# Decoherence control of a single-photon optomechanical system in phase-
sensitive reservoirs.

Vaibhav N Prakash1 and Aranya B Bhattacherjee2,3 1School of Physical Sciences,
Jawaharlal Nehru University, New Delhi-110067, India  2Department of Physics,
Birla Institute of Technology and Science, Pilani, Hyderabad Campus, Hyderabad
- 500078, India  3Department of Physics, School of Applied Sciences,
University of Science and Technology, District -Ri Bhoi, 793101, Meghalaya
India.

###### Abstract

With technological advancements to achieve strong single-photon optomechanical
coupling, it becomes necessary to explore open quantum optomechanical systems
in strong single-photon coupling limit. Although some attention has been given
to strongly coupled optomechanical systems in thermal environment, there
behaviour under the influence of non-thermal environments, remains an open
question. We report on the effect of squeezed vacuum/thermal reservoir on the
decoherence of cavity photon Fock states in an optomechanical system in the
strong single-photon coupling regime. We find that an increase in single-
photon coupling strength leads to increase in decoherence under squeezed
thermal reservoir compared to a thermal reservoir, except for a small range of
values of the reservoir phase. We hence also show that having a squeezed
reservoir gives greater control over system decoherence in the strong single-
photon coupling regime, as expected. We perform simulations in the dressed-
state master equation instead of the standard master equation. Our study might
be important in understanding information processing in strongly coupled
photonic systems and quantum memories.

Keywords

Dressed-state master equation, decoherence, optomechanical nonlinearity,
quantum photonics, single-photon strong coupling, squeezed reservoir.

##  I Introduction

There has been a growing interest in using optical photons for quantum
information processing, communications and sensor technologies. Quite a few
components in photonic systems proposed to be used for information processing
for example, single-photon sources(Luc, ; Ina, ; Luk, ; Ama, ), generation of
nonclassical states of light(Gui, ; Lin, ), interferometers(Alb, ; Hyo, ),
sensors(Tia, ; Tor, ; Feng, ; Ali, ), are based on photon-phonon interactions.
In the quantum regime nonlinear photon-photon interactions mediated by
phonons, give rise to phenomena like photon blockade(Hai, ; Dong1, ; kai, ;
Dong2, ; Rabl, ), arising from sub-Poissonian statistics. Because of the
importance of optomechanical non-linearity to future quantum devices, it
becomes necessary to understand the dynamics of an open quantum optomechanical
system in the single-photon strong coupling regime. It has been previously
reported(Hu, ) how vacuum and thermal environments affect the dynamics of an
optomechanical system in the strong single-photon coupling regime. We further
this argument to squeezed vacuum and thermal reservoirs especially since
squeezed environments have been known to provide additional control over the
dynamics of open quantum systems(Hou, ; Ami, ; Rog, ; Chun, ).
In this article we take a look at the effect of single-photon strong coupling
on the time evolution of the system density operator under the influence of
squeezed vacuum and squeezed thermal reservoir(Gon, ). In particular, we show
that the rate of decoherence increases in the strong to ultra-strong single-
photon coupling regime, under squeezed thermal reservoir, except for certain
values of the reservoir parameters wherein there is a drastic decrease in
decoherence rate. We perform these simulations with the help of the Dressed-
state master equation(DSME) which has been previously used in the case of
strongly coupled systems(Pon, ; Cho, ). We also give modifications to the
Standard Master Equation(SME) in the interaction picture under the influence
of a squeezed thermal reservoir. We compare our findings with those from
thermal reservoir.

##  II Non-quadratic Hamiltonian

Dressed states are eigenstates of the system hamiltonian including the
interaction term. In the strong laser drive limit dressed states, formed by
the superposition of photon-phonon product states, give rise to splitting of
the normal modes of the non-interacting hamiltonian(Mar, ). On the other hand
in the weak laser drive, the strong optomechanical coupling leads to the
dressing of cavity photons by mechanical excitations. The dressed states are
then approximate eigenstates of the total hamiltonian(including the
interaction part) in the photon number basis. This can be demonstrated by a
unitary transformation of the system hamiltionian. In this article we consider
an optmechanical system consisting of a single Fabry-Perot cavity with one end
of the system movable under radiation pressure of cavity photons and a simple
harmonic potential. The system hamiltonian in the frame rotating with the
laser frequency $\omega_{L}$, is given by($\hbar=1$);

$\displaystyle\hat{H}=-\Delta\,\hat{a}^{\dagger}\hat{a}+\omega_{m}\,\hat{b}^{\dagger}\hat{b}-g_{0}\,\hat{a}^{\dagger}\hat{a}(\hat{b}+\hat{b}^{\dagger})+\mathcal{E}(\hat{a}^{\dagger}-\hat{a}),$
(1)

where laser detuning $\Delta$=$\omega_{L}$ \- $\omega_{c}$, $\omega_{c}$ and
$\omega_{m}$ are the cavity frequency and natural frequency of the mechanical
oscillator, respectively, $\omega_{L}$ s the laser drive frequency, while
$\hat{a}(\hat{b})$ are annihilation operators of cavity(mechanical) mode.
$g_{0}$ is the single photon optomechanical coupling and $\mathcal{E}$ is the
energy of the laser drive.

##  III Polaron Transform

The presence of the laser electric field term in Hamiltonian in Equation (1)
renders the Hamiltonian non-Hermitian but we nevertheless work in the very
weak drive limit i.e $\mathcal{E}\rightarrow$ 0\. In this limit we can use
polaron transformation to find the approximate eigenstates of the hamiltonian
in the photon number basis.

$\displaystyle\lim_{\mathcal{E}\to
0}\hat{H^{\prime}}=\hat{U}\hat{H}\hat{U}^{\dagger}\approx-\Delta\,\hat{a}^{\dagger}a+\omega_{m}\,\hat{b}^{\dagger}\hat{b}-\frac{g_{0}^{2}}{\omega_{m}}(\hat{a}^{\dagger}a)^{2},$
(2)

where $U=e^{-S}$ is a unitary operator and
$S=(\hat{a}^{\dagger}\hat{a})\frac{g_{0}}{\omega_{m}}(\hat{b}^{\dagger}-\hat{b})$(Nun,
). The total energy of the system described by hamiltonian in Equation (2) is,
$E^{\prime}=-n\Delta+m\omega_{m}-n^{2}g_{0}^{2}/\omega_{m}$, where
$n\equiv\left\langle\hat{a}^{\dagger}\hat{a}\right\rangle$ and
$m\equiv\left\langle\hat{b}^{\dagger}\hat{b}\right\rangle$ are the expectation
values of photon and phonon number operators respectively. In terms of product
states, the eigenstate can be written as;

$\displaystyle\ket{\Psi}=\ket{k}\otimes\ket{l_{(n)}}=\ket{k}\otimes
e^{\hat{N_{c}}\beta_{0}(\hat{b}^{\dagger}-\hat{b})}\ket{l},$ (3)

where $\beta_{0}=g_{0}/\omega_{m}$, $\hat{N_{c}}=\hat{a}^{\dagger}\hat{a}$, is
the photon number operator, $\ket{l_{(n)}}$ is a mechanical state $\ket{l}$
displaced by an amount proportional to the cavity photon number $n\beta_{0}$
and $\ket{k}$ is the photon Fock state. In this regard a dressed state occurs
when a photon number dependent mechanical state gets excited by the cavity
photon itself. This also gives rise to second order non-linearity i.e photon-
photon coupling mediated by phonons. In the single-photon regime, we consider
only a few photons and hence $n$ is small. To increase non-linearity, we must
have strong to ultastrong optomechanical coupling $g_{0}$. This type of
nonlinear coupling is also known as quantum/intrinsic non-linearity. The total
energy of the system hamiltonian in Equation (1) with $\mathcal{E}\rightarrow
0$, can now be written in this dressed state picture as;

$\displaystyle
E^{\prime}=\bra{k,l_{(n)}}\hat{H}\ket{k,l_{(n)}}=\bra{k,l}e^{-\hat{N_{c}}\beta_{0}(\hat{b}^{\dagger}-\hat{b})}\hat{H}e^{\hat{N_{c}}\beta_{0}(\hat{b}^{\dagger}-\hat{b})}\ket{k,l}=\bra{k,l}\hat{H^{\prime}}\ket{k,l},$
(4)

where $\hat{H^{\prime}}$ is given in Equation (2). Without loss of generality
we assume that photon-phonon states ($\ket{k,l}$) are represented by tensor
product of individual Fock states.

##  IV Dressed-State Master Equation

While studying the dynamics of an open quantum system, a common approach is to
use the Standard Master Equation(SME) under the Born-Markov approximation. As
it was shown in Section III, this requires us to define the eigenmodes of the
system. For weak single photon coupling we can ignore the photon-phonon
coupling and approximately define the eigenmodes of the system to derive the
SME. For a strong single photon coupling however the inter-modal interactions
cannot be ignored. This gives rise to another type of master equation which is
known as the Dressed-State Master Equation(DSME)(Hu, ; Aga, ; Tah, ). In DSME
we work within the photon-phonon eigenbasis discussed in Section III. It is
also assumed that the bath attached to the mechanical resonator couples to
both degrees of freedom but the other bath couples only to the optical
resonator. In the interaction picture and considering ohmic reservoirs, the
DSME is given as;

$\displaystyle\frac{d\tilde{\rho}(t)}{dt}=\gamma_{m}(n_{th}+1)\mathcal{D}[\hat{b}-\beta_{0}\hat{N_{c}}]\tilde{\rho}(t)+\gamma_{m}n_{th}\mathcal{D}[\hat{b}^{\dagger}-\beta_{0}\hat{N_{c}}]\tilde{\rho}(t)+\kappa\mathcal{D}[a]\tilde{\rho}(t),$
(5)

where $\tilde{\rho}(t)$ is the density matrix operator in the interaction
picture, $n_{th}$ is the thermal photon occupancy number at temperature T,
$\gamma_{m}$ and $\kappa$ are the mechanical and optical decay rates
respectively, $\hat{N_{c}}$ is the cavity photon number operator and
$\mathcal{D}[\hat{o}]\tilde{\rho}(t)$ is a Lindbladian superoperator which is
the dissipator defined in Section IV and redefined here as;

$\displaystyle\mathcal{D}[\hat{o}]\tilde{\rho}(t)=\hat{o}\tilde{\rho}(t)\hat{o}^{\dagger}-\frac{1}{2}\anticommutator{\hat{o}^{\dagger}\hat{o}}{\tilde{\rho}(t)},$
(6)

where $\hat{o}$ is the system eigen-operator. In the interaction picture and
in dressed eigenbasis,

$\begin{split}\hat{b}(t)&=e^{-i\,\omega_{m}\,t}\hat{b}+\beta_{0}\hat{N}_{c}.\end{split}$
(7)

This is different from the SME approach, where
$\hat{b}(t)=e^{-i\,\omega_{m}\,t}$. By introducing a ”book-keeping” term
$\beta_{0}\hat{N}_{c}$ we re-write Equation (7) as,

$\hat{b}(t)=e^{-i\,\omega_{m}\,t}(\hat{b}-\beta_{0}\hat{N}_{c})+\beta_{0}\hat{N}_{c}.$
(8)

The ”book-keeping” term is added just to be consistent with the interpretation
of $\hat{b}(t)$ in SME and DSME. In SME Equation (8) can be written as,

$\begin{split}\hat{b}(t)=e^{-i\,\omega_{m}\,t}(\hat{b}-\beta_{0}\hat{N}_{c})+e^{-i\,\omega_{m}\,t}\beta_{0}\hat{N}_{c}&=e^{-i\omega_{m}t}\hat{b}\end{split}$
(9)

Comparing Equation (8) and Equation (9) it is easy to see that the DSME gives
terms rotating with different frequencies while in the SME no such distinction
occurs. It is then straightforward to show that for the dressed
basis($\beta_{0}\sim 1$) the SME approach gives rise to additional terms in
the interaction picture;

$\begin{split}\frac{d\tilde{\rho}(t)}{dt}&=\gamma_{m}(n_{th}+1)\mathcal{D}[\hat{b}-\beta_{0}\hat{N_{c}}]\tilde{\rho}(t)+\gamma_{m}n_{th}\mathcal{D}[\hat{b}^{\dagger}-\beta_{0}\hat{N_{c}}]\tilde{\rho}(t)\\\
&+\kappa\mathcal{D}[a]\tilde{\rho}(t)+\gamma_{m}(2n_{th}+1)\beta_{0}^{2}\mathcal{D}[\hat{N_{c}}]\tilde{\rho}(t).\end{split}$
(10)

The last term in Equation (10) induces dephasing since it doesn’t affect any
photon/phonon population but nonetheless contributes to the decoherence of
$\tilde{\rho}(t)$. For $\beta_{0}<<1$ we can ignore terms involving
$\beta_{0}$ in Equation (10) and get back SME for the unperturbed eigenbasis.
For cavity optomechanical systems another important approximation in the DSME
is that the cavity frequency should be much larger than the mechanical
frequency ($\omega_{c}>>\omega_{m}$), which is true in the case of optical
photons. For microwave photons this approximation is no longer valid and we
have to use the Global Master Equation(Tah, ).

##  V DSME for a squeezed thermal reservoir

A non-unitary evolution of the state density matrix in open quantum systems is
due to its interaction with an external reservoir. A common technique to
understand system-reservoir interaction is within the Born-Markov
approximation where the system-reservoir coupling is considered to be weak and
the reservoir correlation functions are ”memory-less” i.e the time-scale at
which the reservoir components interact with each other is much smaller than
that of the system. Most experimental studies are conducted with vacuum or
thermal reservoirs. Non-thermal or squeezed thermal reservoirs, for example,
are difficult to construct but show promising results in different theoretical
studies. They have been shown to reduce decay rates of the quantum coherence
leading to preservation of non-classical effects(Ser, ; Kim, ). Squeezed
thermal reservoirs have also been shown to be useful as a resource for nano-
mechanical engines to go beyond the Carnot limit and improve power and
efficiency of quantum heat engines(Jan, ; Aba, ). Hence the effect of squeezed
reservoirs on the decoherence of individual states(coherent or Fock) of the
system is well understood but the literature lacks knowledge on how squeezed
reservoirs affect the decoherence of strongly coupled systems. It is thus
important to understand their role when they are coupled to an optomechanical
system which is in the strong photon-phonon coupling regime.

###  V.1 Squeezed thermal reservoir

A squeezed thermal reservoir can be realised by applying signal to a nanobeam
resonator in which the signal noise is such that it is suppressed in one of
the quadratures and amplified in another(Jan, ). Phase-space distribution of
such a signal shows a squeezed thermal state. Thermal noise from a laser
subjected onto a parametric amplifier can also give rise to squeezed thermal
noise. A proposal to engineer squeezed vacuum reservoir using interference
between spontaneous decays of two different excited states in atomic systems
was given by Zoller et.al.(Zol, ) but such a method would require multiple
ground states.
Squeezed reservoirs are examples of a non-stationary reservoirs(Hei, ) as
opposed to thermal reservoirs. This is because the correlation functions of
the electric field operator of the radiation modes are no longer homogeneous
in time i.e. they show time dependence. As a result the master equation for
optical and mechanical operators contains additional terms in the rotating
wave approximation. Under squeezed thermal reservoir the SME in the
unperturbed eigenbasis($\beta_{0}<<1$) is given as;

$\begin{split}\frac{d\tilde{\rho}(t)}{dt}&=\gamma_{m}(N_{eff}+1)\mathcal{D}[\hat{b}]\tilde{\rho}(t)+\gamma_{m}(N_{eff})\mathcal{D}[\hat{b}^{\dagger}]\tilde{\rho}(t)\\\
&+\kappa(N+1)\mathcal{D}[\hat{a}]\tilde{\rho}(t)+\kappa(N)\mathcal{D}[\hat{a}^{\dagger}]\tilde{\rho}(t)\\\
&-\gamma_{m}M_{eff}^{*}\hat{b}\tilde{\rho}(t)\hat{b}-\gamma_{m}M_{eff}\hat{b}^{\dagger}\tilde{\rho}(t)\hat{b}^{\dagger}-\kappa
M^{*}\hat{a}\tilde{\rho}(t)\hat{a}-\kappa
M\hat{a}^{\dagger}\tilde{\rho}(t)\hat{a}^{\dagger},\end{split}$ (11)

where $N_{eff}=n_{th}(\cosh^{2}{r}+\sinh^{2}{r})+\sinh^{2}{r}$,
$M_{eff}=-\cosh{r}\sinh{r}e^{i\theta}(2n_{th}+1)$, with $r,\theta$ being the
single mode squeezing parameters. Since $n_{th}(\omega_{c})\approx 0$ for
optical photons, $N_{eff}$ and $M_{eff}$ are replaced with $N=\sinh^{2}{r}$
and $M=-\cosh{r}\sinh{r}e^{i\theta}$ for operator
$\hat{a}(\hat{a}^{\dagger})$. To understand what happens in the DSME for a
squeezed reservoir we follow procedure similar to Hu et. al.(Hu, ). We first
write down the correlation function for a squeezed thermal reservoir. It is
straightforward to show that the correlation function includes additional
terms having time dependence and is given as;

$\displaystyle\mathcal{R}(s,t)$
$\displaystyle=\sum_{k}|G^{\lambda}_{k}|^{2}(N_{k}-M_{k}\exp(-2i\omega_{k}t))e^{i\omega_{k}s}$
(12)
$\displaystyle+|G^{\lambda}_{k}|^{2}((N_{k}+1)-M_{k}^{*}\exp(2i\omega_{k}t))e^{-i\omega_{k}s},$

where
$<O_{\lambda}^{\dagger}(\vec{k})O_{\lambda^{\prime}}(\vec{k^{\prime}})>=\delta_{\lambda\lambda^{\prime}}\delta_{\vec{k}\vec{k^{\prime}}}N_{k}$,
$<O_{\lambda}(\vec{k})O_{\lambda^{\prime}}^{\dagger}(\vec{k^{\prime}})>=\delta_{\lambda\lambda^{\prime}}\delta_{\vec{k}\vec{k^{\prime}}}(N_{k}+1)$,
$<O_{\lambda}(\vec{k})O_{\lambda^{\prime}}(\vec{k^{\prime}})>=\delta_{\lambda\lambda^{\prime}}\delta_{\vec{k}\vec{k^{\prime}}}M_{k}$,
$<O_{\lambda}^{\dagger}(\vec{k})O_{\lambda^{\prime}}^{\dagger}(\vec{k^{\prime}})>=\delta_{\lambda\lambda^{\prime}}\delta_{\vec{k}\vec{k^{\prime}}}M_{k}^{*}$
and $G^{\lambda}_{k}$ is the system-reservoir coupling for the $k^{th}$
reservoir mode having polarization $\lambda$. As shown in Appendix A, the
correlation function in Equation (12) leads to additional terms proportional
to operators $\hat{a}$, $\hat{a}^{\dagger}$, $(\hat{b}-\beta_{0}\hat{N_{c}})$
and $(\hat{b}-\beta_{0}\hat{N_{c}})^{\dagger}$ in the dressed state. Also for
since $n_{th}(\omega_{c})\approx 0$, $N_{k}$ and $M_{k}$ are replaced by $N$
and $M$ respectively for the cavity mode, while for the mechanical mode they
are replaced by $N_{eff}$ and $M_{eff}$ respectively. The DSME for a squeezed
thermal reservoir can then be written as;

$\displaystyle\frac{d\tilde{\rho}(t)}{dt}$
$\displaystyle=\gamma_{m}(N_{eff}+1)\mathcal{D}[\hat{b}-\beta_{0}\hat{N_{c}}]\tilde{\rho}(t)+\gamma_{m}N_{eff}\mathcal{D}[\hat{b}^{\dagger}-\beta_{0}\hat{N_{c}}]\tilde{\rho}(t)$
(13)
$\displaystyle+\kappa(N+1)\mathcal{D}[a]\tilde{\rho}(t)+\kappa(N)\mathcal{D}[\hat{a}^{\dagger}]\tilde{\rho}(t)$
$\displaystyle-\gamma_{m}M_{eff}^{*}(\hat{b}-\beta_{0}\hat{N_{c}})\tilde{\rho}(t)(\hat{b}-\beta_{0}\hat{N_{c}})-\gamma_{m}M_{eff}(\hat{b}-\beta_{0}\hat{N_{c}})^{\dagger}\tilde{\rho}(t)(\hat{b}-\beta_{0}\hat{N_{c}})^{\dagger}$
$\displaystyle-\kappa M^{*}\hat{a}\tilde{\rho}(t)\hat{a}-\kappa
M\hat{a}^{\dagger}\tilde{\rho}(t)\hat{a}^{\dagger}.$

To derive the above expression we assume that the reservoir degrees of freedom
remain Markovian in nature even at low temperatures. This is clearly not a
good assumption since at low temperatures the reservoir show long correlation
times between the reservoir degrees of freedom which gives rise to non-
Markovianity. Still we do this to differentiate it from what happens at high
temperatures, next.

###  V.2 Cavity dephasing in the DSME

At high enough temperature($k_{B}T>>\omega_{m}$) additional terms get included
in the DSME in Equation (13). This is shown in Appendix A. These terms are
proportional to $\mathcal{D}[\hat{N_{c}}]$ and hence contribute to the
dephasing of cavity photons. It has been previously shown(Hu, ) that for a
thermal reservoir, the DSME at high enough temperatures also includes cavity
dephasing terms,

$\displaystyle\frac{d\tilde{\rho}(t)}{dt}$
$\displaystyle=\gamma_{m}(n_{th}+1)\mathcal{D}[\hat{b}-\beta_{0}\hat{N_{c}}]\tilde{\rho}(t)+\gamma_{m}n_{th}\mathcal{D}[\hat{b}^{\dagger}-\beta_{0}\hat{N_{c}}]\tilde{\rho}(t)$
(14)
$\displaystyle+\kappa\mathcal{D}[a]\tilde{\rho}(t)+4\gamma_{m}(\frac{k_{B}T}{\omega_{m}})\beta_{0}^{2}\mathcal{D}[\hat{N_{c}}]\tilde{\rho}(t),$

in the units of $\hbar=1$. For a squeezed thermal reservoir, on the other
hand, this dephasing term is different due to the presence of squeezing
parameters $r,\theta$. The DSME in Equation (13) then becomes,

$\displaystyle\frac{d\tilde{\rho}(t)}{dt}$
$\displaystyle=\gamma_{m}(N_{eff}+1)\mathcal{D}[\hat{b}-\beta_{0}\hat{N_{c}}]\tilde{\rho}(t)+\gamma_{m}N_{eff}\mathcal{D}[\hat{b}^{\dagger}-\beta_{0}\hat{N_{c}}]\tilde{\rho}(t)$
(15)
$\displaystyle+\kappa(N+1)\mathcal{D}[a]\tilde{\rho}(t)+\kappa(N)\mathcal{D}[\hat{a}^{\dagger}]\tilde{\rho}(t)$
$\displaystyle-\gamma_{m}M_{eff}^{*}(\hat{b}-\beta_{0}\hat{N_{c}})\tilde{\rho}(t)(\hat{b}-\beta_{0}\hat{N_{c}})-\gamma_{m}M_{eff}(\hat{b}-\beta_{0}\hat{N_{c}})^{\dagger}\tilde{\rho}(t)(\hat{b}-\beta_{0}\hat{N_{c}})^{\dagger}$
$\displaystyle-\kappa M^{*}\hat{a}\tilde{\rho}(t)\hat{a}-\kappa
M\hat{a}^{\dagger}\tilde{\rho}(t)\hat{a}^{\dagger}$
$\displaystyle+4\gamma_{m}(\frac{k_{B}T}{\omega_{m}})(\cosh^{2}{r}+\sinh^{2}{r}+2\cosh{r}\sinh{r}\cos{\theta})\beta_{0}^{2}\mathcal{D}[\hat{N_{c}}]\tilde{\rho}(t).$

Note that from the definitions of $N_{eff}$, $M_{eff}$, $N$ and $M$ we see
that for $r=0$(no squeezing), Equation (15) reduces to Equation (14). On the
other hand for different values of $r$ and $\theta$ we can control the amount
of photon dephasing contribution to the decoherence of the density operator
$\tilde{\rho}(t)$. We have exploited this control to demonstrate that the
decoherence rates of coherent superposition states can be lowered for certain
values of $r$ and $\theta$.

###  V.3 Cavity dephasing in the SME

In Section IV we showed that for a strong single-photon optomechanical
coupling($\beta_{0}\sim 1$), the SME in the interaction picture for a thermal
reservoir contains an additional cavity dephasing term(Equation (10)). This is
true even for a squeezed thermal reservoir. Infact two additional terms should
be included for SME as shown in Appendix B,

$\displaystyle\frac{d\tilde{\rho}(t)}{dt}$
$\displaystyle=\gamma_{m}(N_{eff}+1)\mathcal{D}[\hat{b}-\beta_{0}\hat{N_{c}}]\tilde{\rho}(t)+\gamma_{m}N_{eff}\mathcal{D}[\hat{b}^{\dagger}-\beta_{0}\hat{N_{c}}]\tilde{\rho}(t)$
(16)
$\displaystyle+\kappa(N+1)\mathcal{D}[a]\tilde{\rho}(t)+\kappa(N)\mathcal{D}[\hat{a}^{\dagger}]\tilde{\rho}(t)$
$\displaystyle-\gamma_{m}M_{eff}^{*}(\hat{b}-\beta_{0}\hat{N_{c}})\tilde{\rho}(t)(\hat{b}-\beta_{0}\hat{N_{c}})-\gamma_{m}M_{eff}(\hat{b}-\beta_{0}\hat{N_{c}})^{\dagger}\tilde{\rho}(t)(\hat{b}-\beta_{0}\hat{N_{c}})^{\dagger}$
$\displaystyle-\kappa M^{*}\hat{a}\tilde{\rho}(t)\hat{a}-\kappa
M\hat{a}^{\dagger}\tilde{\rho}(t)\hat{a}^{\dagger}$
$\displaystyle+2\gamma_{m}\cosh{r}\sinh{r}\cos{\theta}(2n_{th}+1)\beta_{0}^{2}\hat{N_{c}}\tilde{\rho}(t)\hat{N_{c}}+\gamma_{m}(2N_{eff}+1)\beta_{0}^{2}\mathcal{D}[\hat{N_{c}}]\tilde{\rho}(t).$

From Equation (16) we see that in the case of a squeezed thermal reservoir,
there is presence of cavity dephasing terms. Comparing with Equation (10), in
the last term $n_{th}$ is replaced by $N_{eff}$ while the second last term in
Equation (16) is proportional to $\bm{cosine}$ of reservoir phase(Appendix B).
Both these terms add to cavity depashing and vanish in the weak optomechanical
coupling limit($\beta_{0}<<1$).

##  VI Results and discussion

In this section we provide a comparative analysis for the decoherence of the
system density operator for phase-sensitive(squeezed) and phase-insensitive
reservoirs in the DSME for weak to strong to ultrastrong single-photon
optomechanical coupling. We point out how these reservoirs affect the
decoherence of cavity photon Fock state in coherent superpostion in an
optomechanical system. We also show the influence of squeezing parameters $r$
and $\theta$ on the decoherence rate of the system density operator for cavity
photon Fock states. To perform these simulations we have used the mesolve
routine in quantum toolbox in Python(QuTiP)(Qut1, ) with appropriate Lindblad
superoperators and hamiltonian in Equation (2).
To simulate the evolution of the cavity photons we first define a general
initial state of the optomechanical system as;

$\begin{split}\ket{\psi_{c,m}(0)}&=(\cos{(\zeta/2)}\ket{p}+e^{i\phi}\sin{(\zeta/2)}\ket{q})_{c}\otimes\ket{u}_{m},\end{split}$
(17)

where $c$ and $m$ denote cavity photon and mechanical Fock states
respectively. As demonstrated in Section III, these are the eigenstates of the
Hamiltonian $\hat{H}^{\prime}$. Using superposition of coherent states instead
of Focks states does not change the final result and are hence experimentally
more feasible. The use of Fock states here is done purely for technical
clarity. In the interaction picture, the coherence of the cavity photon state
is represented by the corresponding off-diagonal element of the partial trace
over mechanical Fock state of the time-evolved density operator and in this
case given as;

$\begin{split}P_{pq}(t)&=|\bra{p}\tilde{\rho_{c}}(t)\ket{q}|\\\
&=|\bra{p}Tr_{m}[\tilde{\rho}_{c,m}(t)]\ket{q}|,\end{split}$ (18)

where $\tilde{\rho}_{c,m}(0)=\ket{\psi_{c,m}(0)}\bra{\psi_{c,m}(0)}$ is the
initial density operator.

Figure 1: (Color online) Evolution of the off-diagonal element
$|\bra{0}Tr_{m}[\tilde{\rho}_{c,m}(t)]\ket{3}|$ in a vacuum/thermal reservoir
with $n_{th}=0$/$n_{th}=20$. The optomechanical system parameters are,
$g_{0}=0.8\,\omega_{m}$, $\kappa=0.05\,\omega_{m}$ and $\gamma_{m}=\kappa/3$.

$Tr_{m}[...]$ is the partial trace over the mechanical state. For brevity we
choose $\zeta/2=\pi/4$ and $\phi=0$, $\ket{p}=\ket{0}$ and $\ket{q}=\ket{3}$.
This state is similar to the one used in Hu et. al.(Hu, ). Figure 1 shows the
evolution of $P_{03}(t)$ defined above, as a function of dimensionless unit
$\kappa t$ for depahsed-SME and DSME in vacuum and thermal reservoirs
corresponding to master equations given in Equation (10) and Equation (14)
above. It is qualitatively similar to the one produced in Hu et al.(Hu, ). It
is seen that the effect of cavity dephasing becomes dominant at high
temperatures in the DSME than that for the SME in the interaction picture. A
similar trend can be seen Figure 2 where the evolution occurs under the
influence of a squeezed reservoir. It is also observed from Figure 2(a) that
for the same value of $g_{0}=0.8\omega_{m}$, there is an increase in
decoherence rate of the optical field under the influence of a squeezed
reservoir.

---
Figure 2: (Color online) Evolution of the off-diagonal element
$|\bra{0}Tr_{m}[\tilde{\rho}_{c,m}(t)]\ket{3}|$ in a squeezed vacuum/thermal
reservoir with $r=0.5$ and $n_{th}=0$/$n_{th}=20$. The optomechanical system
parameters are, $g_{0}=0.8\,\omega_{m}$, $\kappa=0.05\,\omega_{m}$ and
$\gamma_{m}=\kappa/3$.

But whereas on the one hand thermal reservoirs do not provide any control over
this decoherence, phase-sensitive(squeezed) reservoirs provide a control over
the decoherence by changing the squeezing parameters $r$ and $\theta$. For a
system weakly interacting with a squeezed thermal reservoir, it is seen from
Figure 2 that for the same value of $r$, the decoherence is drastically
reduced for different values of $\theta$. This is due to the dependence of
decoherence on the $cosine$ of the phase of the reservoir. By setting up an
appropriate reservoir phase one can increase the coherence of optical field.
This explicitly shown in Figure 3.
In order to show the dependence of squeezing parameters, $r$ and $\theta$ on
DSME for interaction with thermal and squeezed thermal reservoirs we choose
that value of $\kappa t$ for which the $P_{03}(t)$ goes below $0.1$, as a
function of $\theta$ and $r$ in Figure 3(a) and Figure 3(b) respectively.

---
Figure 3: (Color online) Dimensionless $\kappa t$ at which $P_{03}$ reduces to
$<20\%$ of its initial value under the influence of thermal and squeezed
thermal reservoirs$(n_{th}=20)$. The optomechanical system parameters are,
$g_{0}=0.8\,\omega_{m}$, $\kappa=0.05\,\omega_{m}$ and $\gamma_{m}=\kappa/3$.

We see that the value of $\kappa t$ for a thermal reservoir does not change
with respect to $r$ or $\theta$(shown as solid blue line) which is expected.
For a squeezed reservoir however we can see that a maximum is reached near
$r=1.3$ and $\theta=\pi$ in the ultrastrong coupling regime,
$g_{0}=0.8\omega_{m}$. From Figure 3(b) we also see that for values of $r>1.5$
the coherence time gradually decreases towards that for thermal DSME. This
means that for $r$ values greater than $1.5$, $P_{03}(t)$ decreases faster
towards $20\%$ of its initial value even though we set reservoir phase at
$\theta=\pi$.
Interaction with a thermal reservoir leads to a decrease in coherence as one
goes from weak to ultrastrong single-photon optomechanical coupling regime, as
shown in Figure 4(a). This is also largely mimicked in the case of a squeezed
thermal reservoir Figure 4(b),

---
Figure 4: (Color online)$P_{03}(t)$ as a function of $\kappa t$ for (a).
Thermal reservoir, (b,c). Squeezed thermal reservoir at $n_{th}=20$ and
different optomechanical coupling strengths $g_{0}$, rest of system parameters
being, $\kappa=0.05\,\omega_{m}$ and $\gamma_{m}=\kappa/3$.

although the coherence time drastically increases in the weak coupling regime
and decreases in the strong to ultrastrong regime. This is not the case when
the phase of the squeezed thermal reservoir is set to $\theta=\pi$ as shown in
Figure 4(c). Comparing with Figure 4(b), for this value of reservoir phase
$\theta$ the decoherence is faster in the weak single-photon coupling regime.
As we go towards strong single-photon coupling this decoherence becomes
slower. Due to this the decoherence for $g_{0}=0.01\,\omega_{m}$ and
$g_{0}=0.1\,\omega_{m}$ pretty much overlap each other above $P_{03}(t)\approx
0.1$.

##  VII Conclusion

In this article we have demonstrated the effects squeezed thermal reservoirs
have on a single cavity optomechanical system in the strong single-photon
coupling regime. In the strong single-photon optomechanical coupling regime,
the appropriate master equation is the dressed-state master equation(DSME) as
long as it is applied for the situation where the frequency of cavity photons
is much larger than the natural frequency of the mechanical resonator. Such an
optomechanical system interacting with a squeezed thermal reservoir gives
additional terms in the master equation which depend on the $\bm{cosine}$ of
the reservoir phase. We have shown that by setting the squeezed thermal
reservoir phase to $\theta=\pi$ we can increase coherence times for strongly
coupled optomechanical systems which otherwise is not possible for thermal
reservoir. On the other hand we have also shown that increasing reservoir
squeezing parameter $r$ does not always lead to an increase in coherence time
and that a maximum is reached for particular values of $r$ and $\theta$ upto
an order of magnitude for strongly coupled photon-phonon systems. For values
of $\beta_{0}\sim 1$ there is a drastic increase in coherence under the
influence of squeezed thermal reservoir with its phase near $\theta=\pi$, as
compared to a thermal reservoir. It would be further interesting to analyse
the effect of reservoir phase on the time evolution of Wigner functions of
Fock states used in our study since negative densities of Wigner functions
have been known to be related to coherence and non-classicality. Our study
might be useful in understanding systems which are reservoir engineered to
increase coherence. It might also help in understanding entanglement in
strongly coupled systems under the influence of phase-sensitive reservoirs.
Although the current state-of-the-art experiments are far from reaching such
strong single-photon optomechanical coupling strengths this study becomes
necessary to demonstrate what can be achieved once there is such a technology
in place. Besides this archival importance this study is as much important for
lower single-photon optomechanical coupling strength.

##  Appendix A Mechanical damping.

Calculations for cavity field damping are pretty straightforward and similar
to the case for a squeezed vacuum reservoir for operator $\hat{a}$ and
$\hat{a}^{\dagger}$. Hence we do not present it here again. They can be found
in any any quantum optics textbook. Instead we begin straightaway with
operators representing mechanical oscillation and damping. From Equation (12)
in Section V.1,

$\begin{split}\int_{0}^{\infty}\mathcal{R}_{M}(s,t)\,e^{i\omega_{m}s}\,ds=\pi|G(\omega_{m})|^{2}\Big{[}(N_{eff}+1)-M_{eff}^{*}\,e^{2i\omega_{m}t}\Big{]}\end{split}$
(19)
$\begin{split}\int_{0}^{\infty}\mathcal{R}_{M}(s,t)\,e^{-i\omega_{m}s}\,ds=\pi|G(\omega_{m})|^{2}\Big{[}N_{eff}-M_{eff}\,e^{-2i\omega_{m}t}\Big{]}\end{split}$
(20)
$\begin{split}\int_{0}^{\infty}\mathcal{R}_{M}(s,t)\,ds&\approx\pi|G(\omega_{m})|^{2}\Big{[}(N_{eff}+1)-M_{eff}^{*}\Big{]},\end{split}$
(21)

where we have assumed approximately equal contributions from the spectral
density $J(\omega)$ near $\omega=0$ and $\omega=\omega_{m}$. If the spectral
density $J(\omega=0)\approx 0$, the contribution of Equation (21) is
negligible as compared to Equations (19) and (20). Recalling,
$|G(\pm\omega_{m})|^{2}=\pm\gamma_{m}/(2\pi)$(ohmic reservoir), Equation (21)
for high temperatures ($k_{B}T>>\omega_{m}$) becomes,

$\begin{split}\int_{0}^{\infty}\mathcal{R}_{M}(s,t)\,ds&=\frac{\gamma_{m}}{2}\Big{(}\frac{k_{B}T}{\omega_{m}}\Big{)}\Big{[}\cosh^{2}{r}+\sinh^{2}{r}+2\cosh{r}\sinh{r}\,e^{-i\theta}\Big{]},\end{split}$
(22)

We define a hermitian operator
$\chi_{m}(t)=\hat{b}_{m}(t)+\hat{b}_{m}^{\dagger}(t)$ in the interaction
picture, where
$\hat{b}_{m}(t)=e^{-i\omega_{m}t}\hat{B}_{m}+\beta_{0}\hat{N}_{c}$ and
$\hat{B}_{m}=b_{m}-\beta_{0}\hat{N}_{c}$ in the interaction picture and
dressed basis. The details of these operators can be found here (Hu, ). Our
aim is to calculate terms in,

$\begin{split}\dfrac{d\tilde{\rho}(t)}{dt}=\int_{0}^{\infty}ds\,\mathcal{R}_{M}(s,t)\Big{[}\chi_{m}(t)\tilde{\rho}(t)\chi_{m}(t-s)-\chi_{m}(t)\,\chi_{m}(t-s)\,\tilde{\rho}(t)\Big{]}+h.c,\end{split}$
(23)

for a squeezed thermal reservoir. Let us consider the first term in Equation
(23),

$\begin{split}\int_{0}^{\infty}ds\,\mathcal{R}_{M}(s,t)\,\chi_{m}(t)\tilde{\rho}(t)\chi_{m}(t-s)&=\int_{0}^{\infty}ds\,\mathcal{R}_{M}(s,t)\Big{[}\hat{b}_{m}(t)\tilde{\rho}(t)\hat{b}_{m}^{\dagger}(t-s)\\\
&+\hat{b}_{m}^{\dagger}(t)\tilde{\rho}(t)\hat{b}_{m}(t-s)+\hat{b}_{m}(t)\tilde{\rho}(t)\hat{b}_{m}(t-s)\\\
&+\hat{b}_{m}^{\dagger}(t)\tilde{\rho}(t)\hat{b}_{m}^{\dagger}(t-s)\Big{]}.\end{split}$
(24)

Next we use Equations (19) to (22) to piece-wise calculate the above integral
for a squeezed thermal reservoir and neglect fast oscillating terms under the
rotating wave approximation(RWA),

$\begin{split}\int_{0}^{\infty}ds\,\mathcal{R}_{M}(s,t)\,\hat{b}_{m}(t)\tilde{\rho}(t)\hat{b}_{m}^{\dagger}(t-s)&=\frac{\gamma_{m}}{2}\Big{[}N_{eff}\hat{B}_{m}\tilde{\rho}(t)\hat{B}_{m}^{\dagger}\\\
&+\big{(}(N_{eff}+1)-M_{eff}^{*}\big{)}\beta_{0}^{2}\,\hat{N}_{c}\tilde{\rho}(t)\hat{N}_{c}\Big{]},\end{split}$
(25)
$\begin{split}\int_{0}^{\infty}ds\,\mathcal{R}_{M}(s,t)\,\hat{b}_{m}^{\dagger}(t)\tilde{\rho}(t)\hat{b}_{m}(t-s)&=\frac{\gamma_{m}}{2}\Big{[}(N_{eff}+1)\hat{B}_{m}^{\dagger}\tilde{\rho}(t)\hat{B}_{m}\\\
&+\big{(}(N_{eff}+1)-M_{eff}^{*}\big{)}\beta_{0}^{2}\,\hat{N}_{c}\tilde{\rho}(t)\hat{N}_{c}\Big{]},\end{split}$
(26)
$\begin{split}\int_{0}^{\infty}ds\,\mathcal{R}_{M}(s,t)\hat{b}_{m}(t)\tilde{\rho}(t)\hat{b}_{m}(t-s)&=\frac{\gamma_{m}}{2}\Big{[}M_{eff}^{*}\hat{B}_{m}\tilde{\rho}(t)\hat{B}_{m}\\\
&+\big{(}(N_{eff}+1)-M_{eff}^{*}\big{)}\beta_{0}^{2}\,\hat{N}_{c}\tilde{\rho}(t)\hat{N}_{c}\Big{]},\end{split}$
(27)
$\begin{split}\int_{0}^{\infty}ds\,\mathcal{R}_{M}(s,t)\hat{b}_{m}^{\dagger}(t)\tilde{\rho}(t)\hat{b}_{m}^{\dagger}(t-s)&=\frac{\gamma_{m}}{2}\Big{[}M_{eff}\hat{B}_{m}^{\dagger}\tilde{\rho}(t)\hat{B}_{m}^{\dagger}\\\
&+\big{(}(N_{eff}+1)-M_{eff}^{*}\big{)}\beta_{0}^{2}\,\hat{N}_{c}\tilde{\rho}(t)\hat{N}_{c}\Big{]}.\end{split}$
(28)

Substituting the above equations in Equation (23) and similarly calculating
the Hermitian conjugate, we get the mechanical part of Equation (15).

##  Appendix B Cavity dephasing in SME

Emergence of cavity dephasing terms in Equation (16) can be understood as
follows; The unitary transformation of the dissipator
$\mathcal{D}[b_{m}]\rho(t)$ from Schrodinger picture of the standard master
equation(SME) to interaction its picture in dressed basis is,

$\displaystyle
U^{\dagger}(t)\mathcal{D}[\hat{b}_{m}]\rho(t)U(t)\approx\mathcal{D}[\hat{b}_{m}-\beta_{0}\hat{N}_{c}]\tilde{\rho}(t)+\mathcal{D}[\beta_{0}\hat{N}_{c}]\tilde{\rho}(t),$
(29)

where $U=\exp{-i\,H^{\prime}t}$, $H^{\prime}$ being the system hamiltonian in
Equation (2). Similarly for a squeezed thermal reservoir,

$\displaystyle
U^{\dagger}(t)\Big{(}M_{eff}^{*}\,\hat{b}_{m}\rho(t)\hat{b}_{m}+M_{eff}\,\hat{b}_{m}^{\dagger}\rho(t)\hat{b}_{m}^{\dagger}\Big{)}U(t)$
$\displaystyle\approx
M_{eff}^{*}\hat{B}_{m}\tilde{\rho}(t)\hat{B}_{m}+M_{eff}\hat{B}_{m}^{\dagger}\tilde{\rho}(t)\hat{B}_{m}^{\dagger}$
(30)
$\displaystyle+\gamma_{m}\beta_{0}^{2}\Big{(}M_{eff}^{*}+M_{eff}\Big{)}\hat{N}_{c}\tilde{\rho}(t)\hat{N}_{c},$

where $\hat{B}_{m}=b_{m}-\beta_{0}\hat{N}_{c}$.

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# Ultra-narrowband interference circuits
enable low-noise and high-rate photon counting for InGaAs/InP avalanche
photodiodes

Yuanbin Fan  1,† Tingting Shi  1,2,3,† Weijie Ji  1Lai Zhou  1 Yang Ji 2,3 and
Zhiliang Yuan1  1Beijing Academy of Quantum Information Sciences, Beijing
100193, China
2State Key Laboratory for Superlattices and Microstructures, Institute of
Semiconductors, Chinese Academy of Sciences, Beijing 100083, China
3College of Materials Science and Opto-Electronic Technology, University of
Chinese Academy of Sciences, Beijing 100049, China
†These authors contributed equally<EMAIL_ADDRESS>
###### Abstract

Afterpulsing noise in InGaAs/InP single photon avalanche photodiodes (APDs) is
caused by carrier trapping and can be suppressed successfully through limiting
the avalanche charge via sub-nanosecond gating. Detection of faint avalanches
requires an electronic circuit that is able to effectively remove the gate-
induced capacitive response while keeping photon signals intact. Here we
demonstrate a novel ultra-narrowband interference circuit (UNIC) that can
reject the capacitive response by up to 80 dB per stage with little distortion
to avalanche signals. Cascading two UNIC’s in a readout circuit, we were able
to enable high count rate of up to 700 MC/s and low afterpulsing of 0.5 % at a
detection efficiency of 25.3 % for 1.25 GHz sinusoidally gated InGaAs/InP
APDs. At -30 ∘C, we measured 1 % afterpulsing at a detection efficiency of
21.2 %.

††journal: oe

## Introduction

Semiconductor avalanche photodiodes (APD’s) are versatile for weak light
detection, with applications from remote ranging[1, 2], quantum communication
[3] and fluorescence lifetime imaging[4] to optical time-domain reflectometry
[5, 6]. For practical fiber quantum key distribution (QKD), InGaAs/InP APD’s
are the detector of choice because they are compact and low cost, and allow
cryogenic-free or even room-temperature operation [3]. However, they suffer
from spurious afterpulsing arising from carrier trapping by defects in the
multiplication layer, especially at high detection efficiencies [7, 8]. To
minimise afterpulsing, an APD can be biased on for a sub-nanosecond duration
only when a photon arrival is expected. In doing so, charge per avalanche can
be reduced to the order of 10 fC [9, 10, 11], corresponding to a transient
current of less than 0.1 mA. Such weak avalanches have to be discriminated
through use of a readout circuit that removes the strong capacitive response
to the applied gates. Gated InGaAs detectors are capable of counting photons
at up to 60% efficiencies [12] and 1 GHz rate [13] and with photon number
resolution [14]. Thanks to this success, gating approach has been applied to
traditionally free-running Si devices for performance enhancement [15, 16].

Existing readout circuits include band stop [11, 17, 8] or low-pass[18, 19,
12] filtering under sine-wave gating[11], self-differencing [20, 7], and
transient reference cancellation [10, 21]. While simple for implementation,
frequency filtering distorts the avalanche signals due to its rejection of a
sizeable portion of frequency components, thus increasing time jitter and
temporal errors in photon registrations [18]. Self-differencing [20] and
reference cancellation methods [10] are able to maintain avalanche signal
fidelity but may suffer operational complexities. The former requires wideband
performance for the entire circuitry and thus inconvenience an adjustable
delayline [9] for frequency alignment, while the latter can be unstable
because the transient reference is derived separately from the capacitive
response.

Here we propose and experimentally demonstrate a simple, low-distortion ultra-
narrowband interference circuit (UNIC) that can suppress the capacitive
response for a 1.25 GHz gated InGaAs/InP APD single photon detector. The
circuit is an asymmetric radio-frequency (RF) interferometer. One of its arms
contains a narrow band pass filter (BPF) based on surface acoustic wave
resonator (SAW) to retrieve the fundamental wave of the gating signal. The
filtered wave then interferes destructively with the same frequency component
transmitted via the other arm through a coupling module, thereby eliminating
the capacitive response. This interference occurs over the narrow band, so it
can provide a broad and continuous pass band in frequency domain to maintain
the avalanche signal with little distortion. This allows to achieve ultra-low
afterpulsing probabilities and excellent jitter performance at high detection
efficiencies from two InGaAs APD’s that exhibit capacitive responses of very
different amplitudes.

## Detector characterisation setup

Figure 1:  (a) Single-photon characterisation setup for 1.25 GHz sinusoidally
gated InGaAs/InP APDs using UNICs for avalanche impulse readout; (b) Histogram
of the photon detection events measured by the characterisation setup (a) on
an InGaAs APD detector that was regulated at a temperature of 30 ∘C. The
photon detection peak exhibits a 30 dB width of 650 ps. AMP: amplifier; APD:
avalanche photodiode; BSF: band stop filter; DISC: discriminator; SG: signal
generator; TDC: time-to-digital converter; UNIC: ultra-narrowband interference
circuit; VOA: variable optical attenuator.

Figure 1(a) shows our single photon characterisation setup for InGaAs APDs. A
1550 nm passively mode-locked laser serves as the light source and provides
stable short pulses of 0.5 ps duration at a repetition rate of 10 MHz. The
laser output power is monitored by an optical power meter (EXFO FTB-1750) and
its pulse intensity is set by a variable optical attenuator (VOA, EXFO
FTB-3500) to 0.1 photon/pulse at the fiber input of APD under test. It
provides a 10 MHz reference to a signal generator (SG) for synthesising a 1.25
GHz sinusoidal wave with up to 27 V voltage swing. In combination of a
suitable DC bias, this AC signal periodically gates the APD above its
breakdown voltage ($60-70$ V) to achieve single photon sensitivity. The APD
output is processed by the readout module consisting of two identical 1.25 GHz
UNIC’s, one 2.5 GHz band stop filter (BSF), and three RF amplifiers (AMPs).
Amplification of the raw APD signalis is useful as it prevents weak avalanche
signals from falling below thermal noise by attenuation of the first UNIC. The
readout signal is discriminated by a discriminator for avalanches before
feeding to a time-digital-converter (TDC) with a dead time of 2 ns for time-
resolved photon counting. Figure 1(b) is a typical histogram obtained with
this setup.

APD under test is temperature-regulated using their integrated thermal-
electric cooler, which is driven by a temperature controller (Thorlabs
TED200C). A source-measure unit (Keithley 2635B) provides the DC bias and
simultaneously monitors the current flowing through the APD. In characterising
the maximum count rate, we replace the 10 MHz laser with a continuous-wave
distributed feedback laser (DFB) laser, the output of which is carved into
1.25 GHz, 50 ps pulse train using an intensity modulator. We use a high speed
digital oscilloscope to record the detector output and extract the count rate
through digital discrimination in software. The oscilloscope method is
carefully calibrated at low count rate regimes to be consistent with the
hardware discriminated result using the photon counter (Stanford Research
SR400).

The setup is able to measure dark count probability, afterpulsing probability,
detection efficiency, maximum count rate, avalanche charge and time jitter.
With no performance screening, two fiber-pigtailed APDs from different
manufacturers were used in this study, named APD#1 and APD#2 respectively.

## Ultra-narrowband interference circuit (UNIC)

Figure 2: (a) Schematic for ultranarrow interference circuit (UNIC); (b)
Transmission spectrum of a heroic UNIC PCB; Inset: Magnified view for region
of 1.24 – 1.26 GHz. (c) Raw capacitive responses from APD#1 (top) and APD#2
(bottom) under identical 27.0 V Vp-p gating; (d) Recovered avalanche impulses.
ATT: attenuator; SAW BPF: surface acoustic wave band pass filter.

With sub-nanosecond gating, a photon induced avalanche is an impulse and has a
wide spectrum. On the other hand, the capacitive response is periodic and has
its most energy concentrated at the gating frequency or its higher harmonics.
This spectral difference allows frequency-dependent signal processing to
remove the capacitive response and keep the wide-band impulses intact. Figure
2(a) shows a circuit diagram of UNIC. It is an RF interferometer containing
two couplers of 9:1 power splitting ratio, a $\pi$-resistive attenuator (ATT)
and surface acoustic wave band pass filter. Two of the ports are terminated by
50 $\Omega$ resistors. The SAW BPF features a central frequency of 1.25 GHz,
20-dB passband of 35 MHz, insertion loss of 3 dB, and group delay of 34 ns. It
filters out the fundamental wave of the gating frequency, which then
interferes with the APD signal transmitted through the other arm. The
attenuation and differential delay are set to enable destructive interference
for the 1.25 GHz frequency component at the UNIC output port. The UNIC
differential delay ($\Delta t$) meets the condition below

$\Delta t=T_{g}^{SAW}+\delta t=(N+1/2)/f_{g},$ (1)

where $T_{g}^{SAW}$ is the group delay of the SAW BPF, $\delta t$ the delay
caused by the track length difference between two arms, $f_{g}=1.25$ GHz the
APD gating frequency, and $N$ is an integer number. For a compact circuit, we
choose $\delta t$ to be less than the half-wave of the gating signal. With the
SAW device used, $N=42$ and $\delta t=155$ ps. The resulting UNIC unit has a
small footprint of $38\times 15$ mm2 on printed circuit boards (PCBs).

The large $T_{g}^{SAW}$ brings two additional benefits. Firstly, it
substantially increases the PCB manufacturing tolerance, as a 0.5 mm deviation
in the RF track length will just alter the circuit central frequency by less
than $10^{-4}$. This eliminates the requirement of an adjustable delayline
which is required in a self-differencing circuit for precise frequency
alignment. Secondly, it helps to produce an ultra-narrow band rejection at its
designed frequency. Figure 2(b) shows the measured transmission spectrum (S21
parameter) of our heroic UNIC PCB, and its inset expands the frequency section
of 1.24 – 1.26 GHz to show the narrowness of the insertion loss dip in the
close proximity of the resonance frequency of 1.25 GHz. The dip of the heroic
(typical) PCB features a loss of -95 dB (-80 dB), representing a suppression
of 80 dB (65 dB) as compared with the background loss for other frequencies
under 2 GHz. The dip has a 30 dB linewidth of merely 30 kHz, thus ensuring
crucial suppression of the APD gating signal without overly distorting the
avalanche signals. The background loss of about 14 dB is caused mainly by the
9:1 couplers and can be reduced in future with more balanced splitters.

Cascading two UNIC’s enables a stable 100 dB suppression of the primary gating
frequency and thus provides sufficient performance redundancy. Their
attenuation to the avalanche signal is compensated by using RF amplifiers
(Fig. 1(a)). Second order harmonics (2.5 GHz) is suppressed by a band stop
filter of conventional LC design. Figure 2(c) shows raw outputs from two
different APD’s under identical sinusoidal gating. Their respective capacitive
responses are measured to be 0.42 V and 1.75 V. Despite their 4 times
differences, UNIC’s can successfully reject the sinusoidal responses and
retrieve avalanches with excellent signal-to-background ratio, as shown in
Fig. 2(d). For APD#2, we just adjusted the gain of the first AMP to avoid
saturation and distortion.

## Results and discussion

Time-resolved photon counting allows precise extraction of the net photon
detection efficiency ($\eta_{net}$) and the afterpulsing probability
($P_{A}$), which is defined as the ratio of the total afterpulses per photon
counting event. Figure 1(b) shows a histogram of avalanche events measured for
APD#1 under 10 MHz pulsed excitation of 0.1 photon/pulse. The illuminated peak
has a full-width of 1/1000 maximum (30 dB width) of just 650 ps, which is
shorter than the gating period of 800 ps and thus allows low-error clock
number assignment that is essential for high speed QKD. The count at non-
illuminated gates arise from detector dark count and afterpulses and is more
than 3 orders of magnitude lower than that of the illuminated gate. We extract
quantities of $P_{I}$ and $P_{NI}$, i.e., the respective counting
probabilities for each illuminated and non-illuminated gate. With a separate
measurement of the detector dark count probability ($P_{D}$), we calculate the
afterpulsing probability using the standard method [20, 17],

$P_{A}=\frac{(P_{NI}-P_{D})\cdot R}{P_{I}-P_{NI}},$ (2)

where $R=125$ here is the ratio of the gating frequency (1.25 GHz) to the
laser illumination (10 MHz). Excluding the dark and afterpulse count
probabilities, the net single photon detection efficiency is given by [7]

$\eta_{net}=\frac{1}{\mu}\mathrm{ln}\frac{1-P_{NI}}{1-P_{I}},$ (3)

where $\mu$ is the average incident photon number per illumination pulse.

Figure 3: Dark count probability (top) and afterpulse probability (bottom) as
a function of photon detection efficiency of (a) APD#1 and (b) APD#2 measured
for several different temperatures.

Figure 3 shows the characterisation results for APD#1 and APD#2. We fixed the
amplitude of the 1.25 GHz sinusoidal signal at 27.0 V, and measured the
relevant parameters as a function of the applied direct current (DC) bias, but
for clarity the results are plotted as a function of the net detection
efficiency ($\eta_{net}$). Each device was measured at several different
temperatures, while APD#2 could reach only a narrower temperature range due to
its cooler compatibility with the temperature control driver. Qualitatively,
two devices behave similarly. Both dark count and afterpulsing probabilities
increase with photon detection efficiency, and exhibit opposite dependencies
on temperature. For both APDs at $\eta_{\mathrm{net}}=30~{}\%$, the
afterpulsing probabilities are less than 2.3 % at their lowest measurement
temperatures with corresponding dark count probabilities of $1.25\times
10^{-6}$ and $1.6\times 10^{-6}$ for APD#1 (-30 ∘C) and APD#2 (-20 ∘C),
respectively. Moreover, our UNIC-APDs can offer record low afterpulsing
probabilities, as summarised for APD#1 in Figure 4. At -30 ∘C, APD#1 is able
to achieve 5 % and 21.2 % detection efficiencies at 0.5 % and 1.0 %
afterpulsing probabilities. At these afterpulsing probabilities, the maximum
detection efficiency increases with temperature and reaches 25.3 % and 34.2 %
at 30 ∘C. At 5.9 % $P_{A}$, APD#2 has a detection efficiency of 50 %
efficiency at 30 ∘C and dark count probability of $1.1\times 10^{-4}$.

Figure 4: Temperature dependencies of photon detection efficiency for APD#1 at
the given afterpulsing probabilities of 0.5 % (blue) and 1 % (red).

Figure 5: Maximum count rate (blue) and photoncurrent (red) vs  incident flux
for APD#1.

The maximum count rate is a crucial parameter for a number of applications,
for example, high bit rate QKD [3] and rapid phase tracking in twin-field QKD
[22, 23]. To determine their maximum count rates, we used a DFB laser
transmitting at 1.25 GHz as the illumination source and measure the count rate
as a function of photon flux. Figure 5 shows an exemplar result obtained from
APD#1 at a temperature of 30 ∘C with its detection efficiency set to 25.3 % in
the low flux regime. The detector maintains a linear dependence with incident
flux for count rates exceeding 100 MHz, while a maximum count rate of 700 MHz
is obtained at the few photons/pulse regime. We attribute the high count rate
to the UNIC’s ability of removing the capacitive response and thus allowing
discrimination of faint avalanches. From the accompanying current measurement,
we extract an avalanche charge of 38 fC, comparable to the best value of 35 fC
[9] obtained with the photocurrent measurement method. The ability to detect
such weak avalanches ensures low afterpulsing probabilities in our UNIC-APDs.
APD#2 was measured to have a similar avalanche charge as that of APD#1. When
setting its efficiency to 50 %, APD#2’s avalanche charge rose to 65 fC due to
stronger bias applied. Nevertheless, it was still able to achieve a maximum
count rate of 600 MHz.

Table 1:  Performance comparison of sub-nanosecond gated InGaAs detectors using different types of readout circuits. |  $P_{\mathrm{A}}$(%) |  $\eta_{\mathrm{net}}$ (%) |  $P_{\mathrm{D}}$ ($\mathrm{gate}^{\mathrm{-1}}$) | T (∘C) |  $f_{g}$ (GHz) | Readout Method
---|---|---|---|---|---|---
This work | 1.0 | 21.2 | 5.4$\times 10^{-7}$ | -30 | 1.25 | UNIC
He et al[19] | 1.0 | 20.7 | 7.6$\times 10^{-7}$ | -30 | 1.00 | low-pass filter +
|  |  |  |  |  | variable width discriminator
Tada et al[8] | 1.8 | 27.7 | 8$\times 10^{-7}$ | -35 | 1.27 | band stop filter
Fang et al[12] | 2.5 | 20 | 1.1$\times 10^{-6}$ | -30 | 1.25 | low-pass filter
Comandar et al[7] | 2.9 | 20 | 1.0$\times 10^{-6}$ | -30 | 1.00 | self-differencing
Liang et al[21] | 4.5 | 20 | 3.2$\times 10^{-6}$ | -30 | 1.25 | reference subtraction

Table 1 compares our results with those gigahertz-gated detectors equipped
with different readout circuits. For impartiality, we list just data measured
at a fixed temperature of -30 ∘C whenever possible. Here, our UNIC-APD
achieved an impressive 1% afterpulsing probability at
$\eta_{\mathrm{net}}=21.2$ %, considerably outperforming most other methods
among filtering[12, 8], self-differencing [7] and reference subtraction [21].
In terms of detection efficiency, our result improves marginally over the
previous best[19], but which was achieved with help of an uncommon variable
width discriminator to mitigate signal distortion by excessive filtering. We
attribute the outstanding performance of our detectors to low-distortion
signal processing by UNIC’s.

It is useful to compare our UNIC-APDs with detectors deployed in QKD systems.
In the QKD system optimised for secure key rates (SKRs) [3], the room-
temperature self-differencing detectors featured $f_{g}=1$ GHz,
$\eta_{\mathrm{net}}=31$ %, $P_{A}=4.4$% and $P_{D}=2.25\times 10^{-4}$ and a
SKR of 13.72 Mb/s over a 2 dB channel was obtained. Our UNIC-APD could
outperform in all these parameters. At 30 ∘C and with $P_{A}=4.4$ %, APD#2
offers a higher efficiency of 49 % efficiency and twice lower dark count
probability of $9.4\times 10^{-5}$, see Fig. 3b. Combined with its high count
capability, UNIC detectors are expected to allow a SKR exceeding 25 Mb/s over
the same channel loss. This provides an interesting technological path towards
100 Mb/s via wavelength multiplexing.

## Conclusion

To summarise, we have developed a novel approach of using UNICs for reading
out avalanche signals from 1.25 GHz sinusoidally gated InGaAs APDs. UNIC-APDs
were characterised to exhibit excellent performance across the temperature
range of $-30$ – 30 ∘C, and can offer >20 % detection efficiency at an ultra
low afterpulsing probability of 1 %. This performance, together with the
circuit’s compactness and manufacturing tolerance, will allow UNIC-APDs a
considerable potential in QKD applications.

Disclosures The authors declare that there are no conflicts of interest
related to this article.

Data availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

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