Team representative will check-in the team two-hours prior to first game. Roster will lock on Dec 13, no roster changes after this date.
* Must Manager Book, with age verification on hand for tournament.
AFL's goal is to provide appropriate competition for teams and players. When possible, single year division competition will be designated. When registration closes, teams registered will be reviewed by the Tournament Committee and divisions assigned. The objective will be to have as many viable single year divisions as possible. There must be at least four registered teams to be a division - a division that is less than four teams may be combined with another division; but will have single year finial games if applicable.
Alaska Futsal State Cup "Thank You Letter"
Rules: Alaska Futsal State Cup will abide by the AYSA Tournament, and FIFA Futsal Rules.
Qualification: Only AFL League players and teams currently registered in the 2018 League, registered under the AYSA umbrella, and in good standing with AYSA are eligible to participate. Players and teams must be Alaskan based and Alaska residents to participate and register.
Team registration will be open on November 1, 2018 for the Alaska Futsal State Cup tournament. The team must be in good standing within the league for the application to be accepted. The State Cup fee will be due at the time of registration.
Players and teams must play within the AFL League to qualify to attend State Cup. Teams entered out of District must play atleast three sanctiond games to qualify for State Cup.
Format: Divisions are single year; 10-U through 19-U both Boys and Girls Teams.
Divisions must have at least four (4)- registered teams to be a valid division. AYSA reserves the right to drop/ combine divisions/ age groups. 16-U older team registration will be at the age of the oldest player on the roster. Club Pass will not be allowed for the tournament.
A division that is less than four registered teams may be combined with another division; but will have single year finial games if applicable.
How many players allowed on a team? Roster size will be minimum of 7 players, and a maximum of 12 players. (only 11 dressed out per game); non-participating player may be on the bench but not in team uniform, nor wearing shirts that are the same color of their, or, the opposing team. Field is at 5 players per side, 4 field players plus GK; minimum of 3 players to start or continue play.
No Dual Rostering allowed. Players and teams currently registered in the 2018 League, registered under the AYSA umbrella, and in good standing with AYSA are eligible to participate. Players and teams must be Alaskan based and residents to register.
Where will Game Venues be? Games are scheduled at O'Malley Sports Center and Arctic Rec Center. All teams are required to list an assistant coach/ manager on the roster for conflicts by game time or location.
Will player cards be required for the tournament? Yes. Approved, dated, and stamped rosters and player member cards will be verified by the referee at each game.
Will Event Passes be used? Yes.Team roster will list Technical Area Staff (coach, manager, DOC/ TD) and submit Technical Staff to AYSA by 12/5 for Event Passes. All adults on the futsal court must have an Event Pass.
Will 10-U Standings be posted? Yes. The referee will record the scores, and report them to AYSA for 10-U and older Divisions.
Will competitive club pass be allowed? Yes prior to the roster lock, you may club pass a registered AFL player to a team for the tournament, but once the roster locks they must remain on the roster team that was submitted. The AFL State Cup Roster will be locked prior to the start of the tournament with no additions allowed after that date.
Guest Players Allowed? Yes. Registered AFL players may transfer from a club team not participating in Futsal State Cup to a club team that is participating. Transfer Fees will apply. The team accepting a player from a non-participating team for Futsal State Cup will only be allowed two (2) Temporary Transfer approvals for their Futsal State Cup roster.
Will coaches with multiple teams be accommodated? Alaska Youth Soccer will accommodate coaches with multiple teams by scheduling games within a reasonable timeframe if possible. All teams are required to list an assistant coach/manager on the roster for conflicts by game time or location.
Adult Verification and Code of Conduct: Alaska Youth Soccer requires that each adult working within the youth program have submitted and passed their annual AYSA Adult Background Check. All adults working with youth (coach, manager, chaperone, administrator) will submit a background check application in the Stack Sport system to be processed by AYSA. The adult applicant will also read the AYSA Code of Conduct, and complete the online Abuse Prevention Systems Training provided through their club/ association.
AFL Refund Policy; Cancel registration any time prior to the registration deadline to receive a refund of your fees minus a $200 administrative fee. After the deadline, no refunds.
Mackenzie German- Cook Inlet S.C.
Award Ceremonies will be held at the designated location at the end of each Championship Game. First Place Team and Second Place Team are expected to attend the ceremony. Players and coaches will be awarded individual medals.


Refurbished magnetic card reader for NH 1500 models only. This does not come with a card reader cable or card reader bezel.
Magnetic card reader assembly, only compatible for NH 1500 models. This does not come with the card reader cable or card reader bezel. This is a non EMV card reader.


I'm a very deadline driven person. At work, I usually have no problem doing things I'm obligated to do. I do my work tasks with vigor. My work reviews are usually excellent, and my bosses always know they can trust and count on me.
The catch??? I fall short when it comes to ME. I'm pretty horrible at sticking to things that affect me personally. If it's for other people, sure no problem, I got it .. but me . errrrr .. that's another story altogether.
So it's no surprise that I'm lagging on getting started on the 2018 Fall One Room Challenge. This little project of mine, is for me .. sure everyone who eats dinner in the dining room will get to experience the space .. but let's face it, not too many people really care about it like I do.
Putting it out there in the inter-web gives me a little more skin in the game … I'm creating a faux-deadline for myself. Really, no one will be affected if I don't follow through (except me).
So this post is just a little psychology behind why I haven't started yet. Haha.
To recap, my project is a California Casual Dining Update with a Scandi-Flair.
Color Block Wall: This is the main part of my project! I'm getting rid of the dark wall and replacing it with what is (I hope) a sophisticated, yet casual neutral-toned scandi-inspired geo patterned wall. The pattern will make its way in some form or fashion around the entire room. My hope is that the geometric pattern will tie together with the 1970s angled walls in the other parts of the house.
Modern White Chairs: I'm keeping the head-of-the table-chairs with my Comfort Works teal slipcovers and adding two modern white chairs. I love wood + white together, and combining it with a little teal/blue is my sweet spot.
Brass Chandelier: I'm working with MP Design Shop (see my kitchen pendants!) on a fixture for the space. I love their work and I'm so excited about getting a new fixture!
Window treatments: When I said this used to be a sunroom, that's no joke. This room gets hotter than-you-know-what.
Wish Item: THRIFTED BUFFET. I'm hoping, but not promising, that I can find a good thrifted buffet to DIY. Paint will be involved so it doesn't matter what color it is. I just need to find one!
I'll be back to report on my progress next week! In the meantime, check out the featured ORC challengers. They're an amazing group! And if you want more, Better Homes and Gardens is this year's sponsor. Read about their involement here. Last but not least, the Home Love Network is featuring challengers throughout the event.


The Telegraph The Caribbean is a cruising paradise. With 7,000 islands, gentle trade winds and a year-round warm climate, the region is irresistible to cruise lines and has become the most competitive cruising destination in the world.
Major companies such as Royal Caribbean International, Carnival Cruise Lines, Norwegian Cruise Line and Princess Cruises dominate these waters, but the beauty of the Caribbean means there is also room for smaller lines, as they can call at islands that larger ships cannot reach.


"I do." To Americans those two words carry great meaning. They can even change your life. Especially if you say them at your own wedding. Making wedding vows is like signing a contract. Now Americans don't really think marriage is a business deal. But marriage is serious business.
It all begins with engagement. Traditionally, a young man asks the father of his sweetheart for permission to marry her. If the father agrees, the man later proposes to her. Often he tries to surprise her by "popping the question" in a romantic way.
Sometimes the couple just decides together that the time is right to get married. The man usually gives his fiancee a diamond ring as a symbol of their engagement. They may be engaged for weeks, months or even years. As the big day approaches, bridal showers and bachelor's parties provide many useful gifts. Today many couples also receive counseling during engagement. This prepares them for the challenges of married life.
At last it's time for the wedding. Although most weddings follow long-held traditions, there's still room for American individualism. For example, the usual place for a wedding is in a church. But some people get married outdoors in a scenic spot. A few even have the ceremony while skydiving or riding on horseback! The couple may invite hundreds of people or just a few close friends. They choose their own style of colors, decorations and music during the ceremony. But some things rarely change. The bride usually wears a beautiful, long white wedding dress. She traditionally wears "something old, something new, something borrowed and something blue." The groom wears a formal suit or tuxedo. Several close friends participate in the ceremony as attendants, including the best man and the maid of honor.
As the ceremony begins, the groom and his attendants stand with the minister, facing the audience. Music signals the entrance of the bride's attendants, followed by the beautiful bride. Nervously, the young couple repeats their vows. Traditionally, they promise to love each other "for better, for worse, for richer, for poorer, in sickness and in health." But sometimes the couple has composed their own vows. They give each other a gold ring to symbolize their marriage commitment. Finally the minister announces the big moment: "I now pronounce you man and wife. You may kiss your bride!"
At the wedding reception, the bride and groom greet their guests. Then they cut the wedding cake and feed each other a bite. Guests mingle while enjoying cake, punch and other treats. Later the bride throws her bouquet of flowers to a group of single girls. Tradition says that the one who catches the bouquet will be the next to marry. During the reception, playful friends "decorate" the couple's car with tissue paper, tin cans and a "Just Married" sign. When the reception is over, the newlyweds run to their "decorated" car and speed off. Many couples take a honeymoon, a one- to two-week vacation trip, to celebrate their new marriage.
Almost every culture has rituals to signal a change in one's life. Marriage is one of the most basic life changes for people of all cultures. So it's no surprise to find many traditions about getting married . . . even in America. Yet each couple follows the traditions in a way that is uniquely their own.


text:
Random Musings on Mistura
Like all bands, we do like having a nice crowd at our gigs and having people listening to and liking our music, especially original material. Most of all we expect to be paid for our services and our effort, if nothing not to fuck everyone else and build on the unfortunately common assumption that creating and performing music is some sort of a hobbyist charity free for all.
Other than that, as you might have guessed by now, we don't really give a shit about popularity, fame, fake likes and even being played on the radio as some sort of favour to the Maltese language. We don't give a shit about appearing to be rock'n'roll, hip or trying to change our music to fit into specific market segments that have bigger or better crowds. We don't do music videos showing off expensive gear and will not go on Xarabank shouting some half-assed opinions and playing a song as a comic distraction. We just like music and have some social issues at heart which we would like to highlight whilst playing the sort of music we like to listen and rock to, hoping enough people will stop and listen.
In this context of not giving a shit, we've just decided that our first single from the new album is going to be one that takes one full minute for the first verse to start, doesn't have a chorus, it has a minute and a half long solo/musical interlude in the middle and is a nice direct letter to heartless building contractors. It's against all advice we've received so far. Good sense dictates that "you need to have a radio-friendly song to promote the album!" and "you need to get people to dance and have fun!". But so what – this song is what we think is more relevant at the moment and not some happy go lucky fun song for idiots who will not like the rest of our songs anyway. And I think this song is one of those that makes us…. us, warts and all. And people who get this song will get us as a band.
So keep watching this space for more. Meanwhile have fun at the Isle of MTV or whatever your echo chamber is telling you it's the in-thing at the moment xxx Antonio
← Aġġornament
Mistura b'vidjo mużikali ġdid →

redpajama_set_name: RedPajamaCommonCrawl


A unification fight for the UFC Lightweight Championship may be all-but-confirmed for later this year in September between Khabib Nurmagomedov and Dustin Poirier.
Last night in the main event of UFC 236 in Atlanta, Poirier defeated Max Holloway to win the interim UFC Lightweight Championship via unanimous decision (49-46, 49-46, 49-46).
During his post-fight interview, the new interim UFC Lightweight Champion reflected on his journey to winning gold for the first time in the promotion.
The Diamond said, via MMA Fighting: "Man, this feels amazing. I feel like I'm in a dream right now. I just beat one of the pound-for-pound best in the world.
"To be the best you've got to beat the best and damn I feel good. It's been a long time, this is my 23rd or 24th fight with Zuffa, I've got knocked down, I've climbed back up.
Poirier is now in prime position to challenge Khabib for the UFC Lightweight title and unify the belts. He made sure The Eagle knew he was coming for him by channeling his inner Daniel Cormier to send him a message.
It didn't take long for the actual UFC Lightweight Champion to send his response to his newest challenger.
Khabib gave his thoughts on the fight as well, but ultimately stated he was looking forward to meeting The Diamond inside the Octagon in September later this year.
He said: "I don't agree with the judges 49:46, I got a draw, my respect to both fighters, and Poirier's achievement so far cannot be denied, my congratulations Dustin, see you in September. 🦅 vs 💎"
Khabib is currently serving his suspension for his involvement in the UFC 229 post-fight brawl between his camp and the camp of his opponent that night, Conor McGregor.
His suspension will be up on July 6, so a September fight date against Poirier will give him enough time to put a fight camp together and prepare to defend his UFC Lightweight title for the second time.


text:
Hillary Clinton raises $33 million for DNC joint campaign account
WASHINGTON, April 16 (UPI) — Hillary Clinton raised $33 million for her joint campaign account with the Democratic Party in the first quarter of 2016, a campaign finance filing shows.
The amount goes to the Hillary Victory Fund, which is a unique joint account set up partially to support the Democrat's presidential campaign, but also to support the Democratic National Committee and an array of 32 state Democratic Party funds.
The joint account also allows Clinton to accept larger donations under federal law than she could for the account set up specifically for her presidential campaign, Hillary for America. Donors can make the maximum individual contribution to Clinton, $5,400, plus the maximum to the DNC, $33,400, and state party accounts at $10,000 each. That makes the maximum donation to the Hillary Victory Fund $358,000. However, in a campaign finance loophole, Clinton is not required by law to dispense the funds the same way they're collected.
A Friday night Federal Elections Commission filing shows in the first quarter, the lion's share of money disbursed from the Hillary Victory fund went to the candidate herself. The Hillary Victory Fund transferred $9.5 million to her campaign account, another $3.5 million to the DNC and $2 million to state parties.
Politico reports much of the $9.5 million from the Hillary Victory Fund was spent on Internet advertising, seeking to create a small donor base online, an area where Clinton's opponent Vermont Sen. Bernie Sanders has far outpaced her.
Sanders also has a joint campaign account, but it has been largely inactive, leading Clinton to criticize Sanders for failing to help build the party by sharing his significant online cash donations. Those donations almost exclusively go directly to the candidate's campaign account.
Sanders raised $44 million in donations for his campaign in the month of March alone.


text:
\section{Introduction}

A great deal of effort has been devoted to generate and control entanglement which is the most intrinsic feature of quantum mechanics with a variety of applications in quantum information processing~\cite{Horodecki2009}. A strict subset of entanglement called quantum steering is of particular interest~\cite{Reid2009,PhysRep2017}. The concept was originally discovered by Schr\"{o}dinger~\cite{Schrodinger1935} in response to the freaky ``spooky action-at-a-distance'' predicted by Einstein, Podolsky, and Rosen (EPR) in their famous paradox~\cite{Einstein1935,Reid1989}, describing the local measurements on one of two entangled particles can adjust (steer) the state of the other distant particle. There has been an increasing interest in EPR steering since it was rigorously defined by mathematical formulation from the perspective of quantum information~\cite{Wiseman2007,Jones2007,Cavalcanti2009}, i.e., in a network one can verify the entanglement between Alice and Bob without the requirement of trust of Bob's equipment used to perform local measurements at his node by confirming the presence of steering of Alice's system by Bob.
This feature makes quantum steering substantial to various quantum information protocols which rely on entanglement by providing extra security~\cite{Opanchuk2014}, such as semi-sided device-independent quantum key distribution~\cite{Branciard2012,Gehring2015,Walk2016} and quantum secret sharing~\cite{Armstrong2015,Xiang2017,Kogias2017}, one-way quantum computing~\cite{Li2015}, no-cloning quantum teleportation~\cite{He2015,Reid2013,CMLi2016}, subchannel discrimination~\cite{Piani2015}, and other related protocols.

A second distinctive feature, in contrast to entanglement and Bell nonlocality, is that quantum steering implies a direction between the parties involved. The steering strengths in two directions may not be the same~\cite{Wagner2008}, especially, a special type is known as one-way EPR steering where the entangled states show steering in one direction but not in the other~\cite{Wiseman2007,Handchen2012}. The asymmetric EPR steering has attracted considerable attention recently for both theory~\cite{Olsen2008,Midgley2010,Olsen2013,Schneeloch2013,He2013,He2014,Evans2014,Bowles2014,Skrzypczyk2014,Wang2014,Reidjosab2015,QHe2015,Tan2015,Marco2015,Bowles2016,Olsen2017,Baker2018} and experiment~\cite{Handchen2012,Armstrong2015,Qin2017,Sun2016,Wollmann2016,Xiao2017,Tischler2018,Fadel2018}. One important method used in above studies to produce directional steering is making the states asymmetric by adding different amount of losses or noises on the subsystems. For example, the entangled two-qubit states which are one-way steerable have been experimentally demonstrated by passing one party into a lossy channel with a given probability of obtaining qubit~\cite{Wollmann2016,Tischler2018}. It is important to realize one-way steerable Gaussian continuous variable states referred in original EPR argument. One-way steering in the Gaussian regime (Gaussian states and Gaussian measurements) was first observed by introducing additional amounts of loss to one mode of a two-mode squeezed state (TMSS) (vacuum is coupled into the signal by a beam splitter with transmission efficiency $\eta$)~\cite{Handchen2012}, and then extended to a multipartite optical network~\cite{Armstrong2015}. The direction of Gaussian steering can be also manipulated by adding Gaussian noises to one party of TMSS and transmitting the TMSS in a lossy channel~\cite{Qin2017}. Rather than by adding asymmetric loss or noise to subsystems which in general leads to the reduction of correlation, recently, Olsen proposed a nonlinear optical apparatus to control the asymmetry of EPR steering by controlling the amplitude of signals injected into nondegenerate parametric oscillator such that the effects are intrinsic to the scheme~\cite{Olsen2017}.

In light of the exceptional importance of these developments, we provide in this paper a different mechanism to prepare and control the asymmetric steering with enhanced steerability via quantum interference effects in a general three-mode system where all three modes are mutually coupled with each other. We show that by engineering interference between two different interaction paths, i.e., a direct interaction path and an indirect interaction path induced by both being coupled to the third intermediate damping mode, two identical modes with symmetric decoherence properties can be driven into an entangled steady state with asymmetric and directional steering, meanwhile the entanglement and steerability can be effectively enhanced due to constructive interference. This provides an active method to create and control the direction of asymmetric steering without the cost of reducing steerability. Moreover, the steady-state entanglement and asymmetric steering created by the interference effects appear to be more robust against thermal noise. Finally, we have also demonstrated the opposite interference patterns of entanglement and steering between other pair of modes in the loop, indicating the monogamy constraints for distributing quantum correlations among multipartite. It is worth noticing that new quantum interference effects arising from a closed coupling loop have been recently applied to study optomechanical interferences and phonon correlations~\cite{Restrepo2017,Sun2017}, the phase effects on phonon blockade effects~\cite{Restrepo2017,Wang2017}, the transduction bandwidth of a rf-to-optical transducer ~\cite{MoaddelHaghighi2018}, and the optical nonreciprocal behavior~\cite{Metelmann2015,Xu2015,Xu2016,Xu2017} which has been experimentally implemented~\cite{Ruesink2016,Barzanjeh2017,Bernier2017}.

The remainder of this paper is organized as follows. In Sec.~\ref{Model}, we introduce a general three-mode system with closed-loop in coupling which creates interference effects determined by the relative phase of the coupling strengths between two interaction paths. In Sec.~\ref{CRITERIA}, we recall the criteria adopted in this paper to test quantum entanglement and Gaussian steering. We then investigate in Sec.~\ref{RESULTS}, the creation and control of the steady-state entanglement and quantum steering by the interfering channels and compare its performance with the results achieved by applying only one interaction path. The results show the enhancement of correlations, the effective control of the asymmetric steering, and the robustness to thermal noises via interference effects. Finally, we summarize our results in Sec.~\ref{SUMMARY}.

\section{Model}\label{Model}
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.5\columnwidth]{Fig1.pdf}
\caption{Schematic diagram of a general three-mode system with closed coupling loop. Modes $a$ and $b$ interact directly via a two-mode squeezing interaction with strength $\lambda$; at the same time, they are coupled to an intermediate damped mode $c$ resulting in an indirect interaction path between them indicated by the blue dashed curve. The relative phase $\phi$ between two paths can be engineered to control and improve the correlation between modes $a$ and $b$. }
\label{scheme}
\end{figure}
We consider a three-mode system represented by the annihilation operators $a$, $b$, $c$, in which modes $a$ and $b$ interact with each other in two distinct paths, i.e., the direct coupling path resulting from a two-mode squeezing interaction and the induced indirect interaction path obtained by simply coupling to the third intermediate damping mode $c$. The interaction of the system is thus described by a general form of Hamiltonian (with $\hbar=1$)
\begin{eqnarray}
H=&&\lambda e^{i\phi} a^\dagger b^\dagger+\lambda e^{-i\phi}ab + g_a(e^{i\phi_a}c^\dagger a^\dagger+e^{-i\phi_a}ca)\nonumber \\
&&+g_b(e^{i\phi_b}c^\dagger b+e^{-i\phi_b}cb^\dagger),
\end{eqnarray}
where $\lambda,\ g_a,\ g_b$ are the effective coupling strengths with individual phase $\phi,\ \phi_a,\ \phi_b$ controlled by the driving laser fields. This creates a three-mode closed-loop in the coupling, as illustrated in Fig.~\ref{scheme}, which will give rise to interference effects and phase dependence of the dynamics of the modes. Since the interference is dependent on the relative phase between the two interaction paths, without loss of generality, we can absorb the phases $\phi_a$ and $\phi_b$ into $\phi$ by redefining the operators $a$ and $b$. In the following we will assume $\phi_a=\phi_b=0$ and treat $\phi$ as the relative phase in the interference for simplicity. This coupling loop could be realized in three-mode optomechnical systems~\cite{Sun2017}, or in antiferromagnets with magnon-photon coupling~\cite{Yuan2017}.

Substituting the Hamiltonian to the Heisenberg equation and taking into account the dissipation-fluctuation processes, we get the quantum Langevin equations (QLEs) for each mode
\begin{eqnarray}\label{Langevin}
\dot{a}&=&-\kappa_aa-i\lambda e^{i\phi}b^\dagger-ig_ac^\dagger-\sqrt{2\kappa_a}a^{in},\nonumber \\
\dot{b}&=&-\kappa_bb-i\lambda e^{i\phi} a^\dagger-ig_bc-\sqrt{2\kappa_b}b^{in},\nonumber \\
\dot{c}&=&-\gamma_cc-i(g_bb+g_aa^\dagger)-\sqrt{2\gamma_c}c^{in},
\end{eqnarray}
where $\kappa_a, \ \kappa_b,\ \gamma_c$ are the damping rates of modes $a$, $b$, $c$, respectively, $a^{in}$, $b^{in}$ and $c^{in}$ are the input quantum noises with zero average value which are taken to be statistically independent with nonzero $\delta$ correlated functions, $
\langle a^{in}\left(t\right)a^{in\dagger}\left(t'\right)\rangle=\delta(t-t')$, $\langle b^{in}\left(t\right)b^{in\dagger}\left(t'\right)\rangle =\delta(t-t')$, $\left<c^{in}\left(t\right)c^{in\dagger}\left(t'\right)\right> = (\bar{n}_{th}+1)\delta(t-t')$, $\langle c^{in\dagger}\left(t\right)c^{in}\left(t'\right)\rangle = \bar{n}_{th}\delta(t-t')$, where
$\bar{n}_{th}=[\exp\frac{\hbar\omega_c}{k_BT}-1]^{-1}$ is mean thermal occupation number of the the intermediate mode $c$ at the frequency $\omega_c$, $T$ is the temperature of the surrounding environment, and $k_B$ is the Boltzmann constant. Here we assume that modes $a$ and $b$ are in the ordinary zero-temperature environment ($\bar{n}_{th_a}=\bar{n}_{th_b}=0$), whereas the intermediate mode is in a thermal state with nonzero temperature. We will also show the influence of noises when all three modes are thermally excited.

We are interested in the phase dependent effects on the generation and control of Gaussian entanglement and steering between modes $a$ and $b$, so define the quadrature operators of modes $X_j=(j+j^\dagger)/\sqrt{2}$, $Y_j=(j-j^\dagger)/\sqrt{2}i$, and corresponding Langevin noise operators $X_j^{in}=(j^{in}+j^{in\dagger})/\sqrt{2}$, $Y_j^{in}=(j^{in}-j^{in\dagger})/\sqrt{2}i$, $(j=a,b,c)$. The QLEs (\ref{Langevin}) can be thus written as
\begin{equation}\label{Langevin_quadrature}
\dot{u}=Mu-\Lambda u^{in},
\end{equation}
where $u=(X_{a},Y_{a},X_{b},Y_{b},X_{c},Y_{c})^T$ (the superscript $T$ denotes the transposition), the diagonal damping matrix $\Lambda=\text{diag}(\sqrt{2\kappa_a},\sqrt{2\kappa_a},\sqrt{2\kappa_b},\sqrt{2\kappa_b},\sqrt{2\gamma_c},\sqrt{2\gamma_c})$, and the corresponding vector of noises $u^{in}=(X_{a}^{in},Y_{a}^{in},X_{b}^{in},Y_{b}^{in},X_{c}^{in},Y_{c}^{in})^T$. The $6\times6$ drift matrix $M$ is the coefficient matrix of the system, which reads
 \begin{equation*}
M=-\left(
   \begin{array}{cccccc}
     \kappa_a & 0 & -\lambda\sin{\phi}& \lambda\cos{\phi} & 0 & g_a\\
     0 & \kappa_a & \lambda\cos{\phi}& \lambda\sin{\phi} & g_a & 0 \\
     -\lambda\sin{\phi} & \lambda\cos{\phi} & \kappa_b & 0 & 0 & -g_b \\
     \lambda\cos{\phi} & \lambda\sin{\phi} & 0 & \kappa_b & g_b & 0\\
     0 & g_a & 0 & -g_b & \gamma_c & 0\\
     g_a & 0 & g_b & 0 & 0 & \gamma_c
   \end{array}
 \right).
 \end{equation*}
 %
The system is stable only when the real parts of all the eigenvalues of matrix $M$ are negative, and the stability condition can be derived from Routh-Hurwitz criterion \cite{Dejesus1987}. The general stability condition depending on the relative phase is too complex, which can be simply given for the cases of $\phi=\frac{\pi}{2}+n\pi$ $(n \in \mathbb{Z})$,
 \begin{eqnarray}
&&\kappa_a\kappa_b\gamma_c-g_a^2\kappa_b+g_b^2\kappa_a-\lambda^2\gamma_c >0, \nonumber \\
&&(\kappa_a+\kappa_b)(\kappa_a+\gamma_c)(\kappa_b+\gamma_c)\nonumber\\
&&- g_a^2(\kappa_a+\gamma_c)+g_b^2(\kappa_b+\gamma_c)-\lambda^2(\kappa_a+\kappa_b)>0.
 \label{stability condition}
 \end{eqnarray}
 %

\section{CRITERIA FOR ENTANGLEMENT AND EPR STEERING}\label{CRITERIA}

The steady state of the system is a zero-mean Gaussian state because of the Gaussian nature of the input quantum noises $a^{in}$, $b^{in}$ and $c^{in}$ and the linearized dynamics, which can be fully characterized by its covariance matrix (CM) $V$ with components $V_{lm}=[\langle u_l(\infty)u_m(\infty)+u_m(\infty)u_l(\infty)\rangle]/2$ $(l,m=1,2,\cdots,6)$, where $u_l(t)$ is the $l$th quadrature component of $u(t)$.
The steady-state CM can be determined by solving the Lyapunov equation $
 MV+V M^{T}=-D$, where the diffusion matrix $D$ characterizes the stationary noise correlations and is defined through $\langle v_l(t)v_m(t')+v_m(t')v_l(t)\rangle/2=D_{lm}\delta(t-t')$ ($v=-\Lambda u^{in}$)~\cite{Genes2008,Barzanjeh2011}, such that here $D=\text{diag}(\kappa_a,\kappa_a,\kappa_b,\kappa_b,(2\bar{n}_{th}+1)\gamma_c,(2\bar{n}_{th}+1)\gamma_c)$. The Lyapunov equation is a linear equation for $V$ and can be straightforwardly solved.

Since we are specifically interested in the entanglement and EPR steering created between modes $a$ and $b$, it is enough to consider the reduced CM
 %
\begin{equation}
V=\left(
   \begin{array}{cc}
   V_a & V_{ab}\\
   V_{ab}^T & V_b
   \end{array}
   \right),
\end{equation}
where the submatrices $V_a$ and $V_b$ are corresponding to the reduced states of modes $a$ and $b$, respectively. In order to measure the entanglement, we adopt the logarithmic negativity $E_N=\max{\{0,-\ln{2\eta^{-}}\}}$ \cite{Vidal2002,Adesso2004}, where $\eta^{-}\equiv\sqrt{\Sigma(V)-[\Sigma(V)^2-4\det{V}]^{1/2}}/\sqrt{2}$, with $\Sigma(V)\equiv\det{V_a}+\det{V_b}-2\det{V_{ab}}$. Therefore, a Gaussian state is entangled if and only if $\eta^{-}<1/2$, and larger value of $E_N$ implies stronger entanglement between modes.

To signify EPR steering, we adopt the computable measure of Gaussian steering proposed in Ref.~\cite{Kogias2015} for arbitrary bipartite Gaussian states under Gaussian measurements. The steering in the direction from mode $a$ to mode $b$ ($\mathcal{G}^{a \rightarrow b}$) and in the opposite direction ($\mathcal{G}^{b\rightarrow a}$) are quantified by
 %
 \begin{eqnarray}
&& \mathcal{G}^{a\rightarrow b}=\max{\{0,S(2V_a)-S(2V)\}}, \nonumber \\
&& \mathcal{G}^{b\rightarrow a}=\max{\{0,S(2V_b)-S(2V)\}},
 \end{eqnarray}
 %
 where $S(\sigma)=\frac{1}{2}\ln{\det{\sigma}}$ is the R\'enyi-2 entropy. $\mathcal{G}^{a \rightarrow b}>0$ ($\mathcal{G}^{b \rightarrow a}>0$) implies that mode $a$ ($b$) can steer mode $b$ ($a$) by Gaussian measurements, and its value quantifies the degree of steering, i.e., the higher the value of $\mathcal{G}$ becomes, the stronger the Gaussian steerability appears.

For the present system, the steady-state values $\langle a^2\rangle=\langle b^2\rangle=\langle a^\dagger b\rangle=0$, such that the conditions to satisfy $E_N>0$, $\mathcal{G}^{a\rightarrow b}>0$ and $\mathcal{G}^{b\rightarrow a}>0$ can be also expressed in terms of correlation-based inequalities, respectively~\cite{Hillery2006,Cavalcanti2011},
 \begin{eqnarray}
 |\langle ab\rangle| &>&\sqrt{\langle a^\dagger a\rangle\langle b^\dagger b\rangle}, \nonumber\\
 |\langle ab\rangle|&>&\sqrt{\langle b^\dagger b\rangle(\langle a^\dagger a\rangle+1/2)}, \label{G12}\nonumber\\
 |\langle ab\rangle|&>&\sqrt{\langle a^\dagger a\rangle(\langle b^\dagger b\rangle+1/2)}. \label{G21}
 \end{eqnarray}
 %
The steady-state solution of populations $\langle a^\dagger a\rangle, \ \langle b^\dagger b\rangle $ and correlation $\langle ab\rangle$ at $\phi=\frac{\pi}{2}+n\pi\ (n \in \mathbb{Z})$ is detailed in Appendix~\ref{solution}, and the derivation of the inequalities~(\ref{G21}) is given in Appendix~\ref{derivation}.

\section{Control and enhancement of quantum correlations via interference}\label{RESULTS}
\subsection{Phase-sensitive quantum entanglement and directional steering}

Both the direct two-mode squeezing interaction and the induced indirect interaction by being coupled to the third intermediate mode with beam-splitter-type coupling and parametric-down-conversion-type coupling, respectively, can create entanglement between modes $a$ and $b$. Thus the superposition of these two physical interaction paths may create quantum interference effects which depend on the relative phase of the coupling strengths. In the following we will investigate the phase dependent effects on the creation of quantum entanglement and EPR steering.
\begin{figure}
\includegraphics[width=0.75\columnwidth]{Fig2a.pdf}
\includegraphics[width=0.75\columnwidth]{Fig2b.pdf}
\caption{Stationary entanglement ($E_N$, black solid) and quantum steering between two modes $a$ and $b$ ($\mathcal{G}^{a\rightarrow b}$: blue dashed, $\mathcal{G}^{b\rightarrow a}$: red dotted) as a function of the relative phase $\phi$ of interfering channels, when (a) $\lambda=0.4\kappa$ and (b) $\lambda=0.605\kappa$. Other parameters are $\kappa_a=\kappa_b=\kappa,\ g_a=3.2\kappa,\ g_b=5\kappa,\gamma_c=2\kappa,\ \bar{n}_{th}=0$.}
\label{bbr_phase}
\end{figure}

The measures of entanglement ($E_N$) and EPR steering ($\mathcal{G}$) vary with $\phi$, as depicted in Fig.~\ref{bbr_phase}. Apparently, entanglement evolves with $\phi$ periodically (black solid curve), maximized when phase $\phi=(2n+3/2)\pi$ ($n \in \mathbb{Z}$) by constructive interference and minimized at $\phi=(2n+1/2)\pi$ by destructive interference. Interestingly, Gaussian steering in two directions behave asymmetrically with phase, that is, steering from mode $a$ to mode $b$ quantified by the parameter $\mathcal{G}^{a\rightarrow b}$ (blue dashed curve) exhibits the same phase-depdent behavior as that of the entanglement measure $E_N$, whereas steering in the other direction measured by the parameter $\mathcal{G}^{b\rightarrow a}$ (red dotted curve) behaves in a completely different way. Specifically, steering in this direction doesn't exist for the case of $\lambda=0.4\kappa$ shown in Fig.~\ref{bbr_phase}(a), such that one-way steering in the direction of $a\rightarrow b$ occurs over the whole period of $\phi$. For larger value of $\lambda=0.605\kappa$, $\mathcal{G}^{b\rightarrow a}$ reaches a maximal value at $\phi=(2n+3/2)\pi$ as well, and we can get rich properties of Gaussian steering, e.g., the overall state's asymmetry is stepwise driven through the one-way regime ($\mathcal{G}^{a\rightarrow b}>0$ but $\mathcal{G}^{b\rightarrow a}=0$), no-way regime ($\mathcal{G}^{a\rightarrow b}=0$ and $\mathcal{G}^{b\rightarrow a}=0$), and two-way regime ($\mathcal{G}^{a\rightarrow b}>0$ and $\mathcal{G}^{b\rightarrow a}>0$), and finally one-way regime again over one period of phase, as shown in Fig.~\ref{bbr_phase}(b).

For the parameters studied here, we can find that the steerability from $a$ to $b$ is always larger than the steerability from $b$ to $a$. This can be understood from the inequalities (\ref{G21}) and the solutions given in Eq.~\ref{correlation_term}. When the two modes have same dissipation rates $\kappa_a=\kappa_b=\kappa$ and mode $c$ is at zero temperature $\bar{n}_{th}=0$, from Eq.~\ref{correlation_term} we can see that the photon fluctuation of mode $a$ is always larger than that in mode $b$, i.e., $\langle a^\dagger a\rangle>\langle b^\dagger b\rangle$, and therefore the condition given in Eq.~(\ref{G12}) to confirm steering $a\rightarrow b$ is easier to satisfy than that required for demonstrating steering $b\rightarrow a$.

Now we come to the first important result of this paper. Two identical modes which have completely symmetric decoherence $\kappa_a=\kappa_b=\kappa$ and $\bar n_{th,a}=\bar n_{th,b}=0$ can be still prepared into an entangled state with asymmetric and directional steering by adjusting the phase of the interfering channels. This reveals the inherent asymmetric nature of quantum steering with respect to the two parties involved. Comparing with the way to produce one-way steering in earlier studies by adding losses or thermal noises on one subsystem~\cite{Handchen2012,Armstrong2015,Qin2017} with the cost of reducing correlation, engineering the interference is an active method which creates asymmetric steering and at the same time enhances the steerability.

\subsection{Performance comparison with correlations created by only one interaction path}
\begin{figure}
\includegraphics[width=1.0\columnwidth]{Fig3.pdf}
\caption{The measures of entanglement ($E_N$) and steering ($\mathcal{G}^{a\rightarrow b}$, $\mathcal{G}^{b\rightarrow a}$) between modes $a$ and $b$ versus the damping rate $\gamma_c/\kappa$ for the cases of creating correlations only by the indirect interaction path $\lambda=0$ (blue solid), and by interfering channels when $\lambda=0.5\kappa$ (red dashed), $\lambda=\kappa$ (black dotted), $\lambda=1.5\kappa$ (green dash-dotted). The destructive and constructive interference at phase $\phi=\pi/2$ and $\phi=3\pi/2$ lead to the reduction (a-c) and enhancement (d-f) of correlations, respectively. Other parameters are $g_a=8.3\kappa,\ g_b=10\kappa,\ \bar{n}_{th}=0$.}
\label{bbr_dissipation_gamma}
\end{figure}

To clearly indicate the performance of control and enhancement of entanglement and steering via interfering channels, we compare with the results achieved via only one of two interaction paths.

First, the numerical results of the dependence of entanglement and steering on the damping rate of the third intermediate mode $\gamma_c/\kappa$ are given in Fig.~\ref{bbr_dissipation_gamma} for different coupling strengths of the direct interaction path $\lambda$. Blue solid curves present the results achieved only by the indirect coupling path, i.e., $\lambda=0$, other curves show the correlations achieved by the interfering channels with phase $\phi=\pi/2$ [Figs.~\ref{bbr_dissipation_gamma}(a-c)] and with phase $\phi=3\pi/2$ [Figs.~\ref{bbr_dissipation_gamma}(d-f)] for $\lambda=0.5\kappa$ (red dashed), $\kappa$ (black dotted), $1.5\kappa$ (green dash-dotted), respectively. It can be seen that the correlations are reduced due to destructive interference of two interaction paths at $\phi=\pi/2$. Comparing the steering in two directions shown in Figs.~\ref{bbr_dissipation_gamma}(b) and (c), increasing value of $\lambda$ leads to the stronger destructive interference effects on reducing the correlation between two modes, but it features one-way steering in a wider parameter range. For instance, when $\lambda=\kappa$ shown by the black dotted lines, the steering becomes possible only in one direction $a\rightarrow b$ for almost all values of $\gamma_c$, while for not accounting for the direct interaction path ($\lambda=0$, blue solid), one-way steering is only possible for strong mechanical damping rate $\gamma_c\gg\kappa$.

When the interfering channels with phase $\phi=3\pi/2$, as shown in Figs.~\ref{bbr_dissipation_gamma}(d-f), the degree of entanglement and steering are remarkably enhanced due to the constructive interference of two interaction paths comparing with the result achieved only by the indirect interaction path ($\lambda=0$, blue solid). For this case, we can get stronger two-way steering over a wider parameter range, which has been proven to be a necessary resource required for teleporting a coherent state with fidelity beyond the no-cloning threshold~\cite{He2015}. The different degree of steerability in two directions also provides the asymmetric guaranteed key rate achievable within a practical one-sided device-independent QKD~\cite{Kogias2015}. In the regime of $\gamma_c\gg\kappa$, the steering in the direction $\mathcal{G}^{b\rightarrow a}$ becomes lost. Thus, the constructive interference can also manipulate the direction of steering from asymmetric two-way to one-way. The steady-state steering generated only by the indirect interaction path has been studied in Ref.~\cite{Tan2015}, where the authors concluded that in the regime $\gamma_c\ll\kappa$ the steady entangled states are definitely not steerable in any direction when two optical modes have same damping rates $\kappa_a=\kappa_b=\kappa$, while only the mode with larger dissipation rate can be steered by the other one. Our result, however, shows that by applying the interfering channels the steering in both directions may be achieved in this regime due to the constructive interference, as indicated by the black dotted curves in Fig.~\ref{comparision}.
\begin{figure}
\includegraphics[width=0.8\columnwidth]{Fig4.pdf}
\caption{Comparison of Gaussian steering in the regime of $\gamma_c\ll\kappa$ achieved by the indirect path only ($\lambda=0$, blue solid) and by the interfering channels with constructive interference ($\lambda=\kappa,\ \phi=3\pi/2$, black dotted). Other parameters are used as those in Fig.~\ref{bbr_dissipation_gamma}.}
\label{comparision}
\end{figure}

Second, we compare with the results obtained when only the direct interaction path exists. In this case, the photon fluctuation and correlation expressions given by Eq.~(\ref{correlation_term}) in Appendix can be simplified as
\begin{eqnarray}
\langle a^\dagger a\rangle&=&\frac{\kappa_b\lambda^2}{(\kappa_a+\kappa_b)(\kappa_a\kappa_b-\lambda^2)}, \nonumber \\
\langle b^\dagger b\rangle&=&\frac{\kappa_a\lambda^2}{(\kappa_a+\kappa_b)(\kappa_a\kappa_b-\lambda^2)}, \nonumber \\
\langle ab\rangle&=&\frac{\kappa_a\kappa_b\lambda(\sin{\phi}-i\cos{\phi})}{(\kappa_a+\kappa_b)(\kappa_a\kappa_b-\lambda^2)}.
\end{eqnarray}
The conditions to confirm the presence of steering given in Eq.~(\ref{G12}) then reduce to
\begin{eqnarray}\label{G_coherent}
&&\mathcal{G}^{a\rightarrow b}>0:(\kappa_b-\kappa_a)(\kappa_a\kappa_b-\lambda^2)>0, \nonumber \\
&&\mathcal{G}^{b\rightarrow a}>0:(\kappa_a-\kappa_b)(\kappa_a\kappa_b-\lambda^2)>0.
\end{eqnarray}
We can see from Eq. (\ref{G_coherent}) that it is not possible to create steering in both directions when two modes possess symmetric dissipation rate, $\kappa_a=\kappa_b$. Since the stability condition of the system for this case requires $\lambda^2<\kappa_a\kappa_b$, then $\mathcal{G}^{a\rightarrow b}>0$ when $\kappa_b>\kappa_a$, and $\mathcal{G}^{b\rightarrow a}>0$ when $\kappa_a>\kappa_b$. This means that one can only produce one-way steering from the mode with lower damping rate to the mode with higher decay rate. This observation consists with the conclusion made in earlier studies~\cite{Tan2015,Handchen2012,Reidjosab2015} that the decoherence with no thermal noise on the steering system has substantial effect more than that on the steered system, such that the condition $\kappa_b>\kappa_a$ may destroy (preserve) steering in the direction $b\rightarrow a$ ($a\rightarrow b$) where mode $b$ possessing higher decoherence acts as steering (steered) party, respectively.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{Fig5a.pdf}
\includegraphics[width=0.8\columnwidth]{Fig5b.pdf}
\caption{The comparison of entanglement and steering achieved by only one direct interaction path (blue solid) and by the interfering channels at phase (a) $\phi=\pi/2$ and (b) $\phi=3\pi/2$, respectively. The entanglement $E_N$ (black dash-dotted) and steering in two directions $\mathcal{G}^{a\rightarrow b}$ (red dotted), $\mathcal{G}^{b\rightarrow a}$ (red dashed) are notably enhanced, meanwhile, the range of steady-state solution is also remarkably broadened when two interaction paths superpose and interference happens. Other parameters are $g_a=8.3\kappa,\ g_b=10\kappa,\gamma_c=5\kappa, \bar{n}_{th}=0$.}
\label{bbr_coherent_lambda0}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{Fig6a.pdf}
\includegraphics[width=0.8\columnwidth]{Fig6b.pdf}
\caption{The influence of thermal noise on the entanglement ($E_N$, solid) and steering ($\mathcal{G}^{a\rightarrow b}$ dotted and $\mathcal{G}^{b\rightarrow a}$ dashed) between modes $a$ and $b$ at phase $\phi=3\pi/2$ when (a) $\bar{n}_{th}$ only exists on the intermediate damping mode $c$, (b) $\bar{n}_{th_a}=\bar{n}_{th_b}=\bar{n}_{th}$. Blue curves indicate the results achieved only by the indirect interaction path where $\lambda=0$, and correspondingly red curves represent the enhanced correlations via interference channels when $\lambda=1.5\kappa$. Other parameters are $g_a=8.3\kappa,\ g_b=10\kappa,\gamma_c=5\kappa$.}
\label{bbr_nth}
\end{figure}

However, for two modes possessing completely symmetric decoherence properties we show in Fig.~\ref{bbr_coherent_lambda0} that by applying the interfering channels one can still create asymmetric and directional steering. Meanwhile, the degree of correlations is enhanced even at phase $\phi=\pi/2$ [Fig.~\ref{bbr_coherent_lambda0}(a)] that induces destructive interference, and the enhancement is more remarkable at phase $\phi=3\pi/2$ due to constructive interference  [Fig.~\ref{bbr_coherent_lambda0}(b)]. The blue curves in two plots are the entanglement achieved by the direct interaction path, while steering in both directions cannot be produced ($\mathcal{G}=0$ not plotted in the figure for clearity). When we include the other indirect coupling path, the entanglement (black dash-dotted) is remarkably enhanced, and asymmetric steering in the direction $\mathcal{G}^{a\rightarrow b}$ (red dotted) and in the opposite direction $\mathcal{G}^{b\rightarrow a}$ (red dashed) appear. Note that with destructive interference shown in Fig.~\ref{bbr_coherent_lambda0}(a), although correlations are reduced when $\lambda$ increases, it is useful to achieve one-way steering $\mathcal{G}^{b\rightarrow a}=0$ and $\mathcal{G}^{a\rightarrow b}>0$. With constructive interference shown in Fig.~\ref{bbr_coherent_lambda0}(b), the enhancement of entanglement and steering becomes notable,  and two-way asymmetric steering occurs for all steady-state solution. This observation consists with that shown in Fig.~\ref{bbr_dissipation_gamma} and the reason why $\mathcal{G}^{a\rightarrow b}>\mathcal{G}^{b\rightarrow a}$ has been explained there. Moreover, the parameter range to achieve steady-state solution is considerably broadened via interference channel. The stability condition given in Eq.~(\ref{stability condition}) at phase $\phi=\frac{\pi}{2}+n\pi\ (n \in \mathbb{Z})$ requires $\lambda^2<\text{min}\{\kappa_a\kappa_b+(g_b^2\kappa_a-g_a^2\kappa_b)/\gamma_c, \ \kappa_a\kappa_b+\gamma_c(\kappa_a+\kappa_b+\gamma_c)+[g_b^2(\kappa_b+\gamma_c)-g_a^2(\kappa_a+\gamma_c)]/(\kappa_a+\kappa_b)\}$, for the parameters studied here, the threshold is larger than $\kappa_a\kappa_b$ required by the direct interaction path only, that is to say, the system is more stable by applying our interference approach.

So far we have shown how the entanglement and steering can be enhanced and the direction of steering can be controlled via interference effects when three modes are not thermal excited. In practical implementation using cavity optomechanical systems, the third intermediate mode could be a mechanical mode with nonzero thermal phonons, so we show the influence of thermal noise $\bar{n}_{th}$ only existing in mode $c$ on the correlations at phase $\phi=3\pi/2$ in Fig.~\ref{bbr_nth}(a). In general, thermal noise is detrimental to correlations because it causes decoherence, however, entanglement (red solid) and steering (red dotted and dashed) created by the interference channels are more robust against thermal noise comparing with that achieved only by the indirect interaction path (correspondingly blue curves). Even when all three modes are in thermal excited states, it shows again in Fig.~\ref{bbr_nth}(b) that the entanglement and asymmetric steering created by the interference channels (red curves) are more robust against thermal noise than that achieved by only applying the indirect interaction path (blue curves).

\subsection{The enhancement of entanglement and steering between other two modes in the loop}
\begin{figure}[htbp!]
\includegraphics[width=0.8\columnwidth]
{Fig7.pdf}
\caption{The phase-dependent behavior of steady-state entanglement and steering between pair $a,b$ (blue curves) and between pair $a,c$ (red curves). When entanglement ($E^{ab}_{N}>0$, blue solid) and one-way steering ($\mathcal{G}^{a\rightarrow b}>0$, blue dashed, and $\mathcal{G}^{b\rightarrow a}=0$, blue dotted) between modes $a$ and $b$ reach a maximum, the entanglement ($E^{ac}_{N}>0$, red solid) and one-way steering ($\mathcal{G}^{a\rightarrow c}>0$, red dashed, and $\mathcal{G}^{c\rightarrow a}=0$, red dotted) between modes $a$ and $c$ are at a minimum. Other parameters are $\kappa_a=\kappa_b=\kappa,\ g_a=3.2\kappa,\ g_b=5\kappa,\ \lambda=0.4\kappa,\ \gamma_c=15\kappa,\ \bar{n}_{th}=0$.}
\label{phase_two pairs}
\end{figure}

We focused on the enhancement of entanglement and the control of the asymmetric and directional steering via interference effects between modes $a$ and $b$ in the previous discussion, here, we want to point out that this could be also realized between modes $a$ and $c$, which means the method is general. As shown by the inset plot in Fig.~\ref{phase_two pairs}, there are two interaction paths to create entanglement between modes $a$ and $c$, one is by the direct two-mode squeezing interaction with coupling strength $g_a$, and the other one is by swapping the entanglement shared by $a$ and $b$ to pair $a$ and $c$ via the beam-splitter-type interaction between modes $b$ and $c$. Two physical interaction paths superpose with relative phase and create interfering channels. The phase-dependent behavior of the entanglement ($E^{ac}_N$ red solid) and one-way steering between modes $a$ and $c$ ($\mathcal{G}^{a\rightarrow c}>0$ red dashed, $\mathcal{G}^{c\rightarrow a}=0$ red dotted) is illustrated in Fig.~\ref{phase_two pairs}. We also notice that when the entanglement and steering between modes $a$ and $c$ reaches a maximum (minimum) of the interference pattern at where the corresponding correlations between modes $a$ and $b$ are a minimum (maximum), indicating that there exists a competition to distribute entanglement and steering between different pairs. This can be understood by the monogamy constraints for distributing correlations among multi parties.

\section{Conclusion}\label{SUMMARY}
In summary, we propose an active method to generate and control steady-state entanglement and asymmetric steering between two identical modes by intriguing constructive or destructive quantum interference effects in a general three-mode system with closed-loop in coupling. The direct interaction path between two modes and the indirect interaction path induced by both coupling to the intermediate damping mode superpose and act as a interfering channel with a tunable phase, which can create phase-dependent correlations. We show that the interference effects can not only enhance the degree of the entanglement and steerability, but also produce asymmetric steering when two modes possess completely symmetric decoherence properties. Instead of introducing different amount of losses or noises to subsystems, this provides inspiration for manipulating the direction of the asymmetric quantum steering with enhanced steerability. In addition, the entangled steady states created by the interference channel are more robust against thermal noises. Furthermore, using the interfering channels to entangle the other pair of modes in the loop is also discussed, and the opposite phase-dependent behavior indicates the monogamy constraints for distributing entanglement and steering among multipartite. This work opens up new perspectives for the experimental production and applications of EPR steering as a precious resource for secure quantum communication technologies.

\begin{acknowledgements}
We acknowledge illuminating discussions with Y. Li and H.Y.Yuan. This work is supported by the National Key R\&D Program of China (Grants No. 2018YFB1107200 and No. 2016YFA0301302) and the National Natural Science Foundation of China (Grants No. 11622428, No. 61475007, and No. 61475006). Q.He thanks the Beijing Computational Science Research Center for their hospitality.
\end{acknowledgements}

\section{Introduction}

A great deal of effort has been devoted to generate and control entanglement which is the most intrinsic feature of quantum mechanics with a variety of applications in quantum information processing~\cite{Horodecki2009}. A strict subset of entanglement called quantum steering is of particular interest~\cite{Reid2009,PhysRep2017}. The concept was originally discovered by Schr\"{o}dinger~\cite{Schrodinger1935} in response to the freaky ``spooky action-at-a-distance'' predicted by Einstein, Podolsky, and Rosen (EPR) in their famous paradox~\cite{Einstein1935,Reid1989}, describing the local measurements on one of two entangled particles can adjust (steer) the state of the other distant particle. There has been an increasing interest in EPR steering since it was rigorously defined by mathematical formulation from the perspective of quantum information~\cite{Wiseman2007,Jones2007,Cavalcanti2009}, i.e., in a network one can verify the entanglement between Alice and Bob without the requirement of trust of Bob's equipment used to perform local measurements at his node by confirming the presence of steering of Alice's system by Bob.
This feature makes quantum steering substantial to various quantum information protocols which rely on entanglement by providing extra security~\cite{Opanchuk2014}, such as semi-sided device-independent quantum key distribution~\cite{Branciard2012,Gehring2015,Walk2016} and quantum secret sharing~\cite{Armstrong2015,Xiang2017,Kogias2017}, one-way quantum computing~\cite{Li2015}, no-cloning quantum teleportation~\cite{He2015,Reid2013,CMLi2016}, subchannel discrimination~\cite{Piani2015}, and other related protocols.

A second distinctive feature, in contrast to entanglement and Bell nonlocality, is that quantum steering implies a direction between the parties involved. The steering strengths in two directions may not be the same~\cite{Wagner2008}, especially, a special type is known as one-way EPR steering where the entangled states show steering in one direction but not in the other~\cite{Wiseman2007,Handchen2012}. The asymmetric EPR steering has attracted considerable attention recently for both theory~\cite{Olsen2008,Midgley2010,Olsen2013,Schneeloch2013,He2013,He2014,Evans2014,Bowles2014,Skrzypczyk2014,Wang2014,Reidjosab2015,QHe2015,Tan2015,Marco2015,Bowles2016,Olsen2017,Baker2018} and experiment~\cite{Handchen2012,Armstrong2015,Qin2017,Sun2016,Wollmann2016,Xiao2017,Tischler2018,Fadel2018}. One important method used in above studies to produce directional steering is making the states asymmetric by adding different amount of losses or noises on the subsystems. For example, the entangled two-qubit states which are one-way steerable have been experimentally demonstrated by passing one party into a lossy channel with a given probability of obtaining qubit~\cite{Wollmann2016,Tischler2018}. It is important to realize one-way steerable Gaussian continuous variable states referred in original EPR argument. One-way steering in the Gaussian regime (Gaussian states and Gaussian measurements) was first observed by introducing additional amounts of loss to one mode of a two-mode squeezed state (TMSS) (vacuum is coupled into the signal by a beam splitter with transmission efficiency $\eta$)~\cite{Handchen2012}, and then extended to a multipartite optical network~\cite{Armstrong2015}. The direction of Gaussian steering can be also manipulated by adding Gaussian noises to one party of TMSS and transmitting the TMSS in a lossy channel~\cite{Qin2017}. Rather than by adding asymmetric loss or noise to subsystems which in general leads to the reduction of correlation, recently, Olsen proposed a nonlinear optical apparatus to control the asymmetry of EPR steering by controlling the amplitude of signals injected into nondegenerate parametric oscillator such that the effects are intrinsic to the scheme~\cite{Olsen2017}.

In light of the exceptional importance of these developments, we provide in this paper a different mechanism to prepare and control the asymmetric steering with enhanced steerability via quantum interference effects in a general three-mode system where all three modes are mutually coupled with each other. We show that by engineering interference between two different interaction paths, i.e., a direct interaction path and an indirect interaction path induced by both being coupled to the third intermediate damping mode, two identical modes with symmetric decoherence properties can be driven into an entangled steady state with asymmetric and directional steering, meanwhile the entanglement and steerability can be effectively enhanced due to constructive interference. This provides an active method to create and control the direction of asymmetric steering without the cost of reducing steerability. Moreover, the steady-state entanglement and asymmetric steering created by the interference effects appear to be more robust against thermal noise. Finally, we have also demonstrated the opposite interference patterns of entanglement and steering between other pair of modes in the loop, indicating the monogamy constraints for distributing quantum correlations among multipartite. It is worth noticing that new quantum interference effects arising from a closed coupling loop have been recently applied to study optomechanical interferences and phonon correlations~\cite{Restrepo2017,Sun2017}, the phase effects on phonon blockade effects~\cite{Restrepo2017,Wang2017}, the transduction bandwidth of a rf-to-optical transducer ~\cite{MoaddelHaghighi2018}, and the optical nonreciprocal behavior~\cite{Metelmann2015,Xu2015,Xu2016,Xu2017} which has been experimentally implemented~\cite{Ruesink2016,Barzanjeh2017,Bernier2017}.

The remainder of this paper is organized as follows. In Sec.~\ref{Model}, we introduce a general three-mode system with closed-loop in coupling which creates interference effects determined by the relative phase of the coupling strengths between two interaction paths. In Sec.~\ref{CRITERIA}, we recall the criteria adopted in this paper to test quantum entanglement and Gaussian steering. We then investigate in Sec.~\ref{RESULTS}, the creation and control of the steady-state entanglement and quantum steering by the interfering channels and compare its performance with the results achieved by applying only one interaction path. The results show the enhancement of correlations, the effective control of the asymmetric steering, and the robustness to thermal noises via interference effects. Finally, we summarize our results in Sec.~\ref{SUMMARY}.

\section{Model}\label{Model}
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.5\columnwidth]{Fig1.pdf}
\caption{Schematic diagram of a general three-mode system with closed coupling loop. Modes $a$ and $b$ interact directly via a two-mode squeezing interaction with strength $\lambda$; at the same time, they are coupled to an intermediate damped mode $c$ resulting in an indirect interaction path between them indicated by the blue dashed curve. The relative phase $\phi$ between two paths can be engineered to control and improve the correlation between modes $a$ and $b$. }
\label{scheme}
\end{figure}
We consider a three-mode system represented by the annihilation operators $a$, $b$, $c$, in which modes $a$ and $b$ interact with each other in two distinct paths, i.e., the direct coupling path resulting from a two-mode squeezing interaction and the induced indirect interaction path obtained by simply coupling to the third intermediate damping mode $c$. The interaction of the system is thus described by a general form of Hamiltonian (with $\hbar=1$)
\begin{eqnarray}
H=&&\lambda e^{i\phi} a^\dagger b^\dagger+\lambda e^{-i\phi}ab + g_a(e^{i\phi_a}c^\dagger a^\dagger+e^{-i\phi_a}ca)\nonumber \\
&&+g_b(e^{i\phi_b}c^\dagger b+e^{-i\phi_b}cb^\dagger),
\end{eqnarray}
where $\lambda,\ g_a,\ g_b$ are the effective coupling strengths with individual phase $\phi,\ \phi_a,\ \phi_b$ controlled by the driving laser fields. This creates a three-mode closed-loop in the coupling, as illustrated in Fig.~\ref{scheme}, which will give rise to interference effects and phase dependence of the dynamics of the modes. Since the interference is dependent on the relative phase between the two interaction paths, without loss of generality, we can absorb the phases $\phi_a$ and $\phi_b$ into $\phi$ by redefining the operators $a$ and $b$. In the following we will assume $\phi_a=\phi_b=0$ and treat $\phi$ as the relative phase in the interference for simplicity. This coupling loop could be realized in three-mode optomechnical systems~\cite{Sun2017}, or in antiferromagnets with magnon-photon coupling~\cite{Yuan2017}.

Substituting the Hamiltonian to the Heisenberg equation and taking into account the dissipation-fluctuation processes, we get the quantum Langevin equations (QLEs) for each mode
\begin{eqnarray}\label{Langevin}
\dot{a}&=&-\kappa_aa-i\lambda e^{i\phi}b^\dagger-ig_ac^\dagger-\sqrt{2\kappa_a}a^{in},\nonumber \\
\dot{b}&=&-\kappa_bb-i\lambda e^{i\phi} a^\dagger-ig_bc-\sqrt{2\kappa_b}b^{in},\nonumber \\
\dot{c}&=&-\gamma_cc-i(g_bb+g_aa^\dagger)-\sqrt{2\gamma_c}c^{in},
\end{eqnarray}
where $\kappa_a, \ \kappa_b,\ \gamma_c$ are the damping rates of modes $a$, $b$, $c$, respectively, $a^{in}$, $b^{in}$ and $c^{in}$ are the input quantum noises with zero average value which are taken to be statistically independent with nonzero $\delta$ correlated functions, $
\langle a^{in}\left(t\right)a^{in\dagger}\left(t'\right)\rangle=\delta(t-t')$, $\langle b^{in}\left(t\right)b^{in\dagger}\left(t'\right)\rangle =\delta(t-t')$, $\left<c^{in}\left(t\right)c^{in\dagger}\left(t'\right)\right> = (\bar{n}_{th}+1)\delta(t-t')$, $\langle c^{in\dagger}\left(t\right)c^{in}\left(t'\right)\rangle = \bar{n}_{th}\delta(t-t')$, where
$\bar{n}_{th}=[\exp\frac{\hbar\omega_c}{k_BT}-1]^{-1}$ is mean thermal occupation number of the the intermediate mode $c$ at the frequency $\omega_c$, $T$ is the temperature of the surrounding environment, and $k_B$ is the Boltzmann constant. Here we assume that modes $a$ and $b$ are in the ordinary zero-temperature environment ($\bar{n}_{th_a}=\bar{n}_{th_b}=0$), whereas the intermediate mode is in a thermal state with nonzero temperature. We will also show the influence of noises when all three modes are thermally excited.

We are interested in the phase dependent effects on the generation and control of Gaussian entanglement and steering between modes $a$ and $b$, so define the quadrature operators of modes $X_j=(j+j^\dagger)/\sqrt{2}$, $Y_j=(j-j^\dagger)/\sqrt{2}i$, and corresponding Langevin noise operators $X_j^{in}=(j^{in}+j^{in\dagger})/\sqrt{2}$, $Y_j^{in}=(j^{in}-j^{in\dagger})/\sqrt{2}i$, $(j=a,b,c)$. The QLEs (\ref{Langevin}) can be thus written as
\begin{equation}\label{Langevin_quadrature}
\dot{u}=Mu-\Lambda u^{in},
\end{equation}
where $u=(X_{a},Y_{a},X_{b},Y_{b},X_{c},Y_{c})^T$ (the superscript $T$ denotes the transposition), the diagonal damping matrix $\Lambda=\text{diag}(\sqrt{2\kappa_a},\sqrt{2\kappa_a},\sqrt{2\kappa_b},\sqrt{2\kappa_b},\sqrt{2\gamma_c},\sqrt{2\gamma_c})$, and the corresponding vector of noises $u^{in}=(X_{a}^{in},Y_{a}^{in},X_{b}^{in},Y_{b}^{in},X_{c}^{in},Y_{c}^{in})^T$. The $6\times6$ drift matrix $M$ is the coefficient matrix of the system, which reads
 \begin{equation*}
M=-\left(
   \begin{array}{cccccc}
     \kappa_a & 0 & -\lambda\sin{\phi}& \lambda\cos{\phi} & 0 & g_a\\
     0 & \kappa_a & \lambda\cos{\phi}& \lambda\sin{\phi} & g_a & 0 \\
     -\lambda\sin{\phi} & \lambda\cos{\phi} & \kappa_b & 0 & 0 & -g_b \\
     \lambda\cos{\phi} & \lambda\sin{\phi} & 0 & \kappa_b & g_b & 0\\
     0 & g_a & 0 & -g_b & \gamma_c & 0\\
     g_a & 0 & g_b & 0 & 0 & \gamma_c
   \end{array}
 \right).
 \end{equation*}
 %
The system is stable only when the real parts of all the eigenvalues of matrix $M$ are negative, and the stability condition can be derived from Routh-Hurwitz criterion \cite{Dejesus1987}. The general stability condition depending on the relative phase is too complex, which can be simply given for the cases of $\phi=\frac{\pi}{2}+n\pi$ $(n \in \mathbb{Z})$,
 \begin{eqnarray}
&&\kappa_a\kappa_b\gamma_c-g_a^2\kappa_b+g_b^2\kappa_a-\lambda^2\gamma_c >0, \nonumber \\
&&(\kappa_a+\kappa_b)(\kappa_a+\gamma_c)(\kappa_b+\gamma_c)\nonumber\\
&&- g_a^2(\kappa_a+\gamma_c)+g_b^2(\kappa_b+\gamma_c)-\lambda^2(\kappa_a+\kappa_b)>0.
 \label{stability condition}
 \end{eqnarray}
 %

\section{CRITERIA FOR ENTANGLEMENT AND EPR STEERING}\label{CRITERIA}

The steady state of the system is a zero-mean Gaussian state because of the Gaussian nature of the input quantum noises $a^{in}$, $b^{in}$ and $c^{in}$ and the linearized dynamics, which can be fully characterized by its covariance matrix (CM) $V$ with components $V_{lm}=[\langle u_l(\infty)u_m(\infty)+u_m(\infty)u_l(\infty)\rangle]/2$ $(l,m=1,2,\cdots,6)$, where $u_l(t)$ is the $l$th quadrature component of $u(t)$.
The steady-state CM can be determined by solving the Lyapunov equation $
 MV+V M^{T}=-D$, where the diffusion matrix $D$ characterizes the stationary noise correlations and is defined through $\langle v_l(t)v_m(t')+v_m(t')v_l(t)\rangle/2=D_{lm}\delta(t-t')$ ($v=-\Lambda u^{in}$)~\cite{Genes2008,Barzanjeh2011}, such that here $D=\text{diag}(\kappa_a,\kappa_a,\kappa_b,\kappa_b,(2\bar{n}_{th}+1)\gamma_c,(2\bar{n}_{th}+1)\gamma_c)$. The Lyapunov equation is a linear equation for $V$ and can be straightforwardly solved.

Since we are specifically interested in the entanglement and EPR steering created between modes $a$ and $b$, it is enough to consider the reduced CM
 %
\begin{equation}
V=\left(
   \begin{array}{cc}
   V_a & V_{ab}\\
   V_{ab}^T & V_b
   \end{array}
   \right),
\end{equation}
where the submatrices $V_a$ and $V_b$ are corresponding to the reduced states of modes $a$ and $b$, respectively. In order to measure the entanglement, we adopt the logarithmic negativity $E_N=\max{\{0,-\ln{2\eta^{-}}\}}$ \cite{Vidal2002,Adesso2004}, where $\eta^{-}\equiv\sqrt{\Sigma(V)-[\Sigma(V)^2-4\det{V}]^{1/2}}/\sqrt{2}$, with $\Sigma(V)\equiv\det{V_a}+\det{V_b}-2\det{V_{ab}}$. Therefore, a Gaussian state is entangled if and only if $\eta^{-}<1/2$, and larger value of $E_N$ implies stronger entanglement between modes.

To signify EPR steering, we adopt the computable measure of Gaussian steering proposed in Ref.~\cite{Kogias2015} for arbitrary bipartite Gaussian states under Gaussian measurements. The steering in the direction from mode $a$ to mode $b$ ($\mathcal{G}^{a \rightarrow b}$) and in the opposite direction ($\mathcal{G}^{b\rightarrow a}$) are quantified by
 %
 \begin{eqnarray}
&& \mathcal{G}^{a\rightarrow b}=\max{\{0,S(2V_a)-S(2V)\}}, \nonumber \\
&& \mathcal{G}^{b\rightarrow a}=\max{\{0,S(2V_b)-S(2V)\}},
 \end{eqnarray}
 %
 where $S(\sigma)=\frac{1}{2}\ln{\det{\sigma}}$ is the R\'enyi-2 entropy. $\mathcal{G}^{a \rightarrow b}>0$ ($\mathcal{G}^{b \rightarrow a}>0$) implies that mode $a$ ($b$) can steer mode $b$ ($a$) by Gaussian measurements, and its value quantifies the degree of steering, i.e., the higher the value of $\mathcal{G}$ becomes, the stronger the Gaussian steerability appears.

For the present system, the steady-state values $\langle a^2\rangle=\langle b^2\rangle=\langle a^\dagger b\rangle=0$, such that the conditions to satisfy $E_N>0$, $\mathcal{G}^{a\rightarrow b}>0$ and $\mathcal{G}^{b\rightarrow a}>0$ can be also expressed in terms of correlation-based inequalities, respectively~\cite{Hillery2006,Cavalcanti2011},
 \begin{eqnarray}
 |\langle ab\rangle| &>&\sqrt{\langle a^\dagger a\rangle\langle b^\dagger b\rangle}, \nonumber\\
 |\langle ab\rangle|&>&\sqrt{\langle b^\dagger b\rangle(\langle a^\dagger a\rangle+1/2)}, \label{G12}\nonumber\\
 |\langle ab\rangle|&>&\sqrt{\langle a^\dagger a\rangle(\langle b^\dagger b\rangle+1/2)}. \label{G21}
 \end{eqnarray}
 %
The steady-state solution of populations $\langle a^\dagger a\rangle, \ \langle b^\dagger b\rangle $ and correlation $\langle ab\rangle$ at $\phi=\frac{\pi}{2}+n\pi\ (n \in \mathbb{Z})$ is detailed in Appendix~\ref{solution}, and the derivation of the inequalities~(\ref{G21}) is given in Appendix~\ref{derivation}.

\section{Control and enhancement of quantum correlations via interference}\label{RESULTS}
\subsection{Phase-sensitive quantum entanglement and directional steering}

Both the direct two-mode squeezing interaction and the induced indirect interaction by being coupled to the third intermediate mode with beam-splitter-type coupling and parametric-down-conversion-type coupling, respectively, can create entanglement between modes $a$ and $b$. Thus the superposition of these two physical interaction paths may create quantum interference effects which depend on the relative phase of the coupling strengths. In the following we will investigate the phase dependent effects on the creation of quantum entanglement and EPR steering.
\begin{figure}
\includegraphics[width=0.75\columnwidth]{Fig2a.pdf}
\includegraphics[width=0.75\columnwidth]{Fig2b.pdf}
\caption{Stationary entanglement ($E_N$, black solid) and quantum steering between two modes $a$ and $b$ ($\mathcal{G}^{a\rightarrow b}$: blue dashed, $\mathcal{G}^{b\rightarrow a}$: red dotted) as a function of the relative phase $\phi$ of interfering channels, when (a) $\lambda=0.4\kappa$ and (b) $\lambda=0.605\kappa$. Other parameters are $\kappa_a=\kappa_b=\kappa,\ g_a=3.2\kappa,\ g_b=5\kappa,\gamma_c=2\kappa,\ \bar{n}_{th}=0$.}
\label{bbr_phase}
\end{figure}

The measures of entanglement ($E_N$) and EPR steering ($\mathcal{G}$) vary with $\phi$, as depicted in Fig.~\ref{bbr_phase}. Apparently, entanglement evolves with $\phi$ periodically (black solid curve), maximized when phase $\phi=(2n+3/2)\pi$ ($n \in \mathbb{Z}$) by constructive interference and minimized at $\phi=(2n+1/2)\pi$ by destructive interference. Interestingly, Gaussian steering in two directions behave asymmetrically with phase, that is, steering from mode $a$ to mode $b$ quantified by the parameter $\mathcal{G}^{a\rightarrow b}$ (blue dashed curve) exhibits the same phase-depdent behavior as that of the entanglement measure $E_N$, whereas steering in the other direction measured by the parameter $\mathcal{G}^{b\rightarrow a}$ (red dotted curve) behaves in a completely different way. Specifically, steering in this direction doesn't exist for the case of $\lambda=0.4\kappa$ shown in Fig.~\ref{bbr_phase}(a), such that one-way steering in the direction of $a\rightarrow b$ occurs over the whole period of $\phi$. For larger value of $\lambda=0.605\kappa$, $\mathcal{G}^{b\rightarrow a}$ reaches a maximal value at $\phi=(2n+3/2)\pi$ as well, and we can get rich properties of Gaussian steering, e.g., the overall state's asymmetry is stepwise driven through the one-way regime ($\mathcal{G}^{a\rightarrow b}>0$ but $\mathcal{G}^{b\rightarrow a}=0$), no-way regime ($\mathcal{G}^{a\rightarrow b}=0$ and $\mathcal{G}^{b\rightarrow a}=0$), and two-way regime ($\mathcal{G}^{a\rightarrow b}>0$ and $\mathcal{G}^{b\rightarrow a}>0$), and finally one-way regime again over one period of phase, as shown in Fig.~\ref{bbr_phase}(b).

For the parameters studied here, we can find that the steerability from $a$ to $b$ is always larger than the steerability from $b$ to $a$. This can be understood from the inequalities (\ref{G21}) and the solutions given in Eq.~\ref{correlation_term}. When the two modes have same dissipation rates $\kappa_a=\kappa_b=\kappa$ and mode $c$ is at zero temperature $\bar{n}_{th}=0$, from Eq.~\ref{correlation_term} we can see that the photon fluctuation of mode $a$ is always larger than that in mode $b$, i.e., $\langle a^\dagger a\rangle>\langle b^\dagger b\rangle$, and therefore the condition given in Eq.~(\ref{G12}) to confirm steering $a\rightarrow b$ is easier to satisfy than that required for demonstrating steering $b\rightarrow a$.

Now we come to the first important result of this paper. Two identical modes which have completely symmetric decoherence $\kappa_a=\kappa_b=\kappa$ and $\bar n_{th,a}=\bar n_{th,b}=0$ can be still prepared into an entangled state with asymmetric and directional steering by adjusting the phase of the interfering channels. This reveals the inherent asymmetric nature of quantum steering with respect to the two parties involved. Comparing with the way to produce one-way steering in earlier studies by adding losses or thermal noises on one subsystem~\cite{Handchen2012,Armstrong2015,Qin2017} with the cost of reducing correlation, engineering the interference is an active method which creates asymmetric steering and at the same time enhances the steerability.

\subsection{Performance comparison with correlations created by only one interaction path}
\begin{figure}
\includegraphics[width=1.0\columnwidth]{Fig3.pdf}
\caption{The measures of entanglement ($E_N$) and steering ($\mathcal{G}^{a\rightarrow b}$, $\mathcal{G}^{b\rightarrow a}$) between modes $a$ and $b$ versus the damping rate $\gamma_c/\kappa$ for the cases of creating correlations only by the indirect interaction path $\lambda=0$ (blue solid), and by interfering channels when $\lambda=0.5\kappa$ (red dashed), $\lambda=\kappa$ (black dotted), $\lambda=1.5\kappa$ (green dash-dotted). The destructive and constructive interference at phase $\phi=\pi/2$ and $\phi=3\pi/2$ lead to the reduction (a-c) and enhancement (d-f) of correlations, respectively. Other parameters are $g_a=8.3\kappa,\ g_b=10\kappa,\ \bar{n}_{th}=0$.}
\label{bbr_dissipation_gamma}
\end{figure}

To clearly indicate the performance of control and enhancement of entanglement and steering via interfering channels, we compare with the results achieved via only one of two interaction paths.

First, the numerical results of the dependence of entanglement and steering on the damping rate of the third intermediate mode $\gamma_c/\kappa$ are given in Fig.~\ref{bbr_dissipation_gamma} for different coupling strengths of the direct interaction path $\lambda$. Blue solid curves present the results achieved only by the indirect coupling path, i.e., $\lambda=0$, other curves show the correlations achieved by the interfering channels with phase $\phi=\pi/2$ [Figs.~\ref{bbr_dissipation_gamma}(a-c)] and with phase $\phi=3\pi/2$ [Figs.~\ref{bbr_dissipation_gamma}(d-f)] for $\lambda=0.5\kappa$ (red dashed), $\kappa$ (black dotted), $1.5\kappa$ (green dash-dotted), respectively. It can be seen that the correlations are reduced due to destructive interference of two interaction paths at $\phi=\pi/2$. Comparing the steering in two directions shown in Figs.~\ref{bbr_dissipation_gamma}(b) and (c), increasing value of $\lambda$ leads to the stronger destructive interference effects on reducing the correlation between two modes, but it features one-way steering in a wider parameter range. For instance, when $\lambda=\kappa$ shown by the black dotted lines, the steering becomes possible only in one direction $a\rightarrow b$ for almost all values of $\gamma_c$, while for not accounting for the direct interaction path ($\lambda=0$, blue solid), one-way steering is only possible for strong mechanical damping rate $\gamma_c\gg\kappa$.

When the interfering channels with phase $\phi=3\pi/2$, as shown in Figs.~\ref{bbr_dissipation_gamma}(d-f), the degree of entanglement and steering are remarkably enhanced due to the constructive interference of two interaction paths comparing with the result achieved only by the indirect interaction path ($\lambda=0$, blue solid). For this case, we can get stronger two-way steering over a wider parameter range, which has been proven to be a necessary resource required for teleporting a coherent state with fidelity beyond the no-cloning threshold~\cite{He2015}. The different degree of steerability in two directions also provides the asymmetric guaranteed key rate achievable within a practical one-sided device-independent QKD~\cite{Kogias2015}. In the regime of $\gamma_c\gg\kappa$, the steering in the direction $\mathcal{G}^{b\rightarrow a}$ becomes lost. Thus, the constructive interference can also manipulate the direction of steering from asymmetric two-way to one-way. The steady-state steering generated only by the indirect interaction path has been studied in Ref.~\cite{Tan2015}, where the authors concluded that in the regime $\gamma_c\ll\kappa$ the steady entangled states are definitely not steerable in any direction when two optical modes have same damping rates $\kappa_a=\kappa_b=\kappa$, while only the mode with larger dissipation rate can be steered by the other one. Our result, however, shows that by applying the interfering channels the steering in both directions may be achieved in this regime due to the constructive interference, as indicated by the black dotted curves in Fig.~\ref{comparision}.
\begin{figure}
\includegraphics[width=0.8\columnwidth]{Fig4.pdf}
\caption{Comparison of Gaussian steering in the regime of $\gamma_c\ll\kappa$ achieved by the indirect path only ($\lambda=0$, blue solid) and by the interfering channels with constructive interference ($\lambda=\kappa,\ \phi=3\pi/2$, black dotted). Other parameters are used as those in Fig.~\ref{bbr_dissipation_gamma}.}
\label{comparision}
\end{figure}

Second, we compare with the results obtained when only the direct interaction path exists. In this case, the photon fluctuation and correlation expressions given by Eq.~(\ref{correlation_term}) in Appendix can be simplified as
\begin{eqnarray}
\langle a^\dagger a\rangle&=&\frac{\kappa_b\lambda^2}{(\kappa_a+\kappa_b)(\kappa_a\kappa_b-\lambda^2)}, \nonumber \\
\langle b^\dagger b\rangle&=&\frac{\kappa_a\lambda^2}{(\kappa_a+\kappa_b)(\kappa_a\kappa_b-\lambda^2)}, \nonumber \\
\langle ab\rangle&=&\frac{\kappa_a\kappa_b\lambda(\sin{\phi}-i\cos{\phi})}{(\kappa_a+\kappa_b)(\kappa_a\kappa_b-\lambda^2)}.
\end{eqnarray}
The conditions to confirm the presence of steering given in Eq.~(\ref{G12}) then reduce to
\begin{eqnarray}\label{G_coherent}
&&\mathcal{G}^{a\rightarrow b}>0:(\kappa_b-\kappa_a)(\kappa_a\kappa_b-\lambda^2)>0, \nonumber \\
&&\mathcal{G}^{b\rightarrow a}>0:(\kappa_a-\kappa_b)(\kappa_a\kappa_b-\lambda^2)>0.
\end{eqnarray}
We can see from Eq. (\ref{G_coherent}) that it is not possible to create steering in both directions when two modes possess symmetric dissipation rate, $\kappa_a=\kappa_b$. Since the stability condition of the system for this case requires $\lambda^2<\kappa_a\kappa_b$, then $\mathcal{G}^{a\rightarrow b}>0$ when $\kappa_b>\kappa_a$, and $\mathcal{G}^{b\rightarrow a}>0$ when $\kappa_a>\kappa_b$. This means that one can only produce one-way steering from the mode with lower damping rate to the mode with higher decay rate. This observation consists with the conclusion made in earlier studies~\cite{Tan2015,Handchen2012,Reidjosab2015} that the decoherence with no thermal noise on the steering system has substantial effect more than that on the steered system, such that the condition $\kappa_b>\kappa_a$ may destroy (preserve) steering in the direction $b\rightarrow a$ ($a\rightarrow b$) where mode $b$ possessing higher decoherence acts as steering (steered) party, respectively.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{Fig5a.pdf}
\includegraphics[width=0.8\columnwidth]{Fig5b.pdf}
\caption{The comparison of entanglement and steering achieved by only one direct interaction path (blue solid) and by the interfering channels at phase (a) $\phi=\pi/2$ and (b) $\phi=3\pi/2$, respectively. The entanglement $E_N$ (black dash-dotted) and steering in two directions $\mathcal{G}^{a\rightarrow b}$ (red dotted), $\mathcal{G}^{b\rightarrow a}$ (red dashed) are notably enhanced, meanwhile, the range of steady-state solution is also remarkably broadened when two interaction paths superpose and interference happens. Other parameters are $g_a=8.3\kappa,\ g_b=10\kappa,\gamma_c=5\kappa, \bar{n}_{th}=0$.}
\label{bbr_coherent_lambda0}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{Fig6a.pdf}
\includegraphics[width=0.8\columnwidth]{Fig6b.pdf}
\caption{The influence of thermal noise on the entanglement ($E_N$, solid) and steering ($\mathcal{G}^{a\rightarrow b}$ dotted and $\mathcal{G}^{b\rightarrow a}$ dashed) between modes $a$ and $b$ at phase $\phi=3\pi/2$ when (a) $\bar{n}_{th}$ only exists on the intermediate damping mode $c$, (b) $\bar{n}_{th_a}=\bar{n}_{th_b}=\bar{n}_{th}$. Blue curves indicate the results achieved only by the indirect interaction path where $\lambda=0$, and correspondingly red curves represent the enhanced correlations via interference channels when $\lambda=1.5\kappa$. Other parameters are $g_a=8.3\kappa,\ g_b=10\kappa,\gamma_c=5\kappa$.}
\label{bbr_nth}
\end{figure}

However, for two modes possessing completely symmetric decoherence properties we show in Fig.~\ref{bbr_coherent_lambda0} that by applying the interfering channels one can still create asymmetric and directional steering. Meanwhile, the degree of correlations is enhanced even at phase $\phi=\pi/2$ [Fig.~\ref{bbr_coherent_lambda0}(a)] that induces destructive interference, and the enhancement is more remarkable at phase $\phi=3\pi/2$ due to constructive interference  [Fig.~\ref{bbr_coherent_lambda0}(b)]. The blue curves in two plots are the entanglement achieved by the direct interaction path, while steering in both directions cannot be produced ($\mathcal{G}=0$ not plotted in the figure for clearity). When we include the other indirect coupling path, the entanglement (black dash-dotted) is remarkably enhanced, and asymmetric steering in the direction $\mathcal{G}^{a\rightarrow b}$ (red dotted) and in the opposite direction $\mathcal{G}^{b\rightarrow a}$ (red dashed) appear. Note that with destructive interference shown in Fig.~\ref{bbr_coherent_lambda0}(a), although correlations are reduced when $\lambda$ increases, it is useful to achieve one-way steering $\mathcal{G}^{b\rightarrow a}=0$ and $\mathcal{G}^{a\rightarrow b}>0$. With constructive interference shown in Fig.~\ref{bbr_coherent_lambda0}(b), the enhancement of entanglement and steering becomes notable,  and two-way asymmetric steering occurs for all steady-state solution. This observation consists with that shown in Fig.~\ref{bbr_dissipation_gamma} and the reason why $\mathcal{G}^{a\rightarrow b}>\mathcal{G}^{b\rightarrow a}$ has been explained there. Moreover, the parameter range to achieve steady-state solution is considerably broadened via interference channel. The stability condition given in Eq.~(\ref{stability condition}) at phase $\phi=\frac{\pi}{2}+n\pi\ (n \in \mathbb{Z})$ requires $\lambda^2<\text{min}\{\kappa_a\kappa_b+(g_b^2\kappa_a-g_a^2\kappa_b)/\gamma_c, \ \kappa_a\kappa_b+\gamma_c(\kappa_a+\kappa_b+\gamma_c)+[g_b^2(\kappa_b+\gamma_c)-g_a^2(\kappa_a+\gamma_c)]/(\kappa_a+\kappa_b)\}$, for the parameters studied here, the threshold is larger than $\kappa_a\kappa_b$ required by the direct interaction path only, that is to say, the system is more stable by applying our interference approach.

So far we have shown how the entanglement and steering can be enhanced and the direction of steering can be controlled via interference effects when three modes are not thermal excited. In practical implementation using cavity optomechanical systems, the third intermediate mode could be a mechanical mode with nonzero thermal phonons, so we show the influence of thermal noise $\bar{n}_{th}$ only existing in mode $c$ on the correlations at phase $\phi=3\pi/2$ in Fig.~\ref{bbr_nth}(a). In general, thermal noise is detrimental to correlations because it causes decoherence, however, entanglement (red solid) and steering (red dotted and dashed) created by the interference channels are more robust against thermal noise comparing with that achieved only by the indirect interaction path (correspondingly blue curves). Even when all three modes are in thermal excited states, it shows again in Fig.~\ref{bbr_nth}(b) that the entanglement and asymmetric steering created by the interference channels (red curves) are more robust against thermal noise than that achieved by only applying the indirect interaction path (blue curves).

\subsection{The enhancement of entanglement and steering between other two modes in the loop}
\begin{figure}[htbp!]
\includegraphics[width=0.8\columnwidth]
{Fig7.pdf}
\caption{The phase-dependent behavior of steady-state entanglement and steering between pair $a,b$ (blue curves) and between pair $a,c$ (red curves). When entanglement ($E^{ab}_{N}>0$, blue solid) and one-way steering ($\mathcal{G}^{a\rightarrow b}>0$, blue dashed, and $\mathcal{G}^{b\rightarrow a}=0$, blue dotted) between modes $a$ and $b$ reach a maximum, the entanglement ($E^{ac}_{N}>0$, red solid) and one-way steering ($\mathcal{G}^{a\rightarrow c}>0$, red dashed, and $\mathcal{G}^{c\rightarrow a}=0$, red dotted) between modes $a$ and $c$ are at a minimum. Other parameters are $\kappa_a=\kappa_b=\kappa,\ g_a=3.2\kappa,\ g_b=5\kappa,\ \lambda=0.4\kappa,\ \gamma_c=15\kappa,\ \bar{n}_{th}=0$.}
\label{phase_two pairs}
\end{figure}

We focused on the enhancement of entanglement and the control of the asymmetric and directional steering via interference effects between modes $a$ and $b$ in the previous discussion, here, we want to point out that this could be also realized between modes $a$ and $c$, which means the method is general. As shown by the inset plot in Fig.~\ref{phase_two pairs}, there are two interaction paths to create entanglement between modes $a$ and $c$, one is by the direct two-mode squeezing interaction with coupling strength $g_a$, and the other one is by swapping the entanglement shared by $a$ and $b$ to pair $a$ and $c$ via the beam-splitter-type interaction between modes $b$ and $c$. Two physical interaction paths superpose with relative phase and create interfering channels. The phase-dependent behavior of the entanglement ($E^{ac}_N$ red solid) and one-way steering between modes $a$ and $c$ ($\mathcal{G}^{a\rightarrow c}>0$ red dashed, $\mathcal{G}^{c\rightarrow a}=0$ red dotted) is illustrated in Fig.~\ref{phase_two pairs}. We also notice that when the entanglement and steering between modes $a$ and $c$ reaches a maximum (minimum) of the interference pattern at where the corresponding correlations between modes $a$ and $b$ are a minimum (maximum), indicating that there exists a competition to distribute entanglement and steering between different pairs. This can be understood by the monogamy constraints for distributing correlations among multi parties.

\section{Conclusion}\label{SUMMARY}
In summary, we propose an active method to generate and control steady-state entanglement and asymmetric steering between two identical modes by intriguing constructive or destructive quantum interference effects in a general three-mode system with closed-loop in coupling. The direct interaction path between two modes and the indirect interaction path induced by both coupling to the intermediate damping mode superpose and act as a interfering channel with a tunable phase, which can create phase-dependent correlations. We show that the interference effects can not only enhance the degree of the entanglement and steerability, but also produce asymmetric steering when two modes possess completely symmetric decoherence properties. Instead of introducing different amount of losses or noises to subsystems, this provides inspiration for manipulating the direction of the asymmetric quantum steering with enhanced steerability. In addition, the entangled steady states created by the interference channel are more robust against thermal noises. Furthermore, using the interfering channels to entangle the other pair of modes in the loop is also discussed, and the opposite phase-dependent behavior indicates the monogamy constraints for distributing entanglement and steering among multipartite. This work opens up new perspectives for the experimental production and applications of EPR steering as a precious resource for secure quantum communication technologies.

\begin{acknowledgements}
We acknowledge illuminating discussions with Y. Li and H.Y.Yuan. This work is supported by the National Key R\&D Program of China (Grants No. 2018YFB1107200 and No. 2016YFA0301302) and the National Natural Science Foundation of China (Grants No. 11622428, No. 61475007, and No. 61475006). Q.He thanks the Beijing Computational Science Research Center for their hospitality.
\end{acknowledgements}

redpajama_set_name: RedPajamaArXiv


text:
Shane Keegan appointed Cobh Ramblers boss
Last updated on June 27th, 2022
Cobh Ramblers have appointed Shane Keegan as their new manager.
The former Dundalk, Wexford Youths and Galway United boss has signed a one and a half year deal at St Colman's Park up to the end of the 2023 season.
Keegan replaces Darren Murphy, who departed the club last week.
Speaking on the announcement, Keegan said "I am delighted & excited to come on board as Cobh Ramblers' new manager.
"To play a part in the club, in this, their 100th year anniversary, was an opportunity that I couldn't turn down. I look forward to getting straight to work with the players & staff as we all get to know each other over the course of the month of July.
"It would be a huge boost to us to get support out in numbers over the next few games. Then it is up to us to give people a reason to keep coming back. I really hope that I can play an important role in helping to grow Cobh as a club both on & off the field in both the short & long term.
"I would also like to place on record my thanks to the board at Treaty United and wish them the best of luck for the rest of the season."
Ramblers chairman Bill O'Leary said "We are delighted to have secured the services of a proven, experienced League of Ireland manager. Shane is highly thought of, and we regard him as the right man for the job at this time.
"All of us at the club look forward to working with him."
Keegan's first game in charge of Ramblers will be at home to Treaty United on Friday night – Keegan had been working as Treaty's Academy head.
Tagged as Cobh ramblers shane keegan
Rain / Wind
Weather forecast Cork, Ireland ▸
Fineen Wycherley Returns From Injury For Benetton Trip
All Ireland Final set for replay after Glen lodge appeal
Ireland name squad for Six Nations
www.redfm.ie
info@redfm.ie
University Technology Centre
Curraheen Road, Bishopstown, Cork
106.1FM – Cork City
105.7FM – North Cork
105.4FM – Fermoy
105.4FM – Bantry
105.2FM – Kinsale
105.1FM – Youghal
104.5FM – West Cork
104.2FM – Crosshaven
Siteridge Limited t/a RedFM
Registered in Ireland: 324025. | Siteridge Limited t/a RedFM
Copyright 2018 Siteridge Limited
Powered by waymigo.com
Select Station Red FM Red Hot
RedFM News Extra
Neil Prendeville show
Breakfast on Cork's Red FM
We use cookies to ensure that we give you the best experience on Red Extra. If you continue to use Red Extra we will assume that you are happy with it.OK


text:
Celina Wimbush
So, guess who just binged Netflix's Wednesday? Unfortunately, not on a Wednesday. Centered around Wednesday Addams of the Addams Family, the show starts with her getting expelled from her high school and transferred to Nevermore Academy, the boarding school where her parents fell in love. Spoilers ahead, continue with caution!
So, how was it overall?
Honestly? Pretty good, though I have a few complaints.
First of all, the love triangle. Please, for the love of god, love triangles need to just stop happening. Unless they're directional love triangles. Those can stay. But my god, during half of Episode 4 I was screaming "Can we get back to the monster stuff?" I'm here for the Addams, Enid, and Eugene. That's it. At least, that's what I was thinking when I wrote this paragraph during episode 5 just to complain about that.
And then it got interesting. Really interesting. One last chance, absolutely massive spoilers start here! Click away if you don't want to see them! Because trust me, you don't want to be spoiled on this if you're planning on watching.
Both of Wednesday's love interests get arrested. And honestly? Good. Xavier… actually didn't do anything wrong, I just don't like him because he's an insensitive jerk and I think he needed to be knocked down a peg. But Tyler is the twist villain of the season, revealing that he's a Hyde, a type of monster which transforms from a human into a powerful and angry creature that will attack anything in its path. The villain is Laurel Gates, posing as Wednesday's dorm mom Marilyn Thornhill. I didn't see that coming in the slightest, and I have a whole issue with twist endings that are never foreshadowed, but that's not the point right now.
Anyway, now to my complaints that weren't at the very least slightly addressed. A few lines in one episode about a conversion camp which I have thought way too hard about. This article explains it much better than I can, but the long and short of it is that they try to make not transforming into a werewolf fast enough, or "Wolfing out" as they call it, into a metaphor for being gay when the two things just aren't comparable. And this part isn't mentioned in the article, but at the end of it all, Enid does end up "Wolfing out."
Writers, what the actual heck? So… you try to make a metaphor for conversion therapy saying that she doesn't have to conform, but then she conforms anyway? That's… I don't think it's actively malicious, but it's certainly a very bad look. It's a big issue with the series, honestly. A school for outcasts where no one actually feels like an outcast for long.
And there's another big issue that's been discussed over and over again, and here's another article about it if you want to know more. Once again, the long and short of it is that every single Black character is an antagonist at some point. I'm not kidding. It's… bad, to say the least.
Okay! Complaining over! Now onto the things I liked.
Let's start with the way they handled Wednesday. Every other character seemed a bit like an afterthought, but Wednesday herself was a stellar character. Portrayed by a Latina actress, Jenna Ortega, she embodies the black-hearted Wednesday Addams perfectly. Women of color protagonists are always a treat to see, so this was amazing. And she's non-stereotypical too!
There was also one scene that stood out a lot to me and Senior Matthew Castillo. The dance scene in episode 4. According to him, it's "…Because Wednesday comes out of her shell for the first time." It's a very, very good scene and it's one of the biggest reasons why Wednesday is such a good character.
And the plot, the plot was absolutely stellar. The high point of the series, by far. Complicated, yet still easy to follow. Even though the twist ending wasn't foreshadowed at all from what I could see, I enjoyed it a lot. I loved every twist and turn it took me on, and I can't help but absolutely adore it.
Also Enid. Enid was the best part of this series and I love her so, so much. She's so good and I think she's better for her than both of Wednesday's love interests combined.
So, final thoughts! Wednesday is a good show with a lot of flaws that I can't ignore. If you can ignore these flaws, go ahead and watch on! If you can't, maybe skip it. Final score: 7/10, though I will give it that it inspired me to watch the Addams Family movies which I have never seen before.
Celina Wimbush, Graphics Staff
Hey! My name is Celina Wimbush. I am a senior and this is my first year in Newspaper. I love flag football and horseback riding. I am really excited...