# Nonclassical correlated deterministic single-photon pairs for a trapped atom
in bimodal cavities

Zhong Peng  Guangdong Provincial Key Laboratory of Quantum Metrology and
Sensing $\&$ School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai
Campus), Zhuhai 519082, China  Yuangang Deng<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing $\&$
School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus),
Zhuhai 519082, China

###### Abstract

Single photons and single-photon pairs, inherently nonclassical in their
nature, are fundamental elements of quantum sciences and technologies. Here,
we propose to realize the nonclassical correlated deterministic photon pairs
at the single-photon level for a single atom trapped in bimodal cavities. It
is shown that the photon emissions for bimodal cavities exhibit a high single
photon purity and relatively large intracavity photon numbers, which are
ascribed to the combination of optical Stark shift enhanced energy-spectrum
anharmonicity and quantum interference suppressed two-photon excitation. Our
scheme generates photon-photon pairs with strong photon blockade for cavity
modes via constructing a highly tunable cavity-enhanced deterministic
parametric down-conversion process. Furthermore, we show that the cross-
correlation function between the two cavity modes vastly violates the Cauchy-
Schwarz inequality of the classical boundary, which unambiguously demonstrates
the nonclassicality of the single-photon pairs. Our result reveals a prominent
strategy to generate the high-quality two-mode single-photon sources with
strong nonclassical correlation, which could provide versatile applications in
distribution of quantum networks, long distance quantum communication, and
fundamental tests of quantum physics.

##  I Introduction

The generation of high-quality deterministic single-photon sources is a
crucial building-block for quantum information science, quantum computing and
quantum cryptography Northup and Blatt (2014); Awschalom  _et al._ (2018);
Arakawa and Holmes (2020); Blais  _et al._ (2020). A paradigmatic mechanism
for generation of single photons is utilizing the photon blockade (PB) Zou and
Mandel (1990); Carmichael (1985); Imamoḡlu  _et al._ (1997), in which the
quantum statistics of the photon emissions exhibit a sub-Poissonian
distribution since one photon excitation will block a second photon
excitation. Up to now, the controllable single-photon states have been
extensively studied by using the mechanisms of conventional PB and
unconventional PB. The former mechanism requires a strong optical nonlinearity
with large enough energy-spectrum anharmonicity in the systems, ranging from
cavity quantum electrodynamics (QED) McKeever  _et al._ (2004); Birnbaum  _et
al._ (2005); Bajcsy  _et al._ (2013); Radulaski  _et al._ (2017); Hamsen  _et
al._ (2017); Zhang  _et al._ (2018); Trivedi  _et al._ (2019); Tang  _et al._
(2019, 2021) to circuit QED Lang  _et al._ (2011); Hoffman  _et al._ (2011);
Liu  _et al._ (2014); Felicetti  _et al._ (2018); Wang  _et al._ (2018), Kerr-
type nonlinear systems Imamoḡlu  _et al._ (1997); Liao and Law (2010);
Ferretti  _et al._ (2010); Miranowicz  _et al._ (2013); Li  _et al._ (2017);
Huang  _et al._ (2018); Ghosh and Liew (2019); Lin  _et al._ (2020), and
optomechanical resonators Rabl (2011); Stannigel  _et al._ (2012); Liao and
Nori (2013); Ng and Law (2016); Zhu  _et al._ (2018); Wang  _et al._ (2019);
Xu  _et al._ (2020); Wang  _et al._ (2020). The latter mechanism relies on the
destructive quantum interference by constructing different quantum transition
pathways Liew and Savona (2010); Majumdar  _et al._ (2012); Tang  _et al._
(2015); Snijders  _et al._ (2018); Vaneph  _et al._ (2018); Li  _et al._
(2019); Zubizarreta Casalengua  _et al._ (2020). Thus the two-photon
excitation can be completely eliminated even with a weak atom-cavity coupling.
In addition to these significant advances, the strong PB by utilizing the
optical Stark-shift enhanced energy-spectrum anharmonicity has been proposed
for Jaynes-Cummings model and electromagnetically induced transparency in
Refs. Tang  _et al._ (2019, 2021).

Meanwhile, the availability of nonclassical correlated photon pairs could
offer new resources for exploration of intriguing quantum physics Pan  _et
al._ (2012); Brunner  _et al._ (2014); Fröwis  _et al._ (2018). The
spontaneous parametric down-conversion (SPDC) of nonlinear optical processes
Burnham and Weinberg (1970); Ou and Lu (1999), as a paradigmatic mechanism for
realizing nonclassical correlated photon pairs, has been studied in numerous
quantum applications, such as quantum cryptography Naik  _et al._ (2000);
Lasota and Kolenderski (2020), entangled photons Kwiat  _et al._ (1995); Wang
_et al._ (2004); Zhang  _et al._ (2011); Clausen  _et al._ (2014), quantum
imaging Lemos  _et al._ (2014), quantum memories Heshami  _et al._ (2016);
Seri  _et al._ (2017); Tsai  _et al._ (2020) and quantum networks Kimble
(2008); Hu  _et al._ (2021). The aforementioned schemes based on SPDC are
probabilistic since the generated photon pairs are random, albeit the recent
advances for photon-pair sources have been proposed by employing deterministic
down-conversion Chang  _et al._ (2016); Sánchez-Burillo  _et al._ (2016). By
contrast, a deterministic photon source can be achieved in the isolated single
atom cavity system McKeever  _et al._ (2004). Unfortunately, the deterministic
source reduces to a probabilistic one with including the inevitable losses for
both atom and cavity, corresponding to nonzero multiphoton excitations. In
addition, due to the intrinsic weak nonlinear interactions between photon
pairs, the experimental realization of high-purity photon pairs at high rates
remains a challenge. Therefore, the realization of deterministic photon-pair
sources in single-photon level beyond strong coupling regime will provide
unprecedented opportunities in quantum-physics community and motivate numerous
implementations in photonic quantum information processing Lodahl  _et al._
(2015); Flamini  _et al._ (2018); Slussarenko and Pryde (2019) and many-body
physics Chang  _et al._ (2014, 2008); Tangpanitanon and Angelakis (2020).

Figure 1: (a) Schematic of the system for realizing nonclassical correlated
deterministic single-photon pairs in bimodal cavities coupled to a single
atom. (b) The typical anharmonic energy spectrum with ignoring the weak atomic
pump field and effective cavity-driven field. (c) Quantum transition paths
based on the Hamiltonian (II). The strong PB is generated with eliminating the
occupation probability of two-photon excitation state $|2,2,g\rangle$ (dashed
line) by using destructive quantum interference.

In this paper, we propose to generate nonclassical correlated deterministic
single-photon pairs for a single atom trapped in bimodal cavities. Here the
deterministic cavity-enhanced parametric down-conversion Hamiltonian is
realized with a high tunability. It is shown that the interplay of optical
Stark-shift enhanced energy-spectrum anharmonicity and quantum interference
suppressed two-photon excitation gives rise to strong PB for both the two
cavity modes. In particular, the second-order correlation function with the
optimal phase and driven amplitude of cavities can be low to $10^{-4}$ at a
moderate Stark shift, corresponding to a relatively large intracavity photon
numbers as well. Remarkably, the photon emissions for the two-mode cavities
exhibit a strong nonclassical correlation with violating Cauchy-Schwarz
inequality of classical boundary, which host a high single photon purity with
identical photon numbers. Our proposal will be valuable for realizing
deterministic high-quality single-photon sources and nonclassical correlated
single-photon pairs sources.

This paper is organized as follows. In Sec. II, we introduce the model and
Hamiltonian for realizing a deterministic parametric down-conversion process.
Section III is devoted to derive the Stark-shift enhanced energy-spectrum
anharmonicity. In Sec. IV, we present strong PB with quantum interference. In
Sec. V, we study nonclassical correlated single-photon pairs of the single
atom-cavity system. Finally, we conclude in Sec. VI.

##  II Model and Hamiltonian

Without loss of generality, we consider the high-finesse bimodal cavities
coupled to a single four-level atom, where $\omega_{a}$ ($\omega_{b}$) is the
bare frequency for the single-mode optical cavity A (B). Figure 1(a) shows the
level structure and laser configuration of the single atom-cavity system. The
four relevant energy levels include one electronic ground state $\ket{g}$ and
three excited states $\ket{e}$, $\ket{r}$ and $\ket{r^{\prime}}$, where
$\ket{r}$ is an elaborately long-lived electronic orbital state with the
hyperfine splitting $\hbar\Delta_{3}$ between the states $\ket{r}$ and
$\ket{r^{\prime}}$. As can be seen, the atomic transition $\ket{g}$
$\leftrightarrow$ $\ket{e}$ ($\ket{e}$ $\leftrightarrow$ $\ket{r}$) is coupled
to the cavity mode A (B) with single atom-cavity coupling strength $g_{a}$
($g_{b}$) and atom-cavity detuning $\Delta_{1}$. Moreover, the atomic
transition $\ket{e}$ $\leftrightarrow$ $\ket{r^{\prime}}$ is far-off resonance
driven by the cavity field B with single atom-cavity coupling $g^{\prime}_{b}$
and large atom-cavity detuning $\Delta_{3}$.

In our laser configuration, the single atom is also illuminated by a pair of
classical pump fields with Rabi frequencies $\Omega_{1}$ and $\Omega_{2}$,
corresponding to the atom-pump detuning $\Delta_{2}$. To generate an effective
driven field of bimodal cavities, we consider applying a strong pump field to
drive the atomic transition $\ket{g}$ $\leftrightarrow$ $\ket{r^{\prime}}$
with Rabi frequency $\Omega_{3}e^{i\theta}$ as depicted in Fig. 1(a). We
should emphasize that the tunable phase $\theta$ is gauge invariant and will
play an essential role in realizing strong PB for bimodal cavities by
employing the quantum interference.

In the far dispersive regime,
$|\Delta_{i}|\gg\\{g_{a},g_{b},\Omega_{i=1,2,3}\\}$, the far-off resonance
atomic excited states $\ket{e}$ and $\ket{r^{\prime}}$ can be adiabatically
eliminated. Under the rotating-wave approximation, the relevant time-
independent Hamiltonian for the single atom-cavity system reads (see Appendix
A for details)

$\displaystyle\hat{\cal H}/\hbar$
$\displaystyle=\Delta_{a}\hat{a}^{\dagger}\hat{a}+\Delta_{b}\hat{b}^{\dagger}\hat{b}+(\Delta_{r}-U_{a}\hat{a}^{\dagger}\hat{a}+U_{b}\hat{b}^{\dagger}\hat{b})\hat{\sigma}_{rr}$
$\displaystyle+[(g\hat{a}^{\dagger}\hat{b}^{\dagger}+\Omega)\hat{\sigma}_{gr}+\eta
e^{-i\theta}\hat{a}^{\dagger}\hat{b}^{\dagger}\hat{\sigma}_{gg}+{\rm H.c.}],$
(1)

where $\hat{a}$ $(\hat{b})$ is the operator of cavity mode A (B) with cavity-
light detuning $\Delta_{a}$ ($\Delta_{b}$), $\Delta_{r}$ is the effective two-
photon detuning, and $U_{a}=-g^{2}_{a}/\Delta_{1}$
($U_{b}=-g^{2}_{b}/\Delta_{1}$) is the optical Stark shift for cavity mode A
(B). $\hat{\sigma}_{kl}=\ket{k}\bra{l}(k,l=g,r)$ denotes the atomic transition
operator, $g=-g_{a}g_{b}/\Delta_{1}$
($\Omega=-\Omega_{1}\Omega_{2}/\Delta_{1}$) is cavity (pump) field coupling
strength, and $\eta=\Omega_{3}g_{a}g^{\prime}_{b}/\Delta_{1}\Delta_{3}$ is the
weak cavity-driven amplitude. Here, we neglect terms with the high-frequency
prefactor $e^{\pm i(\Delta_{1}-\Delta_{2})t}$ in Eq. (II) when
$|\Delta_{1}-\Delta_{2}|\gg\\{g_{a},g_{b},\Omega_{i=1,2}\\}$.

Remarkably, the Hamiltonian (II) essentially describes a two-photon Raman-
assisted deterministic parametric down-conversion process, where the photon
excitations for the bimodal cavities are simultaneously created or destroyed.
As we will see below, high-quality single-photon pairs are expected,
corresponding to a strong nonclassical correlation of the single photons
emitted by the separated bimodal cavities simultaneously. In contrast to the
generation of probabilistic photon pair sources with SPDC Fasel  _et al._
(2004); Eckstein  _et al._ (2011); Caspani  _et al._ (2017); Senellart  _et
al._ (2017); Grimau Puigibert  _et al._ (2017), the realized nonclassicality
two-mode photon pairs can be ascribed to the deterministic sources in our
isolated single quantum system. In the presence of atom and cavities
dissipations, the multiphoton excitations can be completely suppressed by
employing the optical Stark-shift enhanced energy spectrum anharmonicity and
quantum interference suppressed two-photon excitation.

To further investigate the quantum statistics and correlations of photon
emissions, we take into account the dissipations of the atomic field and
bimodal cavities by numerically solving the master equation in the steady
state using the quantum optics toolbox Tan (1999). As a result, the time
evolution of the density matrix $\rho$ for the cavities and atomic field obeys
the master equation, ${d\rho}/{dt}=\mathcal{L}\rho$, corresponding the
Liouvillian superoperator defined as

$\displaystyle\mathcal{L}\rho$
$\displaystyle=-i[{\cal{\hat{H}}},\rho]+\sum_{\hat{o}}\frac{\kappa_{o}}{2}\mathcal{D}[\hat{o}]\rho+\frac{\gamma}{2}\mathcal{D}[\hat{\sigma}_{gr}]\rho+\frac{\gamma_{d}}{2}\mathcal{D}[\hat{\sigma}_{rr}]\rho,$
(2)

where $\kappa_{o}$ is the decay rate of cavity field with ${o}={a}$ and ${b}$,
$\gamma$ is the atomic spontaneous emission rate, $\gamma_{d}$ is the pure
dephasing factor of the atomic field, and
$\mathcal{D}[\hat{o}]\rho=2\hat{o}\rho\hat{o}^{\dagger}-\hat{o}^{\dagger}\hat{o}\rho-\rho\hat{o}^{\dagger}\hat{o}$
is the Lindblad type of dissipation. In the steady-state solution,
$\mathcal{L}\rho_{s}=0$, the intracavity photon number for cavity mode
$\hat{o}$ satisfies, $n^{(o)}_{s}={\rm Tr}[\hat{o}^{\dagger}\hat{o}\rho_{s}]$.
In general, the second order correlation function, which is a key physical
quantity for characterizing PB and nonclassical correlation, is defined as
Glauber (1963)

$\displaystyle g_{oo^{\prime}}^{(2)}(\tau)=\frac{{\rm
Tr}[\hat{o}^{\dagger}(t)\hat{o^{\prime}}^{\dagger}(t+\tau)\hat{o^{\prime}}(t+\tau)\hat{o}(t)\rho_{s}]}{{\rm
Tr}[\hat{o}^{\dagger}(t)\hat{o}(t)\rho_{s}]\times{\rm
Tr}[\hat{o^{\prime}}^{\dagger}(t)\hat{o^{\prime}}(t)\rho_{s}]}.$ (3)

For the zero time interval ($\tau=0$), $g_{oo^{\prime}}^{(2)}(\tau)$ is
directly reduced to the equal time second-order auto-correlation function
$g_{aa}^{(2)}(0)$ [$g_{bb}^{(2)}(0)$] for the optical mode A (B) with
$o=o^{\prime}$, whose value is used to characterize the purity of single-
photon emissions Senellart  _et al._ (2017). As to $o\neq o^{\prime}$,
$g_{ab}^{(2)}(0)=g_{ba}^{(2)}(0)$ represents the cross-correlation function
between separated bimodal cavities. Furthermore, the time-dependent second-
order correlation function $g_{oo^{\prime}}^{(2)}(\tau)$ can be
straightforwardly calculated using the quantum regression theorem Carmichael
(1999). Therefore, the quantum statistics for PB should satisfy two
conditions: $g_{oo}^{(2)}(0)<1$ to ensure the sub-Poissonian statistics and
$g_{oo}^{(2)}(0)<g_{oo}^{(2)}(\tau)$ to ensure the photon antibunching.
Moreover, the second-order correlation functions can be experimentally
measured via Hanbury, Brown, and Twiss interferometer Birnbaum  _et al._
(2005).

As to the experimental feasibility, our proposal could be applicable to
alkaline-earth-metal atoms Winchester  _et al._ (2017); Livi  _et al._ (2016);
Kolkowitz  _et al._ (2017); Bromley  _et al._ (2018) and Rydberg atoms Henkel
_et al._ (2010); Madjarov  _et al._ (2020) by employing the advantages of
energy-level structures. In our numerical simulation, we assume the bimodal
cavities have the equal decay rates with $\kappa=\kappa_{a}=\kappa_{b}$ for
simplicity. The cavity decay rate $\kappa=2\pi\times 800$ kHz Léonard  _et
al._ (2017a, b) is fixed as the energy scale of the system, then the
corresponding atomic decay rate for the long-lived excited state
$\gamma/\kappa=0.01$ and pure dephasing $\gamma_{d}/\kappa=0.01$. We take the
single atom-cavity coupling $g/\kappa=2$, the weak atomic pump field
$\Omega/\kappa=0.1$, and $\Delta_{a}=\Delta_{b}=\Delta_{r}/2$. Thus the
tunable parameters in our atom-cavity system are reduced to cavity-light
detuning $\Delta_{a}$ and optical Stark shift $U_{a}$ with satisfying
$U_{a}U_{b}=g^{2}$. Moreover, the effective cavity-driven strength $\eta$ and
its phase $\theta$ are self-consistently determined by applying the quantum
interference conditions in our numerical simulation.

##  III Stark shift enhanced PB

For first ignoring the weak driving field of the bimodal cavities and atomic
classical pump field, the analytical energy spectrum of the system is obtained
by diagonalizing Hamiltonian (II). We note that the difference of the
excitation number between cavity mode A and B is $zero$ due to the
deterministic parametric down-conversion process. Therefore the restricted
Hilbert spaces for the single atom-cavity system are $\ket{n-1,n-1,r}$ and
$\ket{n,n,g}$, where $n$ denotes the photon number excitation of single-cavity
mode. Explicitly, the energy eigenvalues of system are given by

$\displaystyle E_{n,n,\pm}$
$\displaystyle=2n\Delta_{a}-\frac{1}{2}(n-1)U_{0}\pm\frac{1}{2}\sqrt{(n-1)^{2}U_{0}^{2}+4g^{2}n^{2}},$
(4)

corresponding to the energy splittings for the $n$th pair of dressed states
$\ket{n,n,\pm}$

$\displaystyle\Delta_{n,n,\pm}$
$\displaystyle=\pm\frac{1}{2}\sqrt{(n-1)^{2}U_{0}^{2}+4g^{2}n^{2}}-\frac{1}{2}(n-1)U_{0},$
(5)

where $+$ ($-$) represents the higher (lower) branch and $U_{0}=U_{a}-U_{b}$
is Stark-shift difference of bimodal cavities. It is clear that the vacuum
Rabi splitting $\Delta_{1,1,\pm}=\pm g$ is independent on the value of
$U_{0}$, which is contrast to the realization of strong PB proposed in Ref.
Tang  _et al._ (2019) by utilizing the Stark shift to enhance the vacuum Rabi
splitting. However, the anharmonic ladder of dressed states for the
multiphoton excitations ($n\geq 2$) is enhanced by the large value of $U_{0}$.
In Fig. 1(b), we show the typical anharmonicity ladder of the energy spectrum
with $\eta=0$ and $\Omega=0$. We emphasize that the energy spectrum is immune
to the weak atomic pump and cavities driven fields, i.e., $|\Omega/g|\ll 1$
and $|\eta/g|\ll 1$. As expected, the strong energy-spectrum anharmonicity
indicates that the strong PB for bimodal cavities emissions can be realized.

Figure 2:  The second-order correlation function $g_{oo}^{(2)}(0)$ as a
function of $\Delta_{a}$ for (a) $U_{a}/g=2$ and (b) $U_{a}/g=1/2$,
respectively. (c) The steady-state photon number $n_{s}^{(o)}$ as a function
of $\Delta_{a}$ for $U_{a}/g=2$. Here, the solid blue line (dashed red line)
denotes the result of cavity mode A (B). (d) The $U_{a}/g$ dependence of
$g_{oo}^{(2)}(0)$ for cavity mode A at the red sideband with
$\Delta_{a}/g=1/2$ (solid blue line) and for cavity mode B at the blue
sideband with $\Delta_{a}/g=-1/2$ (dashed red line), respectively. In (a)–(d),
the other parameter is $\eta/g=0$.

To investigate the PB enhanced by optical Stark shift, we calculate the photon
quantum statistics for fixing $\eta/g=0$, where the quantum interference
conditions are invalid when the effective cavity-driven field disappears Tang
_et al._ (2015, 2019). Figures. 2(a) and 2(b) display the equal time second-
order correlation function as a function of $\Delta_{a}$ for $U_{a}/g=2$ and
$U_{a}/g=1/2$, respectively. For $U_{a}/g\neq 1$, it is shown that the line
shapes of $g_{aa}^{(2)}(0)$ (solid blue line) and $g_{bb}^{(2)}(0)$ (dashed
red line) exhibit the red-blue asymmetric behavior with respect to cavity-
light detuning $\Delta_{a}$. Interestingly, the steady-state photon numbers
$n_{s}^{(a)}$ and $n_{s}^{(b)}$ for the bimodal cavities with the same
dissipations always coincide exactly due to the parametric down-conversion
process, as displayed in Fig. 2(c). Moreover, it is clear that $n_{s}^{(o)}$
exhibits a red-blue symmetric profile and possesses two peaks at single-photon
resonance with $\Delta_{a}/g=\pm 1/2$. The reason is that the energy
eigenvalues $E_{1,1,\pm}$ in Eq. (4) is independent of the value of Stark
shift $U_{0}$.

As to the photon quantum statistics, we find that the minimum value of
$g_{aa}^{(2)}(0)$ [$g_{bb}^{(2)}(0)$] for cavity mode A (B) occurs at the red-
sideband (blue-sideband) when $U_{a}/g>1$ ($U_{a}/g<1$). Figure. 2(d) displays
the result of $g_{oo}^{(2)}(0)$ versus $U_{a}/g$. We find that the second-
order correlation function for the cavity mode A at the red sideband is equal
to the cavity mode B at the blue sideband when $U_{a}/g=1$ yielding
$U_{a}/U_{b}=1$. For $U_{a}/g>1$, the antibunching amplitude for cavity mode A
(solid blue line) is rapidly enhancing with increasing $U_{a}$. As to the
regime of $U_{a}/g<1$ ($U_{b}/g>1$), it is clear that $g_{bb}^{(2)}(0)$ for
cavity mode B (dashed red line) is monotonically decreasing with decreasing
$U_{a}$. This result can be readily understood through the calculation of the
energy spectrum dominated by the Hamiltonian (4), where the large Stark shift
gives rise to the large energy-spectrum anharmonicity. As can be seen, the
strong PB for cavity emission, which is defined as $g_{oo}^{(2)}(0)<0.01$
requires a large Stark shift with $U_{a}/g\gg 1$ or $U_{a}/g\ll 1$
($U_{b}/g\gg 1$).

##  IV Strong PB with quantum interference

For the conventional PB in cavity QED McKeever  _et al._ (2004); Birnbaum  _et
al._ (2005); Bajcsy  _et al._ (2013); Radulaski  _et al._ (2017); Hamsen  _et
al._ (2017); Zhang  _et al._ (2018); Trivedi  _et al._ (2019); Tang  _et al._
(2019, 2021) , the strong energy spectrum anharmonicity is required in the
strong single atom-cavity coupling limit. By contrast, the unconventional PB
can be achieved by utilizing quantum interference beyond the strong coupling
regime. By employing the destructive quantum interference, the two-photon
excitations can be completely suppressed, which has been proposed in Jaynes-
Cummings model Tang  _et al._ (2015, 2019). Analogously, the present
deterministic parametric down-conversion Hamiltonian (II) for bimodal cavities
should also exist the quantum interference mechanism with combination of
classical atomic pump field and effective cavity driven field. To obtain the
quantum interference conditions, the wave function of the system is formally
given as

$\displaystyle\ket{\psi}=\sum_{n=0}^{\infty}C_{n,n,g}\ket{n,n,g}+\sum_{n=0}^{\infty}C_{n,n,r}\ket{n,n,r},$
(6)

where $|C_{n,n,g}|^{2}$ and $|C_{n,n,r}|^{2}$ denote the atomic occupation
probability of the system for the eigenstate $\ket{n,n,g}$ and $\ket{n,n,r}$,
respectively. Without loss of generality, we can safely neglect the
multiphoton excitations ($n>2$) in PB regime. Thus, the wave function of the
system with truncating the two-photon excitations subspaces reads

$\displaystyle\ket{\psi}=\sum_{n=0}^{2}C_{n,n,g}\ket{n,n,g}+\sum_{n=0}^{1}C_{n,n,r}\ket{n,n,r}.$
(7)

Substituting the wave function $\ket{\psi}$ into the Schrödinger equation, we
can derive a set of evolution equations

$\displaystyle i\dot{C}_{0,0,g}$ $\displaystyle=\Omega C_{0,0,r}+\eta
e^{i\theta}C_{1,1,g},$ $\displaystyle i\dot{C}_{1,1,g}$
$\displaystyle=\tilde{\Delta}_{a}C_{1,1,g}+gC_{0,0,r}+\Omega C_{1,1,r}+\eta
e^{-i\theta}C_{0,0,g}$ $\displaystyle+2\eta e^{i\theta}C_{2,2,g},$
$\displaystyle i\dot{C}_{0,0,r}$
$\displaystyle=\tilde{\Delta}_{r}C_{0,0,r}+gC_{1,1,g}+\Omega C_{0,0,g},$
$\displaystyle i\dot{C}_{2,2,g}$
$\displaystyle=2\tilde{\Delta}_{a}C_{2,2,g}+2gC_{1,1,r}+2\eta
e^{-i\theta}C_{1,1,g},$ $\displaystyle i\dot{C}_{1,1,r}$
$\displaystyle=(\tilde{\Delta}_{a}+\tilde{\Delta}_{r}-U_{0})C_{1,1,r}+2gC_{2,2,g}+\Omega
C_{1,1,g},$ (8)

with $\tilde{\Delta}_{a}=2\Delta_{a}-i2\kappa$ and
$\tilde{\Delta}_{r}=2\Delta_{a}-i\gamma$ for shorthand notations. In the
steady-state solutions, the two-photon excitation can be completely eliminated
with $C_{2,2,g}=0$, which indicates that the single-photon emission is
obtained. As a result, the nontrivial solutions of Eq. (IV) are given by

$\displaystyle\theta_{\rm{opt}}=-\arctan\left(\frac{2\kappa+\gamma}{4\Delta_{a}-U_{0}}\right),$
$\displaystyle\eta_{\rm{opt}}={g\Omega}/{{\cal R}},$ (9)

with ${\cal R}=\sqrt{(4\Delta_{a}-U_{0})^{2}+(2\kappa+\gamma)^{2}}$.

Remarkably, the optimal parameters $\theta_{\rm{opt}}$ and $\eta_{\rm{opt}}$
denote the quantum interference conditions for realization of strong PB. In
contrast to the optimal atomic pump field in Ref. Tang  _et al._ (2019, 2015),
the quantum interference conditions in our system can be highly tuned by the
effective cavity-driven field. For fixing the single-photon resonance, i.e.,
$\Delta_{a}=\pm g/2$, the strong PB with a large cavity output is naturally
expected by using the obtained optimal cavity-driven amplitude
$\eta_{\rm{opt}}$ and its phase $\theta_{\rm{opt}}$ in Eq. (IV).

Figure 3:  (a) $g_{aa}^{(2)}(0)$ and (b) $n^{(a)}_{s}$ as a function of
$\Delta_{a}$ and $\theta$ with $U_{a}/g=2$. Here, the cavity-driven amplitude
$\eta$ is self-consistently calculated by using the quantum interference
conditions in Eq. (IV). The colors with blue-red gradient shading denote the
values of $\log_{10}[g_{aa}^{(2)}(0)]$ in (a) and $n^{(a)}_{s}$ in (b),
respectively.

To process further, the essential mechanism for destructive quantum
interference can be well understood by the quantum transition paths of the
atom-cavity system, as shown in Fig. 1(c). There exist two quantum transition
pathways from single-photon excitation to two-photon excitation. Explicitly,
the transition $\left|1,1,g\right\rangle\xrightarrow{2\eta
e^{-i\theta}}\left|2,2,g\right\rangle$ is induced by the effective driven
field of cavities. Meanwhile, the transition
$\left|1,1,g\right\rangle\xrightarrow{\Omega}\left|1,1,r\right\rangle\xrightarrow{2g}\left|2,2,g\right\rangle$
is related to the atomic pump field and cavity fields. Actually, the optical
Stark-shift induced effective cavity-driven term in Hamiltonian (II) plays an
important role in building the quantum interference. It is clear that the
quantum interference conditions in Eq. (IV) will vanish when $\eta=0$. More
importantly, the forbidden transition of
$\left|0,0,r\right\rangle\rightarrow\left|1,1,r\right\rangle$ will also be
beneficial to suppress the two-photon excitation in our laser configuration,
which is in contrast to the previous proposals Tang  _et al._ (2019, 2015)
that the single-mode cavity is directly driven by the external laser field.

We should emphasize that the quantum interference mechanism for generating
strong PB also requires a sufficient energy-spectrum anharmonicity. In fact,
the multiphoton ($n\geq 3$) excitations could not be well suppressed in the
weak atom-cavity coupling regime. We check that the second-order correlation
function $g_{aa}^{(2)}(0)$ and $g_{bb}^{(2)}(0)$ is immune to the weak optical
Stark shift when $U_{a}/g\ll 1$ and $U_{b}/g\ll 1$, respectively. Without loss
of generality, we will study the photon quantum statistics for cavity mode A
with focusing on $U_{a}/g>1$.

Figures 3(a) and 3(b) show the numerical results of $g_{aa}^{(2)}(0)$ and
$n^{(a)}_{s}$ in the $\Delta_{a}$-$\theta$ parameter plane with quantum
interference conditions in Eq. (IV) for $g/\kappa=2$ and $U_{a}/g=2$. Compared
with the results of $\eta=0$ (without quantum interference), both
$g_{aa}^{(2)}(0)$ and $n^{(a)}_{s}$ for nonzero $\eta$ exhibit obviously
asymmetric behaviors, which depend on the sign of cavity-light detuning
$\Delta_{a}$. In particular, the strong PB with $g_{aa}^{(2)}(0)<0.01$ is
realized near the blue sideband beyond the strong coupling limit. And the
sufficiently large cavity emission $n^{(a)}_{s}$ can also be obtained in a
large parameter region with strong PB [Fig. 3(b)]. Remarkably, the strong PB
existing in a large parameter regime will facilitate the experimental
feasibility even the system parameters deviating from quantum interference
conditions.

Figure 4: $\Delta_{a}$ dependance of (a) $g_{aa}^{(2)}(0)$ and (b)
$n^{(a)}_{s}$. (c) Time interval $\tau$ dependence of $g^{(2)}_{aa}(\tau)$ for
$\Delta_{a}/g=-1/2$. (d) $\Delta_{a}$ dependence of $g_{aa}^{(2)}(0)$ for
different values of pure dephasing $\gamma_{d}$. In (a)–(d), we fix the Stark
shift $U_{a}/g=2$. (e) $g_{aa}^{(2)}(0)$ and (f) $n^{(a)}_{s}$ versus
$U_{a}/g$ for fixing the blue sideband with $\Delta_{a}/g=-1/2$. Excluding
(d), the solid blue line (dashed red line) represents the result with
(without) quantum interference conditions of Eq. (IV).

Furthermore, Figs. 4(a) and 4(b) show $g_{aa}^{(2)}(0)$ and $n^{(a)}_{s}$ as a
function of $\Delta_{a}$ with (without) quantum interference conditions.
Although the steady-state photon number at blue sideband is decreased, the
second-order correlation function $g_{aa}^{(2)}(0)$ of $\eta\neq 0$ (solid
blue line) can be reduced more than one orders of magnitude compared with the
results of $\eta=0$ (dashed red line) even at a moderate Stark shift, i.e.,
$U_{a}/g=2$. As to red sideband $\Delta_{a}/g=1/2$, we find that cavity
emission of $n^{(a)}_{s}$ is obviously growing with $\eta\neq 0$, albeit
$g_{aa}^{(2)}(0)$ is roughly unchanged.

In Fig. 4(c), we further calculate the time interval $\tau$ dependence of the
second-order correlation function $g^{(2)}_{aa}(\tau)$. It is clear that the
transmitted photons exhibit the strong PB, with $g^{(2)}_{aa}(0)=0.01$ and
photon antibunching with $g^{(2)}_{aa}(0)<g^{(2)}_{aa}(\tau)$. Moreover, the
lifetime of the PB is significantly longer than the typical value of $\tau\sim
1/\kappa$ since $\gamma/\kappa\ll 1$ in our system. As to photon quantum
statistics, we check that $g^{(2)}_{aa}(0)$ with respect to $\Delta_{a}$ is
immune to the weak atomic pure dephasing when $\gamma_{d}/\gamma\leq 10$, as
displayed in Fig. 4(d). In particular, we remark that the qualitative results
for both $g_{aa}^{(2)}(0)$ and $n_{s}^{(a)}$ are robust against the variations
of parameters $\kappa_{a}/\kappa_{b}$, albeit the values of them depend on the
specific choice of $\kappa_{a}$ and $\kappa_{b}$ (see Appendix B).

For fixing $\Delta_{a}/g=-1/2$ of the blue sideband, we plot the $U_{a}/g$
dependence of $g_{aa}^{(2)}(0)$ and $n^{(a)}_{s}$ as shown in Figs. 4(e) and
4(f). Interestingly, the single-photon purity can be increased by more than
two orders of magnitude for large value of $U_{a}/g$, corresponding to
$g_{aa}^{(2)}(0)<10^{-4}$ with $U_{a}/g>7.8$. The mechanism for generation of
strong PB is ascribed to the interplay of the Stark-shift enhanced energy-
spectrum anharmonicity and quantum interference suppressed two-photon
excitation. In addition, the steady-state photon number $n^{(a)}_{s}$ is
monotonically growing with increasing $U_{a}/g$, as displayed in Fig. 4(f).
The large increased $n^{(a)}_{s}$ can be readily understood in part by the
additional effective cavity-driven field. Remarkably, we show that the strong
PB with large $n^{(a)}_{s}$ occurs at a moderate Stark shift when $U_{a}/g\geq
2.0$, which is important for experimental feasibility of cavity QED.
Therefore, our proposal can be used to generate the high-quality deterministic
single-photon source, which satisfies the high single-photon purity and large
cavity output, simultaneously.

Figure 5: $\Delta_{a}$ dependance of (a) cross-correlation function
$g^{(2)}_{ab}(0)$ and (b) nonclassicality parameter $\Gamma$ based on quantum
interference with $U_{a}/g=2$.

##  V Nonclassical correlated deterministic single-photon pairs

To gain more insight into single-photon pairs, we study the nonclassical
correlation for the generated photon pairs of the two-mode cavities.
Remarkably, both the photon emissions in bimodal cavities could exhibit the
strong PB by employing the Stark shift and quantum interference, which
indicates that the nonclassical photon-photon correlations in the single-
photon level are realized. In general, the pair of classical correlation
functions for two-mode fields constrained by the well-known Cauchy-Schwarz
inequality is given by Miranowicz  _et al._ (2010)

$\displaystyle g^{(2)}_{ab}(0)\leq\sqrt{g^{(2)}_{aa}(0)g^{(2)}_{bb}(0)},$ (10)

which gives rise to the bounded value of cross-correlation function
$g^{(2)}_{ab}(0)$ in classical origin. A violation of Cauchy-Schwarz
inequality can act as an unambiguous feature for measuring the nonclassicality
of two-mode correlations Kuzmich  _et al._ (2003); Riedinger  _et al._ (2016).
To proceed further, we introduce a dimensionless parameter for unraveling the
nonclassicality, defined as

$\displaystyle\Gamma=\frac{g^{(2)}_{ab}(0)}{\sqrt{g^{(2)}_{aa}(0)g^{(2)}_{bb}(0)}}.$
(11)

Note that $\Gamma>1$ represents the violation of Cauchy-Schwarz inequality for
the nonclassical correlated deterministic single-photon pairs in our system.

Figure 5(a) shows the cross-correlation function $g^{(2)}_{ab}(0)$ as a
function of $\Delta_{a}$ under the quantum interference. Obviously,
$g^{(2)}_{ab}(0)$ exhibits a double-well structure near the single-photon
resonance of red and blue sidebands. In the parameter regime of our numerical
simulation, the values of $g^{(2)}_{ab}(0)$ are much larger than $1$, albeit
the bimodal cavities photons exhibit the strong antibunching for both
$g^{(2)}_{aa}(0)\ll 1$ and $g^{(2)}_{bb}(0)\ll 1$. In Fig. 5(b), we plot the
nonclassicality parameter $\Gamma$ as a function of $\Delta_{a}$. The strong
violation of the Cauchy-Schwarz inequality for single-photon pairs is
observed, corresponding to $\Gamma\gg 1$, which exceeds the classical bound
($\Gamma=1$) clearly. More important, the bimodal cavities with the same
dissipations possess the strong nonclassical correlations and equal photon
emissions due to the deterministic parametric down-conversion process.
Finally, the proposed system can be used as a high-quality two-mode single-
photon source and single-photon pairs sources with hosting the strong
nonclassicality, which could provide a broad physics community for
applications in quantum information science Lodahl  _et al._ (2015); Flamini
_et al._ (2018); Slussarenko and Pryde (2019) and metrology Modi  _et al._
(2011); Girolami  _et al._ (2013); Streltsov  _et al._ (2017); Braun  _et al._
(2018).

##  VI Conclusion

In conclusion, we propose an experimental scheme for realizing nonclassical
correlated deterministic single-photon pairs by constructing cavity-enhanced
deterministic parametric down-conversion in bimodal cavities. We show that the
photon quantum statistics for the bimodal cavities exhibit the red-blue
asymmetric feature by tuning the value of $U_{a}/g$, albeit the photon
emissions are always the same. The strong photon blockade with
$g_{aa}^{(2)}(0)<10^{-4}$ is achieved by employing optical Stark-shift
enhanced energy-spectrum anharmonicity and quantum interference suppressed
two-photon excitation. In particular, the photon antibunching amplitude can be
over two orders of magnitude smaller than the results in the absence of
quantum interference. The advantages of our proposal is that it does not
require strong atom-cavity coupling, which will facilitate the monitoring of
quantum experimental feasibility in cavity QED. Interestingly, the observed
deterministic single-photon pairs exhibit the strong nonclassical correlation
with violating Cauchy-Schwarz inequality. Our study on realizing the
deterministic photon sources can be extended to explore the multimode
$N$-photon bundle states with strong nonclassicality natures Deng  _et al._
(2021); Muñoz  _et al._ (2014, 2018); Chang  _et al._ (2016); Bin  _et al._
(2021).

## acknowledgment

This work was supported by the National Key R$\&$D Program of China (Grant No.
2018YFA0307500), NSFC (Grants No. 11874433 and No. 12135018), and the Key-Area
Research and Development Program of GuangDong Province under Grants No.
2019B030330001.

## Appendix A: time-independent Hamiltonian

In this appendix we present the details on the derivation of the effective
Hamiltonian for the given level diagram and laser configurations in Fig. 1(a)
of the main text. Under the rotating-wave approximation, the relevant
Hamiltonian for a single atom trapped in bimodal cavities can be written as

$\displaystyle{\cal\hat{H}}_{1}/\hbar$
$\displaystyle=\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger}\hat{b}+\omega_{e}|e\rangle\langle
e|+\omega_{r}|r\rangle\langle r|+\omega_{r^{\prime}}|r^{\prime}\rangle\langle
r^{\prime}|$ $\displaystyle+[g_{a}\hat{a}^{\dagger}|g\rangle\langle
e|+g_{b}\hat{b}^{\dagger}|e\rangle\langle
r|+g^{\prime}_{b}\hat{b}^{\dagger}|e\rangle\langle r^{\prime}|+{\rm H.c.}]$
$\displaystyle+[\Omega_{1}e^{i\omega_{1}t}|g\rangle\langle
e|+\Omega_{2}e^{i\omega_{2}t}|e\rangle\langle r|$
$\displaystyle+\Omega_{3}e^{i(\omega_{3}t+\theta)}|g\rangle\langle
r^{\prime}|+{\rm H.c.}],$ (A1)

where $\hat{a}^{\dagger}$ and $\hat{b}$ is the annihilation operator of cavity
mode A and B), respectively, $\ket{m}$ is the electronically atomic excited
state with $m=e,r,r^{\prime}$, and $\hbar\omega_{m}$ is the bare transition
energy of $\ket{g}\leftrightarrow\ket{m}$.

In the rotating frame, the unitary transformation is defined as

$\displaystyle{\cal U}$
$\displaystyle=\exp\\{-i[\omega^{\prime}_{a}(\hat{a}^{\dagger}\hat{a}+|e\rangle\langle
e|)+\omega^{\prime}_{b}\hat{b}^{\dagger}\hat{b}$
$\displaystyle+\omega_{3}(|r\rangle\langle r|+|r^{\prime}\rangle\langle
r^{\prime}|)]t\\},$ (A2)

with fixing $\omega^{\prime}_{a}+\omega^{\prime}_{b}=\omega_{3}$ and
$\omega_{3}=\omega_{1}+\omega_{2}$. Then the internal-state Hamiltonian Eq.
(Appendix A: time-independent Hamiltonian) reduces to

$\displaystyle{\cal\hat{H}}_{2}/\hbar$ $\displaystyle={\cal
U}^{\dagger}{\cal\hat{H}}_{1}{\cal U}-i{\cal
U}^{\dagger}\frac{\partial}{\partial{t}}{\cal U}$
$\displaystyle=\delta_{a}\hat{a}^{\dagger}\hat{a}+\delta_{b}\hat{b}^{\dagger}\hat{b}+\Delta_{1}|e\rangle\langle
e|+\Delta_{r}|r\rangle\langle r|+\Delta_{3}|r^{\prime}\rangle\langle
r^{\prime}|$ $\displaystyle+[g_{a}\hat{a}^{\dagger}|g\rangle\langle
e|+g_{b}\hat{b}^{\dagger}|e\rangle\langle
r|+g^{\prime}_{b}(\hat{b}^{\dagger}|e\rangle\langle r^{\prime}|+{\rm H.c.}]$
$\displaystyle+[\Omega_{1}e^{i(\Delta_{1}-\Delta_{2})t}|g\rangle\langle
e|+\Omega_{2}[e^{-i(\Delta_{1}-\Delta_{2})t}|e\rangle\langle r|+{\rm H.c.}]$
$\displaystyle+[\Omega_{3}e^{i\theta}|g\rangle\langle r^{\prime}|+{\rm
H.c.}],$ (A3)

where $\delta_{a}=\omega_{a}-\omega^{\prime}_{a}$,
$\delta_{b}=\omega_{b}-\omega^{\prime}_{b}$,
$\Delta_{1}=\omega_{e}-\omega^{\prime}_{a}$,
$\Delta_{2}=\omega_{e}-\omega_{1}$, and
$\Delta_{3}=\omega_{r^{\prime}}-\omega_{3}$ are the single-photon detunings,
and $\Delta_{r}=\omega_{r}-\omega_{1}-\omega_{2}$ is the two-photon detuning.

To proceed further, we rewrite the Hamiltonian (Appendix A: time-independent
Hamiltonian) in the second quantized form

$\displaystyle{\cal\hat{H}}_{2}/\hbar$
$\displaystyle=\delta_{a}\hat{a}^{\dagger}\hat{a}+\delta_{b}\hat{b}^{\dagger}\hat{b}+\Delta_{1}\hat{\psi}_{e}^{\dagger}\hat{\psi}_{e}+\Delta_{r}\hat{\psi}_{r}^{\dagger}\hat{\psi}_{r}+\Delta_{3}\hat{\psi}_{r^{\prime}}^{\dagger}\hat{\psi}_{r^{\prime}}$
$\displaystyle+[g_{a}\hat{a}^{\dagger}\hat{\psi}_{g}^{\dagger}\hat{\psi}_{e}+g_{b}\hat{b}^{\dagger}\hat{\psi}_{e}^{\dagger}\hat{\psi}_{r}+g^{\prime}_{b}\hat{b}^{\dagger}\hat{\psi}_{e}^{\dagger}\hat{\psi}_{r^{\prime}}+{\rm
H.c.}]$
$\displaystyle+[\Omega_{1}e^{i(\Delta_{1}-\Delta_{2})t}\hat{\psi}_{g}^{\dagger}\hat{\psi}_{e}+\Omega_{2}e^{-i(\Delta_{1}-\Delta_{2})t}\hat{\psi}_{e}^{\dagger}\hat{\psi}_{r}+{\rm
H.c.}]$
$\displaystyle+[\Omega_{3}e^{i\theta}\hat{\psi}_{g}^{\dagger}\hat{\psi}_{r^{\prime}}+{\rm
H.c.}],$ (A4)

where $\hat{\psi}_{g}$ and $\hat{\psi}_{m}$ denotes the annihilation field
operator for the ground and excited state, respectively.

Taking into account the atomic spontaneous emissions, the Heisenberg equations
of motion for the far-off resonance atomic field operators are given by

$\displaystyle i\dot{\hat{\psi}}_{e}$
$\displaystyle=(\Delta_{1}-i\gamma_{e})\hat{\psi}_{e}+g_{a}\hat{a}\hat{\psi}_{g}+g_{b}\hat{b}^{\dagger}\hat{\psi}_{r}+g_{b}^{\prime}\hat{b}^{\dagger}\hat{\psi}_{r^{\prime}}$
$\displaystyle+\Omega_{1}e^{-i(\Delta_{1}-\Delta_{2})t}\hat{\psi}_{g}+\Omega_{2}e^{-i(\Delta_{1}-\Delta_{2})t}\hat{\psi}_{r}$
$\displaystyle i\dot{\hat{\psi}}_{r^{\prime}}$
$\displaystyle=(\Delta_{3}-i\gamma_{r^{\prime}})\hat{\psi}_{r^{\prime}}+g_{b}^{\prime}\hat{b}\hat{\psi}_{e}+\Omega_{3}e^{-i\theta}\hat{\psi}_{g},$
(A5)

where $\gamma_{e}$ and $\gamma_{r^{\prime}}$ are the atomic spontaneous
emission rate for $|e\rangle$ and $|r^{\prime}\rangle$ states. Under the
conditions
$|\Delta_{i}|\gg\\{g_{a},g_{b},\Omega_{i},\gamma_{e},\gamma_{r^{\prime}}\\}$
with $i=1,2,3$, the excited states $\ket{e}$ and $\ket{r^{\prime}}$ can be
adiabatically eliminated to yield

$\displaystyle\hat{\psi}_{e}$
$\displaystyle=-\frac{1}{\Delta_{1}-i\gamma_{e}}\left[g_{a}\hat{a}\hat{\psi}_{g}+\Omega_{1}e^{-i(\Delta_{1}-\Delta_{2})t}\hat{\psi}_{g}+g_{b}\hat{b}^{\dagger}\hat{\psi}_{r}\right.$
$\displaystyle\left.+\Omega_{2}e^{-i(\Delta_{1}-\Delta_{2})t}\hat{\psi}_{r}\right],$
$\displaystyle\approx-\frac{1}{\Delta_{1}}\left[g_{a}\hat{a}\hat{\psi}_{g}+\Omega_{1}e^{-i(\Delta_{1}-\Delta_{2})t}\hat{\psi}_{g}+g_{b}\hat{b}^{\dagger}\hat{\psi}_{r}\right.$
$\displaystyle\left.+\Omega_{2}e^{-i(\Delta_{1}-\Delta_{2})t}\hat{\psi}_{r}\right],$
$\displaystyle\hat{\psi}_{r^{\prime}}$
$\displaystyle=\frac{1}{(\Delta_{1}-i\gamma_{e})(\Delta_{3}-i\gamma_{r^{\prime}})}\left[g_{a}g^{\prime}_{b}\hat{a}\hat{b}\hat{\psi}_{g}+g_{b}g^{\prime}_{b}\hat{b}^{\dagger}\hat{b}\hat{\psi}_{r}\right.$
$\displaystyle\left.+g^{\prime}_{b}\Omega_{1}e^{-i(\Delta_{1}-\Delta_{2})t}\hat{b}\hat{\psi}_{g}+g^{\prime}_{b}\Omega_{2}e^{-i(\Delta_{1}-\Delta_{2})t}\hat{b}\hat{\psi}_{r}\right.$
$\displaystyle\left.-\Omega_{3}(\Delta_{1}-i\gamma_{e})e^{-i\theta}\hat{\psi}_{g}\right],$
$\displaystyle\approx-\frac{\Omega_{3}\Delta_{1}e^{-i\theta}\hat{\psi}_{g}}{\Delta_{3}},$
(A6)

here the higher-order terms ($\sim 1/\Delta_{1}^{2}$ and $\sim
1/(\Delta_{1}\Delta_{3})$) are safely ignored.

Substituting Eq. (Appendix A: time-independent Hamiltonian) into Eq. (Appendix
A: time-independent Hamiltonian), one can derive an effective Hamiltonian for
single atom-cavity system

$\displaystyle{\cal\hat{H}}_{3}/\hbar$
$\displaystyle=\Delta_{a}\hat{a}^{\dagger}\hat{a}+\Delta_{b}\hat{b}^{\dagger}\hat{b}+(\Delta_{r}-U_{a}\hat{a}^{\dagger}\hat{a}+U_{b}\hat{b}^{\dagger}\hat{b})\hat{\psi}_{r}^{\dagger}\hat{\psi}_{r}$
$\displaystyle+[(g\hat{a}^{\dagger}\hat{b}^{\dagger}+\Omega)\hat{\psi}_{g}^{\dagger}\hat{\psi}_{r}+\eta
e^{-i\theta}\hat{a}^{\dagger}\hat{b}^{\dagger}\hat{\psi}_{g}^{\dagger}\hat{\psi}_{g}+{\rm
H.c.}]$
$\displaystyle-\left[\frac{g_{a}}{\Delta_{1}}\left(\Omega_{1}\hat{\psi}_{g}^{\dagger}\hat{\psi}_{g}+\Omega_{2}\hat{\psi}_{g}^{\dagger}\hat{\psi}_{r}\right)\hat{a}^{\dagger}e^{-i(\Delta_{1}-\Delta_{2})t}+{\rm
H.c.}\right]$
$\displaystyle-\left[\frac{g_{b}}{\Delta_{1}}\left(\Omega_{2}\hat{\psi}_{r}^{\dagger}\hat{\psi}_{r}+\Omega_{1}\hat{\psi}_{g}\hat{\psi}_{r}\right)\hat{b}^{\dagger}e^{-i(\Delta_{1}-\Delta_{2})t}+{\rm
H.c.}\right],$ (A7)

where $\Delta_{a}=\delta_{a}-g^{2}_{a}/\Delta_{1}$ ($\Delta_{b}=\delta_{b}$)
is the effective cavity-light detuning for cavity mode A (B),
$g=-g_{a}g_{b}/\Delta_{1}$ ($\Omega=-\Omega_{1}\Omega_{2}/\Delta_{1}$) is
cavity (pump) field induced two-photon Rabi frequency,
$\eta=\Omega_{3}g_{a}g^{\prime}_{b}/\Delta_{1}\Delta_{3}$ is the effective
cavity-driven amplitude, and $U_{a}=-g^{2}_{a}/\Delta_{1}$
($U_{b}=-g^{2}_{b}/\Delta_{1}$) is the optical Stark shift for cavity mode A
(B). We emphasize that the terms with the high-frequency prefactor $e^{\pm
i(\Delta_{1}-\Delta_{2})t}$ in Eq. (Appendix A: time-independent Hamiltonian)
are safely neglected when
$|(\Delta_{1}-\Delta_{2})|\gg\\{g_{a},g_{b},\Omega_{i=1,2}\\}$. This condition
can be easily satisfied by taking $\Delta_{1}\neq\Delta_{2}$, e.g.,
$\Delta_{1}=-\Delta_{2}$. Finally, we obtain the effective time-independent
Hamiltonian (II) in the main text.

Figure 6: $\Delta_{a}$ dependence of the second-order correlation function
$g_{aa}^{(2)}(0)$ in (a) and steady-state photon number $n_{s}^{(a)}$ in (b)
for different values of $\kappa_{b}$. Here the other parameters are
$\eta/g=0$, $\kappa_{a}/\kappa=1$, and $U_{a}/g=2$. Figure 7: $\Delta_{a}$
dependence of the second-order correlation function $g_{aa}^{(2)}(0)$ in (a)
and steady-state photon number $n_{s}^{(a)}$ in (b) for different values of
$\kappa_{b}$ using quantum interference conditions of Eq. (IV). Here the other
parameters are $\kappa_{a}/\kappa=1$ and $U_{a}/g=2$.

## Appendix B: The results with different bimodal cavities decays

We present the numerical results of the photon quantum statistics with
different bimodal cavities decays. In Figs. 6(a) and 6(b), we plot the second-
order correlation function $g_{aa}^{(2)}(0)$ and steady-state photon number
$n_{s}^{(a)}$ for different values of cavity decay $\kappa_{b}$ with
$\eta/g=0$. It is clear that the values of $g_{aa}^{(2)}(0)$ at both red and
blue sidebands $\Delta_{a}/g=\pm 1/2$ are monotonically increasing with
decreasing the ratio $\kappa_{b}/\kappa_{a}$. Meanwhile, the steady-state
photon number $n_{s}^{(a)}$ with respect to $\Delta_{a}$ remains exhibiting a
red-blue symmetric profile. As can be seen, the peak values of $n_{s}^{(a)}$
increases as $\kappa_{b}/\kappa_{a}$ increases.

To proceed further, we investigate the effect of the decay rate $\kappa_{b}$
on the photon emissions in the presence of the quantum interference conditions
in Eq. (IV). Figures 7(a) and 7(b) shows $g_{aa}^{(2)}(0)$ and $n_{s}^{(a)}$
as a function of $\Delta_{a}$ for different values of cavity decay
$\kappa_{b}$ by using the quantum interference conditions (IV), respectively.
It is cleat that the antibunching amplitude of $g_{aa}^{(2)}(0)$ at the blue
sideband $\Delta_{a}/g=-1/2$ is rapidly increasing with decreasing the ratio
$\kappa_{b}/\kappa_{a}$, albeit $n_{s}^{(a)}$ is slightly increased. As to the
red sideband $\Delta_{a}/g=1/2$, we find that $n^{(a)}_{s}$ is obviously
growing with decreasing $\kappa_{b}/\kappa_{a}$. Finally, we should remark
that the qualitative results (see Fig. 6 and Fig. 7) for both
$g_{aa}^{(2)}(0)$ and $n_{s}^{(a)}$ are robust against the variations of the
parameters $\kappa_{a}$ and $\kappa_{b}$.

## References

  * Northup and Blatt (2014) T. E. Northup and R. Blatt, Quantum information transfer using photons, Nature Photon. 8, 356 (2014).
  * Awschalom  _et al._ (2018) D. D. Awschalom, R. Hanson, J. Wrachtrup, and B. B. Zhou, Quantum technologies with optically interfaced solid-state spins, Nature Photon. 12, 516 (2018).
  * Arakawa and Holmes (2020) Y. Arakawa and M. J. Holmes, Progress in quantum-dot single photon sources for quantum information technologies: A broad spectrum overview, Applied Physics Reviews 7, 021309 (2020).
  * Blais  _et al._ (2020) A. Blais, S. M. Girvin, and W. D. Oliver, Quantum information processing and quantum optics with circuit quantum electrodynamics, Nature Phys. 16, 247 (2020).
  * Zou and Mandel (1990) X. T. Zou and L. Mandel, Photon-antibunching and sub-poissonian photon statistics, Phys. Rev. A 41, 475 (1990).
  * Carmichael (1985) H. J. Carmichael, Photon Antibunching and Squeezing for a Single Atom in a Resonant Cavity, Phys. Rev. Lett. 55, 2790 (1985).
  * Imamoḡlu  _et al._ (1997) A. Imamoḡlu, H. Schmidt, G. Woods, and M. Deutsch, Strongly Interacting Photons in a Nonlinear Cavity, Phys. Rev. Lett. 79, 1467 (1997).
  * McKeever  _et al._ (2004) J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, and H. J. Kimble, Deterministic generation of single photons from one atom trapped in a cavity, Science 303, 1992 (2004).
  * Birnbaum  _et al._ (2005) K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, Photon blockade in an optical cavity with one trapped atom, Nature 436, 87 (2005).
  * Bajcsy  _et al._ (2013) M. Bajcsy, A. Majumdar, A. Rundquist, and J. Vučković, Photon blockade with a four-level quantum emitter coupled to a photonic-crystal nanocavity, New Journal of Physics 15, 025014 (2013).
  * Radulaski  _et al._ (2017) M. Radulaski, K. A. Fischer, K. G. Lagoudakis, J. L. Zhang, and J. Vučković, Photon blockade in two-emitter-cavity systems, Phys. Rev. A 96, 011801 (2017).
  * Hamsen  _et al._ (2017) C. Hamsen, K. N. Tolazzi, T. Wilk, and G. Rempe, Two-photon Blockade in an Atom-Driven Cavity QED System, Phys. Rev. Lett. 118, 133604 (2017).
  * Zhang  _et al._ (2018) X. Zhang, C. Xu, and Z. Ren, High fidelity heralded single-photon source using cavity quantum electrodynamics, Scientific Reports 8, 3140 (2018).
  * Trivedi  _et al._ (2019) R. Trivedi, M. Radulaski, K. A. Fischer, S. Fan, and J. Vučković, Photon Blockade in Weakly Driven Cavity Quantum Electrodynamics Systems with Many Emitters, Phys. Rev. Lett. 122, 243602 (2019).
  * Tang  _et al._ (2019) J. Tang, Y. Deng, and C. Lee, Strong Photon Blockade Mediated by Optical Stark shift in a Single-Atom-Cavity system, Phys. Rev. Applied 12, 044065 (2019).
  * Tang  _et al._ (2021) J. Tang, Y. Deng, and C. Lee, Tunable photon blockade with a single atom in a cavity under electromagnetically induced transparency, Photon. Res. 9, 1226 (2021).
  * Lang  _et al._ (2011) C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, Observation of Resonant Photon Blockade at Microwave Frequencies Using Correlation Function Measurements, Phys. Rev. Lett. 106, 243601 (2011).
  * Hoffman  _et al._ (2011) A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, Dispersive Photon Blockade in a Superconducting Circuit, Phys. Rev. Lett. 107, 053602 (2011).
  * Liu  _et al._ (2014) Y.-x. Liu, X.-W. Xu, A. Miranowicz, and F. Nori, From blockade to transparency: Controllable photon transmission through a circuit-qed system, Phys. Rev. A 89, 043818 (2014).
  * Felicetti  _et al._ (2018) S. Felicetti, D. Z. Rossatto, E. Rico, E. Solano, and P. Forn-Díaz, Two-photon quantum rabi model with superconducting circuits, Phys. Rev. A 97, 013851 (2018).
  * Wang  _et al._ (2018) Y. Wang, G.-Q. Zhang, and W.-L. You, Photon blockade with cross-kerr nonlinearity in superconducting circuits, Laser Physics Letters 15, 105201 (2018).
  * Liao and Law (2010) J.-Q. Liao and C. K. Law, Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity, Phys. Rev. A 82, 053836 (2010).
  * Ferretti  _et al._ (2010) S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation, Phys. Rev. A 82, 013841 (2010).
  * Miranowicz  _et al._ (2013) A. Miranowicz, M. Paprzycka, Y.-x. Liu, J. c. v. Bajer, and F. Nori, Two-photon and three-photon blockades in driven nonlinear systems, Phys. Rev. A 87, 023809 (2013).
  * Li  _et al._ (2017) A. Li, Y. Zhou, and X.-B. Wang, Cascaded kerr photon-blockade sources and applications in quantum key distribution, Scientific Reports 7, 7309 (2017).
  * Huang  _et al._ (2018) R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, Nonreciprocal Photon Blockade, Phys. Rev. Lett. 121, 153601 (2018).
  * Ghosh and Liew (2019) S. Ghosh and T. C. H. Liew, Dynamical Blockade in a Single-Mode Bosonic System, Phys. Rev. Lett. 123, 013602 (2019).
  * Lin  _et al._ (2020) H. Lin, X. Wang, Z. Yao, and D. Zou, Kerr-nonlinearity enhanced conventional photon blockade in a second-order nonlinear system, Opt. Express 28, 17643 (2020).
  * Rabl (2011) P. Rabl, Photon Blockade Effect in Optomechanical Systems, Phys. Rev. Lett. 107, 063601 (2011).
  * Stannigel  _et al._ (2012) K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, Optomechanical Quantum Information Processing with Photons and Phonons, Phys. Rev. Lett. 109, 013603 (2012).
  * Liao and Nori (2013) J.-Q. Liao and F. Nori, Photon blockade in quadratically coupled optomechanical systems, Phys. Rev. A 88, 023853 (2013).
  * Ng and Law (2016) K. H. Ng and C. K. Law, Single-photon scattering in an optomechanical jaynes-cummings model, Phys. Rev. A 93, 043834 (2016).
  * Zhu  _et al._ (2018) G.-L. Zhu, X.-Y. Lü, L.-L. Wan, T.-S. Yin, Q. Bin, and Y. Wu, Controllable nonlinearity in a dual-coupling optomechanical system under a weak-coupling regime, Phys. Rev. A 97, 033830 (2018).
  * Wang  _et al._ (2019) D.-Y. Wang, C.-H. Bai, S. Liu, S. Zhang, and H.-F. Wang, Distinguishing photon blockade in a $\mathcal{PT}$-symmetric optomechanical system, Phys. Rev. A 99, 043818 (2019).
  * Xu  _et al._ (2020) X. Xu, Y. Zhao, H. Wang, H. Jing, and A. Chen, Quantum nonreciprocality in quadratic optomechanics, Photon. Res. 8, 143 (2020).
  * Wang  _et al._ (2020) D.-Y. Wang, C.-H. Bai, S. Liu, S. Zhang, and H.-F. Wang, Photon blockade in a double-cavity optomechanical system with nonreciprocal coupling, New Journal of Physics 22, 093006 (2020).
  * Liew and Savona (2010) T. C. H. Liew and V. Savona, Single Photons from Coupled Quantum Modes, Phys. Rev. Lett. 104, 183601 (2010).
  * Majumdar  _et al._ (2012) A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, Loss-Enabled Sub-Poissonian Light Generation in a Bimodal Nanocavity, Phys. Rev. Lett. 108, 183601 (2012).
  * Tang  _et al._ (2015) J. Tang, W. Geng, and X. Xu, Quantum interference induced photon blockade in a coupled single quantum dot-cavity system, Scientific Reports 5, 9252 (2015).
  * Snijders  _et al._ (2018) H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, Observation of the Unconventional Photon Blockade, Phys. Rev. Lett. 121, 043601 (2018).
  * Vaneph  _et al._ (2018) C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, Observation of the Unconventional Photon Blockade in the Microwave Domain, Phys. Rev. Lett. 121, 043602 (2018).
  * Li  _et al._ (2019) B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, Nonreciprocal unconventional photon blockade in a spinning optomechanical system, Photon. Res. 7, 630 (2019).
  * Zubizarreta Casalengua  _et al._ (2020) E. Zubizarreta Casalengua, J. C. López Carreño, F. P. Laussy, and E. d. Valle, Conventional and unconventional photon statistics, Laser & Photonics Reviews 14, 1900279 (2020).
  * Pan  _et al._ (2012) J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, Multiphoton entanglement and interferometry, Rev. Mod. Phys. 84, 777 (2012).
  * Brunner  _et al._ (2014) N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
  * Fröwis  _et al._ (2018) F. Fröwis, P. Sekatski, W. Dür, N. Gisin, and N. Sangouard, Macroscopic quantum states: Measures, fragility, and implementations, Rev. Mod. Phys. 90, 025004 (2018).
  * Burnham and Weinberg (1970) D. C. Burnham and D. L. Weinberg, Observation of simultaneity in parametric production of optical photon pairs, Phys. Rev. Lett. 25, 84 (1970).
  * Ou and Lu (1999) Z. Y. Ou and Y. J. Lu, Cavity enhanced spontaneous parametric down-conversion for the prolongation of correlation time between conjugate photons, Phys. Rev. Lett. 83, 2556 (1999).
  * Naik  _et al._ (2000) D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat, Entangled state quantum cryptography: Eavesdropping on the ekert protocol, Phys. Rev. Lett. 84, 4733 (2000).
  * Lasota and Kolenderski (2020) M. Lasota and P. Kolenderski, Optimal photon pairs for quantum communication protocols, Scientific Reports 10, 20810 (2020).
  * Kwiat  _et al._ (1995) P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, New high-intensity source of polarization-entangled photon pairs, Phys. Rev. Lett. 75, 4337 (1995).
  * Wang  _et al._ (2004) H. Wang, T. Horikiri, and T. Kobayashi, Polarization-entangled mode-locked photons from cavity-enhanced spontaneous parametric down-conversion, Phys. Rev. A 70, 043804 (2004).
  * Zhang  _et al._ (2011) H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion, Nature Photonics 5, 628 (2011).
  * Clausen  _et al._ (2014) C. Clausen, F. Bussières, A. Tiranov, H. Herrmann, C. Silberhorn, W. Sohler, M. Afzelius, and N. Gisin, A source of polarization-entangled photon pairs interfacing quantum memories with telecom photons, New Journal of Physics 16, 093058 (2014).
  * Lemos  _et al._ (2014) G. B. Lemos, V. Borish, G. D. Cole, S. Ramelow, R. Lapkiewicz, and A. Zeilinger, Quantum imaging with undetected photons, Nature 512, 409 (2014).
  * Heshami  _et al._ (2016) K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, Quantum memories: emerging applications and recent advances, Journal of Modern Optics 63, 2005 (2016).
  * Seri  _et al._ (2017) A. Seri, A. Lenhard, D. Rieländer, M. Gündoğan, P. M. Ledingham, M. Mazzera, and H. de Riedmatten, Quantum correlations between single telecom photons and a multimode on-demand solid-state quantum memory, Phys. Rev. X 7, 021028 (2017).
  * Tsai  _et al._ (2020) P.-J. Tsai, Y.-F. Hsiao, and Y.-C. Chen, Quantum storage and manipulation of heralded single photons in atomic memories based on electromagnetically induced transparency, Phys. Rev. Research 2, 033155 (2020).
  * Kimble (2008) H. J. Kimble, The quantum internet, Nature 453, 1023 (2008).
  * Hu  _et al._ (2021) X.-M. Hu, C.-X. Huang, Y.-B. Sheng, L. Zhou, B.-H. Liu, Y. Guo, C. Zhang, W.-B. Xing, Y.-F. Huang, C.-F. Li, and G.-C. Guo, Long-distance entanglement purification for quantum communication, Phys. Rev. Lett. 126, 010503 (2021).
  * Chang  _et al._ (2016) Y. Chang, A. González-Tudela, C. Sánchez Muñoz, C. Navarrete-Benlloch, and T. Shi, Deterministic down-converter and continuous photon-pair source within the bad-cavity limit, Phys. Rev. Lett. 117, 203602 (2016).
  * Sánchez-Burillo  _et al._ (2016) E. Sánchez-Burillo, L. Martín-Moreno, J. J. García-Ripoll, and D. Zueco, Full two-photon down-conversion of a single photon, Phys. Rev. A 94, 053814 (2016).
  * Lodahl  _et al._ (2015) P. Lodahl, S. Mahmoodian, and S. Stobbe, Interfacing single photons and single quantum dots with photonic nanostructures, Rev. Mod. Phys. 87, 347 (2015).
  * Flamini  _et al._ (2018) F. Flamini, N. Spagnolo, and F. Sciarrino, Photonic quantum information processing: a review, Reports on Progress in Physics 82, 016001 (2019).
  * Slussarenko and Pryde (2019) S. Slussarenko and G. J. Pryde, Photonic quantum information processing: A concise review, Applied Physics Reviews 6, 041303 (2019).
  * Chang  _et al._ (2014) D. E. Chang, V. Vuletić, and M. D. Lukin, Quantum nonlinear optics - photon by photon, Nature Photonics 8, 685 (2014).
  * Chang  _et al._ (2008) D. E. Chang, V. Gritsev, G. Morigi, V. Vuletić, M. D. Lukin, and E. A. Demler, Crystallization of strongly interacting photons in a nonlinear optical fibre, Nature Physics 4, 884 (2008).
  * Tangpanitanon and Angelakis (2020) J. Tangpanitanon and D. G. Angelakis, Many-body physics and quantum simulations with strongly interacting photons, Nanoscale Quantum Optics 204, 169 (2020).
  * Fasel  _et al._ (2004) S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, High-quality asynchronous heralded single-photon source at telecom wavelength, New Journal of Physics 6, 163 (2004).
  * Eckstein  _et al._ (2011) A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, Highly efficient single-pass source of pulsed single-mode twin beams of light, Phys. Rev. Lett. 106, 013603 (2011).
  * Caspani  _et al._ (2017) L. Caspani, C. Xiong, B. J. Eggleton, D. Bajoni, M. Liscidini, M. Galli, R. Morandotti, and D. J. Moss, Integrated sources of photon quantum states based on nonlinear optics, Light: Science & Applications 6, e17100 (2017).
  * Senellart  _et al._ (2017) P. Senellart, G. Solomon, and A. White, High-performance semiconductor quantum-dot single-photon sources, Nature Nanotechnology 12, 1026 (2017).
  * Grimau Puigibert  _et al._ (2017) M. Grimau Puigibert, G. H. Aguilar, Q. Zhou, F. Marsili, M. D. Shaw, V. B. Verma, S. W. Nam, D. Oblak, and W. Tittel, Heralded single photons based on spectral multiplexing and feed-forward control, Phys. Rev. Lett. 119, 083601 (2017).
  * Tan (1999) S. M. Tan, A computational toolbox for quantum and atomic optics, J. Opt. B: Quantum Semiclass. Opt. 1, 424 (1999).
  * Glauber (1963) R. J. Glauber, The quantum theory of optical coherence, Phys. Rev. 130, 2529 (1963).
  * Carmichael (1999) H. J. Carmichael,  _Statistical methods in quantum optics 1: master equations and Fokker-Planck equations_ , Vol. 1 (Springer Science & Business Media, Berlin, 1999).
  * Winchester  _et al._ (2017) M. N. Winchester, M. A. Norcia, J. R. K. Cline, and J. K. Thompson, Magnetically induced optical transparency on a forbidden transition in strontium for cavity-enhanced spectroscopy, Phys. Rev. Lett. 118, 263601 (2017).
  * Livi  _et al._ (2016) L. F. Livi, G. Cappellini, M. Diem, L. Franchi, C. Clivati, M. Frittelli, F. Levi, D. Calonico, J. Catani, M. Inguscio, and L. Fallani, Synthetic dimensions and spin-orbit coupling with an optical clock transition, Phys. Rev. Lett. 117, 220401 (2016).
  * Kolkowitz  _et al._ (2017) S. Kolkowitz, S. L. Bromley, T. Bothwell, M. L. Wall, G. E. Marti, A. P. Koller, X. Zhang, A. M. Rey, and J. Ye, Spin corbit-coupled fermions in an optical lattice clock, Nature 542, 66 (2017).
  * Bromley  _et al._ (2018) S. L. Bromley, S. Kolkowitz, T. Bothwell, D. Kedar, A. Safavi-Naini, M. L. Wall, C. Salomon, A. M. Rey, and J. Ye, Dynamics of interacting fermions under spin corbit coupling in an optical lattice clock, Nature Physics 14, 399 (2018).
  * Henkel  _et al._ (2010) N. Henkel, R. Nath, and T. Pohl, Three-dimensional roton excitations and supersolid formation in rydberg-excited bose-einstein condensates, Phys. Rev. Lett. 104, 195302 (2010).
  * Madjarov  _et al._ (2020) I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, High-fidelity entanglement and detection of alkaline-earth rydberg atoms, Nature Physics 16, 857 (2020).
  * Léonard  _et al._ (2017a) J. Léonard, A. Morales, P. Zupancic, T. Esslinger, and T. Donner, Supersolid formation in a quantum gas breaking a continuous translational symmetry, Nature 543, 87 (2017a).
  * Léonard  _et al._ (2017b) J. Léonard, A. Morales, P. Zupancic, T. Donner, and T. Esslinger, Monitoring and manipulating higgs and goldstone modes in a supersolid quantum gas, Science 358, 1415 (2017b).
  * Miranowicz  _et al._ (2010) A. Miranowicz, M. Bartkowiak, X. Wang, Y.-x. Liu, and F. Nori, Testing nonclassicality in multimode fields: A unified derivation of classical inequalities, Phys. Rev. A 82, 013824 (2010).
  * Kuzmich  _et al._ (2003) A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. M. Duan, and H. J. Kimble, Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles, Nature 423, 731 (2003).
  * Riedinger  _et al._ (2016) R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gr ö blacher, Non-classical correlations between single photons and phonons from a mechanical oscillator, Nature 530, 313 (2016).
  * Modi  _et al._ (2011) K. Modi, H. Cable, M. Williamson, and V. Vedral, Quantum correlations in mixed-state metrology, Phys. Rev. X 1, 021022 (2011).
  * Girolami  _et al._ (2013) D. Girolami, T. Tufarelli, and G. Adesso, Characterizing nonclassical correlations via local quantum uncertainty, Phys. Rev. Lett. 110, 240402 (2013).
  * Streltsov  _et al._ (2017) A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys. 89, 041003 (2017).
  * Braun  _et al._ (2018) D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, and S. Pirandola, Quantum-enhanced measurements without entanglement, Rev. Mod. Phys. 90, 035006 (2018).
  * Deng  _et al._ (2021) Y. Deng, T. Shi, and S. Yi, Motional n-phonon bundle states of a trapped atom with clock transitions, Photon. Res. 9, 1289 (2021).
  * Muñoz  _et al._ (2014) C. S. Muñoz, E. del Valle, A. G. Tudela, K. Müller, S. Lichtmannecker, M. Kaniber, C. Tejedor, J. J. Finley, and F. P. Laussy, Emitters of n-photon bundles, Nature Photonics 8, 550 (2014).
  * Muñoz  _et al._ (2018) C. S. Muñoz, F. P. Laussy, E. del Valle, C. Tejedor, and A. González-Tudela, Filtering multiphoton emission from state-of-the-art cavity quantum electrodynamics, Optica 5, 14 (2018).
  * Bin  _et al._ (2021) Q. Bin, Y. Wu, and X.-Y. Lü, Parity-symmetry-protected multiphoton bundle emission, Phys. Rev. Lett. 127, 073602 (2021).


# Fourier series for nonperiodic functions

Rodrigo López Pouso111Partially supported by Agencia Estatal de Investigación,
Spain, project PID2020-113275GB-I00.

###### Abstract

We introduce a small change in the definition of the Fourier series so that we
can guarantee the coincidence with the given function at the endpoints of the
interval even if the function does not assume the same value at the endpoints.
This definition of the Fourier series also wipes out the Gibbs phenomenom at
the endpoints of the interval and proves useful in the resolution of
antiperiodic boundary value problems with linear partial differential
equations.

CITMAga, 15782, Santiago de Compostela, Spain

and

Departamento de Análise Matemática, Estatística e Optimización

Universidade de Santiago de Compostela

15782, Faculty of Mathematics, Campus Vida

Santiago de Compostela, Spain.

Email<EMAIL_ADDRESS>
2020 MSC: 42A16; 42A20; 35C10.

Keywords and phrases: Fourier series; antiperiodic functions; antiperiodic
boundary conditions.

##  1 Introduction

The Fourier series of a function $f:[-\pi,\pi]\longrightarrow\mathbb{R}$ is

$Sf(x)=\dfrac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n}\cos nx+b_{n}\sin
nx\right),\quad x\in\mathbb{R},$ (1.1)

where

$\displaystyle a_{n}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{f(x)\cos
nx\,dx},\,\,n=0,1,2,\dots,$ (1.2) $\displaystyle
b_{n}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{f(x)\sin nx\,dx},\,\,n=1,2,3,\dots,$
(1.3)

and provided that the integrals exist (they do exist if $f$ is just Lebesgue
integrable on $[-\pi,\pi]$, although the series $Sf(x)$ need not converge for
every $x\in\mathbb{R}$).

We refer readers to [3, 4] for the basic results on Fourier series and their
applications.

For instance, the Fourier series of $f(x)=x$, $x\in[-\pi,\pi]$, is

$Sf(x)=2\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{n}\sin nx,\,\,x\in\mathbb{R}.$

This simple example shows two disadvantages of the Fourier series which we
overcome here. First, although $f=Sf$ on $(-\pi,\pi)$, we have $f(\pm\pi)\neq
Sf(\pm\pi)=0$ (see Figure 1); second, the partial sums of $Sf(x)$ exhibit the
so–called Gibbs phenomenon around $\pm\pi$, i.e. the error of approximating
$f(x)$ by partial sums of $Sf(x)$ is greater around $\pm\pi$ than around any
other point in $(-\pi,\pi)$, see Figure 2.

Figure 1: Observe that $f(\pm\pi)\neq Sf(\pm\pi)=0$. Figure 2: Observe the
Gibbs phenomenom around $\pm\pi$.

These two drawbacks essentially come from the fact that $f(x)=x$ is not
periodic on $[-\pi,\pi]$, in the sense222We will say $f$ is periodic on
$[-\pi,\pi]$ if $f(-\pi)=f(\pi)$, but this is an abuse of language because $f$
is not defined on the whole of $\mathbb{R}$, and therefore it cannot be
periodic in the strict sense. However, notice that if $f(-\pi)=f(\pi)$, then
$f$ can be extended as a $2\pi$–periodic function on $\mathbb{R}$. A similar
abuse of language will occur in Section 2 when defining antiperiodic
functions. that $f(-\pi)\neq f(\pi)$, and Fourier series are always periodic.

There exist many convergence criteria for Fourier series. The following
result, whose proof can be looked up in [4, Corollary 8.48], is especially
appropriate for the purposes of this paper.

###### Theorem 1.1

Let $f:[-\pi,\pi]\longrightarrow\mathbb{R}$ be such that $f(-\pi)=f(\pi)$.

If $f$ is continuous and has bounded variation on $[-\pi,\pi]$, then $Sf(x)$
is uniformly convergent on $[-\pi,\pi]$ and $Sf(x)=f(x)$ for all
$x\in[-\pi,\pi]$.

Remember that continuity alone is not enough for the convergence of $Sf(x)$,
see [4, Remark 8.26].

On the other hand, recall that any of the following conditions implies that
$f$ is continuous and has bounded variation on $[-\pi,\pi]$:

  1. (i)

$f$ is absolutely continuous on $[-\pi,\pi]$;

  2. (ii)

$f$ Lipschitz continuous on $[-\pi,\pi]$;

  3. (iii)

$f$ is continuous and there is a partition
$-\pi=x_{0}<x_{1}<\cdots<x_{p}=\pi,$ such that the restrictions
$f_{|[x_{k-1},x_{k}]}$ are continuously differentiable on $[x_{k-1},x_{k}]$
for each $k=1,2,\dots,p$;

  4. (iv)

$f$ is continuously differentiable on $[-\pi,\pi]$.

Of course, our first example $f(x)=x$, $x\in[-\pi,\pi]$, is not periodic, so
it falls outside the scope of Theorem 1.1.

Can we have an alternative definition of $Sf$ so that Theorem 1.1 holds true
if we remove the condition that $f(-\pi)=f(\pi)$? The answer is positive and
the alternative definition of $Sf$ is pretty simple: we only have to replace
the basic trigonometric functions $\cos nx$ and $\sin nx$ by $\cos(nx+x/2)$
and $\sin(nx+x/2)$.

##  2 Fourier series for antiperiodic and nonperiodic functions

We start with the somewhat opposite case to periodic functions, namely, that
of antiperiodic functions. Then we will reduce the general (nonperiodic) case
to the antiperiodic case.

Let $f:[-\pi,\pi]\longrightarrow\mathbb{R}$ be continuous and have bounded
variation on $[-\pi,\pi]$. Assume also that $f$ is antiperiodic in the mild
sense that $f(-\pi)=-f(\pi)$.

The fundamental ideas in this paper lean on the functions

$f_{c}(x)=f(x)\cos\dfrac{x}{2}\quad\mbox{and}\quad
f_{s}(x)=f(x)\sin\dfrac{x}{2},\quad x\in[-\pi,\pi],$

which are continuous and have bounded variation on $[-\pi,\pi]$ and, moreover,
they are periodic because $f$ is antiperiodic, i.e. $f_{c}(-\pi)=f_{c}(\pi)$
and $f_{s}(-\pi)=f_{s}(\pi)$.

Hence, Theorem 1.1 can be applied to $f_{c}$ and $f_{s}$ and we get

$f_{c}(x)=Sf_{c}(x)=\dfrac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n}\cos
nx+b_{n}\sin nx\right),\quad x\in[-\pi,\pi],$ (2.4)
$f_{s}(x)=Sf_{s}(x)=\dfrac{\tilde{a}_{0}}{2}+\sum_{n=1}^{\infty}\left(\tilde{a}_{n}\cos
nx+\tilde{b}_{n}\sin nx\right),\quad x\in[-\pi,\pi],$ (2.5)

where, according to (1.2) and (1.3),

$\displaystyle a_{n}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{f_{c}(x)\cos
nx\,dx}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{f(x)\cos\dfrac{x}{2}\cos
nx\,dx},\,\,n=0,1,2,\dots,$ (2.6) $\displaystyle
b_{n}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{f_{c}(x)\sin
nx\,dx}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{f(x)\cos\dfrac{x}{2}\sin
nx\,dx},\,\,n=1,2,3,\dots,$ (2.7)

and

$\displaystyle\tilde{a}_{n}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{f_{s}(x)\cos
nx\,dx}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{f(x)\sin\dfrac{x}{2}\cos
nx\,dx},\,\,n=0,1,2,\dots,$ (2.8)
$\displaystyle\tilde{b}_{n}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{f_{s}(x)\sin
nx\,dx}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{f(x)\sin\dfrac{x}{2}\sin
nx\,dx},\,\,n=1,2,3,\dots$ (2.9)

Obviously, for all $x\in[-\pi,\pi]$ we have

$f(x)=f_{c}(x)\cos\dfrac{x}{2}+f_{s}(x)\sin\dfrac{x}{2},$

and we use (2.4) and (2.5) to obtain that

$\displaystyle f(x)$
$\displaystyle=\dfrac{a_{0}}{2}\cos\dfrac{x}{2}+\dfrac{\tilde{a}_{0}}{2}\sin\dfrac{x}{2}+\sum_{n=1}^{\infty}\left(a_{n}\cos
nx\cos\dfrac{x}{2}+b_{n}\sin nx\cos\dfrac{x}{2}\right)$
$\displaystyle\quad+\sum_{n=1}^{\infty}\left(\tilde{a}_{n}\cos
nx\sin\dfrac{x}{2}+\tilde{b}_{n}\sin nx\sin\dfrac{x}{2}\right)$
$\displaystyle=\dfrac{a_{0}}{2}\cos\dfrac{x}{2}+\dfrac{\tilde{a}_{0}}{2}\sin\dfrac{x}{2}$
$\displaystyle\quad+\lim_{m\to\infty}\sum_{n=1}^{m}\left(a_{n}\cos
nx\cos\dfrac{x}{2}+\tilde{a}_{n}\cos nx\sin\dfrac{x}{2}+b_{n}\sin
nx\cos\dfrac{x}{2}+\tilde{b}_{n}\sin nx\sin\dfrac{x}{2}\right).$ (2.10)

Now we use the identities

$\displaystyle 2\cos
nx\,\cos(x/2)=\cos\left(nx+x/2\right)+\cos\left(nx-x/2\right),$ $\displaystyle
2\cos nx\,\sin(x/2)=\sin\left(nx+x/2\right)-\sin\left(nx-x/2\right),$
$\displaystyle 2\sin
nx\,\cos(x/2)=\sin\left(nx+x/2\right)+\sin\left(nx-x/2\right),$ $\displaystyle
2\sin nx\,\sin(x/2)=\cos\left(nx-x/2\right)-\cos\left(nx+x/2\right),$

and we obtain the following expression for each fixed $m\in\mathbb{N}$:

$\displaystyle\sum_{n=1}^{m}$ $\displaystyle\left(a_{n}\cos
nx\cos\dfrac{x}{2}+\tilde{a}_{n}\cos nx\sin\dfrac{x}{2}+b_{n}\sin
nx\cos\dfrac{x}{2}+\tilde{b}_{n}\sin nx\sin\dfrac{x}{2}\right)$
$\displaystyle=\sum_{n=1}^{m}\dfrac{a_{n}-\tilde{b}_{n}}{2}\cos\left(nx+\dfrac{x}{2}\right)+\sum_{n=1}^{m}\dfrac{\tilde{a}_{n}+b_{n}}{2}\sin\left(nx+\dfrac{x}{2}\right)$
$\displaystyle\quad+\sum_{n=1}^{m}\dfrac{a_{n}+\tilde{b}_{n}}{2}\cos\left(nx-\dfrac{x}{2}\right)+\sum_{n=1}^{m}\dfrac{b_{n}-\tilde{a}_{n}}{2}\sin\left(nx-\dfrac{x}{2}\right)$
$\displaystyle=\sum_{n=1}^{m}\dfrac{a_{n}-\tilde{b}_{n}}{2}\cos\left(nx+\dfrac{x}{2}\right)+\sum_{n=1}^{m}\dfrac{\tilde{a}_{n}+b_{n}}{2}\sin\left(nx+\dfrac{x}{2}\right)$
$\displaystyle\quad+\dfrac{a_{1}+\tilde{b}_{1}}{2}\cos\dfrac{x}{2}+\dfrac{b_{1}-\tilde{a}_{1}}{2}\sin\dfrac{x}{2}$
$\displaystyle\quad+\sum_{n=1}^{m}\dfrac{a_{n+1}+\tilde{b}_{n+1}}{2}\cos\left(nx+\dfrac{x}{2}\right)+\sum_{n=1}^{m}\dfrac{b_{n+1}-\tilde{a}_{n+1}}{2}\sin\left(nx+\dfrac{x}{2}\right).$

We return to (2.10) and we deduce that for all $x\in[-\pi,\pi]$ we have

$\displaystyle f(x)$
$\displaystyle=\dfrac{a_{0}+a_{1}+\tilde{b}_{1}}{2}\cos\dfrac{x}{2}+\dfrac{\tilde{a}_{0}-\tilde{a}_{1}+b_{1}}{2}\sin\dfrac{x}{2}$
$\displaystyle\qquad+\lim_{m\to\infty}\sum_{n=1}^{m}\left(\dfrac{a_{n}+a_{n+1}-\tilde{b}_{n}+\tilde{b}_{n+1}}{2}\cos\left(nx+\dfrac{x}{2}\right)+\dfrac{\tilde{a}_{n}-\tilde{a}_{n+1}+{b}_{n}+{b}_{n+1}}{2}\sin\left(nx+\dfrac{x}{2}\right)\right).$

Finally, we use the definitions (2.6), (2.7), (2.8) and (2.9) and we arrive at
the desired alternative Fourier series expansion for $f$: for all
$x\in[-\pi,\pi]$ we have

$f(x)=\sum_{n=0}^{\infty}\left(\alpha_{n}\cos\left(nx+\dfrac{x}{2}\right)+\beta_{n}\sin\left(nx+\dfrac{x}{2}\right)\right),$
(2.11)

where

$\alpha_{n}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{f(x)\cos\left(nx+\dfrac{x}{2}\right)\,dx},\,\,\beta_{n}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{f(x)\sin\left(nx+\dfrac{x}{2}\right)\,dx}\quad(n=0,1,2,3,\dots).$

Moreover, the series in (2.11) converges uniformly on $[-\pi,\pi]$ thanks to
uniform convergence of $Sf_{c}$ and $Sf_{s}$ on $[-\pi,\pi]$.

Let us define the antiperiodic Fourier series of $f$ as

$ASf(x)=\sum_{n=0}^{\infty}\left(\alpha_{n}\cos\left(nx+\dfrac{x}{2}\right)+\beta_{n}\sin\left(nx+\dfrac{x}{2}\right)\right).$

We have proven the following result, which is the analogue to Theorem 1.1 for
anti–periodic functions.

###### Theorem 2.1

Let $f:[-\pi,\pi]\longrightarrow\mathbb{R}$ be an antiperiodic function, i.e.
$f(-\pi)=-f(\pi)$.

If $f$ is continuous and has bounded variation on $[-\pi,\pi]$, then the
antiperiodic Fourier series $ASf(x)$ is uniformly convergent on $[-\pi,\pi]$
and $ASf(x)=f(x)$ for all $x\in[-\pi,\pi]$.

For $f(x)=x$, $x\in[-\pi,\pi]$, the antiperiodic Fourier series is

$AS\,f(x)=\dfrac{8}{\pi}\sum_{n=0}^{\infty}\dfrac{(-1)^{n}}{(2n+1)^{2}}\sin\left(nx+\dfrac{x}{2}\right),$

which agrees with $f$ also at $\pm\pi$ and does not exhibit the Gibbs
phenomenon around these points. See Figures 3 and 4.

Figure 3: Observe that $f(\pm\pi)=ASf(\pm\pi)$. Figure 4: No Gibbs phenomenom
around $\pm\pi$.

Let us consider now the general nonperiodic case.

The main idea is that any function can be shifted upwards or downwards to
become antiperiodic, so the antiperiodic Fourier series is useful regardless
of the behaviour of $f$ at the boundary.

Indeed, let $f:[-\pi,\pi]\longrightarrow\mathbb{R}$ be an arbitrary function.
It suffices to take $\gamma\in\mathbb{R}$ so that $F(x)=f(x)-\gamma$ be
antiperiodic, i.e. we define

$F(x)=f(x)-\gamma\quad(x\in[-\pi,\pi])\quad\mbox{for}\quad\gamma=\dfrac{f(-\pi)+f(\pi)}{2},$

and $F$ is antiperiodic.

Now, if $f$ is continuous and has bounded variation on $[-\pi,\pi]$ then so
$F$ is, and then Theorem 2.3 ensures that

$f(x)-\gamma=F(x)=ASF(x)=\sum_{n=0}^{\infty}\left(\alpha_{n}\cos\left(nx+\dfrac{x}{2}\right)+\beta_{n}\sin\left(nx+\dfrac{x}{2}\right)\right),\quad
x\in[-\pi,\pi],$

where

$\alpha_{n}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{(f(x)-\gamma)\cos\left(nx+\dfrac{x}{2}\right)\,dx},\,\,\beta_{n}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{(f(x)-\gamma)\sin\left(nx+\dfrac{x}{2}\right)\,dx}\quad(n=0,1,2,3,\dots).$

Hence,

$f(x)=\gamma+\sum_{n=0}^{\infty}\left(\alpha_{n}\cos\left(nx+\dfrac{x}{2}\right)+\beta_{n}\sin\left(nx+\dfrac{x}{2}\right)\right),\quad
x\in[-\pi,\pi].$

Finally, we put everything in a definition and a theorem. For the reader’s
convenience, we rescale formulas so that our results apply to intervals of the
form $[-L,L]$, $L>0$.

###### Definition 2.2

The antiperiodic Fourier series of a function
$f:[-L,L]\longrightarrow\mathbb{R}$, $L>0$, is

$ASf(x)=\gamma+\sum_{n=0}^{\infty}\left(\alpha_{n}\cos\left(\dfrac{n\pi
x}{L}+\dfrac{\pi x}{2L}\right)+\beta_{n}\sin\left(\dfrac{n\pi x}{L}+\dfrac{\pi
x}{2L}\right)\right),$

where $\gamma=(f(-L)+f(L))/2$ and for $n=0,1,2,3,\dots$

$\alpha_{n}=\dfrac{1}{L}\int_{-L}^{L}{(f(x)-\gamma)\cos\left(\dfrac{n\pi
x}{L}+\dfrac{\pi
x}{2L}\right)\,dx},\,\,\beta_{n}=\dfrac{1}{L}\int_{-L}^{L}{(f(x)-\gamma)\sin\left(\dfrac{n\pi
x}{L}+\dfrac{\pi x}{2L}\right)\,dx},$

provided that the integrals exist.

###### Theorem 2.3

Let $f:[-L,L]\longrightarrow\mathbb{R}$ be continuous and have bounded
variation on $[-L,L]$, $L>0$.

Then the antiperiodic Fourier series $ASf(x)$ introduced in Definition 2.2 is
uniformly convergent on $[-L,L]$ and $ASf(x)=f(x)$ for all $x\in[-L,L]$.

As an example, we consider $f(x)=x^{2}+2x+1$ for $x\in[-1,1]$. Its
antiperiodic Fourier series is

$ASf(x)=2-\dfrac{32}{\pi^{3}}\sum_{n=0}^{\infty}\dfrac{(-1)^{n}}{(2n+1)^{3}}\cos\left(n\pi
x+\dfrac{\pi
x}{2}\right)+\dfrac{16}{\pi^{2}}\sum_{n=0}^{\infty}\dfrac{(-1)^{n}}{(2n+1)^{2}}\sin\left(n\pi
x+\dfrac{\pi x}{2}\right),$

and, according to Theorem 2.3, we have $f(x)=ASf(x)$ for all $x\in[-1,1]$. The
usual Fourier series is

$Sf(x)=\dfrac{4}{3}+\dfrac{4}{\pi^{2}}\sum_{n=0}^{\infty}\dfrac{(-1)^{n}}{n^{2}}\cos\left(n\pi
x\right)-\dfrac{4}{\pi}\sum_{n=0}^{\infty}\dfrac{(-1)^{n}}{n}\sin\left(n\pi
x\right).$

Observe that, unlike the usual Fourier series, the antiperiodic Fourier series
does not exhibit the Gibbs phenomenom at $\pm 1$, see Figures 5 and 6.
Moreover, it is clear from the expressions of $ASf$ and $Sf$, or from Figures
5 and 6, that the antiperiodic Fourier series has a faster rate of convergence
than the usual Fourier series. This is not a surprise, because the
antiperiodic Fourier series uses regular Fourier series for periodic functions
(namely, the functions $f_{c}$ and $f_{s}$ that we introduced at the beginning
of this section) which accelerates convergence.

Figure 5: No Gibbs phenomenom around $\pm 1$ and higher rate of convergence.
Figure 6: Gibbs phenomenom around $\pm 1$ and lower rate of convergence.

We have only adapted Theorem 1.1 to the new setting of antiperiodic Fourier
series, but surely every result on classical Fourier series has its
corresponding analogue for antiperiodic Fourier series. To illustrate this
possibility, and due to the importance of the theorem itself, we show how to
prove Jordan’s convergence test for the antiperiodic Fourier series as a
consequence of the same result for the usual Fourier series. Further theorems
on the Fourier series can be adapted for the antiperiodic Fourier series
through analogous arguments.

###### Theorem 2.4

(Jordan’s convergence test) Let $f\in L^{1}(-L,L)$, $L>0$, and let $\bar{f}$
denote its $2L$–periodic extension to $\mathbb{R}$, i.e. $\bar{f}=f$ a.e. on
$(-L,L)$ and $\bar{f}(x+2L)=\bar{f}(x)$ for a.e. $x\in\mathbb{R}$.

Let $x_{0}\in\mathbb{R}$ be fixed. If $\bar{f}$ is well defined and it has
bounded variation on a neighborhood of $x_{0}$, then the antiperiodic Fourier
series of $f$ converges for $x=x_{0}$ and

$ASf(x_{0})=\dfrac{\bar{f}(x_{0}^{+})+\bar{f}(x_{0}^{-})}{2}.$

Proof. We assume, without loss of generality, that $f$ is antiperiodic (so
$\gamma=0$ in Definition 2.2). Adjusting the ideas by means of which we
deduced the identity (2.11), we can prove that

$ASf(x)=Sf_{c}(x)\cos\dfrac{\pi x}{2L}+Sf_{s}(x)\sin\dfrac{\pi x}{2L},\quad
x\in\mathbb{R},$ (2.12)

where $Sf_{c}$ and $Sf_{s}$ are the usual Fourier series of

$f_{c}(x)=f(x)\cos\dfrac{\pi x}{2L}\quad\mbox{and}\quad
f_{s}(x)=f(x)\sin\dfrac{\pi x}{2L},\quad\mbox{for a.a. $x\in[-L,L]$.}$

The validity of (2.12) depends on the convergence of $Sf_{c}(x)$ and
$Sf_{s}(x)$, which we prove below for $x=x_{0}$.

Let $\bar{f}_{c}$ and $\bar{f}_{s}$ denote the $2L$–periodic extensions of
$f_{c}$ and $f_{s}$, respectively. The assumptions imply that $\bar{f}_{c}$
and $\bar{f}_{s}$ satisfy the conditions in Jordan’s test (see [4, Theorem
8.47]) and therefore $Sf_{c}(x_{0})$ and $Sf_{s}(x_{0})$ are convergent and

$\displaystyle ASf(x_{0})$ $\displaystyle=Sf_{c}(x_{0})\cos\dfrac{\pi
x_{0}}{2L}+Sf_{s}(x_{0})\sin\dfrac{\pi x_{0}}{2L}$
$\displaystyle=\dfrac{\bar{f}_{c}(x_{0}^{+})+\bar{f}_{c}(x_{0}^{-})}{2}\cos\dfrac{\pi
x_{0}}{2L}+\dfrac{\bar{f}_{s}(x_{0}^{+})+\bar{f}_{s}(x_{0}^{-})}{2}\sin\dfrac{\pi
x_{0}}{2L}$ $\displaystyle=\dfrac{\bar{f}(x_{0}^{+})+\bar{f}(x_{0}^{-})}{2}.$

$\hbox to0.0pt{$\sqcap$\hss}\sqcup$

###### Remark 2.5

The following sufficient condition is very useful in applications of the
Jordan’s test: if $\bar{f}$ is differentiable on
$(x_{0}-r,x_{0})\cup(x_{0},x_{0}+r)$ for some $r>0$ and there exist

$\lim_{x\to x_{0}^{\pm}}\bar{f}^{\prime}(x)\in\mathbb{R}\quad\mbox{(they need
not be equal),}$

then $\bar{f}$ has bounded variation on a neighborhood of $x_{0}$.

##  3 Application to PDEs and separation of variables

Obviously, the antiperiodic Fourier series can be used to solve PDE problems
with antiperiodic boundary conditions, but not only these ones.

As a sample, consider the following problem with the heat equation:

$\displaystyle u_{t}-k\,u_{xx}=0,\quad(x,t)\in(-\pi,\pi)\times(0,\infty),$
(3.13) $\displaystyle u(x,0)=f(x),\quad x\in[-\pi,\pi],$ (3.14) $\displaystyle
u(-\pi,t)+u(\pi,t)=2,\quad t\geq 0,$ (3.15) $\displaystyle
u_{x}(-\pi,0)+u_{x}(\pi,0)=0,\quad t\geq 0.$ (3.16)

We assume that $k>0$ and $f:[-\pi,\pi]\longrightarrow\mathbb{R}$ is
continuously differentiable on $[-\pi,\pi]$. We also assume that $f$ is
compatible with the boundary conditions, i.e. $f(-\pi)+f(\pi)=2$ and there
exist $f^{\prime}(\pm\pi)$ and $f^{\prime}(-\pi)+f^{\prime}(\pi)=0$.

This problem is a mathematical model for the evolution of the temperature
$u(x,t)$ at the point $x$ and time $t$ of a thin rod that we identify with the
interval $[-\pi,\pi]$. For physical reasons, we should assume that $f(x)\geq
0$ for all $x\in[-\pi,\pi]$, because $f(x)$ is the temperature of the point
$x$ at the initial time $t=0$.

The boundary conditions can be interpreted as follows. Condition (3.15) means
that the mean value of the temperature at the endpoints is kept constantly
equal to $1$. In turn, condition (3.16) means that the heat flux at the
endpoints is equal in absolute value but it has opposite sign, so we either
have heat entering the rod from both ends at the same rate or leaving the rod
from both ends at the same rate.

One can prove, either by energy methods as in [3] or by Fourier series as in
[2], that (3.13)–(3.16) has a unique solution $u\in{\cal
C}^{1}([-\pi,\pi]\times[0,\infty))\cap{\cal
C}^{2}((-\pi,\pi)\times(0,\infty))$. However, we are more interested in
showing how the previous theory can be used in combination with the method of
separation of variables for the computation of the solution.

Observe that (3.15) is not an homogeneous boundary condition, so we cannot use
the method directly. Clearly, the solution can be obtained as

$u(x,t)=v(x,t)+1,\quad(x,t)\in[-\pi,\pi]\times[0,\infty),$ (3.17)

where $v(x,t)$ is the solution of the following problem with homogeneous
boundary data:

$\displaystyle v_{t}-k\,v_{xx}=0,\quad(x,t)\in(-\pi,\pi)\times(0,\infty),$
(3.18) $\displaystyle v(x,0)=f(x)-1,\quad x\in[-\pi,\pi],$ (3.19)
$\displaystyle v(-\pi,t)+v(\pi,t)=0,\quad t\geq 0,$ (3.20) $\displaystyle
v_{x}(-\pi,0)+v_{x}(\pi,0)=0,\quad t\geq 0.$ (3.21)

Now we use separation of variables to compute $v(x,t)$. As usual, we start by
computing nontrivial solutions of (3.18), (3.20) and (3.21) which have the
form $v(x,t)=X(x)T(t)$. Inserting this $v(x,t)$ in (3.18), (3.20) and (3.21)
leads to finding all possible values of $\lambda\in\mathbb{R}$ for which we
have nontrivial solutions of

$X^{\prime\prime}(x)=\lambda X(x),\quad
X(-\pi)=-X(\pi),\,\,X^{\prime}(-\pi)=-X^{\prime}(\pi),$ (3.22)

and

$T^{\prime}(t)=\lambda T(t),\,\,t\geq 0.$ (3.23)

It is easy to prove that (3.22) has only the zero solution if $\lambda\geq 0$.
For $\lambda<0$, the general solution of the ODE in (3.22) is

$X(x)=A\cos\left(\sqrt{-\lambda}x\right)+B\sin\left(\sqrt{-\lambda}x\right),\quad
A,B\in\mathbb{R},$ (3.24)

and then we impose the boundary conditions in (3.22) to get

$X(-\pi)=-X(\pi)\Leftrightarrow 2A\cos\left(\sqrt{-\lambda}\pi\right)=0,$

and

$X^{\prime}(-\pi)=-X^{\prime}(\pi)\Leftrightarrow
2B\cos\left(\sqrt{-\lambda}\pi\right)=0.$

Therefore, if $\cos\left(\sqrt{-\lambda}\pi\right)\neq 0$, then we have
$A=B=0$, so we only get the zero solution. If, on the other hand, we have
$\cos\left(\sqrt{-\lambda}\pi\right)=0$, then (3.24) solves (3.22) for any
values of $A$ and $B$. Observe that $\sqrt{-\lambda}\pi>0$, so

$\cos\left(\sqrt{-\lambda}\pi\right)=0\Leftrightarrow\sqrt{-\lambda}=n+\dfrac{1}{2}\,\,(n=0,1,2,\dots)\Leftrightarrow\lambda=-\left(n+\dfrac{1}{2}\right)^{2}\,\,(n=0,1,2,\dots),$

and, in each of these cases, we have the following nontrivial solutions of
(3.22):

$X_{n}(x)=\cos\left(nx+\dfrac{x}{2}\right)\quad\mbox{and}\quad\tilde{X}_{n}(x)=\sin\left(nx+\dfrac{x}{2}\right)\,\,(n=0,1,2,\dots).$

Now we solve (3.23) for $\lambda=-(n+1/2)^{2}$ and we get the basic nontrivial
solutions

$T_{n}(t)={\rm e}^{-(n+1/2)^{2}kt},\,\,t\geq 0.$

Finally, we solve (3.18)–(3.21) via superposition principle with the basic
solutions $T_{n}(t)X_{n}(x)$ and $T_{n}(t)\tilde{X}_{n}(x)$ as follows:

$v(x,t)=\sum_{n=0}^{\infty}{\rm
e}^{-(n+1/2)^{2}kt}\left(A_{n}\cos\left(nx+\dfrac{x}{2}\right)+B_{n}\sin\left(nx+\dfrac{x}{2}\right)\
\right),$

for some adequate constants $A_{n},\,B_{n}\in\mathbb{R}$ ($n=0,1,2,\dots$)
such that (3.19) is fulfilled, i.e. for all $x\in[-\pi,\pi]$ we must have

$f(x)-1=v(x,0)=\sum_{n=0}^{\infty}\left(A_{n}\cos\left(nx+\dfrac{x}{2}\right)+B_{n}\sin\left(nx+\dfrac{x}{2}\right)\
\right).$

We deduce from Theorem 2.3 applied to $f(x)-1$ (which is an antiperiodic
function) that we must take

$A_{n}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{(f(x)-1)\cos\left(nx+\dfrac{x}{2}\right)\,dx}$

and

$B_{n}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}{(f(x)-1)\cos\left(nx+\dfrac{x}{2}\right)\,dx}.$

To sum up, according to (3.17) the solution of (3.13)–(3.16) is

$u(x,t)=1+\sum_{n=0}^{\infty}{\rm
e}^{-(n+1/2)^{2}kt}\left(A_{n}\cos\left(nx+\dfrac{x}{2}\right)+B_{n}\sin\left(nx+\dfrac{x}{2}\right)\right).$

In particular, the solution for $k=1$ and $f(x)=(x/\pi)^{2}$ is

$u(x,t)=1-\dfrac{32}{\pi^{3}}\sum_{n=0}^{\infty}\dfrac{(-1)^{n}}{(2n+1)^{3}}{\rm
e}^{-(n+1/2)^{2}t}\cos\left(nx+\dfrac{x}{2}\right),$

which tends to $1$ as $t$ tends to infinity; see Figures 7 and 8.

Figure 7: Tenth partial sum of $u(x,t)$ for some values of $t$. Figure 8:
Tenth partial sum of $u_{x}(x,t)$ for some values of $t$.

###### Remark 3.1

All the calculations and graphs in this paper were done with Maple but many
other types of software (even open–source software) could have been used
instead. Readers are referred to our elementary paper [1], which contains some
useful Sage codes for the Fourier series and their applications.

## References

  * [1] D. Cao Labora, M. Ferreiro Subrido and R. López Pouso, Series de Fourier e introdución ás ecuacións en derivadas parciais (Galician), USC Editora, Esenciais USC, 2023. DOI: 10.15304/9788419679239.
  * [2] R. López Pouso, Fourier method revised to solve partial differential equations and prove uniqueness at one stroke, Real Anal. Exchange 44 (2019), 2, 383–390. DOI: 10.14321/realanalexch.44.2.0383.
  * [3] T. Myint-U and L. Debnath, Linear Partial Differential Equations for Scientists and Engineers, Fourth Edition, Birkhäuser, Boston, 2007.
  * [4] K. R. Stromberg, Introduction to Classical Real Analysis, Wadsworth Inc., Belmont, 1981.