# Single-photon frequency conversion via a giant $\Lambda$-type atom

Lei Du  Beijing Computational Science Research Center, Beijing 100193, China
Yong Li<EMAIL_ADDRESS>Beijing Computational Science Research Center,
Beijing 100193, China  Synergetic Innovation Center for Quantum Effects and
Applications, Hunan Normal University, Changsha 410081, China

###### Abstract

We study single-photon scattering via a giant $\Lambda$-type atom, where both
atomic transitions are coupled with the modes of a single waveguide at two
separated points. The giant-atom structure introduces phase-dependent
interference effects to both elastic (frequency-preserving) and inelastic
(frequency-converting) scattering processes, which modify the corresponding
decay rates (as well as the transition frequencies) such that the giant atom
is capable of accessing the various limits of a small one. The condition of
the optimal frequency conversion is also identified and shown to be phase
dependent. Moreover, we consider the combination of the giant-atom
interference and the Sagnac quantum interference by further inserting a Sagnac
interferometer at each of the coupling points. It is shown that the two kinds
of interference effects are compatible and play independent roles, such that
efficient frequency conversion with unit efficiency can be achieved in
addition to the phase-dependent phenomena induced by the giant-atom structure.

##  I Introduction

Waveguide quantum electrodynamics (QED) is an emerging field that studies the
interactions between atoms and electromagnetic fields confined in one-
dimensional open waveguides. It provides an alternative platform for enhancing
light-matter interactions, where the electromagnetic modes can interact
strongly with atoms due to the transverse confinement of the waveguides
Wreview1 ; Wreview2 . Different from cavity QED systems, waveguides support in
general a continuum of modes such that the bandwidth limitation that is
typical for cavities (due to their finite linewidths) can be greatly relaxed
Wreview1 . Moreover, long-range interactions between remote atoms
(resonators), which are vital for studying many-body physics and realizing
large-scale quantum networks, can be mediated by the traveling photons in
waveguides fan1999 ; xiao2008 ; xiao2010 ; JPan ; BBLi ; Solano . To date,
several candidates of waveguide QED systems have been developed, such as
trapped (natural) atoms coupled with optical fibers fiber1 ; fiber2 ; fiber3
or photonic crystal waveguides crystal1 ; crystal2 ; crystal3  and
superconducting qubits coupled with transmission lines tline1 ; tline2 ;
tline3 ; tline4 ; tline5 . The technological improvements and growing research
interests have led to progress in waveguide QED, e.g., chiral photon-atom
interactions chiral1 ; chiral2 ; chiral3 , single-photon routers rout1 ; rout2
; rout3 ; rout4 , topologically induced unconventional quantum optics tuqo ;
PRXtopo , and entangled-state preparations entpre1 ; entpre2 ; entpre3  to
name a few. In particular, frequency conversion at the single-photon level can
be achieved with a three-level atom (either V type or $\Lambda$ type) coupled
to a single waveguide, where the nonlinear optical process can be accomplished
with only a single photon efficient ; shenFC ; njpthree ; LSzhw .

Considering that the sizes of atoms (either natural or artificial) are in
general much smaller than the wavelengths of the waveguide modes, dipole
approximation is usually adopted to regard the atoms as single points walls .
Recent experimental progress revealed that the approximation should be
modified when atoms interact with the waveguide modes at multiple points that
are separated by large distances. For example, transmons are able to interact
with surface acoustic waves via interdigital transducers transmon1 ; transmon2
; transmon3 , with the separations between different transmon-waveguide
coupling points comparable to or even much larger than the typical wavelengths
($\sim 10^{-6}\,\textrm{m}$) of the surface acoustic waves therein.
Alternatively, one can couple single atoms with a bent waveguide so that they
can interact many times with the waveguide modes, such as the Xmon version
demonstrated in Refs. Lamb ; braided ; engineer . Such structures, which are
referred to as giant atoms, demonstrate striking interference effects that
depend on both the atomic size (i.e., the separation between different
coupling points) and the photonic frequency Lamb ; braided ; engineer ; Lamb2
; GANori ; GAWang ; oscillate ; EIAT ; DLoe ; tunchiral . Moreover, non-
Markovian retardation effects should also be included if the separations
between different coupling points are comparable to or even larger than the
coherence length of the emitted photons, with which the dynamics can markedly
deviate from the Markovian predictions Lamb2 ; oscillate ; kockumRev ;
longhiGA ; DLretard . Recently, giant-atom structures have also been extended
to higher dimensions by using optical lattices of cold atoms highD . Despite
the seminal works above, investigations of single-photon frequency conversions
with such giant-atom interferences are still lacking.

In this paper, we study single-photon scattering via a giant $\Lambda$-type
atom, which can be excited by the waveguide modes via both transitions and is
coupled with the waveguide at two separated points. The input photon can
either be transmitted or reflected directly or undergo frequency conversion,
depending on which of the two lower-energy states of the atom is finally
occupied. From a comparison with the small-atom case efficient ; shenFC ,
where the scattering behavior is only determined by the ratio of the
waveguide-induced decay rates of the two transitions, we reveal that the
scattering here also depends on two phase factors which are related to the two
transition frequencies respectively as well as the separation between the two
coupling points. The presence of two coupling points results in phase-
dependent interference effects, which are in general different for the two
atomic transitions. Such interference effects thus affect the scattering
behaviors via modifying the transition frequencies and decay rates of the
atom. By tuning the phases, the giant atom is able to demonstrate phenomena
that typically occur in various limits of the small-atom case, such as perfect
transmission over the whole frequency range and total reflection. Moreover, we
insert a Sagnac interferometer at each of the coupling points, which results
in quantum interferences between the counterpropagating waveguide modes and
thereby enables efficient frequency conversion. On the other hand, all the
aforementioned giant-atom effects can still be observed and the parametric
conditions remain unchanged in this case, implying that the two kinds of
interference effects play their roles in parallel.

##  II Models and methods

We consider in this paper a giant $\Lambda$-type three-level atom which
couples twice with a single waveguide. As shown in Fig. 1(a), both transitions
$|g\rangle\leftrightarrow|e\rangle$ and $|f\rangle\leftrightarrow|e\rangle$ of
the atom are coupled with the waveguide modes via two coupling points located
at $x_{1}=-d/2$ and $x_{2}=d/2$, where $|g\rangle$, $|f\rangle$, and
$|e\rangle$ are the ground, middle, and excited states, respectively. In the
following, we assume that $|f\rangle$ is a metastable state, which can be
achieved if $|g\rangle$ and $|f\rangle$ are two hyperfine levels generated by,
e.g., the Zeeman effects. The real-space Hamiltonian of the system can be
written as ($\hbar=1$)

$\begin{split}H&=H_{\textrm{w}}+H_{\textrm{a}}+H_{\textrm{int}},\\\
H_{\textrm{w}}&=iv_{g}\int_{-\infty}^{+\infty}dx\Big{[}a_{L}^{{\dagger}}(x)\frac{\partial}{\partial
x}a_{L}(x)-a_{R}^{{\dagger}}(x)\frac{\partial}{\partial x}a_{R}(x)\Big{]},\\\
H_{\textrm{a}}&=\omega_{f}|f\rangle\langle f|+\omega_{e}|e\rangle\langle
e|,\\\
H_{\textrm{int}}&=\int_{-\infty}^{+\infty}dxP(x)\\{g_{1}[a_{R}^{{\dagger}}(x)+a_{L}^{{\dagger}}(x)]|g\rangle\langle
e|\\\ &\quad+g_{2}[a_{R}^{{\dagger}}(x)+a_{L}^{{\dagger}}(x)]|f\rangle\langle
e|+\text{H.c.}\\},\end{split}$ (1)

where $P(x)=\delta(x+d/2)+\delta(x-d/2)$; $H_{\textrm{w}}$ is the free
Hamiltonian of the waveguide modes with $v_{g}$ the group velocity;
$a_{R}^{{\dagger}}$ ($a_{R}$) and $a_{L}^{{\dagger}}$ ($a_{L}$) are the
creation (annihilation) operators of the right-moving and left-moving photons
in the waveguide, respectively; $H_{\textrm{a}}$ is the free Hamiltonian of
the $\Lambda$-type atom, where $\omega_{f}$ and $\omega_{e}$ are the energies
of states $|f\rangle$ and $|e\rangle$ with respect to the ground state,
respectively; and $H_{\textrm{int}}$ describes the interactions between the
atom and the waveguide, where $g_{1}$ and $g_{2}$ are the coupling strengths
of transitions $|g\rangle\leftrightarrow|e\rangle$ and
$|f\rangle\leftrightarrow|e\rangle$ with the waveguide modes, respectively. We
have assumed that the coupling strengths are identical for both coupling
points, i.e., $g_{j}(x_{1})=g_{j}(x_{2})$ ($j=1,2$), with which some limit
phenomena can be achieved, as will be discussed below. In the single-photon
manifold, the eigenstate of Hamiltonian (1) can be written as

$\begin{split}|\psi\rangle&=\sum_{\alpha=g,f}\int_{-\infty}^{+\infty}dx[R_{\alpha}(x)a^{{\dagger}}_{R}(x)|0,\alpha\rangle\\\
&\quad+L_{\alpha}(x)a^{{\dagger}}_{L}(x)|0,\alpha\rangle]+u_{e}|0,e\rangle,\end{split}$
(2)

where $R_{\alpha}(x)$ [$L_{\alpha}(x)$] is the probability amplitude of
creating a right-moving (left-moving) photon in the waveguide at position $x$
and the atom finally in state $|\alpha\rangle$. In addition, $u_{e}$ is the
probability amplitude of the atom in the excited state, finding no photon in
the waveguide. By solving the Schrödinger equation
$H|\psi\rangle=E|\psi\rangle$, one can obtain the following equations of the
probability amplitudes

$\begin{split}ER_{g}(x)&=-iv_{g}\frac{\partial}{\partial
x}R_{g}(x)+g_{1}P(x)u_{e},\\\ EL_{g}(x)&=iv_{g}\frac{\partial}{\partial
x}L_{g}(x)+g_{1}P(x)u_{e},\\\
ER_{f}(x)&=\Big{(}\omega_{f}-iv_{g}\frac{\partial}{\partial
x}\Big{)}R_{f}(x)+g_{2}P(x)u_{e},\\\
EL_{f}(x)&=\Big{(}\omega_{f}+iv_{g}\frac{\partial}{\partial
x}\Big{)}L_{f}(x)+g_{2}P(x)u_{e},\\\
Eu_{e}&=\omega_{e}u_{e}+g_{1}P(x)[R_{g}(x)+L_{g}(x)]\\\
&\quad+g_{2}P(x)[R_{f}(x)+L_{f}(x)].\end{split}$ (3)

Figure 1: (a) Schematic illustration of the single-photon frequency converter,
where both transitions of a $\Lambda$-type three-level atom couple twice with
a waveguide at $x_{1}=-d/2$ and $x_{2}=d/2$. (b) Sketch of the Sagnac-
interference scheme.

We first assume that a single photon of wave vector $k$ ($k>0$) is incident
from the far left of the waveguide and the atom is initialized in the ground
state $|g\rangle$. If the atom is excited by the input photon that is nearly
resonant with the transition $|g\rangle\leftrightarrow|e\rangle$, it can
spontaneously decay to the lower-energy state $|g\rangle$ or $|f\rangle$ and
emit a photon of frequency $v_{g}k$ or $v_{g}k-\omega_{f}$ efficient ; shenFC
. In the latter situation, the photon undergoes an inelastic scattering and
the frequency is down-converted. Equation (3) can be solved according to the
ansatz

$\begin{split}R_{g}(x)&=\Big{\\{}\Theta\Big{(}-x-\frac{d}{2}\Big{)}+A\Big{[}\Theta\Big{(}x+\frac{d}{2}\Big{)}\\\
&\quad-\Theta\Big{(}x-\frac{d}{2}\Big{)}\Big{]}+t_{1}\Theta\Big{(}x-\frac{d}{2}\Big{)}\Big{\\}}e^{ikx},\\\
L_{g}(x)&=\Big{\\{}r_{1}\Theta\Big{(}-x-\frac{d}{2}\Big{)}+B\Big{[}\Theta\Big{(}x+\frac{d}{2}\Big{)}\\\
&\quad-\Theta\Big{(}x-\frac{d}{2}\Big{)}\Big{]}\Big{\\}}e^{-ikx},\\\
R_{f}(x)&=\Big{\\{}M\Big{[}\Theta\Big{(}x+\frac{d}{2}\Big{)}-\Theta\Big{(}x-\frac{d}{2}\Big{)}\Big{]}\\\
&\quad+t_{2}\Theta\Big{(}x-\frac{d}{2}\Big{)}\Big{\\}}e^{iqx},\\\
L_{f}(x)&=\Big{\\{}r_{2}\Theta\Big{(}-x-\frac{d}{2}\Big{)}+N\Big{[}\Theta\Big{(}x+\frac{d}{2}\Big{)}\\\
&\quad-\Theta\Big{(}x-\frac{d}{2}\Big{)}\Big{]}\Big{\\}}e^{-iqx},\end{split}$
(4)

where $q=k-\omega_{f}/v_{g}$ and $\Theta(x)$ is the Heaviside step function.
In addition, $A$ and $B$ ($M$ and $N$) are the probability amplitudes of
finding a right-moving photon and a left-moving one in the region of
$x_{1}<x<x_{2}$, respectively, and the atom finally in state $|g\rangle$
($|f\rangle$). For the frequency-preserving case (i.e., the final atomic state
is $|g\rangle$), we define $t_{1}$ and $r_{1}$ as the transmission and
reflection amplitudes of the input photon, respectively, while for the
frequency-converting case (i.e., the final atomic state is $|f\rangle$) we
define $t_{2}$ and $r_{2}$ as the conversion amplitudes of creating an output
photon with wave vectors $q$ and $-q$, respectively.

Substituting Eq. (4) into Eq. (3), one can obtain

$\begin{split}0&=-iv_{g}(A-1)e^{-ikd/2}+g_{1}u_{e},\\\
0&=-iv_{g}(t_{1}-A)e^{ikd/2}+g_{1}u_{e},\\\
0&=-iv_{g}(r_{1}-B)e^{ikd/2}+g_{1}u_{e},\\\
0&=-iv_{g}Be^{-ikd/2}+g_{1}u_{e},\\\ 0&=-iv_{g}Me^{-iqd/2}+g_{2}u_{e},\\\
0&=-iv_{g}(t_{2}-M)e^{iqd/2}+g_{2}u_{e},\\\
0&=-iv_{g}(r_{2}-N)e^{iqd/2}+g_{2}u_{e},\\\
0&=-iv_{g}Ne^{-iqd/2}+g_{2}u_{e},\\\
0&=\frac{g_{1}}{2}[(A+B+1)e^{-ikd/2}+(A+B\\\
&\quad\,+t_{1}+r_{1})e^{ikd/2}]+\frac{g_{2}}{2}[(M+N)e^{-iqd/2}\\\
&\quad\,+(M+N+t_{2}+r_{2})e^{iqd/2}]-\Delta u_{e},\end{split}$ (5)

where $\Delta=E-\omega_{e}=v_{g}k-\omega_{e}$ is the detuning between the
input photon and the $|g\rangle\leftrightarrow|e\rangle$ transition. Then the
transmission amplitude can be readily obtained as

$t_{1}=\frac{\Delta-2\Gamma_{1}\sin{\phi_{1}}+2i\Gamma_{2}(1+e^{i\phi_{2}})}{\Delta+2i\Gamma_{1}(1+e^{i\phi_{1}})+2i\Gamma_{2}(1+e^{i\phi_{2}})},$
(6)

where $\phi_{1}=kd$ and $\phi_{2}=qd$ are the phases accumulated by photons of
wave vectors $k$ and $q$ propagating between the two coupling points,
respectively. Here $\Gamma_{1}=g_{1}^{2}/v_{g}$ ($\Gamma_{2}=g_{2}^{2}/v_{g}$)
is the radiative decay rate from the excited state $|e\rangle$ to the lower-
energy state $|g\rangle$ ($|f\rangle$) contributed from each atom-waveguide
coupling point. For $d=0$, the transmission amplitude in Eq. (6) can be
simplified as

$t_{1}=\frac{\Delta+4i\Gamma_{2}}{\Delta+4i\Gamma_{1}+4i\Gamma_{2}},$ (7)

which recovers that of a “small” $\Lambda$-type atom shenFC . Note that the
radiative decay rates are quadrupled here due to the two coupling points.
Moreover, we point out that the transmission amplitude becomes

$t_{1}=\frac{\Delta-2\Gamma_{1}\sin{\phi_{1}}}{\Delta+2i\Gamma_{1}(1+e^{i\phi_{1}})}$
(8)

in the case of $\Gamma_{2}=0$ (i.e., $g_{2}=0$), which is exactly identical to
that of a giant two-level atom EITWang . Similarly, one can obtain the other
scattering amplitudes as

$\begin{split}&r_{1}=\frac{-i\Gamma_{1}(1+e^{i\phi_{1}})^{2}}{\Delta+2i\Gamma_{1}(1+e^{i\phi_{1}})+2i\Gamma_{2}(1+e^{i\phi_{2}})},\\\
&t_{2}=\frac{-i\sqrt{\Gamma_{1}\Gamma_{2}}(1+e^{i\phi_{1}})(1+e^{-i\phi_{2}})}{\Delta+2i\Gamma_{1}(1+e^{i\phi_{1}})+2i\Gamma_{2}(1+e^{i\phi_{2}})},\\\
&r_{2}=\frac{-i\sqrt{\Gamma_{1}\Gamma_{2}}(1+e^{i\phi_{1}})(1+e^{i\phi_{2}})}{\Delta+2i\Gamma_{1}(1+e^{i\phi_{1}})+2i\Gamma_{2}(1+e^{i\phi_{2}})}.\end{split}$
(9)

Once again, the amplitudes in Eq. (9) can be simplified to those of a small
$\Lambda$-type atom and those of a giant two-level atom for $d=0$ and
$\Gamma_{2}=0$, respectively. It is clear that
$|t_{1}|^{2}+|r_{1}|^{2}+|t_{2}|^{2}+|r_{2}|^{2}=1$ due to the energy
conservation and $|t_{2}|^{2}\equiv|r_{2}|^{2}$ due to the inherent symmetry.
In the following, we define $T_{1}=|t_{1}|^{2}$ and $R_{1}=|r_{1}|^{2}$ as the
transmission and reflection rates (without frequency conversion),
respectively, and $T_{c}=|t_{2}|^{2}+|r_{2}|^{2}$ as the conversion
efficiency. Note that the intrinsic dissipation $\gamma$ of the excited state
to the environment (other than the waveguide) can be taken into account via
$\Delta\rightarrow\Delta+i\gamma$. In this case the total scattering
probability $T_{1}+R_{1}+T_{c}$ should be smaller than unity. However, we do
not consider such dissipation in this paper, because it only reduces the
scattering probabilities and increases the linewidth of the spectra rather
than introducing any qualitative variation efficient ; shenFC .

##  III Phase controlled photon scattering

Figure 2: Transmission rate $T_{1}$ [(a) and (d)], reflection rate $R_{1}$
[(b) and (e)], and conversion efficiency $T_{c}$ [(c) and (f)] versus detuning
$\Delta$ and phase difference $\Delta\phi$ for $\phi_{1}=2m\pi$ (e.g.,
$\phi_{1}=0$) [(a)-(c)] and $\phi_{1}=(2m+1/2)\pi$ (e.g., $\phi_{1}=\pi/2$)
[(d)-(f)]. The white dotted curves in the pseudo-color maps depict the
trajectories of $\textrm{min}[T_{1}(\Delta)]$, $\textrm{max}[R_{1}(\Delta)]$,
and $\textrm{max}[T_{c}(\Delta)]$ versus $\Delta\phi$. Here we assume
$\eta=1$. Figure 3: Transmission rate $T_{1}$ [(a), (d), (g)], reflection rate
$R_{1}$ [(b), (e), (h)], and conversion efficiency $T_{c}$ [(c), (f), (i)]
versus detuning $\Delta$ and decay ratio $\eta$ for $\Delta\phi=2\pi$
[(a)-(c)], $\Delta\phi=\pi$ [(d)-(f)], and $\Delta\phi=\pi/2$ [(g)-(i)]. Here
we assume $\phi_{1}=2m\pi$ (e.g., $\phi_{1}=0$).

We first study the dependence of the scattering probabilities on $\phi_{1}$
and $\Delta\phi=\phi_{1}-\phi_{2}$, which both can be tuned experimentally as
discussed in detail below. In Fig. 2, the upper plots of each panel are
pseudo-color maps of the scattering probabilities versus detuning $\Delta$ and
phase difference $\Delta\phi$, while the lower ones are the two-dimensional
profiles for some specific values of $\Delta\phi$. It can be found that both
the position and the minimum (maximum) of $T_{1}$ ($R_{1}$ and $T_{c}$) change
periodically with $\Delta\phi$. This is reminiscent of the frequency-dependent
Lamb shift and decay rate of a giant two-level atom Lamb  or a single-mode
self-interference resonator DLoe ; SIDong . For the giant $\Lambda$-type atom,
the effective detuning and linewidth of the spectra are given by the real and
imaginary parts of the denominators in Eqs. (6) and (9), respectively, i.e.,

$\Delta_{\textrm{eff}}=\Delta-2(\Gamma_{1}\sin{\phi_{1}}+\Gamma_{2}\sin{\phi_{2}})$
(10)

and

$\Gamma_{\textrm{eff}}=\Gamma_{1,\textrm{eff}}+\Gamma_{2,\textrm{eff}},$ (11)

with $\Gamma_{1,\textrm{eff}}=2\Gamma_{1}(1+\cos{\phi_{1}})$
[$\Gamma_{2,\textrm{eff}}=2\Gamma_{2}(1+\cos{\phi_{2}})$] the effective decay
rate from $|e\rangle$ to $|g\rangle$ ($|f\rangle$) due to the giant-atom
structure. Equations (10) and (11) clearly show that the giant-atom
interference effects affect the scattering probabilities by modifying the
atomic transition frequencies and decay rates. In Figs. 2(a)-2(c) we find
$T_{1}(\Delta=0)=0$, $R_{1}(\Delta=0)=1$, and $T_{c}(\Delta)\equiv 0$ for
$\phi_{1}=2m\pi$ and $\Delta\phi=\pi$, implying that total reflection is
achieved. In this case, the $|f\rangle\leftrightarrow|e\rangle$ transition is
completely suppressed due to the destructive interference of two corresponding
decay channels, i.e., $\Gamma_{2}[1+\textrm{exp}(i\phi_{2})]=0$; thus the
model reduces to a two-level atom where a resonant incident photon should be
perfectly reflected. For the case of $\phi_{1}=(2m+1/2)\pi$, as shown in Figs.
2(d)-2(f), the patterns of the pseudo-color maps are shifted along the $y$
axis by $\pi/2$, while the linewidth is reduced by $2\Gamma_{1}$ according to
Eq. (11). In this case, the condition of total reflection becomes
$\Delta\phi=3\pi/2$ and $\Delta=2\Gamma_{1}$. In fact, we have checked that
total reflection can always be observed as long as
$\phi_{2}=\phi_{1}-\Delta\phi=(2m+1)\pi$ and $\Delta_{\textrm{eff}}=0$. More
interestingly, frequency-independent perfect transmission (FIPT), i.e.,
$T_{1}(\Delta)\equiv 1$ and $R_{1}(\Delta)=T_{c}(\Delta)\equiv 0$, can be
achieved in the case of $\phi_{1}=(2m+1)\pi$, regardless of the value of
$\Delta\phi$ (not shown here). This is because the two excitation paths of the
$|g\rangle\leftrightarrow|e\rangle$ transition cancel each other completely
such that this transition is decoupled from the waveguide. Such a phenomenon
has also been revealed with a self-interference resonator, which is referred
to as the optical dark states DLoe ; synphonon .

It has been shown that the scattering probabilities are determined solely by
the decay ratio $\eta=\Gamma_{2}/\Gamma_{1}$ in the small-atom case efficient
; shenFC  and the optimal frequency conversion occurs when $\eta=1$. For the
giant-atom case, however, such a condition becomes phase dependent and thus
should be modified. We first demonstrate two special situations, i.e.,
$\phi_{1}=2m\pi$ and $\Delta\phi=2\pi$ in Figs. 3(a)-3(c) and $\phi_{1}=2m\pi$
and $\Delta\phi=\pi$ in Figs. 3(d)-3(f). In the former case, the optimal
frequency conversion ($T_{c}=0.5$) occurs at $\Delta=0$ and $\eta=1$, which is
identical to that of the small-atom case. In the latter case, however, the
frequency conversion is completely suppressed and the input photon that is
resonant with the $|g\rangle\leftrightarrow|e\rangle$ transition can be
totally reflected. In this case, all scattering probabilities are independent
of $\eta$. For a more general case, the condition of the optimal conversion
can be summarized as $\eta=(1+\cos{\phi_{1}})/(1+\cos{\phi_{2}})$ (i.e.,
$\eta_{\textrm{eff}}=\Gamma_{2,\textrm{eff}}/\Gamma_{1,\textrm{eff}}=1$) and
$\Delta=-2(\Gamma_{1}\sin{\phi_{1}}+\Gamma_{2}\sin{\phi_{2}})$ (i.e.,
$\Delta_{\textrm{eff}}=0$). For example, as shown in Figs. 3(g)-3(i), the
optimal frequency conversion occurs at $\eta=2$ and $\Delta=-4\Gamma_{1}$ for
$\phi_{1}=2m\pi$ and $\Delta\phi=\pi/2$, which is in good agreement with the
analytical condition above.

##  IV Efficient frequency conversion with Sagnac interferometers

It can be found that the conversion efficiency is still at most $\frac{1}{2}$
in the giant-atom case (see Figs. 2 and 3), although the giant-atom
interference effects provide additional tunability for the scattering
behaviors. This can be understood from the fact that the giant-atom effects
modify the atomic transition frequencies and decay rates such that the giant
atom behaves like a small one with renormalized parameters. In this section,
we show the feasibility of realizing efficient frequency conversions in the
giant-atom case with the assistance of the quantum interferences between
counterpropagating modes efficient ; shenFC ; YanSag ; zubairy . In
experiments, this can be achieved by coupling the atom with the waveguide
through two identical Sagnac interferometers, each of which is connected with
the waveguide via a $50:50$ coupler (beam splitter) SagExp , as shown in Fig.
1(b). At each coupler, the input photon is split equally into two
counterpropagating parts (i.e., the clockwise and counterclockwise fields of
the Sagnac interferometer), which form photonic superposition states in the
interferometers and exhibit quantum interference effects. Moreover, one can
also introduce a phase shifter in each Sagnac loop, which can be used for
tuning the relative phase $\theta$ between the counterpropagating modes and
thus changing their superposition exp2 . For superconducting qubits, one can
also introduce only one Sagnac loop and couple the qubit with it at two
different points zubairy . Supposing that the whole system is symmetric with
respect to the spatial inversion, the waveguide modes can be described in
terms of the even and odd modes

$\begin{split}&a_{e}(x)=\frac{1}{\sqrt{2}}[a_{R}(x)\pm a_{L}(-x)],\\\
&a_{o}(x)=\frac{1}{\sqrt{2}}[a_{R}(x)-a_{L}(-x)],\end{split}$ (12)

which correspond to $\theta=0$ and $\theta=\pi$ (in the absence of the giant
atom), respectively. In this way, $H_{\textrm{w}}$ and $H_{\textrm{int}}$ in
Eq. (1) can be rewritten as

$\begin{split}H_{\textrm{w}}^{\prime}&=-iv_{g}\int_{-\infty}^{+\infty}dx\Big{[}a_{e}^{{\dagger}}(x)\frac{\partial}{\partial
x}a_{e}(x)+a_{o}^{{\dagger}}(x)\frac{\partial}{\partial x}a_{o}(x)\Big{]},\\\
H_{\textrm{int}}^{\prime}&=\int_{-\infty}^{+\infty}dxP(x)(\tilde{g}_{1}a_{e}^{{\dagger}}|g\rangle\langle
e|+\tilde{g}_{2}a_{e}^{{\dagger}}|f\rangle\langle e|+\text{H.c.}),\end{split}$
(13)

where $\tilde{g}_{j}=\sqrt{2}g_{j}$ ($j=1,\,2$). Clearly, the odd mode $a_{o}$
does not interact with the atom and thus can be discarded in the following
calculations. In this way, the single-photon state of the system can be given
by

$\begin{split}|\psi\rangle=&\int_{-\infty}^{+\infty}dx[\psi_{1}(x)a^{{\dagger}}_{e}(x)|0,g\rangle\\\
&+\psi_{2}(x)a^{{\dagger}}_{e}(x)|0,f\rangle]+u_{e}|0,e\rangle,\end{split}$
(14)

where $\psi_{1}(x)=[R_{g}(x)+L_{g}(-x)]/\sqrt{2}$ and
$\psi_{2}(x)=[R_{f}(x)+L_{f}(-x)]/\sqrt{2}$. Then one can obtain

$\begin{split}E\psi_{1}(x)&=-iv_{g}\frac{\partial}{\partial
x}\psi_{1}(x)+\tilde{g}_{1}P(x)u_{e},\\\
E\psi_{2}(x)&=\Big{(}\omega_{f}-iv_{g}\frac{\partial}{\partial
x}\Big{)}\psi_{2}(x)+\tilde{g}_{2}P(x)u_{e},\\\
Eu_{e}&=\omega_{e}u_{e}+P(x)[\tilde{g}_{1}\psi_{1}(x)+\tilde{g}_{2}\psi_{2}(x)]\end{split}$
(15)

by solving the stationary Schrödinger equation. In this case, the spatial
dependence of $\psi_{1}(x)$ and $\psi_{2}(x)$ can be given by

$\begin{split}\psi_{1}(x)&=\Big{\\{}\Theta\Big{(}-x-\frac{d}{2}\Big{)}+\tilde{A}\Big{[}\Theta\Big{(}x+\frac{d}{2}\Big{)}\\\
&\quad-\Theta\Big{(}x-\frac{d}{2}\Big{)}\Big{]}+\tilde{t}_{1}\Theta\Big{(}x-\frac{d}{2}\Big{)}\Big{\\}}e^{ikd},\\\
\psi_{2}(x)&=\Big{\\{}\tilde{B}\Big{[}\Theta\Big{(}x+\frac{d}{2}\Big{)}-\Theta\Big{(}x-\frac{d}{2}\Big{)}\Big{]}\\\
&\quad+\tilde{t}_{2}\Theta\Big{(}x-\frac{d}{2}\Big{)}\Big{\\}}e^{iqd}.\end{split}$
(16)

Here $\tilde{A}$ and $\tilde{B}$ are the wave amplitudes in the region of
$x_{1}<x<x_{2}$ for the frequency-preserving and frequency-converting cases,
respectively, while $\tilde{t}_{1}$ and $\tilde{t}_{2}$ denote in this case
the transmission (without frequency conversion) and conversion amplitudes,
respectively. Combining Eqs. (15) and (16), we have

$\begin{split}0&=-iv_{g}(\tilde{A}-1)e^{-i\phi_{1}/2}+\tilde{g}_{1}u_{e},\\\
0&=-iv_{g}(\tilde{t}_{1}-\tilde{A})e^{i\phi_{1}/2}+\tilde{g}_{1}u_{e},\\\
0&=-iv_{g}\tilde{B}e^{-i\phi_{2}/2}+\tilde{g}_{2}u_{e},\\\
0&=-iv_{g}(\tilde{t}_{2}-\tilde{B})e^{i\phi_{2}/2}+\tilde{g}_{2}u_{e},\\\
0&=\frac{\tilde{g}_{1}}{2}[(\tilde{A}+1)e^{-i\phi_{1}/2}+(\tilde{A}+\tilde{t}_{1})e^{i\phi_{1}/2}]\\\
&\quad\,+\frac{\tilde{g}_{2}}{2}[\tilde{B}e^{-i\phi_{2}/2}+(\tilde{B}+\tilde{t}_{2})e^{i\phi_{2}/2}]-\Delta
u_{e},\end{split}$ (17)

which results in

$\begin{split}&\tilde{t}_{1}=\frac{\Delta-i\tilde{\Gamma}_{1}(1+e^{-i\phi_{1}})+i\tilde{\Gamma}_{2}(1+e^{i\phi_{2}})}{\Delta+i\tilde{\Gamma}_{1}(1+e^{i\phi_{1}})+i\tilde{\Gamma}_{2}(1+e^{i\phi_{2}})},\\\
&\tilde{t}_{2}=\frac{-4i\sqrt{\tilde{\Gamma}_{1}\tilde{\Gamma}_{2}}\cos{\frac{\phi_{1}}{2}}\cos{\frac{\phi_{2}}{2}}}{\Delta+i\tilde{\Gamma}_{1}(1+e^{i\phi_{1}})+i\tilde{\Gamma}_{2}(1+e^{i\phi_{2}})}\end{split}$
(18)

with $\tilde{\Gamma}_{j}=\tilde{g}_{j}^{2}/v_{g}$. Likewise,
$|\tilde{t}_{1}|^{2}+|\tilde{t}_{2}|^{2}=1$ in the case of $\gamma=0$. In the
following, we will show that efficient frequency conversion (i.e.,
$|\tilde{t}_{2}|^{2}=1$) can also be achieved in the giant-atom case by
exploiting the Sagnac quantum interferences.

Figure 4: Conversion efficiency $\tilde{T}_{c}$ versus detuning $\Delta$ and
phase difference $\Delta\phi$ for (a) $\phi_{1}=2m\pi$ (e.g., $\phi_{1}=0$)
and (b) $\phi_{1}=(2m+1/2)\pi$ (e.g., $\phi_{1}=\pi/2$). The white dotted
curves in the pseudo-color maps depict the trajectories of
$\textrm{max}[\tilde{T}_{c}(\Delta)]$ versus $\Delta\phi$. Here we assume
$\eta=1$.

We plot in Fig. 4 the conversion efficiency
$\tilde{T}_{c}=|\tilde{t}_{2}|^{2}$ versus detuning $\Delta$ and phase
difference $\Delta\phi$ for different values of $\phi_{1}$ (similar to Fig. 2,
the upper and lower plots are pseudocolormaps and two-dimensional profiles,
respectively). In this case, the transmission rate
$\tilde{T}_{1}=|\tilde{t}_{1}|^{2}$ includes both the forward and backward
scattering components. It can be simply calculated according to
$\tilde{T}_{1}\equiv 1-\tilde{T}_{c}$ due to the energy conservation and thus
shows inverse patterns with respect to $\tilde{T}_{c}$. Clearly, it can be
seen from Fig. 4 that efficient frequency conversion with $\tilde{T}_{c}=1$
can be achieved in this case owing to the quantum interference between the
counterpropagating modes in the Sagnac loops. Once again, both the position
and the maximum of $\tilde{T}_{c}$ are $\Delta\phi$ dependent with the period
of $2\pi$. For $\phi_{1}=2m\pi$, as shown in Fig. 4(a), the maximum of
$\tilde{T}_{c}$ decreases gradually with $\Delta\phi$ until
$\tilde{T}_{c}(\Delta)\equiv 0$ [i.e., $\tilde{T}_{1}(\Delta)\equiv 1$] at
$\Delta\phi=\pi$ (this phenomenon is in fact the total reflection shown in
Fig. 2, which is frequency dependent if we divide $\tilde{T}_{1}$ into the
forward and backward components). The largest frequency shift with a value of
$\tilde{\Gamma}_{2}$ is achieved when $\Delta=\pi/2$. In this case, the
effective detuning and linewidth can be given by
$\Delta-(\tilde{\Gamma}_{1}\sin{\phi_{1}}+\tilde{\Gamma}_{2}\sin{\phi_{2}})$
and
$\tilde{\Gamma}_{1}(1+\cos{\phi_{1}})+\tilde{\Gamma}_{2}(1+\cos{\phi_{2}})$,
respectively, which are in fact identical to those in Eqs. (10) and (11) due
to $\tilde{\Gamma}_{j}=2\Gamma_{j}$. As shown in Fig. 4(b), the patterns (see
the pseudocolormaps) are shifted along the $y$ axis by $\pi/2$ and the
linewidth is reduced by $\tilde{\Gamma}_{1}$ by changing $\phi_{1}$ from
$2m\pi$ to $(2m+1/2)\pi$, which is similar to the case without the Sagnac
interferences (see Fig. 2). Note that FIPT is also achievable in this case as
long as $\phi_{1}=(2m+1)\pi$, with which $\tilde{T}_{c}(\Delta\phi,\eta)\equiv
0$ and $\tilde{T}_{1}(\Delta\phi,\eta)\equiv 1$ are independent of all other
parameters and input photons that are off-resonance with the
$|f\rangle\leftrightarrow|e\rangle$ transition undergo no scattering.

Figure 5: Conversion efficiency $\tilde{T}_{c}$ versus detuning $\Delta$ and
decay ratio $\eta$ for (a) $\Delta\phi=2\pi$, (b) $\Delta\phi=\pi$, and (c)
$\Delta\phi=\pi/2$. Here we assume $\phi_{1}=2m\pi$ (e.g., $\phi_{1}=0$).

Moreover, we demonstrate in Fig. 5 that the condition of the optimal frequency
conversion obtained in Sec. III still holds in the presence of the Sagnac
interferences. Once again, the optimal conversion ($\tilde{T}_{c}=1$ in this
case) occurs at $\eta=1$ and $\Delta=0$ if both $\phi_{1}$ and $\Delta\phi$
are integer multiples of $2\pi$, as shown in Fig. 5(a), while the frequency
conversion is completely suppressed over the whole frequency range if
$\phi_{2}=\phi_{1}-\Delta\phi=(2m+1)\pi$, as shown in Fig. 5(b). In addition,
the optimal frequency conversion occurs at $\eta=2$ and
$\Delta=-2\tilde{\Gamma}_{1}$ in a more general case of $\phi_{1}=2m\pi$ and
$\Delta\phi=\pi/2$, as shown in Fig. 5(c), which is in fact the same condition
as that in Figs. 2(g)-2(i) due to $\tilde{\Gamma}_{1}=2\Gamma_{1}$.

Finally, we summarize in Table 1 the analytical conditions of some
aforementioned scattering phenomena. We point out that these conditions are in
fact identical for the cases with and without the Sagnac interferences, which
implies that the interference effects induced by the giant-atom structure and
the Sagnac interferometers are compatible and play their roles independently.
We conclude that FIPT or total reflection can be achieved if $\phi_{1}$ or
$\phi_{2}$ is an odd multiple of $\pi$, while the optimal frequency conversion
demands an appropriate decay ratio $\eta$ for given phases. As a side note, we
point out that the intrinsic dissipation $\gamma$ does not change the
conditions in Table 1. In other words, such dissipation does not affect the
scattering phenomena qualitatively. The only influence is that the scattering
probabilities (linewidth) decrease overall (increases) as $\gamma$ increases
in the total reflection and optimal frequency conversion cases, whereas the
transmission rate remains unity over the whole frequency range in the FIPT
case.

Table 1: Conditions of some scattering phenomena. Phenomena | Conditions
---|---
Total reflection | $\phi_{1}-\Delta\phi=(2m+1)\pi$
| [i.e., $\phi_{2}=(2m+1)\pi$]
| $\Delta=2\Gamma_{1}\sin{\phi_{1}}$
FIPT | $\phi_{1}=(2m+1)\pi$
Optimal frequency |  $\eta=(1+\cos{\phi_{1}})/(1+\cos{\phi_{2}})$,
conversion | $\Delta=-(\tilde{\Gamma}_{1}\sin{\phi_{1}}+\tilde{\Gamma}_{2}\sin{\phi_{2}})$

##  V Experimental implementations of a giant $\Lambda$-type atom

In this section, we briefly discuss the feasibility of the giant
$\Lambda$-type atom considered in this paper. Experimentally, such a model can
be achieved by coupling a GaAs quantum dot (which can be confined in a fiber-
coupled semiconductor channel waveguide confine ) to a U-type bent waveguide
(e.g., optical fiber). In this case, $\omega_{e}/2\pi$ and $\omega_{f}/2\pi$
can be tuned at the order of $10^{14}$ and $10^{9}\,\textrm{Hz}$,
respectively, depending on the strength of the external magnetic field. For
example, one has $\omega_{e}/2\pi=3.7\times 10^{14}\,\textrm{Hz}$ and
$\omega_{f}/2\pi=6\,\textrm{GHz}$ for the $D^{0}-DX^{0}$ transition where the
$g$-factor is measured as $0.44$ efficient ; gfactor . By tuning the
separation $d$ between the two coupling points (i.e., the relative position of
the channel waveguide and the fiber) and the strength of the external magnetic
field, one can tune the phase difference $\Delta\phi$ within $[0,\,2\pi]$ and
the phase $\phi_{1}$ within $[2m,\,2m+2]\pi$ ($m$ is an integer, which is of
the order of $10^{5}$ in this case) explain . One can also implement the model
with an artificial $\Lambda$-type atom coupled twice with a transmission line,
where both $\phi_{1}$ and $\Delta\phi$ can be tuned within $[0,\,2\pi]$
readily by adjusting the external parameters such as the voltages and
currents, or the electric and magnetic fields (such that the energy levels can
be reconfigured) Wreview2 . Note that the non-Markovian retardation effect can
be safely neglected in both implementations due to $d/v_{g}\sim
10^{-10}\,\textrm{s}\ll 1/(\Gamma_{1}+\Gamma_{2})\sim 10^{-7}\,\textrm{s}$
Solano .

##  VI Conclusions

In summary, we have considered a giant $\Lambda$-type atom which is coupled
with a waveguide at two separated points and studied the single-photon
scattering at it. A single input photon can either be transmitted or reflected
directly without frequency conversion or undergo an inelastic scattering
process with converted frequency, depending on which of the two lower-energy
states is finally occupied. For the small-atom case, it is known that the
scattering behavior is determined only by the ratio of the two waveguide-
induced radiative decay rates. For the giant-atom case, however, both elastic
and inelastic scattering processes are also dependent on the phase factors
which are related to the two transition frequencies as well as the separation
between the two coupling points. Similar to a giant two-level atom or a
single-mode self-interference resonator, the scattering processes (both
elastic and inelastic) of the $\Lambda$-type atom exhibit a phase-dependent
frequency shift and linewidth. In particular, each of the two transitions can
be completely suppressed when the corresponding coupling channels interfere
destructively with each other. The underlying physics can be simply
interpreted with the effective frequency shift and linewidth induced by the
giant-atom interference effects. In this way, the giant atom is capable of
accessing various limits of a small one and thus exhibits a series of limit
phenomena such as FIPT and total reflection. To further increase the
efficiency of the frequency conversion, we have also introduced quantum
interferences between counterpropagating modes by inserting Sagnac
interferometers at both coupling points. It was shown that efficient frequency
conversion with unity efficiency can be achieved with the assistance of the
Sagnac interferences and all the phenomena that arise from the giant-atom
interferences can still be observed, which implies that the two kinds of
interference effects can play their roles independently. Finally, we have
summarized the analytical conditions of some limit phenomena, which have been
shown to be identical for the cases with and without the Sagnac interferences.

It is known that a V-type three-level atom with one transition coupled with
the waveguide modes and the other one driven by an external field can be
effectively described by a $\Lambda$-type energy-level structure in terms of
the dressed states njpthree ; LSzhw . This implies that our proposal in this
paper can be naturally extended to the V-type giant atom, where the decay
ratio can be tuned flexibly. Moreover, one can implement a $\Delta$-type giant
atom by driving the magnetic dipole transition between the two lower-energy
states via a microwave field magdipole ; jhli , where phase-dependent
nonreciprocal frequency conversion can be expected due to the closed cyclic
energy level. We believe that the results in this paper have potential
applications in quantum communication and quantum information processing with
single photons.

## Acknowledgments

L. D. thanks Da-Wu Xiao, Peng Zhang, Zhihai Wang, and Yao-Tong Chen for
helpful discussions. This work was supported by the Science Challenge Project
(Grant No. TZ2018003) and the National Natural Science Foundation of China
(Grants No. 11774024, No. 12074030, and No. U1930402).

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#  Remote sensing of backward reflection from stimulated axion decay

Kensuke Homma111corresponding author Graduate School of Advanced Science and
Engineering, Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526,
Japan  International Center for Quantum-field Measurement Systems for Studies
of the Universe and Particles (QUP), KEK, Tsukuba, Ibaraki 305-0801, Japan

###### Abstract

We propose a method for remotely detecting backward reflection via induced
decay of cold dark matter such as axion in the background of a propagating
coherent photon field. This method can be particularly useful for probing
concentrated dark matter streams by Earth’s gravitational lensing effect.
Formulae for the stimulated reflection process and the expected sensitivities
in local and remote experimental approaches are provided for testing eV scale
axion models using broad band lasers. The generic axion-photon coupling is
expected to be explorable up to ${\cal O}(10^{-12})$ GeV-1 and ${\cal
O}(10^{-22})$ GeV-1 for the idealized local and remote setups, respectively.

Despite that the topological nature in quantum Chromodynamics (QCD) rather
naturally allows $CP$ violation, why is the $CP$-symmetry conserved in QCD ?
To resolve this strong $CP$ problem, Peccei and Quinn have proposed the global
$U(1)_{PQ}$ symmetry [1, 2]. As a result of the spontaneous PQ-symmetry
breaking [3, 4], axion, a hypothetical particle, could appear as a kind of
Nambu-Goldstone bosons with non-zero mass. If the energy scale for the
symmetry breaking is much higher than that of the electroweak symmetry, the
coupling of axion to ordinary matter can be weak. This invisible axion can be
a candidate for cold dark matter [5, 6, 7]. Typically viable QCD axion models
predict its mass much below the eV scale. In contrast, there are several
recent models which predict the mass range around eV in the context of
astrophobic QCD axion [8, 9] which are motivated to solve the anomalous
cooling in several stellar environments [10, 11, 12]. Moreover, an axion-like
particle (ALP) model relevant to inflation predicts the eV range ALP [13].
Therefore, it is worth testing the eV scale QCD axion and ALP scenarios as in
general as possible.

So far any of axion halo scopes have utilized static magnetic fields to induce
axion decay via the inverse Primakoff process [14]. A giant magnet is
typically not movable and thus it is practically difficult to bring the
searching system to a far region in the universe where the axion density might
be enhanced compared to the average density. In concrete, a possibility that
the dark matter density is locally concentrated by the gravitational lensing
from Earth has been evaluated [15]. The study predicts that fine-grained
streams of dark matter form the hair-like structure above the surface of Earth
and the typical distance to the roots of hairs is estimated as $\sim 10^{9}$ m
from Earth, where the concentration factor to the average density can reach
$10^{7}-10^{9}$ depending on impact parameters of dark matter streams with
respect to Earth [15]. If these hair-roots exist around Earth, we may expect
that these positions can be stationary with respect to Earth’s motion. We are
thus led to consider methods for detecting such remote stationary objects.

Figure 1:  Concept for detecting backward reflection from stimulated decay of
pseudoscalar dark matter such as axion. This concept is applicable to A)
laboratory based experiments and B) remote sensing of dark matter hair-roots
due to the gravitational lensing effect by Earth with respect to dark matter
streams [15]. $C$ expresses the local concentration factor with respect to the
average cold dark matter density around the Sun [19]. ${\cal L}_{2}$
corresponds to the Lagrange point where the sun light can be shadowed by
Earth. The detailed explanations are found in the main text.

As illustrated in Fig.1 we consider the stimulated decay process of axion in
the coherent state into two photons under an inducing photon beam in the
coherent state. This process accompanies generation of signal photons with
different momenta and polarization states from those of the inducing field in
general. However, if axions are non-relativistic and effectively at rest with
respect to an observer, due to energy-momentum conservation in the axion’s
rest frame, the signal photons from stimulated decays must look as if
reflection occurs from moving mirrors whose directions are determined by the
wave vectors of the inducing photons. Thanks to the reversible nature of
focusing and diverging wave vectors in the focused inducing beam, the
stimulated emission is expected to be fully collectable at the incident point
of the inducing beam. In addition, since axion is a pseudoscalar particle, the
decayed two photons must have the characteristic correlation in the
polarization states. If either one photon is in a linearly polarization state,
the other one’s polarization direction tends to be orthogonal. Therefore, with
respect to the polarization direction of the inducing field($y$-axis),
selecting orthogonally polarized backward photons by a polarization beam
splitter (PBS) will help the discrimination from the trivial background
process in the search. In the following we formulate this stimulated
reflection process and discuss the expected sensitivity to search for
pseudoscalar-type dark matter based on local and remote experimental
approaches, A) and B), respectively.

We discuss the generic interaction Lagrangian between a pseudoscalar field
$\sigma$ and two photons

$-{\cal L}=gM^{-1}\frac{1}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}\sigma$ (1)

where $g$ is a dimensionless coupling strength and $M$ is an energy scale at
which a symmetry breaking occurs with the photon field strength tensor
$F_{\mu\nu}$ and its dual tensor $\tilde{F}_{\mu\nu}$ whose definitions are
given below. The first order perturbation of S-matrix for the interaction
Lagrangian is expressed as

$\displaystyle
S^{(1)}=\left(-\frac{1}{4}\frac{g}{M}\right)\frac{i^{1}}{1!}\int
d^{4}xN[F_{\mu\nu}(x)\tilde{F}^{\mu\nu}(x)\sigma(x)],$ (2)

where $N$ denotes the normal-ordering of operators with the following Fourier
expansions of the relevant fields

$\sigma(x)\equiv\int\frac{d^{3}\bm{q}}{(2\pi)^{3}2q^{0}}(e^{-iqx}b_{\bm{q}}+e^{iqx}b^{\dagger}_{\bm{q}}),$
(3)
$F^{\mu\nu}\equiv(-i)\int\frac{d^{3}\bm{p}}{(2\pi)^{3}2p^{0}}\Sigma_{\lambda=1,2}(P^{\mu\nu}e^{-ipx}a_{\bm{p},\lambda}+\hat{P}^{\mu\nu}e^{ipx}a^{\dagger}_{\bm{p},\lambda}),$
(4)

and

$\displaystyle\tilde{F}^{\mu\nu}\equiv\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}\mbox{\hskip
227.62204pt}$ (5)
$\displaystyle=(-i)\int\frac{d^{3}\bm{p}}{(2\pi)^{3}2p^{0}}\Sigma_{\lambda=1,2}(\tilde{P}^{\mu\nu}e^{-ipx}a_{\bm{p},\lambda_{p}}+\hat{\tilde{P}}^{\mu\nu}e^{ipx}a^{\dagger}_{\bm{p},\lambda_{p}}),\mbox{\hskip
56.9055pt}$

where we introduce the following momentum-polarization tensors as capitalized
symbols for an arbitrary four-momentum $p$ of the electromagnetic field with
the polarization state $\lambda_{p}$:

$\displaystyle P^{\mu\nu}$ $\displaystyle\equiv$ $\displaystyle
p^{\mu}e^{\nu}(\lambda_{p})-e^{\mu}(\lambda_{p})p^{\nu},$ (6)
$\displaystyle\hat{P}^{\mu\nu}$ $\displaystyle\equiv$ $\displaystyle
p^{\nu}e^{*\mu}(\lambda_{p})-e^{*\nu}(\lambda_{p})p^{\mu},$

and

$\displaystyle\tilde{P}^{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}\left(p_{\alpha}e_{\beta}(\lambda_{p})-e_{\alpha}(\lambda_{p})p_{\beta}\right),$
(7) $\displaystyle\hat{\tilde{P}}^{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}\left(p_{\alpha}e^{*}_{\beta}(\lambda_{p})-e^{*}_{\alpha}(\lambda_{p})p_{\beta}\right)$

with the Levi-Civita symbol $\varepsilon^{ijkl}$. The commutation relations
between the photon creation and annihilation operators, $a_{\bm{p},\lambda}$
and $a^{\dagger}_{\bm{p},\lambda}$ are as follows

$\displaystyle[a_{\bm{k},\lambda},a^{\dagger}_{\bm{k}^{{}^{\prime}},\lambda^{{}^{\prime}}}]$
$\displaystyle=$
$\displaystyle(2\pi)^{3}2k^{0}\delta^{3}(\bm{k}-\bm{k}^{{}^{\prime}})\delta(\lambda-\lambda^{{}^{\prime}})$
$\displaystyle\equiv$
$\displaystyle\tilde{\delta}^{3}(\bm{k}-\bm{k}^{{}^{\prime}})\delta(\lambda-\lambda^{{}^{\prime}}),$
$\displaystyle[a_{\bm{k},\lambda},a_{\bm{k}^{{}^{\prime}},\lambda^{{}^{\prime}}}]$
$\displaystyle=$
$\displaystyle[a^{\dagger}_{\bm{k},\lambda},a^{\dagger}_{\bm{k}^{{}^{\prime}},\lambda^{{}^{\prime}}}]=0.$

Henceforth we omit the polarization index $\lambda_{p}$ and the sum over it
because we require fixed beam polarizations in assumed experiments.
Substituting Eqs.(3–7) into Eq.(2), we get

$\displaystyle S^{(1)}=\left(-\frac{1}{4}\frac{g}{M}\right)\frac{i^{1}}{1}\int
d^{4}x(-i)^{2}\times\mbox{\hskip 170.71652pt}$ (9)
$\displaystyle\int\frac{d^{3}\bm{s}}{(2\pi)^{3}2s^{0}}\int\frac{d^{3}\bm{t}}{(2\pi)^{3}2t^{0}}\int\frac{d^{3}\bm{u}}{(2\pi)^{3}2u^{0}}\times\mbox{\hskip
128.0374pt}$ $\displaystyle
N[\bigl{(}S_{\mu\nu}\tilde{T}^{\mu\nu}e^{-i(s+t)x}a_{\bm{s}}a_{\bm{t}}+S_{\mu\nu}\hat{\tilde{T}}^{\mu\nu}e^{-i(s-t)x}a_{\bm{s}}a^{\dagger}_{\bm{t}}+\mbox{\hskip
102.43008pt}$
$\displaystyle\hat{S}_{\rho\sigma}\tilde{T}^{\rho\sigma}e^{-i(t-s)x}a^{\dagger}_{\bm{s}}a_{\bm{t}}+\mbox{\hskip
5.69054pt}\hat{S}_{\rho\sigma}\hat{\tilde{T}}^{\rho\sigma}e^{i(s+t)x}a^{\dagger}_{\bm{s}}a^{\dagger}_{\bm{t}}\bigr{)}\mbox{\hskip
108.12054pt}$
$\displaystyle\times\bigl{(}e^{-iux}b_{\bm{u}}+e^{iux}b^{\dagger}_{\bm{u}}\bigr{)}\mbox{\hskip
5.69054pt}]\mbox{\hskip 199.16928pt}$

where four momenta $s$, $t$ and $u$ correspond to those in $F^{\mu\nu}$,
$\tilde{F}^{\mu\nu}$ and $\sigma$, respectively.

The definitions of the coherent state [16] for a given mean number of photons
$N_{\bm{p}}$ with a three dimensional momentum $\bm{p}$ are summarized as
follows:

$|N_{\bm{p}}\rangle\rangle\equiv\exp\left(-N_{\bm{p}}/2\right)\sum_{n=0}^{\infty}\frac{N^{n/2}_{\bm{p}}}{\sqrt{n!}}|n_{\bm{p}}\rangle,$
(10)

where $|n_{\bm{p}}\rangle$ is the normalized state of $n$ photons

$|n_{\bm{p}}\rangle=\frac{1}{\sqrt{n!}}\left(a^{\dagger}_{\bm{p}}\right)^{n}|0\rangle,$
(11)

with the creation operator $a^{\dagger}_{\bm{p}}$ of photons that share a
common momentum $\bm{p}$ and a common polarization state over different number
states. The following relations on the coherent state

$\langle\langle N_{\bm{p}}|N_{\bm{p}}\rangle\rangle=1$ (12)

and

$\langle\langle N_{\bm{p}}|n|N_{\bm{p}}\rangle\rangle=\langle\langle
N_{\bm{p}}|\left(a^{\dagger}_{\bm{p}}a_{\bm{p}}\right)|N_{\bm{p}}\rangle\rangle=N_{\bm{p}},$
(13)

give us basic properties with respect to the creation and annihilation
operators to the coherent state:

$a_{\bm{p}}|N_{\bm{p}}\rangle\rangle=\sqrt{N_{\bm{p}}}|N_{\bm{p}}\rangle\rangle,\quad\mbox{and}\quad\langle\langle
N_{\bm{p}}|a^{\dagger}_{\bm{p}}=\sqrt{N_{\bm{p}}}\langle\langle N_{\bm{p}}|.$
(14)

For the coherent state of a pseudoscalar field we apply the same properties to
the annihilation and creation operators $b_{\bm{p}}$ and
$b^{\dagger}_{\bm{p}}$, respectively,

$b_{\bm{p}}|{\cal N}_{\bm{p}}\rangle\rangle=\sqrt{{\cal N}_{\bm{p}}}|{\cal
N}_{\bm{p}}\rangle\rangle,\quad\mbox{and}\quad\langle\langle{\cal
N}_{\bm{p}}|b^{\dagger}_{\bm{p}}=\sqrt{{\cal N}_{\bm{p}}}\langle\langle{\cal
N}_{\bm{p}}|$ (15)

with

$\langle\langle{\cal N}_{\bm{p}}|{\cal N}_{\bm{p}}\rangle\rangle=1.$ (16)

We consider the stimulated decay process of a pseudoscalar field in the
coherent state $|{\cal N}_{\bm{q}}\rangle\rangle$ into two photons via the
four-momentum exchange: $q\rightarrow p_{i}+p_{s}$ under an inducing fields
$|N_{\bm{p}_{i}}\rangle\rangle$ in the coherent state of photons accompanying
generation of a signal photon $p_{s}$. The initial and final states,
respectively, are thus defined as follows:

$\displaystyle|\Omega\rangle$ $\displaystyle\equiv$ $\displaystyle|{\cal
N}_{\bm{q}}\rangle\rangle|N_{\bm{p}_{i}}\rangle\rangle|0\rangle\quad\mbox{and}$
(17) $\displaystyle\langle\Omega^{{}^{\prime}}|$ $\displaystyle\equiv$
$\displaystyle\langle\langle{\cal N}_{\bm{q}}|\langle\langle
N_{\bm{p_{i}}}|\langle 1_{\bm{p_{s}}}|=\langle\langle\Omega|a_{\bm{p_{s}}}.$

The transition amplitude is then expressed as follows

$\displaystyle\langle\Omega^{{}^{\prime}}|S^{(1)}|\Omega\rangle=\left(-\frac{1}{4}\frac{g}{M}\right)\frac{i^{1}}{1}\int
d^{4}x(-i)^{2}\times\mbox{\hskip 199.16928pt}$ (18)
$\displaystyle\int\frac{d^{3}\bm{s}}{(2\pi)^{3}2s^{0}}\int\frac{d^{3}\bm{t}}{(2\pi)^{3}2t^{0}}\langle\langle
N_{\bm{p_{i}}}|\langle 1_{\bm{p_{s}}}|\mbox{\hskip 199.16928pt}$
$\displaystyle\bigl{(}S_{\mu\nu}\tilde{T}^{\mu\nu}e^{-i(s+t)x}a_{\bm{s}}a_{\bm{t}}+S_{\mu\nu}\hat{\tilde{T}}^{\mu\nu}e^{-i(s-t)x}a^{\dagger}_{\bm{t}}a_{\bm{s}}+\mbox{\hskip
156.49014pt}$
$\displaystyle\hat{S}_{\rho\sigma}\tilde{T}^{\rho\sigma}e^{-i(t-s)x}a^{\dagger}_{\bm{s}}a_{\bm{t}}+\hat{S}_{\rho\sigma}\hat{\tilde{T}}^{\rho\sigma}e^{i(s+t)x}a^{\dagger}_{\bm{s}}a^{\dagger}_{\bm{t}}\bigr{)}|N_{\bm{p_{i}}}\rangle\rangle|0\rangle\times\mbox{\hskip
122.34692pt}$
$\displaystyle\int\frac{d^{3}\bm{u}}{(2\pi)^{3}2u^{0}}\langle\langle{\cal
N}_{\bm{q}}|\bigl{(}e^{-iux}b_{\bm{u}}+e^{iux}b^{\dagger}_{\bm{u}}\bigr{)}|{\cal
N}_{\bm{q}}\rangle\rangle.\mbox{\hskip 170.71652pt}$

Results of photon annihilation-creation operators to the given states are
summarized as follows

$\displaystyle
a_{\bm{k}}|N_{\bm{p_{i}}}\rangle\rangle|0\rangle=\tilde{\delta}^{3}(\bm{k}-\bm{p_{i}})\sqrt{N_{\bm{p_{i}}}}|N_{\bm{p_{i}}}\rangle\rangle|0\rangle+|N_{\bm{p_{i}}}\rangle\rangle
a_{\bm{k}}|0\rangle\mbox{\hskip 17.07164pt}$ (19)
$\displaystyle=\tilde{\delta}^{3}(\bm{k}-\bm{p_{i}})\sqrt{N_{\bm{p_{i}}}}|N_{\bm{p_{i}}}\rangle\rangle|0\rangle.\mbox{\hskip
73.97716pt}$
$a_{\bm{l}}a_{\bm{k}}|N_{\bm{p_{i}}}\rangle\rangle|0\rangle=\tilde{\delta}^{3}(\bm{k}-\bm{p_{i}})\sqrt{N_{\bm{p_{i}}}}|N_{\bm{p_{i}}}\rangle\rangle
a_{\bm{l}}|0\rangle=0\mbox{\hskip 113.81102pt}$ (20)

because of $\bm{l}\neq\bm{k}$.

$\displaystyle\langle\langle N_{\bm{p_{i}}}|\langle
1_{\bm{p_{s}}}|a^{\dagger}_{\bm{k}}=\langle\langle N_{\bm{p_{i}}}|\langle
1_{\bm{p_{s}}}|\tilde{\delta}^{3}(\bm{k}-\bm{p_{i}})\sqrt{N_{\bm{p_{i}}}}+\mbox{\hskip
28.45274pt}$ (21) $\displaystyle\langle\langle N_{\bm{p_{i}}}|\langle
0|\tilde{\delta}^{3}(\bm{k}-\bm{p_{s}}).\mbox{\hskip 76.82243pt}$
$\displaystyle\langle\langle N_{\bm{p_{i}}}|\langle
1_{\bm{p_{s}}}|a^{\dagger}_{\bm{k}}a^{\dagger}_{\bm{l}}$ $\displaystyle=$
$\displaystyle\langle\langle N_{\bm{p_{i}}}|\langle
0|\tilde{\delta}^{3}(\bm{k}-\bm{p_{i}})\sqrt{N_{\bm{p_{i}}}}\tilde{\delta}^{3}(\bm{l}-\bm{p_{s}})\mbox{\hskip
17.07164pt}$ $\displaystyle+$ $\displaystyle\langle\langle
N_{\bm{p_{i}}}|\langle
0|\tilde{\delta}^{3}(\bm{k}-\bm{p_{s}})\tilde{\delta}^{3}(\bm{l}-\bm{p_{i}})\sqrt{N_{\bm{p_{i}}}}$

because of $\bm{l}\neq\bm{k}$. Combination of Eqs.(19) and (21) gives

$\displaystyle\langle\langle N_{\bm{p_{i}}}|\langle
1_{\bm{p_{s}}}|a^{\dagger}_{\bm{l}}a_{\bm{k}}|N_{\bm{p_{i}}}\rangle\rangle|0\rangle=\tilde{\delta}^{3}(\bm{l}-\bm{p_{s}})\tilde{\delta}^{3}(\bm{k}-\bm{p_{i}})\sqrt{N_{\bm{p_{i}}}}.\mbox{\hskip
17.07164pt}$ (23)

By substituting Eqs.(20),( Remote sensing of backward reflection from
stimulated axion decay) and (23) into Eq.(18), we obtain the invariant
amplitude kinematically consistent with the stimulated decay process:
$q\rightarrow p_{i}+p_{s}$ as follows

$\displaystyle\langle\Omega^{{}^{\prime}}|S^{(1_{dcy})}|\Omega\rangle=(-i)(2\pi)^{4}\delta^{4}(q-p_{i}-p_{s}){\cal
M}_{i}$ (24)

with

${\cal M}_{i}\equiv 2\sqrt{N_{\bm{p_{i}}}}\sqrt{{\cal
N}_{\bm{q}}}\left(-\frac{1}{4}\frac{g}{M}\right)\hat{P_{i}}_{\rho\sigma}\hat{\tilde{P_{s}}}^{\rho\sigma}$
(25)

and

$\displaystyle\hat{P_{i}}_{\rho\sigma}\hat{\tilde{P_{s}}}^{\rho\sigma}=\varepsilon^{\mu\nu\alpha\beta}p_{i\mu}p_{s\alpha}e^{*}_{i\nu}e^{*}_{s\beta}=-2\omega^{2}_{i}\cos\Phi$
(26)

where $\omega_{i}(=\omega_{s})=m/2$ with $\omega_{i}\equiv p^{0}_{i}$ and
$\omega_{s}\equiv p^{0}_{s}$ is the photon energy of the inducing field
corresponding to half of the mass of a pseudoscalar particle, $m$, and $\Phi$
is a relative angle between the direction of the electric field component of
the linear polarization of a decayed photon with respect to that of the
magnetic field component of the linearly polarized inducing photons.

Given an invariant amplitude ${\cal M}$ for the spontaneous decay rate of a
rest particle with its four momentum $p_{0}$ and mass $m$ into two photons :
$p_{0}\rightarrow p_{1}+p_{2}$, the differential decay rate $d\Gamma$ is
expressed as

$d\Gamma=\frac{1}{2m}\frac{1}{2!}\int\frac{d^{3}\bm{p_{1}}}{(2\pi)^{3}2\omega_{1}}\int\frac{d^{3}\bm{p_{2}}}{(2\pi)^{3}2\omega_{2}}(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-p_{0})|{\mathcal{M}}|^{2}$
(27)

where we introduced the factor of $1/2!$ because the two photons with energy
$\omega_{1}$ and $\omega_{2}$ are identical particles. Then the differential
rate with respect to a solid angle of $p_{1}$ is eventually expressed as

$\frac{d\Gamma}{d\Omega_{1}}=\frac{|{\mathcal{M}}|^{2}}{128\pi^{2}m}.$ (28)

By substituting Eq.(26) into Eq.(25), we can express the invariant amplitude
for the stimulated decay process as

${\cal M}_{i}=\sqrt{N_{\bm{p_{i}}}}\sqrt{{\cal
N}_{\bm{q}}}\left(\frac{g}{M}\right)\omega^{2}_{i}\cos\Phi.$ (29)

In order to equate ${\cal M}_{i}$ in Eq.(29) with ${\cal M}$ in Eq.(28), we
must require that momentum and polarization of either one of decayed photons
coincides with those of an inducing beam. This requirement originates from the
delta function in the commutation relation in Eq.( Remote sensing of backward
reflection from stimulated axion decay) as explicitly shown in the derivation
of the amplitude. If a pseudoscalar particle exists at rest, $p1$ and $p_{2}$
photons must be emitted back-to-back to each other with equal energies.
Therefore, if $p_{1}$ photon in the spontaneous decay process coincides with
$p_{i}$ within a range of the inducing beam, the stimulated emission of
$p_{2}$, that is, the signal photon $p_{s}$ must be exactly backward with
respect to the $p_{1}$ direction. For later convenience, we define the induced
decay rate as

$\frac{d\Gamma_{i}}{d\Omega}\equiv N_{i}{\cal
N}_{q}\frac{d\Gamma_{0}}{d\Omega}$ (30)

with

$\frac{d\Gamma_{0}}{d\Omega}\equiv\frac{1}{128\pi^{2}m}\left(\frac{g}{M}\right)^{2}\omega^{4}_{i}\cos^{2}\Phi$
(31)

in order to factor out the field strength parameters.

Based on the above picture, we evaluate the stimulated reflection yield. For a
laser field a Gaussian beam as a function of spacetime point $(t,x,y,z)$ is
typically used to express a focused field [17]. We adopt the spherical wave
propagation which is interpreted as the approximated wave propagation of a
Gaussian beam at a far distance from the focal point [17]. Since in this case
the wave front simply form a sphere, the entire solid angle of the inducing
laser propagating from and to the focal point is available as the inducing
decay area. Thus backward signal photons are, in principle, fully collectable
at the incident point of the inducing field if dark matter particles are at
rest with respect to the incident point, wherever signal photons are
generated. This is thanks to the reversible nature of the coherent light
propagation.

We consider the case in the upper part of Fig.1 where a pulsed inducing field
with duration of $\tau$ is focused once and the searching beam is generated
from the focal point. The central pulse position $z$ of the inducing field is
moving with the velocity of light $c$ along the $z$-axis in free space which
is occupied by coherent axions with the number density $\rho_{a}$. The
stimulated reflection yield is then parametrized as follows

${\cal
Y}=\int^{L_{1}/c}_{L_{0}/c}dt\int\rho_{i}dV_{i}\int\rho_{a}dV_{i}\int^{\Omega_{i}}_{0}\frac{d\Gamma_{0}}{d\Omega}d\Omega,$
(32)

where $V_{i}$ specifies the volume scanned by the inducing field while it
propagates over the searching region from $z=L_{0}$ to $L_{1}$ with the solid
angle
$\Omega_{i}=\int^{\theta_{0}}_{0}d\theta\sin\theta\int^{2\pi}_{0}d\phi=2\pi(1-\cos\theta_{0})$
defined by the divergence angle $\theta_{0}$ determined by geometrical optics.
The number density of the inducing field $\rho_{i}$ is parametrized as
$\rho_{i}(z)=\frac{N_{i}}{c\tau z^{2}\Omega_{i}}$ as a function of $z$
reflecting the situation that the pulse center position moves along the z-axis
with the surface area of $z^{2}\Omega_{i}$ at $z$. As for the volume integral
on the axion density, we express it as ${\cal
N}_{q}\equiv\int\rho_{a}dV_{i}=\int^{\infty}_{-\infty}dz^{{}^{\prime}}\rho_{a}c\tau{z^{{}^{\prime}}}^{2}\Omega_{i}\delta(z^{{}^{\prime}}-z),$
where the delta function requires local overlaps with inducible ranges along
the propagation of the inducing field. The stimulated reflection yield while
the inducing field propagates is eventually estimated as

$\displaystyle{\cal
Y}=\int^{L_{1}/c}_{L_{0}/c}dt\int^{L_{1}}_{L_{0}}dzz^{2}\int^{\theta_{0}}_{0}d\theta\sin\theta\int^{2\pi}_{0}d\phi\frac{N_{i}}{c\tau
z^{2}\Omega_{i}}\times\mbox{\hskip 170.71652pt}$ (33)
$\displaystyle\int^{\infty}_{-\infty}dz^{{}^{\prime}}\rho_{a}c\tau{z^{{}^{\prime}}}^{2}\Omega_{i}\delta(z^{{}^{\prime}}-z)\frac{d\Gamma_{0}}{d\Omega}\Omega_{i}\mbox{\hskip
170.71652pt}$
$\displaystyle=\frac{L_{1}-L_{0}}{c}N_{i}\cdot\rho_{a}\frac{2}{3}\pi(L^{3}_{1}-L^{3}_{0})\Omega_{i}\cdot\frac{d\Gamma_{0}}{d\Omega}\Omega_{i}\mbox{\hskip
213.39566pt}$
$\displaystyle=\frac{\pi}{48}\left[\frac{1}{m}\left(\frac{g}{M}\right)^{2}\omega^{3}_{i}\right]\times\mbox{\hskip
290.21802pt}$
$\displaystyle\left(\frac{L_{1}-L_{0}}{c}\frac{c}{\lambda_{i}}\right)\cos^{2}\Phi
N_{i}\rho_{a}(L^{3}_{1}-L^{3}_{0})(1-\cos\theta_{0})^{2}\mbox{\hskip
170.71652pt}$

where the factor enclosed by $\left[\quad\right]$ partially keeps natural
units while $\omega_{i}$ out of $\omega^{4}_{i}$ in Eq.(31) is replaced with
frequency using the wavelength of the inducing field $\lambda_{i}$. It is
important to note that the stimulated reflection yield eventually does not
depend on whether the inducing field is provided as a pulsed wave or a
continuous wave.

As the inducing field in the coherent state, we consider broad-band pulsed
lasers since target axions in this paper are in the eV mass range. In order to
scan a wide mass range, pulsed lasers from the optical comb [18] would be
useful. In the following evaluation we assume availability of optical comb
covering $0.35$ $\mu$m - $4.4$ $\mu$m [18] by assuming a flat wavelength
spectrum for simplicity.

We provide sensitivity evaluations in the following two experimental
approaches: A) a laboratory based experiment targeting on the mean number
density of axions around the Sun as cold dark matter [19], B) a remote
detection aiming at the locally concentrated number density in hair-roots due
to the gravitational lensing effect by Earth. We thus introduce the local
concentration factor $C$ as follows

$\rho_{a}\sim C\frac{0.4\times
10^{9}\quad[\mbox{eV/cm}^{3}]}{m\quad[\mbox{eV}]}.$ (34)

The local velocity of cold dark matter on Earth is traditionally assumed to
follow the Maxwellian velocity distribution by taking the Sun’s velocity is
stationary with respect to the average velocity of cold dark matter [15].
However, there should also be a fraction of cold dark matter whose velocity
distribution has non-thermal natures [20]. In A) with $C=1$, we thus
conservatively assume that dark matter particles are moving with the escape
velocity from the Galaxy $\sim 300$ km/s [19] at maximum to any observation
points, that is, the maximum relative velocity of $\beta={\cal O}(10^{-3})$ in
units of the velocity of light $c$. In order to allow deviations of induced
emission angles from the exact backward direction due to the slight transverse
Lorentz boost, the radius of the signal collector, $R_{c}$, must satisfy
$R_{c}\sim\beta L_{1}$ with the depth of the searching volume $L_{1}$.
Therefore, if $R_{c}=1$ m is assumed, the searchable depth is limited to
$L_{1}\sim 10^{3}$ m. In this case the acceptance factor to all of signal
photons can be $A_{cc}\sim 1.0$.

In B) with $C=10^{7}-10^{9}$, we consider shooting a coherent pulsed inducing
beam to a remote cold dark matter target such as concentrated dark matter
hair-roots localized at around $10^{9}$ m from Earth [15]. Since these dark
matter hair-roots originate from the gravitational lensing effect by Earth,
the target should be comoving with Earth. As illustrated in Fig.1, instead of
observing signal photons on the ground, the most ideal location of the laser
injection and the signal photon detector would be the ${\cal L}_{2}$ point,
$1.5\times 10^{9}$ m from Earth, known as one of five Lagrange points which is
on the straight line between the Sun and Earth. This is because the detection
system can be contained within the shadowed volume by Earth allowing for
mitigating background photons from the Sun and also absorption of signal
photons in the air. The length of a hair-root is estimated by using Eq.(36) in
[15] where the focal length $F$ for cold dark matter streams incident with the
velocity $v=220$ km/s as a function of the impact parameter $b$ is expressed
as $F(b)=\frac{\xi R^{2}_{E}}{3}(1+\frac{b^{2}}{4R^{2}_{E}})$ with $\xi\equiv
v^{2}/(GM_{E})=1.2\times 10^{-4}$ m-1, Earth’s radius $R_{E}$ and mass
$M_{E}$, and the gravitational constant $G$. By changing $b$ from 0 to
$R_{E}$, the expected hair-root range from $L_{0}=(1.6-1.5)\times 10^{9}$ m to
$L_{1}=(2.1-1.5)\times 10^{9}$ m is obtained where the offset is the location
of ${\cal L}_{2}$. We conservatively take a constant $C\sim 10^{7}$ over the
hair-root range. Given $F(b=R_{E})=5\xi R^{2}_{E}/12$, the indent angle of
dark matter streams at the focal point is estimated as $\sim
R_{E}/F(b=R_{E})=12/(5\xi R_{E})=3.1\times 10^{-3}$ rad. Therefore, the
transverse velocity in units of $c$ in a focused hair-root with respect to the
propagation direction of the inducing beam is estimated as $\beta=2.3\times
10^{-6}$ given the incident velocity $v=220$ km/s. It is practically
impossible to collect all of signal photons in this case. The acceptance
factor of the signal collector with the radius $R_{c}=1$ m is estimated as
$A_{cc}\sim(R_{c}/(\beta L_{1}))^{2}=6.2\times 10^{-7}$.

In the both approaches, we consider measuring the arrival time distribution of
signal photons with the orthogonally rotated linear polarization state to the
linear polarization direction of the inducing laser by requiring the same
photon energy as that of the inducing laser. We assume a single photon
sensitive device such as a photomultiplier whose random thermal noise rate can
be typically suppressed at $r_{thn}=1$ Hz. The overall signal collection
factor including the detection efficiency is assumed to be
$\epsilon=0.1A_{cc}$. The repetition rate of the inducing pulse laser is
assumed to be $f=1$ Hz. Within the scanning time window of $(L_{1}-L_{0})/c$
per shot, the accidental coincidence rate between the thermal noise rate and
the laser repetition rate is expressed as $R_{bkg}\equiv
2r_{thn}f(L_{1}-L_{0})/c$. We thus can discuss a null hypothesis as Gaussian
fluctuations after subtracting the total background statistics $TR_{bkg}$ with
the data acquisition time $T=10^{6}$ sec from the observed number of photons
consistent with the signal feature. In order to evaluate the sensitivity, we
require that the observed yield $fT\epsilon{\cal Y}$ based on Eq.(33) is
greater than five standard deviations of the subtraction-associated
fluctuation over the total statistics, that is, $\delta
N_{bkg}=5\sqrt{2TR_{bkg}}$. Therefore, the upper limit of the coupling
strength when no signal photon is observed is express as

$\frac{g}{M}=\sqrt{\frac{\delta
N_{bkg}}{fT\epsilon\frac{\pi\omega^{3}_{i}}{48m}\left(\frac{L_{1}-L_{0}}{\lambda_{i}}\right)N_{i}\rho_{a}(L^{3}_{1}-L^{3}_{0})(1-\cos\theta_{0})^{2}}}$
(35)

by requiring the orthogonal polarization state $\cos\Phi=1$.

Figure 2:  Sensitivity projections based on backward reflection from
stimulated axion decay by surveying with a spherical inducing laser field. The
details of symbols and used parameters can be found in the main text.

The magenta solid and dashed horizontal curves in Fig.2 show expected
reachable coupling strengths obtained from Eq.(35) for A) and B),
respectively, as a function of axion mass $m=2\omega_{i}$ covered by the laser
spectrum range. For simplicity we commonly assume the inducing laser pulse
energy of 1 J and the divergence angle at the incident points, $\theta_{0}=1$
mrad. The range of photon wavelengths is $\lambda_{i}=0.35-4.4\quad\mu$m. The
searching depths of $L_{0}=0\sim L_{1}=10^{3}$ m and $L_{0}=0.1\times
10^{9}\sim L_{1}=0.6\times 10^{9}$ m are assumed for A) and B), respectively.
The upper solid brown line and yellow band are the predictions from the
benchmark QCD axion model: the KSVZ model [21] with the model parameters
$E/N=0$ and with $0.07<\left|E/N-1.95\right|<7$, respectively, while the
bottom dashed brown line is the prediction from the DFSZ model [22] with
$E/N=8/3$. The cyan lines are the predictions from the ALP miracle model
relevant to inflation[13] with the model parameters $c_{\gamma}=1,0.1,0.01$.
The gray area shows the excluded region by the vacuum magnetic birefringence
experiment (PVLAS [23]). The purple areas are excluded regions by the Light-
Shining-through-a-Wall (LSW) experiments (ALPS [24] and OSQAR [25]). The
light-cyan horizontal solid line indicates the upper limit from the search for
eV (pseudo)scalar penetrating particles in the SPS neutrino beam (NOMAD) [26].
The blue areas are exclusion regions from the optical MUSE-faint survey [28].
The green area indicates the excluded region by the helioscope experiment CAST
[29]. The red shaded area shows the excluded range based on a stimulated
resonant photon collider, SRPC, in quasi-parallel collision geometry
(SAPPHIRES01) [30]. The red filled area indicates the excluded range by a
pilot search with the fixed angle three-beam SRPC($\mathrm{{}^{t}SRPC}$00)
[31].

In conclusion, we have formulated the stimulated decay process of a
pseudoscalar particle into two photons in the background of an inducing
coherent electromagnetic field and applied the formula to obtain the
stimulated backward reflection yield based on the spherical wave propagation
of the inducing laser field. For the laboratory based experiment targeting on
locally moving cold dark matter, the sensitivity can reach $f_{a}\sim 10^{7}$
GeV, for instance, based on the concrete relation $g/M=\alpha/(2\pi
f_{a})C_{\gamma}=1.2\times 10^{-11}$ GeV-1 with $C_{\gamma}=0.1$ in the model
[9]. We also have evaluated the yield in the ideal case where an observer
stays at ${\cal L}_{2}$ Lagrange point so that positions of dark matter hair-
roots can be stationary with respect to ${\cal L}_{2}$. The 4th power
dependence on the search depth of the signal yield can significantly improve
the sensitivity to the weak coupling domain. This evaluation indeed indicates
very high sensitivity even beyond the gravitational coupling strength
$\alpha/M_{Planck}=6\times 10^{-22}$ GeV-1.

We have focused on the eV mass range as the concrete application of the
proposed method in this paper, however, the method is in principle extendable
to any frequency ranges of inducing beams. Given intense coherent photon
sources over many orders of magnitude in frequency, it is also in principle
possible to perform B)-type experiments from the ground, since locations of
reflection signals are predictable based on the rotation speed of Earth at the
detector location, though the air and clouds become obvious background
sources. However, if time-of-flight is properly combined with pulsed inducing
fields, we can separate long-traveled reflection signals from the distant dark
matter target against such short distance background components. In practice,
ground based experiments have advantage in capability of supplying high power
to the sources. Once we could identify axion mass, this method can also
provide a measure for local and remote dark matter velocity distributions
based on divergence angles of reflected signal photons.

## Acknowledgments

The author acknowledges Tomohiro Inagaki, Taiyo Nakamura and Tsunefumi Mizuno
for useful discussions and also the support of the Collaborative Research
Program of the Institute for Chemical Research at Kyoto University (Grant Nos.
2018–83, 2019–72, 2020–85, 2021–88, 2022–101, and 2023–101) and Grants-in-Aid
for Scientific Research Nos. 17H02897, 18H04354, 19K21880, and 21H04474 from
the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of
Japan.

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