# Dirac equation in exotic spacetimes

Javier Faba García  Departamento de Física Teórica, Universidad Complutense de
Madrid, Plaza de Ciencias, 1 28040 Madrid, Spain  Carlos Sabín  Instituto de
Física Fundamental, CSIC, Serrano, 113-bis, 28006 Madrid, Spain

###### Abstract

We find solutions of the Dirac equation in curved spacetime. In particular, we
consider 1+1 dimensional sections of several exotic metrics: the Alcubierre
metric, which describes a scenario that allows faster-than-light (FTL)
velocity; the Gödel metric, that describes a universe containing closed
timelike curves (CTC); and the Kerr metric, which corresponds to the metric of
a rotating pointlike source, for instance a rotating black hole. Moreover, we
also show that the techniques that we use in these cases can be extended to
nonstatic metrics.

###### pacs:

Valid PACS appear here

##  I Introduction

In 1928, Paul Dirac formulated a special relativistic version of the wave
equation in quantum mechanics ($\hbar=1,c=1$):

$i\gamma^{\mu}\partial_{\mu}\psi-m\psi=0,$

where, in the standard representation,

$\begin{split}&\gamma^{0}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{pmatrix}\qquad\gamma^{1}=\begin{pmatrix}0&0&0&1\\\ 0&0&1&0\\\
0&-1&0&0\\\ -1&0&0&0\end{pmatrix}\\\ &\gamma^{2}=\begin{pmatrix}0&0&0&-i\\\
0&0&i&0\\\ 0&i&0&0\\\
-i&0&0&0\end{pmatrix}\qquad\gamma^{3}=\begin{pmatrix}0&0&1&0\\\ 0&0&0&-1\\\
-1&0&0&0\\\ 0&1&0&0\end{pmatrix}.\end{split}$

This equation was born as an attempt of linearization of the Klein-Gordon
equation, that comes from the application of the Einstein relation
$E^{2}=p^{2}+m^{2}$ to the Schrödinger equation Hamiltonian. The Dirac
equation was a turning point in physics. Joining Quantum Mechanics and
Relativity, this equation predicts the spin of the electron, the existence of
antimatter etc. and it is considered as the natural transition between
relativistic quantum mechanics and quantum field theory thallerdir . Moreover,
a renewed interest in the Dirac equation has come from the field of quantum
simulations, which has enabled the experimental observation of some of its
more interesting features in tabletop experiments gerritsma ; lamata ; klein1
; klein2 ; potwopot ; salger2011klein ; zitterbecbegung . Nevertheless, even
though the general solutions of this equation are well-known, this is not the
case of the Dirac equation in curved spacetime. In general, obtaining
solutions in curved space is more complicated. For a detailed analysis of the
3+1 Dirac equation in Riemannian spacetimes, see for instance ref1 . Some
particular 1+1 D cases can be found, for instance, in angelakis  and sabin .

In this work, the method explained by one of us in sabin  will be applied to
obtain solutions of the Dirac equation in curved spacetime for different
exotic metrics in 1+1 dimensional sections, with the help of known solutions
in Minkowski spacetime. If we take a 1+1 dimensional section of the spacetime,
the Dirac equations reduces to angelakis :

$i\bigg{(}\partial_{t}+\frac{\dot{\Omega}}{2\Omega}\bigg{)}\psi=-i\sigma_{x}\bigg{(}\partial_{x}+\frac{\Omega^{\prime}}{2\Omega}\bigg{)}\psi+\sigma_{z}\Omega
m\psi$

where $\Omega$ is the conformal factor – note that any metric in 1+1
dimensions allow a coordinate change such that it acquires the form
$ds^{2}=\Omega^{2}(-dt^{2}+dx^{2})$ – the dot and prime stand for time and
spatial derivatives, respectively and $\sigma_{x}$ and $\sigma_{z}$ are the
Pauli matrices:

$\sigma_{x}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\qquad\sigma_{z}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$

Note that if $\Omega=1$, we obtain the well-known Dirac equation in 1+1
dimensional flat spacetime (see, for instance ref2 ). This equation has been
studied in detail, including striking properties such as Zitterbewegung or
Klein paradox ref3 ; ref4 . Given $\Omega$, we could try to find solutions of
this partial differential equation system. However, that procedure could be
very involved, depending on the particular form of $\Omega$. The method
presented in sabin  allows to find the solution of this partial differential
equation system through a coordinate change and an existing solution in
Minkowski spacetime. Thus, the method will be applied to several exotic
metrics of interest, such as the Alcubierre metric, which allows FTL velocity;
to the Gödel metric, which allows CTC; and the Kerr metric, which describes
the spacetime geometry generated by a rotating body (rotating black holes).
Furthermore, we will see that, although in sabin  $\dot{\Omega}=0$ (static
spacetime) is an apparent requirement for the validity of the method, in fact
the method is also valid for nonstatic spacetimes.

Our results might be of interest in the context of quantum simulation of the
Dirac equation. Several ideas have been proposed for simulating the Dirac
equation in Minkowski spacetime for 1+1 dimensional sections: for example,
experiments have been already realized with trapped ions gerritsma ; klein2
and Bose-Einstein condensates salger2011klein ; zitterbecbegung , while
realistic proposals exist with superconducting circuits julen  . Also, some
ideas have been proposed for simulating the Dirac equation in curved spacetime
for 1+1 dimensional sections: for example, through a mapping between this
equation and a multiphoton quantum Rabi model pedernales  . However, since we
obtain solutions in curved spacetime through transformations of Minkowski
solutions, the results obtained in this work could be helpful for simulations
of Dirac equation in non trivial spacetimes with currently existing setups.

The structure of the paper is the following. We start in Section II by
recalling the procedure for obtaining solutions of the Dirac equation in
curved spacetimes out of flat-spacetime solutions, first introduced in sabin .
Then, we proceed to apply it in Section III to the aforementioned curved
spacetimes of interest. Finally, we introduce a generalization of the method
to nonstatic spacetimes in Section IV, which might be useful for further
applications. We conclude in Section V with a summary of our results.

##  II Method

We summarize here the procedure to obtain solutions of the Dirac equation in
curved spacetimes, first discussed in sabin . In general, the Dirac equation
in curved spacetime is obtained by replacing the standard partial derivatives
in the Minkowski equation by the corresponding covariant ones, where the
affine connection would carry the dependence on the particular form of the
metric. In 1+1 D, using the conformally-flat form of the metric and with a
little algebra (see the details in angelakis ), we can get:

$i\bigg{(}\partial_{t}+\frac{\dot{\Omega}}{2\Omega}\bigg{)}\psi=-i\sigma_{x}\bigg{(}\partial_{x}+\frac{\Omega^{\prime}}{2\Omega}\bigg{)}\psi.$
(1)

For the static case, in which the conformal factor is time-independent
$\dot{\Omega}=0$, the equation is:

$i\partial_{t}\psi=-i\sigma_{x}\bigg{(}\partial_{x}+\frac{\Omega^{\prime}}{2\Omega}\bigg{)}\psi.$
(2)

If we make the transformation $\psi=\Omega^{-1/2}\phi$, the equation (2)
becomes:

$i\partial_{t}\phi=-i\sigma_{x}\partial_{x}\phi$

which corresponds to the Dirac equation for a massless particle in Minkowski
spacetime, whose solution is well-known (see, for instance, thaller ). Thus,
using this transformation and the conformal factor of the 1+1 dimensional
metric, we can find analytical solutions for the Dirac equation in curved
spacetime. So, for a given metric, we have to follow this three-step
procedure:

1) Find a change of coordinates $(t,x)\rightarrow(\bar{t},\bar{x})$ such that,
in the new coordinates, the metric acquires the form
$ds^{2}=\Omega^{2}(-d\bar{t}^{2}+d\bar{x}^{2})$.

2) Using those new coordinates $(\bar{t},\bar{x})$, apply
$\psi(\bar{t},\bar{x})=\Omega^{-1/2}(\bar{x})\phi(\bar{t},\bar{x})$, where
$\phi(\bar{t},\bar{x})$ is the solution of the Minkowski spacetime Dirac
equation.

3) Once having $\psi(\bar{t},\bar{x})$, apply the coordinate change
$(\bar{t},\bar{x})\rightarrow(t,x)$ to finally find the solution of the curved
spacetime Dirac equation in $(t,x)$ coordinates.

Please note that, in spite of its apparent simplicity, in general it might not
be necessarily straightforwerd to perform the steps 1) and 3). To write the
metric in a conformally-flat form, it might be needed to express the change of
coordinates in a differential form, which could generate an equation that
might still not be straightforward to solve, as we will see in the (III.2)
section.

This method was applied in sabin  to find solutions of the Dirac equation in a
1+1 dimensional section of a traversable wormhole spacetime. Let us now
consider other examples of interest.

##  III Results

###  III.1 Gödel and Alcubierre metric

Both metrics will be discussed in the same section because of their
similarity.

In 1994, Alcubierre proposed a metric that, in principle, allows FTL motion
alcubierre . Note that, in general relativity, FTL speed is only forbidden
locally. This is not as exotic as it might seem at first glance: for instance,
the expansion of the Universe can make that two distant galaxies move at FTL
speed between them, while each one is moving locally inside its light-cone.
While the oposite might be possible too: if the spacetime were contracting
fast enough, each galaxy were moving near the speed of light locally (inside
its light-cone) in oposite directions, but globally both were getting closer.
With these considerations in mind, Alcubierre’s idea is simple: to create, in
the front of an object, a spacetime contraction, and in the back, a spacetime
dilation. Thus, the contraction will pull the object forward, and the dilation
will push the object forward too. Locally the object will be inside its light-
cone, but due to this spacetime manipulation, it would move FTL– as compared
with $c$, the speed of light in flat-spacetime vacuum.

The Alcubierre metric inside the “bubble” created by the space-temporal
contraction/dilation, under the limit $\sigma\rightarrow\infty$ alcubierre ,
and taking a spatial section $y=y_{0}$ and $z=z_{0}$ (with constant
$y_{0},z_{0}$), acquires the form alcubierre ; sabin2 :

$ds^{2}=-(1-v_{s}^{2})dt^{2}+dx^{2}-2v_{s}dxdt.$ (3)

As expected, Eq. (3) becomes the Minkowski metric when $v_{s}=0$. Now we make
a coordinate change $(t,x)\rightarrow(\bar{t},\bar{x})$, such that:

$\begin{split}dt&=d\bar{t}\\\ dx&=d\bar{x}+v_{s}d\bar{t}.\end{split}$

The Alcubierre metric, using those new coordinates, acquires the form
$ds^{2}=-d\bar{t}^{2}+d\bar{x}^{2}$, that is simply the Minkowski metric, so
the conformal factor will be $\Omega^{2}=1$, and the solution of the curved
spacetime Dirac equation, $\psi(\bar{t},\bar{x})$, will be equal to the
Minkowski spacetime solution, $\phi(\bar{t},\bar{x})$. Now, due to the big
similarity – regarding to the application of our techniques – between the
Alcubierre and Gödel metric, we first discuss the latter case and then we
compare both results. It is relevant to mention that, although the Dirac
equation in a Gödel universe has already been considered in the study of
tachyon’s stability konoplya , our objective is to find explicit analytical
solutions for this equation in the Gödel’s background, using our procedure.

K. Gödel, in 1949, found a solution of the Einstein equations Godel
corresponding to an homogeneous mass distribution that rotates at each point
of the space kajari . That distribution of matter causes unusual effects, such
as the existence of CTC. In cylindrical coordinates kajari , the metric is
given by the following expression:

$\begin{split}ds^{2}=&dt^{2}-\frac{dr^{2}}{1+(\frac{r}{2a})^{2}}-r^{2}\bigg{(}1-\Big{(}\frac{r}{2a}\Big{)}^{2}\bigg{)}d\phi^{2}-dz^{2}\\\
&+\frac{2r^{2}}{a\sqrt{2}}dtd\phi\\\ \end{split}$ (4)

where $a$ is a parameter with units of length, that represents a
characteristic distance. In particular, $r_{G}=2a$ represents the critical
radius from which CTC can exist kajari .

Now, taking a radial section with $\phi=\phi_{0}$ and $z=z_{0}$ sabin2 , the
Gödel metric becomes:

$ds^{2}=dt^{2}-\frac{1}{1+\big{(}\frac{r}{2a}\big{)}^{2}}dr^{2}$

By making the coordinate change $(t,r)\rightarrow(\bar{t},\bar{r})$, with

$\begin{split}d\bar{r}^{2}&=\frac{1}{1+\big{(}\frac{r}{2a}\big{)}^{2}}dr^{2}\\\
d\bar{t}^{2}&=dt^{2}\end{split}$ (5)

the previous metric transforms into the Minkowski metric
$ds^{2}=d\bar{t}^{2}-d\bar{r}^{2}$. Furthermore, we can find the relationship
between the $(t,r)$ and $(\bar{t},\bar{r})$ coordinates by performing the
following integration:

$\bar{r}(r)=\int\frac{dr}{\sqrt{1+\big{(}\frac{r}{2a}\big{)}^{2}}}$

whose solution – setting $\bar{r}(r=0)=0$– is:

$\bar{r}(r)=2a\sinh^{-1}\big{(}\frac{r}{2a}\big{)}.$

With these new coordinates, the conformal factor becomes $\Omega^{2}=1$, and
the Minkowski spacetime solution will be the same as the curved spacetime
solution, as in the case of the Alcubierre metric. Thus, the only difference
between Gödel and Alcubierre metric is the relationship between the original
coordinates and the new coordinates.

Now, if we assume that the wave function has a gaussian initial form:

$\phi(\bar{x},0)=Ne^{-\frac{(\bar{x}-\bar{x}_{0})^{2}}{\sigma^{2}}}\begin{pmatrix}1\\\
1\end{pmatrix}$

where $\bar{x}$ is the conformally flat coordinate and $N$ is a normalization
constant, the solution for the Dirac equation in Minkowski’s spacetime will be
sabin ; thaller :

$\phi(\bar{x},\bar{t})=Ne^{-\frac{(\bar{t}-(\bar{x}-\bar{x}_{0}))^{2}}{\sigma^{2}}}\begin{pmatrix}1\\\
1\end{pmatrix}$ (6)

Since $\psi=\phi$, to find the curved spacetime solution we only have to apply
the coordinate change for each metric. In the Alcubierre case we have:

$\phi\Big{(}\bar{x}(x),\bar{t}(t)\Big{)}=Ne^{-\frac{(t-(x-v_{s}t-\bar{x}_{0}))^{2}}{\sigma^{2}}}\begin{pmatrix}1\\\
1\end{pmatrix},$ (7)

while, for the Gödel metric:

$\phi\Big{(}\bar{r}(r),\bar{t}(t)\Big{)}=Ne^{-\frac{(t-(2a\sinh^{-1}(r/2a)-\bar{x}_{0}))^{2}}{\sigma^{2}}}\begin{pmatrix}1\\\
1\end{pmatrix}.$ (8)

Both solutions are represented in Figure 1.

Figure 1: spacetime diagrams for the probability density of Dirac equation
solutions in a-c) Alcubierre metric with a) $v_{s}=0$ (equivalent to Minkowski
spacetime), b) $v_{s}=0.5$ (subluminal spacetime bubble), c) $v_{s}=2$
(superluminal spacetime bubble); d-f) Gödel metric with d) $a=2$, e) $a=1$, f)
$a=0.5$. Please note that the wave function is not normalized.

For a), b) and c), as the parameter $v_{s}$ increases, the slope of the
gaussian wavepacket’s “trajectory” decreases. In other words, the particle’s
velocity increases as $v_{s}$ is bigger. In fact, if we take a look to the
expression (7), we can see its equivalence to the solution of the Minkowski
spacetime Dirac equation (6) if we substitute $t$ with $(1+v_{s})t$. This
means that, physically, the particle moves with a FTL velocity
$\bar{c}=1+v_{s}$, as expected for a massless particle since this is precisely
the speed of light in this spacetime.

For d), e) and f), we can see that the particle velocity is no longer
constant. This variation is inversely proportional to the value of $a$, as
expected, since in the limit $a\rightarrow\infty$ the Gödel metric (4) becomes
the Minkowski metric in cylindrical coordinates. In addition, the slope
decreases with the propagation of the particle, implying FTL velocity. Of
course, in general the existence of CTC –which is the most genuine feature of
Gödel spacetime– is intimately linked with the possibility of FTL (see, for
instance, mallary ).

###  III.2 Rotating black hole

The main motivation to study the Kerr metric is the fact that it describes the
spacetime geometry of a rotating body, so it can be used to analyse the
physics of black holes in a more realistic scenario than the one provided by
the Schwarzschild metric. Using Boyer-Lindquist coordinates
$(t,r,\theta,\phi)$, the Kerr metric is kerr bambi :

$\begin{split}ds^{2}=&-\Big{(}1-\frac{2Mr}{\Sigma}\Big{)}dt^{2}-\frac{4Mar\sin^{2}\theta}{\Sigma}dtd\theta+\frac{\Sigma}{\Delta}dr^{2}\\\
&+\Sigma
d\theta^{2}+\Big{(}r^{2}+a^{2}+\frac{2Ma^{2}r\sin^{2}\theta}{\Sigma}\Big{)}\sin^{2}\theta
d\phi^{2}\end{split}$

where

$\Sigma(r,\theta)=r^{2}+a^{2}\cos^{2}\theta\qquad\Delta(r)=r^{2}+a^{2}-2Mr$

If we take a radial section ($\theta=\theta_{0}$, $\phi=\phi_{0}$), we obtain
sabin2 :

$ds^{2}=-\Big{(}1-\frac{2Mr}{\Sigma(r,\theta_{0})}\Big{)}dt^{2}+\frac{\Sigma(r,\theta_{0})}{\Delta(r)}dr^{2},$

where $\Sigma(r,\theta_{0})$ is a function of the coordinate $r$ (for
simplicity, $\Sigma(r,\theta_{0})=\Sigma(r)$). So, if we make the coordinate
change $(t,r)\rightarrow(\bar{t},\bar{x})$ with:

$\begin{split}\frac{\Sigma(r)}{\Delta(r)}dr^{2}&=\Big{(}1-\frac{2Mr}{\Sigma(r)}\Big{)}d\bar{x}^{2}\\\
dt^{2}&=d\bar{t}^{2}\\\ \end{split}$ (9)

the metric acquires the form

$ds^{2}=\Omega^{2}(r)(-d\bar{t}^{2}+d\bar{x}^{2})$

with

$\Omega^{2}(r)=1-\frac{2Mr}{\Sigma(r)}.$

Due to the conditions (9), $r$ will be a function of $\bar{x}$, whereas
$t=\bar{t}$, so the conformal factor $\Omega(r)$ will be a function of
$r(\bar{x})$:

$\Omega^{2}\big{(}r(\bar{x})\big{)}=1-\frac{2Mr(\bar{x})}{\Sigma\big{(}r(\bar{x})\big{)}}$
(10)

To find the conformal factor $\Omega\big{(}r(\bar{x})\big{)}$ we need to
obtain $r(\bar{x})$. Using the first equation in (9) and, after doing some
algebraic manipulations, we obtain that $\bar{x}(r)$ is given by:

$\bar{x}(r)=\int
dr\Sigma(r)\sqrt{\frac{1}{\Delta(r)\big{(}\Sigma(r)-2Mr\big{)}}}+C$ (11)

In general, solving this integral is not straightforward. Later, we will
analyze some particular cases in which the integral can be solved. By now, let
us just assume that $r(\bar{x})$ is known. Thus, the conformal factor is be
given by the equation (10) and, therefore, the relation between the solution
to Dirac equation in curved spacetime ($\psi$) and flat spacetime ($\phi$)
will be:

$\begin{split}\absolutevalue{\psi(\bar{x},\bar{t})}^{2}&=\absolutevalue{\Omega^{-1/2}\big{(}r(\bar{x})\big{)}}^{2}\absolutevalue{\phi(\bar{x},\bar{t})}^{2}\\\
&=\absolutevalue{\sqrt{\frac{\Sigma\big{(}r(\bar{x})\big{)}}{\Sigma\big{(}r(\bar{x})\big{)}-2Mr(\bar{x})}}}\absolutevalue{\phi(\bar{x},\bar{t})}^{2}\end{split}$
(12)

All the wave function properties caused by the spacetime curvature are
contained in the factor $\Omega^{-1}\big{(}r(\bar{x})\big{)}$, so it is
interesting to analyze this function. First, there exist regions in which the
probability density of the wave function in curved spacetime becomes infinite.
It happens when the following condition is fulfilled – we relax the notation
and write $r$ instead of $r(\bar{x})$:

$\Sigma(r)-2Mr=0$ (13)

This condition is satisfied whenever $r=r_{\pm}$, with:

$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}\cos^{2}\theta_{0}},$ (14)

which are the well-known apparent singularities where the temporal component
of the metric changes sign, defining the ergosphere gravitation . On the other
hand, there exist points in which the wave function is null, independently of
the form of the solution in Minkowski’s spacetime. This will happen whenever
$\Omega^{-1}(r)=0$, thats is, when:

$\Sigma(r)=0\qquad\rightarrow\qquad r^{2}+a^{2}\cos^{2}\theta_{0}=0,$

which will be satisfied when at least one of those conditions apply:

1) $r=0$, $a=0$ (no rotation)

2) $r=0$, $\theta_{0}=\frac{\pi}{2}$.

Finally, there is a region of the space in which the conformal factor is an
imaginary number. This occurs if:

$\Sigma(r)-2Mr<0.$

This second-degree inequality is verified for $r_{-}<r<r_{+}$, namely, within
the ergosphere.

####  III.2.1 Radial section with $\theta_{0}=0$

Let u consider $\theta_{0}=0$. Then $\Sigma(r)=r^{2}+a^{2}$,
$\Delta(r)=\Sigma-2Mr$, and the integral becomes:

$\bar{x}(r)=\int dr\frac{\Sigma(r)}{\Delta(r)}+C=\int
dr\frac{r^{2}+a^{2}}{r^{2}+a^{2}-2Mr}+C$

whose solution is

$\begin{split}\bar{x}(r)=&\frac{2M^{2}}{\sqrt{a^{2}-M^{2}}}\tan^{-1}\Big{(}\frac{r-M}{\sqrt{a^{2}-M^{2}}}\Big{)}\\\
&+M\log(a^{2}-2Mr+r^{2})+r+C\end{split}$

Bearing in mind that, for a rotating black hole, $M^{2}>a^{2}$ is satisfied
(otherwise, it would not be a black hole, but a highly rotating body stephani
), and using the relationship $\tan^{-1}(ix)=i\tanh^{-1}(x)$, we can write the
solution as:

$\begin{split}\bar{x}(r)=&\frac{2M^{2}}{\sqrt{M^{2}-a^{2}}}\tanh^{-1}\Big{(}\frac{M-r}{\sqrt{M^{2}-a^{2}}}\Big{)}\\\
&+M\log(a^{2}-2Mr+r^{2})+r+C\end{split}$ (15)

The constant of integration can be chosen in such a way that the constant
imaginary part of Eq. (15) cancels out in each of the three spacetime regions
defined by the apparent singularities $r_{\pm}$. Thus:

$C=\left\\{\begin{split}-i\frac{\pi M^{2}}{\sqrt{M^{2}-a^{2}}}\qquad&if\qquad
r<r_{-}\\\ -iM\pi\qquad&if\qquad r_{-}<r<r_{+}\\\ -i\frac{\pi
M^{2}}{\sqrt{M^{2}-a^{2}}}\qquad&if\qquad r_{+}<r\\\ \end{split}\right.$

In Figure 2, we plot $\bar{x}(r)$ for different values of $M$ and $a$.

Figure 2: $\bar{x}(r)$ for different values of the parameter $a$. In absence
of rotation ($a=0$), the singular points are in $r=0$ and $r=2M$
(Schwarzschild’s black hole). As the rotation increases, both points
($r=r_{\pm}$) come closer to $r=M$.

Finally, assuming again that our particle is characterized by a Gaussian
wavepacket, the solution in Kerr spacetime is:

$\begin{split}\absolutevalue{\psi\big{(}\bar{x}(r),t\big{)}}^{2}&=\sqrt{\frac{\Sigma(r)}{\Sigma(r)-2Mr}}\absolutevalue{\phi\big{(}\bar{x}(r),t\big{)}}^{2}\\\
&=2N^{2}\sqrt{\frac{\Sigma(r)}{\Sigma(r)-2Mr}}e^{-\frac{2(t-(\bar{x}(r)-\bar{\bar{x}}_{0}))^{2}}{\sigma^{2}}}\end{split}$

This solution is represented in Figure (3) for several values of $a$ and $M$.

Figure 3: Dirac equation solution for Kerr metric and different values of the
parameters $a$ and $M$. a)-d), $a=0$ (no rotation, Schwarzschild metric) and
$M$ ranging from $0$ to $2$. The horizon is at $r=2M$. e)-h) $M=2$ and $a$
ranging from $0$ to $2$, two singularities appear according to Eq. (14) with
$\theta_{0}=0$ giving rise to an ergosphere which splits the full spacetime
into three separate regions . Notice that as $a$ increases $r_{+}$, $r_{-}$
move from $2M$ and $0$, respectively, to $M$, where they would merge in the
limit $M=a$. Please note that the wave function is not normalized.

Naturally, when we set $M=a=0$, the metric becomes the Minkowski one,
$\Omega^{2}=1$, and the expression (15) reduces to $\bar{x}(r)=r$. In other
words, the solution corresponds to a free particle in flat spacetime, as it is
observed in Figure (3 a). If we slightly increase the value of $M$ (with
$a=0$, thus we are in Schwarzschild spacetime), we see that the free solution
is slightly disturbed, increasing the probability density near the horizon, as
can be seen in Figure (3 b). As the mass increases, the probability density
accumulates nearby the horizon, placed in $r=2M$. In addition, we see that the
width of the wavepacket is drastically reduced (the particle is more localized
in space), and the particle’s speed diminishes, vanishing in the horizon.
Finally, if we set the mass to a constant value and we change $a$ (Figures (3
e-h), we see that, as $a$ increases, the inner horizon is generated, around
which, again, the probability density accumulates in the same way as before.
In addition, in the region between both singular points –the ergosphere– we
see how the particle tends to be expelled towards the interior or exterior
horizon.

##  IV Nonstatic spacetimes

So far, we have only analyzed static metrics, that is, $\dot{\Omega}=0$. We
now show that our techniques are also valid for the case of nonstatic metrics
($\dot{\Omega}\neq 0$). Performing the substitution $\psi=\Omega^{-1/2}\phi$
in Equation (1) and with a little algebra, we find:

$\begin{split}i\bigg{(}\partial_{t}+\frac{\dot{\Omega}}{2\Omega}\bigg{)}(\Omega^{-1/2}\phi)&=-i\sigma_{x}\bigg{(}\partial_{x}+\frac{\Omega^{\prime}}{2\Omega}\bigg{)}(\Omega^{-1/2}\phi)\\\
\partial_{t}(\Omega^{-1/2}\phi)+\frac{\dot{\Omega}}{2\Omega}\Omega^{-1/2}\phi&=-\sigma_{x}\bigg{[}\partial_{x}(\Omega^{-1/2}\phi)+\frac{\Omega^{\prime}}{2\Omega}\Omega^{-1/2}\phi\bigg{]}\end{split}$

Notice that

$\partial_{t}(\Omega^{-1/2}\phi)=-\frac{1}{2}\Omega^{-3/2}\dot{\Omega}\phi+\Omega^{-1/2}\partial_{t}\phi.$
(16)

Therefore the terms not containing $\partial_{t}\phi$ cancel out. The same
thing happens with the spatial partial derivative: the terms without
$\partial_{x}\phi$ cancel out. Thus, putting everything together we recover
the equation:

$i\partial_{t}\phi=-i\sigma_{x}\partial_{x}\phi$

that is the Dirac equation for Minkowski spacetime. Therefore, our techniques
could be also applied to nonstatic spacetimes, such as Friedmann-Lemaître-
Robertson-Walker spacetime, as in angelakis .

##  V Conclusions

Following sabin , we have shown that is possible to obtain solutions of the
Dirac equation in curved spacetime by means of a transformation
$\psi(\bar{x},\bar{t})=\Omega^{-1/2}(\bar{x})\phi(\bar{x},\bar{t})$ where
$\psi(\bar{x},\bar{t})$ is the curved spacetime solution,
$\phi(\bar{x},\bar{t})$ is the Minkowski spacetime solution and
$\Omega(\bar{x})$ is the conformal factor. Then, in order to be able to apply
this transformation, it is necessary to perform first a coordinate change
$(x,t)\rightarrow(\bar{x},\bar{t})$, such that in the new coordinates, the
metric is conformally flat. We have applied this technique to three different
metrics: 1+1 dimensional sections of Alcubierre, Gödel and Kerr metrics. For
Alcubierre and Gödel, the conformal factor turns out to be $\Omega^{2}=1$, so
the only difference between both solutions is the coordinate transformation
employed in each case. Setting a gaussian solution in Minkowski spacetime, and
after undoing the coordinate transformation, we obtain the solutions (7) and
(8), for Alcubierre and Gödel spacetimes, respectively. In this way, we have
been able to analyse the FTL behaviour of massless particles in these exotic
spacetimes. In the case of the Kerr metric, the conformal factor is non-
trivial $\Omega^{2}(r)=1-\frac{2Mr}{\Sigma(r)}$. We find an analytical
solution in a spatial section with $\theta_{0}=0$. We have discussed the
dependence of this solution on the black-hole mass and rotation, as well as
the behaviour of the particle near the singularities and within the
ergosphere. Finally, we have shown that our technique is valid also for
nonstatic spacetimes.

Our results might, in principle, be useful in the context of quantum
simulations of the Dirac equation. In particular, following the spirit of
sabin , notice that our technique entails that curved spacetimes can be
encoded into a transformation realized onto a flat-spacetime Dirac wavepacket.
Therefore, in principle, our results could be tested in already existing
quantum simulators of the Dirac equation in flat spacetime.

## Acknowledgements

C. S. has received financial support through the Junior Leader Postdoctoral
Fellowship Programme from “la Caixa” Banking Foundation and Fundación General
CSIC (ComFuturo Programme)

## References

  * (1) B. Thaller, The Dirac equation Springer-Verlag (Berlin-Heidelberg, 1992).
  * (2) R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt and C. F. Roos, Nature 463, 68 (2010)
  * (3) L. Lamata, J. León, T. Schätz and E. Solano, Phys. Rev. Lett. 98, 253005 (2007)
  * (4) J. Casanova, J. J. García-Ripoll, R. Gerritsma, C. F. Roos, E. Solano, Phys. Rev. A 82, 020101 (2010).
  * (5) R. Gerritsma et al. Phys. Rev. Lett. 106, 060503 (2011).
  * (6) C. Sabín et al. Phys. Rev. A 85, 052301 (2012).
  * (7) T. Salger, C. Grossert, S. Kling, and M. Weitz, Phys. Rev. Lett. 107, 240401 (2011).
  * (8) L . J. LeBlanc, M. C. Beeler, K. Jiménez-García, A. R. Perry, S. Sugawa, R. A. Williams, and I. B. Spielman New J. Phys 15 073011 (2013).
  * (9) M. Gasperini, Theory of gravitational interactions, 2nd ed, Springer (Berlin,2017).
  * (10) Christian Koke, Changsuk Noh and Dimitris Angelakis, Annals of Physics 374, 162 (2016).
  * (11) Carlos Sabín, Sci. Rep. 7,40346 (2017).
  * (12) H. Nitta, T. Kudo, and H. Minowa, Am. J. Phys. 67, 966 (1999).
  * (13) F. M. Toyama, Y. Nogami and F. A. B. Coutinho, J. Phys. A: Math. Gen. 30, 2585 (1997).
  * (14) V. Alonso and S. De Vincenzo, Phys. Scr. 61, 396 (2000).
  * (15) J. S. Pedernales, R. di Candia, D. Ballester, and E. Solano, New J. Phys. 15, 055008 (2013).
  * (16) J. S. Pedernales, M. Beau, S. M. Pittman, I. L. Egusquiza, L. Lamata, E. Solano, A. del Campo, Phys. Rev. Lett. 120, 160403 (2018).
  * (17) B. Thaller, Advanced Visual Quantum Mechanics. Springer, (New York, 2005).
  * (18) M. Alcubierre, Class. Quant. Grav. 11, L73 (1994).
  * (19) Carlos Sabín, New J. Phys. 20, 053028 (2018).
  * (20) R. A. Konoplya, Phys. Lett. B 706, 451-455 (2012).
  * (21) K. Gödel, Rev. Mod. Phys. 21, 447 (1949).
  * (22) E. Kajari, R. Walser, W. P. Schleich, A. Delgado, Gen. Rel. Grav. 36, 2289 (2004).
  * (23) C. Mallary, G. Khanna and R.H. Price, Class. Quantum Grav. 35, 175020 (2018).
  * (24) R. P. Kerr, Phys. Rev. Lett. 11, 237 (1963).
  * (25) C. Bambi, Rev. Mod. Phys. 89, 025001 (2017).
  * (26) S. W. Hawking, G. F. R. Ellis, The large scale structure of spacetime, Cambridge University Press (Cambridge, 1973).
  * (27) Hans Stephani, Relativity. An Introduction to Special and General Relativity. Third Edition. Cambridge University Press (2004).


††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.††thanks: These authors contributed
equally to this work.

# Entanglement-Enhanced Quantum Metrology in Colored Noise by Quantum Zeno
Effect

Xinyue Long  Shenzhen Institute for Quantum Science and Engineering and
Department of Physics, Southern University of Science and Technology, Shenzhen
518055, China  Wan-Ting He  Department of Physics, Applied Optics Beijing Area
Major Laboratory, Beijing Normal University, Beijing 100875, China  Na-Na
Zhang  School of Optoelectronic Engineering, Chongqing University of Posts and
Telecommunications, Chongqing 400065, China  Department of Physics, Applied
Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing
100875, China  Kai Tang  Shenzhen Institute for Quantum Science and
Engineering and Department of Physics, Southern University of Science and
Technology, Shenzhen 518055, China  Zidong Lin  Shenzhen Institute for Quantum
Science and Engineering and Department of Physics, Southern University of
Science and Technology, Shenzhen 518055, China  Hongfeng Liu  Shenzhen
Institute for Quantum Science and Engineering and Department of Physics,
Southern University of Science and Technology, Shenzhen 518055, China  Xinfang
Nie  Shenzhen Institute for Quantum Science and Engineering and Department of
Physics, Southern University of Science and Technology, Shenzhen 518055, China
Guangdong Provincial Key Laboratory of Quantum Science and Engineering,
Southern University of Science and Technology, Shenzhen 518055, China  Guanru
Feng  Shenzhen SpinQ Technology Co., Ltd., Shenzhen, China  Jun Li  Shenzhen
Institute for Quantum Science and Engineering and Department of Physics,
Southern University of Science and Technology, Shenzhen 518055, China
Guangdong Provincial Key Laboratory of Quantum Science and Engineering,
Southern University of Science and Technology, Shenzhen 518055, China  Tao Xin
Shenzhen Institute for Quantum Science and Engineering and Department of
Physics, Southern University of Science and Technology, Shenzhen 518055, China
Guangdong Provincial Key Laboratory of Quantum Science and Engineering,
Southern University of Science and Technology, Shenzhen 518055, China  Qing Ai
<EMAIL_ADDRESS>Department of Physics, Applied Optics Beijing Area Major
Laboratory, Beijing Normal University, Beijing 100875, China  Dawei Lu
<EMAIL_ADDRESS>Shenzhen Institute for Quantum Science and Engineering and
Department of Physics, Southern University of Science and Technology, Shenzhen
518055, China  Guangdong Provincial Key Laboratory of Quantum Science and
Engineering, Southern University of Science and Technology, Shenzhen 518055,
China

(August 28, 2024)

###### Abstract

In open quantum systems, the precision of metrology inevitably suffers from
the noise. In Markovian open quantum dynamics, the precision can not be
improved by using entangled probes although the measurement time is
effectively shortened. However, it was predicted over one decade ago that in a
non-Markovian one, the error can be significantly reduced by the quantum Zeno
effect (QZE) [Chin, Huelga, and Plenio, Phys. Rev. Lett. 109, 233601 (2012)].
In this work, we apply a recently-developed quantum simulation approach to
experimentally verify that entangled probes can improve the precision of
metrology by the QZE. Up to $n=7$ qubits, we demonstrate that the precision
has been improved by a factor of $n^{1/4}$, which is consistent with the
theoretical prediction. Our quantum simulation approach may provide an
intriguing platform for experimental verification of various quantum metrology
schemes.

Introduction.—Quantum metrology utilizes quantum entanglement and coherence to
enhance the measurement precision over classical metrology Caves (1981);
Giovannetti  _et al._ (2004); Pezzè  _et al._ (2018). In the absence of noise,
the precision of classical metrology scales as $n^{-1/2}$, which is limited by
the central-limit theorem with $n$ being the number of resources implemented
in the measurements. In contrast, quantum metrology can reach the Heisenberg
limit, scaling as $n^{-1}$. However, in practice, any quantum system is
inevitably subject to decoherence which results from the interaction with the
environment Breuer and Petruccione (2002). Due to decoherence, there will be
no improvement in the measurement precision over classical metrology, although
the measurement time may be effectively shortened Huelga  _et al._ (1997). In
order to overcome this shortcoming, various methods have been put forward,
e.g., squeezing Ma  _et al._ (2011), purification Yamamoto  _et al._ (2022),
and one-way quantum-computing-based teleportation Matsuzaki  _et al._ (2018).
It was shown that instead of maximally entangled states, partially entangled
initial states may reduce the error to some extent Huelga  _et al._ (1997).
When the system interacts with a specific bath with a band structure, ideal
precision may be retrieved as a result of the existence of the bound state Bai
_et al._ (2019); Ai and Liao (2010). When there is a correlation between the
baths for individual qubits, an auxiliary qubit can be introduced to reach the
Heisenberg limit Monz  _et al._ (2011); He  _et al._ (2021); Kukita  _et al._
(2021). Interestingly, the nonlinear interaction between the system and the
physical quantity to be measured is used to surpass the Heisenberg limit Boixo
_et al._ (2007); Nie  _et al._ (2015); Choi and Sundaram (2008); Roy and
Braunstein (2008); Napolitano  _et al._ (2011). However, the nonlinear
interaction may not be easily realized in practical measurements.

Then, a question naturally comes to our minds: Can we figure out some
practical method to improve the measurement precision in the presence of
noise? It is shown that there are generally three stages in open quantum
dynamics, including the well-known exponential decay in the intermediate stage
Nakazato  _et al._ (1996). However, in the first stage, the open quantum
system decays in a Gaussian way, where the quantum Zeno effect (QZE) Ai  _et
al._ (2010, 2013); Harrington  _et al._ (2017); Virzì  _et al._ (2022);
Kiilerich and Mølmer (2015); Do  _et al._ (2019); Burgarth  _et al._ (2014)
happens. It has been pointed out that due to the QZE, metrology using
maximally entangled states is superior to the one using product states by a
factor of $n^{1/4}$ Chin  _et al._ (2012); Matsuzaki  _et al._ (2011). On the
other hand, recently we have theoretically proposed and experimentally
demonstrated in a nuclear magnetic resonance (NMR) platform, to simulate the
quantum dynamics for various Hamiltonians and spectral densities Wang  _et
al._ (2018); Zhang  _et al._ (2021); Chen  _et al._ (2022). In these works,
the bath-engineering technique Soare  _et al._ (2014a, b); Khaneja  _et al._
(2005); Li  _et al._ (2017) enables a theoretically exact, fully controllable,
and practically efficient quantum simulation Buluta and Nori (2009); Georgescu
_et al._ (2014) approach. Using the approach, we show that the non-
Markovianity of the open quantum dynamics should be essentially characterized
from both aspects of global and local points of view, e.g., quantum mutual
information v.s. quantum Fisher information flows Chen  _et al._ (2022); Luo
_et al._ (2012); Lu  _et al._ (2010); Altherr and Yang (2021). It is this
theoretically exact and fully controllable characteristic that enables us to
experimentally verify the quantum metrology scheme proposed in Refs. Chin  _et
al._ (2012); Matsuzaki  _et al._ (2011), which requires homogeneity of the
qubits.

Protocol.—The bath-engineering technique offers a way to engineer arbitrary
environments by modulating the control field Soare  _et al._ (2014a, b). By
applying a time-dependent magnetic field to the total system, we can
artificially add a spectral characteristic noise, so as to simulate an
arbitrary noisy environment. The Hamiltonian of the total system can be
written as $H(t)=H_{\textrm{S}}+H_{\textrm{SB}}(t)$, where
$H_{\textrm{S}}=\sum_{m}\varepsilon_{m}|m\rangle\langle m|+\sum_{m\neq
n}J_{mn}|m\rangle\langle n|$ is the system Hamiltonian, and
$H_{\textrm{SB}}(t)$ is the noise Hamiltonian to simulate the system-bath
couplings. When simulating the pure-dephasing noise, cf. Fig. 1(a), the noise
Hamiltonian is $H_{\textrm{SB}}(t)=\sum_{m}\beta_{m}(t)|m\rangle\langle m|$.
$\beta_{m}(t)$’s are stochastic errors generated by performing amplitude and
phase modulations on a carrier Wang  _et al._ (2018); Zhang  _et al._ (2021)

$\beta_{m}(t)=\alpha_{z}\sum_{j=1}^{J}\omega_{j}F(\omega_{j})\cos\left(\omega_{j}t+\psi_{j}^{(m)}\right),$
(1)

where $\alpha_{z}$ is the amplitude of the dephasing noise,
$\omega_{\textrm{b}}$ ($J\omega_{\textrm{b}}$) is the base (cutoff) frequency
with $\omega_{j}=j\omega_{\textrm{b}}$, $F(\omega_{j})$ is the modulation
function of a specific spectral characteristic noise, and $\psi_{j}^{(m)}$’s
are a set of random numbers. For a noise of Gaussian type, the average over
the ensemble is equivalent to the average over the time Goodman (2015). The
ensemble-averaged decoherence factor reads
$\Gamma(t)=\alpha_{z}^{2}\sum_{j=1}^{J}F(\omega_{j})^{2}\sin^{2}\frac{\omega_{j}t}{2}$,
as determined by the power spectral density of the noise, which is the Fourier
transform of the second-order correlation function
$\langle\beta_{m}(t+\tau)\beta_{m}(t)\rangle$ Soare  _et al._ (2014a, b).

Figure 1: (a) Single-qubit Ramsey experiment for quantum metrology under non-
Markovian noise. The free precession is governed by non-Markovian dynamics,
which can be simulated using the ensemble-averaging technique. This averaging
can be visualized in the Bloch sphere: A large number of identical initial
probes undergo phase diffusion owing to their different modulated
Hamiltonians, leading to an attenuation in amplitude for the overall spin
signal. This attenuation gives rise to the error in quantum-metrology
experiments. (b) Quantum circuits for multi-qubit Ramsey experiment under non-
Markovian noise with unentangled and entangled probes, respectively. The
evolution $U$ is the simulation of non-Markovian dynamics. For the case of
entangled probes, the readout is typically some collective measurement Ma  _et
al._ (2011); Huelga  _et al._ (1997).

We consider a system Hamiltonian
$H_{\textrm{S}}=\sum_{k=1}^{n}\omega_{0}\sigma_{k}^{z}/2$, which is subject to
a pure-dephasing noise Chin  _et al._ (2012). We present an illustration for
the case with a single qubit in Fig. 1(a). Initially, the qubit is prepared at
$(|0\rangle+|1\rangle)/\sqrt{2}$. Afterwards, it unitarily evolves at the
equator of the Bloch sphere with rate $\omega_{0}+\beta_{1}(t)$. In order to
mimic the pure dephasing, we generate a large number of realizations with the
same initial state and obtain the final result by averaging over the random
ensemble with different rate $\beta_{1}(t)$ after a second $\pi/2$ pulse. We
assume a Drude-Lorentz spectral density, i.e.,
$J_{k}(\omega)=J(\omega)=2\lambda\gamma\omega/(\omega^{2}+\gamma^{2})$, where
$\lambda$ is the reorganization energy and $\gamma$ is the relaxation rate. A
generic spectral density can be decomposed into a summation of Lorentzian
forms. The modulation function then becomes
$F(\omega_{j})=[\lambda\gamma\omega_{0}\coth(\beta\omega_{j})/(\omega_{j}^{3}+\gamma^{2}\omega_{j})]^{1/2}$.
The quantum dynamics of the system is exactly described by the time-local
master equation Breuer and Petruccione (2002); Tao  _et al._ (2020)

$\frac{\partial}{\partial
t}\rho(t)=-i[H_{\textrm{S}},\rho]+\sum_{k}\gamma_{k}(t)(\sigma^{z}_{k}\rho\sigma^{z}_{k}-\rho).$
(2)

Note that $\gamma_{k}(t)$ is explicitly dependent on time. According to the
bath-engineering technique, the ensemble-averaged decoherence rate is
$\gamma_{k}(t)=\frac{\alpha_{z}^{2}}{2}\sum_{j=1}^{J}\omega_{j}F(\omega_{j})^{2}\sin(\omega_{j}t)$.

In Fig. 1(b), we present the quantum circuit for the quantum metrology scheme
for the product and entangled states, respectively. For the former case, the
probes are prepared at $[(|0\rangle+|1\rangle)/\sqrt{2}]^{\otimes n}$ and then
evolve under the unitary evolution
$U=\exp(-i\int_{0}^{t}d\tau[H_{\textrm{S}}+\sum_{k}\beta_{k}(\tau)\sigma_{k}^{z}])$,
which will not induce entanglement between different qubits but mimic the
noise. Then, we perform the individual measurements on each qubit,
respectively, since all qubits are disentangled. However, for the latter case,
since the probes are initialized at the maximally entangled state
$(|0\rangle^{\otimes n}+|1\rangle^{\otimes n})/\sqrt{2}$, we only perform a
collective measurement after the same $U$. For open quantum dynamics governed
by the master equation (2), the variances of the measured frequency for the
unentangled and entangled probes are respectively
$\delta\omega^{2}_{0}|_{u}=\exp[2\Gamma(t_{u})]/(nTt_{u})$ and
$\delta\omega^{2}_{0}|_{e}=\exp[2n\Gamma(t_{e})]/(n^{2}Tt_{e})$, where
$\Gamma(t)=\int_{0}^{t}\gamma_{k}(t^{\prime})dt^{\prime}$ is the decoherence
factor for a single qubit, and the subscript $u$ ($e$) refers to the
unentangled (entangled) state. The optimal times to perform the measurement
are determined by $2t_{u}\gamma_{k}(t_{u})=1$ and
$2nt_{e}\gamma_{k}(t_{e})=1$, respectively. When the decoherence rate
$\gamma_{k}(t)=c$ is time-independent, the variances
$\delta\omega^{2}_{0}=2ce/(nT)$ for both initial states are the same, although
the optimal times are different, i.e., $t_{u}=1/(2c)$ v.s. $t_{e}=1/(2nc)$.
However, when the QZE occurs, i.e., $\gamma_{k}(t)=2ct$, the variance for the
unentangled probes is inferior to that for the entangled ones, as
$\delta\omega^{2}_{0}|_{u}=2\sqrt{ce}/(nT)>\delta\omega^{2}_{0}|_{e}=2\sqrt{nce}/(n^{2}T)$
with $t_{u}=(4c)^{-1/2}>t_{e}=(4nc)^{-1/2}$. Hereafter, we shall give an
introduction to the experimental details.

Figure 2: (a) Molecular structure and relevant parameters of the 7-qubit NMR
processor. Chemical shifts (diagonal, Hz), scalar coupling strengths (off-
diagonal, Hz), and relaxation times ($T_{1}$ and $T_{2}$) are all listed in
the table. (b) Pulse sequence for the 7-qubit Ramsey experiment using the
entangled probe under non-Markovian noise. To simulate the non-Markovian
dynamics, the rotating angles $\theta_{m}^{j}$ are carefully designed by
ensemble-averaging modulation. A step-by-step description of the sequence can
be found in SM sup .

_Experiment._ —The experiments are performed on a Bruker 600 MHz spectrometer
at room temperature. The sample is 13C-labeled trans-crotonic acid dissolved
in $d_{6}$-acetone, which consists of three 1H and four 13C nuclear spins,
forming a 7-qubit quantum processor. The molecular structure of the sample is
shown in Fig. 2(a). The internal Hamiltonian of this system can be described
as

$\mathcal{H}_{\rm{NMR}}=-\sum_{i=1}^{7}\frac{\omega_{i}}{2}\sigma^{z}_{i}+\sum_{i<j,=1}^{7}\frac{\pi}{2}J_{ij}\sigma^{z}_{i}\sigma^{z}_{j},$
(3)

where $\omega_{i}/2\pi$ is the Larmor frequency of the $i$-th spin, and
$J_{ij}$ is the scalar coupling strength between spins $i$ and $j$. The
corresponding parameters are listed in Fig. 2(a), as well as the relaxation
times $T_{1}$ and $T_{2}$.

Overall, the experiment aims to observe the enhancement of quantum metrology
with entangled probes under the non-Markovian circumstance. This is
demonstrated using the Ramsey magnetometry, which typically contains three
major stages: (i) preparation of initial probe states, (ii) accumulation of an
energy-splitting dependent phase, and (iii) measurement of that phase. In the
following, we will describe each stage in detail.

(i) Prepare the two types of probe states: the unentangled probe
$|\psi_{0}\rangle_{u}=[(|0\rangle+|1\rangle)/\sqrt{2}]^{\otimes n}$, and the
maximally entangled probe $|\psi_{0}\rangle_{e}=(|0\rangle^{\otimes
n}+|1\rangle^{\otimes n})/\sqrt{2}$. For the unentangled probe, the processor
is initialized to the pseudo-pure state $|0\rangle^{\otimes n}$ (up to an
identity matrix) from the thermal equilibrium state $\rho_{0}$ using the cat-
state method Knill  _et al._ (2000). Followed by a 2-ms-shaped pulse to
perform a collective single-qubit rotation $R_{y}(\pi/2)$ on all qubits, the
system is prepared into $|\psi_{0}\rangle_{u}$ with over $99\%$ fidelity. The
pulse sequence can be found in Supplemental Material (SM) sup .

For the entangled probe, we directly start from the thermal equilibrium
$\rho_{0}$, and eventually prepare the state $\rho_{e}=|0\rangle^{\otimes
n}\mbox{$\langle 1|$}+|1\rangle^{\otimes n}\mbox{$\langle 0|$}$ with the aid
of nearest-neighbour scalar couplings and gradient-echo techniques sup , as
shown in Fig. 2(b). Despite not the exact form of $|\psi_{0}\rangle_{e}$, the
dynamics of $\rho_{e}$ for quantum metrology purpose is the same since the
frequency information is just encoded in the phase accumulation of coherent
terms sup . In addition, the creation of $\rho_{e}$ avoids regular
initialization of the pseudo-pure state, which reduces the sequence complexity
remarkably.

(ii) For the target system
$H_{\text{S}}=\frac{\omega_{0}}{2}\sum_{i=1}^{n}\sigma^{z}_{i}$ with
$\omega_{0}$ being the energy splitting to be measured, let the probes precess
unperturbedly for a duration $t$ under the non-Markovian noise. This non-
Markovian circumstance is simulated by the aforementioned bath-engineering
technique. Explicitly, we introduce an additional time-dependent Hamiltonian
term in the form of
$H_{\text{SB}}^{m}(t)=\sum_{i=1}^{n}\beta^{m}_{i}(t)\sigma^{z}_{i}$, where
$\beta_{i}^{m}(t)$’s are stochastic errors subject to Drude-Lorentz spectral
density as shown in Eq. (1), and $m$ is the sampling number so as to
approximate the non-Markovian noise via the ensemble-averaging approach sup .
So the total Hamiltonian in the $m$-th experiment can be written as
$H_{\text{S}}+H_{\text{SB}}^{m}(t)$. Note that the evolution of this
Hamiltonian corresponds to a collective single-qubit rotation about the
$z$-axis, which can be easily implemented in most physical systems. In
addition, this time-dependent Hamiltonian can be discretized into time-
independent slices for experimental realization (here we use $10^{3}$ slices
sup ). Regarding the experimental parameters, we set $\omega_{0}=10$ kHz and
choose $m=20$ samples for ensemble average, which can already approximate the
non-Markovian environment with a high precision sup .

Figure 3: (a) Dynamics of population $P(t)$ for the unentangled (top panel)
and entangled (bottom panel) probes. Blue curves are obtained by numerically
fitting the experimental data (yellow triangles) using Eq. (4), and red lines
are envelopes with the form $(1-e^{-\Gamma(t)})/2$, which depicts the time-
dependent decoherence factor $\Gamma(t)$. The star symbol represents the
optimal measurement point determined by the experiment. (b) Standard deviation
$\delta\omega_{0}$ of the measured frequency with respect to the variation of
$t$. Dashed (solid) curves are fitting results corresponding to the
unentangled (entangled) probes for different number of qubits. Triangles and
dots are the optimal measurement points. The scaling behavior of these optimal
times when changing $n$ is shown in the inset.

(iii) Read out the accumulated phase. For the unentangled probes, the final
phase is encoded in each individual qubit, so the readout only involves
single-qubit measurements, which is similar to the standard Ramsey
interferometry experiment. For the entangled probes, the final phase is
retained in the relative phase between $|00...0\rangle$ and $|11...1\rangle$,
and we apply the multi-coherence measurement (MCM) technique Laflamme  _et
al._ (2002) to extract that phase. As shown in Fig. 2(b), this technique
involves a series of elementary quantum gates, which can be extended to other
systems as well. Subsequently, the transition probabilities are computed by
averaging over the ensemble with $m=20$ samples to simulate the non-Markovian
decay.

_Results._ —We have conducted Ramsey experiments for $n=2\sim 7$ qubits.
Taking the 7-qubit case as an example, the transition probability $P(t)$ is
shown in Fig. 3(a) for both the unentangled (top panel) and entangled (bottom
panel) probes. Compared to the unentangled case, $P(t)$ in the entangled case
manifests a much faster oscillation owing to the rapid accumulation ($\sim 7$
times) of phase. We use the theoretical form

$P(t)=\frac{1}{2}[1-\cos(n\omega_{0}t)e^{-n\Gamma(t)}]$ (4)

to fit the experimental data which gives the energy-splitting frequency
$\omega_{0}$ and decoherence factor $\Gamma(t)$. For the unentangled probes,
the measured frequency is $\omega_{0}|_{u}=9.871$ kHz and the decoherence
factor is $\Gamma_{u}=(0.370\pm 0.029)t^{2}$. For the entangled probes, the
measured frequency is $\omega_{0}|_{e}=10.142$ kHz and the decoherence factor
is $\Gamma_{e}=(2.701\pm 0.138)t^{2}$. The minimum standard deviations of the
frequencies are $\delta\omega_{0}|_{u}=0.169\pm 0.003$ kHz and
$\delta\omega_{0}|_{e}=0.105\pm 0.001$ kHz at the optimal measurement times
$t_{u}=0.797\pm 0.032$ ms and $t_{e}=0.288\pm 0.003$ ms, respectively. Hence,
the ratio of the sensitivity enhancement is
$r=\delta\omega_{0}|_{u}/\delta\omega_{0}|_{e}=1.608\pm 0.035\approx
7^{\frac{1}{4}}$, indicating that the entangled probes are indeed superior to
the unentangled ones under the non-Markovian environment. Note that the
readout of the NMR experiment is from the ensemble average result, which means
the statistical error by repeating experiments is almost negligible. So, in
this work, the error bars represent experimental uncertainties by error
propagation from the fitting uncertainties in Fig. 3(a); see SM for details
sup .

Figure 4: Experimental results of the ratio
$r=\delta\omega_{0}|_{u}/\delta\omega_{0}|_{e}$ (dots) and the minimum
standard deviation $\delta\omega_{0}|_{e}$ (triangles) for different number of
qubits $n$. Under the non-Markovian noise, the sensitivity is enhanced by
$r=n^{1/4}$ (solid red) with the entangled probes, reaching the QZE limit;
$\delta\omega_{0}|_{e}$ scales as $n^{-3/4}$ (solid blue). In the absence of
noise, the sensitivity enhancement is governed by the Heisenberg limit
$r=n^{1/2}$ (dashed red), and $\delta\omega_{0}|_{e}$ scales as $n^{-1}$
(dashed blue).

For the other numbers of qubits, the experimentally measured frequencies
$\omega_{u}$ and $\omega_{e}$, in association with their respective
decoherence factors $\Gamma_{u}$ and $\Gamma_{e}$, can be found in SM sup .
The standard deviation of the estimated frequency is obtained by processing
the experimental data, as shown in Fig. 3(b), where the dashed and solid lines
are for the unentangled and entangled probes, respectively. For the
unentangled probes, the optimal measurement time $t_{u}$ is almost the same
for different $n$’s and the minimum standard deviation $\delta\omega_{0}|_{u}$
is proportional to $n^{-1/2}$. For the entangled probes, $t_{e}$ and
$\delta\omega_{0}|_{e}$ decrease with the growth of $n$. By varying the number
of qubits, we compute the ratio
$r=\delta\omega_{0}|_{u}/\delta\omega_{0}|_{e}$ to estimate the improvement of
sensitivity with the entangled probes; see the solid red curve in Fig. 4. As
predicted, the experimentally-observed $r$ scales as $r\sim n^{1/4}$, which is
the QZE limit. For comparison, we have performed another group of experiments
in the absence of noise sup . In this noiseless scenario, the ratio of
sensitivity $r$ reaches the Heisenberg limit, scaling as $r\sim n^{1/2}$ by
the dashed red curve in Fig. 4.

In addition, we depict distinct tendencies of the minimum standard deviation
$\delta\omega_{0}|_{e}$ of the entangled probes for non-Markovian and
noiseless environments, respectively. As shown in Fig. 4, with the growth in
number of qubits, experimental results show that $\delta\omega_{0}|_{e}\propto
n^{-3/4}$ for the non-Markovian noise by solid blue curve, while
$\delta\omega_{0}|_{e}\propto n^{-1}$ for the noiseless case by dashed blue
curve. Definitely, our high-precision experiment is sufficient to verify that
the measurement accuracy of the entangled probes can reach the QZE limit under
the non-Markovian noise.

Conclusion.—We experimentally demonstrate that the entangled probes can
enhance the sensitivity of quantum metrology by the QZE. When the coherence
decays quadratically with time, the entangled probes optimize the time for
performing the measurement. On the contrary, when the decoherence rate is
time-independent, the entangled probes cannot effectively improve the
precision but shortens the measurement time. We remark that since our quantum
simulation approach can subtly engineer the parameters of both the system and
bath, it may provide an effective platform for experimentally verifying
various quantum metrology schemes, e.g., achieving ideal precision by coupling
to a bath with a band structure Bai  _et al._ (2019). And it can also be
implemented in other physical systems Soare  _et al._ (2014a); Potočnik  _et
al._ (2018).

Acknowledgement.—This work is supported by the National Key Research and
Development Program of China (2019YFA0308100), Beijing Natural Science
Foundation (1202017), Beijing Normal University (2022129), National Natural
Science Foundation of China (11674033, 12075110, 11975117, 11905099, 11875159,
11905111 and U1801661), Guangdong Basic and Applied Basic Research Foundation
(2019A1515011383), Guangdong International Collaboration Program
(2020A0505100001), Science, Technology and Innovation Commission of Shenzhen
Municipality (ZDSYS20190902092905285, KQTD20190929173815000,
JCYJ20200109140803865, and JCYJ20180302174036418), Pengcheng Scholars,
Guangdong Innovative and Entrepreneurial Research Team Program (2019ZT08C044),
and Guangdong Provincial Key Laboratory (2019B121203002).

## References

  * Caves (1981) C. M. Caves, Quantum-mechanical noise in an interferometer, Phys. Rev. D 23, 1693 (1981).
  * Giovannetti  _et al._ (2004) V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced measurements: Beating the standard quantum limit, Science 306, 1330 (2004).
  * Pezzè  _et al._ (2018) L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018).
  * Breuer and Petruccione (2002) H. P. Breuer and F. Petruccione,  _The Theory of Open Quantum Systems_ (Oxford University Presss, 2002).
  * Huelga  _et al._ (1997) S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, Improvement of frequency standards with quantum entanglement, Phys. Rev. Lett. 79, 3865 (1997).
  * Ma  _et al._ (2011) J. Ma, X. Wang, C. P. Sun, and F. Nori, Quantum spin squeezing, Phys. Rep. 509, 89 (2011).
  * Yamamoto  _et al._ (2022) K. Yamamoto, S. Endo, H. Hakoshima, Y. Matsuzaki, and Y. Tokunaga, Error-mitigated quantum metrology via virtual purification, arXiv:2112.01850 (2022).
  * Matsuzaki  _et al._ (2018) Y. Matsuzaki, S. Benjamin, S. Nakayama, S. Saito, and W. J. Munro, Quantum metrology beyond the classical limit under the effect of dephasing, Phys. Rev. Lett. 120, 140501 (2018).
  * Bai  _et al._ (2019) K. Bai, Z. Peng, H.-G. Luo, and J.-H. An, Retrieving ideal precision in noisy quantum optical metrology, Phys. Rev. Lett. 123, 040402 (2019).
  * Ai and Liao (2010) Q. Ai and J.-Q. Liao, Quantum anti-Zeno effect in artificial quantum systems, Commun. Theor. Phys. 54, 985 (2010).
  * Monz  _et al._ (2011) T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, 14-qubit entanglement: Creation and coherence, Phys. Rev. Lett. 106, 130506 (2011).
  * He  _et al._ (2021) W.-T. He, H.-Y. Guang, Z.-Y. Li, R.-Q. Deng, N.-N. Zhang, J.-X. Zhao, F.-G. Deng, and Q. Ai, Quantum metrology with one auxiliary particle in a correlated bath and its quantum simulation, Phys. Rev. A 104, 062429 (2021).
  * Kukita  _et al._ (2021) S. Kukita, Y. Matsuzaki, and Y. Kondo, Heisenberg-limited quantum metrology using collective dephasing, Phys. Rev. Applied 16, 064026 (2021).
  * Boixo  _et al._ (2007) S. Boixo, S. T. Flammia, C. M. Caves, and J. M. Geremia, Generalized limits for single-parameter quantum estimation, Phys. Rev. Lett. 98, 090401 (2007).
  * Nie  _et al._ (2015) X. Nie, J. Li, J. Cui, Z. Luo, J. Huang, H. Chen, C. Lee, X. Peng, and J. Du, Quantum simulation of interaction blockade in a two-site Bose–Hubbard system with solid quadrupolar crystal, New J. Phys. 17, 053028 (2015).
  * Choi and Sundaram (2008) S. Choi and B. Sundaram, Bose-Einstein condensate as a nonlinear Ramsey interferometer operating beyond the Heisenberg limit, Phys. Rev. A 77, 053613 (2008).
  * Roy and Braunstein (2008) S. M. Roy and S. L. Braunstein, Exponentially enhanced quantum metrology, Phys. Rev. Lett. 100, 220501 (2008).
  * Napolitano  _et al._ (2011) M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. J. Sewell, and M. W. Mitchell, Interaction-based quantum metrology showing scaling beyond the Heisenberg limit, Nature 471, 486 (2011) .
  * Nakazato  _et al._ (1996) H. Nakazato, M. Namiki, and S. Pascazio, Temporal behavior of quantum mechanical systems, Int. J. Mod. Phys. B 10, 247 (1996).
  * Ai  _et al._ (2010) Q. Ai, Y. Li, H. Zheng, and C. P. Sun, Quantum anti-Zeno effect without rotating wave approximation, Phys. Rev. A 81, 042116 (2010).
  * Ai  _et al._ (2013) Q. Ai, D. Xu, S. Yi, A. G. Kofman, C. P. Sun, and F. Nori, Quantum anti-Zeno effect without wave function reduction, Sci Rep 3, 1752 (2013).
  * Harrington  _et al._ (2017) P. M. Harrington, J. T. Monroe, and K. W. Murch, Quantum Zeno effects from measurement controlled qubit-bath interactions, Phys. Rev. Lett. 118, 240401 (2017).
  * Virzì  _et al._ (2022) S. Virzì, A. Avella, F. Piacentini, M. Gramegna, T. Opatrný, A. G. Kofman, G. Kurizki, S. Gherardini, F. Caruso, I. P. Degiovanni, and M. Genovese, Quantum Zeno and anti-Zeno probes of noise correlations in photon polarization, Phys. Rev. Lett. 129, 030401 (2022).
  * Kiilerich and Mølmer (2015) A. H. Kiilerich and K. Mølmer, Quantum Zeno effect in parameter estimation, Phys. Rev. A 92, 032124 (2015).
  * Do  _et al._ (2019) H.-V. Do, C. Lovecchio, I. Mastroserio, N. Fabbri, F. S. Cataliotti, S. Gherardini, M. M. Müller, N. D. Pozza, and F. Caruso, Experimental proof of quantum Zeno-assisted noise sensing, New J. Phys. 21, 113056 (2019).
  * Burgarth  _et al._ (2014) D. K. Burgarth, P. Facchi, V. Giovannetti, H. Nakazato, S. Pascazio, and K. Yuasa, Exponential rise of dynamical complexity in quantum computing through projections, Nat. Commun. 5, 5173 (2014).
  * Chin  _et al._ (2012) A. W. Chin, S. F. Huelga, and M. B. Plenio, Quantum metrology in non-Markovian environments, Phys. Rev. Lett. 109, 233601 (2012).
  * Matsuzaki  _et al._ (2011) Y. Matsuzaki, S. C. Benjamin, and J. Fitzsimons, Magnetic field sensing beyond the standard quantum limit under the effect of decoherence, Phys. Rev. A 84, 012103 (2011).
  * Wang  _et al._ (2018) B.-X. Wang, M.-J. Tao, Q. Ai, T. Xin, N. Lambert, D. Ruan, Y.-C. Cheng, F. Nori, F.-G. Deng, and G.-L. Long, Efficient quantum simulation of photosynthetic light harvesting, npj Quantum Inf. 4, 52 (2018).
  * Zhang  _et al._ (2021) N.-N. Zhang, M.-J. Tao, W.-T. He, X.-Y. Chen, X.-Y. Kong, F.-G. Deng, N. Lambert, and Q. Ai, Efficient quantum simulation of open quantum dynamics at various Hamiltonians and spectral densities, Front. Phys. 16, 51501 (2021).
  * Chen  _et al._ (2022) X.-Y. Chen, N.-N. Zhang, W.-T. He, X.-Y. Kong, M.-J. Tao, F.-G. Deng, Q. Ai, and G.-L. Long, Global correlation and local information flows in controllable non-Markovian open quantum dynamics, npj Quantum Inf. 8, 22 (2022).
  * Soare  _et al._ (2014a) A. Soare, H. Ball, D. Hayes, J. Sastrawan, M. C. Jarratt, J. J. McLoughlin, X. Zhen, T. J. Green, and M. J. Biercuk, Experimental noise filtering by quantum control, Nat. Phys. 10, 825 (2014a).
  * Soare  _et al._ (2014b) A. Soare, H. Ball, D. Hayes, X. Zhen, M. C. Jarratt, J. Sastrawan, H. Uys, and M. J. Biercuk, Experimental bath engineering for quantitative studies of quantum control, Phys. Rev. A 89, 042329 (2014b).
  * Khaneja  _et al._ (2005) N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms, J. Magn. Reson. 172, 296 (2005).
  * Li  _et al._ (2017) J. Li, X. Yang, X. Peng, and C.-P. Sun, Hybrid quantum-classical approach to quantum optimal control, Phys. Rev. Lett. 118, 150503 (2017).
  * Buluta and Nori (2009) I. Buluta and F. Nori, Quantum simulators, Science 326, 108 (2009).
  * Georgescu  _et al._ (2014) I. M. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Rev. Mod. Phys. 86, 153 (2014).
  * Luo  _et al._ (2012) S. Luo, S. Fu, and H. Song, Quantifying non-Markovianity via correlations, Phys. Rev. A 86, 044101 (2012).
  * Lu  _et al._ (2010) X.-M. Lu, X. Wang, and C. P. Sun, Quantum Fisher information flow and non-Markovian processes of open systems, Phys. Rev. A 82, 042103 (2010).
  * Altherr and Yang (2021) A. Altherr and Y. Yang, Quantum metrology for non-Markovian processes, Phys. Rev. Lett. 127, 060501 (2021).
  * Goodman (2015) J. W. Goodman,  _Statistical Optics_ (John Wiley & Sons, 2015).
  * Tao  _et al._ (2020) M.-J. Tao, N.-N. Zhang, P.-Y. Wen, F.-G. Deng, Q. Ai, and G.-L. Long, Coherent and incoherent theories for photosynthetic energy transfer, Sci. Bull. 65, 318 (2020).
  * (43) See the supplemental information for the complete description of the theory, experimental details and numerical simulations .
  * Knill  _et al._ (2000) E. Knill, R. Laflamme, R. Martinez, and C. H. Tseng, An algorithmic benchmark for quantum information processing, Nature 404, 368 (2000).
  * Laflamme  _et al._ (2002) R. Laflamme, D. Cory, C. Negrevergne, and L. Viola, NMR quantum information processing and entanglement, Quantum Inf. Comput. 2, 166 (2002).
  * Potočnik  _et al._ (2018) A. Potočnik, A. Bargerbos, F. A. Y. N. Schröder, S. A. Khan, M. C. Collodo, S. Gasparinetti, Y. Salathé, C. Creatore, C. Eichler, H. E. Türeci, A. W. Chin, and A. Wallraff, Studying light-harvesting models with superconducting circuits, Nat. Commun 9, 904 (2018).