# Time-ordering effects in a one-atom laser based on electromagnetically-
induced transparency

Dmitri B. Horoshko<EMAIL_ADDRESS>B. I. Stepanov Institute of
Physics, NASB, Nezavisimosti Ave. 68, Minsk 220072 Belarus  Chang-shui Yu
School of Physics, Dalian University of Technology, Dalian 116024, China
Sergei Ya. Kilin  B. I. Stepanov Institute of Physics, NASB, Nezavisimosti
Ave. 68, Minsk 220072 Belarus

###### Abstract

One-atom laser based on electromagnetically-induced transparency, suggested
recently [Phys. Rev. Lett. 124, 093603 (2020)], is capable of generating
Schrödinger cat states in the regime of strong ground-state coupling. In this
regime, we find the exact solution for the Schrödinger equation with a time-
dependent effective Hamiltonian by considering the Magnus expansion of the
time-ordered exponential and calculating analytically the time-ordering terms,
omitted in the previous study. We show that the time-ordering term affects the
relative phase of two coherent components of the generated Schrödinger cat
state. We show this influence by calculating various nonclassicality
indicators for the cavity field, such as total noise, average parity and
relative total noise. We find, that time-ordering becomes important at the
average photon number in the cavity below 1, in striking contrast to the case
of single-pass parametric downconversion, where it becomes important at
average photon number in one optical mode above 4.

##  I Introduction

One atom laser (OAL) is a laser, using just one atom as its active medium,
which was shown to be enough for light amplification by stimulated emission of
radiation [1]. Realizations of OAL use a neutral atom [2] or an ion [3]
trapped in a definite position inside a high-finesse optical cavity. Closely
related concepts are one-atom maser [4, 5], realizing a similar atom-field
interaction in the microwave domain, and single-quantum-dot nanolaser [6],
where the single emitter is represented by a quantum dot in a semiconductor
matrix. All these microscopic laser devices share one feature: the state of
the generated cavity field may be highly nonclassical [7, 8, 9, 10, 11] and
therefore a correct description of the atom-field interaction requires field
quantization, in contrast to the conventional laser, where a semiclassical
description is often sufficient. Studying of these devices is important for
better understanding the fundamental laws of interaction of single emitters
with quantized electromagnetic field. Besides, the generated nonclassical
states of light are prospective for applications in quantum metrology and
quantum information processing [12].

One of the main limiting factors for producing highly nonclassical optical
states in OAL is spontaneous decay from the excited atomic level. Recently, it
was suggested [13] to use the technique of electromagnetically-induced
transparency (EIT) [14, 15, 16, 17, 18, 19, 20] in OAL, employing an atom in
$\Lambda$-configuration with two ground and one excited state. In this
configuration, under conditions of EIT, the population of the excited state of
the atom is very low, which dramatically decreases the influence of its
spontaneous decay on the operation of the laser. Two regimes of the EIT-based
OAL (EIT-OAL) are distinguished. In the regime of weak ground-state coupling,
which was recently realized in an experiment [21], OAL works as a single-
photon source. In the strong ground-state coupling an elaborated theoretical
model predicts generation of Schrödinger-cat states of the optical cavity
field. We note that the model of Ref. [13] disregards the time ordering in the
evolution operator, which is a good approximation for a small number of
photons in the cavity only. Generation of superpositions of two
macroscopically (or at least mesoscopically) distinguishable states, known as
optical analogs of the Schrödinger cat state, may thus require considering the
time-ordering effects in the evolution of a quantum system with a time-
dependent Hamiltonian [22, 23]. Indeed, it was shown for the squeezed states
of light both numerically [24] and analytically [25] that time-ordering
becomes important at squeezing above 12.5 dB, which corresponds, for a
squeezed vacuum state, to approximately 4 photons on average in one optical
mode. On the other hand, a superposition of two coherent states with opposite
phases becomes a Schrödnger cat state at the average photon number above 2
[26].

The main aim of this article is to consider the evolution operator of EIT-OAL
in the regime of strong ground-state coupling in a form of Magnus expansion
taking into account the time-ordering terms of all orders. Fortunately, the
Magnus series breaks off at the second term, allowing us thus to obtain an
exact solution for the atom-field dynamics. A secondary aim is to analyze the
state of the generated cavity field with the help of various measures of
nonclassicality and to observe the effect of time ordering on these measures.
In Sec. II we give a general description of the considered system and deduce
the effective Hamiltonian and the atom-field state without time-ordering,
mainly following Ref. [13]. In Sec. III we provide the exact solution for the
evolution operator with a time-dependent Hamiltonian and show the effect of
time ordering on the state of the generated optical field by calculating
various nonclassicality indicators for the cavity field, such as total noise,
average parity and relative total noise. Sec. IV concludes the article.

##  II Temporal evolution of atom and field in EIT-OAL

###  II.1 System description in the Schrödinger picture

The energy levels of EIT-OAL [13, 21] are shown in Fig. 1. In the strong
ground-state coupling regime, considered here, we assume
$\Omega_{12}\gg(g,\Omega_{23})\gg\kappa$, where $\kappa$ is the cavity-field
decay rate. Smallness of the latter with respect to the Rabi frequencies of
atomic transitions means that the atom makes many cycles of transitions before
a cavity photon is lost. Therefore, one can disregard this loss mechanism for
the atom-field evolution at interaction times shorter than $\kappa^{-1}$,
which we assume in the following. One can also disregard the spontaneous
emission from the exited state because, as we will see later, the population
of this level is negligible in the same period of time.

Figure 1: Energy levels of EIT-OAL. Two ground states $|1\rangle$ and
$|2\rangle$ are connected to the excited state $|3\rangle$ via coupling to
optical modes. One of them, in the transition
$|1\rangle\leftrightarrow|3\rangle$, is the quantized mode of the cavity
field, $g$ being the coupling constant. Another, in the transition
$|2\rangle\leftrightarrow|3\rangle$ is a classical coherent wave with an
amplitude corresponding to Rabi frequency $2\Omega_{23}$. Two ground states
are connected via a Raman transition with an intermediate state (not shown)
pumped by two additional lasers, such that the corresponding Rabi frequency is
$2\Omega_{12}$. All fields are assumed to be resonant with the corresponding
transitions.

Thus, we describe the atom-field evolution by a unitary evolution operator
$\mathcal{U}(t)$, satisfying the Schrödinger equation ($\hbar=1$)

$i\frac{d}{dt}\mathcal{U}(t)=H\mathcal{U}(t),$ (1)

with the Hamiltonian $H=H_{0}+V_{0}$, where

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{3}E_{n}\sigma_{nn}+\omega_{13}a^{\dagger}a+\omega_{12}b^{\dagger}b+\omega_{23}c^{\dagger}c,$
(2) $\displaystyle V_{0}$ $\displaystyle=$ $\displaystyle
ga\sigma_{31}+d_{12}b\sigma_{21}+d_{23}c\sigma_{32}+\textrm{H.c.}$ (3)

Here $E_{n}$ is the energy of the $n$th atomic level,
$\omega_{nm}=E_{m}-E_{n}$ is the circular frequency of the field mode resonant
with the atomic transition $|n\rangle\rightarrow|m\rangle$, $a$, $b$, and $c$
are the photon annihilation operators of the corresponding optical modes,
$\sigma_{nm}=|n\rangle\langle m|$, and $d_{nm}$ is the coupling constant for
the corresponding transition. Below, we assume that the modes $b$ and $c$ are
strong classical fields, but here we need their quantum versions for writing
rigorously the Schrödinger picture Hamiltonian, which has to be time-
independent.

###  II.2 First and second interaction pictures

We define the first interaction picture by introducing the evolution operator
$\mathcal{U}_{I}(t)=\mathcal{U}_{0}^{\dagger}(t)\mathcal{U}(t)$, where
$\mathcal{U}_{0}(t)=\exp\left(-iH_{0}t\right)$. This operator satisfies the
equation

$i\frac{d}{dt}\mathcal{U}_{I}(t)=H_{I}\mathcal{U}_{I}(t),$ (4)

where

$H_{I}=\mathcal{U}_{0}^{\dagger}(t)H\mathcal{U}_{0}(t)-H_{0}=ga\sigma_{31}+d_{12}b\sigma_{21}+d_{23}c\sigma_{32}+\textrm{H.c.}$
(5)

Now we assume that the modes $b$ and $c$ are strong classical fields and we
can disregard their commutators, replacing the operators $b$ and $c$ by
c-numbers $B$ and $C$ respectively, and defining the Rabi frequencies
$2\Omega_{12}=2d_{12}B$ and $2\Omega_{23}=2d_{23}C$, which we assume to be
real and positive.

Introducing the new atomic ground-state basis
$|\pm\rangle=\left(|1\rangle\pm|2\rangle\right)/\sqrt{2}$, we rewrite Eq. (5)
as $H_{I}=H_{1}+V_{1}$, where

$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle\Omega_{12}\left(\sigma_{++}-\sigma_{--}\right),$ (6)
$\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle\frac{g}{\sqrt{2}}a\left(\sigma_{3+}+\sigma_{3-}\right)+\frac{\Omega_{23}}{\sqrt{2}}\left(\sigma_{3+}-\sigma_{3-}\right)+\textrm{H.c.}$
(7)

The second interaction picture is defined by introducing the evolution
operator $\mathcal{U}_{J}(t)=\mathcal{U}_{1}^{\dagger}(t)\mathcal{U}_{I}(t)$,
where $\mathcal{U}_{1}(t)=\exp\left(-iH_{1}t\right)$. This operator satisfies
the equation

$i\frac{d}{dt}\mathcal{U}_{J}(t)=H_{J}(t)\mathcal{U}_{J}(t),$ (8)

where

$H_{J}(t)=\mathcal{U}_{1}^{\dagger}(t)H_{I}\mathcal{U}_{1}(t)-H_{1}=\frac{ga+\Omega_{23}}{\sqrt{2}}\sigma_{3+}e^{-i\Omega_{12}t}+\frac{ga-\Omega_{23}}{\sqrt{2}}\sigma_{3-}e^{i\Omega_{12}t}+\textrm{H.c.}$
(9)

###  II.3 Effective Hamiltonian

In the strong ground-state coupling regime, where
$\Omega_{12}\gg(g,\Omega_{23})$, all terms of the interaction Hamiltonian, Eq.
(9), are fast oscillating and the state of the system at times far exceeding
the period of these oscillations $2\pi/\Omega_{12}$ can be approximately found
by replacing $H_{J}(t)$ by an effective Hamiltonian $H_{\mathrm{eff}}$, which
is found in the following way [27].

We rewrite Eq. (9) in a harmonic form grouping together the positive and
negative frequency parts, as
$H_{J}(t)=he^{-i\Omega_{12}t}+h^{\dagger}e^{i\Omega_{12}t}$, where

$h=\frac{ga+\Omega_{23}}{\sqrt{2}}\sigma_{3+}+\frac{ga^{\dagger}-\Omega_{23}}{\sqrt{2}}\sigma_{-3}.$
(10)

Now the effective Hamiltonian is

$H_{\mathrm{eff}}=\frac{\left[h^{\dagger},h\right]}{\Omega_{12}}=H_{2}+V_{2}+V_{2}^{\prime},$
(11)

where

$\displaystyle H_{2}$ $\displaystyle=$ $\displaystyle\left(\delta
a^{\dagger}a+r^{2}\delta\right)\left(\sigma_{++}-\sigma_{--}\right),$ (12)
$\displaystyle V_{2}$ $\displaystyle=$ $\displaystyle
r\delta\left(a+a^{\dagger}\right)\left(\sigma_{++}+\sigma_{--}\right),$ (13)
$\displaystyle V_{2}^{\prime}$ $\displaystyle=$
$\displaystyle-2r\delta\left(a+a^{\dagger}\right)\sigma_{33},$ (14)

with the shortcuts $\delta=g^{2}/2\Omega_{12}$, $r=\Omega_{23}/g$.

In terms of the effective Hamiltonian the evolution operator of the second
interaction picture is
$\mathcal{U}_{J}(t)=\exp\left(-iH_{\mathrm{eff}}t\right)$. We see that the
effective Hamiltonian is diagonal in the basis
$\left\\{|+\rangle,|-\rangle,|3\rangle\right\\}$. It means that if the
population of some of these basis states is zero initially, it remains zero
with the course of time, which is a characteristic trait of the phenomenon of
EIT [15]. In the following, we assume that the atom is prepared initially in a
combination of its two ground states, and therefore, the operator
$V_{2}^{\prime}$ may be disregarded.

###  II.4 Third interaction picture

The third interaction picture is defined by introducing the evolution operator
$\mathcal{U}_{K}(t)=\mathcal{U}_{2}^{\dagger}(t)\mathcal{U}_{J}(t)$, where
$\mathcal{U}_{2}(t)=\exp\left(-iH_{2}t\right)$. This operator satisfies the
equation

$i\frac{d}{dt}\mathcal{U}_{K}(t)=H_{K}(t)\mathcal{U}_{K}(t),$ (15)

where

$H_{K}(t)=\mathcal{U}_{2}^{\dagger}(t)H_{\mathrm{eff}}\mathcal{U}_{2}(t)-H_{2}=r\delta\left(ae^{-i\delta
t}+a^{\dagger}e^{i\delta t}\right)\sigma_{++}+r\delta\left(ae^{i\delta
t}+a^{\dagger}e^{-i\delta t}\right)\sigma_{--}.$ (16)

The solution of Eq. (15) with the initial condition
$\mathcal{U}_{K}(0)=\mathbf{1}$ can be written in the form of Dyson time-
ordered series [28]

$\mathcal{U}_{K}(t)=\mathcal{T}e^{-i\int_{0}^{t}H_{K}(t^{\prime})dt^{\prime}},$
(17)

where $\mathcal{T}$ is the time-ordering operator, placing the operators
$H_{K}(t)$ with higher time to the left in the Taylor expansion of the
exponential. For sufficiently small $r$ this time ordering can be disregarded,
leading to the solution $\mathcal{\tilde{U}}_{K}(t)\approx e^{\Xi_{1}(t)}$,
where

$\Xi_{1}(t)=r\left[a\left(e^{-i\delta t}-1\right)-a^{\dagger}\left(e^{i\delta
t}-1\right)\right]\sigma_{++}-r\left[a\left(e^{i\delta
t}-1\right)-a^{\dagger}\left(e^{-i\delta t}-1\right)\right]\sigma_{--}.$ (18)

Coming back to the first interaction picture we obtain the evolution operator

$\mathcal{\tilde{U}}_{I}(t)=\mathcal{U}_{1}(t)\mathcal{U}_{2}(t)\mathcal{\tilde{U}}_{K}(t)=e^{-i\tilde{\varphi}(t)\left(\sigma_{++}-\sigma_{--}\right)}D\left(\alpha_{+}\sigma_{++}\right)D\left(\alpha_{-}\sigma_{--}\right)e^{-i\delta
a^{\dagger}at\left(\sigma_{++}-\sigma_{--}\right)},$ (19)

where $\tilde{\varphi}(t)=(\Omega_{12}+r^{2}\delta)t$, $\alpha_{\pm}=\mp
r\left(1-e^{\mp i\delta t}\right)$, and $D(x)=e^{xa^{\dagger}-x^{\dagger}a}$
is an extension of Glauber’s displacement operator to the case of operator-
valued argument. The rightmost factor in the right-hand side of Eq. (19) can
be omitted if the initial state of the field is vacuum, which we assume in the
following.

The evolution operator, Eq. (19), acting on the initial state
$|+\rangle|\mathrm{vac}\rangle$, where $|\mathrm{vac}\rangle$ is the vacuum
state of the field, produces a coherent state of the field with the amplitude
$\alpha_{+}$, leaving the atomic state unchanged. Similarly, acting on the
initial state $|-\rangle|\mathrm{vac}\rangle$, it produces a coherent state of
the field with the amplitude $\alpha_{-}$, leaving the atomic state unchanged.
If the atom is initially prepared in a superposition of the states $|+\rangle$
and $|-\rangle$, then the state of the field and atom becomes entangled. For
instance, choosing the initial state $|1\rangle|\mathrm{vac}\rangle$, we
obtain the state at moment $t$

$|\Psi(t)\rangle=\frac{1}{2}|1\rangle\left(e^{-i\tilde{\varphi}(t)}|\alpha_{+}\rangle+e^{i\tilde{\varphi}(t)}|\alpha_{-}\rangle\right)+\frac{1}{2}|2\rangle\left(e^{-i\tilde{\varphi}(t)}|\alpha_{+}\rangle-e^{i\tilde{\varphi}(t)}|\alpha_{-}\rangle\right),$
(20)

where $|\alpha_{\pm}\rangle$ is a coherent state with the amplitude
$\alpha_{\pm}$.

Detecting the atom in the state $|1\rangle$, one obtains the field in a
superposition of two coherent states with different amplitudes. Unfortunately,
the approximation of time-ordering lifting implies small $r$ and,
consequently, small $\alpha_{\pm}$. Thus, the state, Eq. (20), belongs to the
class of “Schrödinger kitten” states [29], which differ from the Schrödinger
cat states by having an amplitude so small that the two components are not
macroscopically distinguishable.

At arbitrarily high values of $r$, the time-ordering effects should be taken
into account in the solution, Eq. (17), which is the subject of the next
section.

##  III Time-ordering effects in EIT-OAL

###  III.1 Magnus expansion

The interaction Hamiltonian $H_{K}(t)$, Eq. (16), is proportional to $r$, and
at small value of this parameter can be treated as a small perturbation. The
traditional approach to treating a small perturbation in a Hamiltonian system
consists in limiting the consideration to one or several first terms of the
Dyson expansion, Eq. (17). This approach leads to the standard time-dependent
perturbation theory, having found numerous applications in the description of
evolution of various quantum systems. However, this standard approach has an
important limitation: the obtained evolution operator is non-unitary in
general and, as a consequence, does not preserve the commutation relations of
the system variables. An alternative representation of the evolution operator
is the Magnus expansion [30], which results from a Taylor decomposition of
$\ln\mathcal{U}_{K}(t)$ in the small parameter $r$:

$\mathcal{U}_{K}(t)=\exp\Big{(}\Xi_{1}(t)+\Xi_{2}(t)+\Xi_{3}(t)+...\Big{)},$
(21)

where $\Xi_{k}$ is of order $r^{k}$. The three first terms of the Magnus
expansion are

$\displaystyle\Xi_{1}(t)$ $\displaystyle=$
$\displaystyle-i\int_{0}^{t}H_{K}(t_{1})dt_{1},$ (22)
$\displaystyle\Xi_{2}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\left[H_{K}(t_{1}),H_{K}(t_{2})\right],$
(23) $\displaystyle\Xi_{3}(t)$ $\displaystyle=$
$\displaystyle\frac{i}{6}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\int_{0}^{t_{2}}dt_{3}\left\\{\left[H_{K}(t_{1}),\left[H_{K}(t_{2}),H_{K}(t_{3})\right]\right]\right.$
$\displaystyle+\left.\left[\left[H_{K}(t_{1}),H_{K}(t_{2})\right],H_{K}(t_{3})\right]\right\\}.$

When the infinite series is limited to any finite number of initial terms, the
approximate evolution operator remains unitary, which is a great advantage of
the Magnus expansion over a more conventional Dyson expansion.

The first term of this expansion, $\Xi_{1}(t)$, has been found in the previous
section. Substituting Eq. (16) into Eq. (23), applying the commutation
relation $\left[a,a^{\dagger}\right]=1$, and performing the integration, we
find

$\Xi_{2}(t)=i\Delta\varphi(t)\left(\sigma_{++}-\sigma_{--}\right),$ (25)

where $\Delta\varphi(t)=r^{2}\left(\delta t-\sin\delta t\right)$. We see that
at interaction times small compared to the period $t_{0}=2\pi/\delta$, the
second term of Magnus expansion is close to zero, but at larger times it may
become significant, especially with growing $r$.

We find also that $\Xi_{3}(t)$ and higher order terms of the Magnus expansion
equal zero identically, because the nested commutators of the Hamiltonian
$H_{K}(t)$ vanish. Thus, the evolution operator of the third interaction
picture is exactly
$\mathcal{U}_{K}(t)=\exp\Big{[}\Xi_{1}(t)+\Xi_{2}(t)\Big{]}$.

Thus, we have demonstrated another advantage of the Magnus expansion: in the
case of the interaction Hamiltonian having zero nested commutator of some
order $k$, the Magnus series reduces to a finite sum of $k-1$ terms and can be
calculated exactly.

###  III.2 Solution with time-ordering effects

Coming back to the first interaction picture we obtain the evolution operator
with the time-ordering term

$\mathcal{U}_{I}(t)=\mathcal{U}_{1}(t)\mathcal{U}_{2}(t)\mathcal{U}_{K}(t)=e^{-i\varphi(t)\left(\sigma_{++}-\sigma_{--}\right)}D\left(\alpha_{+}\sigma_{++}\right)D\left(\alpha_{-}\sigma_{--}\right)e^{-i\delta
a^{\dagger}at\left(\sigma_{++}-\sigma_{--}\right)},$ (26)

where
$\varphi(t)=\tilde{\varphi}(t)-\Delta\varphi(t)=\Omega_{12}t+r^{2}\sin\delta
t$. We see that the only effect of time-ordering is the change of the phase
$\tilde{\varphi}(t)\rightarrow\varphi(t)$. As a result, the state of the atom
and field, obtained from the initial state $|1\rangle|\mathrm{vac}\rangle$
after interaction time $t$ is given by Eq. (20) with the replacement
$\tilde{\varphi}(t)\rightarrow\varphi(t)$. Detecting the atom at time $t$ in
the state $|1\rangle$ or $|2\rangle$ projects the field onto the state
$|\psi_{+}\rangle$ or $|\psi_{-}\rangle$ respectively, where

$|\psi_{\pm}(t)\rangle=\frac{\mathcal{N}_{\pm}}{\sqrt{2}}\left(e^{-i\varphi(t)}|\alpha_{+}\rangle\pm
e^{i\varphi(t)}|\alpha_{-}\rangle\right),$ (27)

with $\mathcal{N}_{\pm}$ being normalization coefficients guaranteeing
$\langle\psi_{\pm}|\psi_{\pm}\rangle=1$. The trajectories of the generated
coherent states in the phase space of the cavity mode are shown in Fig. 2.
Each amplitude $\alpha_{+}(t)$ or $\alpha_{-}(t)$ traverses a circular
trajectory of radius $r$ with a period $t_{0}=2\pi/\delta$. The relative phase
between the quantum states $|\alpha_{+}\rangle$ and $|\alpha_{-}\rangle$,
$\varphi(t)$ changes very fast, because in the considered regime
$\Omega_{12}/\delta=2\Omega_{12}^{2}/g^{2}\gg 1$.

Figure 2: Trajectories of the generated coherent states in the phase space of
the cavity mode (dashed lines) for $r=1.8$, and the trajectory traversed to
the moment $t$ (red arrows). Blue circles show the two standard deviation
areas of the Wigner functions of the coherent states.

At the beginning of the interaction, at $t=0$,
$|\psi_{+}(0)\rangle=|\mathrm{vac}\rangle$, while $|\psi_{-}(0)\rangle$ does
not exist, since the probability of detecting the atom in the state
$|2\rangle$ is zero. We see from Fig. 2, that with the course of interaction
the amplitudes $\alpha_{+}$ and $\alpha_{-}$ diverge on the phase plane. If
the circle radius $r$ is sufficiently large, at certain moment the Wigner
functions of these states become well separated, and a Schrödinger cat state
is formed. It happens approximately when the distance between the coherent
states $\left|\alpha_{+}-\alpha_{-}\right|$ surpasses four standard deviations
of the Wigner function of each coherent state, as shown in Fig. 2. We recall
that the Wigner function of a coherent state $|\alpha_{0}\rangle$ is [31]

$W(\alpha)=2\exp\left(-2\left|\alpha-\alpha_{0}\right|^{2}\right),$ (28)

and its standard deviation in any direction is $\sigma=\frac{1}{2}$.

Since the relative phase $\varphi$ between the two coherent components changes
very fast, the state $|\psi_{+}\rangle$ passes through all possible variants
of the cat state: (displaced) even coherent state ($\varphi=0$), odd coherent
state ($\varphi=\pi/2$), Yurke-Stoler coherent state ($\varphi=\pi/4$) many
times during one period $t_{0}$ of moving on the circle on the phase plane.
The same concerns the state $|\psi_{-}\rangle$ at different moments. The
maximal separation of the coherent components is reached, obviously, at
$t=t_{0}/2$, where $\alpha_{\pm}=\mp 2r$. At this moment the cat state has the
maximal size and, for sufficiently high $r$, the maximal nonclassicality.

To observe how a nonclassical cat state appears from the classical vacuum
state, we use a nonclassicality measure, known as “total noise” [32], which is
defined for arbitrary pure state $|\psi\rangle$ of an optical mode as

$T(\psi)=\langle\psi|a^{\dagger}a|\psi\rangle-\left|\langle\psi|a|\psi\rangle\right|^{2}.$
(29)

This measure is zero for all coherent states, which are the only classical
states among the pure states. For all other pure states $T(\psi)$ is positive
and gives a good estimate of how nonclassical the state is. Total noise has
important extensions to the class of mixed states of one optical mode, such
the maximal quadrature quantum Fisher information [33, 34] having an explicit
operational meaning, and the ordering sensitivity [35] directly related to the
decoherence scale [36].

Substituting Eq. (27) into Eq. (29) we obtain

$T(\psi_{\pm})=\frac{\left|\alpha_{+}-\alpha_{-}\right|^{2}\left(1-e^{-\left|\alpha_{+}-\alpha_{-}\right|^{2}}\right)}{\left[2\pm
q\pm q^{*}\right]^{2}},$ (30)

where

$q=\langle\alpha_{+}|\alpha_{-}\rangle
e^{2i\varphi}=e^{-\frac{1}{2}|\alpha_{+}|^{2}-\frac{1}{2}|\alpha_{-}|^{2}+\alpha_{+}^{*}\alpha_{-}+2i\varphi}.$
(31)

This function is shown in Fig. 3 for various values of $r$. We see that at
$r=0.25$, where the maximal separation of the coherent amplitudes, $4r=1$,
equals two standard deviations of the coherent state Wigner function, the
maximal value of the total noise is about 1. This value is reached at the
relative phase of $\varphi=\pi/2$, i.e. for a displaced odd coherent state.
The odd coherent state for the case of low amplitude is very close to the one-
photon Fock state, for which the total noise is equal to 1. We note that the
total noise is invariant with respect to displacement of the state on the
phase plane. Thus, the total noise of a displaced odd coherent state is also
close to unity. For other values of $\varphi$ the nonclassicality is close to
zero, because the state has a dominant vacuum component. We see that the
states $|\psi_{+}\rangle$ and $|\psi_{-}\rangle$ reach the maximal
nonclassicality at different moments, because the relative phase of coherent
components differs for these states by $\pi$. In this low-$r$ regime the main
factor influencing the nonclassicality is the relative phase between the
coherent components, while their separation is not important.

Figure 3: Total noise of the conditional state of the cavity field under
condition that the atom is detected in one of its ground states for different
values of the parameter $r=\Omega_{23}/g$. The ratio $\Omega_{12}/\delta$,
which should be much greater than 1, is just 8 here for illustration.

At a higer value of $r=0.5$ we see that a mesoscopic cat state is formed
around $t=t_{0}/2$. This time corresponds to half-period of the motion along
the trajectory in the phase space (dashed circle in Fig. 2), when the
separation of the coherent amplitudes is maximal and equal to $4r$. For the
considered value of $r=0.5$ the separation is $4r=2$, corresponding to 4
standard deviations of the coherent state Wigner function. Total noise is
still oscillating, having higher values for a displaced odd coherent state
than for other states, but there is also a distinct dependence on the
separation of the coherent components. At even higher value of $r=1$ the
separation of components becomes the main factor influencing the
nonclassicality. At times close to $t=t_{0}/2$ a cat state is formed and the
relative phase plays no role in the state nonclassicality.

Analyzing the evolution of the field state we need to remember that the
applicability of the effective Hamiltonian approach, introduced in Sec. II.3,
requires an interaction time much higher than $2\pi/\Omega_{12}$. It means
that the conditional state of the cavity field is given by Eq. (27) after the
relative phase has made many periods of its evolution. Consequently, the
initial oscillations in Fig. 3 should be disregarded. It should be noted that
the frequency of oscillations shown in this figure is rather low for the
illustration purpose. Applicability of the condition
$\Omega_{12}\gg(g,\Omega_{23})$ of the strong ground-state coupling regime
requires that the oscillation frequency is at least 5 times higher.

###  III.3 Effect of time ordering

It is interesting and instructive to observe the effect of time ordering on
the generated state and to determine the minimal value of $r$ where it becomes
important. Total noise is not the best tool for this purpose because, as we
saw in Sec. III.2, the net effect of the time ordering is a change of the
relative phase between the two coherent components, while the total noise
becomes insensitive to this phase at a good separation of the components. To
observe the effect of time ordering, we consider the task of generation of odd
and even coherent states of light, having a paramount importance for optical
quantum computation [37]. For this purpose we calculate the average parity of
state

$P(\psi)=\langle\psi|(-1)^{a^{\dagger}a}|\psi\rangle.$ (32)

The odd coherent state
$|\psi_{\mathrm{o}}\rangle=\left(|\alpha\rangle-|-\alpha\rangle\right)[2(1-e^{-2|\alpha|^{2}})]^{-1/2}$
has only odd number states in its Fock-state decomposition, and its average
parity is $P(\psi_{\mathrm{o}})=-1$. Similarly, the even coherent state
$|\psi_{\mathrm{e}}\rangle=\left(|\alpha\rangle+|-\alpha\rangle\right)[2(1+e^{-2|\alpha|^{2}})]^{-1/2}$
has only even number states in its Fock-state decomposition, and its average
parity is $P(\psi_{\mathrm{e}})=1$. For other values of the relative phase the
average parity lies between -1 and 1.

The average parity of the state $|\psi_{+}\rangle$, generated at time $t$ is
shown in Fig. 4. We consider only a time interval around $t=t_{0}/2$, where
the state amplitude is maximal. We see that at $r=0.25$ the effect of time
ordering on the time of odd (or even) state generation is negligible, while at
$r=0.5$ it should be taken into account.

Figure 4: Comparison of the average parity of the cavity field state
$|\psi_{+}\rangle$ calculated with and without time ordering. The ratio
$\Omega_{12}/\delta=50$. Average parity of 1 corresponds to generation of even
coherent states, while that of -1 corresponds to generation of odd ones.

We can determine a critical value of $r_{c}$ above which the time ordering
becomes important by the following argument. The period of the average parity
corresponds to the phase $\varphi(t)$ growing by $\pi$. The phase
$\Delta\varphi(t)$, appearing because of time ordering, has a value of $\pi
r^{2}$ around the time $t=t_{0}/2$, most interesting for applications. We
assume that the time ordering is non-negligible if this correction is at least
one tenth of the period. Thus we arrive at the critical value
$r_{c}=\sqrt{0.1}=0.32$. At this value of the trajectory radius, odd and even
coherent states are generated around $t=t_{0}/2$ with the amplitude $2r_{c}$.
It is interesting to note, that the average number of photons in the odd
coherent state with the amplitude $2r_{c}$ is $\langle
n\rangle_{\mathrm{o}}=1.05$. This is much less than the value of $\langle
n\rangle\approx 4$, at which time ordering becomes important in single-pass
generation of squeezed states of light [24, 25]. The even coherent state of
this amplitude has even lesser average number of photons $\langle
n\rangle_{\mathrm{e}}=0.15$. However, this state hardly has a practical value,
having a dominant vacuum component.

We ascribe the influence of time ordering on the field state at that low
average photon number to the strong-coupling regime of the atom-field
interaction, where the field and the atom exchange an excitation many times
within the interaction time without losing photons to the environment. This
situation would correspond to the very long crystal limit in the single-pass
parametric downconversion, where the pump and the signal fields would exchange
photons many times. In this limit the time ordering may become important even
at moderate squeezing.

Another state, potentially interesting for applications, is the Yurke-Stoler
coherent state [38] $|\psi_{\mathrm{YS\pm}}\rangle=(|\alpha\rangle\pm
i|-\alpha\rangle)2^{-1/2}$. This state is an eigenstate of the modified photon
annihilation operator $A=e^{i\pi a^{\dagger}a}a$ [39] and its decoherence can
be slowed down by monitoring the cavity output light by photon counting and
changing the phase after a count by a feedback loop [40, 41]. Thus, these
states may be a good test bed for studying decoherence of the cat states. To
determine the times of formation of the Yurke-Stoler coherent states, we
introduce a measure similar to the total noise, but with the operator $a$
replaced by $A$:

$T_{A}(\psi)=\langle\psi|A^{\dagger}A|\psi\rangle-\left|\langle\psi|A|\psi\rangle\right|^{2}.$
(33)

This quantity, which we call “relative total noise”, is vanishing for a Yurke-
Stoler coherent state, similar to the vanishing of total noise for an ordinary
coherent state. Thus, $T_{A}(\psi)=0$ means a creation of a state of our
interest. The dependence of $T_{A}(\psi_{+})$ on time is shown in Fig. 5.

Figure 5: Comparison of the relative total noise of the cavity field state
$|\psi_{+}\rangle$ calculated with and without time ordering. The ratio
$\Omega_{12}/\delta=50$. The relative total noise vanishes when a Yurke-Stoler
coherent state is generated.

As in the case of even and odd coherence states, we see that time ordering is
negligible at low $r$, but becomes important at $r=0.5$. The critical value of
$r$, above which time ordering effect should be calculated, is the same by the
same argument. The average number of photons, corresponding to a Yurke-Stoler
coherent state with the amplitude $2r_{c}$ is $\langle
n\rangle_{\mathrm{YS}}=0.4$. Thus, we see again that time ordering becomes
important in a cavity configuration at very low average photon number.

##  IV Conclusions

We have obtained the exact solution of the Schrödinger equation with a time-
dependent effective Hamiltonian for the EIT-OAL in the regime of strong
ground-state coupling. The net effect of the time-ordering correction has
turned out to be in a modification of the relative phase between the two
coherent components of an optical Schrödinger cat state generated in this
regime. We have analyzed the parameters, corresponding to a non-negligible
shift of the times of generation of definite variants of the cat state, as the
odd and even coherent states and the Yurke-Stoler coherent state. We have
found, that time-ordering becomes important at the average photon number in
the cavity below 1, which is in striking contrast to the case of single-pass
parametric downconversion, where it becomes important at average photon number
in one optical mode above 4. We attribute this effect to multiple excitation
exchanges taking place between the field and atom during the interaction time
in the strong coupling regime. In this regime, inclusion of time-ordering
terms in the evolution operator is necessary for the correct treatment of the
stimulated emission.

These results shed new light on the importance of time-ordering for weak
optical fields and may find their applications in the development of OALs and
generation of nonclassical states of light for the tasks of quantum metrology
and quantum information processing.

## Funding

D.B.H. and S.Y.K. are supported by Belarusian Republican Foundation for
Fundamental Research, grant F20KI-035. C.-S.Y. is supported by the National
Natural Science Foundation of China under Grant No.12011530014.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

All data in Figs. 3-5 are obtained from explicit analytical formulas given in
the text of the article. No other data were generated or analyzed in the
presented research.

## References

  * Mu and Savage [1992] Y. Mu and C. M. Savage, One-atom lasers, Phys. Rev. A 46, 5944 (1992).
  * McKeever  _et al._ [2003] J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, and H. J. Kimble, Experimental realization of a one-atom laser in the regime of strong coupling, Nature 425, 268 (2003).
  * Dubin  _et al._ [2010] F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, Quantum to classical transition in a single-ion laser, Nature Phys. 6, 350 (2010).
  * Raimond  _et al._ [1982] J. Raimond, P. Goy, M. Gross, C. Fabre, and S. Haroche, Statistics of millimeter-wave photons emitted by a Rydberg-atom maser: An experimental study of fluctuations in single-mode superradiance, Phys. Rev. Lett. 49, 1924 (1982).
  * Meschede  _et al._ [1985] D. Meschede, H. Walther, and G. Müller, One-atom maser, Phys. Rev. Lett. 54, 551 (1985).
  * Strauf and Jahnke [2011] S. Strauf and F. Jahnke, Single quantum dot nanolaser, Laser Phot. Reviews 5, 607 (2011).
  * Pellizzari and Ritsch [1994] T. Pellizzari and H. Ritsch, Preparation of stationary Fock states in a one-atom Raman laser, Phys. Rev. Lett. 72, 3973 (1994).
  * Kilin and Karlovich [2002] S. Y. Kilin and T. B. Karlovich, Single-atom laser: Coherent and nonclassical effects in the regime of a strong atom-field correlation, J. Exper. Theor. Phys. 95, 805 (2002).
  * Kilin and Mikhalychev [2012] S. Y. Kilin and A. B. Mikhalychev, Single-atom laser generates nonlinear coherent states, Phys. Rev. A 85, 063817 (2012).
  * Larionov and Kolobov [2013] N. V. Larionov and M. I. Kolobov, Quantum theory of a single-emitter nanolaser, Phys. Rev. A 88, 013843 (2013).
  * Stefanov and Kilin [2019] V. Stefanov and S. Y. Kilin, Squeezed states engineering by coherent pulses train acting on single atom laser, Nonlin. Phenom. Complex Syst. 22, 64 (2019).
  * Kilin [2001] S. Y. Kilin, Quanta and information, in  _Progress in Optics_, Vol. 42, edited by E. Wolf (Elsevier, 2001) pp. 1–91.
  * Villas-Boas  _et al._ [2020] C. J. Villas-Boas, K. N. Tolazzi, B. Wang, C. Ianzano, and G. Rempe, Continuous generation of quantum light from a single ground-state atom in an optical cavity, Phys. Rev. Lett. 124, 093603 (2020).
  * Bermel  _et al._ [2006] P. Bermel, A. Rodriguez, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, Single-photon all-optical switching using waveguide-cavity quantum electrodynamics, Phys. Rev. A 74, 043818 (2006).
  * Mücke  _et al._ [2010] M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, Electromagnetically induced transparency with single atoms in a cavity, Nature 465, 755 (2010).
  * Slodička  _et al._ [2010] L. Slodička, G. Hétet, S. Gerber, M. Hennrich, and R. Blatt, Electromagnetically induced transparency from a single atom in free space, Phys. Rev. Lett. 105, 153604 (2010).
  * Kampschulte  _et al._ [2010] T. Kampschulte, W. Alt, S. Brakhane, M. Eckstein, R. Reimann, A. Widera, and D. Meschede, Optical control of the refractive index of a single atom, Phys. Rev. Lett. 105, 153603 (2010).
  * Witthaut and Sørensen [2010] D. Witthaut and A. S. Sørensen, Photon scattering by a three-level emitter in a one-dimensional waveguide, New J. Phys. 12, 043052 (2010).
  * Gu  _et al._ [2016] X. Gu, S.-N. Huai, F. Nori, and Y.-x. Liu, Polariton states in circuit qed for electromagnetically induced transparency, Phys. Rev. A 93, 063827 (2016).
  * Mirza and Schotland [2018] I. M. Mirza and J. C. Schotland, Influence of disorder on electromagnetically induced transparency in chiral waveguide quantum electrodynamics, J. Opt. Soc. Am. B 35, 1149 (2018).
  * Tolazzi  _et al._ [2021] K. N. Tolazzi, B. Wang, C. Ianzano, J. Neumeier, C. J. Villas-Boas, and G. Rempe, Continuous quantum light from a dark atom, Commun. Phys. 4, 1 (2021).
  * Quesada and Sipe [2015] N. Quesada and J. Sipe, Time-ordering effects in the generation of entangled photons using nonlinear optical processes, Phys. Rev. Lett. 114, 093903 (2015).
  * Horoshko  _et al._ [2018] D. B. Horoshko, M. M. Eskandary, and S. Y. Kilin, Quantum model for traveling-wave electro-optical phase modulator, J. Opt. Soc. Am. B 35, 2744 (2018).
  * Christ  _et al._ [2013] A. Christ, B. Brecht, W. Mauerer, and C. Silberhorn, Theory of quantum frequency conversion and type-II parametric down-conversion in the high-gain regime, New J. Phys. 15, 053038 (2013).
  * Lipfert  _et al._ [2018] T. Lipfert, D. B. Horoshko, G. Patera, and M. I. Kolobov, Bloch-Messiah decomposition and Magnus expansion for parametric down-conversion with monochromatic pump, Phys. Rev. A 98, 013815 (2018).
  * Ourjoumtsev  _et al._ [2009] A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, Preparation of non-local superpositions of quasi-classical light states, Nature Phys. 5, 189 (2009).
  * James and Jerke [2007] D. James and J. Jerke, Effective Hamiltonian theory and its applications in quantum information, Canad. J. Phys. 85, 625 (2007).
  * Berestetskii  _et al._ [1982] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii,  _Quantum Electrodynamics_ , 2nd ed. (Butterworth-Heinemann, Oxford, 1982).
  * Paris [1999] M. G. A. Paris, Generation of mesoscopic quantum superpositions through Kerr-stimulated degenerate downconversion, J. Opt. B 1, 662 (1999).
  * Blanes  _et al._ [2009] S. Blanes, F. Casas, J.-A. Oteo, and J. Ros, The Magnus expansion and some of its applications, Phys. Reports 470, 151 (2009).
  * Cahill and Glauber [1969] K. E. Cahill and R. J. Glauber, Density operators and quasiprobability distributions, Phys. Rev. 177, 1882 (1969).
  * Hillery [1989] M. Hillery, Total noise and nonclassical states, Phys. Rev. A 39, 2994 (1989).
  * Yadin  _et al._ [2018] B. Yadin, F. C. Binder, J. Thompson, V. Narasimhachar, M. Gu, and M. S. Kim, Operational resource theory of continuous-variable nonclassicality, Phys. Rev. X 8, 041038 (2018).
  * Kwon  _et al._ [2019] H. Kwon, K. C. Tan, T. Volkoff, and H. Jeong, Nonclassicality as a quantifiable resource for quantum metrology, Phys. Rev. Lett. 122, 040503 (2019).
  * De Bièvre  _et al._ [2019] S. De Bièvre, D. B. Horoshko, G. Patera, and M. I. Kolobov, Measuring nonclassicality of bosonic field quantum states via operator ordering sensitivity, Phys. Rev. Lett. 122, 080402 (2019).
  * Hertz and De Bièvre [2020] A. Hertz and S. De Bièvre, Quadrature coherence scale driven fast decoherence of bosonic quantum field states, Phys. Rev. Lett. 124, 090402 (2020).
  * Guillaud and Mirrahimi [2019] J. Guillaud and M. Mirrahimi, Repetition cat qubits for fault-tolerant quantum computation, Phys. Rev. X 9, 041053 (2019).
  * Yurke and Stoler [1986] B. Yurke and D. Stoler, Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion, Phys. Rev. Lett. 57, 13 (1986).
  * Horoshko and Kilin [1997a] D. B. Horoshko and S. Y. Kilin, Decoherence slowing via feedback, J. Mod. Opt. 44, 2043 (1997a).
  * Horoshko and Kilin [1997b] D. B. Horoshko and S. Y. Kilin, Direct detection feedback for preserving quantum coherence in an open cavity, Phys. Rev. Lett. 78, 840 (1997b).
  * Horoshko and Kilin [1998] D. B. Horoshko and S. Y. Kilin, Multimode unraveling of master equation and decoherence problem, Opt. Express 2, 347 (1998).


11institutetext: National Centre for Nuclear Research, ul. Andrzeja Sołtana 7,
05-400 Otwock, Poland 22institutetext: Department of Physics, Florida State
University, Tallahassee, Florida 32306, USA 33institutetext: Heavy Ion
Laboratory, University of Warsaw, ul. Pasteura 5a, 02-093 Warsaw, Poland

# Important Influence of Entrance Channel Reorientation Coupling on Proton
Stripping

N. Keeley 11 K. W. Kemper 2233 K. Rusek 33

(Received: date / Revised version: date)

###### Abstract

While it is well established that the ground state reorientation coupling can
have a significant influence on the elastic scattering of deformed nuclei, the
effect of such couplings on transfer channels has been much less well
investigated. In this letter we demonstrate that the 208Pb(7Li,6He)209Bi
proton stripping reaction at an incident energy of 52 MeV can be well
described by the inclusion of the 7Li ground state reorientation coupling
within the coupled channels Born approximation formalism. Full finite-range
distorted wave Born approximation calculations were previously found to be
unable to describe these data. Addition of coupling to the 0.478-MeV $1/2^{-}$
excited state of 7Li, together with the associated two-step transfer path, has
little or no influence on the shape of the angular distributions (except that
for stripping leading to the 1.61-MeV $13/2^{+}$ level of 209Bi which is
significantly improved) but does affect appreciably the values of the
${}^{209}\mathrm{Bi}\rightarrow{{}^{208}\mathrm{Pb}}+p$ spectroscopic factors.
Implications for experiments with weakly-bound light radioactive beams are
discussed.

While the distorted wave Born approximation in its full finite-range version
(FR-DWBA) has been applied to the analysis of a wide body of heavy-ion
reaction data with considerable success, it has been known since the early
1970s that certain classes of reaction, in particular (but by no means
restricted to) single-proton transfers, cannot be satisfactorily described
using it without recourse to such undesirable expedients as ad hoc adjustments
of the exit channel optical potential parameters. These adjustments are
usually such that the resulting parameters no longer describe the relevant
elastic scattering, thus violating one of the fundamental tenets of the DWBA.
Many of these reactions are poorly matched, either in terms of Q value or
transferred angular momentum, and at the time this led to the conjecture that
the inclusion of two-step reaction paths via excited states of the projectile
and/or ejectile within the coupled channels Born approximation (CCBA)
formalism would enable good descriptions to be obtained without the need
arbitrarily to modify the exit channel distorting potentials. However, the
prohibitive computational overhead of such calculations with then available
resources precluded the testing of this hypothesis.

There the question has remained in many cases for the intervening four decades
as interest in heavy-ion reactions waned. However, with the advent of
radioactive beams of sufficient intensity and optical quality there has been a
resurgence in this field and it seems timely to revisit some of the stable
beam data with a view to establishing whether they can, in fact, be
satisfactorily described by including likely two-step reaction paths. Current
easy availability of significant computing power makes the routine application
of the full arsenal of direct reaction theory to the interpretation of such
data a practical possibility. In a recent article Kee20  we showed that data
Tot76 ; Oer84  for the 208Pb(12C,11B)209Bi single-proton stripping reaction
which the FR-DWBA failed to explain could be well described by CCBA
calculations including two-step transfer via the 4.44-MeV $2^{+}_{1}$ excited
state of 12C using shell model spectroscopic amplitudes for the projectile-
like overlaps and distorting potentials that fitted the relevant elastic
scattering data in both entrance and exit channels. Similar calculations for
the 208Pb(12C,13C)207Pb single-neutron pickup improved the already good
description of the corresponding data Tot76 ; Oer84  by the FR-DWBA.

Here we show that the 52-MeV 208Pb(7Li,6He)209Bi single-proton stripping data
of Zeller et al. Zel79 , which the FR-DWBA also failed to describe
satisfactorily, may be fitted simply by including the ground state
reorientation coupling of the 7Li projectile, provided the description of the
entrance channel elastic scattering data is maintained when the coupling is
included. While the influence of ground state reorientation coupling on the
elastic scattering and excited state reorientation couplings on the inelastic
scattering is well established (see, e.g., Hni81  and Vid76 ), we are not
aware of any previous studies that have demonstrated a significant effect of
entrance channel ground state reorientation coupling on transfer channels.
Addition of the excitation of the $1/2^{-}$ first excited state of 7Li and the
two-step transfer path via this state produces a barely perceptible
improvement in the description of the stripping data, with the exception of
that for the 1.61-MeV $13/2^{+}$ level of 209Bi which exhibits a significant
further improvement when this coupling is included. We discuss some of the
implications of these results for experiments involving weakly-bound light
radioactive beams.

All calculations were carried out with the code fresco Tho88 . Inputs to the
CCBA calculations were kept as similar as possible to those of the original
FR-DWBA calculations of Ref. Zel79 , with the exception that the spectroscopic
amplitudes for the $\left<{}^{7}\mathrm{Li}\mid{{}^{6}\mathrm{He}}+p\right>$
overlaps were taken from Rud05  rather than Cohen and Kurath Coh67  in order
to have a consistent set for all the overlaps required. However, the
spectroscopic amplitudes of Refs. Rud05  and Coh67  for the
${}^{7}\mathrm{Li}(3/2^{-})\rightarrow{{}^{6}\mathrm{He}(0^{+})}+p$ transition
only differ by about 4%. The ${}^{6}\mathrm{He}+p$ and ${}^{208}\mathrm{Pb}+p$
binding potentials were as in Ref. Zel79 . We also retained the exit channel
${}^{6}\mathrm{He}+{{}^{209}\mathrm{Bi}}$ optical potential of Ref. Zel79 .
While this was obtained from a fit to 6Li + 209Bi elastic scattering data at
the appropriate energy, the comparative study of Ref. Kuc09  suggests that for
incident energies this far above the Coulomb barrier 6He elastic scattering
should be adequately described by 6Li optical potential parameters, even for
heavy targets like 209Bi. Coupled discretised continuum channels (CDCC)
calculations using the modified dineutron model of Moro et al. Mor07
confirmed this, since the calculated elastic scattering angular distribution
was well described by the exit channel potential parameters of Ref. Zel79 .

The 7Li couplings in the entrance partition were included using standard
rotational model form factors with the nuclear multipoles calculated by
numerically deforming the radii of the diagonal potential and projecting by
Gaussian quadrature onto the required multipoles. Use of the more approximate
derivative method to calculate the nuclear multipoles gave almost identical
results provided that the nuclear deformation length was re-adjusted as well
as the optical potential parameters to give the same inelastic and elastic
scattering angular distributions as the calculations using the more accurate
Gaussian projection method. The $3/2^{-}$ ground state and 0.478-MeV $1/2^{-}$
first excited state were treated as members of a prolate K = 1/2 band, with
the Coulomb coupling strength $B(E2;3/2^{-}\rightarrow 1/2^{-})=8.3\pm 0.5$
e2fm4 taken from Ref. Wel85  and the nuclear deformation length
$\delta_{2}=2.0$ fm from Ref. Wei19 . Two CCBA calculations were carried out,
the first including the 7Li ground state reorientation coupling only and the
second the ground state reorientation plus excitation of the $1/2^{-}$ state
together with the two-step transfer path via this state. In both cases the
optical potential parameters were readjusted to recover the fit to the 7Li +
208Pb elastic scattering data of Ref. Zel79  and the resulting values are
listed in Table 1 as sets B and C respectively, together with the original
parameters of Ref. Zel79  (set A) for ease of reference.

Table 1: Parameters of the 7Li + 208Pb optical model potentials used in the FR-DWBA (Set A) and CCBA calculations (sets B and C). All radii use the convention: $R_{x}=r_{x}\times\mathrm{A_{T}}^{1/3}$ fm. The Coulomb radius parameter $r_{\mathrm{C}}=1.40$ fm in all cases. Set | $V$ | $r_{V}$ | $a_{V}$ | $W$ | $r_{W}$ | $a_{W}$
---|---|---|---|---|---|---
| (MeV) | (fm) | (fm) | (MeV) | (fm) | (fm)
A Zel79  | 293.8 | 1.253 | 0.785 | 18.99 | 1.602 | 0.743
B | 321.7 | 1.253 | 0.785 | 14.65 | 1.602 | 0.802
C | 352.0 | 1.253 | 0.785 | 17.88 | 1.602 | 0.759

The relevant coupling schemes are presented schematically in Fig. 1.

Figure 1:  Schematic representations of the coupling schemes corresponding to
FR-DWBA (set A), CCBA with 7Li ground state reorientation coupling only (set
B) and CCBA with 7Li ground state reorientation plus coupling to the 0.478-MeV
$1/2^{-}$ level (set C).

All calculations used the post form of the DWBA and included the full complex
remnant term.

In Figs. 2 – 4 we compare the results of FR-DWBA calculations with the data of
Ref. Zel79 .

Figure 2:  The 208Pb(7Li,6He)209Bi single-proton stripping data of Ref. Zel79
for transitions populating the 0.0-MeV $9/2^{-}$ (a) and 0.90-MeV $7/2^{-}$
(b) levels of 209Bi (solid circles) compared with the results of FR-DWBA
calculations (dotted curves), CCBA calculations including the 7Li ground state
reorientation coupling only (dashed curves) and ground state reorientation
plus excitation of the 0.478-MeV $1/2^{-}$ state (solid curves). The latter
also include the two-step transfer via the 7Li excited state.  Figure 3:  As
for Fig. 2 but for transitions populating the 1.61-MeV $13/2^{+}$ (a) and
2.82-MeV $5/2^{-}$ (b) levels of 209Bi.  Figure 4:  As for Fig. 2 but for
transitions populating the 3.12-MeV $3/2^{-}$ (a) and 3.64-MeV $1/2^{-}$ (b)
levels of 209Bi.

While the data for stripping to the 0.0-MeV $9/2^{-}$ and 3.12-MeV $3/2^{-}$
levels are reasonably well described the calculations for the other levels
show significant shifts in the position of the peak to larger angles compared
with the data. This is particularly striking in the case of the 1.61-MeV
$13/2^{+}$ level, analysis of which was not pursued in Ref. Zel79  due to the
prohibitive computing time required with the then available resources.

Figures 2 – 4 also compare the stripping data with the results of CCBA
calculations including the 7Li ground state reorientation only (dashed curves)
and ground state reorientation plus excitation of the 0.478-MeV $1/2^{-}$
excited state (solid curves), the latter also including the two-step transfer
path via the 7Li excited state. Simply including the 7Li ground state
reorientation coupling enables a good description of all the data, with the
exception of that for stripping to the $13/2^{+}$ level of 209Bi which, while
considerably improved, could be better. Addition of the coupling to the
0.478-MeV $1/2^{-}$ state and the associated two-step transfer path further
improves the description of this level, with a relatively minor influence on
the description of the other levels. Note that the good description of the
stripping data by the CCBA calculations is conditional on the refitting of the
entrance channel elastic scattering data with the relevant 7Li couplings
included. While the extraction of spectroscopic factors is not the goal of
this work, we also note that in addition to improving the description of the
angular distributions the inclusion of the ground state reorientation coupling
alone and the reorientation plus inelastic excitation couplings (together with
the relevant two-step transfer path) both have a significant effect on the
extracted values of the spectroscopic factors for the
$\left<{}^{209}\mathrm{Bi}\mid{{}^{208}\mathrm{Pb}}+p\right>$ overlaps, see
Table 2.

Table 2: Spectroscopic factors (C2S) for the $\left<{}^{209}\mathrm{Bi}\mid{{}^{208}\mathrm{Pb}}+p\right>$ overlaps extracted from the FR-DWBA (set A) and CCBA calculations including the 7Li ground state reorientation coupling only (set B) and ground state reorientation plus excitation of the 0.478-MeV $1/2^{-}$ excited state together with the relevant two-step transfer path (set C). Set | 0.0-MeV | 0.90-MeV | 1.61-MeV | 2.82-MeV | 3.12-MeV | 3.64-MeV
---|---|---|---|---|---|---
| $9/2^{-}$ | $7/2^{-}$ | $13/2^{+}$ | $5/2^{-}$ | $3/2^{-}$ | $1/2^{-}$
A | 1.40 | 1.19 | 0.81 | 1.07 | 0.89 | 0.64
B | 1.51 | 1.31 | 0.88 | 1.15 | 1.09 | 0.73
C | 1.68 | 1.55 | 0.98 | 1.39 | 1.34 | 0.90

It has long been known that the ground state reorientation coupling has an
important influence on the analysing powers in reactions induced by polarised
beams of 7Li, the elastic scattering second-rank tensor analysing powers in
particular being mainly due to this coupling. It has also been demonstrated to
have a significant effect on the elastic scattering differential cross section
angular distribution Hni81 . However, in this work we present a case where the
reorientation coupling alone has an important influence on a transfer channel;
in previous work of this type the 7Li ground state reorientation has usually
been associated with coupling to the 0.478-MeV $1/2^{-}$ excited state, so
that although significant effects on transfer reaction angular distributions
were observed, the contribution of the reorientation coupling alone was not
established.

We conclude with a number of observations concerning the results of this study
and their bearing on the analysis of transfer reactions induced by radioactive
ion beams. We have demonstrated, for possibly the first time, a significant
influence on the transfer reaction differential cross section angular
distributions calculated within the CCBA framework of the ground state
reorientation coupling in a deformed projectile alone. Since many light
radioactive nuclei have similar or larger quadrupole moments to that of 7Li
($Q=-40.6\pm 0.8$ mb Die88 ), in particular 8Li ($Q=+32.7\pm 0.6$ mb Min93 ),
9Li ($Q=-27.4\pm 1.0$ mb Arn92 ) and 11Li ($Q=-33.3\pm 0.5$ mb Neu08 ) but
especially 8B ($Q=68.3\pm 2.1$ mb Min92 ; Min93a ), this suggests that
inclusion of the ground state reorientation coupling in the analysis of
transfer reactions involving such nuclei is advisable. Also, the need to
ensure that the elastic scattering data remained well described by the
calculations including the reorientation and inelastic couplings in order to
be able to describe the proton stripping data, underlines the importance of
measuring at least the entrance channel elastic scattering as well as the
transfer process(es) of interest. This can be particularly important in
radioactive beam work where, owing to the peculiarities of individual nuclei
of this type, the use of “near enough” optical potential parameters may be
misleading.

It is also clear from Table 2 that inclusion of both ground state
reorientation and excitation of the 0.478-MeV $1/2^{-}$ state of 7Li has a
significant effect on the
${}^{209}\mathrm{Bi}\rightarrow{{}^{208}\mathrm{Pb}}+p$ spectroscopic factors
extracted from the fits to the data. We particularly wish to note that while
the inclusion of the excitation of the $1/2^{-}$ state has little or no effect
on the shapes of the transfer angular distributions—with the exception of that
for stripping populating the 1.61-MeV $13/2^{+}$ level of
${}^{209}\mathrm{Bi}$—its influence on the extracted spectroscopic factors is
far from negligible. Thus, even though including the ground state
reorientation coupling alone provides a good description of the stripping
data, if the extraction of spectroscopic factors is the aim of the analysis
stopping at this point could lead to erroneous conclusions. Indeed, it may
well be that the addition of further couplings, e.g. an explicit treatment of
the ${}^{7}\mathrm{Li}\rightarrow\alpha+t$ breakup via the CDCC method, will
be required to obtain a “converged” set of values for the
${}^{209}\mathrm{Bi}\rightarrow{{}^{208}\mathrm{Pb}}+p$ spectroscopic factors
using this reaction. This illustrates one aspect of a question raised by
Satchler in the introduction to his book Direct Nuclear Reactions Sat83 : “At
this point one may ask, ‘Where does one stop?’ …If the shape (angular
distribution) of the [DWBA cross section] agrees with experiment, it is
trivial to extract a structure factor from the magnitude needed to match the
data. This is no longer true when multistep processes are studied; the
structure amplitudes (and their signs) are involved in an intimate way.”

In summary, we have shown that including the 7Li ground state reorientation
coupling enabled a good description of the 208Pb(7Li,6He)209Bi single-proton
stripping data of Ref. Zel79  which FR-DWBA calculations were unable to fit.
Addition of coupling to the 0.478-MeV $1/2^{-}$ state of 7Li plus the
corresponding two-step transfer path had only a minor effect on the shapes of
the angular distributions, except for that leading to the 1.61-MeV $13/2^{+}$
level of 209Bi, but did have a significant impact on the
${}^{209}\mathrm{Bi}\rightarrow{{}^{208}\mathrm{Pb}}+p$ spectroscopic factors.
A good description of the elastic scattering was necessary to obtain the good
fit to the stripping data. Since many weakly-bound light radioactive nuclei
have similar properties, these results strongly suggest that such couplings
should be included in analyses of transfer data obtained with beams of these
nuclei if erroneous conclusions are not to be drawn. They also underline the
need to obtain as complete a data set as possible—ideally elastic and
inelastic scattering as well as the transfer process(es) of interest—to
facilitate the correct inclusion of such couplings.

## References

  * (1) N. Keeley, K. W. Kemper and K. Rusek, Phys. Rev. C 102, 014617 (2020).
  * (2) K. S. Toth, J. L. C. Ford, Jr., G. R. Satchler, E. E. Gross, D. C. Hensley, S. T. Thornton and T. C. Schweizer, Phys. Rev. C 14, 1471 (1976).
  * (3) W. von Oertzen, H. Lettau, H. G. Bohlen and D. Fick, Z. Phys. A 315, 81 (1984).
  * (4) A. F. Zeller, D. C. Weisser, T. R. Ophel and D. F. Hebbard, Nucl. Phys. A 332, 515 (1979).
  * (5) V. Hnizdo, K. W. Kemper and J. Szymakowski, Phys. Rev. Lett. 46, 590 (1981).
  * (6) F. Videbæk, P. R. Christensen, Ole Hansen and K. Ulbak, Nucl. Phys. A 256, 301 (1976).
  * (7) I. J. Thompson, Comput. Phys. Rep. 7, 167 (1988).
  * (8) A. A. Rudchik, A. T. Rudchik, G. M. Kozeratska, O. A. Ponkratenko, E. I. Koshchy, A. Budzanowski, B. Czech, S. Kliczewski, R. Siudak, I. Skwirczyńska, A. Szczurek, S. Yu. Mezhevych, K. W. Kemper, J. Choiński, T. Czosnyka and L. Głowacka, Phys. Rev. C 72, 034608 (2005).
  * (9) S. Cohen and D. Kurath, Nucl. Phys. A 101, 1 (1967).
  * (10) Y. Kucuk, I. Boztosun and N. Keeley, Phys. Rev. C 79, 067601 (2009).
  * (11) A. M. Moro, K. Rusek, J. M. Arias, J. Gómez-Camacho and M. Rodríguez-Gallardo, Phys. Rev. C 75, 064607 (2007).
  * (12) A. Weller, P. Egelhof, R. C̆aplar, O. Karban, D. Krämer, K. H. Möbius, Z. Moroz, K. Rusek, E. Steffens, G. Tungate, K. Blatt, I. Koenig and D. Fick, Phys. Rev. Lett. 55, 480 (1985).
  * (13) W. D. Weintraub, N. Keeley, K. W. Kemper, K. Kravvaris, F. Maréchal, D. Robson, B. T. Roeder, K. Rusek and A. Volya, Phys. Rev. C 100, 024604 (2019).
  * (14) G. H. F. Diercksen, A. J. Sadlej, D. Sundholm and P. Pyykko, Chem. Phys. Lett. 143, 163 (1988).
  * (15) T. Minamisono, T. Ohtsubo, Y. Nakayama, S. Fukuda, T. Izumikawa, M. Tanigaki, M. Matsui, S. Takeda, N. Nakamura, M. Fukuda, K. Matsuka and Y. Nojiri, Hyperfine Interactions 78, 159 (1993).
  * (16) E. Arnold, J. Bonn, A. Klein, R. Neugart, M. Neuroth, E. W. Otten, P. Lievens, H. Reich, W. Widdra and the ISOLDE Collaboration, Phys. Lett. B 281, 16 (1992).
  * (17) R. Neugart, et al., Phys. Rev. Lett. 101, 132502 (2008).
  * (18) T. Minamisono, T. Ohtsubo, I. Minami, S. Fukuda, A. Kitagawa, M. Fukuda, K. Matsuta, Y. Nojiri, S. Takeda, H. Sagawa et al., Phys. Rev. Lett. 69, 2058 (1992).
  * (19) T. Minamisono, T. Ohtsubo, I. Minami, A. Kitagawa, S. Fukuda, Y. Nakayama, M. Fukuda, K. Matsuta, Y. Nojiri, S. Takeda, H. Sagawa and H. Kitagawa, Hyperfine Interactions 78, 165 (1993).
  * (20) G. R. Satchler, Direct Nuclear Reactions, (Clarendon Press, Oxford, 1983), p. 25.