangle I_{-,j}^{*})\end{pmatrix}.$ (30)

We can also have the von Neumann entropy of the qubit remains unchanged, and
its energy changes just the opposite of the cavity QFT. Now we have

$\displaystyle\Delta Q=2\lambda x\omega_{j}\text{Re}(\langle\hat{a}_{j}\rangle
I_{-,j}^{*}).$ (31)

In this case we need to have $\text{Re}(\langle\hat{a}_{j}\rangle)=0$. Since
$\mu(\tau)$ can be any linear combination of $\hat{\sigma}_{x}$ and
$\hat{\sigma}_{y}$, in order to have $\Delta Q\geq 0$ under any condition, we
can only make $\langle\hat{a}_{j}\rangle=0$, which also makes $\Delta Q=0$.
$\hfill\qed$

###  2.2 The order of $\lambda^{2}$

In this subsection we will prove the following theorem:

Theorem 2: _In the order of $\lambda^{2}$, for a positive quantity $T$, in
order to have $\Delta Q\geq T\Delta S$, the initial state of the cavity QFT
has to satisfy
$\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle:=\sum_{i}c_{i}\langle\psi_{i}|\hat{a}_{j}^{\dagger}\hat{a}_{j}|\psi_{i}\rangle=\frac{1}{e^{\omega_{j}/T}-1}$._

Proof: In this case

$\displaystyle\hat{U}^{(1)}\rho_{\text{in }}\hat{U}^{(1)\dagger}$
$\displaystyle=\lambda^{2}\left(\hat{\sigma}^{-}\hat{a}_{j}^{\dagger}I_{-,j}+\hat{\sigma}^{+}\hat{a}_{j}I_{-,j}^{*}\right)\left(\begin{array}[]{cc}p&x\\\
x&1-p\end{array}\right)\otimes\sum_{i}c_{i}\left|\psi_{i}\right\rangle\left\langle\psi_{i}\right|\left(\hat{\sigma}^{+}\hat{a}_{j}I_{-,j}^{*}+\hat{\sigma}^{-}\hat{a}_{j}^{\dagger}I_{-,j}\right)$
(34)
$\displaystyle=\lambda^{2}\left|I_{-,j}\right|^{2}\left[\left(\begin{array}[]{ll}0&0\\\
0&p\end{array}\right)\hat{a}_{j}^{\dagger}\sum_{i}c_{i}\left|\psi_{i}\right\rangle\langle\psi_{i}|\hat{a}_{j}+\left(\begin{array}[]{cc}1-p&0\\\
0&0\end{array}\right)\hat{a}_{j}\sum_{i}c_{i}|\psi_{i}\rangle\left\langle\psi_{i}\right|\hat{a}_{j}^{\dagger}\right]$
(39)

and

$\displaystyle\hat{U}^{(2)}\rho_{\text{in }}+\rho_{\text{in
}}\hat{U}^{(2)\dagger}$
$\displaystyle=-\frac{\lambda^{2}}{2}\left|I_{-,j}\right|^{2}\bigg{[}\left(\begin{array}[]{cc}0&0\\\
x&1-p\end{array}\right)\hat{a}_{j}^{\dagger}\hat{a}_{j}\sum_{i}c_{i}\left|\psi_{i}\right\rangle\langle\psi_{i}|+\left(\begin{array}[]{cc}p&x\\\
0&0\end{array}\right)\hat{a}_{j}\hat{a}_{j}^{\dagger}\sum_{i}c_{i}|\psi_{i}\rangle\left\langle\psi_{i}\right|$
(44) $\displaystyle+\left(\begin{array}[]{cc}0&x\\\
0&1-p\end{array}\right)\sum_{i}c_{i}\left|\psi_{i}\right\rangle\langle\psi_{i}|\hat{a}_{j}^{\dagger}\hat{a}_{j}+\left(\begin{array}[]{cc}p&0\\\
x&0\end{array}\right)\sum_{i}c_{i}\left|\psi_{i}\right\rangle\langle\psi_{i}|\hat{a}_{j}\hat{a}_{j}^{\dagger}\bigg{]}.$
(49)

Tracing out the field part we can have the correction to the qubit

$\delta
p=\lambda^{2}\left|I_{-,j}\right|^{2}\left((1-2p)\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle-p\right),$
(50)

and

$\delta
d=\lambda^{2}\left|I_{-,j}\right|^{2}x\left(\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle+\frac{1}{2}\right).$
(51)

Tracing out the qubit part we have

$\displaystyle\lambda^{2}\left|I_{-,j}\right|^{2}\bigg{[}p\hat{a}_{j}^{\dagger}\sum_{i}c_{i}|\psi_{i}\rangle\langle|\psi_{i}|\hat{a}_{j}+(1-p)\hat{a}_{j}\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|\hat{a}_{j}^{\dagger}-\frac{1-p}{2}\hat{a}_{j}^{\dagger}\hat{a}_{j}\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|$
$\displaystyle-\frac{p}{2}\hat{a}_{j}\hat{a}_{j}^{\dagger}\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|-\frac{1-p}{2}\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|\hat{a}_{j}^{\dagger}\hat{a}_{j}-\frac{p}{2}\sum_{i}c_{i}|\psi_{i}\rangle\langle\psi_{i}|\hat{a}_{j}\hat{a}_{j}^{\dagger}\bigg{]}.$
(52)

The energy change of the field is

$\displaystyle\Delta
Q=\lambda^{2}\left|I_{-,j}\right|^{2}\omega_{j}\left((2p-1)\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle+p\right)$
(53)

The $\Delta Q$ is independent of $x$, while $\Delta S$ is $x$-dependent. If
for some constant quantity $T$ we want $\Delta Q/T\geq\Delta S$, this implies
that we need $\Delta Q/T$ to be larger than the maximum value of $\Delta S$.
We can find that 333Here we need to insert (50) and (51) into (24) and
calculate the final Von Neumann entropy, then we can have the $\Delta S$ and
its derivative.

$\displaystyle\frac{\partial\Delta S}{\partial x}=0,$ (54)

and the value of $\Delta S$ is maximized at $x=0$. We have

$\displaystyle\Delta S(x=0)=-\ln{\frac{1-p}{p}}\delta
p=\lambda^{2}\left|I_{-,j}\right|^{2}\ln{\frac{1-p}{p}}\left((2p-1)\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle+p\right).$
(55)

If

$\displaystyle(2p-1)\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle+p\geq 0,$
(56)

the $\Delta Q/T\geq\Delta S$ implies

$\displaystyle\frac{\omega_{j}}{T}\geq\ln{\left(\frac{1-p}{p}\right)}.$ (57)

Combining the two equations above, if we have

$\displaystyle\frac{\omega_{j}}{T}\geq\ln{\left(\frac{\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle+1}{\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle}\right)},$
(58)

it will imply

$\displaystyle\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle\geq\frac{1}{e^{\omega_{j}/T}-1},$
(59)

and the (57) must hold.

On the other hand, if
$(2p-1)\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle+p\leq 0$, then all the
$\geq$ in the above inequalities become $\leq$ and we end up with

$\displaystyle\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle\leq\frac{1}{e^{\omega_{j}/T}-1}.$
(60)

Obviously, the only one that satisfies both cases is
$\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle=\frac{1}{e^{\omega_{j}/T}-1}$,
which is also known as the average number of particles in Bose-Einstein
distribution. $\hfill\qed$

##  3 Discussion on Canonical Thermal Pure Quantum State

Our question now is which type of initial state satisfies the two theorems
proved in section 2. The thermal state definitely matches the requirements. We
can write the density matrix of the thermal state as Olivares2012

$\bigotimes_{j=1}^{\infty}\sum_{n_{j}=0}^{\infty}\frac{\bar{n}_{j}^{n_{j}}}{(1+\bar{n}_{j})^{1+n_{j}}}|n_{j}\rangle\langle
n_{j}|,$ (61)

where for each integral value $j$, $n_{j}\in[0,\infty)$, and
$\bar{n}_{j}:={1}/{\left(e^{\frac{\omega_{j}}{T}}-1\right)}$. Since (61)
contains only diagonal terms, we have $\langle\hat{a}_{j}\rangle=0$, and for
each $j$

$\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle=\sum_{n_{j}=0}^{\infty}\frac{\bar{n}_{j}^{n_{j}}n_{j}}{(1+\bar{n}_{j})^{1+n_{j}}}=\bar{n}_{j}.$
(62)

The result corresponds exactly to Theorem 1 and Theorem 2 we proved.

So is a pure state possible? One idea is to construct the following kind of
pure state, called canonical thermal pure quantum (CTPQ) state(unnormalized)
Sugiura2012 ; Sugiura2013 ; Sugiura2014

$|\beta,\hat{H}\rangle=\sum_{i}z_{i}\exp{\Big{(}-\frac{\beta\hat{H}}{2}}\Big{)}|i\rangle,$
(63)

where $z_{i}$ is a set of random complex numbers, whose real and imaginary
parts satisfy the unit normal distribution, $\beta$ is the inverse
temperature, and $|i\rangle$ is a set of orthogonal normalized bases in a
Hilbert space. We have

$\displaystyle\langle\hat{a}_{j}\rangle$
$\displaystyle=\frac{\sum_{k=0}\langle
k|\exp{}(-\beta\omega_{j}k/2)z_{k}^{*}\hat{a}_{j}\sum_{i=0}z_{i}\exp{(-\beta\omega_{j}i/2)}|i\rangle}{\sum_{i=0}|z_{i}|^{2}\exp{(-\beta\omega_{j}i)}}$
$\displaystyle=\frac{\sum_{k=0}\langle
k|\exp{}(-\beta\omega_{j}k/2)z_{k}^{*}\sum_{i=0}z_{i}\exp{(-\beta\omega_{j}i/2)}\sqrt{i}|i-1\rangle}{\sum_{i=0}|z_{i}|^{2}\exp{(-\beta\omega_{j}i)}}$
$\displaystyle=\frac{\sum_{i=0}z_{i}^{*}z_{i+1}\sqrt{i+1}\exp{(-\beta\omega_{j}(2i+1)/2)}}{\sum_{i=0}|z_{i}|^{2}\exp{(-\beta\omega_{j}i)}}$
(64)

and

$\displaystyle\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle=\frac{\sum_{i=0}|z_{i}|^{2}\exp{(-\beta\omega_{j}i)}i}{\sum_{i=0}|z_{i}|^{2}\exp{(-\beta\omega_{j}i)}}.$
(65)

For an arbitrary set of $z_{i}$-sequences, the above result does not support
the Theorems 1 and 2. This is because even if we are taking values with unit
normal distribution, one may have the stochastic error from $z_{i}$ that
cannot be ignored. However, if we take enough sets of $z_{i}$ and then average
them, we will have $\overline{z_{i}^{*}z_{i+1}}=0$ and
$\overline{|z_{i}|^{2}}=1$, thus satisfying the conditions.

It is necessary to emphasize the difference between the CTPQ state and the
thermal state. It is well known that the pure state is a vector in a Hilbert
space, while the mixed state is not. If we compute the expectation value of an
operator $\hat{A}$ in some mixed state
$c_{i}|\psi_{i}\rangle\langle\psi_{i}|$, we have

$\langle\hat{A}\rangle=\text{Tr}\left(c_{i}\langle\psi_{i}|\hat{A}|\psi_{i}\rangle\right).$
(66)

Here we have to do two averages. The first is to find the expectation value of
the operator $\hat{A}$ on the pure state $|\psi_{i}\rangle$, since the state
$|\psi_{i}\rangle$ may not be the eigenstate of the operator $\hat{A}$. The
second is to evaluate the distribution contributed by the coefficient $c_{i}$.

For the pure state, there is no second averaging, since it is a definite
vector in Hilbert space. The CTPQ state, on the other hand, does not choose a
definite vector at the beginning, but mixes in some randomness. By averaging
multiple random vectors again, it has the same results as the thermal state.
For most systems in nature there are typically a large number of particles and
the dimension of a Hilbert space grows exponentially, and the averages of mean
and variance of the operator will converge to the thermal result Sugiura2017 .
But we should also note that this similarity is based on probability, and when
the particle number is not large enough or we take the value in a single run,
the final result may still be very different from the thermal state.

In our model, although there are infinite modes in the cavity QFT, only one
mode actually contributes to the result. Theorems 1 and 2 strictly hold when
this mode is in a thermal state, but if we construct a pure state, theorems 1
and 2 may not hold because the randomization itself introduces errors. We can
get the result of a thermal state by selecting multiple pure states and then
averaging them, but this would probably no longer be the true definition of a
pure state.

##  4 Conclusions and outlook

In this work, we revisit the thought experiment proposed by Polchinski on
whether pure and thermal states can be distinguished. Since the thermal state
appears as a necessary condition in Landauer’s principle, we construct a model
to study the interaction between the qubit and the cavity QFT, where the qubit
serves as the system $S$ and the cavity QFT serves as the reservoir $R$. To
better understand the significance of the thermal state, we did not specify
the initial state of the cavity QFT, but rather asked what conditions the
cavity QFT would have to satisfy if Landauer’s principle holds. In a sense,
this work is not written to discover new things, but to explain what we
already know in a new way. That is, assuming that certain results hold, to use
a operational way to trace back to what we need to find. This may lead to a
better understanding of certain physical conditions. Using the perturbation
method, we obtain two theorems:

  * •

_In the order of $\lambda$, the von Neumann entropy of the qubit remains
unchanged, and its energy changes just the opposite of the cavity QFT. For
Landauer’s principle to hold under all possible conditions, we must have
$\langle\hat{a}_{j}\rangle:=\sum_{i}c_{i}\langle\psi_{i}|\hat{a}_{j}|\psi_{i}\rangle=0$
for the mode $\omega_{j}=\Omega$, and as a result, $\Delta Q=0$._

  * •

_In the order of $\lambda^{2}$, for a positive quantity $T$, in order to have
$\Delta Q\geq T\Delta S$, the initial state of the cavity QFT has to satisfy
$\langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle:=\sum_{i}c_{i}\langle\psi_{i}|\hat{a}_{j}^{\dagger}\hat{a}_{j}|\psi_{i}\rangle=\frac{1}{e^{\omega_{j}/T}-1}$._

These two theorems are valid for the thermal state. Do they hold for the pure
state? We discuss one possible candidate, the CTPQ state. It is defined as a
set of orthogonal normalized bases in a Hilbert space with the operator
$\exp{\big{(}-\frac{\beta\hat{H}}{2}}\big{)}$ acting on them, and mixed in
some random numbers. Averaging these random numbers gives the same result as
the thermal state. When the number of particles in the system is large, the
dimensions of the Hilbert space also become large, and the average errors
generated by random numbers become smaller. It probably more like a mixed
state, but it still gives us a good clue to explore the difference between the
pure and mixed states.

Returning to the original question, can a pure state be operationally
indistinguishable from a thermal state? Our answer is it depends on the
specific system and the definition of the pure state. For most of the systems
in nature, there are typically a large number of particles and the dimension
of a Hilbert space. The thermal state corresponds to a specific distribution.
For the pure state, if all particles are in a particular pure state, or the
total system is in a pure state, it may not give the same result as the
thermal state. Its similarity to the thermal state is based on probability.
However, if each particle is in a random pure state seperately, or we are
considering a large number of pure systems as the ensemble, using a detector
to interact with them may obtain the same result as the thermal state.

We conclude with a brief discussion of possible applications of this work.
First, we can return to the more fundamental physics problem of how the
thermal state and temperature are actually defined. If we consider a large
system that is in a microcanonical distribution and the internal interactions
are weakly coupled, the temperature emerges as a parameter that fixes the
equilibrium condition, and the canonical partition function can be obtained by
saddle-point approximation Martin1959 . The canonical thermal system can be
considered as a subsystem of the microcanonical system, and this allows us to
work on the definition and properties of the thermal state from the source,
e.g. by considering stronger couplings Moreno2019 ; Xu:2021ihm  or the
genalization of microcanonical systems Iyoda2017 .

Second, the qubit in this work is at a fixed position in an inertial reference
system. We can also consider the cases of accelerating detectors Unruh:1976db
; Unruh:1983ms  and detectors in curved spacetime Davies ; Hodgkinson:2014iua
; Ng:2016hzn , which will help us further understand the quantum information
in non-inertial systems. For example, in this work the variation of the off-
diagonal elements of the qubit corresponds to the decoherence effect, and
there are also some recent literatures discussing the connection between
decoherence and accelerating detectors Nesterov:2020exl ; Xu:2023tdt . The
accelerating detectors may also be able to shed some light on the problem of
quantum gravity Alkofer:2016utc .

Finally, Polchinski’s motivation for proposing the thought experiment was to
better understand the black hole information loss paradox, so investigating
the time evolution of the CFT at the boundary corresponding to the matter
collapse process in the bulk is a more straightforward choice Anous:2016kss .
It would be interesting to try to compute observables of this process, such as
the two-point correlation function, and compare them with the results in
AdS/CFT. We will pursue these directions in our future work.

## Acknowledgements

Hao Xu thanks National Natural Science Foundation of China (No.12205250) for
funding support.

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