# Many-body systems with $\mathrm{SU}(1,1)$ dynamical symmetry: from dynamics
to thermodynamics based on the trace formula

Zhaoyu Fei<EMAIL_ADDRESS>Graduate School of China Academy of Engineering
Physics, No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China
C. P. Sun<EMAIL_ADDRESS>Graduate School of China Academy of Engineering
Physics, No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China
Beijing Computational Science Research Center, Beijing, 100193, China

###### Abstract

For a quantum (many-body) system with dynamical symmetry described by a given
Lie group, we study the trace of exponential operators with complex
coefficients in one of the irreducible subspaces in terms of the boson
realization. By using this approach, for compact groups, we obtain the result
of the trace that is consistent with the well-known Weyl character formula.
For non-compact groups (with $\mathrm{SU}(1,1)$ group as an application),
convergent condition of the trace is also obtained. This approach may be a
powerful tool to study the thermodynamics of quantum (many-body) systems in
equilibrium states or nonequilibrium processes.

##  I Introduction

For quantum (many-body) systems with dynamical groups, both the dynamics
(time-evolution operator) and the thermodynamics (partition function) can be
approached based on the trace of exponential operator (trace formula).
Extending the domain of the partition function to the complex plane, the trace
formula corresponds to the Lee-Yang zeros (grand-canonical ensemble) yang1952
; lee1952  or Fisher zeros (canonical ensemble) fisher1965  which establish a
connection between the properties (the zeros) of a partition function for a
finite size system and phase transitions that may occur in the thermodynamic
limit. Moreover, such a trace formula is also involved in the study of quantum
(many-body) systems in nonequilibrium processes, such as the work statistics
aq2000 ; ja2000 , full-counting statistics in electric circuit kinder2003 ,
Loschmidt echoes goussev2012 , and dynamical phase transition. It is worth
mentioning that both the Lee-Yang zeros and the characteristic function of
work have been observed in experiments by measuring quantum coherence of a
probe spin peng2015  and the qubit-assisted Ramsey-like scheme mazzola2013
respectively.

Since the spectra and evolution of the system are solvable, it is reasonable
to expect that the trace formula is also solvable. Actually, the Levitov’s
formula levitov1996 ; klick2003  and the trace formula for quadratic
Hamiltonian fei2019  have already been discovered for identical-particle
systems. However, these formulas, which are valid when the Hilbert space
consists of Fock states, can not be applied when the trace formula in an
irreducible subspace is concerned with. For compact Lie groups, the
irreducible representation is finite-dimensional. Thus, the trace formula is
equivalent to the characters of Lie groups. By using the Weyl character
formula weyl1953 ; fulton2004 , there is a systematical approach to
calculating the characters fall2015 . For non-compact Lie groups, thing are
quite different due to the infinite dimensions of the Hilbert space. The trace
formula is not conjugate-invariant or not convergent in some cases. Then, the
trace formula is not equivalent to the (global) characters of Lie groups
knapp2016 ; kirillov2012  as the latter are always conjugate-invariant and
convergent.

In this paper, we want to overcome this conundrum by using the boson
realization of Lie groups. The boson realization starts from Schwinger’s
representation of $\mathrm{SU}(2)$ and is further applied to other Lie groups
such as $\mathrm{SU}(1,1)$ gerry1991  and $\mathrm{SU}(N)$ mathur2010 . In
addition, with the help of the projection operator and the coherent-
representation technology, we transform the trace formula into a gaussian
integral in a complex plane whose result and convergent condition are easy to
obtain. As an application, we obtain the result and the convergent condition
of the trace formula for the non-compact group: $\mathrm{SU}(1,1)$. This group
is of importance in the study of quantum many-body systems since it is a broad
family of interacting many-body systems that are analytically solvable
beau2020 .

##  II Preliminaries

We study a boson realization of group $G$ means that the corresponding Lie
algebra $\mathfrak{g}$ can be realized by the quadratic form of bosonic
operators. Let $a_{i},a^{\dagger}_{j},i,j=1,\cdots,r$ denote the number of $r$
bosonic annihilation and creation operators, i.e.,
$[a_{i},a_{j}^{\dagger}]=\delta_{ij}$. The associated Hilbert space
$\mathcal{H}$ is the Fock space
$\\{|n_{1},\cdots,n_{r}\rangle\\}\equiv\\{|\bm{n}\rangle\\}$. In the boson
realization of group $G$, for the elements in the Lie algebra ($\forall
ix\in\mathfrak{g}$), there is a $2r\times 2r$ symmetric matrix $M(x)$ such
that

$\hat{x}=\bm{\alpha}M(x)\bm{\alpha}^{T}$ (1)

is a representation of $x$ in the Hilbert space, where $T$ denotes the matrix
transpose, $\bm{\alpha}=(\bm{a},\bm{a}^{\dagger})$ and
$\bm{a}=(a_{1},\cdots,a_{r})$,
$\bm{a}^{\dagger}=(a_{1}^{\dagger},\cdots,a_{r}^{\dagger})$ are the vectors of
annihilation and creation operators. In physics, Eq. (1) is the second
quantization of one-body operators. We also set

$\omega=\left(\begin{matrix}0&1_{r}\\\ 1_{r}&0\end{matrix}\right),$ (2)

with the $r\times r$ identity matrix $1_{r}$ and require that $\omega M(x)$ is
hermitian, which preserves the unitarity of this representation. In addition,
according to Refs. balian1969 , there is a symplectic matrix $2\tau M(x)$
(called the symplectic representation of $x$) associated with $M(x)$, where

$\tau=\left(\begin{matrix}0&1_{r}\\\ -1_{r}&0\end{matrix}\right).$ (3)

Since the Hilbert space is the representation space of group $G$, it can be
decomposed as the direct sum of the irreducible representation of $G$, i.e.,
$\mathcal{H}=\bigoplus_{\mu}\mathcal{H}_{\mu}$. Here,
$\mathcal{H}_{\mu}=\\{|\mu,h\rangle\\}$ is one of the irreducible subspaces,
$\mu$ indicates different irreducible representations and $h$ denotes the
eigenvalues of the Cartan subalgebra of group $G$ (denoted by $\mathfrak{h}$),
i.e., $\hat{y}|\bm{\mu},\bm{h}\rangle=y(\bm{h})|\bm{\mu},\bm{h}\rangle$ for
$\hat{y}\in\mathfrak{h}$ and some function $y(\bm{h})$.

Due to the linearity of Lie algebra, the Hilbert space $\mathcal{H}$ can be
also regarded as the representation space of the complexification of the Lie
algebra $\mathfrak{g}$ (denoted by $\mathfrak{g}_{\mathbb{C}}$). If the
corresponding complex group $G_{\mathbb{C}}$ is a complex connected Lie group,
$G_{\mathbb{C}}$ is weakly exponential, namely
$\bigcup_{x\in\mathfrak{g}_{\mathbb{C}}}\exp(x)$ is a dense set in
$G_{\mathbb{C}}$ hofmann1978 . In this case, the Hilbert space $\mathcal{H}$
is also regarded as the representation space of $G_{\mathbb{C}}$. In this
paper, we aim to calculate the trace formula of $G_{\mathbb{C}}$ in the
subspace $\mathcal{H}_{\bm{\mu}}$, namely to obtain the expression of

$\mathrm{Tr}_{\bm{\mu}}[\exp(\hat{x})]=\sum_{\bm{h}}\langle\bm{\mu},\bm{h}|\exp(\hat{x})|\bm{\mu},\bm{h}\rangle$
(4)

and the convergent condition of Eq. (4). When $\hat{x}$ is skew-Hermitian,
$\exp(\hat{x})$ denotes the time-evolution operator of a quantum system. When
$\hat{x}$ is Hermitian, Eq. (4) is the partition function in the irreducible
subspace. In general, $\hat{x}$ is a combination of Hermitian operator and
skew-Hermitian operator. Then, Eq. (4) corresponds to the Lee-Yang zeros
yang1952 ; lee1952  or Fisher zeros fisher1965 . Moreover, since the
representation matrices preserve the group multiplication, $\exp(\hat{x})$ is
also able to denote the product of several exponential operators. Thus, Eq.
(4) corresponds to the generator of correlation functions, or the distribution
for some statistics of dynamical systems, such as the work statistics aq2000 ;
ja2000 , full-counting statistics in electric circuit kinder2003 .

##  III projection onto the irreducible subspace

Because the trace in the irreducible subspace is hard to deal with, we
introduce the projection operator $\hat{P}_{\bm{\mu}}$

$\hat{P}_{\bm{\mu}}|\bm{\mu}^{\prime},\bm{h}^{\prime}\rangle=\delta_{\bm{\mu}\bm{\mu}^{\prime}}|\bm{\mu},\bm{h}^{\prime}\rangle,$
(5)

and rewrite Eq. (4) as the trace in the Fock states:

$\displaystyle\begin{split}\mathrm{Tr}_{\bm{\mu}}[\exp(\hat{x})]&=\sum_{\bm{\mu}^{\prime}}\sum_{\bm{h}^{\prime}}\langle\bm{\mu}^{\prime},\bm{h}^{\prime}|\hat{P}_{\bm{\mu}}\exp(\hat{x})|\bm{\mu}^{\prime},\bm{h}^{\prime}\rangle\\\
&=\sum_{\bm{n}}\langle\bm{n}|\hat{P}_{\bm{\mu}}\exp(\hat{x})|\bm{n}\rangle\\\
&\equiv\mathrm{Tr}[\hat{P}_{\bm{\mu}}\exp(\hat{x})].\end{split}$ (6)

Furthermore, we assume that the projection operator $\hat{P}_{\bm{\mu}}$ can
be generated from a exponential quadratic form of bosonic operators, i.e.,
$\sum_{\bm{\mu}}t^{\bm{\mu}}\hat{P}_{\bm{\mu}}=\hat{P}(t)$ for $t\in D$ ($D$
is some subset of the complex plane) and

$\hat{P}(t)=f(t)\exp\left(\bm{\alpha}P(t)\bm{\alpha}^{T}\right)$ (7)

for some $2r\times 2r$ symmetric matrix $P(t)$ and some scalar $f(t)$. This
assumption is satisfied for examples considered below. Then, due to Eqs. (1,
7) the trace formula in the subspace is transformed to the trace of an
exponential operator, namely

$\displaystyle\begin{split}\mathrm{Tr}[\hat{P}(t)\exp(\hat{x})]=f(t)\mathrm{Tr}\left[\exp\left(\bm{\alpha}A\bm{\alpha}^{T}\right)\right],\end{split}$
(8)

where $A=P(t)+M(x)$. In the Eq. (8), we have used the property that
$\hat{P}_{\bm{\mu}},\hat{P}(t)$ commute with $\hat{x}$, since the projection
operator is a number in the irreducible subspace.

##  IV coherent-state representation

Coherent-state representation is a well-applied technique to calculate the
trace of operators involved bosonic or fermionic operators gazeau2009 . It
transforms the summation over Fock states to a integral over a complex plane
(the latter is convenient to analyze). In order to simplify Eq. (8), we
introduce the coherent representation of the bosonic operators scully1997 ;
gazeau2009  in the following, i.e., $\bm{z}=(z_{1},\cdots,z_{n})$,

$\displaystyle\begin{split}&a_{i}|\bm{z}\rangle=z_{i}|\bm{z}\rangle,\\\
&\langle\bm{z}|\bm{z}^{\prime}\rangle=\prod_{i=1}^{r}\exp\left[-\frac{1}{2}(|z_{i}|^{2}+|z^{\prime}_{i}|^{2}-2z_{i}^{*}z^{\prime}_{i})\right],\\\
&\int\frac{\mathrm{d}^{2r}\bm{z}}{\pi^{r}}|\bm{z}\rangle\langle\bm{z}|=\sum_{\bm{n}}|\bm{n}\rangle\langle\bm{n}|,\end{split}$
(9)

where $*$ denotes the complex conjugate, and we have used the abbreviation

$\mathrm{d}^{2r}\bm{z}\equiv\prod_{i=1}^{r}\mathrm{d}\mathcal{R}(z_{i})\mathrm{d}\mathcal{J}(z_{i}),$
(10)

where $\mathcal{R}(z_{i}),\mathcal{J}(z_{i})$ denote the real part and
imaginary part of $z_{i}$. Since $|\bm{z}\rangle$ is the eigenstate of the
annihilation operators, by using the normal-ordered form of exponential
operators in agrawal1977  and the overcompleteness of the coherent states
(Eq.(LABEL:e10)), we rewrite Eq. (8) as a gaussian integral in the coherent
representation,

$\displaystyle\begin{split}\mathrm{Tr}\left[\exp\left(\bm{\alpha}A\bm{\alpha}^{T}\right)\right]&=\int\frac{\mathrm{d}^{2r}\bm{z}}{\pi^{r}}\langle\bm{z}|\exp\left(\bm{\alpha}A\bm{\alpha}^{T}\right)|\bm{z}\rangle\\\
&=[\det(\cosh(\tau A)+\omega\tau\sinh(\tau
A))]^{-\frac{1}{2}}\int\frac{\mathrm{d}^{2r}\bm{z}}{\pi^{r}}\exp[-\bm{Z}\omega\tau(\coth(\tau
A)+\omega\tau)^{-1}\bm{Z}^{\dagger}],\end{split}$ (11)

where $\bm{Z}=(\bm{z}^{*},\bm{z})$ and $\bm{Z}^{\dagger}$ is the complex
conjugate of $\bm{Z}$.

Now, the trace formula is transformed to a complex gaussian integral (Eq.
(11)) which is fully solvable (see Appendix A). Hence we obtain the result of
Eq. (11)

$\displaystyle\begin{split}&f(t)\mathrm{Tr}\left[\exp\left(\bm{\alpha}A\bm{\alpha}^{T}\right)\right]\\\
&=f(t)\\{(-1)^{r}\det[\exp(2\tau A)-1]\\}^{-\frac{1}{2}}\\\
&=f(t)\\{(-1)^{r}\det[\exp(2\tau P(t))\exp(2\tau M(x))-1]\\}^{-\frac{1}{2}}\\\
&=\sum_{\bm{\mu}}t^{\bm{\mu}}\mathrm{Tr}_{\bm{\mu}}[\exp(\hat{x})]\end{split}$
(12)

with the convergent condition: the hermitian part of $\omega\tau(\coth(\tau
A)+\omega\tau)^{-1}|_{t\in D}$ is positive definite. In the derivation of Eq.
(LABEL:e13), we have used the property $\det[\exp(2\tau A)]=1$ ($\exp(2\tau
A)$ is a symplectic matrix). The sign of the square root in Eq. (LABEL:e13) is
determined by two conditions: (1) for $iy\in\mathfrak{h}$, $\hat{y}$ is
hermitian. Its eigenvalues are real. Thus,
$\mathrm{Tr}_{\bm{\mu}}[\exp(\hat{y})]>0$; (2) for
$x\in\mathfrak{g}_{\mathbb{C}}$ and $\mathrm{Tr}_{\bm{\mu}}[\exp(\hat{x})]$
converges, $\mathrm{Tr}_{\bm{\mu}}[\exp(\hat{x})]$ is a continuous function of
$\hat{x}$. Since the determinant is invariant under similar transformation,
the trace formula can also be obtained by using the diagonalization of the
representation matrices. That is to say for
$x,s\in\mathfrak{g}_{\mathbb{C}},y\in\mathfrak{h}_{\mathbb{C}}$, and
$x=\exp(-s)y\exp(s)$, if both
$\mathrm{Tr}[\hat{P}(t)\exp(\hat{x})],\mathrm{Tr}[\hat{P}(t)\exp(\hat{y})]$
are convergent (see the convergent condition below Eq. (LABEL:e13)), we have

$\displaystyle\begin{split}\mathrm{Tr}[\hat{P}(t)\exp(\hat{x})]&=f(t)\\{(-1)^{r}\det[\exp(2\tau
P(t))\exp(2\tau M(x))-1]\\}^{-\frac{1}{2}}\\\ &=f(t)\\{(-1)^{r}\det[\exp(2\tau
P(t))\exp(-2\tau M(s))\exp(2\tau M(y))\exp(2\tau M(s))-1]\\}^{-\frac{1}{2}}\\\
&=f(t)\\{(-1)^{r}\det[\exp(2\tau P(t))\exp(2\tau M(y))-1]\\}^{-\frac{1}{2}}\\\
&=\mathrm{Tr}[\hat{P}(t)\exp(\hat{y})],\end{split}$ (13)

and
$\mathrm{Tr}_{\bm{\mu}}[\exp(\hat{x})]=\mathrm{Tr}_{\bm{\mu}}[\exp(\hat{y})]=\sum_{\bm{h}}\exp(y(\bm{h}))$.
In the derivation, we have used the property: $\exp(2\tau M(x))=\exp(-2\tau
M(s))\exp(2\tau M(y))\exp(2\tau M(s))$ because $2\tau M$ is the symplectic
representation of $\mathfrak{g}_{\mathbb{C}}$ balian1969 . To illustrate our
approach, we calculate the known trace formula of compact groups: the spin
system ($\mathrm{SU}(2)$) and the quark system ($\mathrm{SU}(3)$). Besides, as
an application, we obtain the result and the convergent condition of the trace
formula for $\mathrm{SU}(1,1)$ in Sec. (VI).

##  V Examples

###  V.1 $\mathrm{SU}(2)$

Let $iJ_{j},j=1,2,3$ denote the three generators of $\mathfrak{su}(2)$. We
choose $J_{3}$ as the generator of the Cartan subalgebra. The fundamental
representation of $\mathfrak{su}(2)$ (denoted by $\rho$) is given by the Pauli
matrices $\sigma_{j},j=1,2,3$, i.e., $\rho(J_{i})=\sigma_{i}/2$.

Following Schwinger’s construction schw2000 , the boson realization of
$\mathfrak{su}(2)$ is given by
$\hat{J}_{i}=\bm{\alpha}^{\dagger}\rho(J_{i})\bm{\alpha}$, where the dimension
of $\bm{\alpha}$ is 2. Then, the irreducible decomposition of
$\mathcal{H}=\\{|n_{1},n_{2}\rangle\\}$ is
$\bigoplus_{2j=0,1,2\cdots}\\{|j,m\rangle\\}$, where
$2j=n_{1}+n_{2},m=-j,-j+1,\cdots,j$, and

$\displaystyle\begin{split}(\hat{J}_{1}^{2}+\hat{J}_{2}^{2}+\hat{J}_{3}^{2})|j,m\rangle&=j(j+1)|j,m\rangle\\\
\hat{J}_{3}|j,m\rangle&=m|j,m\rangle.\end{split}$ (14)

The projection operator onto the subspace $2j$,
$\hat{P}_{2j}=\sum_{m=-j}^{j}|j,m\rangle\langle j,m|$, is generated from a
Taylor series

$\hat{P}(t)=\sum_{2j=0}^{\infty}t^{2j}\hat{P}_{2j}=\exp[(\bm{\alpha}^{\dagger}\cdot\bm{\alpha})\ln
t],$ (15)

where $t\in[0,\tau]$ for some positive $\tau$ near 0. Then, it follows from
Eq. (7) that $f(t)=t^{-1}$ and

$2\tau A=\left(\begin{matrix}\ln t+[\rho(x)]^{T}&0\\\ 0&-\ln
t-\rho(x)\end{matrix}\right).$ (16)

For the gaussian integral in Eq. (11), the converge condition is always
satisfied because $\omega\tau(\coth(\tau
A)+\omega\tau)^{-1}|_{t\in[0,\tau]}=1/2+O(\tau)$ is positive definite, which
is consistent with the fact that $\mathcal{H}_{2j}$ is a finite-dimensional
representation of $\mathfrak{su}(2)$.

Finally, from Eq. (LABEL:e13), we obtain the trace formula of
$\mathrm{SU}(2)$:

$\displaystyle\begin{split}\mathrm{Tr}[\hat{P}(t)\exp(\hat{x})]&=\\{\det[e^{-\rho(x)}-t]\\}^{-1}\\\
&=\sum_{2j=0}^{\infty}t^{2j}\frac{\varepsilon^{2j+1}-\varepsilon^{-2j-1}}{\varepsilon-\varepsilon^{-1}},\end{split}$
(17)

where $\varepsilon,\varepsilon^{-1}$ are the eigenvalues of $e^{\rho(x)}$.

###  V.2 $\mathrm{SU}(3)$

Let $iT_{j},j=1,2,\cdots,8$ denote the eight generators of $\mathfrak{su}(3)$.
We choose $J_{3},J_{8}$ as the generators of the Cartan subalgebra. There are
two inequivalent fundamental representations of $\mathfrak{su}(3)$ (denoted by
$\rho,\rho^{\prime}$): (1) $\rho(T_{i})=\lambda_{i}/2$, where
$\lambda_{j},j=1,2,\cdots,8$ are the Gell-Mann matrices. This is called the
“$3$” representation; (2) $\rho^{\prime}(T_{i})=-\lambda_{i}^{*}/2$. This is
called the “$\bar{3}$” representation.

According to Ref. mathur2001 , the boson realization of $\mathfrak{su}(3)$ is
given by

$\hat{T}_{i}=\bm{\alpha}^{\dagger}\left(\begin{matrix}\rho(T_{i})&0_{r}\\\
0_{r}&\rho^{\prime}(T_{i})\end{matrix}\right)\bm{\alpha},$ (18)

where the dimension of $\bm{\alpha}$ is 6. Then, the irreducible decomposition
of $\mathcal{H}=\\{|n_{1},\cdots,n_{6}\rangle\\}$ is
$\bigoplus_{p,q=0,1,2\cdots}[\mathcal{H}_{(p,0)}\bigotimes\mathcal{H}_{(0,q)}]$,
where

$p=\sum_{i=1}^{3}n_{i},\ \ \ q=\sum_{i=4}^{6}n_{i},$ (19)

and $(p,q)$ denotes the Dynkin labels of the irreducible representation.

The projection operator onto the subspace
$\mathcal{H}_{(p,0)}\bigotimes\mathcal{H}_{(0,q)}$ is

$\hat{P}_{p,q}=\sum_{n_{1}+n_{2}+n_{3}=p,\atop
n_{4}+n_{5}+n_{6}=q}|n_{1},\cdots,n_{6}\rangle\langle n_{1},\cdots,n_{6}|.$
(20)

It is generated from a Taylor series

$\displaystyle\begin{split}\hat{P}(t,t^{\prime})&=\sum_{p,q=0}^{\infty}t^{p}t^{\prime
q}\hat{P}_{p,q}\\\
&=\exp\left[\left(\sum_{i=1}^{3}a^{\dagger}_{i}a_{i}\right)\ln
t+\left(\sum_{i=4}^{6}a^{\dagger}_{i}a_{i}\right)\ln
t^{\prime}\right],\end{split}$ (21)

where $t,t^{\prime}\in[0,\tau]$ for some positive $\tau$ near 0. it follows
from Eq. (7) that $f(t)=t^{-1}$ and

$2\tau A=\left(\begin{matrix}1_{3}&0&0&0\\\ 0&0&1_{3}&0\\\ 0&1_{3}&0&0\\\
0&0&0&1_{3}\end{matrix}\right)\left(\begin{matrix}\ln t+[\rho(x)]^{T}&0&0&0\\\
0&-\ln t-\rho(x)&0&0\\\ 0&0&\ln t+[\rho^{\prime}(x)]^{T}&0\\\ 0&0&0&-\ln
t-\rho^{\prime}(x)\end{matrix}\right)\left(\begin{matrix}1_{3}&0&0&0\\\
0&0&1_{3}&0\\\ 0&1_{3}&0&0\\\ 0&0&0&1_{3}\end{matrix}\right).$ (22)

For the gaussian integral in Eq. (11), similar to the $\mathrm{SU}(2)$
example, the convergent condition is always satisfied.

Finally, from Eq. (LABEL:e13), we obtain the trace formula of
$\mathrm{SU}(3)$:

$\displaystyle\begin{split}\mathrm{Tr}[\hat{P}(t)\exp(\hat{x})]&=\\{\det[e^{-\rho(x)}-t]\det[e^{-\rho^{\prime}(x)}-t^{\prime}]\\}^{-1}\\\
&=\sum_{p,q=0}^{\infty}t^{p}t^{\prime
q}h_{p}(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})h_{q}(\varepsilon_{1}^{-1},\varepsilon_{2}^{-1},\varepsilon_{3}^{-1}),\end{split}$
(23)

where $\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}$ are the eigenvalues of
$e^{\rho(x)}$, and $h_{k}$ denotes the complete homogeneous symmetric
polynomial of degree $k$. Moreover, since coleman1964 ; chat2002

$\mathcal{H}_{(p,0)}\bigotimes\mathcal{H}_{(0,q)}=\bigoplus_{r=0,1,\cdots,\min(p,q)}\mathcal{H}_{(p-r,q-r)},$
(24)

it follows from Eq. (23) that

$\displaystyle\begin{split}&\mathrm{Tr}_{(p,q)}[\exp(\hat{x})]\\\
&=\mathrm{Tr}_{(p,0)\bigotimes(0,q)}[\exp(\hat{x})]-\mathrm{Tr}_{(p-1,0)\bigotimes(0,q-1)}[\exp(\hat{x})]\\\
&=h_{p}(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})h_{q}(\varepsilon_{1}^{-1},\varepsilon_{2}^{-1},\varepsilon_{3}^{-1})-\\\
&\ \ \ \
h_{p-1}(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})h_{q-1}(\varepsilon_{1}^{-1},\varepsilon_{2}^{-1},\varepsilon_{3}^{-1}).\end{split}$
(25)

In comparison with the Weyl character formula weyl1953 ; fulton2004 , Eq.
(LABEL:e25) indicates some identity relations between the complete homogeneous
symmetric polynomials.

##  VI Application: trace formula of $\mathrm{SU}(1,1)$

$\mathrm{SU}(1,1)$ is the most elementary non-compact group perelomov1986 .
Its irreducible unitary representation is infinite dimensional with
equidistant basis vectors, which is just the energy level of quantum harmonic
oscillators. Besides, the quantum many-body systems with $\mathrm{SU}(1,1)$ as
the dynamical group is of particular interest beau2016 ; deng2018 ; myers2021
since it is a broad family of interacting many-body systems that are
analytically solvable. As an example of this family, the Calogero–Sutherland
model clearly shows the transformation of statistical behaviors between bosons
and fermions through changing the strength of interaction, which is known as
the fractional exclusion statistics murthy1994 . Other examples include the
non-interacting bosons or fermions, the Tonks-Girardeau gas, two-dimensional
Bose-Einstein condensates with s-wave contact interactions pita1997 , and the
unitary Fermi gas werner2006 , which have been realized in ultracold atom
laboratories kino2004 ; pare2004 ; hara2002 ; saint2019 ; deng2018 .

Let $iK_{j},j=1,2,3$ denote the three generators of $\mathfrak{su}(1,1)$. We
choose $K_{3}$ as the generators of the Cartan subalgebra. The fundamental
representation of $\mathfrak{su}(1,1)$ (denoted by $\rho$) is given by the
Pauli matrices:
$\rho(K_{1})=i\sigma_{2}/2,\rho(K_{2})=-i\sigma_{1}/2,\rho(K_{3})=\sigma_{3}/2$.

According to Ref. gerry1991 , the two-mode realization of $\mathfrak{su}(2)$
is given by

$\displaystyle\begin{split}\hat{K}_{1}&=\frac{a^{\dagger}_{1}a^{\dagger}_{2}+a_{1}a_{2}}{2}\\\
\hat{K}_{2}&=\frac{a^{\dagger}_{1}a^{\dagger}_{2}-a_{1}a_{2}}{2i}\\\
\hat{K}_{3}&=\frac{a^{\dagger}_{1}a_{1}+a_{2}^{\dagger}a_{2}+1}{2}\\\
\end{split}$ (26)

Then, the irreducible decomposition of $\mathcal{H}=\\{|n_{1},n_{2}\rangle\\}$
is

$\mathcal{H}=\\{|k=1/2,m,s=0\rangle\\}\bigoplus_{2k=1,2,\cdots\atop
s=\pm}\\{|k,m,s\rangle\\},$ (27)

where $2k=|n_{1}-n_{2}|+1,m=0,1,\cdots$. And the following relations are
satisfied

$\displaystyle\begin{split}(\hat{K}_{1}^{2}-\hat{K}_{2}^{2}-\hat{K}_{3}^{2})|k,m,s\rangle&=k(k-1)|k,m,s\rangle\\\
\hat{K}_{3}|k,m,s\rangle&=(k+m)|k,m,s\rangle\\\
\hat{K}_{+}|k,m,s\rangle&=\sqrt{(m+1)(m+2k)}|k,m+1,s\rangle\\\
\hat{K}_{-}|k,m,s\rangle&=\sqrt{m(m+2k-1)}|k,m-1,s\rangle,\end{split}$ (28)

where $\hat{K}_{\pm}=\hat{K}_{1}\pm i\hat{K}_{2}$. Here, we set $n_{1}>n_{2}$
for $|k,m,+\rangle$ and $n_{1}<n_{2}$ for $|k,m,-\rangle$.

The projection operator onto the subspace $\mathcal{H}_{k,s}$ is

$\hat{P}_{k,s}=\sum_{m=0}^{\infty}|k,m,s\rangle\langle k,m,s|.$ (29)

It is generated from a Laurent series (different from the Taylor series in the
compact groups $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$)

$\displaystyle\begin{split}\hat{P}(t)=&t^{-1}\hat{P}_{1/2,0}+\sum_{2k=1}^{\infty}t^{-2k-1}\hat{P}_{k,-}+\sum_{2k=1}^{\infty}t^{2k-1}\hat{P}_{k,+}\\\
=&\exp[(a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2})\ln t],\end{split}$ (30)

where the domain of $t$ is shown below. Then, it follows from Eq. (7) that
$f(t)=1$ and

$2\tau A=\left(\begin{matrix}1&0&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
0&1&0&0\end{matrix}\right)\left(\begin{matrix}\ln t+\rho(x)&0\\\ 0&-\ln
t+\rho(x)\end{matrix}\right)\left(\begin{matrix}1&0&0&0\\\ 0&0&0&1\\\
0&1&0&0\\\ 0&0&1&0\end{matrix}\right).$ (31)

Finally, from Eq. (LABEL:e13), we obtain the trace formula of
$\mathrm{SU}(1,1)$:

$\mathrm{Tr}[\hat{P}(t)\exp(\hat{x})]=\frac{-t}{\det[e^{\rho(x)}-t]}=\frac{-t}{(t-\varepsilon)(t-\varepsilon^{-1})},$
(32)

where $\varepsilon,\varepsilon^{-1}$ are the eigenvalues of $e^{\rho(x)}$. For
$|\varepsilon|=1$, the Laurent series of Eq. (32) does not exist and our
approach is invalid. For $|\varepsilon|\neq 1$, without loss of generality,
setting $|\varepsilon|<1$, the Laurent series of Eq. (32) reads

$\frac{-t}{(t-\varepsilon)(t-\varepsilon^{-1})}=\sum_{n=-\infty}^{\infty}t^{n}\frac{\varepsilon^{n+1}}{1-\varepsilon^{2}}$
(33)

which is valid for $|\varepsilon|<|t|<|\varepsilon|^{-1}$. Hence, the
convergent condition of Eq. (32) reads: the hermitian part of
$\omega\tau(\coth(\tau
A)+\omega\tau)^{-1}|_{|\varepsilon|<|t|<|\varepsilon|^{-1}}$ is positive
definite. According to Eq. (30), the trace formula of $\mathrm{SU}(1,1)$ is

$\mathrm{Tr}_{2k}[\exp(\hat{x})]=\frac{\varepsilon^{2k}}{1-\varepsilon^{2}},$
(34)

which is also generalized to the case that $2k$ is a positive real number by
using the Barut-Girardello coherent states (see Appendix B).

##  VII Conclusion

Based on the boson realization of (complex) Lie groups, we propose a unified
approach to the calculation of the trace of exponential operators. Besides the
known trace formula of compact groups (Weyl character formula), we also obtain
the result and the convergent condition of the trace for non-compact groups by
using this approach. A useful conclusion is that under convergent conditions,
the trace of exponential operators in the Hilbert space is equal to a
determinant involving the lower-dimensional representation of the group
element. Since two elementary operators in quantum mechanics (the time
evolution operator and the density operator of equilibrium state) are
exponential operators, we hope our approach will be widely applied in the
future in the study of equilibrium thermodynamics and nonequilibrium process
of quantum (many-body) systems.

###### Acknowledgements.

We thank Jinpeng An for helpful discussions. This work is supported by the
National Natural Science Foundation of China (NSFC) (Grants No. 12088101),
NSAF (No. U1930402, and No. U1930403). Z. Y. Fei acknowledges support from
China Postdoctoral Science Foundation (Grant No. 2021M700359).

## Appendix A: evaluation of the complex gaussian integral

Let

$\Lambda(B)=\int\frac{\mathrm{d}^{2r}\bm{z}}{\pi^{r}}\exp\left(-\frac{1}{2}\bm{Z}B\bm{Z}^{{\dagger}}\right),$
(A.1)

where $\omega B$ is a $2r\times 2r$ complex symmetric matrix.

Every $B$ has a unique decomposition $B=B^{\prime}+iB^{\prime\prime}$, with
hermitian $B^{\prime}$ and $B^{\prime\prime}$, and we call $B^{\prime}$ the
hermitian part of $B$. Then, for invertible $B^{\prime}$, the integral in Eq.
(A.1) converges if and only if $B^{\prime}$ is positive definite and the
result reads

$\Lambda(B)=[\det(B)]^{-\frac{1}{2}}.$ (A.2)

_Proof_. Let $z_{k}=x_{2k-1}+ix_{2k},\bm{X}=(x_{1},\cdots,x_{2r})$. Then, Eq.
(A.1) is rewritten as

$\displaystyle\begin{split}&\Lambda(B)=\pi^{-r}\int\prod_{i=1}^{2r}\mathrm{d}x_{i}\exp\left(-\bm{X}UBU^{{\dagger}}\bm{X}^{T}\right)\\\
&=\pi^{-r}\int\prod_{i=1}^{2r}\mathrm{d}x_{i}\exp\left(-\bm{X}UB^{\prime}U^{{\dagger}}\bm{X}^{T}-i\bm{X}UB^{\prime\prime}U^{{\dagger}}\bm{X}^{T}\right),\end{split}$
(A.3)

where $\bm{Z}=\sqrt{2}\bm{X}U$, and both
$UB^{\prime}U^{{\dagger}},UB^{\prime\prime}U^{{\dagger}}$ are real symmetric
matrices (notice that $U^{{\dagger}}=U^{-1}=\omega U^{T}$).

We proceed in three steps: (1) If $B=1$, then $\Lambda(1)=\det(1)=1$; (2) Eq.
(LABEL:a3) is convergent if and only if it is absolutely convergent
beesack1970 , i.e., $B^{\prime}$ is positive definite; (3) If
$B^{\prime\prime}=0$, $B^{\prime}$ is positive definite,
$UB^{\prime}U^{{\dagger}}$ can be factorized as
$UB^{\prime}U^{{\dagger}}=LL^{T}$, where $L$ is a real lower triangular matrix
with positive diagonal entries (the Cholesky decomposition horn1985 ). Setting
$\bm{X}^{\prime}=(x^{\prime}_{1},\cdots,x^{\prime}_{r})=\bm{X}L$, we find that
the Jacobian of the transformation is $[\det(L)]^{-1}$. Hence,
$\Lambda(B)=[\det(L)]^{-1}\Lambda(1)=[\det(B)]^{-\frac{1}{2}}$. This result is
still valid when $B^{\prime\prime}\neq 0$ by following the similar procedure
of analytic continuation in the appendix of Ref. bargmann1962 .

## Appendix B: the trace formula of $\mathrm{SU}(1,1)$ using the Barut-
Girardello coherent states

For $k$ is a positive real number, let the states $|k,m\rangle$ which satisfy
the relations in Eq. (28) denote the (projective) representation of
$\mathrm{SU}(1,1)$. Then, we define eigenstates $|k,z\rangle$ of the lowering
operator $\hat{K}_{-}$ as:

$\hat{K}_{-}|k,z\rangle=z|k,z\rangle,$ (B.1)

where $z$ is an arbitrary complex number. Thus, $\\{|k,z\rangle\\}$ are called
the Barut-Girardello states barut1971  and can be decomposed over the
orthonormal state basis $\\{|k,m\rangle\\}$,

$|k,z\rangle=\frac{z^{k-1/2}}{\sqrt{I_{2k-1}(2|z|)}}\sum_{m=0}^{\infty}\frac{z^{m}}{m!\Gamma(m+2k)}|k,m\rangle$
(B.2)

where $\Gamma$ denotes the Euler gamma function and $I_{2k-1}$ denotes the
modified Bessel function of the first kind. The Barut-Girardello coherent
states are overcomplete, i.e.,

$\displaystyle\begin{split}&\langle
k,z|k,z^{\prime}\rangle=\frac{I_{2k-1}(2\sqrt{z^{*}z^{\prime}})}{\sqrt{I_{2k-1}(2|z|)I_{2k-1}(2|z^{\prime}|)}},\\\
&\int\mathrm{d}\mu(k,z)|k,z\rangle\langle
k,z|=\sum_{m=0}^{\infty}|k,m\rangle\langle k,m|\end{split}$ (B.3)

with the measure

$\displaystyle\begin{split}\mathrm{d}\mu(k,z)&=\frac{2}{\pi}K_{2k-1}(2|z|)I_{2k-1}(2|z|)\mathrm{d}^{2}z,\\\
\mathrm{d}^{2}z&=\mathrm{d}\mathcal{R}(z)\mathrm{d}\mathcal{J}(z).\end{split}$
(B.4)

Here, $K_{2k-1}$ denotes the modified Bessel function of the second kind.

In general, the exponential operator $\exp(x)$ can be rewritten in the
following form:

$e^{x}=e^{\lambda_{+}K_{+}}e^{\lambda_{3}K_{3}}e^{\lambda_{-}K_{-}},$ (B.5)

where $\lambda_{3},\lambda_{\pm}$ are obtained by using the composition
formula of $\mathrm{SU}(1,1)$ in Ref. martinez2020 . Then, the trace formula
for $\mathrm{SU}(1,1)$ reads

$\displaystyle\begin{split}&\mathrm{Tr}_{2k}[e^{\lambda_{+}K_{+}}e^{\lambda_{3}K_{3}}e^{\lambda_{-}K_{-}}]\\\
&\equiv\sum_{m=0}^{\infty}\langle
k,m|e^{\lambda_{+}K_{+}}e^{\lambda_{3}K_{3}}e^{\lambda_{-}K_{-}}|k,m\rangle\\\
&=\int\mathrm{d}\mu(k,z)\langle
k,z|e^{\lambda_{+}K_{+}}e^{\lambda_{3}K_{3}}e^{\lambda_{-}K_{-}}|k,z\rangle\\\
&=\int\mathrm{d}\mu(k,z)\langle k,z|e^{\lambda_{3}K_{3}}|k,z\rangle
e^{\lambda_{+}z^{*}+\lambda_{-}z}\\\
&=\int\mathrm{d}\mu(k,z)\frac{I_{2k-1}(2e^{\lambda_{3}/2}|z|)}{I_{2k-1}(2|z|)}e^{\lambda_{3}/2+\lambda_{+}z^{*}+\lambda_{-}z}\end{split}$
(B.6)

where we have used Eq. (B.2) and the series expansion of $I_{\nu}(2x)$:

$I_{\nu}(2x)=\sum_{n=0}^{\infty}\frac{x^{2n+\nu}}{n!\Gamma(n+\nu+1)}$ (B.7)

in the derivation.

Using the polar coordinate of the complex number
$z=re^{i\phi},r\in[0,\infty),\phi\in[0,2\pi)$ and first integrate Eq.
(LABEL:b6) over $\phi$, we finally obtain the integral of the product of three
Bessel functions:

$\displaystyle\begin{split}&\mathrm{Tr}_{2k}[e^{\lambda_{+}K_{+}}e^{\lambda_{0}K_{0}}e^{\lambda_{-}K_{-}}]\\\
&=4e^{\lambda_{0}/2}\int_{0}^{\infty}\mathrm{d}rrI_{0}(2\sqrt{\lambda_{+}\lambda_{-}}r)I_{2k-1}(2e^{\lambda_{0}/2}r)K_{2k-1}(2r).\end{split}$
(B.8)

This integral converges when each of the four $\mathrm{Re}(2\pm
2\sqrt{\lambda_{+}\lambda_{-}}\pm 2e^{\lambda_{0}/2})$ is positive bailey1936
. For the convergent case, using Eq. (3.1) in Ref. bailey1936  and Eq. (44a)
in Ref. Schlosser2013 , we obtain the result of the integral

$\mathrm{Tr}_{2k}[e^{\lambda_{+}K_{+}}e^{\lambda_{3}K_{3}}e^{\lambda_{-}K_{-}}]=\frac{\varepsilon^{2k}}{1-\varepsilon^{2}},$
(B.9)

where $\varepsilon$ satisfies

$\varepsilon^{2}+e^{-\lambda_{0}/2}(\lambda_{+}\lambda_{-}-e^{\lambda_{0}}-1)\varepsilon+1=0.$
(B.10)

Connecting Eq. (B.10) with the fundamental representation matrix
$e^{\lambda_{+}\rho(K_{+})}e^{\lambda_{3}\rho(K_{3})}e^{\lambda_{-}\rho(K_{-})}=e^{\rho(x)}$,
we find that Eq. (B.10) is just the characteristic polynomial of
$e^{\rho(x)}$. Hence we have

$\mathrm{Tr}_{2k}[e^{x}]=\frac{\varepsilon^{2k}}{1-\varepsilon^{2}},$ (B.11)

where $\varepsilon$ is one of the eigenvalues of $e^{\rho(x)}$.

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††thanks: These authors co-supervised the project††thanks: These authors co-
supervised the project

# Harnessing two-photon dissipation for enhanced quantum measurement and
control

A. Marquet  Alice & Bob, 53 Bd du Général Martial Valin, 75015 Paris, France
Ecole Normale Supérieure de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon,
France  S. Dupouy  Alice & Bob, 53 Bd du Général Martial Valin, 75015 Paris,
France  Ecole Normale Supérieure de Lyon, CNRS, Laboratoire de Physique,
F-69342 Lyon, France  U. Réglade  Alice & Bob, 53 Bd du Général Martial Valin,
75015 Paris, France  Laboratoire de Physique de l’Ecole normale supérieure,
ENS-PSL, CNRS, Sorbonne Université, Université Paris Cité, Centre Automatique
et Systèmes, Mines Paris, Université PSL, Inria, Paris, France  A. Essig
Alice & Bob, 53 Bd du Général Martial Valin, 75015 Paris, France  J. Cohen
Alice & Bob, 53 Bd du Général Martial Valin, 75015 Paris, France  E.
Abertinale  Alice & Bob, 53 Bd du Général Martial Valin, 75015 Paris, France
A. Bienfait  Ecole Normale Supérieure de Lyon, CNRS, Laboratoire de Physique,
F-69342 Lyon, France  T. Peronnin  Alice & Bob, 53 Bd du Général Martial
Valin, 75015 Paris, France  S. Jezouin  Alice & Bob, 53 Bd du Général Martial
Valin, 75015 Paris, France  R. Lescanne  Alice & Bob, 53 Bd du Général Martial
Valin, 75015 Paris, France  B. Huard  Ecole Normale Supérieure de Lyon, CNRS,
Laboratoire de Physique, F-69342 Lyon, France

###### Abstract

Dissipation engineering offers a powerful tool for quantum technologies.
Recently, new superconducting devices demonstrated an engineered two-photon
dissipation rate exceeding all other relevant timescales. In particular, they
have proven most useful to prevent transitions between the logical states
$|\pm\alpha\rangle$ of a cat qubit. Here, we present three key applications of
strong two-photon dissipation for quantum measurement and control, beyond cat
qubit stabilization. Firstly, we demonstrate its efficacy in overcoming
limitations encountered in Wigner tomography at high photon numbers. Secondly,
we showcase its potential for realizing universal gates on cat qubits,
exploiting the coherent mapping between cat qubit states and superpositions of
0 and 1 photons. Finally, we harness the transient dynamics of a cat state
under two-photon dissipation to prepare squeezed cat states with a squeezing
factor exceeding 3.8 dB.

The perception of dissipation in quantum systems has undergone a drastic
change. Once seen as a relentless foe, it is now being recognized as a
potential ally, capable of being ‘tamed’ to become a valuable tool for quantum
measurements, control, and quantum error correction [1, 2, 3, 4, 5]. Within
the framework of bosonic codes, dissipation offers a strategy to remove
entropy from a _memory_ mode (e.g., superconducting resonator, trapped ions,
spin ensemble) and thereby stabilize a logical qubit [6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21]. This is achieved by carefully tailoring
the coupling between the memory and its environment, inducing an effective
dissipation that confines information within the desired qubit subspace and
prevents leakage into the broader Hilbert space. This technique has proven
particularly successful in stabilizing _cat qubits_ , [8, 9, 11, 12, 16, 18,
19], whose computational basis comprises two coherent states
$\ket{\pm\alpha}$. A key resource in this process is two-photon dissipation,
where the memory loses pairs of photons to a low-temperature environment at a
rate $\kappa_{2}$. A two-photon drive acting on the memory completes the
stabilization process of the cat qubit.

This work explores the broader potential of two-photon dissipation beyond cat
qubit stabilization. We leverage the _autoparametric cat_ device [19], which
boasts an engineered two-photon dissipation rate $\kappa_{2}$ exceeding the
residual single-photon loss rate $\kappa_{1}=(11~{}\mu\mathrm{s})^{-1}$ by two
orders of magnitude. Following proposals in [22, 23], we harness two-photon
dissipation to extend Wigner tomography protocols to large photon numbers, and
achieve universal control of cat qubits within a timescale $\kappa_{2}^{-1}$.
Furthermore, we demonstrate the preparation of squeezed cat states with a
squeezing parameter reaching 3.9 dB.

Existing devices [9, 11, 12, 16, 18, 19] that demonstrate two-photon
dissipation use a dedicated mode, denoted as the buffer, to mediate the
interaction between the memory and its environment. In our experiment, this
buffer dissipates photons at a rate $\kappa_{b}/2\pi\approx 40~{}\mathrm{MHz}$
(Fig. 1a). The memory and buffer modes are coupled through a nonlinear element
implementing a two-to-one photon exchange Hamiltonian $\hat{H}=\hbar
g_{2}\hat{m}^{2}\hat{b}^{\dagger}+h.c.$ at a rate $g_{2}$. Here, $\hat{m}$ and
$\hat{b}$ represent the annihilation operators of the memory and buffer modes,
respectively. This interaction is realized in our experiment using a 3-wave
mixing interaction under the condition $2\omega_{m}=\omega_{b}$,
characteristic of autoparametric systems [19, 24]

Operating in the regime $\kappa_{b}\geq 8g_{2}|\alpha|$ ensures that photons
in the buffer mode are evacuated into the environment at a faster rate than
they are converted into pairs of photons in the memory. This allows for the
adiabatic elimination of the buffer mode, leading to an effective dissipation
operator $\sqrt{\kappa_{2}}\hat{m}^{2}$ where
$\kappa_{2}=4g_{2}^{2}/\kappa_{b}$. In the autoparametric cat device [19], an
external flux bias $\phi_{\mathrm{ext}}$ can be used to activate or deactivate
two-photon dissipation. We have identified a specific flux point,
$\phi_{\mathrm{on}}$, at which $\omega_{b}=2\omega_{m}$, resulting in
$\kappa_{2}/2\pi=2.16~{}\mathrm{MHz}$. Another flux point,
$\phi_{\mathrm{off}}$ with negligible $\kappa_{2}/2\pi$ serves to disable two-
photon dissipation (details in [24]). It is chosen at a point where the self-
Kerr effect is weak.

Figure 1: (a) Schematic of a typical device for two-photon dissipation. The
memory mode (blue) interacts with the buffer mode (green) via a 2-to-1 photon
exchange with coupling strength $g_{2}$. The buffer can be driven with an
amplitude $\epsilon_{d}$. Additionally, a dispersively coupled tomography
transmon (orange) with frequency $\omega_{q}$ can be read out through a
dedicated readout resonator at $\omega_{r}$. (b) Photon number distribution
$\mathbb{P}_{n}$ of an arbitrary quantum state before and after undergoing
two-photon dissipation.

Let us begin by demonstrating how two-photon dissipation can address
limitations in Wigner tomography, as proposed in [23]. The Wigner function of
a density matrix $\hat{\rho}$ is proportional to the expectation value of the
parity operator $\hat{\mathcal{P}}=e^{i\pi\hat{m}^{\dagger}\hat{m}}$after the
memory has undergone a displacement operation
$\hat{\mathcal{D}}\left(-\beta\right)=e^{-(\beta\hat{m}^{\dagger}-\beta^{*}\hat{m})}$:

$W_{\hat{\rho}}\left(\beta\right)=\frac{2}{\pi}\mathrm{Tr}\left(\hat{\mathcal{D}}\left(\beta\right)^{\dagger}\hat{\rho}\,\hat{\mathcal{D}}\left(\beta\right)\mathcal{\hat{P}}\right).$
(1)

The displacement is achieved by driving the memory on resonance with a pulse
whose complex amplitude scales with $-\beta$. In our system, the pulse
duration of $20~{}\mathrm{ns}$ is chosen to be significantly shorter than the
inverse self-Kerr rate of the memory
$\chi_{\mathrm{m,m}}\approx(720~{}\mathrm{ns})^{-1}$. Wigner tomography then
boils down to measuring the average value of $\hat{\mathcal{P}}$.

Various parity measurement methods have been demonstrated in superconducting
circuits, usually relying on coupling the memory at resonance [25, 26] or
dispersively[27, 28, 29, 30, 31] to an auxiliary qubit. However, all these
methods deteriorate as the average photon number $\bar{n}$ after displacement
increases. Let us focus on the Ramsey-like sequence proposed in [27]. The
qubit is first prepared in an equal superposition of ground and excited state
$(|g\rangle+|e\rangle)/\sqrt{2}$ by applying an unconditional $\pi/2$ pulse,
much shorter than the inverse dispersive coupling, which is
$(\chi_{\mathrm{q,m}})^{-1}\approx 0.9~{}\mu\mathrm{s}$ in our experiment.
After a time $T_{\mathrm{parity}}=\pi/\chi_{\mathrm{q,m}}$, the superposition
accumulates a phase $n\pi$ with $n$ the number of photons of the memory. Even
and odd numbers of photons are then respectively mapped onto the states
$\left(\ket{g}+\ket{e}\right)/\sqrt{2}$ and
$\left(\ket{g}-\ket{e}\right)/\sqrt{2}$ of the qubit. A final unconditional
$\pi/2$ pulse is applied, mapping the memory parity operator
$\hat{\mathcal{P}}$ onto the Pauli operator $\hat{\sigma}_{z}=|e\rangle\langle
e|-|g\rangle\langle g|$ of the qubit, which can then be readout.

Firstly, this protocol underestimates the amplitude of the Wigner function
since the parity outcome is flipped when a photon is lost during the idle time
$T_{\mathrm{parity}}$, which occurs with probability
$1-e^{-\kappa_{1}T_{\mathrm{parity}}\bar{n}}$. Additionally, for large
$\bar{n}$, the superconducting qubit state can leak out of its subspace
$\mathrm{span}(|g\rangle,|e\rangle)$, especially after the qubit gets excited
with the $\pi/2$-pulse [32, 33, 34, 35, 36, 37]. Finally, the upper bound on
the $\pi/2$ pulse duration scales as $(\chi_{\mathrm{q,m}}\bar{n})^{-1}$,
requiring impractically high driving power at large photon numbers. These
limitations at high photon numbers can be overcome using the protocol proposed
in Ref. [23]. The key lies in leveraging two-photon dissipation’s ability to
preserve parity while transforming a state with an arbitrary photon number
distribution into a state with at most $1$ photon (Fig. 1b).

To illustrate typical errors of Wigner tomography, we prepare a cat state
$\ket{C_{\alpha}^{+}}\propto|\alpha\rangle+|-\alpha\rangle$ and realize its
tomography. The preparation is performed by tuning the flux to
$\phi_{\mathrm{on}}$, which turns on the two-photon dissipation, and driving
the buffer mode on resonance for 300 ns with a drive amplitude
$\epsilon_{d}=-g_{2}^{*}\alpha^{2}$. As demonstrated in Refs. [9, 19], this
implements an effective dissipation operator
$\hat{L}_{2}=\sqrt{\kappa_{2}}\left(\hat{m}^{2}-\alpha^{2}\right)$. Fig. 2b
shows the obtained Wigner tomography for $\alpha=3.3$. The typical fringes of
cat states can be seen close to $\beta=0$ but their contrast is much smaller
than expected for $\ket{C_{\alpha}^{+}}$. Part of this loss of contrast comes
from single photon losses during preparation. However, in the cut
$W_{\hat{\rho}}(\beta)$ along $\beta\in i\mathbb{R}$ (Fig. 2.d), one can see
that the measured Wigner function (orange dots) does not faithfully reproduce
numerical simulations only accounting for the preparation infidelity (grey
line). Here, most of the tomography errors can be attributed to single photon
loss during parity measurement. Indeed, the probability of losing a photon
during $T_{\mathrm{parity}}\approx 2.7~{}\mu\mathrm{s}$ reads
$1-e^{-0.26\bar{n}}$ when the memory has $\bar{n}$ photons on average,
reaching $50~{}\%$ for 2.7 photons. The error is therefore dramatic for the
cat state in Fig. 2.b which has more than 10 photons on average. This effect
is reproduced by simulations (orange line in Fig. 2.d). In contrast, the
measurement of $W_{\hat{\rho}}(\pm\alpha)$ is only marginally affected by
single photon loss. Indeed, after a displacement
$\hat{\mathcal{D}}\left(\pm\alpha\right)$, the memory ends up in a
superposition of $|0\rangle$ and $|\pm 2\alpha\rangle$. The parity of
$|0\rangle$ does not depart from 1 during measurement and the parity of $|\pm
2\alpha\rangle$ is negligible, leading to a measured
${\hat{\rho}}(\pm\alpha)\approx 1/\pi$ as expected.

We solve this issue by quickly turning the two-photon dissipation back on by
tuning the flux to $\phi_{\mathrm{on}}$ after the displacement (Fig. 2a).
Idling at $\phi_{\mathrm{on}}$ for $300~{}\mathrm{ns}$ (a few
$\kappa_{2}^{-1}$) then ensures that the memory decays to a state with at most
1 photon while preserving its parity (Fig. 1b). Tuning the flux back to
$\phi_{\mathrm{off}}$ enables parity measurement. In our example of a cat
state above, a better measurement of the Wigner fringes can then be achieved
(Fig. 2c,d). Simulations corresponding to this parity measurement protocol
(blue line) predict a drastic improvement of the fringe reconstruction,
reproducing what is expected when only accounting for preparation infidelity
(grey line). This is confirmed by the measured Wigner function (blue dots)
which shows an increase in contrast compared to the usual parity measurement
protocol. It should be noted that avoiding single-photon loss while removing
pairs of photons requires a large $\kappa_{2}/\kappa_{1}$ ratio. This
condition can be met with other devices than the autoparametric cat, such as a
simple parametrically pumped transmon qubit [11], making this method
accessible to a wide range of experiments using superconducting qubits.

Figure 2: (a) Pulse sequence of the dissipation enhanced Wigner tomography of
$\ket{C_{\alpha}^{+}}$ including an extra 300 ns idle time at
$\phi_{\mathrm{ON}}$ after memory displacement. (b) Left: Wigner tomography of
$\ket{C_{\alpha}^{+}}$, $\alpha=3.3$, without the extra idle time. (c) Wigner
tomography of $\ket{C_{\alpha}^{+}}$ using the dissipation enhancement. (d)
Cuts of the Wigner function along the imaginary axis $\mathrm{Re(\beta)}=0$ in
various scenarios. Solid gray line: simulated memory state at the end of the
300 ns buffer pulse $\epsilon_{d}$. Orange points: Wigner tomography in (b).
Solid orange line: simulation of the measurement protocol without the extra
idle time. Blue points: Wigner tomography in (c). Solid blue line: simulation
of the dissipation enhanced measurement protocol.

Two-photon dissipation can also be used to control cat qubits. For instance,
driving the memory on resonance while the dissipation $\hat{L}_{2}$ is turned
on performs a gate $\hat{Z}(\theta)$ [11, 19, 18] . In contrast,
$\hat{X}(\theta)$ and $\hat{Y}(\theta)$ gates are seemingly incompatible with
the engineered dissipation $\hat{L}_{2}$, which prevents population transfer
between $|\alpha\rangle$ and $|-\alpha\rangle$. However, one can use the
coherent mapping between the manifolds
$\mathrm{span}\left(\ket{0},\ket{1}\right)$ and
$\mathrm{span}\left(\ket{\alpha},\ket{-\alpha}\right)$ to perform these gates
while the memory transits through the $\\{|0\rangle,|1\rangle\\}$ manifold
(Fig. 3a) [22]. These _deflation_ and _inflation_ processes are performed by
adiabatically modulating the buffer drive between $|\epsilon_{d}|=0$ and
$|\epsilon_{d}|=g_{2}\left|\alpha\right|^{2}$, so that the memory follows a
steady state of the dissipation. From this point of view, the preparation of
$\ket{C_{\alpha}^{+}}$ in Fig. 2 corresponds to the inflation of the vacuum
state, and the Wigner tomography is enhanced by deflation before parity
measurement.

Figure 3: (a) Inflation and deflation processes coherently map the manifolds
$\mathrm{span}\left(\ket{\alpha},\ket{-\alpha}\right)$ and
$\mathrm{span}\left(\ket{0},\ket{1}\right)$ by turning on or off the buffer
drive $\epsilon_{d}$ [22, 18]. (b) Pulse sequence used to prepare an initial
state $\ket{\alpha}$, apply the holonomic $\hat{X}\left(\theta\right)$ gate,
and measure the memory Wigner function. (c) Schematic evolution in the memory
phase space of a coherent state while applying the holonomic
$\hat{X}\left(\theta\right)$ gate. (d) Measured evolution of the memory Wigner
function, starting from the coherent state $\ket{-2.27}$, during the
$\hat{X}\left(\pi/2\right)$ gate. The last snapshot occurs right before the
final virtual rotation $\hat{R}(-\pi/2)$. An animation showing every measured
time can be found as supplementary material [24].

We implement a holonomic $\hat{X}\left(\theta\right)$ gate by modulating the
buffer drive amplitude $\epsilon_{d}(\alpha)$ while remaining at
$\phi_{\mathrm{on}}$ (Fig. 3b). We first turn off the buffer drive to deflate
the memory state into the $\\{|0\rangle,|1\rangle\\}$ manifold. The buffer
drive is then turned back on with an additional phase $2\theta$,
$\epsilon_{d}\rightarrow\epsilon_{d}\left(\alpha\right)e^{2i\theta}$ [24],
mapping the memory into the $\\{|\alpha e^{i\theta}\rangle,|-\alpha
e^{i\theta}\rangle\\}$ manifold. This additional phase implements a virtual
$\hat{X}(\theta)$ gate in the $\\{|0\rangle,|1\rangle\\}$ manifold (see Fig.
3a). A counter-rotation $\hat{R}\left(-\theta\right)$ is finally applied to
bring the memory back into the cat code while preserving the phase of the
superposition. This final rotation could be performed by adiabatically
sweeping the phase of the buffer drive back to $0$. Equivalently, we choose to
apply an instantaneous virtual operation by changing the phase of the local
oscillator of the memory drive by $-\theta$. The evolution of the memory in
phase space starting from $\ket{-\alpha}$ is represented in Fig. 3c. We probe
the evolution of the memory during the $\hat{X}\left(\pi/2\right)$ gate
starting from the coherent state $\ket{-\alpha}$ with $\alpha=2.27$. This
particular gate has been used for tomography without a transmon in [18].
Wigner tomographies of the memory are shown in Fig. 3d at $6$ different times.
The memory state can first be seen evolving from the coherent state towards
$\left(\ket{0}-\ket{1}\right)/\sqrt{2}$ (third snapshot) before being mapped
into $\left(\ket{\alpha e^{i\pi/2}}-i\ket{-\alpha e^{i\pi/2}}\right)/\sqrt{2}$
(last snapshot).

We show in Figs. 5 of [24] the state of the memory after applying the
$\hat{X}\left(\theta\right)$ gate on the same coherent state $\ket{-\alpha}$
for an arbitrary angle $\theta$. Note that the shape of the buffer drive
during the gate was optimized with numerical simulations shown in Fig. S3 of
[24]. The gate error being substantial, we settle for characterizing the gate
properties by its effect on the coherent state $|-\alpha\rangle$ only. We
compute the trace distance
$T\left(\hat{\rho},|\psi(\theta)\rangle\langle\psi(\theta)|\right)$ [38, 24]
between the density matrix $\hat{\rho}$ reconstructed from the measured Wigner
function and the state
$|\psi(\theta)\rangle=\hat{X}\left(\theta\right)|-\alpha\rangle$. The largest
distance occurs at $\theta=\pm\pi/2$, where it reaches $0.23$ (see Fig. S4 in
[24]). We attribute part of this apparent error to the remaining measurement
infidelity (deviations between blue dots and grey line in Fig. 2d). For
$\theta$ close to multiples of $\pi$, the Wigner tomography of a coherent
state does not lead to such measurement errors and simulations of the memory
evolution during the gate reproduce the observed trace distance. Additionally,
simulations with and without the dephasing rate
$\kappa_{\varphi}^{\mathrm{m}}/2\pi=0.08~{}$MHz reveal it has a major
contribution to the gate error owing to the induced decoherence when the
memory is in the $\\{|0\rangle,|1\rangle\\}$ manifold. The remaining gate
errors can be attributed to a non-adiabatic evolution of the memory during the
drive, or single photon loss.

Figure 4: Preparation of a squeezed cat state. (a) Wigner functions of the
memory at early times (0, 16 ns, or 24 ns) during the deflation of
$\ket{C_{\alpha}^{+}}$, where $\alpha=2.9$. The measurements (left) are
compared with numerical simulations made without free parameters (right). (b)
Schematic representation of the buffer and memory states, in the non-adiabatic
regime, at the beginning of deflation. An effective displacement of mode
$\hat{b}$ creates an effective squeezing Hamiltonian acting on the memory
during its decay to the $\left\\{\ket{0},\ket{1}\right\\}$ manifold.

Interestingly, probing the deflation dynamics at short times (a few
$\kappa_{b}^{-1}$) reveals a potential resource for state preparation. The
deflation spontaneously prepares squeezed cat states. After preparing the
state $\ket{C_{\alpha}^{+}}$ in the memory, the drive $\epsilon_{d}(\alpha)$
is turned off while the flux remains at $\phi_{\mathrm{on}}$. The measured
Wigner functions of the memory state are shown in Fig. 4a for three different
times between 0 and 24 ns. As the memory releases photon pairs to the buffer,
it acts as an effective drive on the buffer with an amplitude
$\left<g_{2}\hat{m}^{2}\right>=g_{2}\alpha^{2}$. This drive, which should not
have an impact on the buffer when $\kappa_{b}\gg 8g_{2}\alpha$, here displaces
the buffer into a state whose mean amplitude $\langle\hat{b}\rangle$ is
proportional to $\alpha^{2}$. In turn an effective squeezing Hamiltonian

$\hat{H}_{\mathrm{m,eff}}/\hbar=g_{2}\langle\hat{b}\rangle\hat{m}^{\dagger
2}+h.c$ (2)

acts on the memory, which compresses the superposition of coherent states
(Fig. 4b). As the memory deflates, it reaches a maximal squeezing at a time
that decreases with the initial photon number. This observation is compatible
with the expected evolution of the squeezing parameter $r$ at small time in
$(g_{2}|\alpha|t)^{2}$ [24]. From there, the buffer decays to vacuum, hence
suppressing the effective squeezing Hamiltonian. To extract the squeezing
ratio $e^{2r}$, we fit the two measured peaks in the Wigner function with
Gaussian functions of identical variance. We observe squeezing ratios as high
as $e^{2r}\approx 3.9~{}\mathrm{dB}$ [39], obtained after a time $t=16$ ns,
starting from $\ket{C_{\alpha}^{+}}$ with $\alpha=2.9$ (Fig. 4a). Note that
during the evolution of the memory, the product of the largest and smallest
variances remains close to that of the vacuum state. The predicted evolution
using the bipartite buffer-memory system, without the adiabatic elimination,
reproduces qualitatively the measured Wigner functions (right panels). A
rounding of the Wigner function appears in the theory owing to two-photon
dissipation rate, which ultimately overcomes the squeezing rate as the buffer
is depleted. While bigger compression is expected starting with larger values
of $\left|\alpha\right|$, an uncontrolled deformation of the initial cat state
in the experiment limits the squeezing factor. We attribute this deformation
and the discrepancy between simulations and experiments to the self-Kerr, the
dephasing of the buffer or a detuning between $2\omega_{m}$ and $\omega_{b}$.

This work illustrates how two-photon dissipation is a remarkable tool for
quantum measurement and control. Let us put in perspective the three
applications we demonstrated. The enhancement of Wigner tomography is
straightforward for any device with $\kappa_{2}\gg\kappa_{1}$ and should be
useful for a broad range of experiments. Furthermore, the demonstration of an
arbitrary $\hat{X}(\theta)$ gate together with the known $\hat{Z}(\theta)$
gate [11, 19, 18] provides universal control of cat qubits. We note that since
$\hat{X}(\pi)$ gate can be performed virtually by changing the phase of the
local oscillator of the memory, such a control is not required for quantum
error correcting codes [40]. However, our experiment in Fig. 3 provides a
useful tool to explore and optimize the dynamics of the holonomic gate, which
is an essential component of cat qubit readout without transmon qubits [18].
Finally, two-photon dissipation appears to be an interesting approach for the
preparation of squeezed cat states, whose coherence properties are promising
for quantum information processing [41, 42, 43, 17, 44].

###### Acknowledgements.

This research was supported by the QuantERA grant QuCos ANR-19-QUAN-0006, the
Plan France 2030 through the project ANR-22-PETQ-0003. We acknowledge IARPA
and Lincoln Labs for providing a Josephson Traveling-Wave Parametric
Amplifier. We thank Zaki Leghtas and Mazyar Mirrahimi for inspiring
discussions and feedback.

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