# One-component quantum mechanics and dynamical leakage-free paths

Jun Jing  Department of Physics, Zhejiang University, Hangzhou 310027,
Zhejiang, China  Lian-Ao Wu  Email address<EMAIL_ADDRESS>Department of
Physics, The Basque Country University (EHU/UPV), PO Box 644, 48080 Bilbao,
Spain and Ikerbasque, Basque Foundation for Science, 48011 Bilbao, Spain

###### Abstract

We derive an exact one-component equation of motion for the probability
amplitude of a target time-dependent state, and use the equation to
reformulate quantum dynamics and control for both closed and open systems.
Using the one-component equation, we show that an unexpected time-dependent
leakage-free path can be induced and we capture the essential quantity in
determining the effect of decoherence suppression. Our control protocol based
on the nonperturbative leakage elimination operator provides a unified
perspective connecting some subtle, popular and important concepts of quantum
control, such as dynamical decoupling, quantum Zeno effect, and adiabatic
passage. The resultant one-component equation will promise significant
advantages in both quantum mechanics and control.

##  I Introduction

Quantum mechanics is based on postulates, such as (a) quantum dynamics
governed by the Schrödinger equation, (b) quantum measurement WisemanBook  by
projection or postselection, and (c) boundary conditions of the system wave-
function or density matrix that induce interesting phenomenon such as
topological insulator LaughlinPRB81 . Most of the existing quantum control
protocols BrumerBook  involve with the first two postulates. Dynamical control
AgarwalPRL01 ; AgarwalPRA01  that is usually realized by laser pulse sequence,
includes, such as bang-bang (BB) control ViolaPRL99 ; KurizkiPRL04 ;
UhrigPRL09 ; LidarPRL10 , when the evolution operator of the system and
environment is intersected by unitary and instantaneous control operations;
the nonperturbative dynamical decoupling JingPRA13 ; JingSR13 ; JingSR14 ;
JWPRL15 , when the system Hamiltonian is modified by the time-dependent pulse
sequence; and the adiabatic passage BornZPhys28 ; WuzhaoyanPRA05 ;
ShapiroRMP07  based on the slowly-varying Hamiltonian and the avoiding of the
level crossings. Nonunitary quantum control relies on the projection methods,
such as quantum Zeno-like effect SudarshanJMP77 ; KurizkiNAT00 ; BrionPRA05 ,
when the total evolution operator of the system and environment is frequently
interrupted by an operation projecting the system to the desired state or
subspace. These methods seem to be dramatically different from each other.

In this paper, we show that the aforementioned control protocols and the
corresponding dynamics of the interested system can be uniformly formulated by
an exact one-component equation that properly traces the probability amplitude
of the target time-dependent state $|A(t)\rangle$ or $|A(t)\rangle\rangle$,
which is a vector living in a normal Hilbert space or a superoperator space
JingPRA14 ; LidarPRA05 . Using this equation, we can obtain a sufficient
condition for creating a leakage-free path (LFP), that conceptually
generalizes the conventional decoherence-free subspace KnightPRA97 ;
LidarBook03 . Also, this condition extends the existing conditions for
realizing dynamical decoupling.

The paper is outlined as follows. In Sec. II, we present the theoretical
framework of one-component quantum mechanics via the one-variable dynamical
equation based on the P-Q partition. In Sec. III, we propose a universal
control protocol towards LFP. We show in Sec. IV that the conventional bang-
bang control, quantum Zeno effect, and the adiabatic passage could be united
under the condition for LFP. In Sec. V, our control protocol is applied to
various pedagogical models, including the two-level system, the spin-spin-bath
model, the multiple level system under non-Markovian environment. And we
conclude our work in Sec. VI.

##  II One-component quantum mechanics

We consider a generic linear equation of motion,
$\partial_{t}\mathcal{X}=\mathcal{MX}$, exemplified by matrix representations
of the Schrödinger equation, the stochastic Schrödinger equation that combines
the Schrödinger equation of the entire system and quantum measurements on its
environment or bath, such as quantum-state-diffusion equation GardinerBook ;
DiosiPLA97 ; DiosiPRA98 ; StrunzPRL99 , the perturbative master equation
BreuerBook , and even a classical equation for linear-interacting harmonic
oscillators. We derive the one-component equation via the Feshbach P-Q
partitioning technique WuPRL09 ; Wuonline ; JingPRA12 . The $n$-dimensional
vector $\mathcal{X}$ and the $n\times n$ dynamic matrix $\mathcal{M}$ can be
partitioned as

$\mathcal{X}=\left[\begin{array}[]{c}P\\\ \hline\cr
Q\end{array}\right],\quad\mathcal{M}=\left(\begin{array}[]{c|c}h&R\\\
\hline\cr W&D\end{array}\right),$ (1)

where $P$ is the probability amplitude of the one-dimensional target state
$|A(t)\rangle$, and the $(n-1)$-dimensional $Q$ resides in the subspace
orthogonal to $|A(t)\rangle$.

| $\mathcal{M}$ | $\mathcal{X}$
---|---|---
Classical harmonic oscillator | $\left[\begin{array}[]{cc}0&1/m\\\ -m\omega^{2}&0\end{array}\right]$ | $\left(\begin{array}[]{c}q\\\ p\end{array}\right)$
Closed quantum system | $-iH$ | $|\psi\rangle$
Open quantum system I | $\mathcal{L}$ | $|\rho\rangle\rangle$
Open quantum system II | $H_{\rm eff}$ | $|\psi(z^{*})\rangle$
Table 1: A list of examples of linear dynamical equations in physics. (1)
Hamiltonian mechanics for classical harmonic oscillator model, where $p$ and
$q$ are generalized coordinates and momentum. (2) Schrödinger equation for
closed quantum system. (3) Liouville equation for open quantum system Lindblad
, where $\mathcal{L}$ is a Liouvillian super-operator, where $\rho$ is the
density matrix of the system. For a $d$-dimensional system,
$|\rho\rangle\rangle\equiv(\rho_{1},\rho_{2},\cdots,\rho_{d^{2}})^{\prime}$.
In this case, the dimensionality of $\mathcal{X}$ is $n=d^{2}$ and
$|A(t)\rangle\rangle\equiv P(t)|A(t)\rangle\langle A(t)|$ in normal
representation. (4) Stochastic Schrödinger equation for open quantum system,
e.g., the quantum-state-diffusion equation, where $H_{\rm eff}$ is the non-
Hermitian effective Hamiltonian and $|\psi(z^{*})\rangle\equiv\langle
z|\Psi\rangle$ is a state of system obtained by the inner product of a
stochastic environment state $|z\rangle$ and the whole system state
$|\Psi\rangle$ DiosiPLA97 ; DiosiPRA98 ; StrunzPRL99 .

Table 1 shows various physical realizations of the linear equation of motion.
Suppose that initially $P(0)=1$ and $Q(0)=0$. Equation (1) could be decomposed
into

$\partial_{t}P=hP+RQ,\quad\partial_{t}Q=WP+DQ.$ (2)

The formal solution is

$\partial_{t}P(t)=h(t)P(t)+\int_{0}^{t}dsg(t,s)P(s),$ (3)

where $g(t,s)=R(t)G(t,s)W(s)$ with a propagator
$G(t,s)=\mathcal{T}_{\leftarrow}\\{\exp[\int_{s}^{t}ds^{\prime}D(s^{\prime})]\\}$.
Alternatively, we can set $P(t)=p(t)e^{\int_{0}^{t}dsh(s)}$ and obtain a more
compact formal solution

$\partial_{t}p(t)=e^{iC(t)}\int_{0}^{t}dse^{-iC(s)}g(t,s)p(s)=\int_{0}^{t}dsg^{\prime}(t,s)p(s),$
(4)

where $C(t)\equiv i\int_{0}^{t}dsh(s)$ and
$g^{\prime}(t,s)\equiv\exp[\int_{s}^{t}h(s^{\prime})ds^{\prime}]g(t,s)$. This
one-component integro-differential equation is the core result of this work
and will serve as a powerful tool in studying both dynamics and control of a
given target state. If $g^{\prime}(t,s)=-k^{2}$ with $k$ a constant number,
then we recover the equation of motion for a harmonic oscillator, i.e.,
$\ddot{p}+k^{2}p=0$. And $p(t)=\cos(kt)$ when $p(0)=1$. If
$g^{\prime}(t,s)=-2\lambda\delta(t-s)$, then we find the ideal Markovian
dynamics with $\dot{p}(t)=-\lambda p(t)$. And then $p(t)=e^{-\lambda t}p(0)$,
implying one can not keep the system on the target state.

To walk on a desired target path $|A(t)\rangle$, we need to eliminate the
leakage between $P$ and $Q$ parts, which is equivalently to have a vanishing
integral

$\int_{0}^{t}dse^{-iC(s)}g(t,s)p(s)=0.$ (5)

In the case of quantum mechanics, $h(t)$ is a purely imaginary number so that
$C(t)$ is a real function and then either real or imaginary part of
$e^{-iC(s)}$ is a time-dependent function oscillating between $-1$ to $1$. In
what follows, we will focus on quantum mechanical system and show that the
vanishing of the integral over $e^{-iC(s)}g(t,s)p(s)$ leads to a time-
dependent LFP. A trivial LFP emerges in a special case when $g(t,s)=0$. Note
that $g(t,s)=0$ might hold even if none of $R(t)$, $G(t,s)$, and $W(s)$ is
vanishing. In general, LFP is realized when $h(t)$ can be so manipulated that
a rapid-oscillating $e^{-iC(s)}$ could cancel the effect raised by smoother
functions of both $g(t,s)$ and $p(s)$ through

$\int_{0}^{t}dse^{-iC(s)}g(t,s)p(s)\approx\sum_{k=0}^{n}(-1)^{k}g(t,k\tau)p(k\tau)\tau\rightarrow
0,$ (6)

where $\tau=t/n$ with $n\rightarrow\infty$. Note here smooth means
$g(t,k\tau)\approx g(t,(k+1)\tau)$ and $p(k\tau)\approx p((k+1)\tau)$. This
result is supported by the Riemann$-$Lebesgue lemma JWPRL15  provided that the
characteristic frequency for the function $e^{-iC(s)}$ is larger than the cut-
off frequency of $g(t,s)p(s)$.

##  III A universal quantum control protocol

Consider an either closed or open quantum system $\mathcal{S}$, whose Hilbert
space is spanned by a set of time-independent or time-dependent bases
$|\phi_{m}\rangle$ ($m=0,1,2,\cdots,n-1$) and $|A(0)\rangle=|\phi_{0}\rangle$.
Now we aim at control of the system to evolve along a desired quantum path
characterized by a unitary transformation $\mathcal{U}(t)$ in the system
space, i.e., holding the system on the path
$|A(t)\rangle=\mathcal{U}(t)|\phi_{0}\rangle$. Thus the total Hamiltonian
$H_{\rm tot}$ can be expressed in terms of the time-independent or time-
dependent basis states
$|\tilde{\phi}_{m}\rangle=\mathcal{U}(t)|\phi_{m}\rangle$, where
$|\tilde{\phi}_{0}\rangle=|A(t)\rangle$. Under the Schrödinger equation,
$\partial_{t}|\psi\rangle=-iH_{\rm tot}|\psi\rangle$, we have

$h(t)=-i\langle\tilde{\phi}_{0}|\tilde{H}_{\rm
tot}|\tilde{\phi}_{0}\rangle\equiv-i\langle\tilde{\phi}_{0}|[\mathcal{U}(t)H_{\rm
tot}\mathcal{U}^{\dagger}(t)+i\dot{\mathcal{U}}(t)\mathcal{U}^{\dagger}(t)]|\tilde{\phi}_{0}\rangle,$
(7)

in terms of P-Q partitioning given in Eq. (1). Note $\mathcal{U}(t)$ is
irrelevant to $H_{\rm tot}$. For example in the scenario of open quantum
system, the whole system could be decomposed into the system part that we are
interested and the remainder part that are called environment. In the rotating
frame with respect to the environment Hamiltonian (assumed to be time-
independent), i.e., $\mathcal{U}(t)=\exp(-iH_{\rm env}t)$, we always have

$\tilde{H}_{\rm tot}=H_{\rm sys}(t)+\sum_{k}S_{k}(t)B_{k}(t),$ (8)

where $S_{k}(t)$ and $B_{k}(t)$ are Hermition operators in the space of system
and environment, respectively. It turns out immediately that if
$\langle\tilde{\phi}_{0}|S_{k}(t)|\tilde{\phi}_{0}\rangle=0$ holds for each
$k$, then $h(t)=-i\langle\tilde{\phi}_{0}|H_{\rm
sys}(t)|\tilde{\phi}_{0}\rangle$, which is irrespective to the operators and
the size of environment. The well-known instance is the conventional leakage-
free subspace for collective dephasing and dissipation DFSLidar . It implies
therefore that $h(t)$ could be under a full control without unpractically
invoking control over the environment and the remainder of the system space.

Under the bases $\\{|\phi_{n}(t)\rangle\\}$, the rotating representation
Hamiltonian can always be partitioned into a similar form as $\mathcal{M}$ in
Eq. (1). Suppose $\tilde{H}_{\rm tot}=H_{d}+L$, where the block-diagonal part
$H_{d}=h\oplus D$ and $L$ is the block-off-diagonal part consisted by $R$ and
$W$. To maintain the system in the state of $|A(t)\rangle$, i.e., to create a
rapid-oscillating exponential function $e^{-iC(t)}$ by manipulating $h(t)$, a
leakage elimination operation (LEO) in rotating framework or a rotating LEO
WML

$\tilde{R}_{L}=c(t)\left[|\tilde{\phi}_{0}\rangle\langle\tilde{\phi}_{0}|-\sum_{n>0}|\tilde{\phi}_{n}\rangle\langle\tilde{\phi}_{n}|\right]=c(t)\left[2|\tilde{\phi}_{0}\rangle\langle\tilde{\phi}_{0}|-\mathcal{I}\right]$
(9)

has been introduced to cancel the off-diagonal term (in charge of leakage) $L$
in the time evolution. Here $c(t)$ is the control function, which can be
absorbed into $h(t)$ when Eq. (4) is considered in the control protocol. It is
clear to see $\\{\tilde{R}_{L},L\\}=0$ and $[\tilde{R}_{L},H_{d}]=0$, so that
the LEO serves to parity-kick out the leakage $L$. Ideally, one can easily
prove that if $c(t)\propto\delta(t-n\tau)$ at given times $n\tau$
$(n=1,2,\cdots)$, then

$e^{-i\tilde{H}_{\rm
tot}[(n+1)\tau]\tau}\tilde{R}^{\dagger}_{L}e^{-i\tilde{H}_{\rm
tot}(n\tau)\tau}\tilde{R}_{L}\approx
e^{-i[H_{d}((n+1)\tau)+H_{d}(n\tau)]\tau}$ (10)

when $\tau\rightarrow 0$ and $t\approx n\tau$. Nonperturbatively, it is shown
that the nonideal pulse $c(t)$ JWPRL15  does also allow to achieve the same
result. Furthermore, the general leakages, such as $\sum_{k}S_{k}(t)B_{k}(t)$
in the open quantum system, can be eliminated by $\tilde{R}_{L}$.

##  IV Discussion

###  IV.1 Bang-bang control and quantum Zeno effect

Many existing protocols targeting on decoherence-suppression or leakage
elimination are found to be subsets of the control framework we proposed
through the one-dimensional dynamical equation (4). Moreover, our control
strategy is state-independent.

That could be interpreted using a model of a two-level system subjected to
unwanted disturbances from the uncontrollable Hilbert space orthogonal to that
of the $P$-part in Eq. (1). To cancel the effect from the leakage Hamiltonian
$L=\hat{X}(t)B_{x}(t)+\hat{Y}(t)B_{y}(t)$, where $\hat{X}(t)$ and $\hat{Y}(t)$
can flip the desired state $|A(t)\rangle$ into an orthogonal state, e.g.,
$|A^{\perp}(t)\rangle$, and $B_{x,y}(t)$ is an arbitrary environmental
operator. By Trotter formula and using Eq. (8), in a short time interval
$\delta_{1}$, the system approximately evolves into
$|A(\delta_{1})\rangle-i\delta_{1}|A^{\perp}(\delta_{1})\rangle$, where
$|A(\delta_{1})\rangle=\mathcal{U}(\delta_{1})|A(0)\rangle=e^{-iH_{\rm
sys}(0)\delta_{1}}|A(0)\rangle$. In order to stabilize the passage of the
system along the desired path, one has to insert a BB control pulse indicated
by $\hat{Z}(t)$ to the system evolution after the period of free-evolution
$\delta_{1}$. Similar to $\hat{X}(t)$ and $\hat{Y}(t)$ given before,
$\hat{Z}(t)$ is not necessary the Pauli matrix along the $z$-direction. It is
merely required that these three operators constitute a set of generators of
$SU(2)$. For instance, $\hat{Z}(t)$ can be chosen as
$|A(\delta_{1})\rangle\langle
A(\delta_{1})|-|A^{\perp}(\delta_{1})\rangle\langle A^{\perp}(\delta_{1})|$.
To the first-order perturbation, the state of the system is now approximated
as $|A(\delta_{1})\rangle$ by virtue of $\\{L,\hat{Z}(t)\\}=0$. This process
could be repeated many times until to the desired moment
$t=\sum_{j}\delta_{j}$. BB control always works as long as each interval
$\delta_{j}$ is short enough. However, it amounts to taking the zero-order
perturbation for an effective Hamiltonian $H_{\rm eff}=c\hat{Z}+\tilde{H}_{\rm
tot}$, where $c\delta=\pi/2$ and the system-bath Hamiltonian is effectively
turned off when the pulses are applied to the system. Under this condition,
the control-strength $c$ has to approach infinity when $\delta$ goes to zero,
which gives rise to both inconsistency in theory JingPRA13  and
inaccessibility in experiment. The underlying mechanism of BB control has been
partially justified by the nonperturbative control JWPRL15  when the integral
of control pulses over time domain is sufficient large to enhance the survival
probability of the system under control. It is a solution to attain a high-
frequent exponential function $e^{-iC(s)}$. In the line of using longitudinal
control to cancel the transversal error caused by the flip-flop Hamiltonian
$L$, it is straightforward to extend the above protocol into a multi-level
one.

Different from the BB control, quantum Zeno effect takes on the projection
strategy, instead of an ideal unitary transformation, interpolating the
unitary evolution of the whole system. After each period $\delta_{j}$ of free
evolution, the system is projected upon $|A(\delta_{j})\rangle\langle
A(\delta_{j})|$ canceling all the errors caused by the system-environment
interaction at the cost of a nondeterministic postselection Zeno . Similar to
BB control, this protocol also does not depend on the sequence arrangement of
the projection but the projection frequency. In reality, a particular
realization of $\delta_{j}$ could be random, noisy and even chaos JingPRA13 ;
JingSR13 ; JingSR14 ; JWPRL15 .

###  IV.2 Adiabaticity induced by control

Through maintaining the system along the target path of $|A(t)\rangle$, our
control-activated LFP elevates the condition on achieving adiabatic passage of
the system, where the quantum channel is realized through the time-dependent
quantum eigenstate. Now the Universal quantum control protocol is applied in
the following way. We first construct a time-dependent Hamiltonian for a
nondegenerate system $H(t)=\sum_{n}E_{n}(t)|E_{n}(t)\rangle\langle E_{n}(t)|$
and let $|A(t)\rangle=|E_{0}(t)\rangle$ in lab frame up (at most) to a
geometrical phase, where $E_{n}(t)$ and $|E_{n}(t)\rangle$ are instantaneous
eigenvalues and eigenvectors of $H(t)$, respectively. To obtain the one-
component equation of motion for the amplitude of $|A(t)\rangle$, it is
instructive to rewrite the Hamiltonian in the adiabatic frame by
$\mathcal{U}=\sum_{n}e^{i\theta_{n}(t)}|E_{n}(0)\rangle\langle E_{n}(t)|$ with
$\theta_{n}(t)\equiv\int_{0}^{t}dsE_{n}(s)$. We thus have

$\displaystyle\tilde{H}(t)=\mathcal{U}H(t)\mathcal{U}^{\dagger}+i\dot{\mathcal{U}}\mathcal{U}^{\dagger}$
$\displaystyle=\sum_{n,m,k}e^{i\theta_{n}(t)}|E_{n}(0)\rangle\langle
E_{n}(t)|\cdot E_{m}(t)|E_{m}(t)\rangle\langle E_{m}(t)|\cdot
e^{-i\theta_{k}(t)}|E_{k}(t)\rangle\langle E_{k}(0)|$ $\displaystyle+$
$\displaystyle
i\sum_{m,n}\left[i\dot{\theta}_{m}(t)e^{i\theta_{m}(t)}|E_{m}(0)\rangle\langle
E_{m}(t)|+e^{i\theta_{m}(t)}|E_{m}(0)\rangle\langle\dot{E}_{m}(t)|\right]\cdot
e^{-i\theta_{n}(t)}|E_{n}(t)\rangle\langle E_{n}(0)|$ $\displaystyle=$
$\displaystyle\sum_{n}E_{n}(0)|E_{n}(0)\rangle\langle
E_{n}(0)|-\sum_{n}E_{n}(0)|E_{n}(0)\rangle\langle E_{n}(0)|$
$\displaystyle+i\sum_{m,n}e^{i\theta_{m}(t)-i\theta_{n}(t)}\langle\dot{E}_{m}(t)|E_{n}(t)\rangle|E_{m}(0)\rangle\langle
E_{n}(0)|$ $\displaystyle=$ $\displaystyle-i\sum_{n}\langle
E_{n}(t)|\dot{E}_{n}(t)\rangle|E_{n}(0)\rangle\langle E_{n}(0)|$
$\displaystyle-i\sum_{m\neq n}e^{i\theta_{m}(t)-i\theta_{n}(t)}\frac{\langle
E_{m}(t)|\dot{H}(t)|E_{n}(t)\rangle}{E_{n}(t)-E_{m}(t)}|E_{m}(0)\rangle\langle
E_{n}(0)|.$

So that in language of Eq. (1),
$\mathcal{X}(t)=[\psi_{0}(t),\psi_{1}(t),\psi_{2}(t),\cdots]^{\prime}$, and

$\mathcal{M}_{m\neq n}=-e^{-i[\theta_{n}(t)-\theta_{m}(t)]}\frac{\langle
E_{m}(t)|\dot{H}(t)|E_{n}(t)\rangle}{E_{n}(t)-E_{m}(t)}.$ (11)

In particular, $P(t)=\psi_{0}(t)$ and $h(t)=-\langle
E_{0}(t)|\dot{E}_{0}(t)\rangle$. In the case of unitary transformation into
the adiabatic frame, the rotating LEO in Eq. (9) reads
$\tilde{R}_{L}=c(t)[2|E_{0}(0)\rangle\langle E_{0}(0)|-\mathcal{I}]$ and one
can inversely derive the LEO in the lab frame as

$\displaystyle R_{L}=\mathcal{U}^{\dagger}\tilde{R}_{L}\mathcal{U}$
$\displaystyle=$ $\displaystyle
c(t)\left[2\sum_{m,n}e^{-i\theta_{m}(t)}|E_{m}(t)\rangle\langle
E_{m}(0)|\cdot|E_{0}(0)\rangle\langle E_{0}(0)|\cdot
e^{i\theta_{n}(t)}|E_{n}(0)\rangle\langle E_{n}(t)|-\mathcal{I}\right]$
$\displaystyle=$ $\displaystyle c(t)\left[2|E_{0}(t)\rangle\langle
E_{0}(t)|-\mathcal{I}\right]$ $\displaystyle=$ $\displaystyle
c(t)\left\\{\left[2\sum_{m,n}\langle E_{m}(0)|E_{0}(t)\rangle\langle
E_{0}(t)|E_{n}(0)\rangle|E_{m}(0)\rangle\langle
E_{n}(0)|\right]-\mathcal{I}\right\\}.$

Applying LEO into controlling the system, the traditional condition for
adiabaticity $|\langle E_{m}|\dot{E}_{n}\rangle|\ll|E_{n}-E_{m}|$ could be
generalized to the vanishing accumulation of the product of $e^{-iC(s)}$ and
$g(t,s)p(s)$ in Eq. (4). With sufficiently fast-oscillating exponential
function $e^{-iC(s)}$ via manipulating $h(t)\Rightarrow h(t)+c(t)$, i.e.,
$\mathcal{M}\Rightarrow\mathcal{M}+(-i\tilde{R}_{L})$, the adiabatic passage
could be realized by rescaling the energy difference between the target state
$|A(t)\rangle$ and the other eigenstates of the system over time, and this
control does not necessary boost this energy gap on time average. Our theorem
is thus consistent with the original formalism of the adiabatic theorem and
relaxes the slowly-varying condition. Additionally, it avoids the practical
difficulty in some accelerated adiabatic passage, such as transitionless
quantum driving BerryJPA09  that requires to add a counter-adiabatic term into
the original Hamiltonian (see also Wang18  for the LEO in an experimental
framework).

Beyond the adiabatic passage, any desired time-dependent LFP could be
transformed into a time-independent one in the rotating frame after performing
a proper unitary transformation. As long as the manipulation is highly
frequent, there are unlimited numbers of strategies through which the LFP can
be achieved since the control is fully determined by $e^{-iC(t)}$, rather than
$C(t)\equiv i\int_{0}^{t}dsh(s)$ (a large amplitude of this integral surely
supports a high-frequent $e^{-iC(t)}$, but clearly it does not exhaust all the
solutions) and even the details of the shape and arrangement of these pulses
$c(t)$ presented in $h(t)$ under control. For example, if
$D(t)=a(t)|W\rangle\langle W|+\sum_{j}b_{j}(t)|W^{\perp}_{j}\rangle\langle
W^{\perp}_{j}|$ [assuming both $W$ and $R$ in Eq. (1) are time-independent],
where $\langle W|W^{\perp}_{j}\rangle=0$, i.e., $W$ is an eigenstate of $D$
and the eigenvalue is $a(t)$. Then in the case when $h^{\prime}\equiv h-a$ is
a constant (pure imaginary) number, it is found

$p(t)=\frac{-h^{\prime}+\Delta}{2\Delta}e^{\frac{-h^{\prime}-\Delta}{2}t}+\frac{h^{\prime}+\Delta}{2\Delta}e^{\frac{-h^{\prime}+\Delta}{2}t},$
(12)

where $\Delta\equiv\sqrt{h^{\prime 2}+4\langle R|W\rangle^{2}}$. When
$|h^{\prime}|$ could be so tuned that $|h^{\prime}|t=2k\pi$, $k$ is an
integer, the two oscillation frequencies in the expression of $p(t)$ will
become sufficiently close to each other. Then $|p(t)|$ could be maintained as
unit and a LFP emerges. It is consistent with and extends the previous result
that $h(t)$ with a sufficient large magnitude JWPRL15  overwhelming $2|\langle
R|W\rangle|$ will suppress the decoherence.

##  V Applications

###  V.1 A two-level system in accelerated adiabatic passage

Following the conventions given by Sec. IV.2, a two-level-system state can be
written as
$|\psi(t)\rangle=\psi_{0}(t)e^{-i\theta_{0}(t)}|E_{0}(t)\rangle+\psi_{1}(t)e^{-i\theta_{1}(t)}|E_{1}(t)\rangle$
in lab frame, where $|E_{n}(t)\rangle$’s, $n=0,1$, are instantaneous
eigenstates. Choosing $|A(t)\rangle=|E_{0}(t)\rangle$ and suppose
$E_{1}=-E_{0}=E/2$ without loss of generality, we have

$\mathcal{M}=\left(\begin{array}[]{c|c}h=-\langle
E_{0}|\dot{E}_{0}\rangle&R=-\langle E_{0}|\dot{E}_{1}\rangle e^{i\theta}\\\
\hline\cr W=-\langle E_{1}|\dot{E}_{0}\rangle e^{-i\theta}&D=-\langle
E_{1}|\dot{E}_{1}\rangle\end{array}\right)$ (13)

in the adiabatic frame, where
$\theta(t)\equiv\int_{0}^{t}dsE(s)=\theta_{1}(t)-\theta_{0}(t)$. Setting
$p(t)=\psi_{0}(t)B(t)$ with $B(t)\equiv e^{\int_{0}^{t}ds\langle
E_{0}(s)|\dot{E}_{0}(s)\rangle}$, it yields a one-component dynamical equation
as Eq. (4):

$\partial_{t}p(t)=e^{B(t)}\int_{0}^{t}dse^{-B(s)}g(t,s)p(s)=\int_{0}^{t}dsg^{\prime}(t,s)p(s),$
(14)

where

$\displaystyle g^{\prime}(t,s)$ $\displaystyle=$ $\displaystyle\langle
E_{0}(t)|\dot{E}_{1}(t)\rangle\langle E_{1}(s)|\dot{E}_{0}(s)\rangle$
$\displaystyle\times$
$\displaystyle\exp\left[\int_{s}^{t}ds^{\prime}\left(iE(s^{\prime})+\langle
E_{0}(s^{\prime})|\dot{E}_{0}(s^{\prime})\rangle-\langle
E_{1}(s^{\prime})|\dot{E}_{1}(s^{\prime})\rangle\right)\right].$

For the control $H(t)\rightarrow[1+c(t)]H(t)$, in which the LEO reads

$\tilde{R}_{L}(t)=\frac{c(t)E(t)}{2}\left(\begin{array}[]{cc}-1&0\\\
0&1\end{array}\right)=c(t)H(t).$ (15)

It will change the eigenvalues but not the eigenvectors. Once the frequency of
the exponential function $e^{i\int_{s}^{t}ds^{\prime}E(s^{\prime})}$ is
effectively enhanced by $c(t)$, the integral in Eq. (14) could vanish and then
this two-level system would walk in an accelerated adiabatic path.

###  V.2 A spin-spin-bath model

Consider an electron spin coupled to a nuclear spin-$1/2$ bath through
hyperfine interaction hyperfine :

$H_{\rm
tot}=\Omega(t)S^{z}+\sum_{n}\omega_{n}I_{n}^{z}+\sum_{n\alpha}J_{n}^{\alpha}(t)S^{\alpha}I_{n}^{\alpha}+\sum_{nm\alpha}B_{nm}^{\alpha}I_{n}^{\alpha}I_{m}^{\alpha},$
(16)

where $\alpha=x,y,z$. In the single-exciton subspace and in the absence of the
last term representing the inner coupling between the nuclear spins, we have

$\displaystyle\mathcal{M}=-iH_{\rm tot}$ $\displaystyle=$
$\displaystyle-i\left(\begin{array}[]{c|ccc}\Omega(t)-\sum_{n}\frac{J_{n}^{z}(t)}{2}&J_{1}^{\bot}(t)&J_{2}^{\bot}(t)&\cdots\\\
\hline\cr J_{1}^{\bot}(t)&\omega_{1}-\frac{J_{1}^{z}(t)}{2}&0&0\\\
J_{2}^{\bot}(t)&0&\omega_{2}-\frac{J_{2}^{z}(t)}{2}&0\\\
\cdots&\cdots&\cdots&\cdots\end{array}\right),$

where $J_{n}^{\bot}(t)\equiv J_{n}^{x}(t)+J_{n}^{y}(t)$. Now the $D$ matrix
presents in the diagonal form. So that in the framework of our one-component
quantum mechanics [see P-Q partitioning in Eq. (1) and the dynamical equation
(4)], when the target state is chosen as the electron in the excited state and
the nuclear spins in the ground state $|100\cdots\rangle$,
$g(t,s)=\sum_{n}J_{n}^{\bot}(t)J_{n}^{\bot}(s)e^{-i\omega_{n}(t-s)-\int_{s}^{t}ds^{\prime}J_{n}^{z}(s^{\prime})/2}$,
and
$C(s)=i\int_{0}^{s}ds^{\prime}[\Omega(s^{\prime})-\sum_{n}J_{n}^{z}(s^{\prime})/2]$.
One can then manipulate $\Omega(t)$ to hold the population in the
$P$-subspace. The control efficiency is relevant to the longitudinal
Overhauser field $J_{n}^{z}(t)$ spinde . In the presence of the inner coupling
terms in nuclear spin bath, $h$, $W$ and $R$ do not vary while the diagonal
terms of $D$ become $D_{nn}=\omega_{n}-\frac{J_{n}^{z}(t)}{2}-\sum_{m\neq
n}\frac{B_{nm}^{z}}{2}$ and the off-diagonal terms become
$D_{nm}=B_{nm}^{x}+B_{nm}^{y}$. In this case, a protocol is to adjust the
transversal Overhauser field (flip-flop term) $J_{n}^{\bot}(t)$ until
$D|W\rangle=a|W\rangle$. Then one would achieve a similar result as Eq. (12),
so the electron spin is ensured to be maintained as the target state. It means
that partial control on both system and bath promises to defeat decoherence.

###  V.3 A multi-level system under non-Markovian environment

For an $n$-level atom coupled to a non-Markovian bosonic bath whose
correlation function is $\beta(t,s)$, we consider a genuine multilevel atomic
system $H_{\rm sys}=\sum_{j=0}^{n-1}E_{j}(t)|j\rangle\langle j|$ and the
system coupling operator, $S=\sum_{j=0}^{n-1}\kappa_{j}|j\rangle\langle 0|$.
Using the non-Markovian quantum trajectory method DiosiPLA97 ; DiosiPRA98 ;
StrunzPRL99 , one can get an exact quantum-state-diffusion equation for this
model:

$\partial_{t}\psi_{t}(z^{*})=\mathcal{M}\psi_{t}(z^{*}),\quad\mathcal{M}=-iH_{\rm
sys}+Sz_{t}^{*}-S^{\dagger}\bar{O}(t),$ (18)

where $\bar{O}(t)=\sum_{j=1}^{n-1}F_{j}(t)|j\rangle\langle 0|$. Coefficient
functions $F_{j}(t)\equiv\int_{0}^{t}ds\beta(t,s)f_{j}(t,s)$ satisfy
$f_{j}(s,s)=\kappa_{j}$ and
$\partial_{t}f_{j}(t,s)=[i(E_{0}-E_{j})+\sum_{k=1}^{n-1}\kappa_{k}^{*}F_{k}(t)]f_{j}(t,s)$.
The target state can be arbitrarily chosen as
$|A\rangle=\sum_{j=0}^{n-1}a_{j}|j\rangle$, where
$\sum_{j=0}^{n-1}|a_{j}|^{2}=1$. The fidelity $\mathcal{F}\equiv\langle
A|\rho|A\rangle=M[\langle A|\psi(z^{*})\rangle\langle\psi(z^{*})|A\rangle]$,
where $M[\cdot]$ means ensemble average, measuring the control efficiency, is
then found to depend on the population rather than the amplitude of the
initial state:

$\displaystyle\mathcal{F}(t)$ $\displaystyle=$
$\displaystyle|a_{0}|^{4}e^{-\sum_{j=1}^{n-1}[\bar{F}_{j}(t)+\bar{F}^{*}_{j}(t)]}+\sum_{j,k\geq
1,k\neq j}|a_{j}|^{2}|a_{k}|^{2}$ $\displaystyle+$
$\displaystyle|a_{0}|^{2}\sum_{k=1}^{n-1}|a_{k}|^{2}\left[e^{-\sum_{j=1}^{n-1}\bar{F}_{j}(t)}+e^{-\sum_{j=1}^{n-1}\bar{F}^{*}_{j}(t)}\right]$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n-1}|a_{j}|^{2}\left\\{|a_{j}|^{2}+\int_{0}^{t}ds|a_{0}|^{2}[F_{j}(s)+F_{j}^{*}(s)]e^{-\sum_{j=1}^{n-1}[\bar{F}_{j}(s)+\bar{F}^{*}_{j}(s)]}\right\\},$

where $\bar{F}_{j}(t)\equiv\int_{0}^{t}dsF_{j}(s)$. It implies that the
control fidelity depends only on the populations.

Figure 1: Fidelity dynamics of a $10$-level atomic system under control of
regular and noisy pulse sequences. The target state
$|A\rangle=\sum_{j=0}^{n-1}a_{j}|j\rangle$ is such chosen as
$|a_{j}|^{2}=1/10$. The parameters are chosen as $\kappa_{j}=0.1\omega$,
$E_{j\neq 0}=0$ and $E_{0}(t)=\omega$. For regular rectangular pulse,
$|c(t)|=\Phi/\Delta$ for $m\tau-\Delta\leq t\leq m\tau$, where $m\geq 1$ is an
integer; otherwise, $|c(t)|=0$. $\Phi$, $\Delta$, and $\tau$ are the strength,
duration and period of pulse, respectively. Here $\tau=0.02\omega t$ and
$\Delta/\tau=0.5$. For noisy pulse,
$|c(t)|\rightarrow|c(t)|[1+G\mathcal{N}(t)]$ where $G$ (here $G=50\%$) is a
dimensionless parameter measuring the white noise strength
$\mathcal{N}(t)\in(-1,1)$. The environment correlation function is taken as
$\beta(t,s)=\frac{\gamma}{2}e^{-\gamma|t-s|}$, where $\gamma$ is inversely
proportional to the environmental memory time. A smaller $\gamma$ indicates a
stronger non-Markovian environment.

In Figs. 1 and 1, we plot the fidelity of a multilevel system under control of
regular and noisy sequences of pulse in environments with different
environmental memory parameters. It is shown that the fidelity is enhanced
with increasing pulse strength $\Phi$, which is linearly proportional to the
absolute value of control integral $C(t)$. Note the average of control
integral is kept vanishing for the sign of $c(t)$ is switched periodically
(see the lines with no markers) or randomly (see the lines with markers). One
can find that the effect of regular control is nearly the same as the noisy
sequence, which does not change $e^{-iC(t)}$ in terms of ensemble average and
long time simulation. Comparing Figs. 1 and 1, it is reliable to estimate that
any control will gradually lose its effect when the environment becomes more
and more memoryless (note $\gamma\rightarrow\infty$ indicates a Markovian
environment). The LFP of Figs. 1 and 1 is represented by $|a_{j}|^{2}=1/n$,
while for an arbitrary target state, $\mathcal{M}$ in our control protocol can
be found by a corresponding rotating as
$\mathcal{U}|A\rangle=|\tilde{A}\rangle$.

##  VI Conclusion

In this work, we set up a general framework that allows one to follow and
control a quantum system, open or closed, along a desired leakage-free path,
presented by a one-dimensional dynamical equation addressing the time-
dependent target state. A common mechanism subtly underlying existing quantum
control protocols, including BB control, Zeno effect and adiabatic passage, is
brought to light as a general condition for dynamical leakage-free paths. As
long as the exponential function of a phase provided by the integral or
cumulation over the control pulse is featured with a sufficient large
frequency, the LFP can be realized by leakage elimination operation. As an
active protocol that is not confined by the structure of the total
Hamiltonian, it would be versatile to accommodate arbitrary linear non-
Markovian equations of motion for open quantum systems. Moreover, upon proper
control over system Hamiltonian or partial control over system-environment
interaction Hamiltonian, the target state $|A\rangle$ or $|A\rangle\rangle$
could be extended into a more general LFP in multi-dimensional space that is
able to perform more quantum processing, such as the non-Abelian geometric
quantum gate in degenerate subspace, the shortcut to quantum state
transmission Wang16 , the speeding up holonomic quantum computation in
decoherence-free subspace Pavel16 , the quantum search algorithm and almost-
exact state transfer in non-Markovian environments Ren20 ; Wang20 ; M20 .

## Acknowledgments

We acknowledge grant support from the National Science Foundation of China
(Grants No. 11974311 and No. U1801661), the Basque Government IT986-16,
Spanish Government PGC2018-095113-B-I00 (MCIU/AEI/FEDER, UE).

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