# Schrödinger cats in quantum-dot–cavity systems

M. Cosacchi<EMAIL_ADDRESS>Theoretische Physik III,
Universität Bayreuth, 95440 Bayreuth, Germany  T. Seidelmann  Theoretische
Physik III, Universität Bayreuth, 95440 Bayreuth, Germany  J. Wiercinski
Theoretische Physik III, Universität Bayreuth, 95440 Bayreuth, Germany  M.
Cygorek  Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom  A. Vagov
Theoretische Physik III, Universität Bayreuth, 95440 Bayreuth, Germany  ITMO
University, St. Petersburg, 197101, Russia  D. E. Reiter  Institut für
Festkörpertheorie, Universität Münster, 48149 Münster, Germany  V. M. Axt
Theoretische Physik III, Universität Bayreuth, 95440 Bayreuth, Germany

###### Abstract

A Schrödinger-cat state is a coherent superposition of macroscopically
distinguishable quantum states, in quantum optics usually realized as
superposition of coherent states. Protocols to prepare photonic cats have been
presented for atomic systems. Here, we investigate in what manner and how well
the preparation protocols can be transferred to a solid state platform, namely
a semiconductor quantum-dot–cavity system. In quantum-dot–cavity systems there
are many disruptive influences like cavity losses, the radiative decay of the
quantum dot, and the coupling to longitudinal acoustic phonons. We show that
for one of the protocols these influences kill the quantum coherence between
the states forming the cat, while for a second protocol a parameter regime can
be identified where the essential characteristics of Schrödinger-cat states
survive the environmental influences under conditions that can be realized
with current equipment.

##  I Introduction

Schrödinger cats are probably the most popular example of highly nonclassical,
purely quantum mechanical states. Realizing a coherent superposition of two
macroscopically distinct states, where in analogy to Schrödinger’s
Gedankenexperiment Schroedinger1935  the cat is simultaneously dead and alive,
remains a challenge due to the inevitable decoherence induced by the
environmental coupling. A general Schrödinger-cat state with two
macroscopically distinguishable states $|\textrm{alive}\rangle$ and
$|\textrm{dead}\rangle$ can be written as

$\displaystyle|\textrm{cat}\rangle=\mathcal{N}(|\textrm{alive}\rangle+e^{i\varphi}|\textrm{dead}\rangle)$
(1)

with normalization $\mathcal{N}$ and phase $\varphi$. These states find
numerous applications in advanced quantum metrology Joo2011 ;
facon2016sensitive , quantum teleportation Enk2001 , quantum computation
Ralph2003 ; Lund2008 ; Mirrahimi2014 , and quantum error correction algorithms
Ofek2016 . Cat states being a superposition of more than two states
leibfried2005creation ; gao2010experimental ; Sedov2020 , as well as phononic
cat states Reiter2011  have been investigated. Schrödinger-cat states are a
suitable platform to study the decoherence between two superposed quantum
objects, in other words, to observe the quantum-to-classical transition
Brune1996 . Therefore, Schrödinger-cat states are of fundamental interest in
understanding the very foundations of quantum mechanics.

Schrödinger-cat states have to be sharply distinguished from incoherently
superposed macroscopically distinct states, which are described by the density
matrix

$\displaystyle\rho_{\textrm{mixture}}=\frac{1}{2}(|\textrm{alive}\rangle\langle\textrm{alive}|+|\textrm{dead}\rangle\langle\textrm{dead}|)\,,$
(2)

where the interference terms are missing. This discrimination is best visible
in the Wigner function. While in the cat state there are negative parts, which
imply nonclassicality, the Wigner function of the classical incoherent mixture
$\rho_{\textrm{mixture}}$ is strictly positive.

Figure 1: A Schrödinger cat state appearing in a QDC system. Photons are
created by recombination of the QD exciton. The photoemission is controlled
such that cats are created on demand.

Because of the fundamental and technological importance of Schrödinger cat
states, their preparation has long been a research target. Earlier efforts in
this direction focused mostly on atom-based systems, where atoms are placed in
an optical cavity Gerry2004 ; Law1996 ; Gea-Banacloche1990 . Recently, many
concepts of atom quantum optics have been transferred to solid state systems,
in particular to semiconductor quantum-dot–cavity systems (QDC) giving rise to
the field of semiconductor quantum electrodynamics. QDCs have been already
shown to be highly integrable on-demand emitters of photons in nonclassical
states including high-quality single photons Michler2000 ; Santori2001 ;
Santori2002 ; He2013 ; Wei2014Det ; Ding2016 ; Somaschi2016 ; Schweickert2018
; Hanschke2018 ; Cosacchi2019  and entangled photon pairs Akopian2006 ;
Stevenson2006a ; Hafenbrak2007 ; Dousse2010 ; Schumacher2012 ; delvalle2013dis
; Mueller2014 ; Munoz2015 ; Heinze2017 ; Orieux2017 ; Seidelmann2019 . QDC-
based protocols for generating higher-order Fock states have also been
developed Cosacchi2020b . An easy to use solid state-based source of
Schrödinger cats would be a highly attractive extension of these achievements.

As atomic systems, QDCs have loss channels, namely the energetic loss channels
provided by the radiative decay of a quantum dot (QD) and the finite cavity
quality factor. The main difference between QDC and atomic systems is the
presence of longitudinal acoustic phonons, which is a well-known source of
decoherence even at cryogenic temperatures of $T=4\,$K Ramsay2010a ;
Reiter2019 . Hence, it is an open question whether also in the QDC platform
the generation of Schrödinger-cat states is possible.

In this paper, we consider two protocols to prepare Schrödinger-cat states in
a QDC. For both protocols we adapt existing preparation schemes and apply them
to the QDC system. The first protocol is based on the proposal by Law and
Eberly Law1996 . To use this protocol in the QDC we solely rely on driving the
QD by external laser pulses. To create a Schrödinger-cat state we need a
precise timing of the arrival time of the pulses steering the QD-cavity
coupling. We call the protocol DOD (dot driven protocol). The second protocol
is adapted from Gea-Banacloche Gea-Banacloche1990  and can be transferred to
the QDC system by producing a coherent initial state. This can be achieved by
driving the cavity, hence we call this protocol CAD (cavity driven protocol).

For both protocols, we analyze the impact of losses and phonon coupling in
detail using realistic parameters that have been realized in current
experiments. For the DOD we show that the losses are a detrimental influence
on the sensitive coherence in a Schrödinger cat and the protocol can only
produce mixed states [cf. Eq. (2)] under realsitic conditions. In contrast,
for the CAD we find that even under realistic conditions it is possible to
create a cat state and we identify experimentally accessible parameter regimes
favorable for generating photonic Schrödinger cats.

Our work demonstrates that also in QDCs the preparation of Schrödinger-cat
states is possible and we propose a protocol and specify a suitable parameter
regime to prepare them.

##  II Theoretical Model

The QDC system can be well modeled by a driven two-level QD coupled to a
single photon mode Reiter2019 ; Cosacchi2020b . The corresponding Hamiltonian
then reads

$\displaystyle H=\,$ $\displaystyle
H_{\textrm{QDC}}+H_{\textrm{driving}}+H_{\textrm{AC-Stark}}\,.$ (3)

The QDC is described by the Jaynes-Cummings model

$\displaystyle H_{\textrm{QDC}}=\hbar\omega_{X}|X\rangle\langle
X|+\hbar\omega_{C}a^{\dagger}a+\hbar
g\left(a\sigma_{X}^{\dagger}+a^{\dagger}\sigma_{X}\right)\,,$ (4)

where $|X\rangle$ is the exciton state at energy $\hbar\omega_{X}$.
$\sigma_{X}:=\,|G\rangle\langle X|$ is the transition operator between the
excited state $|X\rangle$ and the ground state $|G\rangle$. The energy of
$|G\rangle$ is set to zero. $a$ ($a^{\dagger}$) denotes the photonic
annihilation (creation) operator. The cavity frequency is denoted by
$\omega_{C}$ and its coupling to the QD by $g$. We consider two different
forms of driving Hamiltonians for the two protocols

$\displaystyle H_{\textrm{driving}}=\,$
$\displaystyle\begin{cases}-\frac{\hbar}{2}\left(f_{\textrm{p}}^{*}(t)\sigma_{X}+f_{\textrm{p}}(t)\sigma_{X}^{\dagger}\right)&\textrm{DOD}\\\
-\frac{\hbar}{2}\left(f_{\textrm{p}}^{*}(t)a+f_{\textrm{p}}(t)a^{\dagger}\right)&\textrm{CAD}\,.\end{cases}$
(5)

Effectively decoupling the QD from the cavity can be achieved by an AC-Stark
pulse driving the QD Unold2004 ; Cosacchi2020b , described by

$\displaystyle H_{\textrm{AC-Stark}}=-\frac{\hbar}{2}\left(f_{\textrm{AC-
Stark}}^{*}(t)\sigma_{X}+f_{\textrm{AC-
Stark}}(t)\sigma_{X}^{\dagger}\right)\,.$ (6)

The exciting and Stark laser pulses are represented by the functions
$f_{\textrm{p}}(t)$ and $f_{\textrm{AC-Stark}}(t)$, which are specified in
Appendix A.

We also account for the coupling to longitudinal acoustic (LA) phonons
Besombes2001 ; Borri2001 ; Krummheuer2002 ; Axt2005 ; Reiter2019  (as detailed
in Appendix A) as well as the radiative decay of the QD exciton and cavity
losses. Whenever we consider phonon effects, the phonons are assumed to be
initially in thermal equilibrium at a temperature of $T=4\,$K. The
corresponding Liouville equation is solved in a numerically complete manner by
employing a path-integral formalism (for details see Refs. Vagov2011, ;
Barth2016, ; Cygorek2017, ). The parameters used in the calculations are given
in Appendix C.

##  III Dot driven protocol - DOD

In the optical realm, coherent states $|\alpha\rangle$ are the most classical
states. A general coherent superposition of two coherent states of the form

$\displaystyle\mathcal{N}(|\alpha\rangle+e^{i\varphi}|-\alpha\rangle)$ (7)

with normalization $\mathcal{N}$ and phase $\varphi$ is one of the most common
realizations of Schrödinger-cat states in quantum optics Gerry2004 . Hence, we
choose this realization as the target state for the DOD and set $\alpha=\pi/2$
and $\varphi=0$. This choice ensures that the corresponding coherent states
are distinct, while their average photon number is low enough that we expect
the influence of cavity losses to be limited.

To prepare this target state we adapt the protocol from Ref. Law1996, , which
is proposed to create arbitrary photonic states in a single-mode microcavity.
However, the originally proposed protocol does not account for any loss
channels. The requirements are a driven Jaynes-Cummings model with
controllable driving $f(t)$ and coupling $g(t)$ between the two-level system
and the cavity. To transfer the proposal to QDCs, a few obstacles have to be
overcome. While time-dependent driving of a QDC is possible by applying
appropriate laser pulses, controlling the QD-cavity coupling time-dependently
remains a challenge. In particular, the protocol in Ref. Law1996,  relies on a
stepwise switching between $f(t)$ and $g(t)$, i.e., one has to be off,
whenever the other one is on.

Accordingly, the challenge of implementing this protocol in a QDC is twofold:
(i) the magnitude of the QD-cavity coupling has to be varied and (ii) the QD
and cavity are supposed to be decoupled during the time the driving is on. In
Ref. Law1996, , the time intervals $\tau$, when either $f(t)$ or $g(t)$ are
on, are kept constant. Only the products $f_{i}\tau$ and $g_{i}\tau$ in the
$i$th interval are relevant for the success of the protocol. Therefore,
problem (i) can be solved by varying the time interval while keeping the
coupling constant, i.e., we use $g\tau_{i}$. Concerning problem (ii), the
decoupling suggested in Ref. Law1996,  can in principle be realized by
inducing suitable Stark shifts, which is, however, highly demanding
experimentally. As demonstrated for protocols to prepare higher-order Fock
states Cosacchi2020b , it can be advantageous to avoid the decoupling provided
the desired goal can be achieved by short enough pulses. Indeed, when the
switching induced by the laser driving $f(t)$ takes place on time scales
shorter than the dynamics induced by the cavity, the action of the latter
cannot interfere noticeably with the switching, even though the QD and the
cavity are coupled. Furthermore, on such fast time scales, the precise shape
of the pulse becomes irrelevant (see, e.g., Appendix A 1 of Ref.
Cosacchi2020b, ). Therefore, a Gaussian pulse with the same area as the
rectangular $f_{i}\tau$ can be used, which vastly reduces the experimental
demand. Its full width at half maximum (FWHM) is chosen to be $100\,$fs. Note
that the corresponding spectral width of the pulse is quite large.
Nonetheless, for typical energetic spacings to higher lying exciton states
Vagov2004  the two-level approximation to the QD still holds well
Cosacchi2020b .

Figure 2: Dynamics of the QDC in the DOD. Panels from bottom to top: External
pulses as well as the Stark pulse, exciton occupation $n_{X}$, average photon
number $\langle n\rangle$ and time-dependent fidelity $\mathcal{F}(t)$ to the
target Schrödinger-cat state in Eq. (7). Dashed lines: ideal case without
phonons and losses. Dashed-dotted lines: without phonons but with losses.
Solid lines: with phonons and losses.

###  III.1 The ideal case

Figure 2 shows the dynamics of the QDC in the DOD. The lowest panel displays
the sequence of laser pulses proposed to prepare the Schrödinger-cat state in
Eq. (7). The pulse sequence is derived by solving the set of equations
determining the protocol to prepare arbitrary states in Ref. Law1996, . After
adapting the solution to pulses as explained above, one obtains the pulse
areas and central peak times necessary to prepare the target state. In total,
a series of ten $\pi$ pulses is applied. Note that relative phases of $\pi$ of
the pulses are absorbed into the definition of the pulse areas and that the
time difference between two subsequent pulses is the time $\tau_{i}$ where the
cavity coupling $g$ takes effect. The arrival times and pulse areas are listed
in Tab. 1. After reaching the target state, the cavity needs to be decoupled
from the QD in order to preserve the preparation, which is achieved by an AC-
Stark pulse as shown in the lower panel of Fig. 2.

Table 1: Pulse sequence for the DOD in Sec. III. The times $t_{c}$ of their maxima and their pulse areas $\Theta$ are given. Number of the pulse |  $t_{c}$ (ps) |  $\Theta$ ($\pi$)
---|---|---
1 | 0.1 | -1
2 | 14.2 | -1
3 | 24.1 | -1
4 | 34.6 | 1
5 | 40.0 | -1
6 | 46.5 | -1
7 | 53.7 | -1
8 | 60.7 | -1
9 | 67.2 | -1
10 | 72.1 | -1

The resulting time evolution of the exciton occupation $n_{X}$ is shown in
blue in Fig. 2, where the dashed lines correspond to the ideal case without
phonons and losses. Each laser pulse partially excites the exciton, which then
decays by photon creation. Accordingly, the average photon number $\langle
n\rangle$, as shown in the second panel from top in Fig. 2, increases after
each pulse.

To see whether we have created a Schrödinger-cat state, we consider the
fidelity defined as

$\mathcal{F}(\rho_{1},\rho_{2})=\left[\textrm{Tr}\left(\sqrt{\sqrt{\rho_{1}}\rho_{2}\sqrt{\rho_{1}}}\right)\right]^{2}$
(8)

for two arbitrary density matrices $\rho_{1}$ and $\rho_{2}$ Jozsa1994 .
Setting one density matrix to the target Schrödinger-cat state in Eq. (7), we
get a measure how close we are to this specific cat state. The results are
shown in the topmost panel of Fig. 2. We see that after the pulse sequence the
fidelity reaches unity, implying that in the purely Hamiltonian ideal case
without losses and phonons, this protocol is able to perfectly prepare the
target state.

Figure 3: Wigner functions of the photonic states prepared by the DOD (a) for
the ideal case, (b) including losses, and (c) taking both phonons and losses
into account. The Wigner functions are calculated at the start time of the
decoupling Stark pulse, i.e., at $t=75.5\,$ps.

This is confirmed by looking at the Wigner function (definition in Appendix B)
in Fig. 3 (a). We see all relevant features of a Schrödinger cat where the two
Gaussians indicate the two macroscopically distinct states, here, coherent
states, and the oscillations between them point to a coherent character of
their superposition.

An important property of a cat state is its nonclassicality. In general, the
nonclassicality $\delta$ of a state can be measured by considering the
negative part of its Wigner function $W(\alpha)$, since all Wigner functions
corresponding to classical states are positive. The doubled volume of the
integrated negative part of the Wigner function was introduced in Ref.
Kenfack2004,  as such a nonclassicality measure:

$\displaystyle\delta=\frac{\int(|W(\alpha)|-W(\alpha))d\alpha}{\int
W(\alpha)d\alpha}\,.$ (9)

where $\delta=0$ implies a classical state. For the DOD we obtain
$\delta=0.51$ for this Schrödinger cat, thus, indeed implying quantum features
of the state.

The results in Figs. 2 and 3 as well as the value of $\delta$ indicate that
the adaption of the protocol in Ref. Law1996,  to ideal QDCs with constant
cavity coupling $g$ is accomplished successfully.

###  III.2 Loss and phonon effects

Next, we account for the loss channels. When taking the radiative decay of the
QD and the cavity losses into account, the preparation fidelity drops to
$\mathcal{F}=48.7\,\%$ (cf., dashed-dotted lines in Fig. 2). Interestingly,
the nonclassicality measure $\delta$ drops to $0.03$. Looking at the
corresponding Wigner function in Fig. 3 (b) reveals that the domains of
negative values of the Wigner function have practically disappeared.

Phonons destroy even the remaining nonclassicality. While the fidelity in the
case including all loss and phonon effects still yields $\mathcal{F}=25.0\,\%$
(see solid lines in Fig. 2), $\delta$ is identically zero, thus indicating a
classical state. Indeed, the Wigner function in Fig. 3 (c) shows two blurred
macroscopically distinct states, here again coherent states, but no
oscillations between them. Thus, they are superimposed incoherently and,
therefore, closely resemble the statistical mixture in Eq. (2).

The reason for this behavior lies in the nature of a Schrödinger-cat state,
which involves the formation of a quantum mechanical superposition of distinct
states with a well defined relative phase. All processes diminishing this
phase relation lead to a degradation of the cat state. For typical QDCs, the
cavity loss rate is much larger than the radiative decay rate, thus having
greater impact on the preparation scheme. Furthermore, the effective cavity
loss rate is proportional to the photon occupation, thus degrading especially
states with large multiphoton contribution, such as cat states. The phonons
then destroy any nonclassicality left after taking the other two processes
into account.

From the analysis in this section it becomes clear that the fidelity alone is
not sufficient to characterize a Schrödinger-cat state, since it may miss the
essential feature of oscillations of the Wigner function to negative values.
Therefore, it is necessary to simultaneously consider a nonclassicality
measure like $\delta$.

In summary, the DOD is not suitable to prepare the Schrödinger-cat state in a
QDC under realistic conditions, because the interference terms of a cat state
do not survive the environmental coupling. Even at $T=4\,$K, phonon effects
destroy the coherent superposition of the two macroscopically distinct states.
Note that it is interesting from a fundamental point of view that the phonons
provide sufficiently strong environmental coupling necessary to facilitate the
quantum-to-classical transition, which in this case is a transition from a
Schrödinger cat as in Eq. (7) to an incoherent mixture as in Eq. (2).

##  IV Cavity driven protocol - CAD

Figure 4: Dynamics of the QDC in the CAD. Panels from bottom to top: the
exciting laser pulse, the occupation of the exciton $n_{X}$, the average
photon number $\langle n\rangle$, $\textrm{Tr}(\rho_{\textrm{photon}}^{2})$
which indicates how close the photonic system is to a pure state, and the
fidelity to the Schrödinger-cat state in Eq. (10). Dashed lines: ideal case
without phonons and losses. Dashed-dotted lines: without phonons but with
losses. Solid lines: with phonons and losses. The temporal range where the
fidelity reaches its maximum value in all three considered cases is shaded in
gray.

The CAD protocol is based on the ideas of Refs. Gea-Banacloche1990, ; Gea-
Banacloche1991, , which showed that the textbook collapse-and-revival example
in the Jaynes Cummings model has two additional very striking features: (i) at
half the revival time the QD and photonic subspaces factorize and (ii) at
precisely this time a Schrödinger-cat state appears in the cavity mode.

The main requirement for this cat state preparation scheme is a coherent state
in the cavity mode as the initial state of the Jaynes-Cummings dynamics Gea-
Banacloche1990 . The photonic state prepared at half the revival time is Gea-
Banacloche1991

$\displaystyle\mathcal{N}\left(|\Phi_{+}\rangle+|\Phi_{-}\rangle\right)\,,$
(10)

which is another realization of a Schrödinger-cat state as defined in Eq. (1).
Here, $\mathcal{N}$ is a normalization constant and

$\displaystyle|\Phi_{\pm}\rangle=e^{-\frac{1}{2}\langle
n\rangle}\sum_{n=0}^{\infty}\frac{\langle
n\rangle^{n/2}}{\sqrt{n!}}e^{-in\phi}e^{\mp i\pi\sqrt{\langle n\rangle
n}}|n\rangle$ (11)

with $\langle n\rangle=|\alpha|^{2}$ the average photon number of the initial
coherent state and $\phi$ the phase of $\alpha$. When using a real envelope
Gaussian pulse as in our case, this phase is determined to be $3\pi/2$ as a
short analytical calculation shows (cf., Appendix D). These two states are
macroscopically distinguishable, but they are not coherent states as in Eq.
(7) because of the $\langle n\rangle$\- and $n$-dependent phase in Eq. (11).
However, it should be noted that $|\Phi_{\pm}\rangle$ approaches in the limit
$\langle n\rangle\to\infty$ the coherent state $|\Phi_{\pm}\rangle\to e^{\mp
i\langle n\rangle\pi/2}|\mp i\alpha\rangle$ (cf., Ref. Gea-Banacloche1991, )
and therefore the state in Eq. (10) becomes
$\mathcal{N}(|i\alpha\rangle+e^{-i\langle n\rangle\pi}|-i\alpha\rangle)$,
which has the form given in Eq. (7). The fidelity with which the state in Eq.
(10) is reached in the Jaynes-Cummings dynamics at half the revival time is
unity only in the limit $\langle n\rangle\to\infty$ Gea-Banacloche1991 . Thus,
in the ideal phonon and loss free case higher average photon numbers are
favorable in this scheme. However, since the revival time is longer for higher
mean photon numbers, the preparation time rises with increasing mean photon
numbers. The system is thus exposed a longer time to losses before the end of
the protocol. As a result of this trade-off situation, there is an optimal
photon number when losses are accounted for.

Figure 5: Wigner functions of the photonic states prepared by the CAD (a) for
the ideal case, (b) including losses, and (c) taking both phonons and losses
into account. The Wigner functions are calculated at the respective times of
maximum fidelity (cf., Fig. 4): $t=39.5\,$ps (ideal case), $t=38.5\,$ps
(including losses), and $t=41.0\,$ps (including losses and phonons).

Using this idea for a preparation protocol of Schrödinger-cat states in the
microcavity mode, we have to face two tasks: (i) preparing a suitable initial
state in the cavity and (ii) finding a suitable $\langle n\rangle$ to achieve
maximum fidelity to the cat state in Eq. (10). Task (i) is solved easily,
since it is textbook-knowledge Gerry2004  that driving an empty cavity with a
classical laser yields a coherent state in the cavity. Now, the cavity in a
QDC is not empty, but as long as the laser pulse driving the cavity mode is
short enough compared with the dynamics induced by the coupling $g$, it may be
approximated as such. We use a $1\,$ps-Gaussian pulse (cf., bottom panel of
Fig. 4) to drive the cavity mode and vary its pulse area $\Theta$ to analyze
the success of the protocol depending on $\langle n\rangle$ to tackle the
second task (ii). Note that the average photon number of the prepared coherent
state is connected to the laser pulse area via $\langle
n\rangle=|\alpha|^{2}=(\Theta/2)^{2}$.

###  IV.1 The ideal case

Figure 4 shows the dynamics of the QDC in the CAD when driving the cavity with
a pulse of area $1.2\pi$. As we show later this is the optimal pulse area when
losses are taken into account. The dashed lines in Fig. 4 correspond to the
ideal case.

For the chosen pulse area the average photon number, as shown in the middle
panel in Fig. 4, is about $3$. While this is not high enough to lead to a
clear-cut collapse-and-revival signature, the exciton dynamics (blue curves)
shows hints of this feature with a revival at about $80$ ps. The fidelity
$\mathcal{F}$ shows oscillations within a bell-like envelope to reach its
maximum of $\mathcal{F}=88.1\,\%$ at $39.5\,$ps after the pulse. At the same
time, the photonic subsystem is close to a pure state as indicated by the
near-unity value of the trace of the squared photonic density matrix. Thus,
the QD and photon subspaces factorize. To preserve the Schrödinger-cat state
as it appears at this point in time, an additional QD-driving pulse is needed
to effectively decouple the QD from the cavity. This is in analogy to the AC-
Stark pulse shown for the DOD in Sec. III.

Figure 5 (a) shows the corresponding Wigner function evaluated at the time of
maximum fidelity when the decoupling is evoked by a Stark pulse for the ideal
case. Two macroscopically distinct states are clearly visible as elongated
Gaussians corresponding to the states $|\Phi_{\pm}\rangle$ in Eq. (11).
Oscillations to negative values between these two structures indicate a
coherent superposition. Therefore, this state is clearly a Schrödinger-cat
state.

As the CAD depends sensitively on the pulse areas, we plot in Fig. 6 (b) the
maximum fidelity to the Schrödinger-cat state in Eq. (10) during the time
evolution after the cavity preparation pulse as a function of the pulse area,
i.e., the average photon number of the initial state. Having seen that the
fidelity is not sufficient as a measure for obtaining a nonclassical photon
state as the cat state, we additionally show in Fig. 6 (a) the nonclassicality
measure $\delta$ at the time of maximum fidelity.

__

Figure 6: (a) Nonclassicality measure $\delta$ at the time of maximum fidelity
and (b) maximum fidelity during the time evolution as a function of the pulse
area, i.e., the average photon number of the initial state. Black: ideal case,
blue: including losses but without phonons, red: with losses and phonons,
orange: solution of the initial value problem .

For the ideal case (black lines in Fig. 6) the nonmonotonic behavior of the
fidelity is prominent. In contrast to the naive expectation that it rises
monotonically with $\Theta$, i.e., $\langle n\rangle$, it increases for
$\langle n\rangle\to 0$ and shows an oscillatory behavior for higher $\langle
n\rangle$. For decreasing $\langle n\rangle$, the prepared state contains a
larger contribution of the vacuum, while the target state in Eq. (10) itself
shows more vacuum characteristics. Therefore, the fidelity approaches unity,
while the nonclassicality measure $\delta$ decreases [cf., Fig. 6(a)]. Thus,
the state for low $\langle n\rangle$ cannot be considered to be a genuine
Schrödinger-cat, since the target state in Eq. (10) ceases to be a
superposition of two macroscopically distinct states.

The nonmonotonic behavior of the maximum fidelity in Fig. 6(b) starting at
$\Theta=\pi$ has its origin in the dynamics of the fidelity as shown exemplary
in Fig. 4. The oscillation frequency within the bell-like envelope increases
with rising $\Theta$ (not shown in the figure), while the maximum of the
oscillation need not coincide with the maximum of the envelope. Therefore,
$\textrm{max}_{t}\mathcal{F}(t)$ does not rise monotonically with $\Theta$,
contrary to the naive expectation.

The proposal in Ref. Gea-Banacloche1990,  is based on the assumption of a
coherent state in the cavity mode as an initial state of the dynamics. The
orange line in Fig. 6(b) shows the fidelity corresponding to the solution of
the initial value problem posed in Ref. Gea-Banacloche1990 , i.e., without
first preparing the initial state with a laser pulse. If such a coherent state
is to be prepared in a cavity mode, an external laser pulse is necessary [cf.,
task (i) in Sec. IV]. For pulse areas greater than $\pi$, the maximum fidelity
obtained after solving the initial value problem perfectly coincides with the
result obtained in the ideal case of the preparation. This implies that indeed
a coherent state is prepared in the cavity mode by the external driving. The
deviations seen at smaller pulse areas have their origin in the finite length
of the preparation pulse.

The overall rise of the nonclassicality measure $\delta$ with $\langle
n\rangle$, on the other hand, and its oscillations [cf., black line in Fig.
6(a)] are a known feature Kenfack2004 . In particular, the oscillation is a
signature of a nonzero phase due to a finite momentum of the cat state in
$(q,p)$-representation that distinguishes ’standing’ and ’moving’ cats
Kenfack2004 . Note that the Wigner function in $(q,p)$\- and
$(\textrm{Re}(\alpha),\textrm{Im}(\alpha))$-representation are connected by a
factor of $2\pi\hbar$ Louisell1973 ; Cahill1969a ; Cahill1969b . Since the
nonclassicality measure $\delta$ is a ratio of volumes of the Wigner function,
it is independent of the representation.

###  IV.2 Loss and phonon effects

The influence of cavity losses and radiative decay on the preparation using
CAD is very strong. The dynamics of all photonic variables shown in Fig. 4 are
damped (orange dashed-dotted line). The effect is even more pronounced when
looking at the protocol as a function of $\langle n\rangle$ in Fig. 6 (blue
lines). In particular, at high $\langle n\rangle$, where the highest fidelity
in the ideal case is achieved, the losses have the greatest impact and the
fidelity drops to almost zero. This is due to the fact that the effective loss
rate for a Fock state $|n\rangle$ is proportional to the photon number $n$,
i.e., $n\kappa$. Likewise, the nonclassicality measure $\delta$ becomes
identically zero when the pulse area exceeds $1.5\pi$. Thus, in stark contrast
to the ideal case, the limit $\langle n\rangle\to\infty$ yields no Schrödinger
cat at all.

Considering phonon effects on top of the loss influence further smoothens out
the dynamics (cf., solid lines in Fig. 4) and lowers both the fidelity and the
nonclassicality of the target state [cf., red lines in Figs. 6], while showing
qualitatively the same behavior as in the case with losses but without
phonons. These findings depend on the considered temperature, the GaAs
material parameters, and the QD geometry and might differ for other parameter
sets. In the case considered in this work, however, the loss effects have the
most detrimental influence on the preparation of the target Schrödinger-cat
state.

This is further underscored by considering the dependence of the protocols’
success on the cavity loss rate, as shown in Fig. 7 for the case without
phonons. For both the DOD and the CAD, the preparation fidelity rises
monotonically with the quality factor, as does the nonclassicality measure
$\delta$. The latter rises faster for the case of controlling the cavity,
implying a comparatively higher robustness of this scheme with respect to
cavity losses. This is due to the fact that the total length of the DOD of
about $~{}80\,$ps is roughly twice the length of the CAD of about $~{}40\,$ps.
Therefore, loss processes have less time to take effect in the CAD.
Furthermore, the DOD presumes ten operations, i.e., pulses which need to be
timed exactly, whereas the CAD relies only on one pulse. This also makes the
CAD more stable against environmental influences and more attractive for
experiments.

Note that it has been shown experimentally that a system with a Purcell factor
roughly three orders of magnitude larger than in our case is well suited for
the preparation of exotic photonic states. Hofheinz et. al. showed
Hofheinz2009  that a so-called Voodoo cat state, a superposition of three
coherent states, can be prepared in a system consisting of a superconducting
qubit in a microwave resonator with a fidelity of $83\,\%$. While such a setup
has drawbacks, such as the wavelength of the emitted photonic state and the
temperature in the mK-regime needed for the qubit to operate, this amazing
result underscores the necessity of a high-quality resonator, both concerning
the quality factor and the Purcell enhancement for the preparation of
Schrödinger-cat or even more complicated Voodoo cat states.

Strikingly, in our QD-cavity system there is still a window of pulse areas,
where the fidelity and the nonclassicality measure are rather high. The
optimum pulse area yielding the maximum preparation fidelity under loss and
phonon influence for our parameters is at $\Theta=1.2\pi$, where both the
fidelity and the nonclassicality measure have a maximum. This is due to a
competition between the rising fidelity for $\langle n\rangle\to\infty$ as
predicted in Ref. Gea-Banacloche1991,  and the inclusion of losses, which also
become stronger for increasing $\langle n\rangle$. As a check, whether we
indeed have created a cat state at $\Theta=1.2\pi$, we have a look at the
Wigner function in Fig. 5 (c). The Wigner function clearly shows two
macroscopically distinct states and oscillations between them, thus indicating
a Schrödinger-cat state even when losses and phonons are accounted for under
realistic conditions.

Figure 7: The maximum fidelity to the Schrödinger-cat states in Eqs. (7) and
(10) in the case of the DOD (black solid line) and the CAD (red dashed line),
respectively, and the nonclassicality measure $\delta$ (open rectangles and
circles) as a function of the cavity loss rate $\kappa$. The cavity quality
factor assuming a mode frequency of $\hbar\omega_{\textrm{C}}=1.5\,$meV is
displayed as a second axis.

##  V Conclusion

We have investigated two protocols for the preparation of photonic
Schrödinger-cat states in the light field mode of a quantum-dot–cavity system
(QDC). While in atomic systems Schrödinger cats have been already prepared, we
here adapted the protocols used in the atomic case to a solid state system. In
the calculations, we considered realistic values for the cavity losses as have
been reported in QDCs, which showed that the radiative decay and cavity losses
can be quite detrimental to the preparation scheme. In contrast to atoms, in
QDC devices also phonons play a role, which have great impact on the
Schrödinger cat preparation. Therefore, a theoretical guidance on the
feasibility to prepare cat states is of high importance.

The first scheme relies on controlling the quantum dot with external laser
pulses (DOD) by adapting Ref. Law1996 . We developed a multi-pulse protocol
for the QDC, where both the precise timing of the pulses and their mutual
phases are of utmost importance. Most detrimental to the fidelity to the
Schrödinger-cat state are cavity and radiative losses. The environmental
coupling to longitudinal acoustic phonons further reduces drastically the
fidelity and completely destroys the coherence between the two states. Only an
incoherent mixture of the two macroscopically distinct states with zero
nonclassicality remains, such that this scheme is not suitable to prepare
Schrödinger cats in realistic QDCs. We mention that this is different for
superconducting qubits in microwave cavities, where similar protocols have
been successfully employed Hofheinz2009 , because the quality factor of
microwave cavities relative to the coupling strength is higher than in QDCs. A
similar boost of the quality factor would be needed to enable a Schrödinger-
cat preparation with this protocol also in QDCs.

The second protocol exploits the internal dynamics of the Jaynes-Cummings
model, where a Schrödinger-cat state can be found naturally in the time
evolution of the system Gea-Banacloche1990 ; Gea-Banacloche1991 . Only one
pulse driving the cavity is necessary to prepare a single coherent state in
the field mode (CAD), which serves as an initial state for the subsequent
Jaynes-Cummings dynamics. While this protocol in the ideal case works best for
high pulse areas, also the losses increase in this case such that no
preparation is possible. Again, the losses are the main cause of destroying
the cat, while the phonon effects are less dramatic than in the first
protocol. Remarkably, for intermediate pulse areas between $\pi$ and $1.5\pi$,
the coherences as well as the nonclassicality of the Schrödinger cats survive
even under the influence of both losses and phonons. Also in the CAD, a boost
of the cavity quality factor would improve the characteristics of the prepared
cat state.

Our results show that Schrödinger cats in QDCs can be prepared under realistic
conditions with an easy to use protocol.

###### Acknowledgements.

M. Co. thanks the cat Alfred for posing in Fig. 1. This work was funded by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project
Nr. 419036043 and the University of Bayreuth in the funding program Open
Access Publishing.

##  Appendix A Coupling Hamiltonian

The external laser pulses are described by

$\displaystyle f_{\textrm{p}}(t)=\,$
$\displaystyle\sum_{m}f_{m}^{\textrm{p}}(t-t_{m})e^{-i\omega_{\textrm{p}}(t-t_{m})}\,.$
(12)

$f_{m}^{\textrm{p}}(t)$ are the envelope functions of the pump fields and
$\omega_{\textrm{p}}$ the corresponding laser frequencies. The product
$\omega_{\textrm{P}}t_{m}$ is chosen as an integer multiple of $2\pi$. The
pump fields are Gaussian pulses with area $\Theta_{j}$

$\displaystyle
f_{j}^{\textrm{p}}(t)=\frac{\Theta_{j}}{\sqrt{2\pi}\sigma}e^{-\frac{t^{2}}{2\sigma^{2}}}\,.$
(13)

where $\sigma$ denotes the standard deviation. It is connected to the full
width at half maximum (FWHM) by FWHM$=2\sqrt{2\ln{2}}\sigma$. Throughout this
work, $\omega_{\textrm{p}}=\omega_{X}$ is assumed. The AC-Stark pulses are
assumed to be of rectangular shape

$\displaystyle f_{\textrm{AC-Stark}}(t)=\,e^{-i\omega_{\textrm{ACS}}t}$
$\displaystyle\times\begin{cases}0&t<-\frac{\tau_{\textrm{length}}}{2}\\\
f_{s}&-\frac{\tau_{\textrm{length}}}{2}\leq
t\leq\frac{\tau_{\textrm{length}}}{2}\\\
0&t>\frac{\tau_{\textrm{length}}}{2}\,,\end{cases}$ (14)

where $f_{s}$ denotes the field strength, i.e., the plateau height of the
rectangular pulse, and $\tau_{\textrm{length}}$ its duration.

The AC-Stark pulses are tuned below the exciton line by
$\omega_{\textrm{ACS,X}}:=\omega_{\textrm{ACS}}-\omega_{\textrm{X}}$ that is
chosen within the range of validity of the RWA. The resulting shift of the
exciton line can be calculated from the energies of the laser dressed states.

The QD is coupled to LA phonons Besombes2001 ; Borri2001 ; Krummheuer2002 ;
Axt2005 ; Reiter2014 ; Reiter2019 .

$\displaystyle
H_{\textrm{Ph}}=\sum_{\mathbf{q}}\hbar\omega_{\mathbf{q}}b_{\mathbf{q}}^{\dagger}b_{\mathbf{q}}+\sum_{\mathbf{q}}\left(\gamma_{\mathbf{q}}b_{\mathbf{q}}^{\dagger}+\gamma_{\mathbf{q}}^{*}b_{\mathbf{q}}\right)|X\rangle\langle
X|\,,$ (15)

$b_{\mathbf{q}}^{\dagger}$ and $b_{\mathbf{q}}$ are the (bulk) phonon
operators with wave vector $\mathbf{q}$ and energy $\hbar\omega_{\mathbf{q}}$.
The deformation potential-type coupling to the electronic state is denoted by
$\gamma_{\mathbf{q}}$. This Hamiltonian is of the so-called pure dephasing-
type Mukamel1995 ; Krummheuer2002 . Many well-known phenomena emerge from the
interaction described by this Hamiltonian, e.g., the phonon sideband in the QD
emission spectrum Besombes2001 ; McCutcheon2016 , the damping of the Rabi
oscillations Machnikowski2004 ; Ramsay2010a  as well as the renormalization of
their frequency Kruegel2005 ; Ramsay2010b . Since our treatment of this
Hamiltonian is numerically complete, all of these phenomena are included in
our results.

Finally, we take radiative recombination of the excitons with rate $\gamma$
and cavity loss processes with rate $\kappa$ into account by introducing
Markovian Lindblad-type operators

$\displaystyle\mathcal{L}_{O,\Gamma}\bullet=\Gamma\left(O\bullet
O^{\dagger}-\frac{1}{2}\left\\{\bullet,O^{\dagger}O\right\\}_{+}\right)\,,$
(16)

where $\\{\cdot,\cdot\\}_{+}$ denotes the anti-commutator. $O$ is a system
operator and $\Gamma$ the decay rate of the associated loss process.

The full Hamiltonian then reads as

$\displaystyle H_{\textrm{full}}=H+H_{\text{Ph}}$ (17)

with the system Hamiltonian $H$ as defined in Sec. II. The dynamics of these
systems are then described by the Liouville-von Neumann equation

$\displaystyle\frac{\partial}{\partial t}\rho=\,$
$\displaystyle-\frac{i}{\hbar}\\{H_{\textrm{full}},\rho\\}_{-}+\mathcal{L}_{a,\kappa}\rho+\mathcal{L}_{\sigma_{X},\gamma}\rho\,,$
(18)

where $\\{\cdot,\cdot\\}_{-}$ denotes the commutator.

A path-integral formalism Makri1995a ; Makri1995b ; Vagov2011 ; Barth2016  is
used to solve Eq. (18) in a numerically complete manner. Tracing out the
phonon degrees of freedom analytically yields a phonon induced memory kernel
for the subsystem of interest $H$ in Eq. (17). We call a solution ”numerically
complete” if a finer time discretization and considering a longer memory do
not change the result noticeably. Since the states considered in this paper
are product states of the QD and number states of the cavity mode and
therefore quite numerous, no solution within the path-integral framework could
be obtained without the advances presented in Ref. Cygorek2017, .

##  Appendix B Definition of the optical Wigner function

The optical Wigner function, which is a function of the complex coherent
amplitude $\alpha$, can be obtained as Louisell1973 ; Cahill1969a ;
Cahill1969b

$\displaystyle
W(\alpha)=2\textrm{Tr}\left[\rho_{\textrm{photon}}D(\alpha)(-1)^{a^{\dagger}a}D(-\alpha)\right]\,,$
(19)

where $\rho_{\textrm{photon}}$ is the photonic density matrix of the system
and $D(\alpha)$ the coherent displacement operator. The photonic density
matrix is obtained by tracing out the phonon and QD degrees of freedom:

$\displaystyle\rho_{\textrm{photon}}=\textrm{Tr}_{\textrm{QD}}\left[\textrm{Tr}_{\textrm{Ph}}(\rho)\right]\,.$
(20)

Using the Fock basis and introducing
$\rho^{\prime}_{\textrm{photon}}(\alpha):=D(-\alpha)\rho_{\textrm{photon}}D(\alpha)$,
this expression simplifies to

$\displaystyle
W(\alpha)=2\sum_{n}[\rho^{\prime}_{\textrm{photon}}(\alpha)]_{nn}(-1)^{n}\,.$
(21)

##  Appendix C Parameters

For the numerical calculations we use typical parameters for self-assembled
strongly confined GaAs/In(Ga)As QDs Krummheuer2005 ; Cygorek2017 . Other
relevant parameters are summarized in Tab. 2. Assuming a mode frequency of
$\hbar\omega_{\textrm{C}}=1.5\,$eV, the cavity loss rate $\kappa$ corresponds
to a quality factor $Q\approx 268,000$, which has been reported in the
experiments in Ref. Schneider2016, . The phonons are assumed to be initially
in thermal equilibrium at a temperature of $T=4\,$K, whenever phonon effects
are considered in this work.

On a time scale of $\approx 3\,$ps, the phonon induced memory kernel for
GaAs/In(Ga)As QDs of $6\,$nm diameter at $T=4\,$K decays to zero Vagov2011 ;
Barth2016 ; Cygorek2017 . To obtain numerically complete converged results, a
two-grid strategy is employed for the time discretization. Details can be
found in Appendix A 3 of Ref. Cosacchi2020b, .

Table 2: Relevant system parameters. QD-cavity coupling (meV) | $\hbar g$ | 0.1
---|---|---
Cavity loss rate (ps-1) | $\kappa$ | 0.0085
QD radiative decay rate (ps-1) | $\gamma$ | 0.001

##  Appendix D Calculation of the phase in the CAD

The phase $\phi$ in Eq. (11) is the phase of the coherent state which is
prepared in the cavity mode by the initial laser pulse driving the cavity.
Since the pulse used to prepare the coherent state is short compared with the
scale of the dynamics induced by the coupling to the QD, we can neglect the
latter in the analysis of the preparation. In a frame co-rotating with the
laser frequency, the Hamiltonian thus reduces to

$\displaystyle H_{\textrm{driving}}(t)=\,$
$\displaystyle-\frac{\hbar}{2}(f_{\textrm{p}}^{*}(t)a+f_{\textrm{p}}(t)a^{\dagger})\,.$
(22)

Up to second order in the time increment $\Delta t$, the time evolution
operator reads

$\displaystyle U(t+\Delta t,t)=$
$\displaystyle\,e^{-\frac{i}{\hbar}H_{\textrm{driving}}(t)\Delta t}$
$\displaystyle=$ $\displaystyle D(u(t)\Delta t)$ (23)

with $u(t):=\frac{i}{2}f_{\textrm{p}}(t)$ and the coherent displacement
operator $D$. Using the relation
$D(\alpha)D(\beta)=\exp{[\textrm{Im}(\alpha\beta^{*})]}D(\alpha+\beta)$ and
noting that the phase vanishes since $u(t)$ is purely imaginary for real pulse
envelopes, one obtains for a pulse with center $t_{c}$ that is chosen such
that at $0$ and $\tau_{\textrm{max}}$ the envelope is essentially zero (again
up to second order in $\Delta t$)

$\displaystyle U(\tau_{\textrm{max}},0)=D(\alpha(\tau_{\textrm{max}}))$ (24)

with

$\displaystyle\alpha(\tau_{\textrm{max}})=$
$\displaystyle\,\int_{0}^{\tau_{\textrm{max}}}u(t)dt$ $\displaystyle=$
$\displaystyle\frac{i}{2}\int_{0}^{\tau_{\textrm{max}}}\frac{\Theta}{\sqrt{2\pi}\sigma}e^{-\frac{(t-t_{c})^{2}}{2\sigma^{2}}}dt\approx\frac{i}{2}\Theta=\frac{\Theta}{2}e^{-i\phi}$
(25)

with $\phi=3\pi/2$.

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