# Note on the Time Dilation for Charged Quantum Clocks

Takeshi Chiba  Department of Physics, College of Humanities and Sciences,
Nihon University,
Tokyo 156-8550, Japan  Shunichiro Kinoshita  Department of Physics, College of
Humanities and Sciences, Nihon University,
Tokyo 156-8550, Japan  Department of Physics, Chuo University, Kasuga, Bunkyo-
ku, Tokyo 112-8551, Japan

###### Abstract

We derive the time dilation formula for charged quantum clocks in
electromagnetic fields. As a concrete example of non-inertial motion, we
consider a cyclotron motion in a uniform magnetic field. Applying the time
dilation formula to coherent state of the charged quantum clock, we evaluate
the time dilation quantum-mechanically.

##  I Introduction

Motivated by the tests of the weak equivalence principle in quantum regime, in
our previous study we derived a formula of the averaged proper time read by
one clock conditioned on another clock reading a different proper time in a
weak gravitational field Chiba and Kinoshita (2022). The time dilation
measured by these quantum clocks is found to have the same form as that in
classical relativity. There, clocks are assumed to be in their inertial motion
and their classical trajectories are the geodesics of the spacetime, that is,
the clocks are always free-falling.

Then, it would also be interesting to study what would happen for clocks in
non-inertial motion. Any non-inertial motion should be caused by other
external force than gravitational interaction. In order to study the effect of
non-inertial motion on the time dilation of quantum clocks, we consider
charged quantum clocks interacting with the external electromagnetic fields.
The study of a quantum charged particle is also interesting in the light of
quantum mechanics in a rotating frame because there exists a close analogy
between the motion in a rotating frame and the motion in a magnetic field
Sakurai (1980). Also, a new class of optical clocks with highly charged ions
has been received interest recent years as references for highest-accuracy
clocks and precision tests of fundamental physics Kozlov  _et al._ (2018);
King  _et al._ (2022). Such an optical clock based on a highly charged ion was
recently realized King  _et al._ (2022). Our study may be applicable to such
clocks.

The paper is organized as follows. In Sec. II, we derive the time dilation
formula for charged particles in electromagnetic fields and weak gravitational
fields as the average of a proper time observable for a quantum clock. In Sec.
III, as a non-inertial motion, we consider the cyclotron motion in a uniform
magnetic field. We evaluate the quantum time dilation by using the coherent
state. In Appendix A, we summarize the several results of the coherent state
for the cyclotron motion in quantum mechanics.

##  II Charged Quantum Clock Particles in Spacetime

###  II.1 Classical Particles

We consider a system of $N$ charged massive particles. Each particle whose
mass and charge are $m_{n}$ and $q_{n}~{}(n=1,\dots,N)$ has a set of internal
degrees of freedom, labeled by the configuration variables $\chi_{n}$ and
their conjugate momenta $P_{\chi_{n}}$Smith and Ahmadi (2020). These internal
degrees of freedom are supposed to represent the quantum clock.

The action of such a system in a curved spacetime with the metric $g_{\mu\nu}$
and an electromagnetic field $A_{\mu}$ is given by

$\displaystyle S=\sum_{n}\int
d\tau_{n}\left(-m_{n}c^{2}+q_{n}A_{\mu}\frac{dx_{n}^{\mu}}{d\tau_{n}}+P_{\chi_{n}}\frac{d\chi_{n}}{d\tau_{n}}-H_{n}^{\rm
clock}\right),$ (1)

where $\tau_{n}$ is the proper time of the $n$th particle and $H_{n}^{\rm
clock}=H_{n}^{\rm clock}(\chi_{n},P_{\chi_{n}})$ is a Hamiltonian for its
internal degrees of freedom.

Let $x_{n}^{\mu}$ denote the spacetime position of the $n$th particle. The
trajectory of the $n$th particle $x_{n}^{\mu}(t)$ is parameterized by an
arbitrary external time parameter $t$. Noting that
$cd\tau_{n}=\sqrt{-g_{\mu\nu}\dot{x}_{n}^{\mu}\dot{x}_{n}^{\nu}}dt\equiv\sqrt{-\dot{x}^{2}_{n}}dt$,
where a dot denotes differentiation with respect to $t$, the action is
rewritten as

$\displaystyle S=\int
dt\sum_{n}\frac{1}{c}\sqrt{-\dot{x}_{n}^{2}}\left(-m_{n}c^{2}+q_{n}A_{\mu}\frac{\dot{x}_{n}^{\mu}c}{\sqrt{-\dot{x}_{n}^{2}}}+P_{\chi_{n}}\frac{\dot{\chi}_{n}c}{\sqrt{-\dot{x}_{n}^{2}}}-H_{n}^{\rm
clock}\right)=:\int dt~{}L.$ (2)

The momentum conjugate to $x_{n}^{\mu}$ is given by

$\displaystyle P_{n\mu}=\frac{\partial
L}{\partial\dot{x}_{n}^{\mu}}=\frac{g_{\mu\nu}\dot{x}_{n}^{\nu}}{c\sqrt{-\dot{x}_{n}^{2}}}\left(m_{n}c^{2}+H_{n}^{\rm
clock}\right)+q_{n}A_{\mu}.$ (3)

Then the Hamiltonian associated with the Lagrangian $L$ is constrained to
vanish:

$\displaystyle
H=\sum_{n}\left(P_{n\mu}\dot{x}_{n}^{\mu}+P_{\chi_{n}}\dot{\chi}_{n}\right)-L\approx
0.$ (4)

In terms of the momentum, the constraints can be expressed in the form

$\displaystyle
C_{H_{n}}:=g^{\mu\nu}(P_{n\mu}-q_{n}A_{\mu})(P_{n\nu}-q_{n}A_{\nu})c^{2}+\left(m_{n}c^{2}+H_{n}^{\rm
clock}\right)^{2}\approx 0.$ (5)

Using the $(3+1)$ decomposition of the metric in terms of the lapse function
$\alpha$, the shift vector $\beta^{i}$ and the three-metric $\gamma_{ij}$ such
that Misner  _et al._ (2017)

$\displaystyle
ds^{2}=-\alpha^{2}c^{2}dt^{2}+\gamma_{ij}(dx^{i}+\beta^{i}cdt)(dx^{j}+\beta^{j}cdt),$
(6)

the constraint is factorized in the form

$\displaystyle C_{H_{n}}$ $\displaystyle=$
$\displaystyle-\alpha^{-2}\left(P_{n0}-q_{n}A_{0}-\beta^{i}\left(P_{ni}-q_{n}A_{i}\right)\right)^{2}c^{2}+\gamma^{ij}(P_{ni}-q_{n}A_{i})(P_{nj}-q_{n}A_{j})c^{2}+\left(m_{n}c^{2}+H_{n}^{\rm
clock}\right)^{2}$ (7) $\displaystyle=$
$\displaystyle-\alpha^{-2}C_{n}^{+}C_{n}^{-}\approx 0,$

where $C_{n}^{\pm}$ is defined by

$\displaystyle C_{n}^{\pm}$ $\displaystyle:=$
$\displaystyle\left(P_{n0}-q_{n}A_{0}-\beta^{i}\left(P_{ni}-q_{n}A_{i}\right)\right)c\pm
h_{n},$ (8) $\displaystyle h_{n}$ $\displaystyle:=$
$\displaystyle\alpha\sqrt{\gamma^{ij}(P_{ni}-q_{n}A_{i})(P_{nj}-q_{n}A_{j})c^{2}+(m_{n}c^{2}+H_{n}^{\rm
clock})^{2}}.$ (9)

Note that we have set $x^{0}=ct$. Hereafter we assume that the spacetime is
stationary. The coordinates $x^{\mu}_{n}$ and their conjugate momenta
$P_{n\mu}$ satisfy the fundamental Poisson brackets:
$\\{x_{m}^{\mu},P_{n\nu}\\}=\delta_{mn}\delta^{\mu}_{\nu}$. The canonical
momentum $P_{n\mu}$ generates translations in the spacetime coordinate
$x_{n}^{\mu}$. Therefore, if $C_{n}^{\pm}\approx 0$, then $\pm
h_{n}-q_{n}A_{0}c-\beta^{i}c\left(P_{ni}-q_{n}A_{i}\right)$ is the generator
of translation in the $n$th particle’s time coordinate and is the Hamiltonian
for both the external and internal degrees of freedom of the $n$th particle.

###  II.2 Quantization

We canonically quantize the system of $N$ particles by promoting the phase
space variables to operators acting on appropriate Hilbert spaces: $x^{0}_{n}$
and $P_{n0}$ become canonically conjugate self-adjoint operators acting on the
Hilbert space ${\cal H}^{0}_{n}\simeq L^{2}(\mathbb{R})$ associated with the
$n$th particle’s temporal degree of freedom; operators $x_{n}^{i}$ and
$P_{ni}$ acting on the Hilbert space ${\cal H}_{n}^{\rm ext}\simeq
L^{2}(\mathbb{R}^{3})$ associated with the particle’s external degrees of
freedom; operators $\chi_{n}$ and $P_{\chi_{n}}$ acting on the Hilbert space
${\cal H}_{n}^{\rm clock}$ associated with the particle’s internal degrees of
freedom. Then the Hilbert space describing the $n$th particle is ${\cal
H}_{n}\simeq{\cal H}^{0}_{n}\otimes{\cal H}_{n}^{\rm ext}\otimes{\cal
H}_{n}^{\rm clock}$.

The constraint equations (7) now become operator equations restricting the
physical state of the theory,

$\displaystyle C_{n}^{+}C_{n}^{-}|\Psi\rangle\rangle=0,~{}~{}~{}~{}\forall n,$
(10)

where $|\Psi\rangle\rangle\in{\cal H}_{\rm phys}$ is a physical state of a
clock $C$ and a system $S$ and lives in the physical Hilbert space ${\cal
H}_{\rm phys}$.

To specify ${\cal H}_{\rm phys}$, the normalization of the physical state in
${\cal H}_{\rm phys}$ is performed by projecting a physical state
$|\Psi\rangle\rangle$ onto a subspace in which the temporal degree of freedom
of each particle (clock $C$) is in an eigenstate $|t_{n}\rangle$ of the
operator $x_{n}^{0}$ associated with the eigenvalue $t\in\mathbb{R}$ in the
spectrum of $x_{n}^{0}$: $x_{n}^{0}|t_{n}\rangle=ct|t_{n}\rangle$. The state
of $S$ by conditioning $|\Psi\rangle\rangle$ on $C$ reading the time $t$ is
then given by

$\displaystyle|\psi_{S}(t)\rangle=\langle t|\otimes I_{S}|\Psi\rangle\rangle,$
(11)

where $|t\rangle=\otimes_{n}|t_{n}\rangle$ and $I_{S}$ is the identity on
${\cal H}\simeq\bigotimes_{n}{\cal H}_{n}^{\rm ext}\otimes{\cal H}_{n}^{\rm
clock}$. We demand that the state $|\psi_{S}(t)\rangle$ is normalized as
$\langle\psi_{S}(t)|\psi_{S}(t)\rangle=1$ for $\forall t\in\mathbb{R}$ on a
spacelike hypersurface defined by all $N$ particles’ temporal degree of
freedom being in the state $|t_{n}\rangle$. The physical state
$|\Psi\rangle\rangle$ is thus normalized with respect to the inner product
Smith and Ahmadi (2020):

$\displaystyle\langle\langle\Psi|\Psi\rangle\rangle_{PW}:=\langle\langle\Psi||t\rangle\langle
t|\otimes I_{S}|\Psi\rangle\rangle=\langle\psi_{S}(t)|\psi_{S}(t)\rangle=1,$
(12)

and the physical state $|\Psi\rangle\rangle$ can be written as

$\displaystyle|\Psi\rangle\rangle=\int dt|t\rangle\langle t|\otimes
I_{S}|\Psi\rangle\rangle=\int dt|t\rangle|\psi_{S}(t)\rangle.$ (13)

Hereafter, we consider physical states that satisfy
$C_{n}^{+}|\Psi\rangle\rangle=0$ for all $n\in\mathbb{N}$. It can be shown
that the conditioned state $|\psi_{S}(t)\rangle$ satisfies the Schrödinger
equation with $t$ as a time parameter Smith and Ahmadi (2020):

$\displaystyle
i\hbar\frac{d}{dt}|\psi_{S}(t)\rangle=H_{S}|\psi_{S}(t)\rangle,$ (14)

where $H_{S}$ is given by

$\displaystyle
H_{S}=\sum_{n}\left(h_{n}-q_{n}A_{0}c-\beta^{i}c\left(P_{ni}-q_{n}A_{i}\right)\right)\otimes
I_{S-n}\equiv\sum_{n}\widetilde{h_{n}}\otimes I_{S-n}$ (15)

with $I_{S-n}$ being the identity on $\bigotimes_{m\neq n}{\cal H}^{\rm
ext}_{m}\otimes{\cal H}^{\rm clock}_{m}$. Therefore, $|\psi_{S}(t)\rangle$ can
be regarded as the time-dependent state of the $N$-particles with the
Hamiltonian $H_{S}$ evolved with the external time $t$.

###  II.3 Probabilistic Time Dilation

Consider two clock particles $\mathrm{A}$ and $\mathrm{B}$ with internal
degrees of freedom. Each clock is defined to be the quadrupole $\\{{\cal
H}^{\rm clock}_{n},\rho_{n},H^{\rm clock}_{n},T_{n}\\}$, where $\rho_{n}$ is a
fiducial state and $T_{n}$ is proper time observable for
$n\in\\{\mathrm{A},\mathrm{B}\\}$. The proper time observable is defined as a
POVM (positive operator valued measure)

$\displaystyle T_{n}:=\left\\{E_{n}(\tau)~{}\forall\tau\in G~{}{\rm
s.t.}\int_{G}d\tau E_{n}(\tau)=I_{n}\right\\},$ (16)

where $E_{n}(\tau)=|\tau\rangle\langle\tau|$ is a positive operator on ${\cal
H}^{\rm clock}_{n}$, $G$ is the group generated by $H^{\rm clock}_{n}$, and
$|\tau\rangle$ is a clock state associated with a measurement of the clock
yielding the time $\tau$.

To probe time dilation effects between two clocks, we consider the probability
that clock $\mathrm{A}$ reads the proper time $\tau_{\mathrm{A}}$ conditioned
on clock $\mathrm{B}$ reading the proper time $\tau_{\mathrm{B}}$ Page and
Wootters (1983); Wootters (1984). This conditional probability is given in
terms of the physical state as

$\displaystyle{\rm
Prob}[T_{\mathrm{A}}=\tau_{\mathrm{A}}|T_{\mathrm{B}}=\tau_{\mathrm{B}}]=\frac{\langle\langle\Psi|E_{\mathrm{A}}(\tau_{\mathrm{A}})E_{\mathrm{B}}(\tau_{\mathrm{B}})|\Psi\rangle\rangle}{\langle\langle\Psi|E_{\mathrm{B}}(\tau_{\mathrm{B}})|\Psi\rangle\rangle}\,.$
(17)

Consider the case where two clock particles $\mathrm{A}$ and $\mathrm{B}$ are
moving in a curved spacetime. Suppose that initial conditioned state is
unentangled,
$|\psi_{S}(0)\rangle=|\psi_{S_{\mathrm{A}}}\rangle|\psi_{S_{\mathrm{B}}}\rangle$,
and that the external and internal degrees of freedom of both particles are
unentangled, $|\psi_{S_{n}}\rangle=|\psi_{n}^{\rm ext}\rangle|\psi_{n}^{\rm
clock}\rangle$. Then, from Eq. (13), the physical state takes the form

$\displaystyle|\Psi\rangle\rangle=\int dt|t\rangle|\psi_{S}(t)\rangle=\int
dt\bigotimes_{n\in\\{\mathrm{A},\mathrm{B}\\}}e^{-i\widetilde{h}_{n}t/\hbar}|\psi_{n}^{\rm
ext}\rangle|\psi_{n}^{\rm clock}\rangle\,,$ (18)

where $\widetilde{h}_{n}$ is defined in Eq. (15). Further suppose that ${\cal
H}^{\rm clock}_{n}\simeq L^{2}(\mathbb{R})$ so that we may consider an ideal
clock such that $P_{n}=H^{\rm clock}_{n}/c$ and $cT_{n}$ are the momentum and
position operators on ${\cal H}^{\rm clock}_{n}$. The canonical commutation
relation yields $[cT_{n},P_{n}]=[T_{n},H^{\rm clock}_{n}]=i\hbar$. Then, the
clock states are orthogonal
$\langle\tau|\tau^{\prime}\rangle=\delta(\tau-\tau^{\prime})$ and satisfy the
covariance relation $|\tau+\tau^{\prime}\rangle=e^{-iH^{\rm
clock}_{n}\tau^{\prime}/\hbar}|\tau\rangle$. The conditional probability (17)
becomes

$\displaystyle{\rm
Prob}[T_{\mathrm{A}}=\tau_{\mathrm{A}}|T_{\mathrm{B}}=\tau_{\mathrm{B}}]=\frac{\int
dt~{}{\rm tr}[E_{\mathrm{A}}(\tau_{\mathrm{A}})\rho_{\mathrm{A}}(t)]{\rm
tr}[E_{\mathrm{B}}(\tau_{\mathrm{B}})\rho_{\mathrm{B}}(t)]}{\int dt~{}{\rm
tr}[E_{\mathrm{B}}(\tau_{\mathrm{B}})\rho_{\mathrm{B}}(t)]},$ (19)

where $\rho_{n}(t)$ is the reduced state of the internal clock degrees of
freedom defined as Smith and Ahmadi (2020)

$\displaystyle\rho_{n}(t)={\rm tr}_{{\cal H}_{S}\backslash{\cal H}_{n}^{\rm
clock}}\left(e^{-iH_{S}t/\hbar}|\psi_{S_{n}}\rangle\langle\psi_{S_{n}}|e^{iH_{S}t/\hbar}\right)$
(20)

with the trace over the complement of the clock Hilbert space.

We assume that the fiducial states of the internal clock degrees of freedom
are the Gaussian wave packets centered at $\tau=0$ with width $\sigma$:

$\displaystyle|\psi_{n}^{\rm clock}\rangle=\frac{1}{\pi^{1/4}\sigma^{1/2}}\int
d\tau~{}e^{-\frac{\tau^{2}}{2\sigma^{2}}}|\tau\rangle\,.$ (21)

Note that in evaluating the conditional probability (19) by using Eq. (20) and
Eq. (21), the terms in the Hamiltonian $H_{S}$ (15) which involve both the
clock Hamiltonian $H_{n}^{\rm clock}$ and the external degrees of freedom
survive. Therefore, as in our previous study Chiba and Kinoshita (2022), the
conditional probability depends only on $h_{n}$ defined in Eq. (9) and is
independent of the terms in Hamiltonian $H_{S}$ which depend only on the
external degrees of freedom (such as $A_{0}$ and $\beta^{i}$).

###  II.4 Time Dilation

In order to find the coupling of the clock Hamiltonian $H_{n}^{\rm clock}$ and
the external degrees of freedom, we expand $\widetilde{h_{n}}$ in the
effective Hamiltonian (15) according to the power of $H_{n}^{\rm clock}$
assuming $H_{n}^{\rm clock}\ll m_{n}c^{2}$

$\displaystyle\widetilde{h_{n}}$ $\displaystyle=$
$\displaystyle\alpha\sqrt{\gamma^{ij}(P_{ni}-q_{n}A_{i})(P_{nj}-q_{n}A_{j})c^{2}+m_{n}^{2}c^{4}}-q_{n}A_{0}c-\beta^{i}c\left(P_{ni}-q_{n}A_{i}\right)$
(22) $\displaystyle+\frac{\alpha
m^{2}c^{4}}{\sqrt{\gamma^{ij}(P_{ni}-q_{n}A_{i})(P_{nj}-q_{n}A_{j})c^{2}+m_{n}^{2}c^{4}}}\frac{H_{n}^{\rm
clock}}{m_{n}c^{2}}+O((H_{n}^{\rm clock}/m_{n}c^{2})^{2}).$

The term in the second line which involves both the clock Hamiltonian
$H_{n}^{\rm clock}$ and the external degrees of freedom is relevant in
calculating the conditional probability. One may recognize that the
coefficient of $H_{n}^{\rm clock}$ is (minus of) the kinetic term of the
$n$-th particle in the Lagrangian (2), that is,
$m_{n}c\sqrt{-\dot{x}_{n}^{2}}=m_{n}c^{2}d\tau_{n}/dt$. This implies that the
average of the time dilation would be given by the same form as the classical
time dilation formula in the leading order of the clock Hamiltonian. In other
words, regardless of inertial or non-inertial motions, the time dilation would
be given by difference of the proper time and distance between trajectories of
each particle.

As a concrete example, in the Newtonian approximation of spacetime, the metric
is given by $g_{00}=-\alpha^{2}=-(1+2\Phi({\bf
x})/c^{2}),\gamma_{ij}=\delta_{ij}$, and $\beta^{i}=0$, where $\Phi({\bf x})$
is the Newtonian gravitational potential. $\widetilde{h_{n}}$ is then further
expanded according to the number of the inverse power of $c^{2}$ as

$\displaystyle\widetilde{h_{n}}=m_{n}c^{2}+H_{n}^{\rm clock}+H_{n}^{\rm
ext}+H_{n}^{\rm int}+O(c^{-4}),$ (23)

where the rest-mass energy term $m_{n}c^{2}$ is a constant and can be
disregarded in $h_{n}$. The external Hamiltonian $H_{n}^{\rm ext}$ and the
interaction Hamiltonian $H_{n}^{\rm int}$ are given by

$\displaystyle H_{n}^{\rm ext}$ $\displaystyle:=$
$\displaystyle\frac{\delta^{ij}(P_{ni}-q_{n}A_{ni})(P_{nj}-q_{n}A_{nj})}{2m_{n}}+m_{n}\Phi_{n}-q_{n}A_{n0}c\equiv\frac{({\bf
P}_{n}-q_{n}{\bf A}_{n})^{2}}{2m_{n}}+m_{n}\Phi_{n}-q_{n}A_{n0}c,$ (24)
$\displaystyle H_{n}^{\rm int}$ $\displaystyle:=$ $\displaystyle-\frac{({\bf
P}_{n}-q_{n}{\bf A}_{n})^{2}H_{n}^{\rm
clock}}{2m_{n}^{2}c^{2}}+\frac{\Phi_{n}H_{n}^{\rm
clock}}{c^{2}}-\frac{1}{2m_{n}c^{2}}\left(\frac{({\bf P}_{n}-q_{n}{\bf
A}_{n})^{2}}{2m_{n}}-m_{n}\Phi_{n}\right)^{2}+O(c^{-4})\,,$ (25)

where $A_{n\mu}:=A_{\mu}({\bf x}_{n}),\Phi_{n}:=\Phi({\bf x}_{n})$.

The reduced state of the internal clock becomes

$\displaystyle\rho_{n}(t)$ $\displaystyle=$ $\displaystyle{\rm tr}_{{\cal
H}_{S}\backslash{\cal H}_{n}^{\rm
clock}}\left[e^{-iH_{S}t/\hbar}|\psi_{S_{n}}\rangle\langle\psi_{S_{n}}|e^{iH_{S}t/\hbar}\right]$
(26) $\displaystyle=$ $\displaystyle\bar{\rho}_{n}(t)-it~{}{\rm tr}_{\rm
ext}\left([H_{n}^{\rm int},\bar{\rho}_{n}^{\rm
ext}(t)\otimes\bar{\rho}_{n}(t)]+O((H_{n}^{\rm int}t)^{2})\right)$
$\displaystyle=$ $\displaystyle\bar{\rho}_{n}(t)+it\left(\frac{\langle({\bf
P}_{n}-q_{n}{\bf
A}_{n})^{2}\rangle}{2m_{n}^{2}c^{2}}-\frac{\langle\Phi_{n}\rangle}{c^{2}}\right)[H_{n}^{\rm
clock},\bar{\rho}_{n}(t)]+O(c^{-4}),$

where $\bar{\rho}_{n}(t)=e^{-iH_{n}^{\rm clock}t/\hbar}\rho_{n}e^{iH_{n}^{\rm
clock}t/\hbar}$ and $\bar{\rho}_{n}^{\rm ext}(t)=e^{-iH_{n}^{\rm
ext}t/\hbar}\rho_{n}^{\rm ext}e^{iH_{n}^{\rm ext}t/\hbar}$. The conditional
probability (17) is evaluated to leading relativistic order as

$\displaystyle{\rm
Prob}[T_{\mathrm{A}}=\tau_{\mathrm{A}}|T_{\mathrm{B}}=\tau_{\mathrm{B}}]$
$\displaystyle=$
$\displaystyle\frac{e^{-\frac{(\tau_{\mathrm{A}}-\tau_{\mathrm{B}})^{2}}{2\sigma^{2}}}}{\sqrt{2\pi}\sigma}\left[1+\left(\frac{\langle({\bf
P}_{\mathrm{A}}-q_{\mathrm{A}}{\bf
A}_{\mathrm{A}})^{2}\rangle}{4m_{\mathrm{A}}^{2}c^{2}}-\frac{\langle({\bf
P}_{\mathrm{B}}-q_{\mathrm{B}}{\bf
A}_{\mathrm{B}})^{2}\rangle}{4m_{\mathrm{B}}^{2}c^{2}}-\frac{\langle\Phi_{\mathrm{A}}\rangle}{2c^{2}}+\frac{\langle\Phi_{\mathrm{B}}\rangle}{2c^{2}}\right)\right.$
(27)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\times\left(1-\frac{\tau_{\mathrm{A}}^{2}-\tau_{\mathrm{B}}^{2}}{\sigma^{2}}\right)\right],$

where $\langle H_{n}^{\rm ext}\rangle=\langle\psi^{\rm ext}_{n}|H_{n}^{\rm
ext}|\psi_{n}^{\rm ext}\rangle$. Then the average proper time read by clock
$\mathrm{A}$ conditioned on clock $\mathrm{B}$ indicating the time
$\tau_{\mathrm{B}}$ is

$\displaystyle\langle T_{\mathrm{A}}\rangle$ $\displaystyle=$
$\displaystyle\int d\tau~{}{\rm
Prob}[T_{\mathrm{A}}=\tau|T_{\mathrm{B}}=\tau_{\mathrm{B}}]\tau$ (28)
$\displaystyle=$
$\displaystyle\tau_{\mathrm{B}}\left[1-\left(\frac{\langle({\bf
P}_{\mathrm{A}}-q_{\mathrm{A}}{\bf
A}_{\mathrm{A}})^{2}\rangle}{2m_{\mathrm{A}}^{2}c^{2}}-\frac{\langle\Phi_{\mathrm{A}}\rangle}{c^{2}}\right)+\left(\frac{\langle({\bf
P}_{\mathrm{B}}-q_{\mathrm{B}}{\bf
A}_{\mathrm{B}})^{2}\rangle}{2m_{\mathrm{B}}^{2}c^{2}}-\frac{\langle\Phi_{\mathrm{B}}\rangle}{c^{2}}\right)\right]\,.$

This is the quantum analog of time dilation formula for the charged particles
in the Newtonian gravity. Noting that the time evolution of the position
$\dot{\bf x}_{n}=\frac{[{\bf x}_{n},H_{n}^{\rm ext}]}{i\hbar}=\frac{{\bf
P}_{n}-q_{n}{\bf A}_{n}}{m_{n}}$ from the Heisenberg equation,111Note that the
equation becomes $\dot{\bf x}_{n}=\frac{{\bf P}_{n}-q_{n}{\bf
A}_{n}}{m_{n}}-\boldsymbol{\beta}c$ in the presence of the shift vector.  one
may recognize that this time dilation formula has the same form as the
classical time dilation in the Newtonian gravity. The time dilation of a
clock, regargless of whether it is in inertial motion or non-inertial motion,
is induced by its velocity and gravitational potential.

##  III Time Dilation in a Uniform Magnetic Field

As an application of the time dilation formula of Eq. (28), we consider the
motion of a charged particle in a uniform magnetic field $B$ along the $z$
direction. The quantum mechanics of the charged particle and the coherent
state are discussed in detail in Appendix A. For the particle moving in the
$xy$-plane in the flat spacetime, the Hamiltonian is given by

$\displaystyle H_{n}^{\rm ext}=\frac{({\bf P}_{n}-q_{n}{\bf
A}_{n})^{2}}{2m_{n}}.$ (29)

The time dilation formula (28) is reduced to the difference of the Hamiltonian

$\displaystyle\langle
T_{\mathrm{A}}\rangle=\tau_{\mathrm{B}}\left(1-\frac{\langle
H_{\mathrm{A}}^{\rm ext}\rangle}{m_{\mathrm{A}}c^{2}}+\frac{\langle
H_{\mathrm{B}}^{\rm ext}\rangle}{m_{\mathrm{B}}c^{2}}\right).$ (30)

Since $H_{n}^{\rm ext}$ does not depend on the external time $t$ explicitly,
the expectation value of $H_{n}^{\rm ext}$ is conserved. Therefore, the time
dilation does not depend on $t$ in contrast to the gravitational time dilation
Chiba and Kinoshita (2022). In the following, we calculate the time dilation
between a charged quantum clock $\mathrm{A}$ (with its charge
$q_{\mathrm{A}}$) and an uncharged ($q_{\mathrm{B}}=0$) quantum clock
$\mathrm{B}$ for coherent state. We also note that as explained in Appendix A
the time dilation formula (30) does not change even if we move to a rotating
frame.

###  III.1 Time Dilation in Coherent State

It is known that the cyclotron motion of a charged particle in a uniform
magnetic field can be quantum mechanically well-described by the coherent
state. We consider the coherent state $|\alpha,\beta\rangle$ defined by Eq.
(56) for the charged clock $\mathrm{A}$.222$\alpha$ should not be confused
with the lapse function in Eq. (6).  Introducing the cyclotron frequency
$\omega_{\mathrm{A}}=q_{\mathrm{A}}B/m_{\mathrm{A}}$ and the radius of the
cyclotron motion $r_{0}$, the center of the cyclotron motion $(X_{0},Y_{0})$
is related to $\beta$ as
$X_{0}-iY_{0}=\sqrt{\frac{2\hbar}{m_{\mathrm{A}}\omega_{\mathrm{A}}}}\beta$
and the relative position $(r_{0}\cos\theta_{0},r_{0}\sin\theta_{0})$ is
related to $\alpha$ as
$r_{0}e^{i\theta_{0}}=\sqrt{\frac{2\hbar}{m_{\mathrm{A}}\omega_{\mathrm{A}}}}\alpha$.
Note that the uniform magnetic field $B$ is given by the vector potential
${\bf A}=\frac{B}{2}(-y,x,0)$ in the symmetric gauge. From Eq. (64), the
expectation value of the external Hamiltonian of the clock $\mathrm{A}$
becomes

$\displaystyle\langle\alpha,\beta|H_{\mathrm{A}}^{\rm
ext}|\alpha,\beta\rangle=\hbar\omega_{\mathrm{A}}\left(|\alpha|^{2}+\frac{1}{2}\right)=\frac{1}{2}m_{\mathrm{A}}\omega_{\mathrm{A}}^{2}r_{0}^{2}+\frac{1}{2}\hbar\omega_{\mathrm{A}}\,.$
(31)

On the other hand, we assume that the state of the uncharged clock
$\mathrm{B}$ is a Gaussian state centered at
$(x_{\mathrm{B}},y_{\mathrm{B}})=(x_{\mathrm{B}0},y_{\mathrm{B}0})$ with width
$\sigma_{\mathrm{B}}$, whose wave function is

$\displaystyle\langle{\bf
x}_{\mathrm{B}}|\psi_{\mathrm{B}}\rangle=\left(\pi\sigma_{\mathrm{B}}^{2}\right)^{-1/2}\exp\left[-\frac{(x_{\mathrm{B}}-x_{\mathrm{B}0})^{2}+(y_{\mathrm{B}}-y_{\mathrm{B}0})^{2}}{2\sigma_{\mathrm{B}}^{2}}\right]\,.$
(32)

Then, the expectation value of the external Hamiltonian of the clock
$\mathrm{B}$ becomes

$\displaystyle\langle\psi_{\mathrm{B}}|H_{\mathrm{B}}^{\rm
ext}|\psi_{\mathrm{B}}\rangle=\frac{\hbar^{2}}{2m_{\mathrm{B}}\sigma_{\mathrm{B}}^{2}}\,.$
(33)

Putting these together, the observed average time dilation between two clocks
is given by

$\displaystyle\langle T_{\mathrm{A}}\rangle$ $\displaystyle=$
$\displaystyle\tau_{\mathrm{B}}\left(1-\frac{\langle H_{\mathrm{A}}^{\rm
ext}\rangle}{m_{\mathrm{A}}c^{2}}+\frac{\langle H_{\mathrm{B}}^{\rm
ext}\rangle}{m_{\mathrm{B}}c^{2}}\right)$ (34) $\displaystyle=$
$\displaystyle\tau_{\mathrm{B}}\left(1-\frac{\omega_{\mathrm{A}}^{2}r_{0}^{2}}{2c^{2}}-\frac{\hbar\omega_{\mathrm{A}}}{2m_{\mathrm{A}}c^{2}}+\frac{\hbar^{2}}{2m_{\mathrm{B}}^{2}c^{2}\sigma_{\mathrm{B}}^{2}}\right)\,.$

###  III.2 Superposition

Figure 1:  The superposition of two coherent states $|\alpha,\beta\rangle$ and
$|\alpha^{*},\beta\rangle$. The radius of the circle is
$r_{0}=\sqrt{2\hbar/m_{\mathrm{A}}\omega_{\mathrm{A}}}|\alpha|$ and the
angular separation is $2\theta_{0}=2\tan^{-1}({\rm Im}~{}\alpha/{\rm
Re}~{}\alpha)$.

Next, we consider two clocks $\mathrm{A}$ and $\mathrm{B}$ and suppose that
initially clock $\mathrm{A}$ is in a superposition of two coherent state
Schleich  _et al._ (1991):

$\displaystyle|\psi_{\mathrm{A}}\rangle=\frac{1}{\sqrt{N}}\left(|\alpha,\beta\rangle+e^{i\phi}|\alpha^{\prime},\beta\rangle\right)\,.$
(35)

Two coherent states are assumed to have the same center of circle, namely the
same $\beta$, but have different positions on the circle as shown in Fig. 1 :

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\sqrt{\frac{m_{\mathrm{A}}\omega_{\mathrm{A}}}{2\hbar}}r_{0}e^{i\theta_{0}},$
(36) $\displaystyle\alpha^{\prime}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{m_{\mathrm{A}}\omega_{\mathrm{A}}}{2\hbar}}r_{0}e^{-i\theta_{0}}=\alpha^{*}\,,$
(37)

which means the angular separation is $2\theta_{0}$ for $0\leq\theta_{0}<\pi$.
The normalization factor $N$ is given by

$\displaystyle N=2+2{\rm
Re}\left(e^{-i\phi}\langle\alpha^{\prime},\beta|\alpha,\beta\rangle\right)=2+2{\rm
Re}\left(e^{-i\phi}e^{\alpha^{2}-|\alpha|^{2}}\right).$ (38)

Then, the average of $H_{\mathrm{A}}^{\rm ext}$ is

$\displaystyle\langle\psi_{\mathrm{A}}|H_{\mathrm{A}}^{\rm
ext}|\psi_{\mathrm{A}}\rangle$ $\displaystyle=$
$\displaystyle\hbar\omega_{\mathrm{A}}\left(|\alpha|^{2}+\frac{1}{2}\right)+\frac{2\hbar\omega_{\mathrm{A}}}{N}{\rm
Re}\left((\alpha^{2}-|\alpha|^{2})e^{-i\phi}e^{\alpha^{2}-|\alpha|^{2}}\right)$
(39) $\displaystyle=$
$\displaystyle\frac{1}{2}m_{\mathrm{A}}\omega_{\mathrm{A}}r_{0}^{2}+\frac{1}{2}\hbar\omega_{\mathrm{A}}+2\sin\theta_{0}\frac{m_{\mathrm{A}}\omega_{\mathrm{A}}^{2}r_{0}^{2}}{N}{\rm
Re}\left(ie^{i(\theta_{0}-\phi)}e^{\alpha^{2}-|\alpha|^{2}}\right).$

Hence the time dilation between two clocks becomes

$\displaystyle\langle
T_{\mathrm{A}}\rangle=\tau_{\mathrm{B}}\left(1-\frac{\omega_{\mathrm{A}}^{2}r_{0}^{2}}{2c^{2}}-\frac{\hbar\omega_{\mathrm{A}}}{2m_{\mathrm{A}}c^{2}}-2\sin\theta_{0}\frac{\omega_{\mathrm{A}}^{2}r_{0}^{2}}{Nc^{2}}{\rm
Re}\left(ie^{i(\theta_{0}-\phi)}e^{\alpha^{2}-|\alpha|^{2}}\right)+\frac{\hbar^{2}}{2m_{\mathrm{B}}^{2}c^{2}\sigma_{\mathrm{B}}^{2}}\right)\,.$
(40)

The term proportional to $\sin\theta_{0}$ arises from quantum interference due
to the superposition and may be regarded as the quantum time dilation.

Figure 2:  $K_{Q}/(\omega_{\mathrm{A}}^{2}r_{0}^{2}/2c^{2})$ as a function of
$\theta_{0}$ for several $\phi$.

To make the effect of quantum time dilation manifest, as in Chiba and
Kinoshita (2022) we split the time dilation formula (40) into $K_{C}$ and
$K_{Q}$ as $\langle T_{\mathrm{A}}\rangle=\tau_{\mathrm{B}}(1-K_{C}-K_{Q})$.
$K_{C}$ is given by the contribution of a statistical mixture of the coherent
states of clock $\mathrm{A}$ and clock $\mathrm{B}$, and $K_{Q}$ is the term
due to the interference effect

$\displaystyle K_{C}$ $\displaystyle=$
$\displaystyle\frac{\omega_{\mathrm{A}}^{2}r_{0}^{2}}{2c^{2}}+\frac{\hbar\omega_{\mathrm{A}}}{2m_{\mathrm{A}}c^{2}}-\frac{\hbar^{2}}{2m_{\mathrm{B}}^{2}c^{2}\sigma_{\mathrm{B}}^{2}},$
(41) $\displaystyle K_{Q}$ $\displaystyle=$ $\displaystyle
2\sin\theta_{0}\frac{\omega_{\mathrm{A}}^{2}r_{0}^{2}}{Nc^{2}}{\rm
Re}\left(ie^{i(\theta_{0}-\phi)}e^{\alpha^{2}-|\alpha|^{2}}\right)\,.$ (42)

Positive $K_{Q}$ implies the enhanced time dilation. In Fig. 2, $K_{Q}$
normalized by the classical time dilation factor
$\omega_{\mathrm{A}}^{2}r_{0}^{2}/2c^{2}$ is shown. In this example, we
assumed $q_{\mathrm{A}}=13e$, $m_{\mathrm{A}}=6.6\times 10^{-26}{\rm kg}$,
$B=1.0{\rm T}$ and $r_{0}=1.0\times 10^{-7}{\rm m}$, so that the classical
time dilation factor becomes
$\omega_{\mathrm{A}}^{2}r_{0}^{2}/2c^{2}=5.5\times 10^{-17}$. The quantum
effect can either enhance or reduce the time dilation and can be as large as
$10\%$ of the classical time dilation.

##  IV Summary

As an extension of the proper time observable proposed in Smith and Ahmadi
(2020) and applied to a weak gravitational field Chiba and Kinoshita (2022),
we studied charged quantum clocks interacting with the external
electromagnetic fields. We derived a formula of the average proper time read
by one clock conditioned on another clock reading a different proper time, Eq.
(28), which has the same form as that in classical relativity consisting of
kinetic part (velocity squared term) and gravitational part (gravitational
redshift term). We found that the time dilation is given by difference of the
proper time and distance between trajectories of each clock, regardless of
whethr the clock is in inertial motion or non-inertial motion.

When applied to a charged quantum clock in a uniform magnetic field, we
considered the case in which the state of one clock is in a superposition. We
found that the effect arising from quantum interference appears in the time
dilation which can be as large as $10\%$ of the classical time dilation.

Optical clocks based on highly charged ions have been considered as a new
class of references for highest-accuracy clocks and precision tests of
fundamental physics Kozlov  _et al._ (2018). Moreover, such an optical clock
based on a highly charged ion was realized recently King  _et al._ (2022). Our
study may be relevant in interpreting the measurements of the time dilation of
a highly charged optical clock.

## Acknowledgments

This work is supported by JSPS Grant-in-Aid for Scientific Research Number
22K03640 (TC), Nos. 16K17704 and 21H05186 (SK), and in part by Nihon
University.

##  Appendix A Quantum Mechanics of a Charged Particle in a Uniform Magnetic
Field

Here, we summerize the basic results on quantum mechanics of a charged
particle in a uniform magnetic field Landau and Lifshits (1991); Schulman
(2005)

###  A.1 Hamiltonian and Relative Coordinate

Consider a particle with the mass $m$ and the charge $q$ moving in a uniform
magnetic field $B$. Take the $z$-axis in the direction of the magnetic field
and assume that the particle moves in the $xy$-plane.

The Hamiltonian in the symmetric gauge

$\displaystyle{\bf A}=\frac{1}{2}{\bf B}\times{\bf x}=\frac{B}{2}(-y,x,0)$
(43)

is given by

$\displaystyle H=\frac{({\bf p}-q{\bf
A})^{2}}{2m}=\frac{1}{2m}\left(p_{x}+\frac{m\omega}{2}y\right)^{2}+\frac{1}{2m}\left(p_{y}-\frac{m\omega}{2}x\right)^{2}\,,$
(44)

where we have introduced the cyclotron frequency $\omega=qB/m$.

Since the time evolution of position operator is given from the Heisenberg
equation by $\dot{x}_{i}=\frac{[x_{i},H]}{i\hbar}=\frac{p_{i}-qA_{i}}{m}$,
considering the classical cyclotron motion, we introduce the position
operators $X$ and $Y$ corresponding to the center of the circle

$X=\frac{p_{y}+m\omega x/2}{m\omega},\quad Y=-\frac{p_{x}-m\omega
y/2}{m\omega}\,,$ (45)

and the operators $\xi$ and $\eta$ corresponding to the relative coordinates

$\displaystyle\xi=x-X=-\frac{p_{y}-m\omega
x/2}{m\omega},~{}~{}~{}~{}~{}\eta=y-Y=\frac{p_{x}+m\omega y/2}{m\omega}\,.$
(46)

Note that both $X$ and $Y$ commute with the Hamiltonian, $[X,H]=0=[Y,H]$, and
hence they are conserved, but $X$ and $Y$ do not commute with each other,
$[X,Y]=-i\hbar/m\omega$.

###  A.2 Creation and Annihilation Operators

We introduce the following creation and annihilation operators

$\displaystyle a$ $\displaystyle=$
$\displaystyle\sqrt{\frac{m\omega}{2\hbar}}\left(\xi+i\eta\right)=\sqrt{\frac{m\omega}{2\hbar}}\left(\left(\frac{x}{2}+i\frac{p_{x}}{m\omega}\right)+i\left(\frac{y}{2}+i\frac{p_{y}}{m\omega}\right)\right),$
(47) $\displaystyle a^{\dagger}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{m\omega}{2\hbar}}\left(\xi-i\eta\right)=\sqrt{\frac{m\omega}{2\hbar}}\left(\left(\frac{x}{2}-i\frac{p_{x}}{m\omega}\right)-i\left(\frac{y}{2}-i\frac{p_{y}}{m\omega}\right)\right),$
(48) $\displaystyle b$ $\displaystyle=$
$\displaystyle\sqrt{\frac{m\omega}{2\hbar}}\left(X-iY\right)=\sqrt{\frac{m\omega}{2\hbar}}\left(\left(\frac{x}{2}+i\frac{p_{x}}{m\omega}\right)-i\left(\frac{y}{2}+i\frac{p_{y}}{m\omega}\right)\right),$
(49) $\displaystyle b^{\dagger}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{m\omega}{2\hbar}}\left(X+iY\right)=\sqrt{\frac{m\omega}{2\hbar}}\left(\left(\frac{x}{2}-i\frac{p_{x}}{m\omega}\right)+i\left(\frac{y}{2}-i\frac{p_{y}}{m\omega}\right)\right),\,$
(50)

where $a$ and $b$ commute with each other and obey the usual commutation
relations

$\displaystyle[a,a^{\dagger}]=1,~{}~{}~{}~{}~{}[b,b^{\dagger}]=1\,.$ (51)

Then, the Hamiltonian and the $z$ component of the angular momentum $L_{z}$
are written in terms of $a$ and $b$ in simple form as

$\displaystyle H$ $\displaystyle=$
$\displaystyle\hbar\omega\left(a^{\dagger}a+\frac{1}{2}\right),$ (52)
$\displaystyle L_{z}$ $\displaystyle=$ $\displaystyle
xp_{y}-yp_{x}=\hbar(-a^{\dagger}a+b^{\dagger}b)\,.$ (53)

From Eqs. (47)-(50), the number operator $a^{\dagger}a$ corresponds to the
squared distance from the center of the circle and $b^{\dagger}b$ corresponds
to the squared distance of the center from the origin of the coordinates.

We also note that the center of the circle and the relative coordinates are
written in terms of creation and annihilation operators as

$\displaystyle X$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\frac{2\hbar}{m\omega}}(b+b^{\dagger}),~{}~{}~{}~{}~{}Y=\frac{i}{2}\sqrt{\frac{2\hbar}{m\omega}}(b-b^{\dagger}),$
(54) $\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\frac{2\hbar}{m\omega}}(a+a^{\dagger}),~{}~{}~{}~{}~{}\eta=\frac{i}{2}\sqrt{\frac{2\hbar}{m\omega}}(-a+a^{\dagger}).$
(55)

###  A.3 Coherent State

As in the case of one-dimensional harmonic oscillator, we introduce the
coherent state $|\alpha,\beta\rangle$ such that
$a|\alpha,\beta\rangle=\alpha|\alpha,\beta\rangle$ and
$b|\alpha,\beta\rangle=\beta|\alpha,\beta\rangle$, which is constructed by
applying the operators $e^{\alpha a^{\dagger}}$ and $e^{\beta b^{\dagger}}$ on
the ground state $|0\rangle$ as

$\displaystyle|\alpha,\beta\rangle=e^{-\frac{|\alpha|^{2}+|\beta|^{2}}{2}}e^{\alpha
a^{\dagger}}e^{\beta b^{\dagger}}|0\rangle\,.$ (56)

Then, from Eq. (47) and Eq. (49), the eigenvalues $\alpha$ and $\beta$
corresponding to the relative coordinate
$(r_{0}\cos\theta_{0},r_{0}\sin\theta_{0})$ and the center of the circle
$(X_{0},Y_{0})$ are given by

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\sqrt{\frac{m\omega}{2\hbar}}r_{0}e^{i\theta_{0}},$ (57)
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\sqrt{\frac{m\omega}{2\hbar}}(X_{0}-iY_{0}).$ (58)

The wave function of the coherent state is given by

$\displaystyle\langle{\bf x}|\alpha,\beta\rangle=$
$\displaystyle\sqrt{\frac{m\omega}{2\pi\hbar}}\exp\left\\{-\frac{m\omega}{4\hbar}\left[(x-r_{0}\cos\theta_{0}-X_{0})^{2}+(y-r_{0}\sin\theta_{0}-Y_{0})^{2}\right]\right\\}$
(59)
$\displaystyle\times\exp\left\\{i\frac{m\omega}{2\hbar}\left[(r_{0}\sin\theta_{0}-Y_{0})x-(r_{0}\cos\theta_{0}-X_{0})y-r_{0}(X_{0}\sin\theta_{0}-Y_{0}\cos\theta_{0})\right]\right\\}.$

$a(t)$ and $b(t)$ evolve according to the Heisenberg equation as

$\displaystyle i\hbar\dot{a}(t)$ $\displaystyle=$
$\displaystyle[a(t),H]=\hbar\omega a(t),$ (60) $\displaystyle
i\hbar\dot{b}(t)$ $\displaystyle=$ $\displaystyle[b(t),H]=0\,.$ (61)

Hence, we have $a(t)=e^{-i\omega t}a$ and $b(t)=b$. Then, from Eq. (55), the
expectation values of $\xi(t)$ and $\eta(t)$ in the coherent state are given
by

$\displaystyle\langle\xi(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\frac{2\hbar}{m\omega}}\langle\alpha,\beta|(a(t)+a^{\dagger}(t))|\alpha,\beta\rangle=\frac{1}{2}\sqrt{\frac{2\hbar}{m\omega}}(\alpha
e^{-i\omega t}+\alpha^{*}e^{i\omega t})=r_{0}\cos(\theta_{0}-\omega t),$ (62)
$\displaystyle\langle\eta(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{i}{2}\sqrt{\frac{2\hbar}{m\omega}}\langle\alpha,\beta|(-a(t)+a^{\dagger}(t))|\alpha,\beta\rangle=\frac{1}{2}\sqrt{\frac{2\hbar}{m\omega}}(-\alpha
e^{-i\omega t}+\alpha^{*}e^{i\omega t})=r_{0}\sin(\theta_{0}-\omega t)\,.$
(63)

This corresponds to the position of a charged particle orbiting clockwise with
the angular velocity $\omega$.333For a negatively charged particle
$\omega=qB/m<0$, the particle orbits counterclockwise. The expectation values
of $X(t)$ and $Y(t)$ do not depend on time: $\langle X(t)\rangle=X_{0}$ and
$\langle Y(t)\rangle=Y_{0}$.

The expectation value of the Hamiltonian becomes

$\displaystyle\langle\alpha,\beta|H|\alpha,\beta\rangle=\hbar\omega\left(|\alpha|^{2}+\frac{1}{2}\right)=\frac{1}{2}m\omega^{2}r_{0}^{2}+\frac{1}{2}\hbar\omega.$
(64)

###  A.4 Time Dilation in a Rotating Frame

We show that the time dilation Eq. (30) is invariant even if we move to a
rotating frame.

Consider a frame $(x^{\prime},y^{\prime})$ which rotates with the angular
velocity $\Omega$ about the $z$ axis with respect the inertial frame $(x,y)$.
The two coordinates are related by

$\displaystyle\begin{pmatrix}x^{\prime}\\\ y^{\prime}\\\
\end{pmatrix}=\begin{pmatrix}\cos\Omega t&\sin\Omega t\\\ -\sin\Omega
t&\cos\Omega t\\\ \end{pmatrix}\begin{pmatrix}x\\\ y\\\ \end{pmatrix}.$ (65)

Then, the shift vector appears in the rotating frame

$\displaystyle-c^{2}dt^{2}+dx^{2}+dy^{2}=-c^{2}dt^{2}+(dx^{\prime}-\Omega
y^{\prime}dt)^{2}+(dy^{\prime}+\Omega x^{\prime}dt)^{2},$ (66)

that is, $\beta^{x^{\prime}}c=-\Omega y^{\prime}$ and
$\beta^{y^{\prime}}c=\Omega x^{\prime}$. In the presence of the shift vector,
the (external) Hamiltonian becomes $H=\frac{({\bf P}-q{\bf
A})^{2}}{2m}-\boldsymbol{\beta}c\cdot({\bf P}-q{\bf A})$, so that the time
evolution of the position vector is given by

$\displaystyle\dot{\bf x}^{\prime}=\frac{[{\bf
x}^{\prime},H]}{i\hbar}=\frac{{\bf P}-q{\bf A}}{m}-\boldsymbol{\beta}c\,.$
(67)

Moreover, from Eq. (65) and Eq. (46), we have

$\displaystyle\dot{x}^{\prime}$ $\displaystyle=$ $\displaystyle\Omega
y^{\prime}+\omega(\eta\cos\Omega t-\xi\sin\Omega t),$ (68)
$\displaystyle\dot{y}^{\prime}$ $\displaystyle=$ $\displaystyle-\Omega
x^{\prime}-\omega(\xi\cos\Omega t+\eta\sin\Omega t)\,.$ (69)

Hence

$\displaystyle\frac{({\bf P}-q{\bf A})^{2}}{m^{2}}$ $\displaystyle=$
$\displaystyle(\dot{x}^{\prime}+\beta^{x^{\prime}}c)^{2}+(\dot{y}^{\prime}+\beta^{y^{\prime}}c)^{2}$
(70) $\displaystyle=$
$\displaystyle\omega^{2}\left(\xi^{2}+\eta^{2}\right)=\dot{x}^{2}+\dot{y}^{2}\,.$

Therefore, the time dilation formula Eq. (30) holds in a rotating frame. This
implies, in particular, that even if we move to a rotating frame with
$\Omega=-\omega$ so that a particle is at rest (classically), the time
dilation does not change.

## References

  * Chiba and Kinoshita (2022) T. Chiba and S. Kinoshita, Phys. Rev. D 106, 124035 (2022), arXiv:2209.07638 [gr-qc] .
  * Sakurai (1980) J. J. Sakurai, Phys. Rev. D 21, 2993 (1980).
  * Kozlov  _et al._ (2018) M. G. Kozlov, M. S. Safronova, J. R. Crespo López-Urrutia, and P. O. Schmidt, Rev. Mod. Phys. 90, 045005 (2018), arXiv:1803.06532 [physics.atom-ph] .
  * King  _et al._ (2022) S. A. King _et al._ , Nature 611, 43 (2022), arXiv:2205.13053 [physics.atom-ph] .
  * Smith and Ahmadi (2020) A. R. H. Smith and M. Ahmadi, Nature Commun. 11, 5360 (2020), arXiv:1904.12390 [quant-ph] .
  * Misner  _et al._ (2017) C. W. Misner, K. S. Thorne, and J. A. Wheeler,  _Gravitation_ (Princeton University Press, Princeton, N.J., 2017).
  * Page and Wootters (1983) D. N. Page and W. K. Wootters, Phys. Rev. D 27, 2885 (1983).
  * Wootters (1984) W. K. Wootters, International Journal of Theoretical Physics 23, 701 (1984).
  * Schleich  _et al._ (1991) W. Schleich, M. Pernigo, and F. L. Kien, Phys. Rev. A 44, 2172 (1991).
  * Landau and Lifshits (1991) L. D. Landau and E. M. Lifshits,  _Quantum Mechanics: Non-Relativistic Theory_ , Course of Theoretical Physics, Vol. v.3 (Butterworth-Heinemann, Oxford, 1991).
  * Schulman (2005) L. S. Schulman,  _TECHNIQUES AND APPLICATIONS OF PATH INTEGRATION_ (Dover, 2005).


# Failed Gamma-Ray Bursts: Thermal UV/Soft X-ray Emission Accompanied by
Peculiar Afterglows

M. Xu11affiliation: Department of Astronomy, Nanjing University, Nanjing
210093, China<EMAIL_ADDRESS>22affiliation: Yukawa Institute for Theoretical
Physics, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan
33affiliation: Department of Physics, Yunnan University, Kunming 650091, China
, S. Nagataki22affiliation: Yukawa Institute for Theoretical Physics, Oiwake-
cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan , Y. F.
Huang11affiliation: Department of Astronomy, Nanjing University, Nanjing
210093, China<EMAIL_ADDRESS>, and S.-H. Lee22affiliation: Yukawa Institute
for Theoretical Physics, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502,
Japan

###### Abstract

We show that the photospheres of “failed” Gamma-Ray Bursts (GRBs), whose bulk
Lorentz factors are much lower than 100, can be outside of internal shocks.
The resulting radiation from the photospheres is thermal and bright in UV/Soft
X-ray band. The photospheric emission lasts for about one thousand seconds
with luminosity about several times $10^{46}$ erg/s. These events can be
observed by current and future satellites. It is also shown that the
afterglows of failed GRBs are peculiar at the early stage, which makes it
possible to distinguish failed GRBs from ordinary GRBs and beaming-induced
orphan afterglows.

gamma-ray bursts: general — radiation mechanisms: thermal

##  1 Intronduction

Gamma-Ray Bursts (GRBs) are the most powerful explosion in the universe. The
origin of prompt emission remains unresolved, owing to the fact that the
prompt emission has a large explosion energy showing non-thermal spectrum with
rapid time-variabilities.

It is widely accepted that the prompt emission is coming from a highly-
relativistic flow, because it can reduce the optical depth of the flow which
makes the radiation spectrum non-thermal (Rees & Mészáros 1994). In fact, in
the internal shock scenario, which is one of the most promising scenarios, the
relativistic shells collide with each other after the system becomes optically
thin (e.g. Piran 1994 for a review). However, it was pointed out by Mészáros &
Rees (2000) that even such a relativistic flow should have a photosphere
inevitably and thermal radiation should be coming from there. Since then,
there have been many theoretical (Daigne & Mochkovitch 2002; Pe′er et al.
2006; Pe′er 2008; Pe′er & Ryde 2011) and observational (Ghirlanda et al. 2003;
Ryde 2004, 2005; Ryde et al. 2010; Guiriec et al. 2010; Ryde et al. 2011)
studies on how the thermal component contributes to the prompt emission (or
the precursor). Nowadays, the internal shock model with a photosphere is
frequently discussed (e.g. Toma et al. 2011; Wu & Zhang 2011). In this
picture, the radius of the photosphere, $R_{\rm PS}$, is usually smaller than
the radius $R_{\rm IS}$ where internal shocks are happening.

The above scenario is based on the assumption that the bulk Lorentz factor of
the jet is as large as 100-1000. But what happens if the bulk Lorentz factor
is not so high? Theoretically, it is natural to consider such a case, because
it is very hard to realize such a clean, highly-relativistic flow. Especially,
in case of long GRBs, some bursts are at least coming from the death of
massive stars where a lot of baryons should be surrounding the central engine
(MacFadyen & Woosley 1999; Proga et al. 2003; Nagataki et al. 2007; Nagataki
2009, 2010). Thus we can expect there are a lot of “failed GRBs” that have
dirty, not so highly-relativistic flows in the universe (e.g. Dermer et al.
1999; Huang et al. 2002; Paragi et al. 2010; Xu et al. 2011). Recently,
Bromberg et al. (2011a, 2011b) suggested the existence of a large population
of failed GRBs if the jets failed to break out of the progenitor stars in
collapsar model.

Qualitatively, $R_{\rm IS}$ becomes smaller if the bulk Lorentz factor of the
flow is smaller, while $R_{\rm PS}$ increases with the decreasing of bulk
Lorentz factor. Thus we can expect that the photosphere will become outside of
the internal shock region for some lower Lorentz factors (see Fig. 1a). In
such a case, $\gamma$-rays from the internal shocks cannot escape. Instead,
softer thermal radiation from the photosphere followed by an afterglow will be
seen.

We note that such a situation was considered in case of a “successful GRB”
(Rees & Mészáros 2005; Lazzati et al. 2009; Mizuta et al. 2011; Nagakura et
al. 2011; Ryde & Pe′er 2009; Ryde et al. 2010, 2011). In the photospheric
model, it is considered that the photospheric emission itself is the origin of
the prompt emission: its spectrum is modified to non-thermal due to the
heating by relativistic electrons that are produced at the internal shocks
inside the photosphere (Beloborodov 2010; Vurm et al. 2011). Thus the
photospheric model can explain normal successful GRBs.

Here, in this study, we consider the case of “failed GRBs”. We show that the
direct emission from the photosphere of a failed-GRB will be a UV/soft X-ray
burst, followed by an afterglow with a peculiar spectrum at early stage. It is
also shown that the afterglows of failed GRBs can be distinguished from
ordinary GRB afterglows and beaming-induced orphan afterglows.

Figure 1:  (a) A sketch illustrating the emission regions of a failed-GRB.
Note that the internal shock radius is smaller than the photosphere radius.
(b) Schematic diagram showing the path of a photospheric photon escaping from
the ejecta in the stellar frame. A photon emitted at point E will escape from
the ejecta at point G.

##  2 Photospheric emission

In this study, we consider axisymmetric jet and assume that the observer is on
the axis $\overline{OD}$ in Fig. 1b. This is the extension of the 1-D
formulation derived by Daigne & Mochkovitch (2002). For a baryon-rich ejecta
in the stellar frame (i.e., burst source frame, and from now on all variables
are defined in this frame), we assume the ejecta has been accelerated at a
distance $r_{\rm acc}$ from the central engine. The mass flux of the ejecta is
written as $\dot{M}=\dot{E}/\Gamma c^{2}$, where $\Gamma$ is the Lorentz
factor of the ejecta and $\dot{E}$ is the energy injection rate. The energy
injection begins at $t_{\rm inj}$=0, and stops at $t_{\rm inj}=t_{\rm w}$,
i.e., the central engine activity lasts for a period of time $t_{\rm w}$.

The ejecta can be subdivided into a series of concentric layers, where each
layer has been injected at $r_{\rm acc}$ at a certain injection time $t_{\rm
j}$. Each ejecta layer becomes transparent when it has expanded to a distance
$r(t_{\rm inj}=t_{\rm j},t)$ at a specific time $t$, where $r(t_{\rm
inj}=t_{\rm j},t)=r_{\rm acc}+\beta c(t-t_{\rm j})$ and
$\beta=\sqrt{1-1/\Gamma^{2}}$. The photons emitted at $r(t_{\rm inj}=t_{\rm
j},t)$ will escape from the ejecta at a time $t_{\rm esc}$ at a distance
$r_{\rm esc}(t_{\rm inj}=t_{\rm j},t)$. Here $r_{\rm esc}$ and $t_{\rm esc}$
are defined where a photon emitted at time t by the shell ejected at time
$t_{\rm inj}=t_{\rm j}$ escapes the outflow, i.e. reaches the first shell
emitted at time $t_{\rm inj}=0$. In other words, the optical depth from
$r(t_{\rm inj}=t_{\rm j},t)$ to $r_{\rm esc}(t_{\rm inj}=t_{\rm j},t)$ is
unity.

A geometric sketch illustrating the escape path of photons inside the ejecta
is shown in Fig. 1b. A photon emitted at point E (at a distance $r(t_{\rm
inj}=t_{\rm j},t)$ and propagation angle $\varphi$) will escape from the
ejecta at point G, such that the optical depth from E to G is

$\tau(t_{\rm j},\varphi)=\int^{r_{\rm esc}(t_{\rm inj}=t_{\rm j},t)}_{r(t_{\rm
inj}=t_{\rm j},t)}d\tau(r),$ (1)

where $d\tau(r)$ can be estimated as (Abramowicz et al. 1991; Daigne &
Mochkovitch 2002; Pe′er 2008)

$d\tau(r)=\frac{\kappa\dot{M}(1-\beta{\rm cos}\varphi)}{4\pi r^{2}}dr.$ (2)

We define the photospheric radius $R_{\rm PS}(t_{\rm j})=r(t_{\rm j},t)$ at
which $\tau$ is equal to unity. In light of Fig. 1b, we choose a cylindrical
coordinate system (Pe′er 2008) with the central point $O$ being the stellar
center, and the observer located along the +z-direction (defined by the
direction of $\overline{OD}$). Photons are emitted at a perpendicular distance
$r_{\rm min}=r(t_{\rm j},t){\rm sin}\varphi$ from the z-axis and a distance
$z_{\rm min}=r(t_{\rm j},t){\rm cos}\varphi$ along the z-axis from point $O$.
The escape radius can be estimated from the triangle $OEG$, i.e., $r_{\rm
esc}(t_{\rm inj}=t_{\rm j},t)=r(t_{\rm j},t)+\beta c(t_{\rm j}-0)+\beta
c(t_{\rm esc}-t)=\\{r(t_{\rm j},t)^{2}+[c(t_{\rm esc}-t)]^{2}-2r(t_{\rm
j},t)c(t_{\rm esc}-t){\rm cos}(\pi-\varphi)\\}^{\frac{1}{2}}$,

The integration for the optical depth can be conveniently rewritten in
cylindrical coordinates as the following

$\tau(t_{\rm j},\varphi)=\int^{z_{\rm max}}_{z_{\rm
min}}\frac{\kappa\dot{M}(1-\beta{\rm cos}\varphi)}{4\pi
r^{2}}\frac{dr}{dz}dz,$ (3)

where $r=\sqrt{z^{2}+r_{\rm min}^{2}}$, $dz/dr=z/\sqrt{z^{2}+r_{\rm min}^{2}}$
and $z_{\rm max}=\sqrt{r_{\rm esc}^{2}-r_{\rm min}^{2}}$. The photospheric
radius ($R_{\rm PS}$), which depends on the propagation angle ($\varphi$), can
be found readily by defining $\tau=1$.

For a relativistic ejecta with Lorentz factor $\Gamma$, the arrival time of
photons emitted at the photosphere in the observer frame are delayed relative
to that measured in the stellar frame

$t_{\rm obs}=t-R_{\rm PS}{\rm cos}\varphi/c=t_{\rm j}+(1-\beta{\rm
cos}\varphi)R_{\rm PS}/\beta c,$ (4)

i.e., the observer time is a function of injection time and propagation angle.
In this equation, we have neglected the effect of the acceleration radius
($r_{\rm acc}$) because it is much smaller than the photosphere radius.

The evolution of photospheric radius with propagation angle is shown in Fig.
2b. The parameters are taken as $\Gamma=10$, $\dot{E}=10^{51}$erg/s and
$t_{\rm w}=2000$s. The solid curve shows its evolution at observer time
$t_{\rm obs}=100$ s and the dashed curve at $t_{\rm obs}=2000$ s. From this
panel, we can see that in the observer frame, the photospheric radius
decreases with propagation angle.

Figure 2:  Evolution of the photospheric radius and temperature with respect
to the observer time and photon propagation angle. The parameters are taken as
$\Gamma=10$, $\dot{E}=10^{51}$erg/s and $t_{\rm w}=2000$s.

(a) The photospheric radius vs. the observer time. The solid curve is plot for
photons propagating along the expansion direction of the ejecta ($\varphi=0$).
The dashed curve corresponds to $\varphi=0.1$ rad. (b) The photospheric radius
vs. the photon propagation angle. The solid and dashed curves correspond to
the observer time of 100 s and 2000 s respectively. (c) The photospheric
temperature vs. observer time. The solid and dashed curves correspond to
photon propagation angles of 0 and 0.1 rad, respectively. (d) The photospheric
temperature vs. the photon propagation angle. The solid and dashed curves are
for $t_{\rm obs}$=100 s and 2000 s, respectively.

We also show the evolution of photospheric radius with observer time in Fig.
2a. The solid, dashed curves present the cases for $\varphi=0$ and
$\varphi=0.1$ rad respectively. As is shown in this panel, the photospheric
radius increases with time, and the duration of the photospheric emission is
prolonged at larger propagation angle. The end points of the two curves
indicate the observer time when the last layer of the ejecta becomes
transparent.

According to the fireball model, the temperature of a layer at its
photospheric radius is given by (Piran 1999)

$kT_{\rm PS}=\frac{D}{\Gamma}kT^{0}\big{(}\frac{R_{\rm PS}}{r_{\rm
acc}}\big{)}^{-2/3},$ (5)

where $D=[\Gamma(1-\beta{\rm cos}\varphi)]^{-1}$ is the Doppler factor,
$r_{\rm acc}$ is the saturation radius and $T^{0}$ is its blackbody
temperature. In Fig. 2c and Fig. 2d, we show the evolution of $T_{\rm PS}$
with respect to the propagation angle and observer time respectively. From the
two panels, we find that the photosperic temperature decreases with the
propagation angle and time in the observer frame.

The evolution of the injection time with respect to the observer time and the
propagation angle is shown in Fig. 3a and 3b, respectively. For the “standard”
parameter set ($\Gamma=10$, $\dot{E}=10^{51}$ erg/s, $t_{\rm w}$=2000 s),
$t_{\rm inj}(t_{\rm obs})$ is mildly smaller than $t_{\rm obs}$ for different
$\varphi$, and $t_{\rm inj}(\varphi)$ is almost independent of $\varphi$ for
different $t_{\rm obs}$. In Fig. 3c and Fig. 3d, we also show the evolution of
the escaping radius $r_{\rm esc}$ with respect to the observer time and the
propagation angle, respectively. The relations of $t_{\rm obs}$ and $\varphi$
with respect to $r_{\rm esc}$ are similar to that of the photospheric radius
$R_{\rm PS}$.

Figure 3:  Evolution of the injection time and escaping radius with respect to
the observer time and photon propagation angle. The parameters are the same as
those in Fig. 2. (a) The injection time vs. the observer time. The solid curve
is plot for photons propagating along the expansion direction of the ejecta
($\varphi=0$). The dashed curve is the evolution of the injection time for
$\varphi=0.1$ rad. (b) The injection time vs. the photon propagation angle.
The solid and dashed curves are observed at 100s and 2000s respectively. (c)
The escaping radius vs. the observer time. The solid and dashed curves are
plot with a propagation angle of 0 and 0.1 rad, respectively. (d) The escaping
radius vs. the photon propagation angle. The solid and dashed curves are for
$t_{\rm obs}$=100 s and 2000 s, respectively.

As for a jet with half-opening angle $\theta=0.1$ rad, constant Lorentz factor
$\Gamma=10$, and energy injection from $t_{\rm inj}=0$ to $t_{\rm inj}=t_{\rm
w}=2000$ s with energy injection rate per solid angle
$\dot{E}/4\pi=10^{51}/4\pi$ erg/s, we can estimate that $r_{\rm acc}\simeq
9\times 10^{7}$cm and $kT^{0}\simeq 0.41$ MeV for a fireball model (Piran
1999; $\rm M\acute{e}sz\acute{a}ros~{}\&~{}Rees$ 2000; Daigne & Mochkovitch
2002). If the line-of-sight is along the jet central axis, we can find that
the photospheric radius is about $1.1\times 10^{14}$ cm when the last layer
becomes transparent, the observer’s time can be calculated from Eq. 4, which
is found to be about 2020 s and corresponds to the end point of the solid
curve in Fig. 2a.

The observed luminosity of photospheric emission can be determined by
integrating over the surface of the photosphere

$L=\int_{0}^{\theta}\sigma T_{\rm PS}^{4}dS{\rm cos}\vartheta$ (6)

where $\vartheta$ is the angle between the tangential direction of the
photosphere surface and the line-of-sight when the propagation angle is
$\varphi$. $dS{\rm cos}\vartheta=2\pi R_{\rm PS}(t_{\rm obs},\varphi){\rm
sin}\varphi[R_{\rm PS}(t_{\rm obs},\varphi+d\varphi){\rm
sin}(\varphi+d\varphi)-R_{\rm PS}(t_{\rm obs},\varphi){\rm sin}\varphi]$ is
the photospheric surface area from propagation angle $\varphi$ to
$\varphi+d\varphi$. The evolution of the photospheric luminosity with observer
time is shown by the solid curve in Fig. 4. There is a break in the light
curve at about $t_{\rm obs}\simeq$ 2020 s, which is attributed to the stop of
energy injection by the central engine and when the last layer of the ejecta
became transparent as the photons propagate along the line-of-sight.
Afterwards, only photospheric emission at high latitude (large propagation
angles) contributes to the observed luminosity. The photospheric emission
ceases when the last layer with propagation angle $\varphi=\theta=0.1$ become
transparent, which is about 2050 s in the observer frame.

Figure 4:  Evolution of the photospheric luminosity (solid curve) and
effective temperature (dashed curve) with observer time for a jet with
parameters of $\theta=0.1$ rad, $\Gamma=10$, $\dot{E}=10^{51}$erg/s and
$t_{\rm w}$=2000 s.

We can also define an effective temperature for the photosphere

$T_{\rm eff}=\frac{\int_{0}^{\theta}TdL}{\int_{0}^{\theta}dL}.$ (7)

This effective temperature is shown as a dashed curve in Fig. 4. We can find
that the photospheric emission of a failed GRB is presented as a short soft
X-ray burst and then becomes a UV burst which lasts for about several thousand
seconds.

We also investigated the parameter effect on the photospheric emission, which
are shown in Fig. 5 and Fig. 6. All the curves are derived when the last layer
of ejecta along the line-of-sight became transparent , i.e., $\varphi=0$ and
$t_{\rm j}=t_{\rm w}$. As is shown in Fig. 5a, the photospheric radii are
decreasing with the increase of Lorentz factor for different sets of
parameters. The solid curve corresponds to the standard parameters
($\dot{E}=10^{51}$erg/s, $t_{\rm w}=2000$ s), while the parameters for the
dashed curve and the dotted curve are $\dot{E}=10^{49}$erg/s, $t_{\rm w}=2000$
s and $\dot{E}=10^{51}$erg/s, $t_{\rm w}=200$ s, respectively. A lower energy
injection rate and shorter injection time will decease the radius of the
photosphere. Fig. 5b shows the evolution of the observer time with Lorentz
factor. The parameters for each curve are the same as Fig. 5a. This time
period can be interpreted as the duration of the photospheric emission. From
this panel, we can find that the duration of photospheric emission decreases
with an increase of Lorentz factor. Lower energy injection rate and shorter
injection time will decease the duration of the photospheric emission.

Figure 5:  Parameter dependence of the photospheric emission. All curves are
obtained when the last layer of the ejecta became transparent along the line-
of-sight, i.e., $\varphi=0$ and $t_{\rm j}=t_{\rm w}$. (a) Parameter
dependence of the photospheric radius. The solid curve is derived using the
standard parameters, i.e., $\dot{E}=10^{51}$erg/s and $t_{\rm w}$=2000 s; the
dashed curve is for $\dot{E}=10^{49}$erg/s and $t_{\rm w}$=2000 s; and the
dotted curve is for $\dot{E}=10^{51}$erg/s and $t_{\rm w}$=200 s. The
evolution of the internal shock radii $R_{\rm IS}$ for a variability timescale
of $\delta t$=1 s, 0.33 s and 0.1 s are shown and marked correspondingly. (b)
Parameter dependence of the observer time. The identities of the curves are
the same as in (a).

In Fig. 6, we show the parameter effect on the photospheric luminosity and
effective temprature. The parameters of each curve are the same as Fig. 5.
From Fig. 6a, we can find that the luminosity of the photospheric emission are
low in both high and low Lorentz factor. Lower energy injection rate results
in lower luminosity. As is shown in Fig. 6b, the effective temperatures are
increasing with the increase of Lorentz factor. Lower energy injection rate
and shorter injection time will decease the radii of the photosphere and hence
results in a higher effective temperature.

Figure 6:  Parameter dependence of the photospheric luminosity (a) and the
effective temperature (b) in the Lorentz factor space. All curves are obtained
when the last layer of ejecta became transparent along the line-of-sight,
i.e., $\varphi=0$ and $t_{\rm j}=t_{\rm w}$. The identities of the curves are
the same as in Figure 5.

As for a jet with half-opening angle of about 0.1 rad, Lorentz factor
$\Gamma=2-20$, energy injection rate $\dot{E}=10^{49-51}$erg/s and injection
time $t_{\rm w}$=200-2000s, from Fig. 5 and 6, we can conclude that the prompt
emission for a failed GRB is thermal soft X-ray or UV photospheric emission,
there will be no significant non-thermal gamma-ray emission. The photospheric
luminosity is about $10^{46}$erg/s and last for about one thousand seconds.
Note that the photospheric luminosity is far lower than the energy injection
power, most of of the energy is re-converted into the ejecta’s kinetic energy.
From Fig. 5a, we find that the radius of the photosphere is about $10^{14}$
cm, which is larger then the prediction for the internal shock’s radius, i.e.,
$R_{\rm IS}\simeq\Gamma^{2}c\delta t\simeq 10^{12}$ cm (Mészáros 2006), where
$\delta t\sim 0.33$ s here is the variability timescale of the prompt
emission. The evolution of $R_{\rm IS}$ with respect to $\Gamma$ for $\delta
t$=1 s, 0.33 s and 0.1 s are shown and marked in Fig. 5a correspondingly. This
radius is consistent with Fig. 1a.

This thermal radiation will be in the UV or soft X-ray band. The lower band of
Swift-XRT ($0.2-10$ keV) may cover this energy range and it is sensitive
enough to detect such a photospheric emission component (Gehrels et al. 2004).
MAXI-SSC monitors all-sky in the energy range of $0.5-10$ keV and also has a
chance to detect such events (Matsuoka et al. 1997). Future UV satellites may
be also have the capability to detect these events, such as TAUVEX (wavelength
range 120nm-350nm) (Safonova et al. 2008).

##  3 Afterglow emission

As the outflow expands outward, it will collide with the surrounding medium
and afterglow will be produced. The dynamical evolution of a relativistic jet
in interstellar medium has been studied by Huang et al. (1999). Their codes
can be used in both ultra-relativistic and non-relativistic phases.

In our model, we consider a jet with the bulk Lorentz factor $\Gamma=10$, the
half-opening angle $\theta=0.1$ and the isotropic energy $E=10^{50}$ erg. The
jet expands laterally at the comoving sound speed and collides with a medium
whose number density is $n_{\rm ISM}=1$ cm-3. We also assume typical values
for some other parameters of the jet, i.e., the electron energy fraction
$\epsilon_{e}=0.1$, the magnetic energy fraction $\epsilon_{B}=0.01$ and the
power-law index of the energy distribution function of electrons $p=2.5$.
Multiband afterglow emission is expected from synchrotron radiation of
relativistic electrons. Using this exquisite model, we numerically calculated
the afterglow light curves and spectra with line of sight parallel to the jet
axis. We assume a redshift $z=1$ and a standard cosmology with
$\Omega_{M}=0.27$, $\Omega_{\Lambda}=0.73$ and with the Hubble constant of
$H_{0}=71$ km s-1 Mpc-1.

Our results for the afterglow spectra of failed GRBs are shown in Fig. 7. The
thick and thin solid curves are the spectra observed at $10^{3}$ s and
$10^{6}$ s respectively. In this figure, we can find that the spectrum becomes
softer with the elapse of the observational time. At the early stage ($10^{3}$
s), the peak flux appears at about $5\times 10^{12}$ Hz, i.e., in the IR band.
Both the peak flux and peak frequency decrease with time. At late time
($10^{6}$ s), the peak flux is more than one magnitude less than that in the
early stage. The peak frequency decreases to about $10^{9}$ Hz at late stage.
It is in the radio band and more than three magnitudes less than the peak
frequency at the early stage.

Figure 7:  Evolution of the afterglow spectra for the three types of GRBs. The
solid, dashed and dotted curves are spectra of a failed GRB afterglow, an
ordinary GRB afterglow and a beaming-induced orphan afterglow respectively.
The thick and thin curves are the spectra observed at $10^{3}$ s and $10^{6}$
s respectively.

For comparison, we also show the afterglow spectra of an ordinary GRB in Fig.
7. Here we choose the same parameters as the failed GRB except for a much
larger bulk Lorentz factor ($\Gamma=300$). The dashed curves show the
afterglow spectra of this GRB with observing angle $\theta_{\rm obs}=0$ (the
line of sight is parallel to the jet axis). As is shown in Fig. 7, the spectra
of the failed GRB and the ordinary GRB are similar at the late stage (thin
solid and thin dashed curves) because their energies and Lorentz factors are
both similar at this moment. But at early stage, they are very different
(thick solid and thick dashed curves) due to their very different initial
Lorentz factors and the corresponding minimum Lorentz factors of electrons
(Sari et al. 1998). The peak frequency of the GRB afterglow is much larger
than that of the failed GRB afterglow.

In Fig. 7, we also show the spectra of a beaming-induced orphan afterglow
(afterglow from an ordinary highly collimated GRB outflow, but with the
observing angle larger than the jet half-opening angle so that no prompt
gamma-rays can be observed in the main burst phase, Rhoads 1997; Huang et al.
2002). Here we assume the same parameters as the ordinary GRB except
$\theta_{\rm obs}=0.125$. The early and late spectra of this orphan afterglow
are shown in Fig. 7 with thick dotted curve and thin dotted curve. From this
figure, we can find the spectra of beaming-induced orphan afterglow are
similar to that of the ordinary GRB. Although it is hard to distinguish a
beaming-induced orphan afterglow from a failed GRB afterglow through their
afterglow light curves (Huang et al. 2002), they can be potentially
distinguished from their spectra at the early stages. Their spectra of early
afterglows are very different: the peak frequency of a failed GRB afterglow is
far lower than that of a beaming-induced orphan afterglow. Another way to
distinguish them is through their early light curves. Early afterglow of a
beaming-induced orphan afterglow will show a rebrightening while failed GRB
will not (Huang et al. 1999, 2002; Xu & Huang 2010).

##  4 Conclusion and Discussions

The analysis in this paper shows that the emission of ejecta with low Lorentz
factors is very different from that expected from ejecta with high Lorentz
factors. Prompt emission of a GRB is non-thermal and bright in the gamma-ray
band. For a failed GRB, however, the emission originates from the photosphere
with a thermal spectrum, and is bright in the UV or soft X-ray band instead of
gamma-rays. This photospheric emission lasts for about a thousand seconds with
a luminosity about several times $10^{46}$ erg/s.

Since the photospheric emission manifests as a UV or soft X-ray transient, it
can be detected by some current and future satellites, such as Swift-XRT,
MAXI-SSC and TAUVEX etc. On 2008 January 9, Swift-XRT discovered a peculiar
X-ray transient 080109 in NGC 2770 (Berger & Soderberg 2008; Page et al.
2008). No gamma-ray emission was detected. This X-ray transient reached its
peak at about $60s$ and lasted for about $600$ s. Its spectrum can be fitted
with an absorbed double blackbody model with temperatures about $0.36$ keV and
$1.24$ keV respectively (Li 2008). This transient may be a candidate of
photospheric emission from a failed GRB. Meanwhile, some unidentified X-ray
transients have been detected by MAXI during its one-year monitoring (Nakajima
et al. 2009; Suzuki et al. 2010). These transients generally showed an
absorbed blackbody spectrum and lasted for tens of seconds. It is possible
that some of them are photospheric emission from failed GRBs.

If we extend the injection time to about $10^{5}$ s and the jet half-opening
to about 0.4 rad in our model, we find that the photospheric radius is about
$10^{15}$ cm and the effective temperature is deceased to lower than 1 eV,
i.e., there will be an optical burst. This kind of optical burst will last for
about several thousand seconds with a luminosity about $10^{42}$ erg/s, which
may be detected by the Hyper-Suprime Camera of the $Subaru$ telescope in the
future.

In this work, we have assumed that the prompt emission is thermal radiation
coming from the photosphere where the optical depth is unity. Due to the low
density of GRB jets, however, it has been pointed out that the last-scattering
positions of the observed photons may not simply coincide with the
photosphere, but instead possess a finite distribution around it (e.g. Pe′er
et al. 2006; Pe′er 2008; Beloborodov 2010; Pe′er & Ryde 2011). This stochastic
effect can lead to differentiation of the observed spectrum from a thermal one
of purely photospheric origin. Such mechanism can work even in failed GRBs,
and it is our future work to study how the spectrum will be reshaped using
Monte-Carlo calculations. We are planning to investigate this effect in the
context of failed GRBs as a next step of our study.

From the comparison of afterglow emissions from failed and ordinary GRBs,
while we find it challenging to distinguish them at their late stage of
evolution, their spectra at the early stage are profoundly different. We
conclude that it is possible to identify failed GRBs by observing their
afterglow emission in the early stage. The typical frequency at peak flux in
the afterglow phase for failed GRBs is much lower than that for ordinary GRBs
(or beaming-induced orphan GRBs). We can thus define a hardness ratio, for
instance, as the flux contrast between $10^{12}$ Hz and $10^{14}$ Hz at an
observed time of $1000$ s, i.e., $f_{\rm 1ks}\equiv F_{10^{12}{\rm
Hz}}/F_{10^{14}{\rm Hz}}$. If $f_{\rm 1ks}>1$, then it is quite likely that
the emission is coming from a failed GRB. If $f_{\rm 1ks}<1$, then it would be
more likely to come from an ordinary GRB afterglow or a beaming-induced orphan
afterglow. In addition, at the early afterglow stage, a rebrightening phase
will be present in the case of a beaming-induced orphan GRB, while it is not
expected for ordinary or failed GRBs. Therefore, the afterglows of failed GRBs
can be distinguished from both ordinary GRB afterglows and beaming-induced
orphan afterglows through observations at the early stages.

We thank the anonymous referee for many of the useful suggestions and
comments. We also would like to thank P. Mészáros, T. Piran and J. Aoi for
helpful discussions. This work was supported by the National Natural Science
Foundation of China (Grant No. 11033002), the National Basic Research Program
of China (973 Program, Grant No. 2009CB824800) and the Grant-in-Aid for the
’Global COE Bilateral International Exchange Program’ of Japan, Grant-in-Aid
for Scientific Research on Priority Areas No. 19047004 and Scientific Research
on Innovative Areas No. 21105509 by Ministry of Education, Culture, Sports,
Science and Technology (MEXT), Grant-in-Aid for Scientific Research (S) No.
19104006 and Scientific Research (C) No. 21540404 by Japan Society for the
Promotion of Science (JSPS).

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