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# Dynamical Transition of Operator Size Growth in Open Quantum Systems

Pengfei Zhang  Department of Physics, Fudan University, Shanghai, 200438,
China  Walter Burke Institute for Theoretical Physics & Institute for Quantum
Information and Matter, California Institute of Technology, Pasadena, CA
91125, USA  Zhenhua Yu  Guangdong Provincial Key Laboratory of Quantum
Metrology and Sensing, School of Physics and Astronomy, Sun Yat-Sen University
(Zhuhai Campus), Zhuhai 519082, China  State Key Laboratory of Optoelectronic
Materials and Technologies, Sun Yat-Sen University (Guangzhou Campus),
Guangzhou 510275, China

###### Abstract

We study the operator size growth in open quantum systems with all-to-all
interactions, in which the operator size is defined by counting the number of
non-trivial system operators. We provide a general argument for the existence
of a transition of the operator size dynamics when the system-bath coupling
$\gamma$ is tuned to its critical value $\gamma_{c}$. We further demonstrate
the transition through the analytical calculation of the operator size
distribution in a solvable Brownian SYK model. Our results show that: (i) For
$\gamma>\gamma_{c}$, the system is in a dissipative phase where the system
operator size decays with a rate $\sim(\gamma-\gamma_{c})$, which indicates
the initial information of the system all dives into the bath eventually. (ii)
For $\gamma<\gamma_{c}$, the system sustains a scrambling phase, where the
average operator size grows exponentially up to the scrambling time
$t_{s}\sim(\gamma_{c}-\gamma)^{-1}\log N$ and saturates to a $O(N)$ value in
the long-time limit. (iii) At the critical point $\gamma=\gamma_{c}$, which
separates the two phases, the operator size distribution at finite size shows
a power-law decay over time.

_Introduction.–_ Information scrambling emerges as a cornerstone in
understanding thermalization in most closed quantum systems. In such systems,
the initial information, though fully preserved under the unitary evolution,
is scrambled from local physical objects into those highly non-local Hayden
and Preskill (2007); Sekino and Susskind (2008); Shenker and Stanford (2015);
Roberts  _et al._ (2015), which gives rise to quantum thermalization of simple
operators in a sufficiently long time Srednicki (1994); Deutsch (1991). The
evolution of the operator size provides a quantitative description of the
information scrambling process, and has been studied in various contexts via
toy models or numerical simulations Roberts  _et al._ (2015); Nahum  _et al._
(2018); von Keyserlingk  _et al._ (2018); Khemani  _et al._ (2018); Hunter-
Jones (2018); Chen and Lucas (2021); Lucas (2019); Lucas and Osborne (2020);
Chen  _et al._ (2020); Lucas and Osborne (2020); Yin and Lucas (2021); Zhou
and Swingle (2021); Dias  _et al._ (2021); Wu  _et al._ (2021); Yao ; Zhang
and Gu . It has become clear that in closed chaotic systems with all-to-all
interactions, the operator size starts with an early-time exponential growth,
and saturates as a maximally scrambled form in which all operators appear with
equal probability. The operator size growth is also proposed as a criterion
for selecting quantum neural networks architectures Wu  _et al._ (2021).

Experimental study of the scrambling dynamics has been carried out in various
systems Islam  _et al._ (2015); Li  _et al._ (2017); Gärttner  _et al._
(2017); Brydges  _et al._ (2019); Sánchez  _et al._ (2019); Landsman  _et al._
(2019); Joshi  _et al._ (2020); Blok  _et al._ (2021); Domínguez  _et al._
(2021); Domínguez and Álvarez (2021); Mi  _et al._ (2021); Cotler  _et al._
(2022); Sánchez  _et al._ (2022). However, couplings between the systems and
external baths are inevitable in experiment. Theoretical exploration of the
effects of external baths on the scrambling dynamics kicks off only recently
Bhattacharya  _et al._ (2022); Liu  _et al._ (2022); Schuster and Yao (2022).
Refs. Liu  _et al._ (2022); Bhattacharya  _et al._ (2022) studied the
information scrambling in open quantum systems from the perspective of the
Krylov complexity Parker  _et al._ (2019), and established a relation between
the complexity and the operator size distribution at least in certain cases.
The authors of Ref. Schuster and Yao (2022) conducted a comprehensive
numerical study on the operator size growth in open quantum systems,
nevertheless, focusing on small system-bath couplings. Present knowledge in
the field is rather limited.

Figure 1: Schematics of the operator size growth in open quantum systems. Here
$J$ represents the interaction between system Majorana fermion modes and
$\gamma$ represents the hopping between the system and the bath. We find
different dynamical phases exist when tunning $\gamma$. For small $\gamma$,
the system is in a scrambling phase where the average operator size
$\overline{n}$ grows to $O(N)$ values at time $\sim\log N$. For large
$\gamma$, the average operator size decays to zero with $O(1)$ rate. Insets
are diagrammatic illustrations of the typical operator size growth history in
different phases.

In this work, based on semi-classical epidemiological models, we propose the
existence of a dynamical transition in open quantum systems with all-to-all
interactions at finite system-bath coupling $\gamma$, above which the
scrambling process halts; the transition separates a scrambling phase at small
$\gamma$ and a dissipative phase at large $\gamma$. We further demonstrate the
transition by solving analytically a Brownian SYK model accompanied with a
bath Kitaev (2015); Maldacena and Stanford (2016); Kitaev and Suh (2018); Saad
_et al._ (2018); Sünderhauf  _et al._ (2019). We derive the analytic
expressions of the operator size distribution both in the early-time and the
late-time regimes. The analytic results indicate a two-stage scenario for the
operator size growth: in the early stage, the bath operators act as an
attractor and continuously absorb the weight from the system operators. If the
intrinsic scrambling rate of the open system is not fast enough, all the
weight eventually falls into the bath. Only if the system intrinsic scrambling
rate is sufficiently large, a finite weight escapes the absorption and
survives in the system, and the growth enters into the second stage where,
though dragged by the bath, the scrambling process persists and the growth
converges to a reduced operator size of the same order as in the closed
system. We argue that the scenario shall be generic to a wide class of
Brownian SYK models in the presence of baths. We also discuss the implications
of our results on the teleportation transition and point out possible
relations between our “bath-induced scrambling transition” to the measurement-
induced entanglement phase transitions Li  _et al._ (2018); Skinner  _et al._
(2019); Chan  _et al._ (2019).

_Operator Size in Open Systems.–_ Our goal is to study the operator size
growth in open quantum systems with all-to-all interactions. We consider a
class of chaotic quantum many-body systems $S$ coupled to an external bath
$B$; the system $S$ and the bath $B$ consist of $N$ Majorana fermions
$\chi_{j}$ ($j=1,2,...,N$) and $M$ Majorana fermions $\psi_{a}$
($a=1,2,...,M$) respectively. We choose the normalization to be
$\chi_{j}^{2}=\psi_{a}^{2}=1$. The total Hamiltonian is given by
$H_{\text{tot.}}=H_{S}[\chi]+H_{B}[\psi]+H_{SB}[\chi,\psi]$. We take $M\gg N$
throughout our study such that once information spreads from the system $S$
into the bath $B$, it will not flow back to the system $S$ Chen  _et al._
(2017); Zhang (2019); Almheiri  _et al._ (2019).

We define the operator size distribution in the open quantum systems in the
following way. The time evolution of a certain operator $O$ is determined by
the Heisenberg equation as
$O(t)=e^{iH_{\text{tot.}}t}Oe^{-iH_{\text{tot.}}t}$. We expand the operator
$O(t)$ in terms of the complete orthonormal operator basis
$\\{\chi_{1}^{p_{1}}...\chi_{N}^{p_{N}}\psi_{1}^{q_{1}}...\psi_{M}^{q_{M}}\\}$
with $p_{j}$, $q_{a}\in\\{0,1\\}$:

$O(t)=\sum_{\\{p_{j}\\}}\sum_{\\{q_{a}\\}}c_{\\{p_{j}\\},\\{q_{a}\\}}(t)\chi_{1}^{p_{1}}...\chi_{N}^{p_{N}}\psi_{1}^{q_{1}}...\psi_{M}^{q_{M}}.$
(1)

Due to the system-bath coupling, $O(t)$ generally involves degrees of freedom
both of the system $S$ and the bath $B$. In previous studies of closed systems
consisting of Majorana fermions $\chi_{j}$, the operator size of a string
operator $\chi_{i_{1}}\chi_{i_{2}}...\chi_{i_{n}}$ is defined to be $n$, equal
to the number of the Majorana operators Roberts  _et al._ (2018); Qi and
Streicher (2019); Zhang and Gu . To generalize to our open quantum systems, we
define the size $n$ of the basis operator
$\chi_{1}^{p_{1}}...\chi_{N}^{p_{N}}\psi_{1}^{q_{1}}...\psi_{M}^{q_{M}}$ by
counting the number of the system Majorana operators $\chi_{j}$, regardless of
the bath operators $\psi_{a}$. The operator size distribution $P(n,t)$ for
$O(t)$ sums the weight of size $n$ basis operators in the expansion of $O(t)$:

$P(n,t)=\sum_{\\{p_{j}\\}}\delta_{n,\sum
p_{j}}\sum_{\\{q_{a}\\}}\left|c_{\\{p_{j}\\},\\{q_{a}\\}}(t)\right|^{2}.$ (2)

We normalize operators $O(t)$ in the way that
$\text{Tr}\,O^{\dagger}O=2^{(M+N)/2}$; resultantly $\sum_{n}P(n,t)=1$. The
moments of the operator size are given by
$\overline{n^{k}}=\sum_{n}n^{k}P(n,t)$.

Epidemiological models have been applied to interpret the results of operator
size growth in closed systems with all-to-all interactions Roberts  _et al._
(2018); Qi and Streicher (2019), where the number of elementary Majorana
operators $\chi_{j}$ is an analog of the number of patients. For a Hamiltonian
$H_{S}$ with $q$-body interactions of strength $J$, the commutator
$i[H_{S},O]$ in the Heisenberg equation of $O(t)$ brings about the infection
of $(q-2)$ previously unexposed people at a rate $\sim J$. In the early-time
regime, the argument based on the epidemiological model leads to the average
operator size $\overline{n}$ satisfying
$d\overline{n}/dt=J(q-2)\overline{n}\equiv\varkappa\overline{n}$, which
reproduces the result obtained by diagrammatic calculations Roberts  _et al._
(2018); Qi and Streicher (2019). Here we have introduced the intrinsic quantum
Lyapunov exponent $\varkappa$.

We generalize the argument based on the epidemiological model to the cases of
our open systems; we include the system-bath coupling of the form
$H_{SB}=i\sum_{ia}\gamma_{ia}\chi_{i}\psi_{a}$, which is due to direct hopping
between the system $S$ and the bath $B$. Two additional commutators show up in
the Heisenberg equation of $O(t)$: $i[H_{B},O]$ does not change the operator
size while $i[H_{SB},O]$ can replace one of the system Majorana operators
$\chi_{i}$ by a bath Majorana operator $\psi_{a}$ 111There is also a process
where some bath Majorana fermion operator is replaced by system Majorana
fermion operators. However, the corresponding contribution vanishes when
taking $M\rightarrow\infty$ for proper scaling of $M$ for $\gamma_{ia}$., and
results in decreasing the operator size by one at rate $\gamma$.
Conrrespondingly, the epidemiological model shall be modified to give

$\frac{d\overline{n}}{dt}=\varkappa\overline{n}-\gamma\overline{n}$ (3)

in the early-time regime. The solution $\overline{n}\sim
e^{(\varkappa-\gamma)t}$ indicates two distinct dynamical phases separated by
the critical point at $\gamma_{c}=\varkappa$. For $\gamma<\gamma_{c}$, the
average operator size increases exponentially in the early-time regime as its
counterparts in the closed systems; we take the exponential increase as the
defining feature of the scrambling phase, which has been numerically studied
in Schuster and Yao (2022). The exponential growth of $\overline{n}$ is
expected to terminate at an enhanced scrambling time
$t_{s}\sim(\gamma_{c}-\gamma)^{-1}\log N$, where additional information is
required to analyze the full operator dynamics. For $\gamma>\gamma_{c}$, the
average operator size decreases to exponentially small value after time
$(\gamma-\gamma_{c})^{-1}$, which suggests only bath operators remain in the
expansion (1); we call this the dissipative phase. At the critical point
$\gamma_{c}=\varkappa$, $\overline{n}$ is expected not to change with time.
However, as we show below through an explicit model calculation, the operator
size distribution $P(n,t)$ is found to exhibit a non-trivial power-law
decrease in time $t$ for any $n\geq 1$. Schematics for the operator size
growth in different phases are presented in FIG. (1).

_Solvable SYK Model.–_ In the following parts, we demonstrate the existence of
the dynamical transition and reveal the dynamics of the operator size
distribution in different phases via solvable Brownian SYK models Saad  _et
al._ (2018); Sünderhauf  _et al._ (2019), which are closely related to
Brownian circuits in spin models Zhou and Chen (2019). The specific class of
the Hamiltonians we consider have the form

$H(t)=\sum_{i_{1}<...<i_{q}}i^{\frac{q}{2}}J_{\\{i_{p}\\}}(t)\chi_{i_{1}}\chi_{i_{2}}...\chi_{i_{q}}+\sum_{j,a}iV_{ja}(t)\chi_{j}\psi_{a}$
(4)

for even $q$, and

$H(t)=\sum_{i_{1}<...<i_{q},a}i^{\frac{q-1}{2}}J_{\\{i_{p}\\},a}(t)\chi_{i_{1}}\chi_{i_{2}}...\chi_{i_{q}}\psi_{a}+\sum_{j,a}iV_{ja}(t)\chi_{j}\psi_{a}$
(5)

for odd $q$. We focus on the cases of $q\geq 3$, in which the first term in
both (4) and (5) leads to a growth of the operator size $n$. The coupling
constants $J_{\\{i_{p}\\}}(t)$, $J_{\\{i_{p}\\},a}(t)$, and $V_{ja}(t)$ with
different indices are independent Brownian variables and have the property

$\displaystyle\overline{J_{\\{i_{p}\\}}(t_{1})J_{\\{i_{p}\\}}(t_{2})}={(q-1)!J}\delta(t_{1}-t_{2})/{N^{q-1}},$
(6)
$\displaystyle\overline{J_{\\{i_{p}\\},a}(t_{1})J_{\\{i_{p}\\},a}(t_{2})}={(q-1)!J}\delta(t_{1}-t_{2})/{N^{q-1}M},$
$\displaystyle\overline{V_{ja}(t_{1})V_{ja}(t_{2})}={V}\delta(t_{1}-t_{2})/{M}.$

In the following, we take the limit of $M\gg N\gg 1$ in which our Brownian SYK
models can be worked out using the $1/N$ expansion Maldacena and Stanford
(2016).

To study the operator size growth, we generalize a method introduced in Qi and
Streicher (2019): We construct auxiliary Majorana fermions $\tilde{\chi}_{j}$
and $\tilde{\psi}_{a}$, which are paired up with $\chi_{j}$ and $\psi_{a}$ to
form the complex fermions $c_{j}=(\chi_{j}+i\tilde{\chi}_{j})/2$ and
$c_{a}^{B}=(\psi_{a}+i\tilde{\psi}_{a})/2$ respectively. We introduce a
normalized EPR state $|\text{EPR}\rangle$ which satisfies
$c_{j}|\text{EPR}\rangle=c_{a}^{B}|\text{EPR}\rangle=0$; in other words,
$|\text{EPR}\rangle$ is the vacuum state of the complex fermion modes.
Consequently, when we apply $O(t)$ to $|\text{EPR}\rangle$, the complex
fermion number can never decrease. In particular, for a string operator of the
Majorana fermions, each Majorana operator $\chi_{j}/\psi_{a}$, when acting on
$|\text{EPR}\rangle$, creates one excitation
$c^{\dagger}_{j}/c^{B,\dagger}_{a}$. Thus the operator size distribution can
be calculated via $P(n,t)=\langle O(t)^{\dagger}\Pi(n)O(t)\rangle$, where
$\Pi(n)$ is the projection operator into the subspace with $n$ excitations of
$c^{\dagger}_{j}$ while leaving the number of $c^{B,\dagger}_{a}$ excitations
arbitrary. The expectation values $\langle\dots\rangle$ are taken with respect
to $|\text{EPR}\rangle$.

It is more convenient to work with the generating function $z(\mu,t)=\sum
e^{-\mu n}P(n,t)$, which takes the form

$z(\mu,t)=\langle
O(t)^{\dagger}e^{-\mu\sum_{j}c^{\dagger}_{j}c_{j}}O(t)\rangle.$ (7)

In particular, the average operator size is equal to the summation of out-of-
time-order commutators between $O$ and $\chi_{j}$:

$\overline{n}=-\partial_{\mu}z(0,t)=\frac{1}{4}\sum_{j}\left\langle\left|[O(t),\chi_{j}]\right|^{2}\right\rangle,$
(8)

where the commutator shall be replaced by the anti-commutator if $O$ is
fermionic.

In the following discussions, we focus on the case $O=\chi_{1}$, which is the
simplest non-trivial operator. The retarded Green’s functions
$G^{R}(t)\equiv-i\theta(t)\left<\\{\chi_{i}(t),\chi_{i}(0)\\}\right>$ can be
worked out by using the path-integral approach on the Keldysh contour as in
Zhang  _et al._ (2020); we find:

$G^{R}(\omega)^{-1}=\omega/2-\Sigma^{R},\ \ \ \ \
\Sigma^{R}=-i(J+V)\equiv-i\Gamma/4,$ (9)

where the quasi-particle decay rate $\Gamma$ is independent of $q$. After the
Fourier transform, we have $G^{R}(t)=-2i\theta(t)e^{-\Gamma t/2}$. The
generating function (7) turns into a two-point function on the double Keldysh
contour Aleiner  _et al._ (2016) with a perturbation source:

$\Delta
I=\frac{1}{4}(1-e^{-\mu})\sum_{j}\left[\chi_{j}^{u_{2}}(0)-\chi_{j}^{d_{2}}(0)\right]\left[\chi_{j}^{d_{1}}(0)-\chi_{j}^{u_{1}}(0)\right].$
(10)

Here $\chi_{j}^{u_{k}/d_{k}}(t)$ labels the Majorana fields on the double
Keldysh contour Zhang  _et al._ (2020): $u/d$ denotes the forward/backward
evolution and $k=1$ or $2$ labels the two worlds. The factor $(1-e^{-\mu})$
appears because for coherent states $|\psi\rangle$, $\langle\bar{\psi}|e^{-\mu
c^{\dagger}c}|\psi\rangle=e^{-e^{-\mu}\bar{\psi}\psi}$.

Figure 2: The plot for the early-time operator size distribution $P(n,t)$ with
$q=3$ and $J=1/4$ in different phases: (a) In the dissipative phase with
$r=V/J=3/2$, the operator weight for $n\geq 1$ decays to zero quickly. (b) In
the scrambling phase with $r=V/J=1/2$, the operator weight at $n=0$ saturates
to $P(0,t)=r$, leaving finite weight on operators with $n\geq 1$.

_Early-time Operator Dynamics.–_ In the early-time regime with $t\ll t_{s}$,
we can derive the two-point function $z(\mu,t)$ using the saddle-point
equation and have SM

$\displaystyle z(\mu,t)=-\int
dt^{\prime}~{}G^{R}(t-t^{\prime})^{2}\Sigma_{W}(t^{\prime}),$ (11)
$\displaystyle\Sigma_{W}(t)=Jz(\mu,t)^{q-1}+V-\frac{1}{4}(1-e^{-\mu})\delta(t).$

For $t<0$, the system is in equilibrium and $z=1$. At $t=0$, the last term in
$\Sigma_{W}$ leads to an instant change of $z$. By applying
$(\partial_{t}+\Gamma)$ to both sides of the first line of (11), we find

$\frac{dz}{dt}=4J(z^{q-1}-z)+4V(1-z),\ \ \ \ \ z(\mu,0^{+})=e^{-\mu}.$ (12)

which determines the early-time operator size growth in the open quantum
system.

We analyze (12) by firstly examining the early-time evolution of
$\overline{n}$. Taking derivative with respective to $\mu$ and set $\mu=0$, we
have

$\frac{d\overline{n}}{dt}=4J(q-2)\overline{n}-4V\overline{n},$ (13)

which matches (3) with identifications $\varkappa=4J(q-2)$ and $\gamma=4V$.
This agreement suggests that the solvable SYK models we constructed show the
dynamical phase transition between the scrambling phase and the dissipative
phase.

To fully characterize the operator dynamics in the early-time regime, we need
to solve the generating function via (12). In the following discussions, we
would focus on $q=3$ where the evolution of the operator size distribution can
be computed in closed-form.

From now on, we set $J=1/4$ and introduce $r\equiv V/J$ for conciseness.
Assuming $r\neq 1$, by integrating the differential e quation (12) for $q=3$,
the solution reads

$z(\mu,t)=1-\frac{(r-1)(1-e^{-\mu})}{e^{(r-1)t}(r-e^{-\mu})-(1-e^{-\mu})}.$
(14)

Expanding in terms of $e^{-\mu}$, we find the operator size distribution at
early time $t\ll t_{s}$:

$P(n,t)=\begin{cases}1-(1-r)\frac{e^{(1-r)t}}{e^{(1-r)t}-r},&(n=0)\\\
(1-r)^{2}\frac{e^{(1-r)t}[e^{(1-r)t}-1]^{n-1}}{[e^{(1-r)t}-r]^{n+1}}.&(n\geq
1)\end{cases}$ (15)

In the dissipative phase with $r>1$, $z(\mu,t)$ increases from $e^{-\mu}$
towards unity as time increases for any $\mu>0$, which indicates that
$\chi_{1}(t)$ only contains the bath operators in the long term. This trend is
because $z=1$ is the stable fixed point in (12). Correspondingly, (15) shows
the operator weight is localized near $n=0$, with an exponentially decaying
tail $P(n,t)\sim e^{-n/\xi(t)}$ and
$\xi(t)=1/\ln[(r-e^{(1-r)t})/(1-e^{(1-r)t})]$. This feature is illustrated in
FIG. 2 (a) for $r=3/2$.

On the other hand, in the scrambling phase $r<1$, $z(\mu,t)$ changes from
$e^{-\mu}$ to a non-zero value $r$ since the fixed point $z=1$ is now
unstable. It is known that in closed systems, the out-of-time-order
correlation decays to zero after a perturbation Aleiner  _et al._ (2016). In
our open quantum systems, a finite residual correlation instead survives due
to the coupling induced by the bath $B$, which is reflected by the saturation
$P(0,t)\approx r$ for $(1-r)t\gg 1$; since the distribution is normalized, the
sum of the operator weight at $n\geq 1$ is always finite. Furthermore, (15)
shows that the operators with size $1\leq n\lesssim e^{(1-r)t}$ are occupied
by the same order, as illustrated in FIG. 2 (b). Outside the early-time
regime, when $t\gtrsim t_{s}$, a typical operator would have size $\sim N$ in
the scrambling phase, and the saddle-point description (11) is no longer
sufficient Gu  _et al._ (2022). It is necessary to go beyond and employ the
full scramblon effective theory Gu  _et al._ (2022); Zhang and Gu  to compute
the operator size distribution 222An alternative approach can be applied to
the Brownian SYK model Yao ..

Before turning to the late-time regime, we comment on the critical phase with
$r=1$. Instead of (14) and (15), we have

$\displaystyle z(\mu,t)$ $\displaystyle=1-\frac{1-e^{-\mu}}{1+t(1-e^{-\mu})},$
(16) $\displaystyle P(n,t)$
$\displaystyle=\begin{cases}1-\frac{1}{1+t},&(n=0)\\\
\frac{t^{n-1}}{(1+t)^{n+1}}.&(n\geq 1)\end{cases}$

As in the scrambling phase, the occupied operators extend to large $n$: the
operators with $1\leq n\lesssim t$ are occupied by the same order. However,
for $t\gg 1$, $z(\mu,t)\to 1$; all the information dives into the bath $B$.
Consequently, there are no non-trivial dynamics in the late-time regime.

Figure 3: The plot for the late-time operator size distribution
$\mathcal{P}_{\text{reg}}(s,t)$ with $q=3$ and $J=1/4$ in the scrambling phase
($r<1$) for different $\lambda=e^{(1-r)t}/C$ with $C=N(1-r)^{2}/2$. When
$\lambda$ is fixed, $\mathcal{P}_{\text{reg}}(s,t)$ for different $r$ collapse
to a universal function of $s/(1-r)$.

_Scrambling at the Late Time.–_ We now study the late-time operator size
distribution in the scrambling phase using the scramblon effective theory Gu
_et al._ (2022). Since in the late-time regime a typical operator has size
$n\propto N$, we consider a continuum limit with $s=n/N\in[0,1]$ and
$\mathcal{P}(s,t)\equiv NP(n,t)$ as in the closed quantum systems Zhang and Gu
. We have $\int_{0}^{1}ds~{}\mathcal{P}(s,t)=1$, and the generating function

$\mathcal{S}(\nu,t)\equiv\int_{0}^{1}ds~{}e^{-\nu
s}\mathcal{P}(s,t)=z({\nu}/{N},t).$ (17)

Our task is to compute $\mathcal{S}(\nu,t)$ for $t\sim t_{s}$ by making use of
$z(\mu,t)$ obtained for $t\ll t_{s}$.

The crucial observation is that important corrections to the saddle-point
solution are only induced by the interaction between fermions that are
mediated by scramblon modes, which have a soft action Stanford  _et al._
(2022). The scramblon propagator and the scattering vertices between fermions
and scramblons can be determined using (14) for $z(\mu,t)$. The late-time
operator-size distribution $\mathcal{S}(\nu,t)$ is then computed by summing up
scramblon diagrams to the leading order in $1/N$ as in Zhang and Gu . The
final result is given by SM

$\displaystyle\mathcal{P}(s,t)=$ $\displaystyle
r\delta(s)+\mathcal{P}_{\text{reg}}(s,t)$ (18)
$\displaystyle\mathcal{P}_{\text{reg}}(s,t)=$
$\displaystyle\theta\left(1-\frac{2s}{1-r}\right)\frac{2(r-1)^{2}e^{\frac{2s}{\lambda(r+2s-1)}}}{\lambda(r+2s-1)^{2}},$

where $\lambda=e^{(1-r)t}/C$ with $C=N(1-r)^{2}/2$ is the propagator of the
scramblons. The singular part $\delta(s)$ is consistent with the saturation
value of $P(0,\infty)$ derived in (15). This part means that in the late-time
regime, the weight of pure bath operators does not change. This is because the
typical operators appearing in the expansion contain $O(N)$ system fermions,
which have negligible coupling to the $n=0$ operators. The regular part
$\mathcal{P}_{\text{reg}}(s,t)$ describes operators scrambled into the large
size, with a total weight $(1-r)$. Interestingly, it is only a function of
$s/(1-r)$. For closed quantum systems, the operator becomes maximally
scrambled in the long-time limit ($t\rightarrow\infty$), which corresponds to
a delta peak at $s=1/2$ Zhang and Gu . However, in our open systems, we see
that the effect of the system-bath coupling is to exert a “drag force” and
renders the analog of the maximally scrambled operator having a typical size
$s=(1-r)/2$. A plot of $\mathcal{P}_{\text{reg}}(s,t)$ is presented in FIG. 3.
Using (18), we could also compute moments of the operator size
$\overline{n^{k}}$. As an example, we find the average operator size reads

$\overline{n}=N\overline{s}=N\frac{(1-r)^{2}}{2}{\left[1-\frac{1}{\lambda}e^{\frac{1}{\lambda}}\Gamma\left(0,\frac{1}{\lambda}\right)\right]},$
(19)

which increases monotonically to the saturation value $N\frac{(1-r)^{2}}{2}$.

_Discussions.–_ In this work, based on the epidemiological models and the
solvable Brownian SYK models, we show that couplings to external baths can
give rise to a dynamical transition of the operator size growth in open
quantum systems. The explicit analytic results obtained for the solvable
Brownian SYK model of $q=3$ suggest that the operator growth consists of two
stages: in the first stage, the bath operators keep absorbing weight from the
system operators. If the intrinsic scrambling of the system is strong enough,
a finite weight would manage to escape the absorption, and in the second
stage, the operator size distribution $\mathcal{P}(s,t)$ has the same scaling
form as when the system is closed. Since (12) is the base for our analysis of
the case of $q=3$, we expect that the existence of the dynamic transition
stand as well for $q>3$ from (12) SM .

The dynamic transition separating the dissipative and scrambling phases
implies that teleportation through the wormhole Gao  _et al._ (2017);
Maldacena  _et al._ (2017); Susskind and Zhao (2018); Gao and Liu (2019);
Brown  _et al._ (2019); Gao and Jafferis (2021); Nezami  _et al._ (2021); Gu
_et al._ is still possible when an environment is added if the scrambling of
the initial operator into the holographic bulk is fast enough. However, the
amount of information that can be teleported is reduced. Our “bath-induced
scrambling phase transition” shall have a close relation with the celebrated
“measurement-induced entanglement phase transition” Li  _et al._ (2018);
Skinner  _et al._ (2019); Chan  _et al._ (2019). In particular, after
integrating out the bath which is assumed to be Markovian, the evolution of
the operator size (7) can be formulated in terms of a generalized master
equation, which was introduced to study the measurement-induced entanglement
phase transition Zhou (2022). The volume-law phase in the presence of
measurement is known to be protected from local quantum errors Gullans and
Huse (2020); Choi  _et al._ (2020); Fan  _et al._ (2021), so is expected the
scrambling phase in open systems. Furthermore, the averaged operator size (8)
conveys information similar as the entropy of a system Hosur  _et al._ (2016);
Fan  _et al._ (2017). We postpone a detailed study to future works.

Acknowledgment. We thank Chang Liu for helpful discussions. PZ is partly
supported by the Walter Burke Institute for Theoretical Physics at Caltech. ZY
is supported by the National Natural Science Foundation of China (Grant No.
12074440), and Guangdong Project (Grant No. 2017GC010613).

Note Added. When completing this work, we noted that a scrambling transition
was also revealed in a random unitary circuit that exchanges qubits with an
environment Weinstein  _et al._ (2022). The study focused on the numerical
evidence from the out-of-time-order correlator.

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Supplementary Material: Dynamical Transition of Operator Size Growth in Open
Quantum Systems

In this Supplementary Material, we present the derivations of (1) the self-
consistent equation for the generating function $z(\mu,t)$; (2) the late-time
operator size distribution $\mathcal{P}(s,t)$; (3) the analysis of the
operator size dynamics for $q\geq 4$.

##  I the self-consistent equation for the generating function

In this section, we present the derivation of

$\displaystyle z(\mu,t)=-\int
dt^{\prime}~{}G^{R}(t-t^{\prime})^{2}\Sigma_{W}(t^{\prime}),\ \ \ \ \ \
\Sigma_{W}(t)=Jz(\mu,t)^{q-1}+V-\frac{1}{4}(1-e^{-\mu})\delta(t),$ (20)

for the generating function $z(\mu,t)=\sum e^{-\mu n}P(n,t)$ as stated in the
main text. The idea is to write out the path-integral representation of Gu
_et al._ (2022).

$z(\mu,t)=\langle\text{EPR}|\chi_{1}(t)^{\dagger}e^{-\mu\sum_{j}c^{\dagger}_{j}c_{j}}\chi_{1}(t)|\text{EPR}\rangle,$
(21)

where $z(\mu,t)$ is a two-point function with a perturbation source, and
derive (20) by using the Schwinger-Dyson equation.

As a warm-up, we first consider the average operator size, which is the first
moment of $z(\mu,t)$:

$\overline{n}=\sum_{j}\langle\text{EPR}|\chi_{1}(t)^{\dagger}c^{\dagger}_{j}c_{j}\chi_{1}(t)|\text{EPR}\rangle=\frac{1}{4}\sum_{j}\left(\langle\chi_{1}(t)\chi_{j}\chi_{1}(t)\chi_{j}\rangle+\langle\chi_{j}\chi_{1}(t)\chi_{1}(t)\chi_{j}\rangle+\langle\chi_{j}\chi_{1}(t)\chi_{j}\chi_{1}(t)\rangle+\langle\chi_{1}(t)\chi_{j}\chi_{j}\chi_{1}(t)\rangle\right).$
(22)

The calculation of $\overline{n}$ can be represented as a four-point function
on the double Keldysh contour Aleiner  _et al._ (2016)

$t$world $1$$u$$d$$u$$d$world $2$ (23)

which contains two forward evolution branches $(u_{1},u_{2})$ and two backward
evolution branches $(d_{1},d_{2})$. The black points represent the insertions
of $\chi_{1}$ and the red points represent the insertions of $\chi_{j}$. More
explicitly, we have

$\overline{n}=\frac{1}{4}\sum_{j}\left<T_{\mathcal{C}}\,\chi^{u_{2}}_{1}(t)\chi^{u_{1}}_{1}(t)\left[\chi_{j}^{u_{2}}(0)-\chi_{j}^{d_{2}}(0)\right]\left[\chi_{j}^{d_{1}}(0)-\chi_{j}^{u_{1}}(0)\right]\right>.$
(24)

Here the minus sign is from the anti-commutation relation of Grassmann fields,
and $T_{\mathcal{C}}$ is the contour ordering operator. Generalizing the
derivation to $z(\mu,t)$, we find

$\displaystyle z(\mu,t)$
$\displaystyle=\left<T_{\mathcal{C}}\,\chi^{u_{2}}_{1}(t)\chi^{u_{1}}_{1}(t)e^{-\Delta
I}\right>,\ \ \ \ \ \ \Delta
I=\frac{1}{4}(1-e^{-\mu})\sum_{j}\left[\chi_{j}^{u_{2}}(0)-\chi_{j}^{d_{2}}(0)\right]\left[\chi_{j}^{d_{1}}(0)-\chi_{j}^{u_{1}}(0)\right].$
(25)

Here $(1-e^{-\mu})$ appears instead of $\mu$ since we have
$\langle\bar{\psi}|e^{-\mu
c^{\dagger}c}|\psi\rangle=e^{-e^{-\mu}\bar{\psi}\psi}$. The two-point function
$z(\mu,t)$ is the equal time limit of the inter-world Wightmann function
$G_{W}(t_{1},t_{2})=\left<\chi^{u_{2}}(t_{1})\chi^{u_{1}}(t_{2})\right>_{\Delta
I}$, which satisfies the Schwinger-Dyson equation Gu  _et al._ (2022)

$G_{W}(t_{1},t_{2})=\int
dt_{3}dt_{4}~{}G_{R}(t_{1},t_{3})\Sigma_{W}(t_{3},t_{4})G_{A}(t_{4},t_{2}).$
(26)

For our Brownian SYK models, the self-energy takes a simple form:

$\Sigma_{W}(t,t^{\prime})=\delta(t-t^{\prime})\left(JG_{W}(t,t)^{q-1}+V-\frac{1}{4}(1-e^{-\mu})\delta(t)\right)\equiv\delta(t-t^{\prime})\Sigma_{W}(t).$
(27)

Using (26), $\eqref{supp:SD2}$, and $G_{R}(t_{1},t_{2})=-G_{A}(t_{2},t_{1})$,
we arrive at (20) after taking $t_{1}=t_{2}=t$.

##  II the late-time operator size distribution

Here we derive the late-time operator size distribution in the scrambling
phase $r<1$:

$\mathcal{P}(s,t)=r\delta(s)+\mathcal{P}_{\text{reg}}(s,t)=r\delta(s)+\theta\left(1-\frac{2s}{1-r}\right)\frac{2(r-1)^{2}e^{\frac{2s}{\lambda(r+2s-1)}}}{\lambda(r+2s-1)^{2}}.$
(28)

In the long-time limit $e^{(1-r)t}\gg 1$, we assume the perturbation source
$\Delta I$ interacts with $\chi_{1}(t)$ only by the exchanging collective
scramblon mode. The scramblon mode is the origin of quantum many-body chaos,
which has an exponential propagator $-e^{(1-r)t}/C$ with $C\propto N$ Gu and
Kitaev (2019); Gu  _et al._ (2022). We define the scattering between the
fermion fields and the scramblons as

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{{}{}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{{}}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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{{}{}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{{}}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-21.6pt}{0.0pt}\pgfsys@curveto{-20.975pt}{0.0pt}{-20.6625pt}{2.0pt}{-20.0375pt}{2.0pt}\pgfsys@curveto{-19.585pt}{2.0pt}{-19.19502pt}{1.02399pt}{-18.7875pt}{0.0pt}\pgfsys@curveto{-18.38pt}{-1.02399pt}{-17.99pt}{-2.0pt}{-17.5375pt}{-2.0pt}\pgfsys@lineto{-14.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-17.733pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\Upsilon^{R,m},\ \ \
\ \ \ \leavevmode\hbox to41.45pt{\vbox
to29.29pt{\pgfpicture\makeatletter\hbox{\hskip-6.93399pt\lower-14.64673pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@fill{.75}\pgfsys@invoke{
}{}\pgfsys@moveto{38.0pt}{0.0pt}\pgfsys@curveto{38.0pt}{4.41833pt}{34.41833pt}{8.0pt}{30.0pt}{8.0pt}\pgfsys@curveto{25.58167pt}{8.0pt}{22.0pt}{4.41833pt}{22.0pt}{0.0pt}\pgfsys@curveto{22.0pt}{-4.41833pt}{25.58167pt}{-8.0pt}{30.0pt}{-8.0pt}\pgfsys@curveto{34.41833pt}{-8.0pt}{38.0pt}{-4.41833pt}{38.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{30.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{30.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}
{{}{}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{{}{}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
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{{}{}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{{}}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{{}}} } \pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{24.0603pt}{5.9397pt}\pgfsys@curveto{23.61836pt}{6.38164pt}{21.98318pt}{5.18839pt}{21.54124pt}{5.63033pt}\pgfsys@curveto{21.0993pt}{6.07227pt}{22.29256pt}{7.70744pt}{21.85062pt}{8.14938pt}\pgfsys@lineto{18.68628pt}{11.31372pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{{}{}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{{}}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{{}}} } \pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{24.0603pt}{-5.9397pt}\pgfsys@curveto{23.61836pt}{-6.38164pt}{24.81161pt}{-8.01682pt}{24.36967pt}{-8.45876pt}\pgfsys@curveto{23.92773pt}{-8.9007pt}{22.29256pt}{-7.70744pt}{21.85062pt}{-8.14938pt}\pgfsys@lineto{18.68628pt}{-11.31372pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.95328pt}{-11.31372pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{{}{}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{{}}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}} } \pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{21.6pt}{0.0pt}\pgfsys@curveto{20.975pt}{0.0pt}{20.6625pt}{-2.0pt}{20.0375pt}{-2.0pt}\pgfsys@curveto{19.4125pt}{-2.0pt}{19.1pt}{0.0pt}{18.475pt}{0.0pt}\pgfsys@lineto{14.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{10.267pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\Upsilon^{A,m}.$ (29)

Here for simplicity, we assumed all the fermion fields are inserted (in the
future for $\Upsilon^{R}$ and in the past for $\Upsilon^{A}$ ) at the same
time. In particular, since our models have time-reversal symmetry, we have
$\Upsilon^{R,m}=\Upsilon^{A,m}=\Upsilon^{m}$.

To determine $\Upsilon^{R/A,m}$, we turn to the early-time form of the
generating function

$z(\mu,t)=1-\frac{(r-1)(1-e^{-\mu})}{e^{(r-1)t}(r-e^{-\mu})-(1-e^{-\mu})}.$
(30)

We consider the situation $\mu\ll 1$ and $1\ll e^{(1-r)t}\lesssim 1/\mu\ll N$,
under which all non-scramblon contribution vanishes; we find

$z(\mu,t)=r+\frac{1-r}{1+e^{(1-r)t}\mu(1-r)^{-1}}.$ (31)

On the other hand, the scramblon contribution to the early-time generating
function takes the form:

$z(\mu,t)=\leavevmode\hbox to69.35pt{\vbox
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} \pgfsys@invoke{\lxSVG@closescope
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par{{}}{}
{{}{}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}
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{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}} } \pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{12.4pt}{0.0pt}\pgfsys@curveto{13.025pt}{0.0pt}{13.3375pt}{2.0pt}{13.9625pt}{2.0pt}\pgfsys@curveto{14.5875pt}{2.0pt}{14.9pt}{0.0pt}{15.525pt}{0.0pt}\pgfsys@lineto{24.88614pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}
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{{{}}} } \pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{8.76813pt}{-8.76813pt}\pgfsys@curveto{9.21007pt}{-9.21007pt}{10.84525pt}{-8.01682pt}{11.28719pt}{-8.45876pt}\pgfsys@curveto{11.60715pt}{-8.77872pt}{11.19276pt}{-9.74461pt}{10.75685pt}{-10.75685pt}\pgfsys@curveto{10.32094pt}{-11.76906pt}{9.90654pt}{-12.73497pt}{10.22652pt}{-13.05495pt}\pgfsys@curveto{10.66847pt}{-13.4969pt}{12.30365pt}{-12.30365pt}{12.74559pt}{-12.74559pt}\pgfsys@lineto{17.59737pt}{-17.59737pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\sum_{m=0}^{\infty}\left(-\frac{e^{(1-r)t}}{C}\right)^{m}\left(\frac{\mu
N}{2}\Upsilon^{1}\right)^{m}\frac{\Upsilon^{m}}{m!}.$ (32)

Here each insertion of $\Delta I$ only creates a single scramblon due to the
suppression $\lambda\sim O(1/N)$. In fact, we have not fixed the normalization
of the scramblon field. The transformation $C\rightarrow a^{2}C$ and
$\Upsilon^{m}\rightarrow a^{m}\Upsilon^{m}$ leaves the expression invariant.
We could make a convenient choice

$\frac{N\Upsilon^{1}}{2C}=(1-r)^{-1},$ (33)

and define

$f(x)=\sum_{m=0}^{\infty}\frac{(-x)^{m}}{m!}\Upsilon^{m}=r+\frac{1-r}{1+x}.$
(34)

By expanding $f(x)$, we can fix $\Upsilon^{m}$. In particular, we find

$\Upsilon^{0}=1,\;\;\;\;\;\Upsilon^{m}=(1-r)\,m!,\;\;\;\;\;C=N(1-r)^{2}/2.$
(35)

Now we consider the late-time operator size generating function
$\mathcal{S}(\nu,t)$ with $\lambda\sim O(1)$. Unlike the early-time result, we
need to sum up all diagrams where each insertion of $\Delta I$ create
arbitrary number of scramblons:

$\mathcal{S}(\nu,t)=\leavevmode\hbox to69.35pt{\vbox
to69.09pt{\pgfpicture\makeatletter\hbox{\hskip 28.9508pt\lower-34.54642pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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0\\}}(-\lambda)^{\sum_{j}m_{j}}\frac{\Upsilon^{\sum_{j}m_{j}}\Upsilon^{m_{1}}\ldots\Upsilon^{m_{n}}}{m_{1}!m_{2}!\ldots
m_{n}!}$ (36)

We introduce $h(y)$ as the inverse Laplace transform of $f(x)$:

$f(x)=\int_{0}^{\infty}dy~{}e^{-xy}h(y),\ \ \ \ \ \
h(y)=(1-r)e^{-y}+r\delta(y).$ (37)

By comparing with (34), we find

$\Upsilon^{m}=\int_{0}^{\infty}dy~{}y^{m}h(y).$ (38)

This leads to

$\displaystyle\mathcal{S}(\nu,t)$
$\displaystyle=\int_{0}^{\infty}dy~{}h(y)~{}e^{-\frac{\nu}{2}}\sum_{n=0}^{\infty}\frac{(\nu/2)^{n}}{n!}\sum_{\\{m_{j}\geq
0\\}}(-\lambda)^{\sum_{j}m_{j}}\frac{y^{\sum_{j}m_{j}}\Upsilon^{m_{1}}\ldots\Upsilon^{m_{n}}}{m_{1}!m_{2}!\ldots
m_{n}!}$ (39)
$\displaystyle=\int_{0}^{\infty}dy~{}h(y)~{}e^{-\frac{\nu}{2}}\sum_{n=0}^{\infty}\frac{(\nu/2)^{n}}{n!}\left[\sum_{\\{m\geq
0\\}}(-\lambda)^{m}\frac{y^{m}\Upsilon^{m}}{m!}\right]^{n}$
$\displaystyle=\int_{0}^{\infty}dy~{}h(y)~{}e^{-\frac{\nu}{2}}\sum_{n=0}^{\infty}\frac{(\nu/2)^{n}}{n!}f(\lambda
y)^{n}$
$\displaystyle=\int_{0}^{\infty}dy~{}h(y)e^{-\frac{\nu}{2}\left(1-f(\lambda
y)\right)},$

Performing the inverse inverse Laplace transform with respect to $\nu$, we
find

$\displaystyle\mathcal{P}(s,t)$
$\displaystyle=\int_{0}^{\infty}dy~{}h(y)\delta\left(s-\frac{1-f(\lambda
y)}{2}\right).$ (40)

Integrating over $y$ in (40) leads to the result (28).

##  III dynamical trnsition of the opertaor size for general ${q}\geq 3$

In the main text, we first use the epidemiological models to predict the
existence of the bath-induced scrambling transition. Later, we work out
explicitly the operator size distribution for the Brownian SYK model of $q=3$;
using the early-time generating function, we find $P(0,\infty)=1$ in the
dissipative phase and $P(0,\infty)<1$ in the scrambling phase. In the
following, we show that certain features of the transition are independent of
$q$ for the Brownian SYK model of general $q\geq 3$, i.e.,

  1. 1.

For $r<q-2$, we have $P(0,\infty)<1$. Together with the exponential growth of
the operator size $\overline{n}$ in the early-time regime, this indicates the
intrinsic scrambling of the system is strong enough, a finite weight would
manage to escape the absorption.

  2. 2.

For $r\geq q-2$, we have $P(0,\infty)=1$. Since we have the normalization
$\sum_{n}P(n,t)=1$, this indicates the system is in the dissipative phase
where a typical operator at late time contains no system operator.

We start with the differential equation satisfied by the early-time generating
function

$\frac{dz}{dt}=(z^{q-1}-z)+r(1-z),\ \ \ \ \ z(\mu,0^{+})=e^{-\mu}.$ (41)

Integrating (41) directly, we find

$t=\int_{e^{-\mu}}^{z(\mu,t)}d\tilde{z}\frac{1}{(\tilde{z}^{q-1}-\tilde{z})+r(1-\tilde{z})}.$
(42)

By definition, we have $P(0,t)=z(\infty,t)$. This relation gives rise to

$t=\int_{0}^{P(0,t)}d\tilde{z}\frac{1}{(\tilde{z}^{q-1}-\tilde{z})+r(1-\tilde{z})}.$
(43)

For $t\rightarrow\infty$, the L.H.S. of (43) diverges. As a result,
$P(0,\infty)$ should coincide with the pole of the integrand, namely,

$P(0,\infty)^{q-1}-(r+1)P(0,\infty)+r=0.$ (44)

Moreover, since $P(0,t=0)=0$, $P(0,\infty)$ should be the smallest real
positive root of $u(x)=x^{q-1}-(r+1)x+r$. It is straightforward to check that
$x_{r}=1$ is always a root of $u(x)$. For $q\geq 3$, $u^{\prime\prime}(x)>0$
for $x>0$. Considering the fact that $u(0)=r>0$ and $u(1)=0$, we have one
additional root $x_{r}^{\prime}<1$ if and only if

$u^{\prime}(1)=(q-2-r)>0.$ (45)

This is satisfied when $r<q-2$. As a result, we have $P(0,\infty)<1$ for
$r<q-2$ and $P(0,\infty)=1$ for $r\geq q-2$, as stated at the beginning of the
section.