text:
\section{Introduction}
There have been many studies on the generation~\cite{zakka2011quantum,clemmen2016ramsey,aguayo2022integrated} or manipulation~\cite{tanzilli2005photonic,karnieli2018frequency,lukens2017frequency,lu2020fully,lu2019controlled,lu2018quantum,olisager2010frequency} of a single photon in a superposition of two frequencies (or `colours'). These frequency-superposition states have been called `color qubits'~\cite{aguayo2022integrated}, bichromatic photons~\cite{clemmen2016ramsey}, or frequency-bin qubits~\cite{lukens2017frequency}. Such states can be generated in the process of quantum frequency conversion, where a photon's frequency is converted while its quantum information is maintained, as experimentally demonstrated three decades ago~\cite{huang1992observation}. These states also arise in quantum frequency mixing (where different superposition states are converted to and from each other) which has been demonstrated using $\chi^{(2)}$ nonlinearities~\cite{kobayashi2016frequency,kobayashi2017coherent}, $\chi^{(3)}$ nonlinearities~\cite{mcguiness2010quantum,clemmen2016ramsey,dmitriev2017quantum} as well as electro-optic modulation~\cite{lu2018quantum,lu2020fully}.

The manipulation of frequency superposition states at the single-photon level has generated interest for its potential applications in quantum information processing~\cite{lukens2017frequency}. Building on the work in linear optical quantum computation~\cite{knill2001scheme}, frequency-bin qubits have been proposed as an alternative to other photonic qubits such as those based on polarization. Polarization-based qubits can only have a two-dimensional Hilbert space, whereas frequency qubits can have a Hilbert space with much higher dimension. Frequency qubits may also be scalable for on-chip photonics~\cite{ultrabroadband2021javid}. In recent years, it has been theoretically proposed and experimentally verified that it is possible to achieve arbitrary single-qubit gates in frequency space~\cite{lu2020fully,lu2019controlled}. Aside from quantum information processing applications, frequency superposition states may also be useful for quantum metrology~\cite{giovannetti2011advances} or phase-spectroscopy~\cite{rinehart2012quantitative} at the low-light level.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{setup.png}
\caption{(a) A $\Lambda$-type 3-level interacting with a photon propagating in single-mode waveguide (highlighted as gray). The photon is in frequency-superposition state. The photon frequencies $k,k-\Delta$ are in resonance with the atomic transition frequencies $E_2-E_1$ and $E_2-E_3$. (b) Transmission and reflection pathways for atom initially in $\ket{k,1}$ and $\ket{k-\Delta,3}$, where $k$ is kept general. Note that both scenarios have two scattering output channels, in which the atoms are in either the $\ket{1}$ or $\ket{3}$ state.}
\label{maysetup}
\end{figure}

Although the generation and manipulation of single-photon frequency superposition states has been well-studied~\cite{yesharim2022observation,zakka2011quantum,clemmen2016ramsey,aguayo2022integrated,tanzilli2005photonic,karnieli2018frequency,lukens2017frequency,lu2020fully,lu2019controlled,lu2018quantum,olisager2010frequency}, there is less attention devoted to the measurement of the relative phase between different frequency components of such states. Most existing works focus on measuring the amplitude spectrum of the state, where the phase information is lost. The relative phase of two frequency components of a single photon was measured in Ref.~\cite{clemmen2016ramsey} using single-photon Ramsey interference. Here, they generated arbitrary frequency-superposition states using Bragg scattering four-wave mixing which is a $\chi^{(3)}$ nonlinear process. The relative phase between different frequency components could be measured because the outcome of their projective measurement depends on this relative phase. A related study uses electro-optic modulators instead of nonlinear processes~\cite{lu2020fully}. Their set-up also allows them to achieve arbitrary frequency qubit rotations on the Bloch sphere and the output state can then be measured using quantum state tomography, where the relative phase between frequency components can be determined using Pauli measurements.

In this study, we propose a novel way to measure the relative phase of different frequency components of a single photon using a setup in waveguide quantum electrodynamics~\cite{sheremet2021waveguide,roy2017colloquium,turschmann2019coherent,chang2018colloquium}. Our proposed set-up consists of a three-level atom coupled to a one-dimensional waveguide as seen in Fig.~\ref{maysetup}(a). The atom is of the $\Lambda$ type with a two-dimensional ground state manifold spanned by eigenstates $\ket{1}$ and $\ket{3}$ at energy $E_1$ and $E_3$, respectively, and an excited state $\ket{2}$ at energy $E_2$.   It has been shown that one can prepare the atom to be in an arbitrary ground state in this two-dimensional manifold~\cite{djotyan2008multiphoton,he2009arbitrary}. Here we scatter a single-photon with two frequency components near $E_2-E_1$ and $E_2-E_3$ (where $\hbar=1$) against the atom. We show that with a suitable choice of the atomic ground state, the outcome of the scattering process can be used to determine the relative phase of the two photon frequency components.
Scattering for a $\Lambda$ atom in a superposition of ground states was studied in Ref.~\cite{martens2013photon}. Quantum state transfer~\cite{chan2022quantum} and quantum frequency conversion~\cite{bradford2012single} has also been studied in a $\Lambda$-type atom. However, none of these studies specifically study the dependence of the relative photon phase on the transport for a $\Lambda$ atom coupled to a waveguide, which is the focus of this work.\\

\section{Model}

To treat the system of Fig.~\ref{maysetup}(a), we use a real space formalism originally developed in Ref.~\cite{shen2005coherent} and applied to three level atoms in Ref.~\cite{witthaut2010photon}. Let $g_1, g_3$ be the atom-photon coupling constants for the $\ket{1} \rightarrow \ket{2}$ and $\ket{3} \rightarrow \ket{2}$ transitions and assume the $\ket{1} \rightarrow \ket{3}$ transition is forbidden. The Hamiltonian for this system is given by~\cite{shen2007strongly,roy2017colloquium,shen2005coherent,bradford2012single,witthaut2010photon}:
\begin{equation}
\hat{H} = \hat{H}_p + \hat{H}_a + \hat{H}_i
\label{hamiltonian}
\end{equation}
where

\begin{align}
\begin{split}
\hat{H}_p &= \int \mathrm{d} x \, \hat{c}_R ^\dagger (x) \left(\omega_0 -i v_g \ddx{}\right) \hat{c}_R  (x)  \\
&+ \int \mathrm{d} x \, \hat{c}_L^\dagger (x) \left(\omega_0 +i v_g \ddx{}\right)  \hat{c}_L (x)\\
\hat{H}_a &= \sum_{n=1,2,3}  E_{n}  \ket{n}\bra{n}\\
\hat{H}_i &= \int \mathrm{d} x \, \delta(x)\big( \sum_{n={1,3}} g_n ( \hat{c}_R ^\dagger(x) + \hat{c}_L^\dagger(x) )\ket{n}\bra{2} + \text{h.c.} \big)
\end{split}
\end{align}

are the photonic, atomic and interaction terms of the Hamiltonian respectively. In this paper we set $\hbar = 1$.  Here, $v_g$ is the group velocity of the photons travelling in the waveguide, $\hat{c}_R ^\dagger (x)$ and $\hat{c}_L^\dag (x)$ are the creation operators for a right-going and left-going photon at position $x$ in the waveguide. We have chosen to linearize the dispersion relation of the waveguide around the frequency $\omega_0 = E_2 - E_1$. The total excitation number operator $\hat{N}= \int d x c_{L}^{\dagger}(x) c_{L}(x)+\int d x c_{R}^{\dagger}(x) c_{R}(x)+|2\rangle\langle 2|$  commutes with the Hamiltonian $\hat{H}$. The basis states corresponding to the eigenstates of the number operator $\hat{N}$ in the single-excitation regime are:
\begin{equation}
\begin{split}
\{\hat{c}_{R}^{\dagger}(x)|1, v a c\rangle, \hat{c}_{L}^{\dagger}(x)|1, v a c\rangle,\\ \hat{c}_{R}^{\dagger}(x)|3, v a c\rangle, \hat{c}_{L}^{\dagger}(x)|3, v a c\rangle,|2, v a c\rangle \}
\end{split}
\label{basis}
\end{equation}
Since $[H, N] = 0$, we also use these basis states in Eq.~\eqref{basis} to expand the scattering eigenstate of the Hamiltonian with a total energy $E$ in the single-excitation regime as~\cite{shen2005coherent}:
\begin{equation}
\begin{split}
\ket{E} &= \int \mathrm{d} x \, \Big( \left(\phi_{1, R}(x) \hat{c}_{R}^{\dagger}(x)+\phi_{1, L}(x) \hat{c}_{L}^{\dagger}(x)\right)|1, v a c\rangle \\
&+     \left(\phi_{3, R}(x) \hat{c}_{R}^{\dagger}(x)+\phi_{3, L}(x) \hat{c}_{L}^{\dagger}(x)\right)|3, v a c\rangle \Big)  \\
 &+ e_k \ket{2, vac} .
\end{split}
\label{ansatz}
\end{equation}
Here the $\phi$'s are the single-photon field amplitudes. The subscripts of $\phi$ denotes whether the photon is right or left-going, and the state of the atom. For example $\phi_{1, R}$ describes the case where the photon is right-going, and the atom is in state $\ket{1}$.  $e_k$ is the probability amplitude of the three-level atom being in the excited $\ket{2}$ state. $\ket{n,vac}$ is a vacuum state with zero photons and the atom is in state $\ket{n}$.  Using Eq.~\eqref{ansatz} and Eq.~\eqref{hamiltonian} with the time-independent Schrodinger equation $H \ket{E} = E\ket{E}$ gives the following set of linear differential equations:
\begin{align}
E \phi_{1, R}(x) &= - i v_g \frac{\partial \phi_{1, R}(x)}{\partial x}+\phi_{1, R}(x) (E_{1}+\omega_0)+e_{k} g_{1} \delta(x)  \label{diff_eqns_a}
\\
E \phi_{1, L}(x) &=  i v_{g} \frac{\partial \phi_{1, L}(x)}{\partial x}+\phi_{1, L}(x) (E_{1}+\omega_0)+e_{k} g_{1} \delta(x)  \\
E \phi_{3, R}(x) &=  -i v_{g} \frac{\partial \phi_{3, R}(x)}{\partial x}+\phi_{3, R}(x) (E_{3}+\omega_0) +e_{k} g_{3} \delta(x)  \label{diff_eqns_c}\\
E \phi_{3, L}(x)&= i v_{g} \frac{\partial \phi_{3, L}(x)}{\partial x}+\phi_{3, L}(x) (E_{3}+\omega_0) +e_{k} g_{3} \delta(x)  \\
E e_{k} &= E_{2} e_{k}+\sum_{n=1,3} g_{n}\left(\phi_{n, R}(0)+\phi_{n, L}(0)\right).
\label{diff_eqns_b}
\end{align}
We first consider a scenario where the atom is initially in the $\ket{1}$ state and a right-going photon is incoming from the left. Our photon wave-function ansatz is:
\begin{align}
&\phi_{1, R}(x)=e^{i k_{1} x}\left(\Theta(-x)+t_{1} \Theta(x)\right) \\
&\phi_{3, R}(x)=e^{i k_{3} x} t_{3} \Theta(x) \\
&\phi_{1, L}(x)=e^{-i k_{1}^{\prime} x} r_{1} \Theta(-x) \\
&\phi_{3, L}(x)=e^{-i k_{3}^{\prime} x} r_{3} \Theta(-x)
\end{align}
$|t_1|^2, |t_3|^2$ ($|r_1|^2, |r_3|^2$) gives the probability that a photon will be transmitted (reflected) leaving the atom in the $\ket{1}$ or $\ket{3}$ state respectively when the atom was initially in $\ket{1}$. Here, $\Theta(x)$ is the Heaviside function with $\Theta(0) = 0.5$. Using our ansatz and Eq.~\eqref{diff_eqns_a} with $x<0$ and Eq.~\eqref{diff_eqns_c} with $x>0$, we find that $k_1 = (E-E_1-\omega_0)/v_g$, and $k_3 = (E-E_3-\omega_0)/v_g$. Substituting these photon wave-functions into Eq.~\eqref{diff_eqns_a} to~\eqref{diff_eqns_b} gives the following solutions:
\begin{align}
t_{1} &=1-\frac{\gamma_{1}}{i\left(E_{2}-E\right) +\gamma_{1}+\gamma_{3}}\label{sol_1a} \\
r_{1} &=\frac{-\gamma_{1}}{i\left(E_{2}-E\right) +\gamma_{1}+\gamma_{3}} \\
t_{3} &=\frac{-\sqrt{\gamma_{1} \gamma_{3}}}{i\left(E_{2}-E\right) +\gamma_{1}+\gamma_{3}} \\
r_{3} &=\frac{-\sqrt{\gamma_{1} \gamma_{3}}}{i\left(E_{2}-E\right) +\gamma_{1}+\gamma_{3}}\\
e_{k}&=\frac{-i \sqrt{\gamma_{1} v_{g}}}{i\left(E_{2}-E\right)+\gamma_{1}+\gamma_{3}}\label{sol_1b}
\end{align}
Here $\gamma_{1,3} = g_{1,3}^2/v_g$ are the decay rates of the state $\ket{2}$ into state $\ket{1}$ and state $\ket{3}$  respectively.

In the second scenario, we consider the case where the atom is initially in the $\ket{3}$ state and a right-going photon incoming from the left. Our photon wave-function ansatz is
\begin{align}
&\phi_{1, R}(x)=e^{i k_{1} x} t_{1}' \Theta(x) \\
&\phi_{3, R}(x)=e^{i k_{3} x}\left(\Theta(-x)+t_{3}' \Theta(x)\right) \\
&\phi_{1, L}(x)=e^{i k_{1}^{\prime} x} r_{1}' \Theta(-x) \\
&\phi_{3, L}(x)=e^{i k_{3}^{\prime} x} r_{3}' \Theta(-x)
\end{align}
$|t_1'|^2, |t_3'|^2$ ($|r_1'|^2, |r_3'|^2$) gives the probability that a photon will be transmitted (reflected) leaving the atom in the $\ket{1}$ or $\ket{3}$ state respectively when the atom was initially in $\ket{3}$. We can obtain our solutions by substituting this ansatz into the linear differential equations in Eq.~\eqref{diff_eqns_a} to~\eqref{diff_eqns_b} or by simply replacing the labels $1 \leftrightarrow 3$ (due to parity symmetry~\cite{shen2009theory}) in Eq.~\eqref{sol_1a} to Eq.~\eqref{sol_1b}.
\begin{align}
t_{1}' &=\frac{-\sqrt{\gamma_{1} \gamma_{3}}}{i\left(E_{2}-E\right)+\gamma_{1}+\gamma_{3}} \\
r_{1}' &=\frac{-\sqrt{\gamma_{1} \gamma_{3}}}{i\left(E_{2}-E\right) +\gamma_{1}+\gamma_{3}} \\
t_{3}' &=1-\frac{\gamma_{3}}{i\left(E_{2}-E\right) +\gamma_{1}+\gamma_{3}} \\
r_{3} '&=\frac{-\gamma_{3}}{i\left(E_{2}-E\right)+\gamma_{1}+\gamma_{3}} \\
e_{k}^{\prime}&=\frac{-i \sqrt{\gamma_{3} v_{g}}}{i\left(E_{2}-E\right)+\gamma_{1}+\gamma_{3}}
\label{ekp}
\end{align}
We can now summarize the scattering process for a right-going photon of energy $\omega_0 + v_g k$ interacting with a $\Lambda$ atom initially in the $\ket{1}$ state:
\begin{equation}
\begin{split}
\ket{k,1}_R \rightarrow t_1(E) \ket{k,1}_R + t_3 (E) \ket{k-\Delta,3}_R + \\r_1(E) \ket{k,1}_L+ r_3 (E) \ket{k-\Delta,3}_L
\label{twowayscat1}
\end{split}
\end{equation}
where $E= E_1 + v_g k+\omega_0, \Delta = (E_3-E_1)/v_g$, and the label $k$ inside the ket denotes a plane wave of the form $e^{ik x}$. Similarly, for a right-going photon of energy $\omega_0 +v_g (k-\Delta)$ interacting with a $\Lambda$ atom initially in the $\ket{3}$ state:
\begin{equation}
\begin{split}
\ket{k-\Delta,3}_R \rightarrow t_1'(E) \ket{k,1}_R + t_3' (E) \ket{k-\Delta,3}_R + \\
r_1'(E) \ket{k,1}_L + r_3' (E) \ket{k-\Delta,3}_L.
\label{twowayscat2}
\end{split}
\end{equation}
The scattering processes for both cases are shown in Fig.~\ref{maysetup}(b). We note that there are two transmission and reflection output channels for both scenarios.\\

\begin{figure*}[t!]
\centering\includegraphics[width=\textwidth]{interference_terms.png}
\caption{Output transmission and reflection for interference terms Eq.~\eqref{int1} to~\eqref{int4} as a function of normalized total energy $(E_2-E)/\gamma$ and initial relative photon phase $\phi_p$ between the two frequency superposition terms. Here, $A= B= C= D = 1/\sqrt{2}, \phi_a= 0, \omega_0/\gamma=10, E_1 = 0, E_2/\gamma = 10, E_3/\gamma = 5,\gamma_1= \gamma_3=\gamma/2$.}
\label{twoway_full}
\end{figure*}
\begin{figure}[h]
\centering
\includegraphics[scale=0.4]{trans_phi.png}
\caption{Transmission for the scattering output terms (a) $ p_{\ket{k, 1}_R}$ and (b) $ p_{\ket{k-\Delta, 3}_R}$ that exhibit interference as a function of normalized total energy $(E -E_2)/\gamma$ where $A= B= C= D = 1/\sqrt{2}, \phi_a= 0, \omega_0/\gamma=10, E_1 = 0, E_2/\gamma = 10, E_3/\gamma = 5, \gamma_1= \gamma_3=\gamma/2$ for $\phi_p = 0$ and $ \pi/2$.}
\label{trans_fortwo_phis}
\end{figure}

\section{Results}
Let us now consider a separable input state where the atom is in a superposition state $A \ket{1} + B e^{i\phi_a} \ket{3}$ and the photon is in a state $C \ket{k} + De^{i\phi_p} \ket{k-\Delta}$. which is a frequency superposition state with two frequencies $\omega_0 + v_g k$ and $\omega_0 + v_g (k-\Delta)$. With our choice of $\omega_0$ the two frequencies are on resonance with the two atomic transitions when $k = 0$.  $A, B, C, D$ are assumed to be real without any loss of generality. $\phi_a$ is the relative phase between the atomic states and $\phi_p$ is the relative phase between the two photon frequencies. Our input state is then made of the tensor product of the photon state $\ket{\psi_p}$ and the atomic state $\ket{\psi_a}$:
\begin{align}
\begin{split}
\left|\psi_{\text {in }}\right\rangle &= \ket{\psi_{p}}\otimes \ket{\psi_{a}}\\
&= (A|k\rangle_R+Be^{i\phi_p}|k-\Delta\rangle_R) \otimes (C|1\rangle+D e^{i\phi_a}|3\rangle) \\
 &=AC|k,1\rangle_R+ AD e^{i\phi_a}|k,3\rangle_R+BCe^{i\phi_p}|k-\Delta,1\rangle_R \\
 &+ BDe^{i(\phi_a+\phi_p)}|k-\Delta,3\rangle_R
\label{input}
\end{split}
\end{align}
as depicted in Fig.~\ref{maysetup}(a). The energies of the four components in Eq.~\eqref{input} are $E_1+\omega_0 + v_g k$, $E_1+\omega_0 + v_g(k-\Delta)$, $E_3+\omega_0 + v_gk$ and $E_3+\omega_0 + v_g(k-\Delta)$ respectively. However, with the choice of the two photon frequencies as described above, the $|k,1\rangle_R$ and $|k-\Delta,3\rangle_R$ components in the input state have the same energy $ E_{int} \equiv E_1+\omega_0 + v_g k =E_3+\omega_0 + v_g(k-\Delta)$ and interference between the scattering processes starting from these two components can occur.

The main result of this paper is that this interference allows us to measure the relative phase relation between $\phi_p$ and $\phi_a$. Since the total energy is conserved for each term after scattering, we focus only on the subspace with energy $E_{int}$. The scattering of the component $\left|\psi_{\mathrm{in}}\right\rangle_{int} = A C|k, 1\rangle_{R} + B D e^{i\left(\phi_{a}+\phi_{p}\right)}|k-\Delta, 3\rangle_{R}$ yields four different outcomes with the following probabilities:
\begin{align}
p_{\ket{k, 1}_R} &=\left|A C t_{1}\left(E_{int}\right)+B D e^{i\left(\phi_{p}+\phi_{a}\right)} t_{1}^{\prime}\left(E_{int}\right)\right|^{2} \label{int1}\\
p_{\ket{k, 1}_L} &=\left|A C r_{1}\left(E_{int}\right)+B D e^{i\left(\phi_{p}+\phi_{a}\right)} r_{1}^{\prime}\left(E_{int}\right)\right|^{2} \label{int2}\\
p_{\ket{k-\Delta, 3}_R} &=\left|A C t_{3}\left(E_{int}\right)+B D e^{i\left(\phi_{p}+\phi_{a}\right)} t_{3}^{\prime}\left(E_{int}\right)\right|^{2} \label{int3}\\
p_{\ket{k-\Delta, 3}_L} &=\left|A C r_{3}\left(E_{int}\right)+B D e^{i\left(\phi_{p}+\phi_{a}\right)} r_{3}^{\prime}\left(E_{int}\right)\right|^{2} \label{int4}
\end{align}
Here, the subscript refers to the outgoing states. We note that these terms all contain information about $\phi_p + \phi_a$ and they can be measured by a joint projective measurement of the atom and the photon onto the appropriate states.  \\

In Fig.~\ref{twoway_full}, we plot the various probabilities in Eq.~\eqref{int1} to \eqref{int4} as a function of the normalized total energy $(E-E_2)/\gamma$ and the input photon phase $\phi_p$. In Fig.~\ref{trans_fortwo_phis}, we plot a cross-section from Fig.~\ref{twoway_full} at $\phi_p=0,\pi/2$ for the two transmission terms Eq.~\eqref{int1} and Eq.~\eqref{int3} to highlight the $\phi_p$ dependence. Here,
\begin{align}
\begin{split}
A&= B= C= D = 1/\sqrt{2}, \phi_a= 0, \omega_0/\gamma=10, E_1 = 0,\\
 E_2/\gamma &= 10, E_3/\gamma = 5, \gamma_1=\gamma_3=\gamma/2
\label{parameter}
\end{split}
\end{align}
The sum of the panels in Fig.~\ref{twoway_full} do not add to 1 because the non-interference terms must also be included. We see that the reflection terms are symmetric about $\phi_p=0$ while the transmission terms are asymmetric about $\phi_p=0$.  This can be understood by decomposing the input photon state into the scattering mode that maximally excites the atom and the non-scattering mode that does not excite the atom. For arbitrary $\phi_a, \phi_p, A, C$ ($B$ and $D$ are constrained to be $B= \sqrt{1-|A|^2}$ and $D= \sqrt{1-|C|^2}$), and in the sub-space with energy $E_{int}$, we can solve for the non-scattering mode by setting the total excitation probability to be zero:
\begin{equation}
e_{k,\text{total}} = AC (e_k(E_{int}) + BD e^{i(\phi_a+\phi_p)}e_k'(E_{int})) =0
\end{equation}
from which one obtains
\begin{equation}
-AC \sqrt{\gamma_1}=\sqrt{1-|A|^2}\sqrt{1-|C|^2} e^{i(\phi_a+\phi_p)}\sqrt{\gamma_3}.
\end{equation}
For our parameter set in Eq.~\eqref{parameter}, this becomes $-1=e^{i\phi_p}$. Hence, the non-scattering mode $\ket{\psi_n}$ is
\begin{equation}
\ket{\psi_n}= \frac{1}{\sqrt{2}} \ket{k,1}_R - \frac{1}{\sqrt{2}} \ket{k-\Delta,3}_R .
\end{equation}
Applying the scattering transformations in Eq.~\eqref{twowayscat1} and Eq.~\eqref{twowayscat2} to $\ket{\psi_n}$ gives back the same state:
\begin{equation}
\ket{\psi_n} \rightarrow  \frac{1}{\sqrt{2}} \ket{k,1}_R - \frac{1}{\sqrt{2}} \ket{k-\Delta,3}_R .
\end{equation}
We see that the non-scattering mode has no contribution to the reflection, and its transmission has no frequency dependence. The scattering mode is orthogonal to the non-scattering mode, so for this parameter set in Eq.~\eqref{parameter},  it corresponds to when $\phi_p=0$ and can be written as
\begin{equation}
\ket{\psi_s}= \frac{1}{\sqrt{2}} \ket{k,1}_R + \frac{1}{\sqrt{2}} \ket{k-\Delta,3}_R .
\end{equation}
The scattering output of $\ket{\psi_s}$ is
\begin{equation}
\begin{split}
\ket{\psi_s}&\rightarrow  \frac{1}{\sqrt{2}}\Big(\left(1- \frac{\gamma}{i(E_2-E)+\gamma}\right)\ket{k,1}_R \\
&- \frac{\gamma}{i(E_2-E)+\gamma}\ket{k,1}_L \\
 &+\left(1- \frac{\gamma}{i(E_2-E)+\gamma}\right)\ket{k-\Delta,3}_R \\
 &- \frac{\gamma}{i(E_2-E)+\gamma}\ket{k-\Delta,3}_L\Big).
\end{split}
\label{psi_s}
\end{equation}
We see that the transmission coefficient of $\ket{\psi_s}$ is a constant minus a Lorentzian and the reflection is a Lorentzian.

In the subspace with the energy $E = E_{int}$, any arbitrary input state can be decomposed into a linear superposition of $\ket{\psi_s}$  and $\ket{\psi_n}$, as they also form a basis of the Hilbert space for an incoming right-going photon. In this parameter set of Eq.~\eqref{parameter} when $\phi_p=0$, the contribution is only from $\ket{\psi_s}$. Thus, the transmission is symmetric about $(E_2-E)$, and the reflection is a Lorentzian as expected from Eq.~\eqref{psi_s}. As $\ket{\psi_n}$ does not contribute to the reflection, the reflection remains a Lorentzian for all $\phi_p$. On the other hand when $\phi_p \neq 0,\pm  \pi$, the transmission has components from both $\ket{\psi_s}$ and $\ket{\psi_n}$. The interference of the transmission from the scattering mode $\ket{\psi_s}$ and the transmission from the non-scattering mode $\ket{\psi_n}$ gives rise to a Fano-lineshape~\cite{fan2002analysis}. This explains the asymmetric transmission about $(E_2-E)=0$ for $\phi_p\neq 0, \pm \pi$. The antisymmetry about $\phi_p$ for the transmission plots in Fig.~\ref{twoway_full} is because changing the sign of $\phi_p$ changes the sign of contribution from $\ket{\psi_s}$.

We note that the existence of a non-scattering input mode that does not excite the atom can be alternatively argued as follows: The atomic excitation $e_{k,\text{total}}$ is a linear function of the amplitudes in the states of $p_{\ket{k, 1}_R}$ and $p_{\ket{k-\Delta, 3}_R}$. Therefore, there is always a particular choice of the amplitudes for which $e_{k,\text{total}} = 0$.

\section{Conclusion}
In summary, we have shown that the scattering of a single photon in a superposition of two photon frequencies with a $\Lambda$ atom in a superposition of two non-degenerate ground states depends on the initial photon phase between the two photon frequencies (and the initial atomic phase between the ground states). Our results show that three-level $\Lambda$ atoms can be used as a relative photon phase detector for single photons in a superposition state, offering a simpler alternative to other experimental platforms using nonlinear optics~\cite{clemmen2016ramsey} and electro-optic modulators~\cite{lu2020fully}.

\section*{Acknowledgements}
This work is supported by a Vannevar Bush Faculty Fellowship from the U. S. Department of Defense (Grant No. N00014-17-1-3030). JZ acknowledges A. Poddubny for useful comments.

\nocite{apsrev41Control}
\bibliographystyle{apsrev4}

\section{Introduction}
There have been many studies on the generation~\cite{zakka2011quantum,clemmen2016ramsey,aguayo2022integrated} or manipulation~\cite{tanzilli2005photonic,karnieli2018frequency,lukens2017frequency,lu2020fully,lu2019controlled,lu2018quantum,olisager2010frequency} of a single photon in a superposition of two frequencies (or `colours'). These frequency-superposition states have been called `color qubits'~\cite{aguayo2022integrated}, bichromatic photons~\cite{clemmen2016ramsey}, or frequency-bin qubits~\cite{lukens2017frequency}. Such states can be generated in the process of quantum frequency conversion, where a photon's frequency is converted while its quantum information is maintained, as experimentally demonstrated three decades ago~\cite{huang1992observation}. These states also arise in quantum frequency mixing (where different superposition states are converted to and from each other) which has been demonstrated using $\chi^{(2)}$ nonlinearities~\cite{kobayashi2016frequency,kobayashi2017coherent}, $\chi^{(3)}$ nonlinearities~\cite{mcguiness2010quantum,clemmen2016ramsey,dmitriev2017quantum} as well as electro-optic modulation~\cite{lu2018quantum,lu2020fully}.

The manipulation of frequency superposition states at the single-photon level has generated interest for its potential applications in quantum information processing~\cite{lukens2017frequency}. Building on the work in linear optical quantum computation~\cite{knill2001scheme}, frequency-bin qubits have been proposed as an alternative to other photonic qubits such as those based on polarization. Polarization-based qubits can only have a two-dimensional Hilbert space, whereas frequency qubits can have a Hilbert space with much higher dimension. Frequency qubits may also be scalable for on-chip photonics~\cite{ultrabroadband2021javid}. In recent years, it has been theoretically proposed and experimentally verified that it is possible to achieve arbitrary single-qubit gates in frequency space~\cite{lu2020fully,lu2019controlled}. Aside from quantum information processing applications, frequency superposition states may also be useful for quantum metrology~\cite{giovannetti2011advances} or phase-spectroscopy~\cite{rinehart2012quantitative} at the low-light level.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{setup.png}
\caption{(a) A $\Lambda$-type 3-level interacting with a photon propagating in single-mode waveguide (highlighted as gray). The photon is in frequency-superposition state. The photon frequencies $k,k-\Delta$ are in resonance with the atomic transition frequencies $E_2-E_1$ and $E_2-E_3$. (b) Transmission and reflection pathways for atom initially in $\ket{k,1}$ and $\ket{k-\Delta,3}$, where $k$ is kept general. Note that both scenarios have two scattering output channels, in which the atoms are in either the $\ket{1}$ or $\ket{3}$ state.}
\label{maysetup}
\end{figure}

Although the generation and manipulation of single-photon frequency superposition states has been well-studied~\cite{yesharim2022observation,zakka2011quantum,clemmen2016ramsey,aguayo2022integrated,tanzilli2005photonic,karnieli2018frequency,lukens2017frequency,lu2020fully,lu2019controlled,lu2018quantum,olisager2010frequency}, there is less attention devoted to the measurement of the relative phase between different frequency components of such states. Most existing works focus on measuring the amplitude spectrum of the state, where the phase information is lost. The relative phase of two frequency components of a single photon was measured in Ref.~\cite{clemmen2016ramsey} using single-photon Ramsey interference. Here, they generated arbitrary frequency-superposition states using Bragg scattering four-wave mixing which is a $\chi^{(3)}$ nonlinear process. The relative phase between different frequency components could be measured because the outcome of their projective measurement depends on this relative phase. A related study uses electro-optic modulators instead of nonlinear processes~\cite{lu2020fully}. Their set-up also allows them to achieve arbitrary frequency qubit rotations on the Bloch sphere and the output state can then be measured using quantum state tomography, where the relative phase between frequency components can be determined using Pauli measurements.

In this study, we propose a novel way to measure the relative phase of different frequency components of a single photon using a setup in waveguide quantum electrodynamics~\cite{sheremet2021waveguide,roy2017colloquium,turschmann2019coherent,chang2018colloquium}. Our proposed set-up consists of a three-level atom coupled to a one-dimensional waveguide as seen in Fig.~\ref{maysetup}(a). The atom is of the $\Lambda$ type with a two-dimensional ground state manifold spanned by eigenstates $\ket{1}$ and $\ket{3}$ at energy $E_1$ and $E_3$, respectively, and an excited state $\ket{2}$ at energy $E_2$.   It has been shown that one can prepare the atom to be in an arbitrary ground state in this two-dimensional manifold~\cite{djotyan2008multiphoton,he2009arbitrary}. Here we scatter a single-photon with two frequency components near $E_2-E_1$ and $E_2-E_3$ (where $\hbar=1$) against the atom. We show that with a suitable choice of the atomic ground state, the outcome of the scattering process can be used to determine the relative phase of the two photon frequency components.
Scattering for a $\Lambda$ atom in a superposition of ground states was studied in Ref.~\cite{martens2013photon}. Quantum state transfer~\cite{chan2022quantum} and quantum frequency conversion~\cite{bradford2012single} has also been studied in a $\Lambda$-type atom. However, none of these studies specifically study the dependence of the relative photon phase on the transport for a $\Lambda$ atom coupled to a waveguide, which is the focus of this work.\\

\section{Model}

To treat the system of Fig.~\ref{maysetup}(a), we use a real space formalism originally developed in Ref.~\cite{shen2005coherent} and applied to three level atoms in Ref.~\cite{witthaut2010photon}. Let $g_1, g_3$ be the atom-photon coupling constants for the $\ket{1} \rightarrow \ket{2}$ and $\ket{3} \rightarrow \ket{2}$ transitions and assume the $\ket{1} \rightarrow \ket{3}$ transition is forbidden. The Hamiltonian for this system is given by~\cite{shen2007strongly,roy2017colloquium,shen2005coherent,bradford2012single,witthaut2010photon}:
\begin{equation}
\hat{H} = \hat{H}_p + \hat{H}_a + \hat{H}_i
\label{hamiltonian}
\end{equation}
where

\begin{align}
\begin{split}
\hat{H}_p &= \int \mathrm{d} x \, \hat{c}_R ^\dagger (x) \left(\omega_0 -i v_g \ddx{}\right) \hat{c}_R  (x)  \\
&+ \int \mathrm{d} x \, \hat{c}_L^\dagger (x) \left(\omega_0 +i v_g \ddx{}\right)  \hat{c}_L (x)\\
\hat{H}_a &= \sum_{n=1,2,3}  E_{n}  \ket{n}\bra{n}\\
\hat{H}_i &= \int \mathrm{d} x \, \delta(x)\big( \sum_{n={1,3}} g_n ( \hat{c}_R ^\dagger(x) + \hat{c}_L^\dagger(x) )\ket{n}\bra{2} + \text{h.c.} \big)
\end{split}
\end{align}

are the photonic, atomic and interaction terms of the Hamiltonian respectively. In this paper we set $\hbar = 1$.  Here, $v_g$ is the group velocity of the photons travelling in the waveguide, $\hat{c}_R ^\dagger (x)$ and $\hat{c}_L^\dag (x)$ are the creation operators for a right-going and left-going photon at position $x$ in the waveguide. We have chosen to linearize the dispersion relation of the waveguide around the frequency $\omega_0 = E_2 - E_1$. The total excitation number operator $\hat{N}= \int d x c_{L}^{\dagger}(x) c_{L}(x)+\int d x c_{R}^{\dagger}(x) c_{R}(x)+|2\rangle\langle 2|$  commutes with the Hamiltonian $\hat{H}$. The basis states corresponding to the eigenstates of the number operator $\hat{N}$ in the single-excitation regime are:
\begin{equation}
\begin{split}
\{\hat{c}_{R}^{\dagger}(x)|1, v a c\rangle, \hat{c}_{L}^{\dagger}(x)|1, v a c\rangle,\\ \hat{c}_{R}^{\dagger}(x)|3, v a c\rangle, \hat{c}_{L}^{\dagger}(x)|3, v a c\rangle,|2, v a c\rangle \}
\end{split}
\label{basis}
\end{equation}
Since $[H, N] = 0$, we also use these basis states in Eq.~\eqref{basis} to expand the scattering eigenstate of the Hamiltonian with a total energy $E$ in the single-excitation regime as~\cite{shen2005coherent}:
\begin{equation}
\begin{split}
\ket{E} &= \int \mathrm{d} x \, \Big( \left(\phi_{1, R}(x) \hat{c}_{R}^{\dagger}(x)+\phi_{1, L}(x) \hat{c}_{L}^{\dagger}(x)\right)|1, v a c\rangle \\
&+     \left(\phi_{3, R}(x) \hat{c}_{R}^{\dagger}(x)+\phi_{3, L}(x) \hat{c}_{L}^{\dagger}(x)\right)|3, v a c\rangle \Big)  \\
 &+ e_k \ket{2, vac} .
\end{split}
\label{ansatz}
\end{equation}
Here the $\phi$'s are the single-photon field amplitudes. The subscripts of $\phi$ denotes whether the photon is right or left-going, and the state of the atom. For example $\phi_{1, R}$ describes the case where the photon is right-going, and the atom is in state $\ket{1}$.  $e_k$ is the probability amplitude of the three-level atom being in the excited $\ket{2}$ state. $\ket{n,vac}$ is a vacuum state with zero photons and the atom is in state $\ket{n}$.  Using Eq.~\eqref{ansatz} and Eq.~\eqref{hamiltonian} with the time-independent Schrodinger equation $H \ket{E} = E\ket{E}$ gives the following set of linear differential equations:
\begin{align}
E \phi_{1, R}(x) &= - i v_g \frac{\partial \phi_{1, R}(x)}{\partial x}+\phi_{1, R}(x) (E_{1}+\omega_0)+e_{k} g_{1} \delta(x)  \label{diff_eqns_a}
\\
E \phi_{1, L}(x) &=  i v_{g} \frac{\partial \phi_{1, L}(x)}{\partial x}+\phi_{1, L}(x) (E_{1}+\omega_0)+e_{k} g_{1} \delta(x)  \\
E \phi_{3, R}(x) &=  -i v_{g} \frac{\partial \phi_{3, R}(x)}{\partial x}+\phi_{3, R}(x) (E_{3}+\omega_0) +e_{k} g_{3} \delta(x)  \label{diff_eqns_c}\\
E \phi_{3, L}(x)&= i v_{g} \frac{\partial \phi_{3, L}(x)}{\partial x}+\phi_{3, L}(x) (E_{3}+\omega_0) +e_{k} g_{3} \delta(x)  \\
E e_{k} &= E_{2} e_{k}+\sum_{n=1,3} g_{n}\left(\phi_{n, R}(0)+\phi_{n, L}(0)\right).
\label{diff_eqns_b}
\end{align}
We first consider a scenario where the atom is initially in the $\ket{1}$ state and a right-going photon is incoming from the left. Our photon wave-function ansatz is:
\begin{align}
&\phi_{1, R}(x)=e^{i k_{1} x}\left(\Theta(-x)+t_{1} \Theta(x)\right) \\
&\phi_{3, R}(x)=e^{i k_{3} x} t_{3} \Theta(x) \\
&\phi_{1, L}(x)=e^{-i k_{1}^{\prime} x} r_{1} \Theta(-x) \\
&\phi_{3, L}(x)=e^{-i k_{3}^{\prime} x} r_{3} \Theta(-x)
\end{align}
$|t_1|^2, |t_3|^2$ ($|r_1|^2, |r_3|^2$) gives the probability that a photon will be transmitted (reflected) leaving the atom in the $\ket{1}$ or $\ket{3}$ state respectively when the atom was initially in $\ket{1}$. Here, $\Theta(x)$ is the Heaviside function with $\Theta(0) = 0.5$. Using our ansatz and Eq.~\eqref{diff_eqns_a} with $x<0$ and Eq.~\eqref{diff_eqns_c} with $x>0$, we find that $k_1 = (E-E_1-\omega_0)/v_g$, and $k_3 = (E-E_3-\omega_0)/v_g$. Substituting these photon wave-functions into Eq.~\eqref{diff_eqns_a} to~\eqref{diff_eqns_b} gives the following solutions:
\begin{align}
t_{1} &=1-\frac{\gamma_{1}}{i\left(E_{2}-E\right) +\gamma_{1}+\gamma_{3}}\label{sol_1a} \\
r_{1} &=\frac{-\gamma_{1}}{i\left(E_{2}-E\right) +\gamma_{1}+\gamma_{3}} \\
t_{3} &=\frac{-\sqrt{\gamma_{1} \gamma_{3}}}{i\left(E_{2}-E\right) +\gamma_{1}+\gamma_{3}} \\
r_{3} &=\frac{-\sqrt{\gamma_{1} \gamma_{3}}}{i\left(E_{2}-E\right) +\gamma_{1}+\gamma_{3}}\\
e_{k}&=\frac{-i \sqrt{\gamma_{1} v_{g}}}{i\left(E_{2}-E\right)+\gamma_{1}+\gamma_{3}}\label{sol_1b}
\end{align}
Here $\gamma_{1,3} = g_{1,3}^2/v_g$ are the decay rates of the state $\ket{2}$ into state $\ket{1}$ and state $\ket{3}$  respectively.

In the second scenario, we consider the case where the atom is initially in the $\ket{3}$ state and a right-going photon incoming from the left. Our photon wave-function ansatz is
\begin{align}
&\phi_{1, R}(x)=e^{i k_{1} x} t_{1}' \Theta(x) \\
&\phi_{3, R}(x)=e^{i k_{3} x}\left(\Theta(-x)+t_{3}' \Theta(x)\right) \\
&\phi_{1, L}(x)=e^{i k_{1}^{\prime} x} r_{1}' \Theta(-x) \\
&\phi_{3, L}(x)=e^{i k_{3}^{\prime} x} r_{3}' \Theta(-x)
\end{align}
$|t_1'|^2, |t_3'|^2$ ($|r_1'|^2, |r_3'|^2$) gives the probability that a photon will be transmitted (reflected) leaving the atom in the $\ket{1}$ or $\ket{3}$ state respectively when the atom was initially in $\ket{3}$. We can obtain our solutions by substituting this ansatz into the linear differential equations in Eq.~\eqref{diff_eqns_a} to~\eqref{diff_eqns_b} or by simply replacing the labels $1 \leftrightarrow 3$ (due to parity symmetry~\cite{shen2009theory}) in Eq.~\eqref{sol_1a} to Eq.~\eqref{sol_1b}.
\begin{align}
t_{1}' &=\frac{-\sqrt{\gamma_{1} \gamma_{3}}}{i\left(E_{2}-E\right)+\gamma_{1}+\gamma_{3}} \\
r_{1}' &=\frac{-\sqrt{\gamma_{1} \gamma_{3}}}{i\left(E_{2}-E\right) +\gamma_{1}+\gamma_{3}} \\
t_{3}' &=1-\frac{\gamma_{3}}{i\left(E_{2}-E\right) +\gamma_{1}+\gamma_{3}} \\
r_{3} '&=\frac{-\gamma_{3}}{i\left(E_{2}-E\right)+\gamma_{1}+\gamma_{3}} \\
e_{k}^{\prime}&=\frac{-i \sqrt{\gamma_{3} v_{g}}}{i\left(E_{2}-E\right)+\gamma_{1}+\gamma_{3}}
\label{ekp}
\end{align}
We can now summarize the scattering process for a right-going photon of energy $\omega_0 + v_g k$ interacting with a $\Lambda$ atom initially in the $\ket{1}$ state:
\begin{equation}
\begin{split}
\ket{k,1}_R \rightarrow t_1(E) \ket{k,1}_R + t_3 (E) \ket{k-\Delta,3}_R + \\r_1(E) \ket{k,1}_L+ r_3 (E) \ket{k-\Delta,3}_L
\label{twowayscat1}
\end{split}
\end{equation}
where $E= E_1 + v_g k+\omega_0, \Delta = (E_3-E_1)/v_g$, and the label $k$ inside the ket denotes a plane wave of the form $e^{ik x}$. Similarly, for a right-going photon of energy $\omega_0 +v_g (k-\Delta)$ interacting with a $\Lambda$ atom initially in the $\ket{3}$ state:
\begin{equation}
\begin{split}
\ket{k-\Delta,3}_R \rightarrow t_1'(E) \ket{k,1}_R + t_3' (E) \ket{k-\Delta,3}_R + \\
r_1'(E) \ket{k,1}_L + r_3' (E) \ket{k-\Delta,3}_L.
\label{twowayscat2}
\end{split}
\end{equation}
The scattering processes for both cases are shown in Fig.~\ref{maysetup}(b). We note that there are two transmission and reflection output channels for both scenarios.\\

\begin{figure*}[t!]
\centering\includegraphics[width=\textwidth]{interference_terms.png}
\caption{Output transmission and reflection for interference terms Eq.~\eqref{int1} to~\eqref{int4} as a function of normalized total energy $(E_2-E)/\gamma$ and initial relative photon phase $\phi_p$ between the two frequency superposition terms. Here, $A= B= C= D = 1/\sqrt{2}, \phi_a= 0, \omega_0/\gamma=10, E_1 = 0, E_2/\gamma = 10, E_3/\gamma = 5,\gamma_1= \gamma_3=\gamma/2$.}
\label{twoway_full}
\end{figure*}
\begin{figure}[h]
\centering
\includegraphics[scale=0.4]{trans_phi.png}
\caption{Transmission for the scattering output terms (a) $ p_{\ket{k, 1}_R}$ and (b) $ p_{\ket{k-\Delta, 3}_R}$ that exhibit interference as a function of normalized total energy $(E -E_2)/\gamma$ where $A= B= C= D = 1/\sqrt{2}, \phi_a= 0, \omega_0/\gamma=10, E_1 = 0, E_2/\gamma = 10, E_3/\gamma = 5, \gamma_1= \gamma_3=\gamma/2$ for $\phi_p = 0$ and $ \pi/2$.}
\label{trans_fortwo_phis}
\end{figure}

\section{Results}
Let us now consider a separable input state where the atom is in a superposition state $A \ket{1} + B e^{i\phi_a} \ket{3}$ and the photon is in a state $C \ket{k} + De^{i\phi_p} \ket{k-\Delta}$. which is a frequency superposition state with two frequencies $\omega_0 + v_g k$ and $\omega_0 + v_g (k-\Delta)$. With our choice of $\omega_0$ the two frequencies are on resonance with the two atomic transitions when $k = 0$.  $A, B, C, D$ are assumed to be real without any loss of generality. $\phi_a$ is the relative phase between the atomic states and $\phi_p$ is the relative phase between the two photon frequencies. Our input state is then made of the tensor product of the photon state $\ket{\psi_p}$ and the atomic state $\ket{\psi_a}$:
\begin{align}
\begin{split}
\left|\psi_{\text {in }}\right\rangle &= \ket{\psi_{p}}\otimes \ket{\psi_{a}}\\
&= (A|k\rangle_R+Be^{i\phi_p}|k-\Delta\rangle_R) \otimes (C|1\rangle+D e^{i\phi_a}|3\rangle) \\
 &=AC|k,1\rangle_R+ AD e^{i\phi_a}|k,3\rangle_R+BCe^{i\phi_p}|k-\Delta,1\rangle_R \\
 &+ BDe^{i(\phi_a+\phi_p)}|k-\Delta,3\rangle_R
\label{input}
\end{split}
\end{align}
as depicted in Fig.~\ref{maysetup}(a). The energies of the four components in Eq.~\eqref{input} are $E_1+\omega_0 + v_g k$, $E_1+\omega_0 + v_g(k-\Delta)$, $E_3+\omega_0 + v_gk$ and $E_3+\omega_0 + v_g(k-\Delta)$ respectively. However, with the choice of the two photon frequencies as described above, the $|k,1\rangle_R$ and $|k-\Delta,3\rangle_R$ components in the input state have the same energy $ E_{int} \equiv E_1+\omega_0 + v_g k =E_3+\omega_0 + v_g(k-\Delta)$ and interference between the scattering processes starting from these two components can occur.

The main result of this paper is that this interference allows us to measure the relative phase relation between $\phi_p$ and $\phi_a$. Since the total energy is conserved for each term after scattering, we focus only on the subspace with energy $E_{int}$. The scattering of the component $\left|\psi_{\mathrm{in}}\right\rangle_{int} = A C|k, 1\rangle_{R} + B D e^{i\left(\phi_{a}+\phi_{p}\right)}|k-\Delta, 3\rangle_{R}$ yields four different outcomes with the following probabilities:
\begin{align}
p_{\ket{k, 1}_R} &=\left|A C t_{1}\left(E_{int}\right)+B D e^{i\left(\phi_{p}+\phi_{a}\right)} t_{1}^{\prime}\left(E_{int}\right)\right|^{2} \label{int1}\\
p_{\ket{k, 1}_L} &=\left|A C r_{1}\left(E_{int}\right)+B D e^{i\left(\phi_{p}+\phi_{a}\right)} r_{1}^{\prime}\left(E_{int}\right)\right|^{2} \label{int2}\\
p_{\ket{k-\Delta, 3}_R} &=\left|A C t_{3}\left(E_{int}\right)+B D e^{i\left(\phi_{p}+\phi_{a}\right)} t_{3}^{\prime}\left(E_{int}\right)\right|^{2} \label{int3}\\
p_{\ket{k-\Delta, 3}_L} &=\left|A C r_{3}\left(E_{int}\right)+B D e^{i\left(\phi_{p}+\phi_{a}\right)} r_{3}^{\prime}\left(E_{int}\right)\right|^{2} \label{int4}
\end{align}
Here, the subscript refers to the outgoing states. We note that these terms all contain information about $\phi_p + \phi_a$ and they can be measured by a joint projective measurement of the atom and the photon onto the appropriate states.  \\

In Fig.~\ref{twoway_full}, we plot the various probabilities in Eq.~\eqref{int1} to \eqref{int4} as a function of the normalized total energy $(E-E_2)/\gamma$ and the input photon phase $\phi_p$. In Fig.~\ref{trans_fortwo_phis}, we plot a cross-section from Fig.~\ref{twoway_full} at $\phi_p=0,\pi/2$ for the two transmission terms Eq.~\eqref{int1} and Eq.~\eqref{int3} to highlight the $\phi_p$ dependence. Here,
\begin{align}
\begin{split}
A&= B= C= D = 1/\sqrt{2}, \phi_a= 0, \omega_0/\gamma=10, E_1 = 0,\\
 E_2/\gamma &= 10, E_3/\gamma = 5, \gamma_1=\gamma_3=\gamma/2
\label{parameter}
\end{split}
\end{align}
The sum of the panels in Fig.~\ref{twoway_full} do not add to 1 because the non-interference terms must also be included. We see that the reflection terms are symmetric about $\phi_p=0$ while the transmission terms are asymmetric about $\phi_p=0$.  This can be understood by decomposing the input photon state into the scattering mode that maximally excites the atom and the non-scattering mode that does not excite the atom. For arbitrary $\phi_a, \phi_p, A, C$ ($B$ and $D$ are constrained to be $B= \sqrt{1-|A|^2}$ and $D= \sqrt{1-|C|^2}$), and in the sub-space with energy $E_{int}$, we can solve for the non-scattering mode by setting the total excitation probability to be zero:
\begin{equation}
e_{k,\text{total}} = AC (e_k(E_{int}) + BD e^{i(\phi_a+\phi_p)}e_k'(E_{int})) =0
\end{equation}
from which one obtains
\begin{equation}
-AC \sqrt{\gamma_1}=\sqrt{1-|A|^2}\sqrt{1-|C|^2} e^{i(\phi_a+\phi_p)}\sqrt{\gamma_3}.
\end{equation}
For our parameter set in Eq.~\eqref{parameter}, this becomes $-1=e^{i\phi_p}$. Hence, the non-scattering mode $\ket{\psi_n}$ is
\begin{equation}
\ket{\psi_n}= \frac{1}{\sqrt{2}} \ket{k,1}_R - \frac{1}{\sqrt{2}} \ket{k-\Delta,3}_R .
\end{equation}
Applying the scattering transformations in Eq.~\eqref{twowayscat1} and Eq.~\eqref{twowayscat2} to $\ket{\psi_n}$ gives back the same state:
\begin{equation}
\ket{\psi_n} \rightarrow  \frac{1}{\sqrt{2}} \ket{k,1}_R - \frac{1}{\sqrt{2}} \ket{k-\Delta,3}_R .
\end{equation}
We see that the non-scattering mode has no contribution to the reflection, and its transmission has no frequency dependence. The scattering mode is orthogonal to the non-scattering mode, so for this parameter set in Eq.~\eqref{parameter},  it corresponds to when $\phi_p=0$ and can be written as
\begin{equation}
\ket{\psi_s}= \frac{1}{\sqrt{2}} \ket{k,1}_R + \frac{1}{\sqrt{2}} \ket{k-\Delta,3}_R .
\end{equation}
The scattering output of $\ket{\psi_s}$ is
\begin{equation}
\begin{split}
\ket{\psi_s}&\rightarrow  \frac{1}{\sqrt{2}}\Big(\left(1- \frac{\gamma}{i(E_2-E)+\gamma}\right)\ket{k,1}_R \\
&- \frac{\gamma}{i(E_2-E)+\gamma}\ket{k,1}_L \\
 &+\left(1- \frac{\gamma}{i(E_2-E)+\gamma}\right)\ket{k-\Delta,3}_R \\
 &- \frac{\gamma}{i(E_2-E)+\gamma}\ket{k-\Delta,3}_L\Big).
\end{split}
\label{psi_s}
\end{equation}
We see that the transmission coefficient of $\ket{\psi_s}$ is a constant minus a Lorentzian and the reflection is a Lorentzian.

In the subspace with the energy $E = E_{int}$, any arbitrary input state can be decomposed into a linear superposition of $\ket{\psi_s}$  and $\ket{\psi_n}$, as they also form a basis of the Hilbert space for an incoming right-going photon. In this parameter set of Eq.~\eqref{parameter} when $\phi_p=0$, the contribution is only from $\ket{\psi_s}$. Thus, the transmission is symmetric about $(E_2-E)$, and the reflection is a Lorentzian as expected from Eq.~\eqref{psi_s}. As $\ket{\psi_n}$ does not contribute to the reflection, the reflection remains a Lorentzian for all $\phi_p$. On the other hand when $\phi_p \neq 0,\pm  \pi$, the transmission has components from both $\ket{\psi_s}$ and $\ket{\psi_n}$. The interference of the transmission from the scattering mode $\ket{\psi_s}$ and the transmission from the non-scattering mode $\ket{\psi_n}$ gives rise to a Fano-lineshape~\cite{fan2002analysis}. This explains the asymmetric transmission about $(E_2-E)=0$ for $\phi_p\neq 0, \pm \pi$. The antisymmetry about $\phi_p$ for the transmission plots in Fig.~\ref{twoway_full} is because changing the sign of $\phi_p$ changes the sign of contribution from $\ket{\psi_s}$.

We note that the existence of a non-scattering input mode that does not excite the atom can be alternatively argued as follows: The atomic excitation $e_{k,\text{total}}$ is a linear function of the amplitudes in the states of $p_{\ket{k, 1}_R}$ and $p_{\ket{k-\Delta, 3}_R}$. Therefore, there is always a particular choice of the amplitudes for which $e_{k,\text{total}} = 0$.

\section{Conclusion}
In summary, we have shown that the scattering of a single photon in a superposition of two photon frequencies with a $\Lambda$ atom in a superposition of two non-degenerate ground states depends on the initial photon phase between the two photon frequencies (and the initial atomic phase between the ground states). Our results show that three-level $\Lambda$ atoms can be used as a relative photon phase detector for single photons in a superposition state, offering a simpler alternative to other experimental platforms using nonlinear optics~\cite{clemmen2016ramsey} and electro-optic modulators~\cite{lu2020fully}.

\section*{Acknowledgements}
This work is supported by a Vannevar Bush Faculty Fellowship from the U. S. Department of Defense (Grant No. N00014-17-1-3030). JZ acknowledges A. Poddubny for useful comments.

\nocite{apsrev41Control}
\bibliographystyle{apsrev4}

redpajama_set_name: RedPajamaArXiv


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The video for "Don't Let the Old Man In" features footage from the movie, which officially comes out December 14th and finds Eastwood playing a senior citizen who falls into a career of drug trafficking.


text:
using System.Collections;
using System.Collections.Generic;
using UnityEngine;
using UnityEngine.Timeline;
using UnityEngine.Playables;

public class AnjuEndLvl1 : Interactable {

    public DialogueOld dialogue;
    public Sprite characterSprite;
    public DialogueManagerOld dManager;
    public Sprite nameSprite;
    public GameObject portal;
    public PlayableDirector wormDirector;

    private void Start()
    {

    }
    public override void Interact()
    {
        base.Interact();
        TriggerDialogue();
    }

    private void FixedUpdate()
    {
        //InteractUpdate();
        Debug.Log(dManager.dialogueActive);
    }

    public virtual void TriggerDialogue()
    {
        interactionAnim.SetBool("IsOpen", false);
        dialogueAnim.SetBool("IsOpen", true);
        if (dManager.dialogueActive)
        {
            dManager.DisplayNextSentence();
        }
        else if (!dManager.dialogueActive)
        {
            dManager.StartDialogue(dialogue);
        }
        //Dialogue.textImage.sprite = characterSprite;
        dialogue.nameImage = nameSprite;
    }

    private void OnTriggerEnter(Collider other)
    {
        if (other.tag == "Player")
        {
            readyToInteract = true;
            Debug.Log("Ready to Interact with" + this.name);
            interactionAnim.SetBool("IsOpen", true);
            TriggerDialogue();

        }
    }
    private void OnTriggerExit(Collider other)
    {
        if (other.tag == "Player")
        {
            readyToInteract = false;
            interactionAnim.SetBool("IsOpen", false);
            portal.SetActive(true);
            wormDirector.Play();
        }

    }
}

redpajama_set_name: RedPajamaGithub


text:
News The Essential Daily Briefing
Check the URL for errors or try our searching inews.co.uk Check your search terms and try again...
iTravel
iEat
ibuys
Travel and outdoor
Phil Foden is fishing for more trophies to add to his overflowing cabinet
Manchester City teenager is targeting U21 Euros success with England this summer - and some more giant catfish when he gets home
Phil Foden at an England U21 Training Session on 27 May 2019 (Getty Images)
Ben Fisher 4 weeks Sunday June 16th 2019
Spurs and Man Utd in bidding war for £67m Lo Celso
Teenagers charged with destruction of Market Deeping model railway
Boris Johnson's idea that hard work cures mental illness is totally ab
England's cricket win is a celebration of our multicultural nation
Best Prime Day 2019 deals on Amazon devices, laptops, fashion and more
We don't need the next PM to unite Britain - we need disagreements
Ante Palaversa: 'Man City's interest was a huge surprise'
When does the Premier League 2019-20 season start?
Bruce's top priorities at Newcastle – the managerial job from hell
What is 'snus' and why are so many footballers using it?
5 contenders to replace Sarri at Chelsea
The i newsletter
News for free thinkers
CESENATICO, ITALY — In the foyer of the England Under-21s' luxury hotel, a stone's throw from a sandy beach jam-packed with Italian holidaymakers on the Adriatic coast, 15 miles up the road from Rimini, Phil Foden is not talking about winning a World Cup competition aged 17 nor lifting his second Premier League title with Manchester City – but fishing.
"I go with my dad when I have a bit of quiet time – it's relaxing," Foden says. "It is something that I've always done." As for his biggest catch, the teenager breaks into the laughter. "I think it was a catfish – it weighed 136 lbs and it took three of us to hold it."
Sometimes stopping Foden is a three-man job, too. The England No 10 – one of the most gifted players at this summer's European Championship in Italy and San Marino – has a knack for leaving defenders weak at the knees. Foden won his first major trophy two years ago, wreaking havoc in India, where England Under-17s were crowned world champions, and the midfielder admits he is fast running out of room to find a home for them all. "I keep them all in a cabinet but it's getting full, I need a new one," he says. "Sometimes I walk past and just have a look and I realise I have won a lot. The World Cup started everything off – that was the first one, the biggest – and I've started winning trophies for City. Every single one of them is special to me."
Winning habit
Phil Foden of Manchester City celebrates with the Premier League Trophy on 20 May 2019 (Getty Images)
Read more: U21 Euros: 7 young players to watch at Italy 2019
By now, Foden knows what it takes to win silverware. It has become a habit, something that the Stockport-born teenager says has been ingrained in him since winning a medal at an Under-9s tournament with Reddish Vulcans as a kid. These days the trophies are proudly positioned in a cabinet in the kitchen at home, where he lives with his parents, Phil and Claire, his partner, Rebecca and his six-month old son, Ronnie. "It's all about keeping winning the next thing, once you win one thing, you want to win the next thing. In my head, I just love winning. Every game I go into I just want to win it. It's the excitement of winning."
Even now Foden says he still has to pinch his self at supporters wearing shirts adorning his name and number.
When Foden, who has sky blue running through his veins and used to go to games with his mum as a kid, tasted his first Premier League title success under Pep Guardiola last year, he politely rejected Vincent Kompany's invite to join the afterparty in Hale in favour of going night fishing with his father in Stoke on Trent. This season, however, Foden joined the celebrations. "I was on the team bus on the parade but didn't really do anything after."
England take on France in their Group C opener on Tuesday in Cesena with plenty of expectation – another thing the teenager is au fait with. At the heart of a team teeming with technical ability, Foden is determined to help Aidy Boothroyd's vibrant side deliver.
"It comes down to Gareth [Southgate], to be honest. Just letting us play free, and let us play the way we want to play, brings a lot of out on you. You can tell England youth, and even the seniors, that are doing that, playing out from the back, just being free, expressing ourselves, there's no fear, no fear to play, it's the best way to play.
Follow us on Twitter: @iPaperSport
"It takes a lot of heart and desire, especially when you are playing a team who haven't got the legs. You have use your mind, your heart and you need a lot of determination. I'm excited, playing against great players, and with great players, such a big challenge for me and I can't wait for the first game."
More Premier League:
Spurs move quickly to hijack Arsenal's deal for £25m William Saliba
Analysis: Arsenal are a club with a crumbling identity
Why Koscielny refused to go on Arsenal tour - and what happens next
Bruce addresses Newcastle job links
Analysis: Man Utd's only good option is to get Pogba back on side
England Football
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redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Applying a class once a div has been slidetoggled and is visible How do i apply a class that i have set in my css (which is the same as the hover state) if a div i have on slidetoggle is visible?
I have 6 buttons, layout of 3 by 2, my slidetoggle div slide down in the middle of hte 2 rows of 3 and pushes the content down with it, this is all working perfectly, but i would like the one button that has been clicked to have its slidetoggled activated to show the same state as what i have defined in the css for the hover state.
Code.
HTML

//etc etc etc to product 6

CSS
#row1 > a:hover > #product1 >#productblueheader > #productlogosblue1
/*#row1 >a.active > #product1 > #productblueheader> #productlogosblue1 */{
    background-image: url(../images/mwprologo.png);
}

#row1 > a:hover > #product1
/*#row1 > a.active > #product1*/ {
    border: 1px solid #e8bb27;
    background-color: #fff;
}

#row1 > a:hover > #product1 > #productstitle
/*#row1 > a.active > #product1 > #productstitle*/ {
    color: #e8bb27;
}

And JS
$(document).ready(function(){
$("a.active").removeClass('active');
            $("#product1").click(function(event){
                if($(this).parent('a').hasClass('active')){
                                $(this).parent('a').removeClass('active');
                         }else{
                                $(this).parent('a').addClass('active');
                            }
                $("#product2box").slideUp('slow', function() {
                $("#product3box").slideUp('slow', function() {
                $("#product4box").slideUp('slow', function() {
                $("#product5box").slideUp('slow', function() {
                $("#product6box").slideUp('slow', function() {
                    event.preventDefault();
                $("#product1box").stop().slideToggle(1000);
                return false;
            });
            });
            });
            });
            });
            });

Updated this js, the color is off at the beginning (correct) and goes on when i click the button (correct) now i just need to unactivate it on the second click when i click off it?

A: According to your JS, I think you can change it into this:
EDIT
$(document).ready(function(){
    $("a.active").removeClass('active');

    $("#product1").click(function(event){
        //Make sure you put this inside every click function
        $("a.active").removeClass('active');

        if($(this).parent('a').hasClass('active')){
                $(this).parent('a').removeClass('active');
         }else{
                $(this).parent('a').addClass('active');
        }
        $("#product2box").slideUp('slow', function() {
            $("#product3box").slideUp('slow', function() {
                $("#product4box").slideUp('slow', function() {
                    $("#product5box").slideUp('slow', function() {
                        $("#product6box").slideUp('slow', function() {
                            event.preventDefault();
                            $("#product1box").stop().slideToggle(1000);
                            return false;
                        });
                    });
                });
            });
        });
    });
});

A: $(document).ready(function(){
    $("a.active").removeClass('active');
                $("#product1").click(function(event){
                    if($(this).parent('a').hasClass('active')){
                                    $(this).parent('a').removeClass('active');
                             }else{
                                    $(this).parent('a').addClass('active');
                                }
                    $("#product2box").slideUp('slow', function() {
                    $("#product3box").slideUp('slow', function() {
                    $("#product4box").slideUp('slow', function() {
                    $("#product5box").slideUp('slow', function() {
                    $("#product6box").slideUp('slow', function() {
                        event.preventDefault();
                    $("#product1box").stop().slideToggle(1000);
                    return false;
                });
                });
                });
                });
                });
                });

This is nearly perfect.. only problem now is that because i have set it up so that when one product is clicked the toggled div slides back up and opens the newly clicked div, now the active class wont go off from unclicking on another product... they turn their own active state off but when i click a different product it the active color state stays on for the first button and opens on the second also then the order is thrown off...

A: $(document).ready(function() {
    $(".producthandler").click(function() {
        var ctx = $(this);
        var productboxId = ctx.children().eq(0).attr("id");

        ctx.toggleClass('active');
        $("#" + productboxId + "box").stop().slideToggle(1000);

        $(".producthandler").each(function() {
            var ctx = $(this);
            var producthandlerId = ctx.children().eq(0).attr('id');

            if (productboxId !== producthandlerId) {
                ctx.removeClass('active');
                $("#" + producthandlerId + "box").slideUp(1000);
            }
        });
    });
});

redpajama_set_name: RedPajamaStackExchange


text:
Victim Of Bizarre Rowland Heights Slayings Remembered By Family, Friends
Filed Under:Fatal Shooting, Mass Shooting, Murder, Rowland Heights, Victim, Violence
ROWLAND HEIGHTS (CBSLA.com) — Earnie Calzadilla was one of four people shot and killed New Years night at a home in Rowland Heights.
His brother, Wilfred, told KCAL9's Crystal Cruz that Earnie just became a new dad eight months ago.
"He was the type of guy that if you have $300 he was the type that would spend on his daughter he didn't care if he stayed broke," said Wilfred. "Even if he had nothing, he would spend it all on his daughter."
Neighbors of the homeowner where the violence took place — the 2200 block of Batson Avenue, — told Cruz there was always a lot of drinking in the home. They said beer bottles on the front porch and lawn were standard.
"Similar event has happened in the past but no one was injured, luckily, but there was shots fired," said neighbor Sean Tubbs.
Darlene Tubbs lives close by and says the 54-year-old homeowner, described as a heavy drinker, shot and killed his bed-ridden wife, his son's girlfriend and his son's friend, Earnie.
"I guess the dad just snapped," said Darlene.
Deputies said the shooter's 33-year-old son wrestled the gun away from his dad and then killed him.
RELATED LINK: 4 Dead, Son Arrested In Connection With Bizarre New Year's Eve Shooting In Rowland Heights
The Sheriff's Department says the son — identified as Christopher Morey — faces a murder charge because once he wrestled the gun away from his father, the threat was eliminated. (Officials later said the homeowner had hundreds of guns.)
Earnie's sister-in-law, Brenda Guiterrez,told Cruz she had been up all night crying.
"[Earnie] was always good to our kids and he was a really good dad and just sad that this had to happen," she said.
Earnie's sister left flowers on the porch, clearly troubled her family is starting the New Year off without a very much loved member of the clan.
[Editor's Note: The family set up a GoFundMe account to help pay for Calzadilla's funeral expenses. Click here for more information.]


text:
Everywhere Resources This category
The Royal Navy's Air Service in the Great War by David Hobbs
Author Seaweed
Creation date Nov 8, 2017
Commander Hobbs, after service as a Fleet Air Arm pilot with several hundred deck landings to his log book, for many years ran the FAA Museum at Yeovilton, and has published several works on naval aviation. He is thus exactly the right person to compile this history of the RNAS' gestation and WW1 service, and he does not disappoint.
Less than a year into the War the three year old RNAS had expanded beyond landplanes, seaplanes, flying boats, airships and balloons, into armoured cars (originally to support flight operations in France and Belgium), its own organic ack-ack, armoured trains, and tanks. It had deployed to the Dardanelles, the Adriatic and East Africa (and later into the Eastern Mediterranean and the Red Sea in support of the Army), and in its armoured car role to Namibia, Russia and Romania. The RFC was markedly inferior to the Admiralty in the matter of procurement and recruitment. The RNAS was repeatedly, to the detriment of necessary maritime operations, bled for aircraft, personnel, aircraft engines and ultimately whole squadrons and was supporting the Western Front with strategic bombing. The response, forensically analysed by Hobbs, was resentment, obstruction and eventually the subordinated absorption of the RNAS into the infant RAF on All Fools' Day 1918. The RNAS transferred 55,066 officers and men (up from 830 at the outbreak of the War), 2,949 aircraft (91),103 airships and 126 coastal stations (up from nine) – massive feat of organisation in only 3 1/2 years, the while swallowing massive leaps in technical sophistication.
The RNAS' surrender was engineered via the Smuts pipe-dream report whose chief features were total ignorance of UK maritime needs – the Admiralty's failure to engage and paucity of advocacy much to blame – together with misrepresentation and argument by false analogy. The failure to challenge this by Churchill's over-promoted protegé Beatty reflects, for me, that officer's notoriously shallow and slapdash approach to detail. Politics and personal ambitions won over the essential maritime interests of the country. Recovery took nearly thirty years and some of the effects are still being felt today.
Besides this enormous expansion the RNAS had allowed junior officers their head in proposing new methods and materiel. At its demise the RNAS' personnel's initiative, creativity and drive was fully bearing fruit. It was experimenting with air-dropped torpedoes and planning a 'Taranto' on the High Seas Fleet and was pioneering the world's first flush-deck aircraft carrier; its flight operations from other ships prefigured the CAM ships of WW2 (with an inevitable ditching) and the need for a Combat Air Patrol had been realised. Anti-submarine warfare using airships and aircraft was well established and successful. Anti-Zeppelin (and Gotha) techniques were mature. The USN was learning from us - so also, later and ironically, our other allies the Japanese. All of this and much more is described in deep detail, interspersed with accounts of individual operations and actions which give a fine flavour of the dedication and courage of RNAS aircrew.
The book is profusely illustrated throughout with photographs and diagrams appropriate to the adjacent text. Many are from the author's extraordinarily extensive own, and other private collections so will not have been previously seen. Citations are carefully noted and an extensive bibliography is provided. My only gripe was having to keep a marker in the notes to pick up on asides as I read along.
I look forward enthusiastically to the author's sequel relating to naval aviation between the Wars.
More resources from Seaweed
The Last British Battleship by RA Burt
The story of HMS Vanguard 1944-1960
Enemy Waters by Cdr David D Bruhn USN(Retd) & Lt Cdr Rob Hoole RN(Retd)
The story of British and Allied mine forces battling the Germans and Italians in WW2
U-Boots' Escape by M Borowiak and P Wytykowski
The sagas of the U-boats who escaped to Argentina in 1945
A Carrier at Risk by Mariano Sciaroni
Argentine aircraft carrier and anti-submarine operations during the Falklands War, 1982
The Battle of the Denmark Strait by Robert J Winklareth
The life and death of the Nazi battleship Bismarck, in extreme detail.


text:
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Draconomicon I: Chromatic Dragons (4e)
Draconomicon™ I: Chromatic Dragons describes several varieties of dragons, including red, blue, green, black, and white dragons, as well as three completely new chromatic dragons. The sourcebook gives details of each dragon's powers, tactics, myths, lairs, servitors, and more. In addition, this book provides new information about draconic nations (such as Arkhosia) and organizations,... [click here for more] Wizards of the Coast $14.99 $11.24
Draconomicon: The Book of Dragons (3.5)
Ancient Creatures of Legend Endless tomes recount epic tales illuminating the might and majesty of dragons. Formidable allies and more fearsome opponents, no other creatures have ever inspired such awe and wonder. This lavishly illustrated supplement for the D&D® game presents a comprehensive overview of the most evocative creatures in fantasy roleplaying. Along with the... [click here for more] Wizards of the Coast $14.99 $11.24
Dungeon Builder's Guidebook (2e)
You are probably wondering: Is this guidebook for me? In a word?yes! Dungeon Builder's Guidebook assembles tried-and -true dungeon building advice, approaches, and tactics gathered over many years. The suggestions and samples in this book are an invaluable resource for initiaing and fleshing out the ruins, mines, caverns, and tombs of your imagination. What if your already design all your own dungeons?... [click here for more] Wizards of the Coast $4.99 $3.74
Dungeon Master's Design Kit (1e)
As any good Dungeon Master knows, preparation is the key to a great adventure. The more work done in advance, the smoother the play session will go. And nothing turns players off faster than a DM fumbling for the right piece of information in the middle of a fast-paced session. That's where the Dungeon Master's Design Kit comes in. Inside its 96 time-saving pages are scores of organizational ideas,... [click here for more] Wizards of the Coast $4.99 $3.74
Dungeon Master's Options: High-Level Campaigns
What's a DM to do when the player characters in his campaign become so powerful that nothing in the 'world is a challenge for them anymore? Just what are the essential ingredients in an adventure geared toward high-level play? Here is a trove of advice and new rules for every AD&D game Dungeon Master who wishes to create adventures for truly legendary heroes. Find everything you need to create... [click here for more] Wizards of the Coast $9.99 $7.49
Evil Tide (2e)
YOU SHOULD BE AFRAID OF THE WATER... When the sun sinks low, the sea devils rise dripping from the dark waves. They swarm like sharks, lethal and unstoppable. Death glints like a promise in their rabid eyes, and the blood of their prey paints their claws and teeth scarlet. Night after night they come, slaying all who stand in their way. Their sudden raids have devastated the island, bringing fishing... [click here for more] Wizards of the Coast $4.99 $3.74
Expedition to the Demonweb Pits (3.5)
Visit the Abyss. Meet Demon Lords. Kill Everything. In the heart of the Demonweb, Lolth surveyed the Grand Council Chamber and the demon lords assembled by her invitation. Throaty growls indicated a dispute amond Yeenoghu's pack. Obox-ob's insect swarms chittered while Demogorgon ranted from the central podium. The other emissaries seemed restless. Across the hall, Graz'zt met Lolth's gaze and... [click here for more] Wizards of the Coast $17.50 $13.13
Fantastic Locations: Fane of the Drow (3.5)
Fane of the Drow features two double-sided poster maps designed for roleplaying and miniatures skirmish play. An accompanying 16-page adventure can be dropped into any campaign. Enemies consist of creatures from various D&D miniatures. Miniatures skirmish scenarios are included for additional play options, written to allow Dungeon Masters to seamlessly integrate D&D Miniatures into the adventure.... [click here for more] Wizards of the Coast $4.99 $3.74
Force of Nature (3.5)
Earthquakes, tidal waves, and volcanic eruptions have suddenly wracked Porthaven, a city normally known for its mild weather and calm seas. The root of the problem is an arcane machine built to protect the city 300 years ago by extraplanar creatures called modrons. Now the machine has run out of fuel, and it's up to the PCs to obtain more by visiting each of the four elemental planes. Can they get... [click here for more] Wizards of the Coast $0.99 $0.74
Haunting Lodge (3.0)
The chill November wind has started to blow and, for many folks out there, that means hunting season has arrived. So, in the "spirit" of the season, Owen K.C. Stephens gives us Haunting Lodge, a miniadventure suitable for four 17th-level characters. This site-based adventure centers around an exclusive getaway for high-powered sportsmen seeking some rather unusual game. Fair warning,... [click here for more] Wizards of the Coast $0.99 $0.74
I9 Day of Al'Akbar (1e)
"The land of Arir - a once peaceful desert country, dotted with oases, teeming with caravans - fell into the hand of infidels. The ruler, the dearly loved Sultan Amhara, was killed in the battle for the capital city of Khaibar. He left behind one of the greatest treasure stores ever amassed - jewels and coins, more than anyone had ever seen before or since - and in addition, the Cup and Talisman... [click here for more] Wizards of the Coast $4.99 $3.74
Ill Wind in Friezford (3.0)
A new mini-module by Skip and Penny Williams challenges PCs with a haunting adventure. What is the wild hunt? Will the PCs survive their visit to Friezford? Drop this adventure into your campaign in any wilderness setting that contains a former trading route. Suitable for four 14th-level characters. Adventure Preview Introduction What haunts the woods surrounding the... [click here for more] Wizards of the Coast $0.99 $0.74
Lochfell's Secret (3.5)
Hollyvyre saw her husband eaten right before her eyes by a giant sea monster that walked on water. That same day, a young man found his mother's grave dug up -- and his mother's body missing. Are the two events connected? What would a sea monster want with corpses? Drop this adventure for D&D v3.5 by Eric Haddock into your campaign world wherever there is a small port town. Suitable for four... [click here for more] Wizards of the Coast $0.99 $0.74
M1 Blizzard Pass (Basic)
A SOLO ADVENTURE FOR THIEVES LEVEL 1-3 or Group Play Level 1-3 The storm sweeps off the mountaintops without warning, as if conjured by some supernatural power. Within minutes, huge drifts of snow have halted the caravan, and the snow falls so hard it is impossible to see more than five feet. You huddle next to the sledge, trying to shelter your numb face from the icy winds. A... [click here for more] Wizards of the Coast $4.99
M2/MSOLO2 Maze of the Riddling Minotaur (Basic)
An expert set solitaire adventure. For a long time you have heard legends about the Island of Vacros; when you were only a child, the stories about evil minotaurs were used to scare you into behaving. Now you are grown, and you no longer believe children's stories. But King Trueon of Cathos is worried: his daughter Princess Lydora has been kidnapped, and he has found clues that she has been taken... [click here for more] Wizards of the Coast $4.99 $3.74
Masters of Eternal Night (2e)
"Rock, and dark, and time not passing." I walk through darkness and a land frozen in the grip of winter. Everywhere frigid gusts of wind send icy needles at my face. I do not even flinch anymore; the pain reminds me that I yet live. I can feel them - their alien thoughts twirling about my mind. They wish me to serve, but I will not . not yet. In Masters of Eternal Night, adventurers... [click here for more] Wizards of the Coast $4.99 $3.74
MC14 Monstrous Compendium Fiend Folio Appendix (2e)
Look to your weapons! Here comes a horde of the strangest creatures ever to inhabit the worlds of the AD&D game. The denizens of the original FIEND FOLIO tome are back, updated and expanded for the AD&D 2nd edition game. Treat yourself to an in-depth look at the peculiar achaierai, vicious stilt-legged birds that roam the infernal planes - and they also hunt closer to civilization. Find out... [click here for more] Wizards of the Coast $4.99 $3.74
MC2 Monstrous Compendium Volume Two (2e)
The other half of the core monsters for AD&D 2e, originally released as looseleaf sheets to fit in the original Monstrous Compendium binder. Product History MC2: Monstrous Compendium Volume Two (1989) is the second monster manual for AD&D 2e. It was published in August 1989. Moving Toward AD&D 2e. During the AD&D 1e era (1977-1989),... [click here for more] Wizards of the Coast $4.99 $3.74
MC8 Monstrous Compendium Outer Planes Appendix (2e)
Welcome to the Monstrous Compendium volume that finally details the powerful creatures unique to the outer planes. These denizens of the planes can be powerful allies or terrible enemies. From aasimon to zoveri, and everything between, this supplement is a must have for any DM planning to run an adventure in the outer planes. Product History MC8: "Monstrous Compendium... [click here for more] Wizards of the Coast $4.99 $3.74
Monstrous Arcana: The Sea Devils (2e)
Evil denizens of the depths: the sahuagin. The sahuagin are known by many names. Some call them the Devil Men of the Deep; to others, they are simply the Sea Devils. This deadly race of aquatic beings has remained a mystery to surfacedwellers - until now. This fully illustrated 96-page accessory details the highly organized structure of sahuagin society, offers insight into the workings of the predatory... [click here for more] Wizards of the Coast $4.99 $3.74
Moonlight Madness (2e)
The moon rises, and you feel yourself start to change. Fur sprouts from your palms, your fingers turn to claws, and a howl rips from your throat. Your clothes shred and tear as your Other Self breaks free.. Moonlight Madness follows the exploits of six unlucky heroes trying to free themselves of the curse of lycanthropy while evading a full-scale werewolf hunt led by a crusading paladin and his... [click here for more] Wizards of the Coast $4.99 $3.74
Night Below: An Underdark Campaign (2e)
A missing mage... A ruthless band of kidnappers... A sinister conspiracy... Night Below, the first epic campaign adventure for the AD&D game, is designed to take the player characters from 1st level to 10th level and beyond. The PCs start as beginning adventures on a routine courier mission who soon become drawn into combating a sinister plot that menaces the pleasant land of Haranshire.... [click here for more] Wizards of the Coast $9.99 $7.49
Night of the Shark (2e)
"When the storm comes in with its teeth, teeth, teeth..." Black clouds build on the horizon with supernatural swiftness as the light of day drains away like a guttering candle. As the wind screams and sea spray blinds the eyes, the masts splinter and the deck heels over, a plaything of the storm. A nightmare shark hunts the murky ocean for all that succumb to the hammer of the winds. Can anyone... [click here for more] Wizards of the Coast $4.99 $3.74
OD&D Supplement III: Eldritch Wizardry (0e)
The book you now hold in your hand represents new dimensions to an already fascinating game system. This is the third supplement to DUNGEONS & DRAGONS, and was produced as a result of an ever increasing demand for new material. This book also represents a new trend in the fine art of Dungeon Mastering. As originally conceived, D & D was limited in scope only by the imagination and devotion... [click here for more] Wizards of the Coast $4.99 $3.74
OD&D Supplement IV: Gods, Demi-gods & Heroes (0e)
Well, here it is: the last D&D supplement. It is with a strange mixture of sadness and relief that I tell you this. My first assignment, fresh out of college, was BLACKMOOR. I came to regard it with a mixture of love and loathing, that has gradually seen the love win out. The loathing grew out of the educational trip that it was for me. They don't teach you in college what to do when the press... [click here for more] Wizards of the Coast $4.99 $3.74
Poor Wizard's Almanac II (Basic)
This new almanac provides a massive collection of world events for the game year of AC1011. It picks up where the previous almanac ended, wrapping up major plots that started in the game year of AC1010, and starting new ones. Battles, skirmishes, and a host of historical events take place in a world in perpetual development, enough to fill an entire game year! ... [click here for more] Wizards of the Coast $7.99 $5.99
Priest's Spell Compendium Vol 1 (2e)
The Priest's Spell Compendium is a three-volume series describing every wizard spell for the AD&D game. Spells have been collected from rule books, campaign settings, supplements, adventures, and magazines dating back to 1975. Early entries have been updated to be immediately usable, making this the player's and DM's definitive source for priest spells. This volume contains spells from A to... [click here for more] Wizards of the Coast $4.99 $3.74
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This recipe is so versatile. You can add dried fruit, make a pie or crumble, serve it with breakfast cereal…its a great staple to have on hand.
Place apples into your saucepan with sugar, water, cloves and the cinnamon.
Simmer this gently for 10 minutes. You want your apples to be soft- Not mushy!
You can serve this hot straight away with lovely vanilla ice cream or greek yogurt!


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Accolades, supports trail African players in Budapest
April 30, 2019 / ATTF Media / World Events
The performance of African players at the 2019 ITTF World Championships in Budapest has attracted accolades and encouragement from the continental ruling body. From the President of African Table Tennis Federation (ATTF), Khaled El-Salhy, the players gave their best and
Mixed fortune for African players in Budapest
Majority of the African players that featured in the 2019 ITTF World Championships failed to make it beyond the group stages. Players from Algeria, Cote d'Ivoire, Congo DRC, Congo Brazzaville, Mauritius and Madagascar all failed to win any match in
Chinese super star stings Aruna Quadri
Aruna Quadri could not find the solution to the ferocious shots of China's Fan Zhendong in the third round of men's singles at the 2019 ITTF World Championships holding in Budapest, Hungary. Despite a good start from the African champion,
African stars tango with heavyweights in Budapest
The journey to glory for African super stars at the ITTF World Championships begins on Tuesday April 23 as the main draw of the men and women singles serves in Budapest, Hungary. Nigeria's Aruna Quadri and Egypt's Omar Assar will
Nigeria debutant sparkles in Budapest
Ajoke Ojomu started her debut at the 2019 World Championships on a good note after recording a convincing 4-1 win over Luxembourg's Sarah Meyer in the group opener in Budapest. The 19-year-old was on the winning side in the explosive
African stars hunt for glory in Budapest
The search for glory by African stars continues as the 55th World Table Tennis Championships serve off on Sunday April 21 at the Hungexpo in Budapest, Hungary. With only three players from Africa making the seeding list in the men
Africa honours Ghana's table tennis legend, Ethel Jacks
April 11, 2019 / ATTF Media / General
When Ethel Jacks walked into the venue of the 2019 ITTF Africa Junior and Cadet Championships – Sports Trust Emporium in Accra, Ghana not many of the players knew who she was. But some of the older coaches quickly recognised
Battle for WJC slots begins in Ghana
April 6, 2019 / ATTF Media / Africa Events
It is a record entry for this year's African Junior and Cadet Championships holding in Ghana and it serves as qualification for the ITTF World Junior Championships (WJC) with 21 African teams slugging it out for the limited slots available
A new entry feat heralds African showpiece in Ghana
For the first time in the history of the African Junior and Cadet Championships, an entry record will be achieved this year in Accra, Ghana as over 180 players from 20 African nations will be competing at the continental tournament.


BRIDGEWATER, Mass. –The Fitchburg State men's outdoor track and field team opened its 2017 campaign this afternoon, competing in the 2017 Bridgewater State University Bears Outdoor Invite at Swenson Field.
Sophomore Cameron Stalters led the way for the Green and Gold, placing third overall in the javelin with a new personal best and Open New England qualifying mark of 61.22 meters. With that toss, Stalters currently ranks third nationally in the javelin.
In the 5000 meter, junior Brian Amaral turned in a second place time of 15.29.95 to qualify for DIII New England's, while classmate Jake Hastings (7th) and sophomores Anthony Darrigo (12th) and Joseph Darrigo (16th) turned in MASCAC qualifying times of 15:51.43, 16:09.82 and 16:17.13 respectively.
Senior Nicholas Levenson posted a pair of post-season qualifying times in the 110m (17.39) and 400m hurdles (59.38) to place seventh and second respectively, while senior Chauncey Beatty turned in a solid time of 52.31 seconds in the 400 meter dash to place sixth overall.
In the field, freshman David Sicinski registered a fifth place effort in the pole vault with a MASCAC qualifying mark of 3.05 meters, while in the shot put freshman Kyle Nelson (8th) and sophomore Brendon Courtney (11th) notched post-season qualifying tosses of 12.65m and 12.17m respectively to round out the top Falcon efforts on the afternoon.
The Fitchburg State men's outdoor track team returns to action next Saturday as it hosts the Jim Sheehan Memorial Alumni Invitational at Elliot Field, starting at 10 a.m.


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