# Ladder of Loschmidt anomalies in the deep strong-coupling regime of a
qubit-oscillator system

J. M. Betancourt  Departamento de Física, Universidad de los Andes, A.A. 4976,
Bogotá Distrito Capital, Colombia  F. J. Rodríguez  Departamento de Física,
Universidad de los Andes, A.A. 4976, Bogotá Distrito Capital, Colombia  N. F.
Johnson  Department of Physics, George Washington University, Washington D.C.
20052, U.S.A.  L. Quiroga  Departamento de Física, Universidad de los Andes,
A.A. 4976, Bogotá Distrito Capital, Colombia<EMAIL_ADDRESS>
###### Abstract

We uncover a remarkably regular array of singularity-like structures within
the deep strong-coupling limit of qubit-oscillator (e.g. light-matter) systems
described by the quantum Rabi model, as a function of time and coupling
strength. These non-analytic anomalies in the Loschmidt amplitude (echoes)
suggest the existence of new forms of dynamical phase transition within this
deep strong-coupling regime. The key feature whereby the initial state
collapses into orthogonal states at select values of the interaction strength
and select times, may be used to enhance – or attack – quantum information
processing or computation schemes that rely on removing – or retaining – a
given quantum state.

##  I Introduction

Qubit-oscillator systems (e.g. confined light-matter interactions or spin-
nanomechanical resonators) play a key role in physics and its applications,
including the burgeoning field of quantum computing [1] as well as quantum
optics [2], molecular physics [3] and hybrid quantum devices [4]. This
motivates the search for a deeper theoretical understanding of their quantum
dynamics. The quantum Rabi model (QRM) [5] describes the fundamental building
block for such systems – a single two-level qubit (e.g. atom, quantum dot or
NV center) coupled to an oscillator (e.g. photon or phonon) mode – whereas the
Dicke model [6] generalizes this to multiple qubits. Although there is an
exact solution for the spectrum of the QRM [7, 8], the resulting expression is
difficult to manipulate for practical dynamical applications.

Perturbative methods have been developed for different values of the qubit-
oscillator (e.g. light-matter) coupling strength $g$, leading to a
classification of the system into regimes depending on the value of $g$ [9].
When $g$ is considerably smaller than the natural frequencies of the qubit and
the bosonic mode, i.e. the weak coupling regime, the QRM is well described by
the exactly solvable Jaynes-Cummings model [10]. When $g$ is a significant
fraction of these frequencies, the system lies in the strong coupling (SC) or
ultrastrong coupling (USC) regimes. When the coupling strength becomes larger
than the natural frequencies of the system, the system lies in the deep strong
coupling (DSC) regime. Although experiments have been performed in the SC
regime for decades via cavity QED [11], recent advances in achieving the USC
[12, 13, 14] and DSC [15, 16, 17] experimentally have brought these regimes to
the forefront of attention.

The current capability to experimentally vary the strength of the coupling $g$
over a wide range of values raises the possibility of exploring non-analytic
behaviors that point toward new phase transitions [18]. Such behaviors could
also arise as a function of time in a driven system, i.e. a dynamical quantum
phase transition (DQPT) [19]. One of the key quantities in the study of DQPTs
is the Loschmidt amplitude (LA) which is the overlap between the initial
quantum state of the system and the time-evolved one. Its zeros play the same
role in the complex time plane as the Yang-Lee zeros of the partition function
do in the complex temperature plane [20]. Sharp, non-analytic features in its
time-evolution suggest dynamical phase transitions [19]. Recent theoretical
work on the single qubit QRM has shown several quantum phase transition
behaviors in the off-resonant limiting case where the ratio between the qubit
splitting energy and the mode goes to infinity at the same time as $g$ goes to
infinity [21]. In this limit, the ground-state of the QRM undergoes a second-
order phase transition whereas excited states show signatures of an excited
state quantum phase transition [22]. Experimental realizations with a single
trapped ion have already been reported [23], and a DQPT has been proposed in
the off-resonant limit [24]. More generally, studies across different types of
qubit-oscillator systems have suggested that the ultra-strong coupling regime
is relevant to wide a range of fundamental physics phenomena such as virtual
photon accessibility in the Casimir effect [25, 26] and novel applications
including optomechanical nanosystems [27, 28, 29].

This paper shows that deep strong coupling generates a new regime of non-
analytic features suggestive of dynamical phase transitions, even for the
simplest of systems (QRM). Our full numerical calculations show that
approximate zeros of the LA (echoes) emerge in a staircase structure as a
function of the coupling strength $g$ and time $t$ after the onset of this
coupling. In addition to being relevant to the fundamental quantum speed limit
for information processing, i.e. minimal time for the evolution to an
orthogonal state [30, 31], these orthogonality-points in $(g,t)$ space can
provide well-defined stages in a quantum computation or information processing
scheme. For situations where they are undesirable, our findings provide the
combinations of $g$ and $t$ to avoid.

This paper is organized as follows. We begin by introducing the system
Hamiltonian in Sec. II, as well as the Loschmidt echo for an initial state
with no photons and no qubit excitation. In Sec. III, we show numerical
results where Loschmidt echo singularities appear. In addition, an
approximation using the zero qubit splitting case and an equation of motion
involving the derivatives of the echo are derived. Concluding remarks are
given in Sec. IV.

##  II Hamiltonian model

We consider arguably the simplest example of a driven qubit-oscillator system:
a two-level system with level splitting $\omega_{0}$ described by Pauli
operators $\hat{\sigma}_{z}$, $\hat{\sigma}^{+}$ and $\hat{\sigma}^{-}$,
coupled with a time dependent strength $g(t)$ to a single boson mode (QRM)
with frequency $\omega_{c}$:

$\hat{H}(t)=\omega_{c}\hat{a}^{{\dagger}}\hat{a}+\frac{\omega_{0}}{2}\hat{\sigma}_{z}+g(t)(\hat{a}^{{\dagger}}+\hat{a})(\hat{\sigma}^{+}+\hat{\sigma}^{-}),$
(1)

where $\hbar=1$. The operators $\hat{a}^{{\dagger}}$ $\hat{a}$ denote boson
ladder operators, which for simplicity we call photon operators.

There is a controversy in the literature about the contribution of the
quadratic term of vector potential $A$, related with the quantum phase
transitions for the Dicke model [32]. By contrast, our paper is focused on the
dynamical overlap between the initial state and the state later in time – it
is not focused on this transition. Therefore, we eliminate a possible $A^{2}$
term by performing a Bogoliugov-Hopfield transformation on the photon
operators.

This Hamiltonian commutes with the parity operator
$\hat{P}=e^{i\pi(\hat{a}^{{\dagger}}\hat{a}+\frac{1}{2}(1+\hat{\sigma}_{z}))}$.
Given an initial state with zero excitations in both qubit and oscillator sub-
systems $|\psi(0)\rangle=|0,0\rangle$ (i.e. even parity) the time-evolution
remains confined to the even Hilbert subspace. Hence a basis can be chosen as
$\\{|n,m\rangle\\}$ with $n+m=$ even and $n=0,1,2,...$ denoting the number of
boson excitations whereas $m=0,1$ corresponds to the number of qubit
excitations. We can denote the basis vectors simply by
$\\{|n\rangle\\}\equiv\\{|n,m\rangle\\}$, noting that $m=0$ when $n$ is even
whereas $m=1$ otherwise. Hence Eq. (1) becomes:

$\hat{H}(t)=\omega_{c}\hat{a}^{{\dagger}}\hat{a}-\frac{\omega_{0}}{2}e^{i\pi\hat{a}^{{\dagger}}\hat{a}}+g(t)(\hat{a}^{{\dagger}}+\hat{a}).$
(2)

This form explicitly shows the strong anharmonicity, mediated by the qubit, of
the oscillator sub-system. The LA

$\displaystyle{\cal L}(t)=\langle\psi(0)|\hat{U}(t,0)|\psi(0)\rangle,$ (3)

calculates the fidelity of the time-evolved state
$|\psi(t)\rangle=\hat{U}(t,0)|\psi(0)\rangle$ with respect to the initial
state $|\psi(0)\rangle$. We calculate the final state $|\psi(t)\rangle$ under
Eq. (1) given the initial ground state $|\psi(0)\rangle\equiv|0,0\rangle$.
Furthermore, throughout the paper we consider the simplest quenching case in
which the qubit-cavity coupling strength jumps suddenly from 0 to a constant
value $g>0$ at time 0, and then maintains that constant value for time $t$.
Although the Rabi model is integrable [7], no analytical results are available
for the case of interest here, i.e. the non-equilibrium resonant Rabi model.

Figure 1: A ladder of singularity peaks in the Loschmidt echo emerges in the
deep strong coupling (DSC) regime, as a function of the coupling $g$ and time
$t$ after the coupling onset. Results are shown in two dimensions (2D) and
three dimensions for the resonant case, $\omega_{c}=\omega_{0}$, and a single
cycle time evolution, $\frac{\omega_{c}t}{2\pi}\leq 1$. Figure 2: Parametric
plots of $\mathcal{L}(g,t)$ for the coupling values $g$ where the first three
zeros of $\mathcal{L}$ arise (the $n$-th lowest value is denoted by $g_{n}$).
Arrows indicate the LA evolution as time progresses. Changes of LA chirality
are evident for $g_{2}$ and $g_{3}$. The inset: LA full evolution for
$0\leq\frac{\omega_{c}t}{2\pi}\leq 1$ in a full scale view, for the same
$g_{n}$ values.

We benchmark tested our full numerical calculations using the
$\omega_{0}\rightarrow 0$ limit for which the exact analytic solution is known
[33, 34], noting that our calculations can be generalized to other time-
dependent quenchings [35]. Although this simple limiting case does not show
any signature of a dynamical phase transition, the situation changes
drastically with a finite qubit splitting $\omega_{0}=\omega_{c}$. For
convenience, we use a logarithmic scale by defining a Loschmidt echo rate
(LER) exponent as [19] $\gamma(g,t)=-\log_{10}|{\cal L}(g,t)|$ whereas
corresponding exponents can be defined for derivatives with respect to $t$ and
$g$: $\beta(g,t)=-\log_{10}\left|\frac{\partial{\cal L}(g,t)}{\partial
t}\right|$ and $\chi(g,t)=-\log_{10}\left|\frac{\partial{\cal
L}(g,t)}{\partial g}\right|$.

Figure 3: Boundary between regions of clockwise and counter- clockwise
rotation of the time derivative of the LA (blue solid line) in the complex
plane shown over a 2D projection of Fig 1. The boundary corresponds to
$\partial_{t}\arg(\partial_{t}\mathcal{L}(g,t))=0$. Singularities of $\gamma$
(black dots), $\beta$ (green triangles) and $\chi$ (magenta diamonds) are also
shown. Changes in chirality are more clearly seen in the right panel, where a
red dashed line indicating $\partial_{t}\arg(\mathcal{L}(g,t))=0$ is included.

##  III Loschmidt singularities

In this section we study the time-dependent Loschmidt echo by solving the
quantum propagator numerically for the Rabi model. For a pulsed coupling of
constant height $g$, the dynamics are contained within the complete set of
even parity eigenstates $|E_{m}\rangle$ with energy $E_{m}$ for the QRM
Hamiltonian (Eq. (1)) where ${\cal L}(g,t)=\sum_{m}e^{-iE_{m}t}|\langle
E_{m}|\psi(0)\rangle|^{2}$. Fig. 1 shows our results for LER ($\gamma(g,t)$)
at $\omega_{0}=\omega_{c}$ and $|\psi(0)\rangle=|0,0\rangle$, across a wide
region of the DSC $(g,t)$ plane. This regime – which is largely unexplored –
can only be understood in those systems, if non-perturbative methods are
correctly included in the calculation. We find that using a basis set of up to
750 photon states guarantees convergence. Although a smooth continuum occupies
the left half of the one-period evolution, a ladder of localized singularities
arises for $\gamma(g,t)$ in the right half (see Appendix A for similar
features in $\beta(g,t)$ and $\chi(g,t)$). We emphasize that this ladder of
LER singularities has nothing to do with quenching through an equilibrium
quantum phase transition, since the QRM at resonance doesn’t show the
emergence of any kind of equilibrium phase transition. Therefore, we conclude
that this non-analytic dynamical behavior of the LER at select points in the
$(g,t)$ plane, can be understood as a DQPT when quenching from the uncoupled
qubit-oscillator ground state to a final DSC coupled system – regardless of
the finite size of the matter sub-system.

So far we have pointed out the existence of a ladder of LA echoes. Deeper
insight is gained by inspecting the trajectories of the full LA in the complex
plane. In Fig. 2, we show these trajectories for the first three values of $g$
where zeros can be located, namely, $g_{1}=1.30$ (blue dotted line),
$g_{2}=1.67$ (orange dashed line) and $g_{3}=1.90$ (red solid line). It is
important to note that the chosen values of $g$ have been achieved
experimentally [15, 16, 17]. These trajectories seem to present increasingly
complex winding behaviors, suggesting that information about the structure of
the LA might be embedded into these geometric aspects. This leads us to
conclude that the ladder of LA zeros is intimately connected with DQPTs, which
are well known to have no local order parameter. Instead, DQPTs have been
associated with dynamical topological order parameters which can be extracted
from the Pancharatnam phase of the LA itself [36]. For critical translational
invariant systems, the dynamical topological order parameter is quantized as
an integer number that jumps at critical times. We do not know of any previous
reports of such signatures in a system such as the QRM that does not have an
associated length scale.

The LA exhibits highly nontrivial behavior in terms of its rotation in the
complex plane, which can be captured by analyzing the temporal evolution of
$\arg(\partial_{t}\mathcal{L}(g,t))$. Specifically,
$\partial_{t}\arg(\partial_{t}\mathcal{L}(g,t))>0$ and
$\partial_{t}\arg(\partial_{t}\mathcal{L}(g,t))<0$ correspond to anti-
clockwise and clockwise rotation of the LA velocity, respectively. Fig. 3
shows the rotation directions of the LA and its time derivative in different
regimes, and the location of the singularities of $\gamma$, $\beta$ and
$\chi$. Notice that the singularities of $\gamma$ must lie on the red (dashed)
boundary, whereas the singularities of $\beta$ must always lie on both
boundaries (see Appendix A for details). In the DSC regime, we find
alternating regions of clockwise and anti-clockwise rotations. Fig. 3 reveals
a rich dynamical phase diagram with additional topological phase transitions
which are a different class from those previously reported for the off-
resonant QRM. The phases separated by these topological transitions are
distinguished from each other by this feature, and each can in principle be
assigned a topological number. Interestingly, the boundary between these
regions seems to follow the location of the discrete ladder of LA zeros for
higher values of $g$ (Fig. 3 right panel) indicating a hidden relationship
between changes in this rotational behavior and the emergence of
singularities. Moreover, zeros in the LA derivatives with respect to time
($\beta(g,t)$) and coupling ($\chi(g,t)$) fall on this same undulating border
that separates different LA-chirality regions. This is a key signature of the
emergence of a topological dynamical phase transition.

For the LA and LER exponents, we find a near perfect collapse of numerical
values of $\gamma(g,t)/g^{2}$ for the QRM, with the corresponding analytical
result for the zero qubit splitting (ZQS) limit result given by

$\displaystyle{\cal
L}_{0}(g,t)=e^{i\theta(g,t)}e^{-2\left(\frac{g}{\omega_{c}}\right)^{2}{\rm
sin}^{2}\left(\frac{\omega_{c}t}{2}\right)\ .}$ (4)

This yields

$\displaystyle\gamma_{0}(g,t)/g^{2}\sim{\rm
sin}^{2}\left(\frac{\omega_{c}t}{2}\right)$ (5)

as a universal value which is independent of the specific coupling strength
$g$ (see Appendix B for the expression of $\theta(g,t)$ and additional
details). In the trailing edge of the coupling pulse, there is good agreement
between the exact numerical results and the expression calculated from the
Loschmidt echo rate for a ZQS system $|{\cal L}_{0}(g,t)|$ in Eq. (4),
especially after the kink in $\gamma(g,t)/g^{2}$ (see Fig. 4). This collapse
suggests that the anharmonicity induced by the qubit in the effective
oscillator in Eq. (2), fades away for long pulse durations that place the
system to the right of the wavy chirality border in Fig. 3. This raises the
possibility of designing tailored pulses for which the LA and the
corresponding exponent become identical in the long time pulse limit for the
highly anharmonic resonant qubit-boson case and for the simple harmonic ZQS
system.

Figure 4:  Numerical and ZQS results for $\gamma(g,t)/(g/\omega_{c})^{2}$, for
coupling strength $g$ values corresponding to the first four LA zeros. In the
trailing edge of the coupling pulse, there is a collapse of the exact
numerical results onto the curve for the ZQS (Eq. (5)).

Further insight into the emergence of this LA ladder comes from the following
exact relationship for an arbitrary quantum system [37]

$\displaystyle{\cal L}(t)={\cal L}(t-\tau){\cal L}(\tau)+M(t,\tau)$ (6)

where the memory term is

$\displaystyle
M(t,\tau)=\langle\psi_{0}|\hat{U}(t,\tau)\,\hat{Q}\,\hat{U}(\tau,0)|\psi_{0}\rangle$
(7)

and $\hat{Q}=1-|\psi_{0}\rangle\langle\psi_{0}|$. Using Eqs. (3) and (6), we
obtain an equation of motion for the LA

$\displaystyle g\frac{\partial\,{\cal L}}{\partial\,g}-t\frac{\partial\,{\cal
L}}{\partial\,t}=-\frac{it}{2}{\cal L}(g,t)+{\cal G}(g,t)$ (8)

where

$\displaystyle{\cal G}(g,t)=$ $\displaystyle
it\sum_{n=1}^{\infty}\left[\frac{(-1)^{n+1}-1}{2}+n\right]$
$\displaystyle\int_{0}^{1}dx\,\langle\psi_{0}|e^{-ix\hat{H}(g)t}|n\rangle\langle
n|e^{-i(1-x)\hat{H}(g)t}|\psi_{0}\rangle$

and $\\{|n\rangle\\}\equiv\\{|n,m\rangle\\}$ (see Appendix C for derivation).
The staircase nature of the anomalies (i.e. LA zeros) as a function of $g$ and
$t$, reflects the fact that the LA equation of motion resembles a generalized
Burger’s shock-wave equation in which $\frac{\partial{\cal L}}{\partial
t}-(g/t)\frac{\partial{\cal L}}{\partial g}$ contains a convolution of ${{\cal
L}(g,t)}$ over time plus correction terms. This suggests the propagation of
short-lived, modified shock waves in ${\cal L}(g,t)$ as a function of $g$ and
$t$, as observed.

##  IV Concluding remarks

Our findings have further implications. First, this discrete staircase of LA-
zeros not only appears for the QRM, but it also shows signs of existing for
multi-qubit systems (Dicke model) and with more complicated time varying
couplings [38]. Although a comprehensive treatment of the scaling of such
complex systems remains to be explored, our results open up a path to
tailoring zeros of the LA using a time-dependent interaction and by increasing
the number of qubits.

Second, photon states lacking the vacuum component are predicted to present
highly non-classical features, such as negative values of the Wigner function
and Schrödinger-cat state generation [39]. Our findings demonstrate that a
simple QRM (e.g. single qubit-photon) system is an ideal platform for
realizing such non-classical light states, simply by varying the coupling $g$
and time since onset $t$.

Third, the key feature whereby the initial state collapses into orthogonal
states in a punctuated way as a function of $g$ and $t$, may be used to
enhance – or attack – future quantum information processing or computation
implementations that rely on removing – or retaining – a given quantum state.
This finding could motivate new schemes that exploit this fact for erasing
memory of an initial state. However, it also flags a potential vulnerability
to external attack in quantum information processing schemes that require
maintaining some memory (i.e. non-zero vacuum component) of the initial state.

Finally, we note that decoherence should have a negligible effect on the
ladder of LA anomalies reported here, since they occur within a single cavity
cycle scale of time and in the DSC regime, placing the system in an ultrafast
regime where the unitary treatment discussed above should be appropriate.

###### Acknowledgements.

We thank F.J. Gómez-Ruiz for discussions. J.M.B, F.J.R. and L.Q. are thankful
to Facultad de Ciencias-UniAndes project “Excited State Quantum Phase
Transitions in Driven Models - Part II: Dynamical QPT” (2019-2020) for partial
financial support. N.F.J. was funded by the AFOSR through Grants No.
FA9550-20-1-0382 and No. FA9550-20-1-0383.

##  Appendix A Chirality and singularities of $\beta$ and $\chi$

Figs. 5 and 6 show plots of the behavior of $\beta(g,t)$ and $\chi(g,t)$,
respectively, for the analyzed values of coupling strength and evolution time.

Figure 5: Values of $\beta(g,t)$ for the analyzed values of coupling strength
and evolution time. Figure 6: Values of $\chi(g,t)$ for the analyzed values of
coupling strength and evolution time.

The radius of curvature for a parametric curve in two dimensions
$\mathbf{r}(t)$ is given by

$\displaystyle\kappa(t)=-\frac{[\mathbf{r}^{\prime}(t)\times\mathbf{r}^{\prime\prime}(t)]_{z}}{\lVert\mathbf{r}^{\prime}(t)\rVert^{3/2}},$
(9)

where $[\mathbf{A}]_{z}$ is the $z$ component of $\mathbf{A}$. This is applied
to our case by taking – for a given value of the coupling strength – the curve
described by the following position vector:

$\displaystyle\mathbf{r}(t)=\begin{pmatrix}\textrm{Re}(\mathcal{L}(g,t))\\\
\textrm{Im}(\mathcal{L}(g,t))\end{pmatrix}.$ (10)

It follows that the time derivatives of $\mathbf{r}(t)$ are given by the real
and imaginary components of the time derivatives of the LA. If
$\mathcal{L}(g,t)$ is expressed as

$\displaystyle\mathcal{L}(g,t)=A(g,t)\exp(i\theta(g,t)),$

then we have that

$\displaystyle\partial_{t}\theta(g,t)=\frac{[\mathbf{r}(t)\times\mathbf{r}^{\prime}(t)]_{z}}{\lVert\mathbf{r}(t)\rVert^{2}}.$

Using a similar argument applied to $\partial_{t}\mathcal{L}$, we obtain that

$\displaystyle\kappa(t)=-\lVert\mathbf{r}(t)\rVert^{1/2}\partial_{t}\arg(\partial_{t}\mathcal{L}(g,t)).$

This means that the calculated chiralities will have the opposite sign to the
defined curvature, giving a geometric interpretation of the results in Fig. 3.

##  Appendix B Zero qubit splitting case

In this $\omega_{0}\rightarrow 0$ limit, we consider the simplest quenching
case in which the qubit-cavity coupling strength jumps suddenly from $0$ to a
constant $g>0$ at time $0$, and then maintains that constant value for time
$t$. The operator

$\hat{U}(t,0)=e^{i\omega_{c}t\hat{a}^{{\dagger}}\hat{a}}e^{-\left[\int_{0}^{t}dsK(s)\frac{dK^{*}(s)}{ds}\right]}e^{iK(t)\hat{a}^{{\dagger}}}e^{iK^{*}(t)\hat{a}},$
(11)

where

$K(t)=\int_{0}^{t}ds\,g(s)e^{i\omega_{c}s}$ (12)

encodes the time variation of the qubit-photon coupling strength. Thus

$\displaystyle\hat{U}(t,0)|\psi(0)\rangle=e^{i\omega_{c}t\hat{a}^{{\dagger}}\hat{a}}e^{i\theta(t)}\hat{D}\left(iK(t)\right)|\psi(0)\rangle,$
(13)

where
$\hat{D}\left(\alpha\right)=e^{-\frac{|\alpha|^{2}}{2}}e^{\alpha\hat{a}^{{\dagger}}}e^{-\alpha^{*}\hat{a}}$
is the displacement operator and
$\theta(t)=\textrm{Im}\\{\int_{0}^{t}ds\,K(s)\frac{dK^{*}(s)}{ds}\\}$. Hence

$\displaystyle{\cal L}_{0}(t)$ $\displaystyle=$
$\displaystyle\langle\psi(0)|\hat{U}(t,0)|\psi(0)\rangle$ (14)
$\displaystyle=$ $\displaystyle
e^{i\theta(t)}\langle\psi(0)|e^{i\omega_{c}t\hat{a}^{{\dagger}}\hat{a}}\hat{D}\left(iK(t)\right)|\psi(0)\rangle.$

Given an initial Fock state with $n$ photons, $|\psi(0)\rangle=|n\rangle$,
this yields

$\displaystyle{\cal
L}_{0}(t)=e^{i\theta(t)}e^{in\omega_{c}t}e^{-\frac{1}{2}|K(t)|^{2}}L_{n}\left(|K(t)|^{2}\right),$
(15)

where $L_{n}\left(x\right)$ is the Laguerre polynomial of order $n$. Starting
from the initial ground state $|\psi(0)\rangle=|g,0\rangle$ yields

$\displaystyle{\cal L}_{0}(t)=e^{i\theta(t)}e^{-\frac{1}{2}|K(t)|^{2}}.$ (16)

which shows that the LA is the projection of the coherent state
$|iK(t)\rangle=\hat{D}\left(iK(t)\right)|0\rangle$ onto the initial vacuum
state. The Loschmidt decay factor $K(\tau)$ becomes

$\displaystyle K(t)=\frac{2g}{\omega_{c}}{\rm
sin}\left(\frac{\omega_{c}t}{2}\right)e^{i\frac{\omega_{c}t}{2}}$ (17)

and so

$\displaystyle{\cal
L}_{0}(g,t)=e^{i\theta(g,t)}e^{-2\left(\frac{g}{\omega_{c}}\right)^{2}{\rm
sin}^{2}\left(\frac{\omega_{c}t}{2}\right)},$ (18)

with the phase

$\displaystyle\theta(g,t)=\left(\frac{g}{\omega_{c}}\right)^{2}\left[\omega_{c}t-{\rm
sin}\left(\omega_{c}t\right)\right].$ (19)

##  Appendix C Equation of motion for the LA

Using $\mathcal{L}(g,t)=\braket{\psi_{0}}{e^{-i\hat{H}t}}{\psi_{0}}$, we
obtain

$\displaystyle\frac{\partial\mathcal{L}}{\partial
t}=-i\braket{\psi_{0}}{\hat{H}e^{-i\hat{H}t}}{\psi_{0}}.$ (20)

For the derivative with respect to $g$, an important aspect is that $\hat{H}$
and $d\hat{H}/dg$ do not commute which makes the computation of this
derivative more complicated. Following previous results on the derivatives for
exponential operators [40]:

$\displaystyle\frac{de^{A(t)}}{dt}=\int_{0}^{1}dx\,e^{xA(t)}\left(\frac{dA(t)}{dt}\right)e^{(1-x)A(t)}.$
(21)

Applying this result to our case, we obtain

$\displaystyle\frac{\partial\mathcal{L}}{\partial g}=$ $\displaystyle-
it\int_{0}^{t}dt^{\prime}\,\braket{\psi_{0}}{e^{-i\hat{H}t^{\prime}}\left(\frac{d\hat{H}}{dg}\right)e^{-i\hat{H}(t-t^{\prime})}}{\psi_{0}}.$
(22)

Using the expression for the Hamiltonian given in Eq. (1), we obtain the
following result for $g\neq 0$:

$\displaystyle-
it\frac{d\hat{H}}{dg}=-i\frac{t}{g}\left(\hat{H}-\hat{N}+\frac{1}{2}\right).$
(23)

Combining this with the result for $\partial\mathcal{L}/\partial t$, we obtain

$\displaystyle\frac{\partial\mathcal{L}}{\partial g}=$
$\displaystyle\frac{t}{g}\frac{\partial\mathcal{L}}{\partial t}$ (24)
$\displaystyle+\frac{i}{g}\int_{0}^{t}dt^{\prime}\,\braket{\psi_{0}}{e^{-i\hat{H}t^{\prime}}\left(\hat{N}-\frac{1}{2}\right)e^{-i\hat{H}(t-t^{\prime})}}{\psi_{0}}.$

Introducing a complete set of states after $\hat{N}+1/2$ yields

$\displaystyle g\frac{\partial\mathcal{L}}{\partial
g}-t\frac{\partial\mathcal{L}}{\partial t}=$ (25)
$\displaystyle+i\sum_{n=0}^{\infty}\left[\frac{(-1)^{n+1}}{2}+n\right]\int_{0}^{t}dt^{\prime}\,\braket{\psi_{0}}{e^{-i\hat{H}t^{\prime}}}{n}\braket{n}{e^{-i\hat{H}(t-t^{\prime})}}{\psi_{0}}.$

Extracting the first term:

$\displaystyle g\frac{\partial\mathcal{L}}{\partial
g}-t\frac{\partial\mathcal{L}}{\partial
t}=-\frac{i}{2}\int_{0}^{t}dt^{\prime}\,\mathcal{L}(g,t^{\prime})\mathcal{L}(g,t-t^{\prime})$
(26)
$\displaystyle+i\sum_{n=1}^{\infty}\left[\frac{(-1)^{n+1}}{2}+n\right]\int_{0}^{t}dt^{\prime}\,\braket{\psi_{0}}{e^{-i\hat{H}t^{\prime}}}{n}\braket{n}{e^{-i\hat{H}(t-t^{\prime})}}{\psi_{0}}.$

Using Eq. (6), we obtain the following result:

$\displaystyle\mathcal{L}(g,t^{\prime})\mathcal{L}(g,t-t^{\prime})=$
$\displaystyle\mathcal{L}(g,t)-\sum_{n=1}^{\infty}\braket{\psi_{0}}{e^{-i\hat{H}(t-t^{\prime})}}{n}\braket{n}{e^{-i\hat{H}t^{\prime}}}{\psi_{0}}.$
(27)

and hence the following, which leads to Eq. (8):

$\displaystyle g\frac{\partial\mathcal{L}}{\partial
g}-t\frac{\partial\mathcal{L}}{\partial t}=-\frac{it}{2}\mathcal{L}(g,t)$ (28)
$\displaystyle+i\sum_{n=1}^{\infty}\left[\frac{(-1)^{n+1}-1}{2}+n\right]\int_{0}^{t}dt^{\prime}\,\braket{\psi_{0}}{e^{-i\hat{H}t^{\prime}}}{n}\braket{n}{e^{-i\hat{H}(t-t^{\prime})}}{\psi_{0}}.$

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# A Bijection Between Strongly Stable and Totally Symmetric Partitions

Seth Ireland  Dept. of Mathematics, Colorado State University, Fort Collins,
CO, USA<EMAIL_ADDRESS>
###### Abstract.

Artinian monomial ideals in $d$ variables correspond to $d$-dimensional
partitions. We define $d$-dimensional strongly stable partitions and show that
they correspond to strongly stable ideals in $d$ variables. We then show a
bijection between strongly stable partitions and totally symmetric partitions
which preserves the side length of the minimal bounding box.

Keywords: Generic Initial Ideal, Borel Group, Borel Ideal, Symmetric Monomial
Ideal, Strongly Stable Ideal, Strongly Stable Partition, Totally Symmetric
Plane Partition, Plane Partition, Enumeration

MSC classification: Primary: 05A17; Secondary: 13F55

##  1\. Introduction

The subgroup of $GL_{d}(K)$ consisting of upper triangular matrices is called
the Borel group. Ideals which are fixed by the Borel group are monomial ideals
called Borel-fixed ideals. Generic initial ideals are always Borel-fixed, and
in characteristic zero, ideals are Borel-fixed if and only if they are
strongly stable [4]. So the generic initial ideals over a field of
characteristic zero are exactly the strongly stable ideals [1].

Artinian monomial ideals in $K[x_{1},\dots,x_{d}]$ correspond naturally to
$d$-dimensional partitions by considering the monomials which are not in the
ideal. We can think of these as $d$-dimensional stacks of blocks in a
($d$-dimensional) corner. Totally symmetric partitions are $d$-dimensional
partitions which are fixed by the symmetric group $S_{d}$.

In this paper, we define strongly stable partitions (Definition 2.5), show
that they naturally correspond to strongly stable ideals (Proposition 4.5),
and finally show a bijection with totally symmetric partitions which preserves
the side length $n$ of the smallest box containing the partitions (Theorem 1).

##  2\. Partitions

Let $d$ be a positive integer and let $\mathbb{N}=\\{0,1,2,3,\dots\\}$.

######  Definition 2.1.

A $d$-dimensional partition is a finite subset $P\subset\mathbb{N}^{d}$ such
that

$(a_{1},a_{2},\dots,a_{j},\dots,a_{d})\in
P\implies(a_{1},a_{2},\dots,a_{j}-1,\dots,a_{d})\in P$

for all $1\leq j\leq d$ if $a_{j}>0$. Refer to the elements of a partition as
cells.

Denote the set of $d$-dimensional partitions with largest coordinate of any
cell equal to $n-1$ by $\mathcal{P}_{d}(n)$. We think of these as
$d$-dimensional partitions which fit inside a $d$-dimensional box of side
length $n$ and also touch at least one edge of the box. Note that the ”boxes”
we refer to are $n\times\cdots\times n$ boxes of dimension $d$ (every side of
the box is the same length). Integer partitions correspond exactly with the
2-dimensional partitions.

######  Example 2.2.

For example, consider the integer partition 7=3+2+1+1. We can represent the
integer partition with the Ferrers diagram below,

&

which depicts the 2-dimensional partition

$P=\\{(0,0),(0,1),(0,2),(1,0),(1,1),(2,0),(3,0)\\}\in\mathcal{P}_{2}(4).$

######  Example 2.3.

A 3-dimensional partition is commonly called a plane partition. An example of
a plane partition of 10 is

$P=\\{(0,0,0),(0,0,1),(0,1,0),(0,1,1),(1,1,0),(1,0,1),(1,2,0),(0,2,0),(2,0,0)\\}\in\mathcal{P}_{3}(3)$

which we can visualize with the diagram below

Plane partitions are also commonly represented in matrix notation. In matrix
notation, the partition above could be written

$\begin{matrix}2&2&1\\\ 2&1\\\ 1&1\end{matrix}$

######  Definition 2.4.

Let $P$ be a $d$-dimensional partition. For a cell
$\alpha=(a_{1},\dots,a_{d})\in P$, define the $j$th arm length to be the
largest integer $h_{j}$ such that
$\alpha=(a_{1},a_{2},\dots,a_{j}+h_{j},\dots,a_{d})\in P$. Denote the vector
of arm lengths by $H(\alpha)=(h_{1},\dots,h_{d})$ and call this the Hook
vector of $\alpha$.

######  Definition 2.5.

A strongly stable partition is a partition for which every cell’s Hook vector
is weakly increasing.

We will denote the set of strongly stable partitions which fit inside a box of
side length $n$ by

$\tilde{\mathcal{B}}_{d}(n):=\\{P\mid P\in\mathcal{P}_{d}(n)\text{ and $P$ is
strongly stable}\\}$

######  Example 2.6.

An example of a 2-dimensional strongly stable partition
$P\in\tilde{\mathcal{B}}_{2}(7)$ is given below with Hook vectors given inside
each cell.

& 0,0

0,1

0,2

1,3 0,0

2,4 1,1 0,0

2,5 1,2 0,1

3,6 2,3 1,2 0,0

Note that 2-dimensional strongly stable partitions are exactly the integer
partitions with no repeats. These are commonly called strict partitions.

######  Example 2.7.

An example of a 3-dimensional strongly stable partition
$P\in\tilde{\mathcal{B}}_{3}(4)$ is given below. One can verify that the Hook
vectors of each cell are weakly increasing.

######  Definition 2.8.

A totally symmetric partition is a partition for which $\alpha\in
P\implies\pi(\alpha)\in P$ for all $\pi\in S_{d}$.

We denote the set of totally symmetric partitions which fit inside a box of
side length $n$ by

$\tilde{\mathcal{T}}_{d}(n):=\\{P\mid P\in\mathcal{P}_{d}(n)\text{ and $P$ is
totally symmetric}\\}$

######  Example 2.9.

An example of a 2-dimensional totally symmetric partition
$P\in\tilde{\mathcal{T}}_{2}(7)$ is given below.

&

Note that 2-dimensional totally symmetric partitions are commonly called self-
conjugate partitions.

######  Example 2.10.

An example of a 3-dimensional totally symmetric partition (totally symmetric
plane partition) $P\in\tilde{\mathcal{T}}_{3}(4)$ is shown below.

##  3\. Monomial Ideals

Fix a positive integer $n$ and let $K$ be a field of characteristic zero. We
will often refer to monomials by their multi-exponent notation
$\textbf{x}^{\alpha}:=x_{1}^{a_{1}}\cdots x_{d}^{a_{d}}\in
K[x_{1},\dots,x_{d}]$. In this notation, let
$deg(\alpha)=\sum_{i=1}^{d}a_{i}$. A monomial ideal is an ideal $I\subset
K[x_{1},\dots,x_{d}]$ which is generated by monomials. Every monomial ideal
$I\subset K[x_{1},\dots,x_{d}]$ has a unique minimal subset of monomial
generators $G(I)$.

Elements $(a_{i,j})\in GL_{d}(K)$ induce a linear automorphism on
$K[x_{1},\dots,x_{d}]$

$f(x_{1},\dots,x_{d})\mapsto
f\big{(}\sum_{i=1}^{d}a_{i,1}x_{i},\dots,\sum_{i=1}^{d}a_{i,d}x_{i}\big{)}$

The Borel group is the subgroup of $GL_{d}(K)$ consisting of upper triangular
matrices. The ideals which are fixed by the action of the Borel group on the
variables $x_{1},\dots,x_{d}$ are called Borel-fixed ideals.

######  Definition 3.1.

Let $I\subset K[x_{1},\dots,x_{d}]$ be a monomial ideal. If
$m\frac{x_{i}}{x_{j}}\in I$ for all $m\in I$ with $x_{j}\mid m$ and $i<j$,
then $I$ is called strongly stable.

We will refer to the defining property of strongly stable ideals as the
variable exchange condition.

######  Proposition 3.2.

Let $I$ be a monomial ideal. If $char(K)=0$, then $I$ is strongly stable iff
$I$ is Borel-fixed.

Note that we are implicitly using the variable order
$x_{1}>x_{2}>\cdots>x_{d}$. By shuffling this order, we would get different
(but equivalent) sets of strongly stable ideals. The next proposition shows
that we can check if $I$ is strongly stable by checking the variable exchange
condition on generators of $I$.

######  Proposition 3.3.

Let $I\subset K[x_{1},\dots,x_{d}]$ be a monomial ideal. Suppose that for all
$m\in G(I)$ and all integers $1\leq i<j\leq d$ such that $x_{j}\mid m$, we
have $m\frac{x_{i}}{x_{j}}\in I$. Then $I$ is strongly stable.

See [4] for proofs of the previous two propositions.

######  Example 3.4.

The ideal $I=(x^{4},x^{3}y,x^{2}y^{3},xy^{4},y^{7})$ is strongly stable,
because $x^{3}y\frac{x}{y}=x^{4}\in I$, $x^{2}y^{3}\frac{x}{y}=x^{3}y^{2}\in
I$, $xy^{4}\frac{x}{y}=x^{2}y^{3}\in I$, and $y^{7}\frac{x}{y}=xy^{6}\in I$.

Given a set of monomials
$A=\\{\textbf{x}^{\alpha_{1}},\dots,\textbf{x}^{\alpha_{r}}\\}$, we can
consider the minimal subset of $A$ which generates the same ideal. We will use
the notation $min(A)$ to refer to this minimal subset.

######  Definition 3.5.

Let $m\in K[x_{1},\dots,x_{d}]$ be a monomial. A Borel move is an operation
that sends $m$ to a monomial
$m\frac{x_{i_{1}}}{x_{j_{1}}}\cdots\frac{x_{i_{s}}}{x_{j_{s}}}$, where
$i_{t}<j_{t}$ and $m$ is divisible by $x_{j_{t}}$ for all $t$.

######  Definition 3.6.

A monomial ideal $I\subset K[x_{1},\dots,x_{d}]$ is a strongly stable ideal if
it is closed under Borel moves.

######  Definition 3.7.

Let $A$ be a subset of monomials. Define $Borel(A)$ to be the smallest
strongly stable ideal containing $A$. We call the monomials in $A$ Borel
generators of $Borel(A)$.

######  Proposition 3.8.

Every strongly stable ideal $I$ has a unique minimal set of Borel generators.
Refer to this set as $Bgens(I)$.

######  Proposition 3.9.

Suppose $I$ is a strongly stable ideal and $m\in I$. Then $m\in Bgens(I)$ iff
for all $x_{q}$ dividing $m$, $\frac{m}{x_{q}}\notin I$ and
$m\frac{x_{q+1}}{x_{q}}\notin I$.

Definitions 3.5, 3.6, 3.7 and Propositions 3.8 and 3.9 are due to [1]. Note
that definitions 3.5 and 3.6 are equivalent to definition 3.1. Also, in [1],
they refer to strongly stable ideals as Borel ideals. We have chosen to use
the term strongly stable in this paper.

######  Example 3.10.

For the ideal $I=(x^{4},x^{3}y,x^{2}y^{3},xy^{4},y^{7})$,

$Bgens(I)=\\{x^{3}y,xy^{4},y^{7}\\}$

######  Definition 3.11.

A monomial ideal $I$ is symmetric if it is closed under the action of the
symmetric group $S_{d}$ on the variables.

######  Proposition 3.12.

A monomial ideal $I$ is symmetric iff $G(I)$ is closed under the action of the
symmetric group $S_{d}$ on the variables.

###### Proof.

($\Rightarrow$) Let $g\in G(I)$ and $\pi\in S_{d}$. Since $I$ is symmetric,
$\pi(g)\in I$. To see that $\pi(g)\in G(I)$, suppose not. Then there exists
$j$ so that $\frac{\pi(g)}{x_{j}}\in I$. It follows that
$\pi^{-1}(\frac{\pi(g)}{x_{j}})=\frac{g}{x_{\pi^{-1}(j)}}$ so $g\notin G(I)$.
Contradiction.

($\Leftarrow$) Let $u\in I$. Then there exists some $g\in G(I)$ so that $g\mid
u$. It follows that $\pi(g)\mid\pi(u)$ for all $\pi\in S_{d}$, so $\pi(u)\in
I$. ∎

Given a set of monomials
$A=\\{\textbf{x}^{\alpha_{1}},\dots,\textbf{x}^{\alpha_{r}}\\}$, we can
symmetrize this set of monomials by letting $S_{d}$ act on the variables.
Denote this set by

$sym(A):=\\{\textbf{x}^{\pi(\alpha_{j})}\mid\textbf{x}^{\alpha_{j}}\in
A;\pi\in S_{d}\\}$

When $A=\\{\textbf{x}^{\alpha}\\}$ consists of a single monomial, we refer to
$sym(A)$ as the orbit of $\textbf{x}^{\alpha}$.

######  Definition 3.13.

A monomial $\textbf{x}^{\alpha}$ is a pure power of $x_{j}$ if
$\alpha=(a_{1},\dots,a_{j},\dots,a_{d})$ with $a_{i}=0$ for $i\neq j$.

######  Definition 3.14.

An ideal $I\subset K[x_{1},\dots,x_{d}]$ is Artinian if the Krull dimension of
$R/I$ is zero.

Note that a monomial ideal is Artinian if and only if $I$ contains a pure
power of $x_{j}$ for $1\leq j\leq d$. Denote the set of Artinian ideals with
the largest degree of any pure power equal to $n$ by $\mathcal{A}_{d}(n)$.
Denote the set of Artinian strongly stable ideals with pure power
$x_{d}^{n}\in G(I)$ by

$\mathcal{B}_{d}(n):=\\{I\in\mathcal{A}_{d}(n)\mid\text{$I$ is strongly
stable}\\}$

Similarly, use

$\mathcal{T}_{d}(n):=\\{I\in\mathcal{A}_{d}(n)\mid\text{$I$ is symmetric}\\}$

to denote the set of Artinian symmetric monomial ideals with pure power
$x_{d}^{n}\in G(I)$.

######  Proposition 3.15.

Let $I\in\mathcal{B}_{d}(n)$ with pure power $x_{d}^{n}\in G(I)$. Then for
every pure power $x_{j}^{k_{j}}\in G(I)$, we have $k_{j}\leq n$.

###### Proof.

We can use Borel moves to get $x_{j}^{n}\in I$ for all $j$. So every pure
power $x_{j}^{k_{j}}\in G(I)$ must satisfy $k_{j}\leq n$. ∎

######  Proposition 3.16.

Let $I\in\mathcal{T}_{d}(n)$ with pure power $x_{d}^{n}\in G(I)$. Then for
every pure power $x_{j}^{k_{j}}\in G(I)$, we have $k_{j}=n$.

###### Proof.

This follows from Proposition 3.12. ∎

##  4\. Partitions Correspond to Artinian Monomial Ideals

######  Proposition 4.1.

If $I\in\mathcal{A}_{d}(n)$, then
$P=\\{\alpha\mid\textnormal{{x}}^{\alpha}\notin I\\}\in\mathcal{P}_{d}(n)$.

###### Proof.

First, we need to show that $P=\\{\alpha\mid\textbf{x}^{\alpha}\notin I\\}$ is
a partition. Just notice that if $\alpha=(a_{1},\dots,a_{d})\in P$, then
$\textbf{x}^{\alpha}\notin I$. It follows that
$\textbf{x}^{(a_{1},\dots,a_{j}-1,\dots,a_{d})}\notin I$ so
$(a_{1},\dots,a_{j}-1,\dots,a_{d})\in P$. Since $I$ is Artinian, $P$ is
finite.

For every cell $\gamma=(c_{1},\dots,c_{d})\in P$, we have $c_{i}<n$, because
$I$ contains a pure power of every variable with degree less than or equal to
$n$. Since $x_{d}^{n}\in G(I)$, $(0,0,\dots,0,n-1)\in P$. ∎

######  Proposition 4.2.

If $P\in\mathcal{P}_{d}(n)$, then
$\\{\textnormal{{x}}^{\alpha}\mid\alpha\notin P\\}\in\mathcal{A}_{d}(n)$.

###### Proof.

Let $I=\\{\textbf{x}^{\alpha}\mid\alpha\notin P\\}$. It is clear that $I$ is a
monomial ideal. Since $(0,0,\dots,n,\dots,0,0)\notin P$, $x_{j}^{n}\in I$ for
$1\leq j\leq d$.

There is some cell $\gamma=(c_{1},\dots,c_{j},\dots,c_{d})\in P$ with
$c_{j}=n-1$ for some $1\leq j\leq d$. By the definition of $d$-dimensional
partition, $(0,0,\dots,n-1,\dots,0)\in P$, so $x_{j}^{n}\in G(I)$. ∎

The two propositions above allow us to define a map
$\varphi:\mathcal{A}_{d}(n)\rightarrow\mathcal{P}_{d}(n)$ by

$\varphi(I)=\\{\alpha\in\mathbb{N}^{d}\mid\textbf{x}^{\alpha}\notin I\\}$

with inverse $\varphi^{-1}:\mathcal{P}_{d}(n)\rightarrow\mathcal{A}_{d}(n)$
given by $\varphi^{-1}(P)=\\{\textbf{x}^{\alpha}\mid\alpha\notin P\\}$.

######  Example 4.3.

Let $I=(x^{4},x^{2}y,xy^{2},y^{3})\in\mathcal{A}_{2}(4)$. Then $\varphi(I)$ is
the partition from Example 2.2 and is shown below with $\bullet$ used to
represent generators of $I$.

& [∙]

[∙]

[∙]

[∙]

######  Example 4.4.

Let $I=(x^{3},x^{2}y,y^{3},x^{2}z,xyz,y^{2}z,z^{2})\in\mathcal{A}_{3}(3)$.
Then $\varphi(I)$ is the plane partition from Example 2.3 shown below.

We will show that we can restrict $\varphi$ to a bijection between strongly
stable ideals and strongly stable partitions and between symmetric monomial
ideals and totally symmetric partitions.

######  Proposition 4.5.

A monomial ideal $I$ is strongly stable iff $\varphi(I)$ is a strongly stable
partition.

###### Proof.

($\Rightarrow$) Let $\beta\in\varphi(I)$ so that $\textbf{x}^{\beta}\notin I$
and consider $H(\beta)=(h_{1},\dots,h_{d})$. Fix $1\leq i<j\leq d$. We have
$(b_{1},\dots,b_{i},\dots,b_{j}+h_{j}+1,\dots,b_{d})\notin\varphi(I)$, and we
can use the fact that $I$ is strongly stable to get

$\textbf{x}^{(b_{1},\dots,b_{i},\dots,b_{j}+h_{j}+1,\dots,b_{d})}\in
I\implies\textbf{x}^{(b_{1},\dots,b_{i}+h_{j}+1,\dots,b_{j},\dots,b_{d})}\in
I$

so $(b_{1},\dots,b_{i}+h_{j}+1,\dots,b_{j},\dots,b_{d})\notin P$. Therefore,
$h_{i}<h_{j}+1\implies h_{i}\leq h_{j}$.

($\Leftarrow$) Let $\textbf{x}^{\alpha}\in G(I)$ such that
$x_{j}\mid\textbf{x}^{\alpha}$. Then
$\frac{\textbf{x}^{\alpha}}{x_{j}}=\textbf{x}^{(a_{1},\dots,a_{j}-1,\dots,a_{d})}\notin
I$, so $\tilde{\alpha}=(a_{1},\dots,a_{j}-1,\dots,a_{d})\in\varphi(I)$. If
$H(\tilde{\alpha})=(h_{1},\dots,h_{d})$, then $h_{j}=1$ because $\alpha\notin
P$. Since $\varphi(I)$ is a strongly stable partition, $h_{i}\leq h_{j}=1$ for
$i<j$. In particular, $(a_{1},\dots,a_{i}+1,\dots,a_{j}-1,\dots,a_{d})\notin
I$. So $\textbf{x}^{\alpha}\frac{x_{i}}{x_{j}}\in I$. ∎

######  Proposition 4.6.

A monomial ideal $I$ is symmetric iff $\varphi(I)$ is a totally symmetric
partition.

###### Proof.

($\Rightarrow$) Let $\beta\in\varphi(I)$ and let $\pi\in S_{d}$. Since
$\textbf{x}^{\beta}\notin I$, $\textbf{x}^{\pi(\beta)}\notin I$, so
$\pi(\beta)\in\varphi(I)$.

($\Leftarrow$) Let $\textbf{x}^{\alpha}\in I$. Then
$\alpha\notin\varphi(I)\implies\pi(\alpha)\notin\varphi(I)$, so
$\textbf{x}^{\pi(\alpha)}\in I$. ∎

In this section, we have shown that
$\varphi\mid_{\mathcal{B}_{d}(n)}:\mathcal{B}_{d}(n)\rightarrow\tilde{\mathcal{B}}_{d}(n)$
and
$\varphi\mid_{\mathcal{T}_{d}(n)}:\mathcal{T}_{d}(n)\rightarrow\tilde{\mathcal{T}}_{d}(n)$
are bijections. In the next section, we show a bijection
$\mathcal{B}_{d}(n)\rightarrow\mathcal{T}_{d}(n)$.

##  5\. Bijection Between Monomial Ideals

Denote the set of all monomials in $K[x_{1},\dots,x_{d}]$ by $M_{d}$ and the
set of monomials with weakly increasing exponent vector by $F_{d}$. Then use
$\mathcal{F}_{d}(n)$ to denote the set of minimal subsets of $F_{d}$ which
contain $x_{d}^{n}$ and no coordinate of any exponent vector exceeding $n$.
Define a map $\psi:M_{d}\rightarrow F_{d}$ by

$\psi(\textbf{x}^{(a_{1},a_{2},\dots,a_{d})})=\textbf{x}^{(a_{1},a_{1}+a_{2},a_{1}+a_{2}+a_{3},\dots,a_{1}+a_{2}+\cdots+a_{d})}$

and notice that $\psi^{-1}:F_{d}\rightarrow M_{d}$ is given by

$\psi^{-1}(\textbf{x}^{(a_{1},\dots,a_{d})})=\textbf{x}^{(a_{1},a_{2}-a_{1},a_{3}-a_{2},\dots,a_{d}-a_{d-1})}$

######  Proposition 5.1.

$\psi(m\frac{x_{q+1}}{x_{q}})=\frac{\psi(m)}{x_{q}}$

###### Proof.

$\displaystyle\psi(m\frac{x_{q+1}}{x_{q}})$
$\displaystyle=\psi(\textbf{x}^{(a_{1},\dots,a_{q}-1,a_{q+1}+1,\dots,a_{d})})$
$\displaystyle=\textbf{x}^{(a_{1},a_{2}-a_{1},a_{3}-a_{2},\dots,a_{q}-a_{q-1}-1,a_{q+1}-a_{q},a_{q+2}-a_{q+1},\dots,a_{d}-a_{d-1})}$
$\displaystyle=\frac{\psi(m)}{x_{q}}$

∎

For a subset of monomials
$A=\\{\textbf{x}^{\alpha_{1}},\dots,\textbf{x}^{\alpha_{r}}\\}$, we will use
$\psi(A):=\\{\psi(\textbf{x}^{\alpha_{1}}),\dots,\psi(\textbf{x}^{\alpha_{r}})\\}$.
For an ideal $I$, we will use $\psi(I)$ to refer to the ideal generated by
$\psi(G(I))$.

######  Proposition 5.2.

For a strongly stable ideal $I$, $m\in I\iff\psi(m)\in\psi(I)$.

###### Proof.

($\Rightarrow$) If $m\in I$, then there exists $g\in G(I)$ so that $g\mid m$.
If $g=\textbf{x}^{(c_{1},\dots,c_{d})}$ and
$m=\textbf{x}^{(a_{1},\dots,a_{d})}$, then $c_{i}\leq a_{i}$ for $1\leq i\leq
d$. It follows that $c_{1}+\cdots+c_{i}\leq a_{1}+\cdots+a_{i}$ for all $i$.
So $\psi(g)\mid\psi(m)$.
($\Leftarrow$) Assume $\psi(m)\in\psi(I)$ and let
$k_{i}:=a_{1}+\cdots+a_{i}-(c_{1}+\cdots+c_{i})$ so that

$\textbf{x}^{(k_{1},\dots,k_{d})}\psi(g)=\psi(m)$

Note that $k_{i}=a_{i}-c_{i}+k_{i-1}$ for $i=2,\dots,d$. Then we have

$\displaystyle\tilde{g}$
$\displaystyle:=g\frac{x_{1}^{k_{1}}}{x_{2}^{k_{1}}}\frac{x_{2}^{k_{2}}}{x_{3}^{k_{2}}}\cdots\frac{x_{d-1}^{k_{d-1}}}{x_{d}^{k_{d-1}}}$
$\displaystyle=g\frac{x_{1}^{k_{1}-k_{2}}}{x_{2}^{k_{1}-k_{2}}}\frac{x_{1}^{k_{2}-k_{3}}}{x_{3}^{k_{2}-k_{3}}}\cdots\frac{x_{1}^{k_{d-2}-k_{d-1}}}{x_{d-1}^{k_{d-2}-k_{d-1}}}\frac{x_{1}^{k_{d-1}}}{x_{d}^{k_{d-1}}}$
$\displaystyle=\textbf{x}^{(c_{1}+k_{1},c_{2}+k_{2}-k_{1},c_{3}+k_{3}-k_{2},\dots,c_{d-1}+k_{d-1}-k_{d-2},c_{d}-k_{d-1})}$
$\displaystyle=\textbf{x}^{(a_{1},a_{2},\dots,a_{d-1},c_{d}-k_{d-1})}$

since $c_{i}+k_{i}-k_{i-1}=c_{i}+(a_{i}-c_{i}+k_{i-1})-k_{i-1}=a_{i}$. For the
last coordinate, we have

$\displaystyle c_{d}-k_{d-1}$
$\displaystyle=c_{d}-(a_{1}+\cdots+a_{d-1}-(c_{1}+\cdots+c_{d-1}))$
$\displaystyle=c_{1}+\cdots+c_{d}-(a_{1}+\cdots+a_{d-1})$ $\displaystyle\leq
a_{d}$

We have shown that $\tilde{g}\mid m$. Now, we claim that $\tilde{g}\in I$
because it is a Borel move from $g$. To see this, notice that
$k_{i-1}-k_{i}=c_{i}-a_{i}$ for $i=2,\dots,d$ and
$k_{d-1}=a_{1}+\cdots+a_{d-1}-(c_{1}+\cdots+c_{d-1})\leq c_{d}$. So $g$ is
divisible by each of the denominators in the second line of $\tilde{g}$. ∎

######  Proposition 5.3.

If $I$ is a strongly stable ideal, then $Bgens(I)=\psi^{-1}(min(\psi(G(I))))$.

###### Proof.

($\subseteq$) Let $m\in Bgens(I)\subset G(I)$. For any $x_{q}\mid\psi(m)$,
$\frac{\psi(m)}{x_{q}}=\psi(m\frac{x_{q+1}}{x_{q}})\notin\psi(I)$ because
$m\frac{x_{q+1}}{x_{q}}\notin I$. Therefore, $\psi(m)\in min(\psi(G(I)))$.

($\supseteq$) Let $m\in\psi^{-1}(min(\psi(G(I))))$. Then $\psi(m)$ is minimal,
so $\frac{\psi(m)}{x_{q}}=\psi(m\frac{x_{q+1}}{x_{q}})\notin\psi(I)$. It
follows that $m\frac{x_{q+1}}{x_{q}}\notin I$. ∎

Note that this gives an algorithm to compute $Bgens(I)$ given a generating set
$G(I)$. As a result of Proposition 3.15, observe that for any
$I\in\mathcal{B}_{d}(n)$, we have

$\textbf{x}^{\alpha}\in G(I)\implies deg(\alpha)\leq n.$

It follows that every coordinate of $\psi(\textbf{x}^{\alpha})$ is less than
or equal to $n$. This observation combined with the previous proposition
allows us to define a map

$\displaystyle\Lambda:\mathcal{B}_{d}(n)$
$\displaystyle\rightarrow\mathcal{F}_{d}(n)$ $\displaystyle I$
$\displaystyle\mapsto\psi(Bgens(I)).$

######  Proposition 5.4.

$\Lambda:\mathcal{B}_{d}(n)\rightarrow\mathcal{F}_{d}(n)$ is a bijection.

###### Proof.

We claim that $\Lambda^{-1}:\mathcal{F}_{d}(n)\rightarrow\mathcal{B}_{d}(n)$
is given by $S\mapsto Borel(\psi^{-1}(S))$. Since $x_{d}^{n}\in S$,
$\psi^{-1}(x_{d}^{n})=x_{d}^{n}\in\psi^{-1}(S)$. There are no other pure
powers of $x_{d}$ in $\psi^{-1}(S)$, so $x_{d}^{n}\in G(I)$. It follows that
$Borel(\psi^{-1}(S))\in\mathcal{B}_{d}(n)$ and

$\displaystyle(\Lambda^{-1}\circ\Lambda)(I)$
$\displaystyle=Borel(\psi^{-1}(\psi(Bgens(I))))$ $\displaystyle=I.$

∎

######  Proposition 5.5.

If $I\subset K[x_{1},\dots,x_{d}]$ is a symmetric monomial ideal, then
$sym(G(I)\cap F_{d})=G(I)$.

###### Proof.

($\subseteq$) If $u\in sym(G(I)\cap F_{d})$, then $u=\pi(m)$ for some $m\in
G(I)\cap F_{d}$ and some $\pi\in S_{d}$. Since $G(I)$ is closed under
operations of $S_{d}$, $u\in G(I)$.
($\supseteq$) If $u\in G(I)$, then there exists some $\pi\in S_{d}$ so that
$\pi(u)\in G(I)\cap F_{d}$, so $u\in sym(G(I)\cap F_{d})$. ∎

For an ideal $I\in\mathcal{T}_{d}(n)$, define a map
$\Omega:\mathcal{F}_{d}(n)\rightarrow\mathcal{T}_{d}(n)$ by

$\Omega(A)=ideal(sym(A))$

######  Proposition 5.6.

$\Omega:\mathcal{F}_{d}(n)\rightarrow\mathcal{T}_{d}(n)$ is a bijection

###### Proof.

We claim that the inverse
$\Omega^{-1}:\mathcal{T}_{d}(n)\rightarrow\mathcal{F}_{d}(n)$ is given by
$\Omega^{-1}(I)=G(I)\cap F_{d}$. Notice that for $I\in\mathcal{T}_{d}(n)$, we
have $x_{d}^{n}\in G(I)$, so $x_{d}^{n}\in G(I)\cap F_{d}$. It follows that
$\Omega^{-1}(I)\in\mathcal{F}_{d}(n)$ and

$\displaystyle\Omega(\Omega^{-1}(I))$ $\displaystyle=\Omega(G(I)\cap F_{d})$
$\displaystyle=ideal(sym(G(I)\cap F_{d}))$ $\displaystyle=I.$

∎

######  Theorem 1.

There is a bijection between $d$-dimensional strongly stable partitions and
$d$-dimensional totally symmetric partitions which preserves the side length
$n$ of the minimal $d$-dimensional box containing the partitions.

###### Proof.

We have the diagram of bijections below. The vertical bijections were given in
section 4 and the horizontal map $\Omega\circ\Lambda$ was shown to be a
bijection by Propositions 5.4 and 5.6.

${\mathcal{B}_{d}(n)}$${\mathcal{T}_{d}(n)}$${\tilde{\mathcal{B}}_{d}(n)}$${\tilde{\mathcal{T}}_{d}(n)}$$\scriptstyle{\Omega\circ\Lambda}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi}$

∎

######  Example 5.7.

Consider the strongly stable ideal
$I=(x^{4},x^{3}y,x^{2}y^{3},xy^{4},y^{7})\in\mathcal{B}_{2}(7)$ from Example
3.10. The corresponding partition $\varphi(I)\in\tilde{\mathcal{B}}_{2}(7)$
from Example 2.6 is shown on the left below. Minimal generators of $I$ which
are not in $Bgens(I)$ are represented with $\bullet$. Elements of $Bgens(I)$
are represented by $*$. The middle diagram depicts $\psi(I)$ and illustrates
that $\psi(Bgens(I))=min(\psi(G(I)))$. The diagram on the right is the
corresponding totally symmetric partition.

######  Remark 5.8.

For any $\textbf{x}^{\alpha}\notin I$, we have
$\psi(\textbf{x}^{\alpha})\notin\Lambda(I)$. Since $\psi(\textbf{x}^{\alpha})$
is the unique representative of its orbit, the number of orbits of monomials
not in $\Omega(\Lambda(I))$ is exactly the total number of monomials in the
original ideal $I$. In partition language, this means that the number of cells
in the strongly stable partition is exactly the number of orbits in the
totally symmetric partition. In the example above, there are 15 cells in the
strongly stable partition on the left which correspond to 15 orbits in the
totally symmetric partition on the right.

######  Example 5.9.

Consider the strongly stable partition $P\in\tilde{\mathcal{B}}_{3}(4)$ from
Example 2.7 shown below.

One can check that
$I=(x^{2},xy^{2},y^{3},xyz,y^{2}z,xz^{2},yz^{3},z^{4})\in\mathcal{B}_{3}(4)$
is the corresponding strongly stable ideal ($\varphi(I)=P$) and that
$Bgens(I)=\\{x^{2},xz^{2},y^{2}z,z^{4}\\}$. To compute the totally symmetric
partition corresponding to $P$, we first compute

$\displaystyle\Omega(\Lambda(I))$ $\displaystyle=\Omega(\psi(Bgens(I)))$
$\displaystyle=\Omega(\\{x^{2}y^{2}z^{2},xyz^{3},y^{2}z^{3},z^{4}\\})$
$\displaystyle=ideal(sym(\\{x^{2}y^{2}z^{2},xyz^{3},y^{2}z^{3},z^{4}\\}))$

Finally, notice that
$\varphi(ideal(sym(\\{x^{2}y^{2}z^{2},xyz^{3},y^{2}z^{3},z^{4}\\})))\in\tilde{\mathcal{T}}_{3}(4)$
is the partition shown below.

##  6\. Enumerations

In this last section, we will explore the enumeration of strongly stable
partitions (equivalently totally symmetric partitions) in boxes. Let

$B_{d}(n):=\sum_{k=0}^{n}|\tilde{\mathcal{B}}_{d}(k)|$

denote the number of $d$-dimensional strongly stable partitions which fit in a
box of side length $n$. Similarly, use

$T_{d}(n):=\sum_{k=0}^{n}|\tilde{\mathcal{T}}_{d}(k)|$

By convention, we count the empty partition. As a result, we have
$B_{d}(0)=T_{d}(0)=1$ since the empty partition fits inside a box of side
length $0$. As a result of Theorem 1,

$B_{d}(n)=T_{d}(n)$

for positive integers $d,n$.

In [3], Hawkes shows a bijection between $d$-dimensional totally symmetric
partitions inside a box of side length $n$ and $(n-1)$-dimensional totally
symmetric partitions inside a box of side length $d+1$. As a result,
$T_{d}(n)=T_{n-1}(d+1)$ for $n\geq 2$. Combining this with the formula above,
we have

$B_{d}(n)=B_{n-1}(d+1)$

for positive integers $d,n$.

Lastly, we consider some formulae for $B_{d}(n)$ when $d$ is fixed. When
$d=1$, partitions are simply nonnegative integers, and every $1$-dimensional
partition is (trivially) strongly stable.

$B_{1}(n)=T_{1}(n)=n+1$

The number of $2$-dimensional strongly stable partitions which fit inside a
box of side length $n$ (strict integer partitions with largest part $n$) is
given by

$B_{2}(n)=T_{2}(n)=2^{n},$

because for $1\leq i\leq n$, we have only the choice of whether or not to
include $i$ in the integer partition.

In [6], Stembridge proved that the number of totally symmetric plane
partitions which fit inside a box of side length $n$ is given by the product
formula

$T_{3}(n)=\prod_{1\leq i\leq j\leq k\leq n}\frac{i+j+k-1}{i+j+k-2}$

and so we have a formula for $B_{3}(n)$.

In fact, in the case $d=3$, an even stronger result has been shown for totally
symmetric plane partitions. The $q$-TSPP formula

$\sum_{P\in\tilde{\mathcal{T}}_{3}(n)}q^{\left|P/S_{3}\right|}=\prod_{1\leq
i\leq j\leq k\leq n}\frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}$

was shown to be the orbit-counting generating function for totally symmetric
plane partitions fitting inside an $n\times n\times n$ box in [5]. Remark 5.8
implies that this same formula is a cell-counting generating function for
strongly stable plane partitions.

There is no known formula for $B_{d}(n)=T_{d}(n)$ for $d\geq 4$ [2].

## References

  * [1] Christopher A Francisco, Jeffrey Mermin and Jay Schweig  “Borel generators”  In _Journal of Algebra_ 332.1 Elsevier, 2011, pp. 522–542
  * [2] Hawkes, Graham  “Number of totally symmetric solid partitions which fit in an n X n X n X n box” Accessed: 2023-01-22, https://oeis.org/A236691, 2014
  * [3] Hawkes, Graham  “Totally Symmetric Partitions in Boxes” Accessed: 2023-01-22, https://oeis.org/A005157/a005157_2.pdf, 2014
  * [4] Jürgen Herzog and Takayuki Hibi  “Monomial ideals”  In _Monomial Ideals_ Springer, 2011, pp. 57
  * [5] Christoph Koutschan, Manuel Kauers and Doron Zeilberger  “Proof of George Andrews’s and David Robbins’ sq-TSPP conjecture”  In _Proceedings of the National Academy of Sciences_ 108.6 National Acad Sciences, 2011, pp. 2196–2199
  * [6] John R Stembridge  “The enumeration of totally symmetric plane partitions”  In _Advances in Mathematics_ 111.2 Elsevier, 1995, pp. 227–243


††thanks: These authors contributed equally.††thanks: These authors
contributed equally.††thanks: These authors contributed equally.

# Blind Ghost Imaging

A. M. Paniagua-Diaz  University of Exeter, Stocker Road, Exeter EX4 4QL,
United Kingdom  I. Starshynov  University of Exeter, Stocker Road, Exeter EX4
4QL, United Kingdom  N. Fayard  ESPCI Paris, PSL Research University, CNRS,
Institut Langevin, 1 rue Jussieu, F-75005, Paris, France  A. Goetschy  ESPCI
Paris, PSL Research University, CNRS, Institut Langevin, 1 rue Jussieu,
F-75005, Paris, France  R. Pierrat  ESPCI Paris, PSL Research University,
CNRS, Institut Langevin, 1 rue Jussieu, F-75005, Paris, France  R. Carminati
<EMAIL_ADDRESS>ESPCI Paris, PSL Research University, CNRS, Institut
Langevin, 1 rue Jussieu, F-75005, Paris, France  J. Bertolotti
<EMAIL_ADDRESS>University of Exeter, Stocker Road, Exeter EX4 4QL,
United Kingdom

###### Abstract

Ghost imaging is an unconventional optical imaging technique that reconstructs
the shape of an object combining the measurement of two signals: one that
interacted with the object, but without any spatial information, the other
containing spatial information, but that never interacted with the object
Pittman  _et al._ (1995); Bennink  _et al._ (2002). Ghost imaging is a very
flexible technique, that has been generalized to the single-photon regime
Aspden  _et al._ (2013), to the time domain Ryczkowski  _et al._ (2016), to
infrared and terahertz frequencies Shrekenhamer  _et al._ (2013), and many
more conditions Padgett and R.W. (2017). Here we demonstrate that ghost
imaging can be performed without ever knowing the patterns illuminating the
object, but using patterns correlated with them, doesn’t matter how weakly. As
an experimental proof we exploit the recently discovered correlation between
the reflected and transmitted light from a scattering layer Fayard  _et al._
(2015); Starshynov  _et al._ (2018), and reconstruct the image of an object
hidden behind a scattering layer using only the reflected light, which never
interacts with the object. This method opens new perspectives for non-invasive
imaging behind or within turbid media.

In its simplest form ghost imaging (GI), also known as single pixel camera
Duarte  _et al._ (2008) or dual photography Sen  _et al._ (2005), is an
imaging technique where, instead of illuminating uniformly an object and then
detect the scattered light with a multipixel camera, the object is illuminated
with a sequence of known patterns, and the scattered light is detected by a
single photodiode Bennink  _et al._ (2002). By using enough illumination
patterns, high quality images can be formed Ferri  _et al._ (2005). Ghost
imaging finds application in all cases where large arrays of detectors are
harder to come by than reliable sources (e.g. THz imaging Shrekenhamer  _et
al._ (2013)), or when only a very small amount of signal is available Aspden
_et al._ (2013). As there is a lot of freedom in the choice of the patterns
used, one can optimize them to increase resolution in the areas of interest
Phillips  _et al._ (2017), or use compressive sensing to speed-up measurement
Katz  _et al._ (2009). Furthermore, as long as the patterns used are known,
they do not need to be deterministically generated or even orthogonal, and
even a set of speckle patterns allow to reconstruct an image Zerom  _et al._
(2012).

A property that is shared by all variants of ghost imaging is that one needs
to know exactly what the set of illumination patterns is. What is effectively
measured with the single pixel detector is proportional to the overlap between
the object $O$ and the illumination pattern $P_{i}$, i.e. the coefficient
$b_{i}=\int P_{i}(\mathbf{r})O(\mathbf{r})d\mathbf{r}$. If the set of
illumination patterns forms a complete basis, one can reconstruct an image of
the object as
$I(\mathbf{r}^{\prime})=\sum_{i}b_{i}P_{i}(\mathbf{r}^{\prime})$, but if the
patterns $P_{i}$ are unknown, this approach breaks down.

In this article we show that, even if the illumination patterns are completely
unknown, one can still use a different set of patterns in the reconstruction
formula, as long as this second set is correlated with the first one. In
particular, we exploit the recently discovered spatial correlation between the
transmitted and reflected speckle patterns generated at both sides of a
scattering medium Fayard  _et al._ (2015); Starshynov  _et al._ (2018), which
allows us to reconstruct the shape of an object hidden behind a turbid medium,
potentially fully opaque, using only the reflected speckle pattern, instead of
the transmitted one. Furthermore, we generalize this technique to a completely
non-invasive geometry, where both the camera measuring the speckle pattern and
the single-pixel detector are on the same side of the scattering layer,
allowing to image a fluorescent object placed on the other side.

Figure 1: a) Experimental apparatus. A cw laser illuminates an opaque
scattering material and an object hidden behind (insets). An imaging system
records the reflected speckle pattern from the surface of the scattering
sample and a bucket detector collects the intensity transmitted by the object.
b) Elements 5 and 6 of Group 4 of the resolution target used as object to
image in this experiment, highlighted by the pink square in the inset of panel
a. c) Typical speckle pattern collected in reflection with the imaging system
presented. d) Retrieved image using BGI with $2.27\times 10^{6}$ disorder
realizations.

When using speckle to perform ghost imaging, one usually sends a laser beam
through a time-varying scattering medium, often a rotating diffuser, and the
resulting transmitted intensity speckle pattern, $T_{i}$, is measured and used
as the illumination pattern $P_{i}$. The transmitted light passing through the
object is then integrated and measured with a single pixel detector, yielding
the coefficient $b_{i}=\int T_{i}(\mathbf{r})O(\mathbf{r})d\mathbf{r}$. Full
knowledge of both $b_{i}$ and $T_{i}$ allows one, for a large enough number
$N$ of patterns, to obtain a faithful representation of the object $O$. In
order to measure directly the transmitted speckle patterns, one needs to have
an imaging system placed behind the scattering layer. In most practical
situations, this is actually not possible, e.g. because access is restricted,
as in biomedical imaging. In these cases one can rely on the reflected speckle
patterns $R_{i}$ only, which share mutual information with the transmitted
ones in the form of a spatial correlation Fayard  _et al._ (2015); Starshynov
_et al._ (2018); Fayard  _et al._ (2018). The simplest approach we can take to
make use of this mutual information is to replace each
$T_{i}(\mathbf{r}^{\prime})$ with $R_{i}(\mathbf{r}^{\prime})$, which results
in the reconstructed image

$\tilde{I}(\mathbf{r}^{\prime})=\sum_{i=1}^{N}b_{i}R_{i}(\mathbf{r}^{\prime}).$
(1)

Identifying the sum $\sum_{i=1}^{N}(\dots)_{i}$ with the ensemble average
$\langle\dots\rangle$ and substituting in the definition of $b_{i}$, we can
express the reconstructed image as

$\displaystyle\tilde{I}(\mathbf{r}^{\prime})=$ $\displaystyle\langle\int
O(\mathbf{r})T(\mathbf{r})R(\mathbf{r}^{\prime})d\mathbf{r}\rangle$ (2)
$\displaystyle=$ $\displaystyle\int O(\mathbf{r})\langle
T(\mathbf{r})R(\mathbf{r}^{\prime})\rangle d\mathbf{r}$ $\displaystyle=$
$\displaystyle\langle T\rangle\langle R\rangle\left[O\ast C^{RT}+\int
O(\mathbf{r})d\mathbf{r}\right]$ $\displaystyle\propto$ $\displaystyle\,O\ast
C^{RT}+\mathcal{A}$

where $C^{RT}(\Delta\mathbf{r})=\langle\delta R(\mathbf{r})\delta
T(\mathbf{r}+\Delta\mathbf{r})\rangle$ is the normalized correlation function
of the reflected and transmitted intensity patterns ($\delta f=f/\langle
f\rangle-1$ denotes the normalized statistical fluctuation of the random
variable $f$) and the constant $\mathcal{A}=\int O(\mathbf{r})d\mathbf{r}$
represents a flat background proportional to the total signal from the object.
Hence, using the reflected speckle patterns instead of the transmitted ones,
we obtain the very same image, but with a lower resolution, given by the range
of the correlation function $C^{RT}$, which acts as a point spread function.
We name this Blind Ghost Imaging (BGI), as it allows to perform ghost imaging
without ever knowing the patterns used to illuminate the object.

To verify our prediction we designed an experiment where we image an object
hidden behind an opaque scattering medium. The experimental apparatus is shown
in Fig. 1a. A $2\text{\,}\mathrm{mW}$ He-Ne laser is incident on a scattering
medium (Fig. 1a, inset) at an angle of approximately 45∘ with respect to the
sample surface. In this way, contributions of the specularly reflected and
ballistically transmitted light, which spoil the correlation $C^{RT}$, are not
collected Starshynov  _et al._ (2018). The scattering layer is made of a
suspension of TiO2 particles in glycerol, with a scattering mean free path
$\ell=$16\text{\,}$\pm$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and a
$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thickness, resulting in an Optical
Density ($OD$) $\simeq 2.5$. The object to image, a Thorlabs USAF 1951
calibration test target (Fig. 1b), is in contact with the scattering layer.
The reflected speckle pattern (Fig. 1c) is imaged on the scattering medium
surface and recorded using a CCD camera. As the scattering layer is liquid,
the speckle patterns change with time which allows us to record a large number
of different speckle patterns without moving or changing the sample. The
transmitted light passing through the object is then integrated by a bucket
detector. For simplicity of alignment, this is done by using an identical CCD
camera and integrating over all pixels. This allows us to measure the
correlation $C^{RT}(\Delta\mathbf{r})$, discussed later on, using the same
apparatus.

In Fig. 1d we show the reconstructed image of the object represented in Fig.
1b, when using the reflected speckle patterns and integrating the transmitted
intensity, according to Eq. (1). Here, we used $N=2.27\times 10^{6}$
realizations of the disorder. Apart from the residual noise, the object is
clearly visible and all features are resolved. We notice that a gaussian
smoothing of the picture would remove most of the noise, producing a more
pleasing image. This experiment demonstrates that it is possible to perform
ghost imaging using a set of patterns different from the illuminating one but
correlated with it. In particular it is possible to use the reflected, instead
of the transmitted speckle, to reconstruct the shape of an object placed
behind an opaque scattering layer. Compared to other ghost imaging schemes
using reflected signal Hardy and Shapiro (2011); Bina  _et al._ (2013), this
method works in the deep multiple scattering regime without making use of any
ballistic light.

Figure 2:  a) Experimental apparatus used for non-invasive BGI. A
$450\text{\,}\mathrm{n}\mathrm{m}$ laser is incident on the scattering sample
at $\approx 45^{\circ}$. The resolution target is placed on the back surface
of the scattering material, and right behind it we have a fluorescent layer
(Cerium-doped YAG), acting as a fluorescent object. The bucket detector is in
this case also in reflection from the sample, filtering the fluorescent light
with a $500\text{\,}\mathrm{n}\mathrm{m}$ long pass filter. b) Elements 5 and
6 of Group 4 of the resolution target used as the object, and the image
retrieved using BGI with $4\times 10^{6}$ disorder realizations. c) Object
representing Groups 5, 6 and 7 from the resolution target, and the image
retrieved using BGI with $1.5\times 10^{6}$ disorder realizations.

As the bucket detector does not have any spatial resolution, there is no
fundamental reason why it should be placed behind the object as in traditional
ghost imaging. This suggest that blind ghost imaging can be adapted to a
completely non-invasive configuration. We modified the apparatus so that all
optical components are on the opposite side of the scattering layer with
respect to the object, as shown in Fig. 2a. The fluorescent sample consists of
the USAF negative target with a fluorescent layer of Cerium-doped YAG just
behind it. The illumination geometry is the same as in the first experiment,
but in this case we used a $100\text{\,}\mathrm{mW}$ blue laser
($450\text{\,}\mathrm{nm}$) producing a white fluorescent emission from the
Cerium-doped YAG layer. Both the reflected speckle and the fluorescence are
collected by a 10x microscope objective, and a plano-convex
$150\text{\,}\mathrm{m}\mathrm{m}$ lens, in an epi configuration. The speckle
pattern is recorded by a CCD camera, and the fluorescence from the object is
collected by the bucket detector after passing through a long-pass
$500\text{\,}\mathrm{nm}$ filter. Again in this case the bucket detector is a
CCD with the intensity integrated over all pixels. In Fig. 2b we show the
retrieved image for this case, obtained with $N=4\times 10^{6}$ disorder
realizations. The image is very well reconstructed, with an outcome very
similar to the one shown in Fig. 1d obtained with the bucket placed on the
transmission side.

Figure 3:  a) 2D map and a 1D cross section along $\Delta y=0$ of the averaged
correlation between the transmitted and reflected speckle patterns. b) and c)
Expected images obtained by numerically convolving the objects shown in Fig.
2b,c with the correlation function shown in a).

In order to evaluate the performances of the blind ghost imaging setup, we
first took an image of the elements 5 and 6 of group 4 of the USAF target, as
shown in Fig. 2b, and found a lateral resolution $\Delta r\simeq$
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$. We then repeated the measurement
with an object with smaller features (groups 5, 6 and 7 of the resolution
target) shown in Fig. 2c, in order to better quantify the resolution of this
method. According to our prediction (Eq. 2), this resolution should be given
by the width of the correlation $C^{RT}(\Delta\mathbf{r})$ which acts as a
point spread function. To confirm that this is indeed the case, we made a
separate measurement of the average intensity correlation between transmitted
and reflected speckle patterns Starshynov  _et al._ (2018), and compared the
blind ghost imaging results of Fig. 2b,c with the numerical convolution of the
object with $C^{RT}$. Results are presented in Fig. 3. The retrieved images
(Fig. 3b,c) resemble very well the expected ones (Fig. 2b,c), resolving the
same elements and thus demonstrating that the resolution of the resulting
image depends on the width and shape of the correlation function $C^{RT}$
(Fig. 3a), as dictated by Eq. (2). In particular, the width of the correlation
function limits the features of the object that can be resolved, even in the
ideal and noise-free case, where it is possible to resolve mainly the first
few elements of group 5.

The shape and the sign of the correlation $C^{RT}$ depend both on the sample
thickness $L$ and the transport mean free path $\ell$ in a non-trivial way
Starshynov  _et al._ (2018). However, in the multiple scattering regime
($L\gtrsim\ell$), it takes a simple form, mostly isotropic and negative, with
a width $\sim L$, as shown in Fig. 3a. The negative sign of the correlation is
the reason why the images appear as a negative signal on top of a bright
background. In addition, the width scaling can be understood from the
microscopic scattering process responsible for the correlation Feng  _et al._
(1988); Stephen and Cwilich (1987); van Rossum and Nieuwenhuizen (1999).
Interferences between scattered waves create a bulk speckle pattern inside the
disordered medium, which acts as an ensemble of local fluctuating sources for
diffusive transport Zyuzin and Spivak (1987); Pnini and Shapiro (1989); Van
Albada  _et al._ (1990); Genack  _et al._ (1990); de Boer  _et al._ (1992).
Two diffusive paths generated by the same source and emerging on opposite
sides of the sample are thus correlated Rogozkin and Cherkasov (1995); Froufe-
Perez  _et al._ (2007). Since diffusive paths explore a domain of transverse
size bounded by $L$, the range of $C^{RT}$ necessarily scales linearly with
$L$. This means that the resolution of the blind ghost imaging scheme is given
by the depth of the target object. This spatial resolution is comparable to
that obtained in diffuse optical imaging, which uses a CCD camera in
transmission instead of a simple bucket detector Durduran  _et al._ (2010).

Another specific feature of the blind ghost imaging scheme is its signal to
noise ratio (SNR), which depends on the amplitude and the range of the
correlation $C^{RT}$, as well as the size of the illuminated object. As
discussed above, $C^{RT}$ has a width of order $L$ and a small amplitude
$\alpha$, so that the useful signal (i.e. first term of Eq. (2)) is always
smaller than the constant background $\mathcal{A}$. In addition, because of
the Rayleigh-like statistics of the speckle patterns used to reconstruct the
image, fluctuations are large and proportional to the full signal. This
results in a SNR $\sim\sqrt{N}\alpha L^{2}/\mathcal{A}$ (see SI for details).
Typically in our experiment $\alpha\sim 10^{-3}$, which imposes a number of
measurements $N\gtrsim 10^{6}$ to get SNR $\gtrsim 1$. In the deep diffusive
regime, $L\gg\ell$, which is not reached in our experiment, it is known that
$\alpha\sim\lambda^{2}/L^{2}$ Starshynov  _et al._ (2018), so that the
correlation $C^{RT}$ becomes independent of the disorder strength parametrized
by the mean free path $\ell$, and the SNR independent of both $L$ and $\ell$
(SNR $\sim\sqrt{N}\lambda^{2}/\mathcal{A}$). This analysis shows that blind
ghost imaging can, in principle, be used to take the image of an object hidden
behind a fully opaque medium in the deep diffusive regime.

In the experiments described above, the object to be imaged was placed right
on the back of the scattering layer and the reflected speckle pattern was
recorded at its front surface. In this configuration, $C^{RT}$ is expected to
be maximally peaked Starshynov  _et al._ (2018). Since the latter originates
from bulk speckle patterns, and thus from interferences, we could wonder how
$C^{RT}$ is modified when the object is further away from the surface. As free
space propagation preserve mutual information, the integral of
$C^{RT}(\Delta\mathbf{r})$ must be constant even when it is measured between
two planes away from the scattering layer. At the same time, we expect that
the mutual information will spread over larger and larger areas, until it
becomes a constant function in the far field. To be more quantitative, we
extended the theoretical analysis of Ref. Starshynov  _et al._ (2018) and
computed analytically $C^{RT}$ on two planes at arbitrary distances, $D$ and
$D^{\prime}$, away from the sample. We found that, in the regime $L\gg\ell$,
one obtains the simple form
$C^{RT}(\Delta\mathbf{r},D,D^{\prime})=C^{RT}(\Delta\mathbf{r},0,0)\ast
h(\Delta\mathbf{r},D)\ast h(\Delta\mathbf{r},D^{\prime})$, where
$h(\Delta\mathbf{r},D)$ is a normalized function of width $\sim D$ (see SI for
details). This means that objects located further away from the scattering
layer can be imaged with almost unaffected resolution and contrast as long as
$D,D^{\prime}\ll L$. It also implies that the image quality does not depend on
the exact position of the disordered sample, but rather on the distance
between the object and the plane where the reflected speckle is imaged. To
test these predictions, we measured the correlation $C^{RT}$ from the same
sample used in the previous experiments, on two planes placed at various
distances from the sample. Representative results are shown in Fig. 4a,b for
planes at $80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$160\text{\,}\mathrm{\SIUnitSymbolMicro m}$ respectively away from the sample
(see SI for a systematic study). As can be seen, the correlation becomes
indeed wider, but does so gradually. Hence, it is possible to use blind ghost
imaging to image objects away from the scattering layer at the price of a
reduced resolution, but without introducing complicated aberrations. This is
illustrated in Fig. 4c, where we show an object and its blind ghost imaging
retrieved image, when the reflected speckle pattern was measured on the
surface of the sample and the scattering medium is
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ away from the object. The number
of measurements needed to retrieve that image was $N=1.5\times 10^{5}$. This
experiment successively mimics a situation where one does not necessarily know
how far away the object is from the scattering layer.

Figure 4:  a) and b) Correlation functions between the reflected and
transmitted speckle patterns measured $80\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ and $160\text{\,}\mathrm{\SIUnitSymbolMicro m}$ respectively away from the
transmission and reflection surfaces. c) Object separated by a cover slip of
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the scattering medium and
retrieved image using BGI with $5.65\times 10^{5}$ disorder realizations.

In conclusion, we have demonstrated ghost imaging through an opaque scattering
medium without measuring the transmitted speckle pattern that illuminates the
target. This blind ghost imaging scheme uses instead a measurement of the
reflected speckle, that is merely spatially correlated with the transmitted
one. The achievable resolution is given by the width of the correlation
function, while the number of realizations of the disorder needed to obtain a
noiseless image depends both on the amplitude of the correlation function and
the total signal received by the bucket detector. Fundamentally, our results
illustrate an important feature of ghost imaging, namely, that one does not
need to measure the illuminating signal, but only a signal weakly correlated
to it. Practically, this broadens the potential range of applications of ghost
imaging, in particular for non-invasive imaging in biological tissues. Several
possible strategies can be used to improve the processing speed, limited by
the large amount of measurements required to reach a viable signal to noise
ratio: fast-moving scattering media in conjunction with fast cameras will
naturally reduce measurement time, but for slow-moving media one can generate
different (unknown) illumination patterns by modulating the incident wavefront
with a spatial light modulator. Alternatively compressive sensing techniques
could reduce the number of necessary measurements, as long as some assumption
(e.g. sparsity) can be made about the object to be imaged Katz  _et al._
(2009).

## Methods

The scattering medium is made of a suspension of TiO2 particles in glycerol
with a concentration of $300\text{\,}\mathrm{m}\mathrm{g}$ of TiO2 for
$10\text{\,}\mathrm{mL}$ of glycerol, which lead to a scattering mean free
path $\ell=$16\text{\,}$\pm$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The
suspension is held between one glass slide and the resolution target that
works as the object to image, and its thickness is controlled using calibrated
feeler gauges. Throughout the experiments described here we used a fixed
$L=$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ thickness.

The reflected speckle pattern (see Fig. 1c for a typical measurement) is
recorded at the surface of the scattering medium using a conventional imaging
system, composed of a 10x microscope objective, a plano-convex
$150\text{\,}\mathrm{mm}$ lens and a CCD camera (Allied Vision Manta G-146).
As glycerol is very viscous, we used a piezoelectric buzzer attached to the
glass slide holding the sample to speed up the movement of the particles and
shorten the decorrelation time of the generated speckle patterns, which
allowed us to record different speckle patterns at the maximal acquisition
speed of the cameras, around 17 frames per second, and thus to perform an
ensemble average.

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