# Qubit Decoherence and Symmetry Restoration through Real-Time Instantons

Foster Thompson1 and Alex Kamenev1,2 1School of Physics and Astronomy,
University of Minnesota, Minneapolis, Minnesota 55455, USA  2William I. Fine
Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota
55455, USA

###### Abstract

A parametrically driven quantum oscillator, stabilized by a nonlinear
dissipation, exhibits a spontaneous breaking of the parity symmetry. It
results in the quantum bi-stability, corresponding to a Bloch sphere of dark
states. This makes such a driven-dissipative system an attractive candidate
for a qubit. The parity symmetry breaking is exact both on the classical level
and within the quantum mechanical perturbation theory. Here we show that non-
perturbative quantum effects lead to the symmetry restoration and result in
exponentially small but finite qubit decoherence rate. Technically the
symmetry restoration is due to real time instanton trajectories of the Keldysh
path integral, which represents the Lindbladian evolution of the driven-
dissipative oscillator.

Open quantum systems subject simultaneously to dissipative effects and a
strong external drive obey effective dynamics described by the Lindblad
equation OpenQM ; QuantumNoise ; DrivenDissipativeLight ;
DrivenDissipativeColdAtoms ; DiehlQuantumSim ; Diehl . Such systems have
gained attention for their ability to realize multiple non-equilibrium steady
states NonAbelianSym ; AlbertGeom ; NonEqSSKeldysh ; AlbertSym ;
LindbladSymBreak ; Prosen ; Clerk ; MultiDarkSpace . Of particular interest
are systems exhibiting quantum bi-stability, in which the space of steady
states has the structure of a Bloch sphere and can be used to encode a qubit
stably in contact with its environment QCompDrivenDissipative ; AlbertCats ;
Albert .

The stationary states of a Lindblad equation are determined by its symmetries
AlbertSym ; NonAbelianSym ; the spontaneous breaking of symmetry can lead to
bi-stability LindbladSymBreak . A broken ‘strong’ symmetry (in the sense of
Ref. Prosen ) leads to a Bloch sphere stationary space Albert , forming a
qubit. Similarly to the unitary quantum mechanics, non-perturbative effects
often restore the symmetry, leading to a slow precession and decoherence of
the would-be qubits. The quantum mechanical symmetry restoration (i.e.
tunneling) is achieved through the instanton solution of the imaginary time
equations of motion (EoM). Remarkably, the strong symmetry restoration in the
dissipative Lindbladian dynamics occurs through real time solutions of
complexified EoM. Consequently, this purely quantum phenomenon is akin to
classical activation in a multidemnsional space DykmanClassical1 ;
DykmanClassical2 .

To exemplify this, consider a harmonic oscillator subject to an external drive
that periodically modulates its resonant frequency
$\hat{H}=\omega_{0}\hat{a}^{\dagger}\hat{a}+2\lambda\cos(2\omega_{p}t)(\hat{a}^{\dagger
2}+\hat{a}^{2})$, where $\hat{a}$ is a Bose annihilation operator,
$[\hat{a},\hat{a}^{\dagger}]=1$. In addition to the coherent drive we allow
the oscillator to dissipate through interaction with its environment, modelled
as a bath of linear oscillators in thermal equilibrium at zero temperature. To
ensure the $\mathbb{Z}_{2}$ parity symmetry
$\hat{a}\to\hat{P}\hat{a}\hat{P}^{\dagger}=-\hat{a}$ is still present when the
oscillator is put in contact with the environment, we choose a nonlinear
coupling between the system and the bath:
$\hat{H}_{\mathrm{int}}=\sum_{s}\frac{g_{s}}{\sqrt{2\omega_{s}}}(\hat{b}^{\dagger}_{s}\hat{a}^{2}+\hat{a}^{\dagger
2}\hat{b}_{s})$, where bath modes $\hat{b}_{s}$ oscillate at frequencies
$\omega_{s}$.

In the rotating frame of the external drive, ignoring counter-rotating terms,
the Hamiltonian becomes effectively time independent:

$\hat{H}(\hat{a},\hat{a}^{\dagger})\simeq\Delta\hat{a}^{\dagger}\hat{a}+\lambda(\hat{a}^{\dagger
2}+\hat{a}^{2}),$ (1)

where $\Delta=\omega_{0}-\omega_{p}$. Integrating out the bath degrees of
freedom, leads to a dissipative dynamics, described by the Lindblad equation
for the reduced density matrix of the oscillator:

$\partial_{t}\rho=\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}}\rho=-i[\hat{H},\rho]+\gamma\Big{(}\hat{a}^{2}\rho\hat{a}^{\dagger
2}-\frac{1}{2}\\{\hat{a}^{\dagger 2}\hat{a}^{2},\rho\\}\Big{)},$ (2)

where the Lindbladian ‘superoperator’
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}}$ acts in
the Hilbert space of operators and the dissipation strength is
$\gamma=\sum_{s}(\pi g_{s}^{2}/2\omega_{s})\delta(\omega_{0}-\omega_{s})$.

Both $\hat{H}$ and the jump operator $\hat{a}^{2}$ commute with $\hat{P}$,
making Eq. (2) invariant under $\hat{a}\to-\hat{a}$ on both the left and the
right of $\rho$ independently. Defining the left/right parity superoperators
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{\pm}$ to
act by left/right multiplication by $\hat{P}$:
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{+}\rho=\hat{P}\rho$,
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{-}\rho=\rho\hat{P}$,
this is made precise by the fact that both commute with the Lindbladian
$[\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{\pm},\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}}]=0$.
In this case the system is said to posses the strong parity symmetry. The
simultaneous diagonalizability of
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}}$ and
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{-},\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{+}$
means that there are at least two stationary density matrices. If this
symmetry is broken, there are four stationary density matrices, isomorphic to
the Bloch sphere of of steady states Albert .

Models of a parametrically driven oscillator linearly coupled to a bath have
been studied extensively in, for example, Dykman1 ; Dykman2 ; Dykman3 . Such
models posses only weak symmetry and so can exhibit only classical, not
quantum, bi-stability. Models with master equations similar to (2) have been
studied using methods based on operator formalism AlbertCats ; Clerk  and
quasi-probability distributions Minganti1 ; Minganti2 . Experimentally, this
is realizable in a pumped system of microwave cavities coupled via Josephson
junction Devoret1 ; Devoret2 .

When the drive amplitude $\lambda$ is sufficiently large, the oscillator
becomes unstable at the level of the Hamiltonian (1). To see this, it is
convenient to go to Hermitian operators $\hat{x},\hat{p}$, such that
$\hat{a}=(\hat{p}-i(\Delta+2\lambda)\hat{x})/\sqrt{(\Delta+2\lambda)}$, in
terms of which Eq. (1) becomes
$\hat{H}=\hat{p}^{2}+(\Delta^{2}-4\lambda^{2})\hat{x}^{2}$. This is a
conventional harmonic oscillator so long as $\Delta>2\lambda$, but for
$\Delta<2\lambda$ the potential flips upside down causing instability. In the
latter case the model is stabilized by the nonlinear dissipation. This can be
seen by, eg., taking $\Delta=0$ and noticing that the Hamiltonian can be
absorbed into the dissipative part of the Lindbladian by a linear shift of the
jump operator:

$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}}\rho=\gamma\Big{(}(\hat{a}^{2}-\phi_{0}^{2})\rho(\hat{a}^{\dagger
2}-\phi_{0}^{*2})-\frac{1}{2}\big{\\{}(\hat{a}^{\dagger
2}-\phi_{0}^{*2})(\hat{a}^{2}-\phi_{0}^{2}),\rho\big{\\}}\Big{)},$ (3)

where $\phi_{0}=\sqrt{2\lambda/i\gamma}$. The two coherent states
$\ket{\pm\phi_{0}}$ are anihilated by the shifted jump operators, so the four
density matrices formed by their products $\ket{\pm\phi_{0}}\bra{\pm\phi_{0}}$
and $\ket{\pm\phi_{0}}\bra{\mp\phi_{0}}$ are all stationary. These operators
are exchanged by the right or left action of $\hat{P}$, meaning the parity
symmetry is spontaneously broken. The even and odd cat states
$\ket{\pm}\propto\ket{\phi_{0}}\pm\ket{-\phi_{0}}$ form an orthonormal basis
of the steady-state space. Pure states made from their linear combinations
form a Bloch sphere.

The model is zero-dimensional, so symmetry breaking cannot occur in the bulk
of the parameter space. When $\Delta$ is small but finite, the symmetry must
be restored. An exponentially small dissipative gap will open in the
Lindbladian spectrum as the eigenvalues of the operators $\ket{\pm}\bra{\mp}$
move into the complex plain as a conjugate pair, denoted as $\Lambda$ and
$\Lambda^{*}$. The stationary-state space is reduced in dimension from four to
two. This phenomenon is non-perturbative in $\gamma$ and occurs due to
instanton contributions to the Keldysh path integral.

To show this, it is useful to study the character-valued “partition functions”
$\operatorname{Tr}(\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{\pm}\exp(T\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}}))$
and
$\operatorname{Tr}(\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{+}\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{-}\exp(T\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}}))$,
where the capital Tr indicates that the trace is over a superoperator and
$T\to\infty$ is the evolution time. The action on the eigenvectors
$\mathcal{P}_{+}\ket{-}\bra{\pm}=-\ket{-}\bra{\pm}$ means the their
eigenvalues come with an additional sign in the $\mathcal{P}_{+}$-twisted
trace, and similarly for the other parity superoperators. Assuming all nonzero
eigenvalues of
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}}$ have a
finite real part, only the smallest eigenvalues contribute in the limit
$T\to\infty$:

$\operatorname{Tr}\big{(}\exp(T\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}})\big{)}\simeq
2+e^{\Lambda T}+e^{\Lambda^{*}T}\simeq 4+T(\Lambda+\Lambda^{*}),$ (4a)
$\operatorname{Tr}\big{(}\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{\pm}\exp(T\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}})\big{)}\simeq\pm(e^{\Lambda
T}-e^{\Lambda^{*}T})\simeq\pm T(\Lambda-\Lambda^{*}),$ (4b)
$\operatorname{Tr}\big{(}\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{+}\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{-}\exp(T\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}})\big{)}\simeq
2-e^{\Lambda T}-e^{\Lambda^{*}T}\simeq-T(\Lambda+\Lambda^{*}),$ (4c)

where the last approximate equalities indicate one instanton contributions.

Using the machinery of Ref. Diehl , the character-valued partition functions
may be represented as coherent state Keldysh path integrals
$\int\text{D}\phi^{\pm}\text{D}\bar{\phi}^{\pm}e^{iS}$ over the spaces of
twisted loops in the Keldysh phase space, Suplemental . Here the fields
$(\phi^{\pm},\bar{\phi}^{\pm})$ reside on the forward/backward part of the
closed time contour and the action is given by

$S=\int\limits_{-T/2}^{T/2}\\!\text{d}t\,\big{(}\bar{\phi}^{+}i\partial_{t}\phi^{+}-\bar{\phi}^{-}i\partial_{t}\phi^{-}-H^{+}+H^{-}-iD\big{)},$
(5)

where $H^{\pm}=H(\phi^{\pm},\bar{\phi}^{\pm})$ from Eq. (1) and

$D(\phi^{\pm},\bar{\phi}^{\pm})=\gamma\Big{(}(\bar{\phi}^{-}\phi^{+})^{2}-\frac{1}{2}(\bar{\phi}^{+}\phi^{+})^{2}-\frac{1}{2}(\bar{\phi}^{-}\phi^{-})^{2}\Big{)}.$
(6)

Notice that the dissipator, $D$, breaks the time-reversal symmetry between the
forward and backward parts of the contour. The strong Lindbladian parity
symmetry is realized as a classical $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
symmetry of the Keldysh action: transforming either of the sets of fields
independently $\phi^{\pm}\to-\phi^{\pm}$ leaves the action invariant.

We will consider a saddle point approximation to the character-valued
partition functions described above. For this, one must find solutions to the
classical EoM with the appropriate boundary conditions for $T\to\infty$.
Expressing the action (5) in terms of the Keldysh rotated fields QFT&NonEq
$\phi^{\pm}=(\phi^{\mathrm{c}}\pm\phi^{\mathrm{q}})/\sqrt{2}$, one finds for
the action:
$S=\int\\!\text{d}t\,\Big{(}\bar{\phi}^{\mathrm{q}}i\partial_{t}\phi^{\mathrm{c}}+\bar{\phi}^{\mathrm{c}}i\partial_{t}\phi^{\mathrm{q}}-K\Big{)}$,
where

$K(\phi^{\mathrm{c}},\phi^{\mathrm{q}},\bar{\phi}^{\mathrm{c}},\bar{\phi}^{\mathrm{q}})=\Delta(\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{c}}+\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{q}})+2\lambda(\bar{\phi}^{\mathrm{q}}\bar{\phi}^{\mathrm{c}}+\phi^{\mathrm{q}}\phi^{\mathrm{c}})\\\
-i\frac{\gamma}{2}\big{(}(\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{c}}-\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{q}})(\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{c}}-\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{q}})+4\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{q}}\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{c}}\big{)}.$
(7)

The classical $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry realized on these
fields is generated by
$\phi^{\mathrm{c},\mathrm{q}}\to-\phi^{\mathrm{c},\mathrm{q}}$ and
$\phi^{\mathrm{c}}\leftrightarrow\phi^{\mathrm{q}}$.

The classical EoM are found by varying the action in each of the four fields.
They acquire the Hamiltonian structure with the conserved “energy” $K$:

$i\partial_{t}\phi^{\mathrm{c}}=\partial_{\bar{\phi}^{\mathrm{q}}}K,\ \ \
-i\partial_{t}\bar{\phi}^{\mathrm{c}}=\partial_{\phi^{\mathrm{q}}}K,$ (8)

and the other EoM are given by exchanging labels
$\mathrm{c}\leftrightarrow\mathrm{q}$. In addition to the trivial fixed point
at the origin of the phase space, EoM admit additional nontrivial fixed points
Suplemental . When $\Delta=0$, these occur when either the $\mathrm{c}$ or
$\mathrm{q}$ fields are equal to $\pm\sqrt{2}\phi_{0}$ from Eq. (3) while the
complimentary fields are zero. At these four fixed points the classical
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry is spontaneously broken. Away
from the $\Delta=0$ limit, the symmetry is restored quantum mechanically due
to instanton solutions to the EoM, connecting the fixed points in the
complexified phase space of the field configurations.

Indeed, EoM (8) do not respect $\bar{\phi}^{\mathrm{c},\mathrm{q}}$ being
complex conjugated of $\phi^{\mathrm{c},\mathrm{q}}$, but rather operate in an
enlarged complexified phase space, where $\phi^{\mathrm{c},\mathrm{q}}$ and
$\bar{\phi}^{\mathrm{c},\mathrm{q}}$ are regarded as independent complex
variables. This should be understood as a complex deformation of the original
integration contours over
$\phi^{\mathrm{c},\mathrm{q}},\bar{\phi}^{\mathrm{c},\mathrm{q}}$ to pass
through proper saddle-point field configurations. The classical motion,
resulting from this complexification, takes place in the eight real-
dimensional phase space spanned by the four holomorphic coordinates
$(\phi^{\mathrm{c}},\phi^{\mathrm{q}},\bar{\phi}^{\mathrm{c}},\bar{\phi}^{\mathrm{q}})$
and their complex conjugates
$(\phi^{\mathrm{c}*},\phi^{\mathrm{q}*},\bar{\phi}^{\mathrm{c}*},\bar{\phi}^{\mathrm{q}*})$.
In this eight-dimensional space, $K+K^{*}$ acts as the Hamiltonian generating
the motion and $K-K^{*}$ is an integral of motion. In addition to this, there
is a hidden symmetry generated by the holomorphic integral of motion $M$. This
can be found by looking for holomorphic polynomials that Poisson–commute with
$K$:

$K(\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\phi^{\mathrm{c}}}\vec{\partial}_{\bar{\phi}^{\mathrm{q}}}-\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\bar{\phi}^{\mathrm{q}}}\vec{\partial}_{\phi^{\mathrm{c}}}+\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\phi^{\mathrm{q}}}\vec{\partial}_{\bar{\phi}^{\mathrm{c}}}-\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\bar{\phi}^{\mathrm{c}}}\vec{\partial}_{\phi^{\mathrm{q}}})M=0.$
(9)

For $\Delta=0$ (see Suplemental  for a more general case), the lowest order
solution (not equal to a multiple of $K$) is given by:

$\displaystyle\\!\\!M\\!\\!=\\!\\!\Big{[}2\phi_{0}^{2}(\phi^{\mathrm{c}2}\\!+\\!\bar{\phi}^{\mathrm{c}2})\\!+\\!(\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{q}}\\!-\\!\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{c}}\\!+\\!2\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{c}})(\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{c}}\\!+\\!\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{q}})\Big{]}$
$\displaystyle\\!\times\Big{[}2\phi_{0}^{2}(\phi^{\mathrm{q}2}\\!+\\!\bar{\phi}^{\mathrm{q}2})\\!-\\!(\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{q}}\\!-\\!\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{c}}\\!-\\!2\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{q}})(\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{c}}\\!+\\!\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{q}})\Big{]}.$

As a consequence $K$, $M$, and their conjugates are four independent conserved
quantities. This makes the corresponding eight-dimensional classical motion
completely integrable.

Figure 1: Phase portrait of the Hamiltonian $K(X,P)$, Eq. (12). Bold directed
lines show separatrix trajectories $K=0$, thinner lines show examples of other
trajectories, and dots show fixed points. Instanton paths that contribute to
the real and imaginary parts of $\Lambda$ highlighted in blue and red,
respectively. The first satisfies the boundary condition
$X(-\infty)=-X(\infty)$ and $P(-\infty)=-P(\infty)$ and the second,
$X(-\infty)=P(\infty)$ and $P(-\infty)=X(\infty)$. The shaded areas show the
instanton action for each path.

In the limit $T\to\infty$, the instantons are given by the separatrix
trajectories connecting the fixed points. Since at the fixed points $K=M=0$,
this is also the case for the separatrix trajectories and thus all four
conserved quantities are equal to zero. The condition $K+K^{*}=0$ dictates
$\bar{\phi}^{\mathrm{c}}=\phi^{\mathrm{c}*}$ and
$\bar{\phi}^{\mathrm{q}}=-\phi^{\mathrm{q}*}$, while the condition $M=0$ fixes
$\phi^{\mathrm{c}}=i\bar{\phi}^{\mathrm{c}}$ and
$\phi^{\mathrm{q}}=-i\bar{\phi}^{\mathrm{q}}$. Each of these conditions
defines a four-dimensional sub-space of the eight-dimensional phase space.
Their intersection is thus a 2D plane parametrized by the real coordinates
$(X,P)$, with

$\phi^{\mathrm{c}}=i\phi^{\mathrm{c}*}=\sqrt{\frac{4\lambda}{i\gamma}}\,\,X;\qquad\bar{\phi}^{\mathrm{q}}=i\bar{\phi}^{\mathrm{q}*}=-\sqrt{\frac{4i\lambda}{\gamma}}\,\,P.$
(10)

At this invariant plain, the Keldysh action acquires an explicit Hamiltonian
structure:

$iS=\frac{8\lambda}{\gamma}\int\limits_{-T/2}^{T/2}\text{d}t\Big{(}P\partial_{t}X-K(X,P)\Big{)},$
(11)

where the effective Hamiltonian is found by the substitution of Eq. (10) into
Eq. (7):

$K(X,P)=2\lambda\,XP(1+P+X)(1-P-X).$ (12)

The parity symmetry manifests itself in the invariance vis-a-vis
$X\leftrightarrow P$ exchange. Notice that Eq. (11) is a real time action.
Unlike in Hermitian quantum mechanics, there is no need to go to imaginary
time to find connecting trajectories. The relevant fixed points in the new
coordinates are $(\pm 1,0)$ and $(0,\pm 1)$. As can be seen from the phase
portrait for the Hamiltonian (12), Fig. 1, there are several instanton
trajectories connecting the various fixed points.

Figure 2: Imaginary and real parts of $\Lambda$ vs. the drive strength
$\lambda$ with $\gamma=1$ and $\Delta=0.1$. Best linear fit lines are plotted
over the data and have slopes of $-3.7$ and $-4.0$, correspondingly. The
insert is log-log plot of $\Lambda$ vs. $\Delta$ for $\lambda=3$. Best fits of
the imaginary and real parts of $\Lambda$ come with the slopes of $1.04(\pm
0.02)$ and $2.04(\pm 0.02)$.

According to Eqs. (4b), (4c) the imaginary and real parts of the smallest non-
zero eigenvalue $\Lambda$ are determined by the one instanton contributions to
the corresponding partition functions (the factor $T$ there originates from
the freedom to choose the instanton position along the time axis). They are
distinguished by the choice of boundary conditions Suplemental , illustrated
in Fig. 1. Notably the action along all such paths is same:

$iS_{\mathrm{inst}}=\frac{8\lambda}{\gamma}\int\text{d}XP=-\frac{4\lambda}{\gamma}.$
(13)

This leads to $\Lambda\sim\exp(iS_{\mathrm{inst}})=\exp(-4\lambda/\gamma)$.
Numerical calculations (see Ref. Suplemental  for details) of $\Lambda$, shown
in Fig. 2, are in a good agreement with this result. The pre-exponential
factors in the imaginary and real parts appear to have different dependence on
$\Delta$, with our numerics being consistent with:

$\mathrm{Im}\,\Lambda\sim\Delta\,e^{-4\lambda/\gamma};\qquad\mathrm{Re}\,\Lambda\sim\Delta^{2}\,e^{-4\lambda/\gamma}.$
(14)

The former represents the qubit precession frequency, while the latter – its
decoherence rate.

To conclude: we have developed the instanton calculus to evaluate non-
perturbative phenomena within the Lindbladian dynamics of the parametrically
driven oscillator with the non-linear dissipation. It exhibits two oscillatory
states, distinguished by their phases relative to the external drive, forming
a qubit basis. Such qubit is, however, subject to a precession and decoherence
due to the non-perturbative symmetry restoration. We found that the
semiclassical description of this phenomenon relies on the real-time instanton
trajectories in the complexified (and Keldysh doubled) phase space of the
oscillator. This put the corresponding decoherence mechanism in the
universality class of the quantum activation Dykman1 ; Dykman2 ; Dykman3 .

We are indebted to Mark Dykman for numerous discussions. We also acknowledge
helpful conversations with Victor Albert. This work was supported by the NSF
grant DMR-2037654.

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Supplemental Material for “Qubit Dephasing and Symmetry Restoration through
Real-Time Instantons”

##  I Equations of Motion Details

The equations of motion for of the Keldysh action written in terms of the
Keldysh-rotated fields are Hamilton’s equation in the Keldysh Hamiltonian $K$:

$i\partial_{t}\phi^{\mathrm{c}}=\partial_{\bar{\phi}^{\mathrm{q}}}K=\Delta\phi^{\mathrm{c}}+2\lambda\bar{\phi}^{\mathrm{c}}-i\frac{\gamma}{2}\big{(}\bar{\phi}^{\mathrm{c}}(\phi^{\mathrm{c}2}+\phi^{\mathrm{q}2})-2\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{q}}\phi^{\mathrm{c}}+4\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{c}}\phi^{\mathrm{q}}\big{)},$
(15a)
$-i\partial_{t}\bar{\phi}^{\mathrm{c}}=\partial_{\phi^{\mathrm{q}}}K=\Delta\bar{\phi}^{\mathrm{c}}+2\lambda\phi^{\mathrm{c}}+i\frac{\gamma}{2}\big{(}\phi^{\mathrm{c}}(\bar{\phi}^{\mathrm{c}2}+\bar{\phi}^{\mathrm{q}2})-2\phi^{\mathrm{q}}\bar{\phi}^{\mathrm{q}}\bar{\phi}^{\mathrm{c}}-4\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{c}}\bar{\phi}^{\mathrm{q}}\big{)},$
(15b)
$i\partial_{t}\phi^{\mathrm{q}}=\partial_{\bar{\phi}^{\mathrm{c}}}K=\Delta\phi^{\mathrm{q}}+2\lambda\bar{\phi}^{\mathrm{q}}-i\frac{\gamma}{2}\big{(}\bar{\phi}^{\mathrm{q}}(\phi^{\mathrm{q}2}+\phi^{\mathrm{c}2})-2\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{c}}\phi^{\mathrm{q}}+4\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{q}}\phi^{\mathrm{c}}\big{)},$
(15c)
$-i\partial_{t}\bar{\phi}^{\mathrm{q}}=\partial_{\phi^{\mathrm{c}}}K=\Delta\bar{\phi}^{\mathrm{q}}+2\lambda\phi^{\mathrm{q}}+i\frac{\gamma}{2}\big{(}\phi^{\mathrm{q}}(\bar{\phi}^{\mathrm{q}2}+\bar{\phi}^{\mathrm{c}2})-2\phi^{\mathrm{c}}\bar{\phi}^{\mathrm{c}}\bar{\phi}^{\mathrm{q}}-4\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{q}}\bar{\phi}^{\mathrm{c}}\big{)}.$
(15d)

Fixed points of the motion are determined by setting the left-hand side of
each expression to zero. There is a trivial fixed point in which all four
fields are zero. Additional fixed points can be determined by setting either
the $\mathrm{c}$ or the $\mathrm{q}$ fields equal to zero and solving the
resulting algebraic equations. For
$\phi^{\mathrm{q}}=0=\bar{\phi}^{\mathrm{q}}$, this is:

$\Delta\phi^{\mathrm{c}}+2\lambda\bar{\phi}^{\mathrm{c}}-i\frac{\gamma}{2}\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{c}2}=0=\Delta\bar{\phi}^{\mathrm{c}}+2\lambda\phi^{\mathrm{c}}-i\frac{\gamma}{2}\bar{\phi}^{\mathrm{c}2}\phi^{\mathrm{c}}.$
(16)

One may look for solutions when
$\phi^{\mathrm{c}}=\phi_{*}=\bar{\phi}^{\mathrm{c}*}$; there are two
solutions:

$\phi_{*}=\pm\sqrt{\frac{4\lambda}{\gamma}\sqrt{1-\Big{(}\frac{\Delta}{2\lambda}\Big{)}^{2}}}e^{i\varphi},\
\ \ \cos(2\varphi)=-\frac{\Delta}{2\lambda},\ \ \ \sin(2\varphi)<0.$ (17)

Alternatively, one may look for solutions
$-\phi^{\mathrm{c}*}=\chi_{*}=\bar{\phi}^{\mathrm{c}}$; there are also two
solutions:

$\chi_{*}=\pm\sqrt{\frac{4\lambda}{\gamma}\sqrt{1-\Big{(}\frac{\Delta}{2\lambda}\Big{)}^{2}}}e^{i\varphi},\
\ \ \cos(2\varphi)=\frac{\Delta}{2\lambda},\ \ \ \sin(2\varphi)>0.$ (18)

Analogously, there are also fixed points for
$\phi^{\mathrm{c}}=0=\bar{\phi}^{\mathrm{c}}$ and either
$\phi^{\mathrm{q}}=\phi_{*}=\bar{\phi}^{\mathrm{q}*}$ or
$-\phi^{\mathrm{q}*}=\chi_{*}=\bar{\phi}^{\mathrm{q}}$. In the $\Delta\to 0$
limit, $\phi_{*}=\pm\sqrt{2}\phi_{0}$ and $\chi_{*}=\pm\sqrt{2}\phi_{0}^{*}$
from (3). As a consequence, the locations of the fixed points of the
forward/backward fields $\phi^{\pm}$ match the steady-state coherent states.
The fixed points important for this problem are those for which
$\phi^{\mathrm{c}}=0=\bar{\phi}^{\mathrm{c}}$,
$-\phi^{\mathrm{q}*}=\chi_{*}=\bar{\phi}^{\mathrm{q}}$ and
$\phi^{\mathrm{q}}=0=\bar{\phi}^{\mathrm{q}}$,
$\phi^{\mathrm{c}}=\phi_{*}=\bar{\phi}^{\mathrm{c}*}$. When $\Delta=0$ all
four of these fixed points exist in the two-dimensional plain in the complex
Keldysh phase space on which $K+K^{*}=0=M$.

Note that one of the sets of fixed points occurs when
$\bar{\phi}^{\mathrm{q}}\neq\phi^{\mathrm{q}*}$, where $(\cdot)^{*}$ is
complex conjugate. Similarly, any non-static solutions to the EoM require some
complexification of the fields. This is apparent when comparing Eq.s (15a) and
(15b): these equations are not complex conjugates of one another when
$\bar{\phi}^{\mathrm{c},\mathrm{q}}=\phi^{\mathrm{c},\mathrm{q}*}$. As a
consequence, any non-static solutions will require
$\phi^{\mathrm{c},\mathrm{q}}$ and $\bar{\phi}^{\mathrm{c},\mathrm{q}}$ to be
regarded as independent complex variables. The four equations of motion from
Eq.s (15) and their four complex conjugates specify eight equations of motion
on an eight real-dimensional complexified phase space with the four
holomorphic and four anti-holomorphic coordinates
$(\phi^{\mathrm{c}},\phi^{\mathrm{q}},\bar{\phi}^{\mathrm{c}},\bar{\phi}^{\mathrm{q}},\phi^{\mathrm{c}*},\phi^{\mathrm{q}*},\bar{\phi}^{\mathrm{c}*},\bar{\phi}^{\mathrm{q}*})$.
These EoM are Hamilton’s equations generated by the Hamiltonian $K+K^{*}$. The
corresponding Poisson structure has a bracket with holomorphic and
antiholomorphic parts:

$\displaystyle\\{A,B\\}=-iA\big{(}\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\phi^{\mathrm{c}}}\vec{\partial}_{\bar{\phi}^{\mathrm{q}}}-\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\bar{\phi}^{\mathrm{q}}}\vec{\partial}_{\phi^{\mathrm{c}}}+\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\phi^{\mathrm{q}}}\vec{\partial}_{\bar{\phi}^{\mathrm{c}}}-\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\bar{\phi}^{\mathrm{c}}}\vec{\partial}_{\phi^{\mathrm{q}}}-\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\phi^{\mathrm{c}*}}\vec{\partial}_{\bar{\phi}^{\mathrm{q}*}}+\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\bar{\phi}^{\mathrm{q}*}}\vec{\partial}_{\phi^{\mathrm{c}*}}-\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\phi^{\mathrm{q}*}}\vec{\partial}_{\bar{\phi}^{\mathrm{c}*}}+\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\bar{\phi}^{\mathrm{c}*}}\vec{\partial}_{\phi^{\mathrm{q}*}}\big{)}B=0.$
(19)

The holomorphic part of the Poisson bracket kills any anti-holomorphic
function and vice-versa. As a consequence, holomorphic and anti-holomorphic
functions always commute. Moreover, two holomorphic functions $A$ and $B$
commute when

$\displaystyle
0=A\big{(}\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\phi^{\mathrm{c}}}\vec{\partial}_{\bar{\phi}^{\mathrm{q}}}-\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\bar{\phi}^{\mathrm{q}}}\vec{\partial}_{\phi^{\mathrm{c}}}+\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\phi^{\mathrm{q}}}\vec{\partial}_{\bar{\phi}^{\mathrm{c}}}-\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\bar{\phi}^{\mathrm{c}}}\vec{\partial}_{\phi^{\mathrm{q}}}\big{)}B.$
(20)

In such cases, there conjugates will also commute with one another, yielding a
set of four commuting functions $A,B,A^{*},B^{*}$. It is this fact that
explains why the complexified motion in the problem considered here is
integrable. There is an additional integral of motion $M$ commutes with $K$:

$M=\bigg{(}2\phi_{0}^{2}(\phi^{\mathrm{c}2}+\bar{\phi}^{\mathrm{c}2})+(\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{q}}-\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{c}}+2\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{c}})\Big{(}\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{c}}+\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{q}}+\frac{2i\Delta}{\gamma}\Big{)}\bigg{)}\\\
\bigg{(}2\phi_{0}^{2}(\phi^{\mathrm{q}2}+\bar{\phi}^{\mathrm{q}2})-(\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{q}}-\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{c}}-2\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{q}})\Big{(}\bar{\phi}^{\mathrm{q}}\phi^{\mathrm{c}}+\bar{\phi}^{\mathrm{c}}\phi^{\mathrm{q}}+\frac{2i\Delta}{\gamma}\Big{)}\bigg{)}.$
(21)

This implies the existence of a total of three additional mutually Poisson-
commuting functions $M,M^{*},K-K^{*}$ alongside the Hamiltonian $K+K^{*}$.

##  II Keldysh Path Integral Details

In the conventional construction of the Keldysh path integral, the object
$1=\operatorname{Tr}(\exp(T\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}})\rho)$
is converted into a functional integral by insertion of compete sets of
coherent states on both the left and the right of the density matrix, as
discussed in Diehl . Instead, we consider the trace of the time evolution
superoperator
$\operatorname{Tr}(\exp(T\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}}))$.
This trace can be expressed as the integral over the Hilbert-Schmidt operator
norm of coherent states:

$\operatorname{Tr}\big{(}\exp(T\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}})\big{)}=\int\frac{\text{d}^{2}\phi_{0}^{+}\text{d}^{2}\phi_{0}^{-}}{\pi^{2}}e^{-\bar{\phi}_{0}^{+}\phi_{0}^{+}-\bar{\phi}_{0}^{-}\phi_{0}^{-}}\operatorname{tr}\Big{(}\ket{\phi_{0}^{+}}\bra{\phi_{0}^{-}}^{\dagger}\exp(T\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}})\ket{\phi_{0}^{+}}\bra{\phi_{0}^{-}}\Big{)}.$
(22)

The trace in right expression can be brought into the form of a path integral
in a conventional way. The remaining insertions
$\ket{\phi_{0}^{+}}\bra{\phi_{0}^{-}}$ will set the boundary conditions at the
beginning and end of the time contour. At this step, the integral in front of
the operator trace in (22) can be absorbed into the path integral measure to
an integral over the loop space of the Keldysh phase space:

$\operatorname{Tr}\big{(}\exp(T\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}})\big{)}=\smashoperator[]{\int_{\phi^{\pm}(-T/2)=\phi^{\pm}(T/2)}^{}}\text{D}\phi^{\pm}\text{D}\bar{\phi}^{\pm}e^{iS}.$
(23)

The three character-valued partition functions
$\operatorname{Tr}(\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{\pm}\exp(T\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}}))$
and
$\operatorname{Tr}(\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{+}\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{P}}}}_{-}\exp(T\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{L}}}}))$
can be massaged into path integrals in the same way. The parity superoperators
will act to the left on the tensor of coherent states, replacing
$\phi_{0}^{\pm}$ with $-\phi_{0}^{\pm}$ on the left or on the right, changing
the boundary conditions on the fields. The corresponding path integral must be
taken over the space twisted loops on the Keldysh phase space. The boundary
conditions corresponding to the different twisting superoperators are:

$\mathcal{P}_{+}\ \longleftrightarrow\ \phi^{\pm}(-T/2)=\mp\phi^{\pm}(T/2)$
(24a) $\mathcal{P}_{-}\ \longleftrightarrow\
\phi^{\pm}(-T/2)=\pm\phi^{\pm}(T/2)$ (24b) $\mathcal{P}_{+}\mathcal{P}_{-}\
\longleftrightarrow\ \phi^{\pm}(-T/2)=-\phi^{\pm}(T/2)$ (24c)

For the rescaled Keldysh-rotated fields $X$ and $P$ from (10), these become:

$\mathcal{P}_{\pm}\ \longleftrightarrow\ X(-T/2)=\mp P(T/2),\ P(-T/2)=\mp
X(T/2)$ (25a) $\mathcal{P}_{+}\mathcal{P}_{-}\ \longleftrightarrow\
X(-T/2)=-X(T/2),\ P(-T/2)=-P(T/2)$ (25b)

Either of the first two can be used to isolate the imaginary part of the
smallest finite Lindbladian eigenvalue $\Lambda$; the third is used to isolate
the real part.

##  III Numerics

Numerical calculations were obtained by truncating and then vectorizing the
operator Hilbert space. The Hilbert space was was truncated to finite
dimension $N$; Boson creation/annihilation operators were replaced with
$N\times N$ matrices. Vectorization was done in the standard way by replacing
simple tensor operators by vectors
$\ket{\psi}\bra{\varphi}\to\ket{\psi}\otimes\ket{\varphi}$ and superoperators
by operators acting on different tensor factors $A\cdot B\to A\otimes
B^{\mathrm{T}}$. After vectorization, the Lindbladian superoperator is
transformed into an $N^{2}\times N^{2}$ matrix. Its spectrum is obtained by
exact diagonalization. All numerical data shown was obtained using $N=50$.
Variation in the data was unnoticeable when going to larger values of $N$ for
the range of parameters studied in the main text, meaning the results shown
are effectively independent of $N$. The low-lying part of the spectrum of the
diagonalized truncated Lindbladian had two zero eigenvalues and two
exponentially small eigenvalues; the plots in the main text show one of the
two small eigenvalues.