# Strong photon antibunching effect in a double cavity optomechanical system
with intracavity squeezed light

M. Amazioug<EMAIL_ADDRESS>LPTHE, Department of Physics, Faculty of
sciences, Ibn Zohr University, Agadir, Morocco  M. Daoud<EMAIL_ADDRESS>Department of Physics, Faculty of Sciences, University Ibn Tofail, Kénitra,
Morocco  Abdus Salam International Centre for Theoretical Physics, Strada
Costiera, 11, 34151 Trieste, Italy  S. K. Singh<EMAIL_ADDRESS>Graphene and Advanced 2D Materials Research Group (GAMRG), School of
Engineering and Technology, Sunway University, Jalan Universiti, Bandar
Sunway, Petaling Jaya, 47500, Selangor, Malaysia  M. Asjad
<EMAIL_ADDRESS>Department of Applied Mathematics and Sciences, Khalifa
University, Abu Dhabi 127788, UAE

###### Abstract

We study the behaviour of the second-order correlation function in a double
cavity optomechanical system and a degenerate optical parametric amplifier
(OPA) is placed in each cavity. The first cavity is additionally driven by a
weak classical laser field. The occurrence of strong photon antibunching
effect in these two coupled cavities is observed. For suitable values of
optomechanical coupling strength as well as photon hopping process, the system
can exhibit a very strong photon antibunching effect. Our study also shows
that the unconventional photon blockade occurs in both coupling, i.e. the weak
coupling as well as in the strong coupling regimes as compared to the
conventional photon blockade which occurs only in the strong coupling regime.
We get a very strong photon antibunching effect under the unconventional
photon blockade mechanism than the conventional photon blockade mechanism. Our
study can be also used for the generation of single photon in coupled
nonlinear optomechanical systems.

##  I Introduction

Cavity optomechanics is a rapidly developing research area for exploring the
radiation-pressure-mediated interaction between optical and mechanical degrees
of freedom [1, 2]. Due to the nonlinear coupling in between the optical mode
to mechanical oscillations, cavity optomechanical system offers a robust
platform for studying many interesting quantum phenomena such as quantum
entangled states [3, 4, 5, 6, 7, 8, 9, 10, 11, 12], weak force sensing [13,
14], squeezing [15, 16, 17, 18], precision measurements [19], quantum
teleportation [20, 21], optomechanical induced transparency [22, 23, 24, 25]
and optomechanical induced absorption [26, 27, 28], ground state cooling of
macroscopic objects [29, 30, 31, 32], photon and phonon blockade [33, 34, 35,
36]. Photon blockade (PB) is a nonlinear optical effect that suppresses
completely the multiple-photon occupancy in a quantum mode and favours only
the single photon state [37]. The photon blockade mechanism also generate sub-
Poissonian light when the system is driven by a classical light field. So far,
the photon blockade effect is also one of the major experimental schemes to
generate on demand single-photon sources and hence plays a significant role in
present day quantum information technologies [38, 39].

Photon blockade phenomenon has been studied theoretically in various nonlinear
quantum optical systems, e.g., Kerr-type nonlinear cavity [40, 41], cavity
optomechanical systems [33, 42, 43, 44] whereas the experimental works
investigated the trapped atom-cavity system [45] and a photonic crystal cavity
coupled to a single quantum dot [46, 47, 48]. Moreover, we have two wells-
known mechanisms to realize a strong PB effect in any quantum system. The
first one is $(i)$ unconventional photon blockade (UCPB) [49, 50, 51] which is
generated due to the destructive quantum interference between different
quantum transition paths from the ground state to a two-excited state, and the
second one is $(ii)$ conventional photon blockade (CPB) [37, 52, 53] which
depends on the larger nonlinearities to change the energy-level structure of
the system. Both UCPB and CPB mechanisms have been implemented experimentally
in [45, 46, 54, 55, 56] as well as theoretically studied in [57, 58, 59, 60,
61, 62, 63, 64].
Motivated by the above-mentioned works, in this paper, we propose a scheme to
generate a strong photon antibunching effect in a double cavity optomechanical
system. Each cavity also contains a degenerate optical parametric amplifier
(OPA). These two cavities are also spatially separated and coupled through the
single photon hopping process. We investigate photon antibunching effect in
both the weak and the strong coupling regimes through studying the second-
order correlation function in this system. Here, the first cavity is also
driven by a weak classical laser field as shown in Fig 1. We also discuss the
occurrence of the both types of photon blockade effect, i.e. unconventional
photon blockade and conventional photon blockade as well as impact of various
physical parameters for achieving a strong photon antibunching effect.
This paper is organised as follows. In Section 2, we have introduced the model
Hamiltonian of our proposed optomechanical system. In Section 3, we have
obtained the analytical and numerical results related to the equal-time
second-order correlation function to discuss PB effect. In this same Section,
we also give the optimal parameter values to achieve the strong photon
antibunching effect and we discuss the CPB and the UCPB mechanisms in both
weak and strong coupling regimes. We conclude our results in Section 4.

##  II The Model Hamiltonian

We consider an optomechanical system consisting of two Fabry-Pérot cavities
coupled via the single photon hopping process (Fig 1). Each cavity $j(j=1,2)$
has a movable end-mirror $M_{j}$ and also contain inside a degenerate optical
parametric amplifier (OPA) [65]. In our scheme, first cavity is also driven by
a weak classical laser field with amplitude $\mathcal{E}$ and frequency
$\omega_{L}$. The mass and the frequency of the $j^{th}$ movable mirror
$M_{j}(j=1,2)$ are respectively denoted by $m_{j}$ and $\omega_{M_{j}}$. The
mirror $M_{j}$ is coupled to the photons inside the cavity $j$ via radiation
pressure. The coupling rate is
$g_{j}=\frac{\omega_{a_{j}}}{L_{j}}\sqrt{\frac{\hbar}{m_{j}\omega{{}_{M}}_{j}}}$
with $L_{j}$ denoting the length of the $j{th}$ cavity [1]. The annihilation
and creation operators of the $j{th}$ cavity mode are denoted by $\hat{a}_{j}$
and $\hat{a}_{j}^{\dagger}$ with $[\hat{a}_{j},\hat{a}_{j}^{\dagger}]=1$
($j=1,2$). The laser-cavity detuning is $\Delta_{j}=\omega_{L}-\omega_{a_{j}}$
where $\omega_{a_{j}}$ is the resonant frequencies of the cavity mode $jth$
cavity. The coupling strength of the photon hopping process is denoted by $J$.
The total Hamiltonian describing the system in the rotating frame
approximation is given by ($\hbar=1$)

$\widehat{\mathcal{H}}=\widehat{\mathcal{H}}_{0}+\widehat{\mathcal{H}}_{PH},$
(1)

where

$\widehat{\mathcal{H}}_{PH}=J(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1})$
(2)

and the Free Hamiltonian $\widehat{\mathcal{H}}_{0}$ is the sum of three terms
given as

$\widehat{\mathcal{H}}_{0}=\widehat{\mathcal{H}}_{free}+\widehat{\mathcal{H}}_{om}+\widehat{\mathcal{H}}_{drive},$
(3)

where the hermitians operators $\mathcal{H}_{free}$, $\mathcal{H}_{om}$ and
$\mathcal{H}_{drive}$ written as

$\displaystyle\widehat{\mathcal{H}}_{free}=\sum_{j=1}^{2}[-\Delta_{j}\hat{a}^{\dagger}_{j}\hat{a}_{j}+\omega_{M_{j}}\hat{b}^{\dagger}_{j}\hat{b}_{j}],$
(4)
$\displaystyle\widehat{\mathcal{H}}_{om}=-\sum_{j=1}^{2}g_{j}\hat{a}^{\dagger}_{j}\hat{a}_{j}(\hat{b}^{\dagger}_{j}+\hat{b}_{j}),$
(5)
$\displaystyle\widehat{\mathcal{H}}_{drive}=\mathcal{E}e^{i\phi}\hat{a}^{\dagger}_{1}+\mathcal{E}e^{-\i\phi}\hat{a}_{1},$
(6)

where $\mathcal{E}=\sqrt{\frac{2\kappa_{1}\wp}{\hbar\omega{{}_{L}}}}$ and
$\phi$ are respectively the amplitude and the phase of the input coherent
laser field. The quantity $\wp$ denotes the drive pump power of the input
field. The cavity decay rate of the $jth$ cavity is $\kappa_{j}$. The
annihilation and creation operators of the phonons mode of the $j{th}$ movable
mirrors are represented as $\hat{b}_{j}$ and $\hat{b}^{\dagger}_{j}$ and
satisfy $[\hat{b}_{j},\hat{b}_{j}^{\dagger}]=1$ ($j=1,2$). We recall that the
tunable photon statistics in parametrically amplified photonic molecules were
studied in Ref. [57] without optomechanical interaction. In addition to
optomechanical interaction we take into account the degenerate optical
parametric amplifier. This coupling is described by

$\widehat{\mathcal{H}}_{OPA}=\sum_{j=1}^{2}\left(i\lambda_{j}(\hat{a}^{\dagger}_{j})^{2}e^{i\theta}-i\lambda_{j}\hat{a}_{j}^{2}e^{-i\theta}\right),$
(7)

where the nonlinear gain and the phase of the field driving the OPA inside the
$jth$ cavity are respectively given by $\lambda_{j}$ and $\theta$. The total
Hamiltonian is now given by the sum of the Hamiltonian $\widehat{\mathcal{H}}$
(Eq.(1)) and the interaction term given by

$\widehat{\mathcal{H}}_{tot}=\widehat{\mathcal{H}}+\widehat{\mathcal{H}}_{OPA}.$
(8)

The degenerate OPA is pumped by another laser field at frequency $2\omega_{L}$
[66, 67].

Figure 1: Schematic of a double cavity optomechanical system.

Using the following unitary transformation
$\widehat{\mathcal{U}}=\exp[\widehat{\mathcal{S}}]$ with
$\widehat{\mathcal{S}}=\sum^{2}_{j=1}\frac{g_{j}}{\omega_{m_{j}}}\hat{a}^{\dagger}_{j}\hat{a}_{j}(\hat{b}^{\dagger}_{j}-\hat{b}_{j})$;
($j=1,2$), the Hamiltonian $\widehat{\mathcal{H}}_{tot}$ transform under this
transformation as
$\widehat{\mathcal{U}}^{\dagger}\widehat{\mathcal{H}}_{tot}\widehat{\mathcal{U}}=\widehat{\mathcal{H}}_{1}$
To derive this equation, we have used the Baker-Campbell-Hausdorff formula
together with the commutation values of the operators $\hat{a}^{\dagger}$,
$\hat{a}$, $\hat{b}^{\dagger}$ and $\hat{b}$. The unitary transformation
$\widehat{\mathcal{U}}$ decouples the two mechanical and the optical modes in
the special case of weak optomechanical coupling, i.e.
$g_{j}/\omega_{m_{j}}\ll 1$ ($j=1,2$) [58]. In fact, as we are interested by
the photon statistic in the system, the Hamiltonian can be written as

$\displaystyle\widehat{\mathcal{H}}_{1}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{2}[-\Delta_{j}\hat{a}^{\dagger}_{j}\hat{a}_{j}-\mu_{j}(\hat{a}^{\dagger}_{j}\hat{a}_{j})^{2}+i\lambda(\hat{a}^{\dagger}_{j})^{2}e^{i\theta}-i\lambda\hat{a}_{j}^{2}e^{-i\theta}]$
(9) $\displaystyle+$
$\displaystyle\mathcal{E}e^{i\phi}\hat{a}^{\dagger}_{1}+\mathcal{E}e^{-i\phi}\hat{a}_{1}+J(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1}),$

where $\mu_{j}=g_{j}^{2}/\omega_{m_{j}}$ is the Kerr-type nonlinear strength.

##  III Photon Statistics

In this section, we focus on the analytical solution of the non-Hermitian
Schrödinger equation as well as numerical solution using Master equation. In
addition, we discuss the generation of a strong photon antibunching effect in
the weak and the strong coupling regimes. The non-Hermitian Hamiltonian is
directly written by adding phenomenologically the imaginary decay terms as

$\widehat{\mathcal{H}}_{2}=\widehat{\mathcal{H}}_{1}-i\frac{\kappa_{1}}{2}\hat{a}^{\dagger}_{1}\hat{a}_{1}-i\frac{\kappa_{2}}{2}\hat{a}^{\dagger}_{2}\hat{a}_{2}.$
(10)

The analytical expression of the correlation function can be obtained by
solving the Schrödinger equation in the weak driving condition
($\mathcal{E}\ll\kappa_{1}$), i.e., $i\frac{\partial|\psi(t)\rangle}{\partial
t}=\mathcal{H}_{2}|\psi(t)\rangle$, where $|\psi(t)\rangle$ is the state of
the system. The evolution space can be limited in the low-excitation subspace,
up to 2, i.e., $\mathcal{B}=\\{|n_{1},n_{2}\rangle/n_{1}+n_{2}\leq 2\\}$. The
state of the system with the bare-state bases are

$|\psi(t)\rangle=\sum_{n_{1},n_{2}}^{n_{1}+n_{2}\leq
2}\mathcal{C}_{n_{1}n_{2}}(t)|n_{1},n_{2}\rangle,$ (11)

where $\mathcal{C}_{n_{1}n_{2}}(t)$ is the probability amplitude of the bare
state $|n_{1},n_{2}\rangle$ with $n_{1}$ represents the photons being in the
cavity $(1)$ and $n_{2}$ represents the photons being in the cavity $(2)$. For
two identical cavities, $\kappa_{1}=\kappa_{2}=\kappa$, $m_{1}=m_{2}=m$,
$\omega_{m_{1}}=\omega_{m_{2}}=\omega_{m}$,
$\omega_{a_{1}}=\omega_{a_{2}}=\omega_{a}$, $L_{1}=L_{2}=L$,
$\lambda_{1}=\lambda_{2}=\lambda$, $\Delta_{1}=\Delta_{2}=\Delta$ and
$g_{1}=g_{2}=g$. Under the weak driving condition, the probabilities
amplitudes satisfy the relation $|\mathcal{C}_{00}|\simeq
1\gg|\mathcal{C}_{01}|,|\mathcal{C}_{10}|\gg|\mathcal{C}_{11}|,|\mathcal{C}_{02}|,|\mathcal{C}_{20}|$.
To simplify our purpose we consider $\theta=\phi=0$. The schrödinger equation
leads to the following set of linear differential equations for the
probability amplitudes $\mathcal{C}_{n_{1}n_{2}}(t)$

$i\frac{\partial\mathcal{C}_{00}}{\partial
t}=\mathcal{E}\mathcal{C}_{10}-i\lambda(\mathcal{C}_{02}+\mathcal{C}_{20}),$
(12) $i\frac{\partial\mathcal{C}_{01}}{\partial
t}=\mathcal{E}\mathcal{C}_{00}-\Lambda\mathcal{C}_{01}+\sqrt{2}\mathcal{E}\mathcal{C}_{02}-J\mathcal{C}_{10},$
(13) $i\frac{\partial\mathcal{C}_{10}}{\partial
t}=\mathcal{E}\mathcal{C}_{00}-\Lambda\mathcal{C}_{10}+\sqrt{2}\mathcal{E}\mathcal{C}_{20}-J\mathcal{C}_{01},$
(14) $i\frac{\partial\mathcal{C}_{11}}{\partial
t}=-2\Lambda\mathcal{C}_{11}+\mathcal{E}\mathcal{C}_{01}+\sqrt{2}J\mathcal{C}_{02}-\sqrt{2}J\mathcal{C}_{20},$
(15) $i\frac{\partial\mathcal{C}_{02}}{\partial
t}=-2\Gamma\mathcal{C}_{02}-\sqrt{2}J\mathcal{C}_{11}+\i\sqrt{2}\lambda\mathcal{C}_{00},$
(16) $i\frac{\partial\mathcal{C}_{20}}{\partial
t}=-2\Gamma\mathcal{C}_{20}+\sqrt{2}\mathcal{E}\mathcal{C}_{10}-\sqrt{2}J\mathcal{C}_{11}+i\sqrt{2}\lambda\mathcal{C}_{00},$
(17)

where $\Lambda=\Delta+i\frac{\kappa}{2}-\mu$ and
$\Gamma=\Delta+i\frac{\kappa}{2}-2\mu$. Eqs.(12)-(17) can be solved
analytically to obtain the dynamical state. Moreover, the steady-state result
can be obtained by solving $\partial\mathcal{C}_{n_{1}n_{2}}/\partial t=0$,
which can be simplified using some appropriate approximations as for instance
ignoring those higher-order terms in the case of weak coupling interaction.
The coefficients associated with one-photon ($n_{1}+n_{2}=1$) state are given
by

$\mathcal{C}_{01}=\frac{J\mathcal{E}}{\Lambda^{2}-J^{2}},$ (18)
$\mathcal{C}_{10}=\frac{\Lambda\mathcal{E}}{\Lambda^{2}-J^{2}},$ (19)

and the coefficients associated with two-photon ($n_{1}+n_{2}=2$) states write
as

$\mathcal{C}_{11}=\frac{J\left(-\mathcal{E}^{2}\Gamma-2iJ^{2}\lambda+\mathcal{E}^{2}\Lambda+2i\lambda\Lambda^{2}\right)}{2(\Gamma\Lambda-J^{2})(\Lambda^{2}-J^{2})},$
(20)
$\mathcal{C}_{02}=\frac{J^{2}\mathcal{E}^{2}\Gamma+J^{2}\mathcal{E}^{2}\Lambda-2iJ^{2}\lambda\Gamma\Lambda+2i\lambda\Gamma\Lambda^{3}}{2\sqrt{2}\Gamma(\Gamma\Lambda-J^{2})(\Lambda^{2}-J^{2})},$
(21)
$\mathcal{C}_{20}=\frac{J^{2}\mathcal{E}^{2}\Gamma-J^{2}\mathcal{E}^{2}\Lambda-i2J^{2}\lambda\Gamma\Lambda+2\mathcal{E}^{2}\Gamma\Lambda^{2}+i2\lambda\Gamma\Lambda^{3}}{2\sqrt{2}\Gamma(\Gamma\Lambda-J^{2})(\Lambda^{2}-J^{2})}.$
(22)

The equal-time second-order correlation functions defined by
$g_{j}^{(2)}(0)=\langle\hat{a}^{\dagger}_{j}\hat{a}^{\dagger}_{j}\hat{a}_{j}\hat{a}_{j}\rangle/\langle\hat{a}^{\dagger}_{j}\hat{a}_{j}\rangle^{2}$
($j=1,2$). It describes the probability to observe two photons in the $jth$
cavity at the same time. Using the results obtained here, are gets

$g_{1}^{(2)}(0)=\frac{2|\mathcal{C}_{20}|^{2}}{\overline{n}^{2}_{1}},\quad\,\quad
g_{2}^{(2)}(0)=\frac{2|\mathcal{C}_{02}|^{2}}{\overline{n}^{2}_{2}},$ (23)

where
$\overline{n}_{1}=|\mathcal{C}_{10}|^{2}+|\mathcal{C}_{11}|^{2}+2|\mathcal{C}_{20}|^{2}\simeq|\mathcal{C}_{10}|^{2}$
and
$\overline{n}_{2}=|\mathcal{C}_{01}|^{2}+|\mathcal{C}_{11}|^{2}+2|\mathcal{C}_{02}|^{2}\simeq|\mathcal{C}_{01}|^{2}$
represent the average photon occupations. The condition $g_{j}^{(2)}(0)>1$
corresponds to the photon bunching effect whereas $g_{j}^{(2)}(0)<1$ leads to
the photon antibunching effect and is characterized by the sub-Poissonian
photon statistics [33, 34, 35]. Furthermore, the strong photon antibunching
effect holds in the cavity (1) when $g_{1}^{(2)}(0)=0$, i.e.,
$\mathcal{C}_{20}=0$. On the other hand, the strong photon antibunching effect
can be realized in the cavity (2) when $g_{2}^{(2)}(0)=0$, i.e.,
$\mathcal{C}_{02}=0$. The Master equation approach which allows us to
numerically calculate has the following expression

$\frac{d\rho}{dt}=-i[\widehat{\mathcal{H}_{1}},\rho]+\widehat{\mathcal{L}}_{\hat{a}_{j}}(\rho),$
(24)

where
$\widehat{\mathcal{L}}_{a_{j}}(\rho)=\frac{\kappa}{2}(2\hat{a}_{j}\rho\hat{a}^{+}_{j}-\hat{a}^{\dagger}_{j}\hat{a}_{j}\rho-\rho\hat{a}^{\dagger}_{j}\hat{a}_{j})$
and $\widehat{\mathcal{H}_{1}}$ is the Hamiltonian given by Eq.(9).

###  III.1 Weak Coupling Regime

In this subsection, we investigate the evolution of the equal-time second-
order correlation function $g_{j}^{(2)}(0)$ ($j=1;2$). We discuss the
realization of the strong photon antibunching effect in weak coupling regime
($J<\kappa$ and $g\ll\omega_{m}$). The achieved photon blockade effect belongs
to the UCPB (destructive quantum interference).

The optimal parameter pairs $\lambda$ (the gain) and $\Delta$ (the cavity
laser detuning) can be obtained together with the other fixed parameters by
solving the equation $g_{j}^{(2)}(0)=0$ ($j=1;2$). Here we also employ the
parameters value considered in Ref. [58] : $J=0.95\kappa$,
$\omega_{m}=2\pi\times 75\times 10^{6}$ Hz, $g=0.042\omega_{m}$,
$\mathcal{E}=0.02\kappa$ and $\kappa=2\pi\times 0.15\times 10^{6}$ Hz.

The solution in the cavity (1) is $\\{\\{\Delta^{(1)}_{opt}=-0.73\times
10^{-4}\omega_{m},\lambda^{(1)}_{opt}=0.93\times
10^{-6}\omega_{m}\\},\\{\Delta^{(1)}_{opt}=0.15\times
10^{-2}\omega_{m},\lambda^{(1)}_{opt}=-0.20\times
10^{-6}\omega_{m}\\},\\{\Delta^{(1)}_{opt}=33\times
10^{-4}\omega_{m}~{}\text{Hz},\lambda^{(1)}_{opt}=1.47\times
10^{-6}\omega_{m}\\}\\}$. Moreover, the real solution of the optimal parameter
pairs value in the cavity (2) is $\\{\\{\Delta^{(2)}_{opt}=-7.4\times
10^{-4}\omega_{m},\lambda^{(2)}_{opt}=0.33\times
10^{-6}\omega_{m}\\},\\{\Delta^{(2)}_{opt}=0.21\times
10^{-2}\omega_{m},\lambda^{(2)}_{opt}=-0.80\times
10^{-6}\omega_{m}\\},\\{\Delta^{(2)}_{opt}=0.47\times
10^{-2}\omega_{m}~{}\text{Hz},\lambda^{(2)}_{opt}=0.40\times
10^{-6}\omega_{m}\\}\\}$.

Figure 2: Plot of the analytical (solid line) and numerical (dot) results in a
double cavities of the equal-time second-order correlation function
$g_{(}j)^{(2)}(0)$ ($j=1,2$) versus the cavity-laser detuning
$\Delta/\omega_{m}$ (we use $\Delta\to-\Delta$ in this figure) for different
values of the parameter $\lambda$ with $J=0.95\kappa$ and $g=0.042\omega_{m}$.
$\lambda=0.93\times 10^{-6}\omega_{m}$ with $\Delta=0.73\times
10^{-4}\omega_{m}$ in figure (a) in the cavity (1), and $\lambda=0.4\times
10^{-6}\omega_{m}$ with $\Delta=-0.47\times 10^{-2}\omega_{m}$ in figure (b)
in the cavity (2). The solid horizontal line delimits the region under which
represents the photon antibunching effect in the cavity.

We plot in Fig 2 the equal-time second-order correlation function
$g_{1}^{(2)}(0)$ as a function of the cavity-laser detuning $\Delta$ for
different values of $\lambda$ by using analytic and numerical solutions. We
note that the analytic and the numerical solution is obtained by solving
master equation (Eq.24) are the same as shown in Figs 2 (a) and (b) which
means that our results are correct. We remark that a strong photon
antibunching effect occurs when we use the optimal values of parameter pairs
$\\{\Delta,\lambda\\}$ listed above, i.e., $g_{j}^{(2)}(0)$ ($j=1,2$) is much
smaller than unity ($g_{j}^{(2)}(0)\ll 1$) at $\Delta^{(1)}_{opt}=-0.73\times
10^{-4}\omega_{m}$ with $\lambda^{(1)}_{opt}=0.93\times 10^{-6}\omega_{m}$ in
Fig 2 (a) and $\Delta^{(2)}_{opt}=0.47\times 10^{-2}\omega_{m}$ with
$\lambda^{(2)}_{opt}=0.4\times 10^{-6}\omega_{m}$ in Fig 2 (b). Moreover, when
the parameter $\lambda$ does not take the optimal value $\lambda_{opt}$, the
correlation function $g^{(2)}(0)$ cannot vanishes or tends to zero. We can
explain the generation of a strong photon antibunching effect in the cavity
(1) by the destructive quantum interference generated by different two photon
between excitation schemes. The two-photon excited state $|0,2\rangle$ can be
completely suppressed due to the ideal destructive quantum interference
between in three different ways in [57]: $(i)$ the direct excitation
$|0,0\rangle\xrightarrow[]{\lambda}|2,0\rangle$ by a degenerate optical
parametric amplifier interaction with the gain $\lambda$, $(ii)$ the direct
excitation
$|0,0\rangle\xrightarrow[]{\mathcal{E}}|1,0\rangle\xrightarrow[]{\mathcal{E}}|2,0\rangle$by
the driving field with the amplitude $\mathcal{E}$, and $(iii)$ the
tunnelling-mediated transition
$|1,0\rangle\overset{J}{\leftrightarrow}|0,1\rangle\xrightarrow[]{\mathcal{E}}(|1,1\rangle\xrightarrow[]{\text{J}}|0,2\rangle)\xrightarrow{\text{J}}|2,0\rangle$.
From Fig 2(b) the strong photon antibunching effects in the cavity (2) can be
understand by the destructive quantum interference between directs the two
paths : $(i)$ the direct excitation
$|0,0\rangle\xrightarrow[]{\lambda}|0,2\rangle$ by the degenerate optical
parametric amplifier interaction with the gain $\lambda$, and $(ii)$ the
tunnelling-mediated transition
$|1,0\rangle\overset{J}{\leftrightarrow}|0,1\rangle\xrightarrow[]{\mathcal{E}}(|1,1\rangle\xrightarrow[]{\text{J}}|2,0\rangle)\xrightarrow[]{\text{J}}|0,2\rangle$.

According to the above analysis and discussions, we find that the strong
photon antibunching effect which occurs in the weak coupling regime
($J<\kappa$ and $g\ll\omega_{m}$) belongs to the class of unconventional
photon blockade mechanism (the destructive quantum interference) [58].

Figure 3: (a) Plot of the equal-time second-order correlation function
$g_{1}^{(2)}(0)$ versus the cavity-laser detuning $\Delta$ for various values
of the optomechanical coupling strength $g$ with
$\lambda^{(1)}_{opt}=0.93\times 10^{-6}\omega_{m}$ Hz and $J=0.95\kappa$. (b)
Plot of the equal-time second-order correlation function $g_{1}^{(2)}(0)$ in
the weak coupling region versus the cavity-laser detuning $\Delta$ for various
values of the photon hopping coupling strength $J$ with
$\lambda^{(1)}_{opt}=0.93\times 10^{-6}\omega_{m}$ and $g=0.042\omega_{m}$ Hz.
The dashed horizontal line delimits the region under which represents the
photon antibunching effect in the cavity (1).

We plot in Fig 3(a) the equal-time second-order correlation function
$g_{1}^{(2)}(0)$ versus the cavity-laser detuning $\Delta/\omega_{m}$ for
various values of the optomechanical coupling strength $g$. This figure shows
that $g_{1}^{(2)}(0)$ is very small when $g=0.042\omega_{m}$, i.e, strong
photon antibunching effect occurs when $g=0.042\omega_{m}$. We remark that a
strong photon antibunching effect is achieved for the exact value of
$\Delta=-34304.2$ Hz as it is precisely predicted by the optimal parameters
when $g=0.042\omega_{m}$ with $\lambda=437.053$ Hz. We notice that the equal-
time second-order correlation function $g_{1}^{(2)}(0)$ is greater than 1 (the
photon bunching effect) when $g=0$ with
$\Delta\in[-400000~{}\text{Hz},300000~{}\text{Hz}]$. This means that the
achieved photon antibunching effect is related to the optomechanical coupling
$g$.
We plot in Fig 3(b) the equal-time second-order correlation function
$g_{1}^{(2)}(0)$ versus the cavity-laser detuning for different values of the
coupling $J$. This figure exhibits a strong photon antibunching effect for the
value $\Delta^{(1)}_{opt}=1.54229\times 10^{6}$ Hz. This exactly the value
expected via the parameter optimization when $J=0.95$ Hz with
$\lambda^{(1)}_{opt}=692.709$ Hz. The equal-time second-order correlation
function $g_{1}^{(2)}(0)$ is less than 1 when $J=0$ (the photon antibunching
effect) for a wide range value of $\Delta$ as reported in Fig 3(b). This can
be explained by the two-photon excitation path of the tunnelling-mediated
transition, i.e.,
$|1,0\rangle\overset{J}{\leftrightarrow}|0,1\rangle\xrightarrow[]{\mathcal{E}}|1,1\rangle\overset{J}{\leftrightarrow}|2,0\rangle$.
This means the coupling $J$ influences the occurrence of the photon
antibunching effect.

###  III.2 Strong Coupling Regime

In this subsection, we will discuss both conventional and unconventional
photon blockade mechanisms in the strong coupling regime. Moreover, the
conventional photon blockade phenomenon would become obvious because the
energy-level splitting induced by the strong coupling causes a large
transition detuning between single-photon and two-photon states. Hence to
discuss the conventional photon blockade in terms of the energy spectrum of
different excitations, we consider the Hamiltonian of the system without the
laser driving term, i.e.,
$\widehat{\mathcal{H}}_{ND}=\sum_{j=1}^{2}[-\Delta_{j}\hat{a}^{\dagger}_{j}\hat{a}_{j}-\mu_{j}(\hat{a}^{\dagger}_{j}\hat{a}_{j})^{2}+i\lambda(\hat{a}^{\dagger}_{j})^{2}e^{i\theta}-i\lambda\hat{a}_{j}^{2}e^{-i\theta}]+J(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1})$.
Eigenvalues of the single excitation $|0,1\rangle$ and $|1,0\rangle$ are
$\epsilon_{1\pm}=J\pm\Delta\mp\mu$. The locations of CPB can be obtained using
the theory of conventional blockade mechanism (resonant transition between the
zero and single excitations) [58]

$\Delta_{+}=\mu+J\quad;\quad\Delta_{-}=\mu-J.$ (25)

Here, we consider the following values [58] : photon hopping coupling
$J=8\kappa$, $\omega_{m}=2\pi\times 75\times 10^{6}$ Hz, $g=0.2\omega_{m}$,
$\mathcal{E}=0.02\kappa$ and $\kappa=2\pi\times 0.15\times 10^{6}$ Hz. The
parameter optimization of the pair of the parameter $\\{\Delta,\lambda\\}$ in
the cavity (1) gives the following results $\\{\\{\Delta^{(1)}_{opt}=2.4\times
10^{-2}\omega_{m},\lambda^{(1)}_{opt}=1.1\times
10^{-6}\omega_{m}\\},\\{\Delta^{(1)}_{opt}=4.0\times
10^{-2}\omega_{m},\lambda^{(1)}_{opt}=-0.8\times
10^{-6}\omega_{m}\\},\\{\Delta^{(1)}_{opt}=6.0\times
10^{-2}\omega_{m},\lambda^{(1)}_{opt}=1.5\times
10^{-6}\omega_{m}\\},\\{\Delta^{(1)}_{opt}=8.0\times
10^{-2}\omega_{m},\lambda^{(1)}_{opt}=-0.12\times
10^{-6}\omega_{m}\\},\\{\Delta^{(1)}_{opt}=8.2\times
10^{-2}\omega_{m},\lambda^{(1)}_{opt}=-0.02\times 10^{-6}\omega_{m}\\}\\}$.
Moreover, in the cavity (2) the optimal parameter pairs are
$\\{\\{\Delta^{(2)}_{opt}=2.4\times
10^{-2}\omega_{m},\lambda^{(2)}_{opt}=0.51\times
10^{-6}\omega_{m}\\},\\{\Delta^{(2)}_{opt}=4.0\times
10^{-2}\omega_{m},\lambda^{(2)}_{opt}=-0.80\times
10^{-6}\omega_{m}\\},\\{\Delta^{(2)}_{opt}=5.6\times
10^{-2}\omega_{m},\lambda^{(2)}_{opt}=0.12\times
10^{-6}\omega_{m}\\},\\{\Delta^{(2)}_{opt}=5.9\times
10^{-2}\omega_{m},\lambda^{(2)}_{opt}=0.01\times 10^{-6}\omega_{m}\\}\\}$.

Figure 4: Plot of the analytic (solid line) and numerical (dot) results in a
double cavities of the equal-time second-order correlation function
$g_{j}^{(2)}(0)$ ($j=1,2$) in the strong coupling regime versus the cavity-
laser detuning $\Delta/\omega_{m}$ (we use $\Delta\to-\Delta$ in this figure)
for different values of the gain $\lambda$, (a) $\lambda^{(1)}_{opt}=1.1\times
10^{-6}\omega_{m}$ and (b) $\lambda^{(1)}_{opt}=0.01\times 10^{-6}\omega_{m}$,
where $J=8\kappa$ and $g=0.2\omega_{m}$. The solid horizontal line delimits
the region under which represents the photon antibunching effect in the cavity
(1) and (2).

We plot in Fig 4 the equal-time second-order correlation function
$g_{j}^{(2)}(0)$ ($j=1,2$) versus the detuning $\Delta/\omega_{m}$ for various
values of the gain $\lambda$. We also make a remark here that the analytical
and the numerical solution is obtained by solving master equation (Eq.24) are
the same as shown in this figure. The photon blockade locations are obtained
from the Fig 4(a) for the values $\Delta=2.4\times 10^{-2}\omega_{m}$ and
$\Delta=\Delta_{+}=\mu+J$. The small dips located is associated the CPB
mechanism and the others dips are associated to the UCPB mechanism. We notice
that the strong photon antibunching effect obtained under the UCPB mechanism
is strong than one obtained under the CPB mechanism. We remark that a strong
antibunching effect is realized when $g_{1}^{(2)}(0)$ takes small values
($g_{j}^{(2)}(0)\ll 1$) ($j=1,2$). This can be explained by the unconventional
photon blockade mechanism (the complete destructive quantum interference
between different paths of two-photon excitation in the cavity (1) and (2)).

Figure 5: (a) Plot of the equal-time second-order correlation function
$g_{1}^{(2)}(0)$ versus the gain $\lambda$ for various values of the
optomechanical coupling $g$ with $J=8\kappa$ and
$\Delta=\Delta^{(1)}_{opt}=2.4\times 10^{-2}\omega_{m}$. (b) Plot of the
equal-time second-order correlation function $g_{1}^{(2)}(0)$ versus the
cavity-laser detuning $\Delta$ for various values of the photon hopping
coupling strength $J$ with $\lambda^{(1)}_{opt}=1.1\times 10^{-6}\omega_{m}$
and $g=0.2\omega_{m}$. The dashed horizontal line delimits the region under
which represents the photon antibunching effect in the cavity (1).

We plot in Fig 5(a) the equal-time second-order correlation function
$g_{1}^{(2)}(0)$ versus the cavity-laser detuning $\Delta$ for various values
of the optomechanical coupling $g$. This figure shows that $g_{1}^{(2)}(0)$
takes small values when $g=0.2\omega_{m}$ in this case a strong photon
antibunching effect is achieved in the cavity (1). A strong photon blockade
effect occurs for $\Delta=2.4\times 10^{-2}\omega_{m}$ as it is predicted by
the parameters optimization when $g=0.2\omega_{m}$ with $\lambda=1.1\times
10^{-6}\omega_{m}$. We remark that when $g=0$ the equal-time second-order
correlation function became greater than 1 ($g_{1}^{(2)}(0)>1$) i.e., the
photon bunching effect for large values of $\lambda$. This means that the
optomechanical coupling influences the generation of the photon antibunching.
We plot in Fig 5(b) the equal-time second-order correlation function
$g^{(2)}(0)$ versus the cavity-laser detuning for different values of a photon
hopping coupling $J$. We remark that a strong photon blockade effect occurs
for various values of $J$ and $\Delta$ as shown in Fig 5(b). For example when
$J=8\kappa$ a strong photon antibunching effect is achieved for
$\Delta=2.4\times 10^{-2}\omega_{m}$. In agreement with the parameters
optimization when $J=8\kappa$ with $\lambda=1.1\times 10^{-6}\omega_{m}$. And
this shows the role of the photon hopping coupling $J$ in the generation of a
strong photon antibunching effect.

##  IV Conclusion

We have discussed a strong photon blockade effect through the analytical and
numerical evaluation of the second-order correlation function in a double-
cavity optomechanical system. Here the two optical cavities are also coupled
through the single photon hopping process. A degenerate optical parametric
amplifier is placed inside each cavity. The first cavity is also driven by a
weak classical laser field. We have obtained the conditions for strong photon
antibunching effect in each optical cavity for which the second-order
correlation function satisfy $g^{(2)}(0)\ll 1$. We have also analysed the
photon blockade mechanisms in the weak as well as the strong coupling regimes
for the conventional and unconventional photon blockade mechanisms. We have
shown that the UCPB is achieved even for the weak coupling regime ($J<\kappa$
and $g\ll\omega_{m}$), whereas CPB occurs only in the strong coupling regime.
We have discussed in details the effect of the optomechanical coupling
strength and the photon hopping coupling on the generation of strong photon
antibunching effects. Our present study on the single-photon blockade and its
generation in coupled optomechanical systems can be of significant interest
for the various applications in quantum information processing and quantum
communication.

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To make the observation more tangible, take a generic “model” of
discrimination. There are two populations of agents who characterized by their
_productivity types_ and their productivity-irrelevant identity types. An
agent’s productivity type is not observed by the decision-maker, who instead
observes some imperfect _productivity signal_ before making a _decision_ that
is relevant for that agent’s outcome. In this setup, an econometrician might
be interested in measuring discrimination according to two different
perspectives: (i) an _ex-post_ perspective: for a given signal realization
observed by the decision-maker, does their decision vary with the agent’s
identity?; (ii) an _ex-ante_ perspective: for a given productivity type,
averaging across their possible signal realizations, how does an agent’s
expected outcome vary with their identity?

To measure (i), the econometrician would need to observe the productivity-
signals that are seen by the decision-maker before choosing the agent’s
outcome. And to measure (ii), the econometrician would need to know agents’
true productivity types – which are not observed by the decision maker. The
main issue addressed by the literature briefly reviewed below is that often
one or both of these objects are often not observable to the econometrician.
In section 7.1, I discuss audit and correspondence studies which measure
discrimination from the ex-post perspective in experiments, where the signal
observed by the decision-maker is controlled by the experimenter.

A criticism of audit and correspondence studies is that it is impossible to
distinguish whether their provided measures of ex-post discrimination reflect
true biases of the decision maker or the fact that accurate productivity
assessments based on observed signals may indeed vary with agent’s identities
(statistical discrimination). In section 7.2 I briefly review the literature
on outcome-based measures of discrimination, focusing mainly on a few recent
developments. Unlike measurements provided by audit and correspondence
studies, these outcome-based measurements can be attributed to underlying
biases of the decision maker, or at times decomposed into bias and
“statistical discrimination” components.

Finally, in sections 7.3 and 7.4 I discuss two other recently proposed
decompositions of discrimination measures. Bohren, Haggag, Imas, and Pope
(2022) propose an identification procedure to distinguish discrimination due
to biases in the decision-maker’s preferences and biases in their (inaccurate)
beliefs. Bohren, Hull and Imas (2022) propose a decomposition of a total
measure of discrimination into a _direct_ component, attributable to the
decision maker, and a _systemic_ component, resulting from cumulative biases
in the agent’s previous interactions with the “economic system.”

### 7.1. Audit and Correspondence Studies

Audit and correspondence studies are two experimental methods used to measure
discrimination. Their methods and produced empirical evidence are thoroughly
surveyed in Bertrand and Duflo (2017); see also Baert (2018), and Lippens,
Vermeiren, and Baert (2022).

In an audit study in the labor market context, two job candidates (the
auditors) belonging two distinct identity groups (say, one white and one black
auditor) are chosen so as to be as equal as possible in terms of all
characteristics potentially relevant their sought jobs, such as their age and
education. The two individuals are then sent to to apply for the same job
(perform the audit), and they are trained to behave equally during the job
application process. Their performance at the application process is measured;
for example, the experimenters will register whether the candidates were
called for interviews, whether they made it to further stages of the search
process, and whether they were offered the job. These performances are then
compared across the individuals, and the difference in their outcomes is then
attributed to discrimination by the potential employer, because all other
possible explanations for the differential treatment were controlled for by
the matching and training of the auditors.

This type of study has been conducted in many contexts beyond the labor
market, such as vehicle purchases, the housing market, and mortgage
applications. In each context, the idea is that the experimenter is able to
observe (and control) the signal observed by the decision maker; and ensure
that this signal is the same across a matched pair of auditors, who differ
from each other only in terms of their identities.

Such studies have been criticized by, for example, Heckman and Siegelman
(1993) and Heckman (1998), who question the experimenter’s ability to create
such matched pairs of auditors. They are skeptical that an experimenter would
be able to know what would be “all the relevant characteristics” for auditors
to be matched on; and remark on how unlikely an experimenter would be to find
pairs of people who would exactly match on all these characteristics. Further,
Heckman and Siegelman (1993) question whether any bias is introduced into the
study through the process by which the auditors are trained to perform the
audit.

Correspondence studies were designed so as to address some of the mentioned
weaknesses of audit studies. In a correspondence study, there are no real job
seekers, and therefore no matching of pairs of auditors. Rather, fictional job
seekers are created, whose resumés are designed to be equal in every way
except for the name of the applicant, which is chosen to have strong identity
associations that vary across the fictional job seekers. Once the fictional
resumés are sent to existing job vacancies, the experimenter observes the
rates at which jobs contact the applicants to schedule interviews. Any
differences in such callback rates across applicants’ identities are then
measured and attributed to discrimination by the potential employers.

Two recent papers on the experimental detection of discrimination, Kline and
Walters (2021) and Kline, Rose, and Walters (2022), further develop the
experimental design and develop econometric tools to measure the
discriminatory behavior not of “the labor market” as a whole, but rather of
particular employers.

In all mentioned experimental designs, the experimenter measures how the
decision-maker’s behavior varies with the agent’s identity, fixing the
relevant signal observed by the decision maker. This concept corresponds to
the _ex-post_ discrimination notion introduced above. Ex-post discrimination
is in itself an object of interest, however the econometrician cannot with
certainty attribute observed ex-post discrimination to employers’ biases
against minority identities. Rather, fixing a particular observed signal,
discriminatory behavior across identities may be (partly or fully) explained
by statistical discrimination, as in the Phelpsian model introduced in section
2.2.

### 7.2. Distinguishing Bias and Statistical Discrimination

An alternative approach to measuring discrimination is the use of marginal
outcome tests. For concreteness, I introduce these tests in the language of
vehicle searches by police officers looking for contraband — this context is
studied in Knowles, Persico and Todd (2001), among others. In this scenario,
an agent’s underlying type is the probability that they are carrying
contraband in their vehicle. The police officer sees relevant characteristics
of the vehicle (a signal about whether it carries contraband), as well as the
identity of the driver, and decides whether to search the vehicle. If the
vehicle is searched, the presence or absence of contraband is revealed;
otherwise, the agent’s true type remains unobserved.

The signal observed by the police officer gives them some idea of how likely
the vehicle is to be carrying contraband. A rational police officer decides to
search a vehicle if the likelihood of contraband is large enough (above some
threshold). We say that the police officer is biased if they use different
threshold rules to decide whether to search a vehicle, depending on the
identity of the driver. The econometrician, who wants to assess the police
officer’s bias, cannot observe the true underlying types of all agents, or the
signals seen by the police officer at the time of the search decision.
However, Becker (1957) notes that it suffices for the econometrician to
compare the incidence of contraband in “the marginal vehicle” stopped by the
police officer for drivers with different identities.

Intuitively, suppose we knew that a set of drivers of identity $A$ produced
signals that made the officer just about indifferent between stopping and not
stopping the car, and suppose they were stopped. And suppose there is a
similar such population of drivers with identity $B$. Now say the
econometrician observes that the average incidence of contraband in the
population $A$ of such marginally stopped vehicles is larger than that in
population $B$. The test then indicates that the officer uses a lower
threshold to stop vehicles with drivers with identity $B$, thereby revealing
that the officer is biased against population $B$. Specifically, the officer
is “more willing” to stop vehicles with identity-$B$ drivers, revealing the
officer’s preference for stopping such vehicles.

While the intuition above is clear, there are many hurdles with actually
implementing marginal outcome tests, the main one being how to identify who
the “marginal stopped driver” is in each population. Some contributions such
as Knowles, Persico and Todd (2001) and Persico and Todd (2006) specify
complete behavioral models for both the drivers and the police officers, in
which the marginal outcome can be identified by the average outcome amongst
stopped drivers (the _infra-marginality problem_ is circumvented). More
recently, Arnold, Dobbie, and Yang (2018) — see also Canay, Mogstad, and
Mountjoy (2020) and Hull (2021) — propose new econometric tools to identify
bias “at the margin.”

Unlike audit and correspondence studies, marginal outcome tests capture the
bias component of discrimination, but not statistical discrimination. This is
the case because marginal outcome tests are leveraging the econometrician’s
(imperfect) observation of the agents’ true types (whether the vehicle carried
contraband or not), which are revealed after the vehicle’s search is conducts.
Contrastingly, remember that in audit and correspondence studies, the
econometrician observes the agent’s signal, but never their true underlying
type.

A recent contribution to this literature, in Arnold, Dobbie, and Hull (2022),
shows that in the context of bail decisions (a context often explored for
marginal outcome tests), the quasi-random assignment of judges to cases can be
used to measure a broader notion of discrimination – including not only biases
in judges preferences and beliefs, but also statistical discrimination. The
authors use the quasi-random assignment of judges to purge the omitted
variable bias which is introduced by the fact that “true types” are not
observed for all the agents (in the vehicle search example, “true types” are
only observed for vehicles that are stopped). By doing so, Arnold, Dobbie, and
Hull (2022) are able to measure differences in treatment of agents with
different identities, but the same underlying “true type” — this notion
corresponds to ex-ante discrimination, as introduced in the beginning of this
section. Arnold, Dobbie, and Hull (2022) are then able to decompose which
portion of the ex-ante discrimination is due to judges’ biases and which
portion is due to statistical discrimination.

A related exercise is pursued by Deb and Renou (2022). In their setup, the
econometrician is assumed to observe _not_ the agents’ true types or their
signals, but rather the _average_ true type in each population. In their
model, there are two populations $A$ and $B$ of workers with the same average
productivity type. There is an employer who sees some signal about an agent’s
productivity and forms a posterior mean $p\in\mathbb{R}$ about their type. The
employer then offers the agent some wage $w_{A}(p)$ if they belong to
population $A$ and wage $w_{B}(p)$ if they belong to population $B$, where
$w_{A}$ and $w_{B}$ are two strictly increasing functions. We say the employer
is biased if the function $w_{A}$ is different from the function $w_{B}$, and
unbiased otherwise.

The econometrician in their paper observes realized wage distributions $G_{A}$
and $G_{B}$ for the respective populations, and wishes to identify whether
differences in these distributions are due to populations $A$ and $B$ having
different signaling technologies (statistical discrimination) or to the
employer being biased. Deb and Renou (2022) find that if neither wage
distribution $G_{A}$ or $G_{B}$ dominates the other in the first order
stochastic, then differences between the distributions can be “justified” by
statistical discrimination. That is, there exist two signaling technologies,
for populations $A$ and $B$ respectively, such that the wage distributions can
be generated by those signals and identical wage functions $w_{A}=w_{B}$. If
instead one of the wage distributions first-order dominates the other, then it
must be that $w_{A}\neq w_{B}$, and therefore the employer is biased.

Martin and Marx (2022) also propose a “robust” test for bias, in that they
characterize situations where differential outcomes across two populations
_cannot_ be explained by differences in signals. Martin and Marx (2022)
consider the decision-maker makes a binary decision (to hire or not to hire
the agent, say), and the agents’ true type is revealed to the econometrician
after either decision is made. Martin and Marx (2022) show that if the average
productivity amongst un-hired agents in population $A$ is larger than the
average productivity amid hired agents in population $B$, then this is a
robust indicator of the decision-maker’s bias.

### 7.3. Measuring Biased Beliefs

Much of the literature reviewed above aims to not only measure discrimination,
but also to decompose it into “statistical” discrimination and discrimination
due to the decision-maker’s bias. A recent literature – Bohren, Imas,
Rosenberg (2019), Bohren, Haggag, Imas, Pope (2022), Hull (2021) — notes that
it is sometimes possible to further distinguish whether the bias component of
discrimination is due to biases in the decision-maker’s preferences or to
inaccuracies in the decision-maker’s beliefs.424242A related decomposition is
pursued by Cunningham and de Quidt (2022), who note that a decision maker’s
discriminatory behavior may reflect an _explicit_ or an _implicit_ bias in
their preferences. In their leading example, they consider a hiring manager
who always chooses to hire a woman over a man with the same qualifications,
but always chooses to hire a man over a woman if their qualifications differ.
They interpret such observation as evidencing an explict bias of the manager
for women — whom they hire when the woman and the men are directly comparable
— but an implicit bias for men, who are hired when the two candidates’
qualifications are not directly comparable.

Bohren, Haggag, Imas, Pope (2022) propose an experiment in a labor market
setting where they are able to disentangle preference biases and belief
biases. Agents (“workers”) with different identities perform some task in a
lab, and their performances in that task are observed – these are their “true
productivity types.” Other individuals, denoted “employers,” are shown
profiles of workers including their identities, but excluding their
performances, and report the maximum “wages” they would offer these workers
(employers are incentivized in the lab so as to truthfully report these
maximum wages). One of their observations is that American workers are
systematically offered lower wages than Indian workers; even though
performances do not systematically differ between American and Indian workers.

Next, the authors elicit the beliefs of the employers in the experiment, and
find that employers mistakenly predicted that American workers perform much
worse than their Indian counterparts. Therefore, the employers’ bias can be
mostly attributed to their inaccurate beliefs, rather than to biases in their
preferences. Bohren, Haggag, Imas, Pope (2022) perform one final treatment,
where they reveal the true performance distributions of both populations to
the employers, and note that after this observation employers significantly
change their wage offers in the direction consistent with correcting their
beliefs. This result highlights that employers’ inaccurate beliefs are due to
lack of information, as opposed to “motivated biased beliefs” fueled by
employers’ underlying preference biases.

### 7.4. Systemic vs. Direct Discrimination

A recent paper by Bohren, Hull and Imas (2022) comments on the various
measures of discrimination in the economic literature, and compare them to the
distinction between _direct_ and _systemic_ discrimination in the broader
social sciences literature. Specifically, they refer to a large body of work
in other social sciences that views discrimination as a systemic phenomenon,
viewing disparate group-based treatment as a “cumulative outcome of both
direct and indirect interactions between outcomes and evaluations across
different stages and domains.”

Bohren, Hull and Imas (2022) propose a simple motivating example to
disentangle the definitions of direct and systemic discrimination. Imagine a
hiring process consisting of two stages. In a first stage, a recruiter meets a
potential hire and issues an evaluation. This recruiter is biased and
consistently evaluates female candidates less favorably than equally able male
candidates. At a second stage, a hiring manager at the firm observes the
recruiter’s evaluation and makes a hiring decision.

Suppose an econometrician is only able to observe the second stage of the
hiring process: they see the evaluations of all potential hires and the hiring
decisions. Analyzing that data, the econometrician may find no statistically
significant differences between hiring probabilities for female candidates and
male candidates, conditional on the recruiter’s evaluation. At that point, the
econometrician may conclude that the hiring process is not discriminatory, as
male and female candidates with equal observable outcomes are equally likely
to be hired. Bohren, Hull and Imas (2022) instead suggest that the
econometrician should conclude that “there is no direct discrimination by the
hiring manager.”

However, this statistical finding does not rule out _systemic discrimination_
in the hiring process as a whole. Indeed, we know that the recruiter
consistently under-evaluates women and so, if the econometrician could observe
the hiring process as a whole, they would find that women are less likely to
be hired, when compared with equally able men. Systemic discrimination takes
place in this example because women and men are treated equally in the second
period, when instead the “overall” non-discriminatory outcome should have men
being discriminated against at that stage.

To connect this to the language developed in this survey, note that the
measurement of _direct discrimination_ corresponds to the measurement of _ex-
post discrimination_ introduced in the beginning of this section. That is,
direct discrimination occurs when the decision-maker treats two individuals
“with the same signals” differently, based on their identities. Contrastingly,
Bohren, Hull and Imas’ (2022) notion of _total discrimination_ corresponds to
my introduced notion of _ex-ante discrimination_ , that is, the differential
treatment of individuals who have “the same underlying types,” based on their
identities. Finally, systemic discrimination corresponds to the difference
between total and expected direct discrimination, or equivalently, the
difference between ex-ante and expected ex-post discrimination.

Beyond noting that it is possible to decompose discrimination into such direct
and systematic components, an important contribution by Bohren, Hull and Imas
(2022) is highlighting that the systemic component can be regarded as
discrimination in previous stages of agents’ lives, or in other spheres. This
interpretation establishes an important connection between the economic
literature and the broader view in social sciences that discrimination is a
cumulative process.

The significance of this systemic discrimination notion indicates a need for
developing dynamic theories of discrimination, that studies discrimination as
a cumulative process throughout people’s careers and in different spheres of
their economic lives. Most of the dynamic theories presented in this survey
already hint at the importance of that systemic component. For example, take
the case of spiralling discrimination in Bardhi, Guo and Strulovici (2020). If
an econometrician were to see data about the interaction between the firm and
workers with different identities at any point in time (except the very first
instant), they would conclude that no discrimination is taking place. Indeed,
any two workers with the same “track record” would be treated equally by the
firm. However, as the authors show, discrimination is a cumulative process,
which spirals out of any very small differences between workers at an early
career stage. In their model, discrimination is a “systemic” process, in the
sense proposed by Bohren, Hull and Imas (2022).

Other models of discrimination arising from learning traps in social learning
dynamics – Che, Kim and Zhong (2019), Komiyama and Noda (2021) and Li, Raymond
and Bergman (2020) – also highlight the systemic nature of discrimination. The
models in Bohren, Imas and Rosenberg (2019) and Fryer (2007), described in
section 4.1, are especially close to the Bohren, Hull and Imas’ (2022)
motivating example, and also view discrimination as arising from a “systemic”
failure of social learning. While the idea that discrimination is not only
direct, but also possibly systemic, is not entirely new in the literature,
Bohren, Hull and Imas (2022) contribute by proposing this useful
classification (and relating it to research in other social sciences), and
more importantly by proposing methods to measure discrimination and decompose
it into its direct and systemic components.

Finally, the observation that the measurement of discrimination may differ
depending on the econometrician’s vantage point hints at the importance of
theories that highlight and contrast their predictions in terms of these
different notions of discrimination. In one recent example, Onuchic ⓡ Ray
(2022) make such predictions in the context of discrimination in credit
assignment for teamwork. In their model, conjecture an equilibrium where women
systematically receive less credit for joint work than their male counterparts
and suppose we want to know what is the relative likelihood that a woman would
attain a certain career outcome (a “target posterior”), relative to a man.
They find that, if this career outcome is sufficiently ambitious (say,
receiving a prestigious award), then women would be relatively more likely to
reach that outcome – the same is true if the target posterior is sufficiently
low, while men are relatively more likely to attain moderate career outcomes.
In that context, if an econometrician were to evaluate the relative likelihood
that members of either group win an award and were to find that women are
statistically more likely to win it, then it would be wrong to view this
result as a “proof” that no discrimination takes place, or that men are the
ones subject to discrimination. According to their model, this outcome is
possibly consistent with a discriminatory equilibrium where women are the dis-
favored group.

## 8\. Conclusion

This article surveys recent contributions to theories of discrimination. It
attempts to contribute to the understanding of the literature by making the
following points:

1\. The expansion of the literature on information design provides some new
language and baseline results with which to regard traditional models of
(statistical) discrimination. For example, Chambers and Echenique (2021) use
this new framework to more fully characterize market conditions that foster
Phelpsian statistical discrimination.

2\. Within the broad framework of statistical discrimination, the analysis of
specific features of learning and signaling environments – for example,
dynamic social learning, learning by myopic algorithms, learning with rational
inattention, or signaling in teams – generates novel empirical predictions and
policy implications.

3\. Recent contributions have expanded traditional models to consider
“behavioral agents,” with behavioral learning heuristics or misspecified
beliefs about their environments. These theories differ crucially from
frameworks with fully rational agents.

4\. Thus far, most of the literature on economic theories of discrimination
has been classified as either taste-based or statistical discrimination. In
contrast, work in sociology often treats the discrimination phenomenon as
arising from neither personal prejudice nor rational guesses based on group
characteristics, but rather from discriminatory institutions (Small and Pager,
2020). While economic theories do not fully describe the sociological
perspective on institutional discrimination, sections 5 and 6 surveys theories
that come closer to that approach – including theories of discriminatory
social norms and the design of discriminatory institutions.

5\. Some new work on the measurement of discrimination has also helped
contextualize the economic perspective on “theories of discrimination” within
the broader scope of social science research on discrimination. In section 7,
amongst other comments about the measurement of discrimination, this survey
highlights Bohren, Hull and Imas’ (2022) categorization of discrimination as
direct and systemic discrimination, and connects it to other definitions of
discrimination in the economic literature.

This paper is still very much a work in progress. I am sure there is a lot of
recent literature I do not know, or failed to fit in the framework of this
survey. Please, email me with any leads and tips on how to make it better and
more complete.

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