text: 7 3 10 numerical (L = 4) x 102 analytical, Faxen (1946) C D 2 C 10 D 1 10 -2 -1 10 10 R 1 10 0 0.2 0.4 R 3 2 1 0 FIG. 3. Upper: Drag coeﬃcient as a function of cylinder radius for Newtonian ﬂow in the biperiodic geometry. Dotted → line: analytical prediction in the limit R 0 [46]. Dashed → line: analytical prediction in limit 2R L of near-touching y cylinders [46]. Symbols: numerical results using immersed FIG. 4. Upper: Comparison of the drag coeﬃcient CD as boundary method (circles), propagator method (squares), obtained using IBM in the channel geometry for large cylinder phase ﬁeld method with l = 0.02 (diamonds) and l = 0.01 spacing Lx/Ly = 4 with Faxen’s analytical result for ﬂow past (triangles). (See [47] for deﬁnition of the parameter l.) a single cyl|ind| er in a channel [49]. Lower: Map of velocity Lower: Streamlines (red lines) and map of velocity magni- magnitude v calculated using IBM (left) and from Ref. [50] | | tude v calculated using IBM (left) and propagator (right) (right) for a cylinder radius R = 0.25, cylinder spacing Lx = methods for a cylinder radius R = 0.25 and overall through- 0.625 and throughput rate Q = 1.0. The colourscheme of this put rate Q = 1.0. ﬂow map is diﬀerent from that of the others in this manuscript to match that of the original work [50] against which we are benchmarking. 51]. Therefore we focus here only on benchmarks to val- idate the second substep, in which the velocity ﬁeld is calculated by solving Eqns. 6 and 7. The presence of vis- obtained in the present study using two diﬀerent numer- coelasticity in this substep is trivial: it appears only as ical methods that are independent of the IBM described ∇ a source term .Σ in Eqn. 6). All the issues of principle above: a phase ﬁeld method and a circular-propagator are already contained in the solution of Eqns. 6 and 7 for method [47]. As can be seen, these numerical results for purely Newtonian ﬂow in the geometries of interest here, the drag coeﬃcient are in excellent agreement between for which known benchmarks exist, as follows. all our three methods across the full range of cylinder Sangani and Acrivos [46] derived analytical expressions radii. For one particular value of the cylinder radius, the for the drag coeﬃcient C as a function of the cylinder full ﬂow ﬁeld as calculated in a biperiodic array is shown D radius R in Newtonian ﬂow past the biperiodic array of in the lower panel of Fig. 3: on the left using the IBM, Fig. 1 (upper), separately for the small cylinder limit and on the right using the propagator method. Excellent → → R 0, and for the limit R L /2 = 1/2 (in our units) agreement is seen between the two methods. y in which adjacent cylinders approach contact. These are In Fig. 4 (upper), we show that numerical results ob- shown by the dotted and dashed lines respectively in the tained using our IBM for the drag coeﬃcient as a func- upper panel of Fig. 3. Numerical results obtained us- tion of cylinder radius R for a ﬁxed (wide) horizontal ing our IBM method are in excellent agreement with cylinder separation L = 4.0 in the channel geometry of x these predictions, as shown by the circles in the same Fig. 1 (lower) are in excellent agreement with the an- panel. Also shown (by the diamonds, squares and tri- alytical prediction of Faxen [49] for ﬂow past a single angles) are results for the drag coeﬃcient additionally cylinder in a channel. Fig. 4 (lower) shows that our re- ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: numerical

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text: Dotted

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text: Symbols:

text: numerical

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text: Comparison

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text: CD

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text: boundary

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text: (triangles).

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text: overall

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text: Q

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text: diﬀerent

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text: circular-propagator

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text: the

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text: excellent

text: agreement

text: between

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text: known

text: benchmarks

text: exist,

text: as

text: follows.

text: all

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text: cylinder

text: Sangani

text: and

text: Acrivos

text: derived

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text: expressions

text: radii.

text: For

text: one

text: particular

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text: of

text: the

text: cylinder

text: radius,

text: the

text: for

text: the

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text: coeﬃcient

text: C

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text: biperiodic

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text: shown

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text: biperiodic

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text: shown

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text: the

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text: tained

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text: our

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text: drag

text: coeﬃcient

text: as

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text: results

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text: R

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text: (wide)

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text: ing

text: our

text: IBM

text: method

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text: excellent

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text: cylinder

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text: L

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text: x

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text: circles

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text: (lower)

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text: panel.

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text: diamonds,

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text: prediction

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text: Faxen

text: for

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text: past

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text: single

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text: are

text: results

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text: drag

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text: cylinder

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text: Fig.

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text: their decision and almost none of the algorithms previously A policy with small Bayesian regret may not be uniformly proposed in the literature is able to provide non-trivial re- good in all P. Since in the Bayesian setting we frequently gret guarantees without the feedback graphs being disclosed. need to discuss posterior probabilities and expectations, we However, Cohen, Hazan, and Koren (2016) argue that the also introduce the notation E f (cid:44) E(f | h ) and P (·) (cid:44) assumption that the entire observation system is revealed P(· | h ) for expectations andtprobabilities ctonditionted on t to the learner on each round, even if only after making her the current history. prediction, is rather unnatural. In principle, the learner need 2.2 The graph feedback model not be even aware of the fact that there is a graph under- In this model, we assume the existence of an undirected lying the feedback model; the feedback graph is merely a technical notion for us to specify a set of observations for graph G = (V, E) with vertices corresponding to arms. By each of the possible arms. Ideally, the only signal we would taking an arm a ∈ V, we not only receive the reward of the arm we played, but we also observe the rewards of all neigh- like the agent to receive following each round is the set of (cid:48) (cid:48) observations that corresponds to the arm she has taken on that bouring arms Na = { a ∈ V | (a, a ) ∈ E }.(cid:48)More precisely, wroournkdin(inthaisddseittiuopn-tothtehyedoobtnaoint endeerdewthaerdw).hOoluergarlagpohrittohmbes aotuerarcehwatirmdei-ssstetiplltrwt =e oRbs(eYrtv,AetY)t.,a(cid:48) for all a ∈ NAt, while disclosed either when selecting the arm or when updating If the graph is empty, then the setting is equivalent to the beliefs - only the local neighborhood is needed. Furthermore, bandit problem. If the graph is fully connected, then it is the underlying graph is allowed to change arbitrarily at each equivalent to the prediction (i.e. full information) problem. step. The detailed proofs of all our main results are available However, many practical graphs, such as those derived from in the full version of this paper. social networks, have an intermediate connectivity. In such 2 Setting cases, the amount of information that we can obtain by pick- ing an arm can be characterised by graph properties, such as 2.1 The stochastic bandit model the clique cover number: The stochastic K-armed bandit problem is a well known Deﬁnition 2.1 (Clique cover number). A clique covering C sequential decision problem involving an agent sequentially of a graph G is a partition of all its vertices into sets S ∈ C choosing among a set of K arms V = {1 . . . K}. At each such that the sub-graph formed by each S is a clique i.e. all round t, the agent plays an arm A ∈ V and receives a reward vertices in S are connected to each other in G. The smallest t r = R(Y ), where Y : Ω → Y is a random variable number of cliques into which the nodes of G can be parti- dteﬁned ont,sAotme probabilti,tAytspace (P, Ω, Σ) and R : Y → R tioned is called the clique cover number. We denote by C(G) is a reward function. the minimum clique cover and χ(G) its size, omitting G Each arm i has mean reward µ (P) = E R(Y ). Our when clear from the context. i P t,i goal is to maximize its expected cumulative reward after T The domination number is another useful similar notion rounds. An equivalent notion is to minimize the expected for the amount of information that we can obtain. regret against an oracle which knows P. More formally, the expected regret Eπ L of an agent policy π for a bandit prob- Deﬁnition 2.2 (Domination number). A dominating set in P lem P is deﬁned as: a graph G = (V, E) is a subset U ⊆ V such that for every (cid:88) vertex u ∈ V , either u ∈ U or (u, v) ∈ E for some v ∈ U. The smallest size of a dominating set in G is called the E E T domination number of G and denoted γ(G). π L = Tµ∗(P) − π r , (2.1) P P At t=1 3 Related work and our contribution where µ∗(P) = max ∈V µ (P) is the mean of the opti- i i Optimal policies for the stochastic multi-armed ban- mal arm and π(A |h ) is the policy of the agent, deﬁning a t t dit problem were ﬁrst characterised by (Lai and Robbins probability distribution on the next arm A given the history t 1985), while index-based optimal policies for general non- h = (cid:104)A − , r − (cid:105) of previous arms and rewards. t 1:t 1 1:t 1 parametric problems were given by (Burnetas and Katehakis The main challenge in this model is that the agent does 1997). Later (Auer, Cesa-Bianchi, and Fischer 2002a) proved not know P, and it only observes the reward of the arm it ﬁnite-time regret bounds for a number of UCB (Upper Con- played. As a consequence, the agent must trade-off exploita- ﬁdence Bound) index policies, while (Garivier and Cappe´ tion (taking the apparently best arm) with exploration (trying 2011) proved ﬁnite-time bounds for index policies similar out other arms to assess their quality). to those of (Burnetas and Katehakis 1997), with problem- The Bayesian setting offers a natural way to model this uncertainty, by assuming that the underlying probability law dependent bounds O(K ln T). Recently, a number of poli- cies based on sampling from the posterior distribution (i.e. P is in some set P = { P | θ ∈ Θ } parametrised by θ, over θ P Thompson sampling(Thompson 1933)) were analysed in both which we deﬁne a prior(cid:90)probability distribution . In that the frequentist (Agrawal and Goyal 2012) and Bayesian set- case, we can deﬁne the Bayesian regret: ting (Russo and Roy 2016) and shown to obtain the same order of regret bound for the stochastic case. For the ad- Eπ L = EπPθ(L) d P(θ). (2.2) versarial bandit problem the bounds are of order O(√KT). Θ The analysis for the full information case generally results ##LTLine## ##LTLine##

text: their

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text: (cid:15)-greedy-D, even when we tune the parameters of the latter Buccapatnam, S.; Eryilmaz, A.; and Shroff, N. B. 2014. post hoc. Stochastic bandits with side observations on networks. ACM It would be interesting to extend our techniques to other SIGMETRICS Performance Evaluation Review 42(1):289– types of feedback. For example, the Bayesian foundations 300. of Thompson sampling render our algorithms applicable to Burnetas, A. N., and Katehakis, M. N. 1997. Optimal adap- arbitrary dependencies between arms. In future work, we will tive policies for markov decision processes. Mathematics of analytically and experimentally consider such problems and Operations Research 22(1):222–255. related applications. Finally, an open question is the existence Caron, S.; Kveton, B.; Lelarge, M.; and Bhagat, S. 2012. of information-theoretic lower bounds in settings with partial Leveraging side observations in stochastic bandits. UAI. feedback. Acknowledgments. This research was partially sup- Cesa-Bianchi, N., and Lugosi, G. 2006. Prediction, Learning ported by the SNSF grants “Adaptive control with ap- and Games. Cambridge Press. proximate Bayesian computation and differential privacy” Chapelle, O., and Li, L. 2011. An empirical evaluation (IZK0Z2 167522), “Swiss Sense Synergy” (CRSII2 154458), of thompson sampling. In Advances in neural information by the the People Programme (Marie Curie Actions) of processing systems, 2249–2257. the European Union’s Seventh Framework Programme Cohen, A.; Hazan, T.; and Koren, T. 2016. Online learning (FP7/2007-2013) under REA grant agreement number with feedback graphs without the graphs. In ICML 2016. 608743, and the Future of Life Institute. Condon, A., and Karp, R. M. 2001. Algorithms for graph par- titioning on the planted partition model. Random Structures and Algorithms 18(2):116–140. David, H. 1968. Miscellanea: Gini’s mean difference redis- 4.0 covered. Biometrika 55(3):573–575. 3.5 Garivier, A., and Cappe´, O. 2011. The kl-ucb algorithm for bounded stochastic bandits and beyond. arXiv preprint 3.0 arXiv:1102.2490. Gini, C., and Pearson, K. 1912. Variabilita` e mutabilita`: 2.5 contributo allo studio delle distribuzioni e delle relazioni statistiche. Fascicolo 1. tipograﬁa di Paolo Cuppini. 2.0 Goh, K.-I.; Kahng, B.; and Kim, D. 2001. Universal behavior UCB-N 1.5 UCB-MaxN of load distribution in scale-free networks. Physical Review TS-N Letters 87(27):278701. 1.0 TS-MaxN Lai, T. L., and Robbins, H. 1985. Asymptotically efﬁcient 20 40 60 80 100 120 Number of groups adaptive allocation rules. Advances in applied mathematics 6(1):4–22. Figure 1: Relative regret in the planted partition setting as the Mannor, S., and Shamir, O. 2011. From bandits to experts: number of groups increases. On the value of side-observations. In Advances in Neural Information Processing Systems, 684–692. McSherry, F. 2001. Spectral partitioning of random graphs. References In Foundations of Computer Science, 2001. Proceedings. Agrawal, S., and Goyal, N. 2012. Analysis of thompson 42nd IEEE Symposium on, 529–537. IEEE. sampling for the multi-armed bandit problem. In COLT Ruan, L.; Du, H.; Jia, X.; Wu, W.; Li, Y.; and Ko, K.-I. 2004. 2012. A greedy approximation for minimum connected dominating Alon, N.; Cesa-Bianchi, N.; Dekel, O.; and Koren, T. 2015. sets. Theoretical Computer Science 329(1):325–330. Online learning with feedback graphs: beyond bandits. In Russo, D., and Roy, B. V. 2016. An information-theoretic Proceedings of the 28th Annual Conference on Learning analysis of thompson sampling. Journal of Machine Learning Theory, 23–35. Research 17(68):1–30. Alon, N.; Matias, Y.; and Szegedy, M. 1996. The space Scott, S. L. 2010. A modern bayesian look at the multi-armed complexity of approximating the frequency moments. In bandit. Applied Stochastic Models in Business and Industry 28th STOC, 20–29. ACM. 26(6):639–658. Auer, P.; Cesa-Bianchi, N.; and Fischer, P. 2002a. Finite Scott, S. L. 2015. Multi-armed bandit experiments in the on- time analysis of the multiarmed bandit problem. Machine line service economy. Applied Stochastic Models in Business Learning 47(2/3):235–256. and Industry 31(1):37–45. Auer, P.; Cesa-Bianchi, N.; and Fischer, P. 2002b. Finite-time Thompson, W. 1933. On the Likelihood that One Unknown analysis of the multiarmed bandit problem. Machine learning Probability Exceeds Another in View of the Evidence of two 47(2-3):235–256. Samples. 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text: 5 FIG. 3: (Color online) The ground-truth and detected partitions in the network of students. (a) The classiﬁcations of middle (blue) and high (red) school. (b) Ethnicity metadata: purple for White, green for Black, and yellow for others. (c) The detected partition recovers high/middle school, q = 2, α = 0.1. (d)-(f) The detected partitions recover ethnicity. (d) q = 2, α = 0.3, (e) q = 3, α = 0.5 and (f) q = 3, α = 0.75. The ﬁgures are drawn with the Gephi network visualization software [22] and ForceAtlas2 layout algorithm [23]. (a ) (b ) 0.50 1.0 1.2 mmoodduullaarriittyy-like function 0.6 modularity 0.45 NNMMII wwiitthh ehtighhn/icmitiyddle school 0.8 mNModI uwlaitrhit ye-tlhikneic fiutynction 1.0 NMI with high/middle school 0.5 0.40 0.8 0.6 0.35 0.4 0.6 0.4 0.30 0.4 0.3 0.2 0.25 0.2 0.20 0.0 0.2 0.0 00..00 00..22 00..44 00..66 00..88 11..00 00..00 00..22 00..44 00..66 00..88 FIG. 4: (Color online) The objective functions and NMI of the network of students. (a) With detected group number q = 2. (b) q = 3. noitcnuf evitcejbo IMN noitcnuf evitcejbo IMN ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 17 0 0 0 −1 −2 −2 −2 25 MHz −4 −3 −4 −4 50 MHz −6 −6 −5 75 MHz −8 −6 100 MHz −8 −10 −7 −8 (a) 1 OPO −10 (b) −12 (c) −9 −12 −14 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 0 −1 −1 −1 −2 −2 −3 −2 −3 −4 −3 −4 −5 −4 −5 −6 −6 −7 −5 −7 −8 −6 (d) −8 (e) −9 (f) −7 −9 −10 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 0 −1 −1 −2 −1 −2 −3 −2 −3 −4 −3 −4 −5 −5 −6 −4 (g) −6 (h) −7 (i) −5 −7 −8 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 ω/2π (MHz) ω/2π (MHz) ω/2π (MHz) FIG. 17. The squeezing spectrum Q−(ω) for the optimal operation of both networks under the minimization of J = P−(ω ) of Eq. (50). B B Each subplot shows four curves corresponding to the optimally operated CQFC network of two coupled OPOs for different values of ω { } B (ω /2π = 25, 75, 50, 100 MHz), along with a curve corresponding to the optimally operated single OPO network for any value of ω , as B B indicated in the legend. The transmission line losses are L = L = 0.01 (subplots (a), (b), (c)), L = L = 0.05 (subplots (d), (e), (f)), tl out tl out and L = L = 0.1 (subplots (g), (h), (i)). The upper limit on the scaled pump amplitude is x = 0.1 (subplots (a), (d), (g)), x = 0.2 tl out u u (subplots (b), (e), (h)), and x = 0.3 (subplots (c), (f), (i)). Other parameters are T = 0.9, L = L = 0.01. u u in X. CONCLUSIONS terms of squeezing achieved under similar conditions, even with higher losses in the CQFC network due to additional components and transmission lines. Also, the CQFC net- We modeled the squeezing spectrum of the output ﬁeld of work is more effective in terms of converting a higher pump the CQFC network of two coupled OPOs and used a suite of power into a stronger squeezing. While this superior per- global optimization methods to examine the limits to which formance of the CQFC network of two coupled OPOs relies this spectrum can be varied under conditions typical for table- on a phase-sensitive interference between multiple ﬁelds, we top experiments. We found that, in contrast to a single OPO, discovered, perhaps surprisingly, that squeezing generated the CQFC network can utilize the interference between the by the optimally operated CQFC network is rather robust to ﬁelds in the plant OPO and the controller OPO to signiﬁcantly small variations of phase parameters. This robustness can be modify the squeezing spectrum of the output ﬁeld in response attributed to the tendency of global optimization algorithms to the selected optimization objective. In particular, when the to avoid solutions that are overly sensitive to small parameter objective is to maximize the degree of squeezing at a high- variations, but the fact that such robust network conﬁgurations frequency sideband ωopt, the CQFC network can operate in do actually exist is quite remarkable. an optimal regime characterized by a high degree of coopera- Overall, our results strongly indicate that CQFC networks tivity between the plant OPO and the controller OPO, as quan- provide a very effective tool for engineering quantum opti- tiﬁed by the ﬂows of light between them and the correlation cal systems with new properties and unprecedented levels of between the phase shifts φ and φ . In this operation regime, 1 2Q− performance. This work also demonstrates the usefulness of the optimized squeezing spectrum (ω) of the CQFC net- advanced optimization methods for analyzing and improving work of two coupled OPOs has the minimum at ω = ω , opt the performance of such networks. while the minimum of the optimized spectrum of the single OPO network is always at zero sideband frequency. ACKNOWLEDGMENTS For both types of optimization objectives considered in this work (maximizing the degree of squeezing at a selected This work was supported by the Laboratory Directed Re- sideband frequency and maximizing the average degree of search and Development program at Sandia National Labora- squeezing over a selected bandwidth), the CQFC network of tories. Sandia is a multi-mission laboratory managed and op- two coupled OPOs signiﬁcantly outperforms a single OPO in erated by Sandia Corporation, a wholly owned subsidiary of )Bd( rewop esioN ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: −1

text: −2

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text: MHz

text: −4

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text: −6

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text: MHz

text: −8

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text: MHz

text: −8

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text: (a)

text: OPO

text: (b)

text: (c)

text: −9

text: −1

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text: (d)

text: −8

text: (e)

text: −9

text: (f)

text: −7

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text: −1

text: −1

text: −2

text: −1

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text: −2

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text: −4

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text: −4

text: −5

text: −5

text: −6

text: −4

text: (g)

text: −6

text: (h)

text: −7

text: (i)

text: −5

text: −7

text: −8

text: ω/2π

text: (MHz)

text: ω/2π

text: (MHz)

text: ω/2π

text: (MHz)

text: FIG.

text: The

text: squeezing

text: spectrum

text: Q−(ω)

text: for

text: the

text: optimal

text: operation

text: of

text: both

text: networks

text: under

text: the

text: minimization

text: of

text: J

text: P−(ω

text: of

text: Eq.

text: B

text: B

text: Each

text: subplot

text: shows

text: four

text: curves

text: corresponding

text: to

text: the

text: optimally

text: operated

text: CQFC

text: network

text: of

text: two

text: coupled

text: OPOs

text: for

text: different

text: values

text: of

text: ω

text: B

text: (ω

text: MHz),

text: along

text: with

text: a

text: curve

text: corresponding

text: to

text: the

text: optimally

text: operated

text: single

text: OPO

text: network

text: for

text: any

text: value

text: of

text: ω

text: as

text: B

text: B

text: indicated

text: in

text: the

text: legend.

text: The

text: transmission

text: line

text: losses

text: are

text: L

text: L

text: (subplots

text: (a),

text: (b),

text: (c)),

text: L

text: L

text: (subplots

text: (d),

text: (e),

text: (f)),

text: tl

text: out

text: tl

text: out

text: and

text: L

text: L

text: (subplots

text: (g),

text: (h),

text: (i)).

text: The

text: upper

text: limit

text: on

text: the

text: scaled

text: pump

text: amplitude

text: is

text: x

text: (subplots

text: (a),

text: (d),

text: (g)),

text: x

text: tl

text: out

text: u

text: u

text: (subplots

text: (b),

text: (e),

text: (h)),

text: and

text: x

text: (subplots

text: (c),

text: (f),

text: (i)).

text: Other

text: parameters

text: are

text: T

text: L

text: L

text: u

text: u

text: in

text: X.

text: CONCLUSIONS

text: terms

text: of

text: squeezing

text: achieved

text: under

text: similar

text: conditions,

text: even

text: with

text: higher

text: losses

text: in

text: the

text: CQFC

text: network

text: due

text: to

text: additional

text: components

text: and

text: transmission

text: lines.

text: Also,

text: the

text: CQFC

text: net-

text: We

text: modeled

text: the

text: squeezing

text: spectrum

text: of

text: the

text: output

text: ﬁeld

text: of

text: work

text: is

text: more

text: effective

text: in

text: terms

text: of

text: converting

text: a

text: higher

text: pump

text: the

text: CQFC

text: network

text: of

text: two

text: coupled

text: OPOs

text: and

text: used

text: a

text: suite

text: of

text: power

text: into

text: a

text: stronger

text: squeezing.

text: While

text: this

text: superior

text: per-

text: global

text: optimization

text: methods

text: to

text: examine

text: the

text: limits

text: to

text: which

text: formance

text: of

text: the

text: CQFC

text: network

text: of

text: two

text: coupled

text: OPOs

text: relies

text: this

text: spectrum

text: can

text: be

text: varied

text: under

text: conditions

text: typical

text: for

text: table-

text: on

text: a

text: phase-sensitive

text: interference

text: between

text: multiple

text: ﬁelds,

text: we

text: top

text: experiments.

text: We

text: found

text: that,

text: in

text: contrast

text: to

text: a

text: single

text: OPO,

text: discovered,

text: perhaps

text: surprisingly,

text: that

text: squeezing

text: generated

text: the

text: CQFC

text: network

text: can

text: utilize

text: the

text: interference

text: between

text: the

text: by

text: the

text: optimally

text: operated

text: CQFC

text: network

text: is

text: rather

text: robust

text: to

text: ﬁelds

text: in

text: the

text: plant

text: OPO

text: and

text: the

text: controller

text: OPO

text: to

text: signiﬁcantly

text: small

text: variations

text: of

text: phase

text: parameters.

text: This

text: robustness

text: can

text: be

text: modify

text: the

text: squeezing

text: spectrum

text: of

text: the

text: output

text: ﬁeld

text: in

text: response

text: attributed

text: to

text: the

text: tendency

text: of

text: global

text: optimization

text: algorithms

text: to

text: the

text: selected

text: optimization

text: objective.

text: In

text: particular,

text: when

text: the

text: to

text: avoid

text: solutions

text: that

text: are

text: overly

text: sensitive

text: to

text: small

text: parameter

text: objective

text: is

text: to

text: maximize

text: the

text: degree

text: of

text: squeezing

text: at

text: a

text: high-

text: variations,

text: but

text: the

text: fact

text: that

text: such

text: robust

text: network

text: conﬁgurations

text: frequency

text: sideband

text: ωopt,

text: the

text: CQFC

text: network

text: can

text: operate

text: in

text: do

text: actually

text: exist

text: is

text: quite

text: remarkable.

text: an

text: optimal

text: regime

text: characterized

text: by

text: a

text: high

text: degree

text: of

text: coopera-

text: Overall,

text: our

text: results

text: strongly

text: indicate

text: that

text: CQFC

text: networks

text: tivity

text: between

text: the

text: plant

text: OPO

text: and

text: the

text: controller

text: OPO,

text: as

text: quan-

text: provide

text: a

text: very

text: effective

text: tool

text: for

text: engineering

text: quantum

text: opti-

text: tiﬁed

text: by

text: the

text: ﬂows

text: of

text: light

text: between

text: them

text: and

text: the

text: correlation

text: cal

text: systems

text: with

text: new

text: properties

text: and

text: unprecedented

text: levels

text: of

text: between

text: the

text: phase

text: shifts

text: φ

text: and

text: φ

text: In

text: this

text: operation

text: regime,

text: performance.

text: This

text: work

text: also

text: demonstrates

text: the

text: usefulness

text: of

text: the

text: optimized

text: squeezing

text: spectrum

text: (ω)

text: of

text: the

text: CQFC

text: net-

text: advanced

text: optimization

text: methods

text: for

text: analyzing

text: and

text: improving

text: work

text: of

text: two

text: coupled

text: OPOs

text: has

text: the

text: minimum

text: at

text: ω

text: ω

text: opt

text: the

text: performance

text: of

text: such

text: networks.

text: while

text: the

text: minimum

text: of

text: the

text: optimized

text: spectrum

text: of

text: the

text: single

text: OPO

text: network

text: is

text: always

text: at

text: zero

text: sideband

text: frequency.

text: ACKNOWLEDGMENTS

text: For

text: both

text: types

text: of

text: optimization

text: objectives

text: considered

text: in

text: this

text: work

text: (maximizing

text: the

text: degree

text: of

text: squeezing

text: at

text: a

text: selected

text: This

text: work

text: was

text: supported

text: by

text: the

text: Laboratory

text: Directed

text: Re-

text: sideband

text: frequency

text: and

text: maximizing

text: the

text: average

text: degree

text: of

text: search

text: and

text: Development

text: program

text: at

text: Sandia

text: National

text: Labora-

text: squeezing

text: over

text: a

text: selected

text: bandwidth),

text: the

text: CQFC

text: network

text: of

text: tories.

text: Sandia

text: is

text: a

text: multi-mission

text: laboratory

text: managed

text: and

text: op-

text: two

text: coupled

text: OPOs

text: signiﬁcantly

text: outperforms

text: a

text: single

text: OPO

text: in

text: erated

text: by

text: Sandia

text: Corporation,

text: a

text: wholly

text: owned

text: subsidiary

text: of

text: )Bd(

text: rewop

text: esioN

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text: 3 − † − × × where V = 1K K iΩ is an m m complex matrix is the network’s transfer-matrix function. The 2n 2n ma- and Y = −K†S i2s an m × n complex matrix. trix Z˘(ω) can be decom(cid:20)posed into the block f(cid:21)orm: To obtain the transfer-matrix function from input to output ﬁelds, we seek the solution of Eq. (8) in the frequency domain. − Z (ω) Z+(ω) Using the Fourier transfor(cid:90)m, we deﬁne: Z˘(ω) = − ∗ − − ∗ , (15) Z+( ω) Z ( ω) ∞ b(t) = √1 (cid:90) dω b(ω)e−iωt, (9a) where Z−(ω) and Z+(ω) are n × n matrices. Correspond- 2π −∞ ingly, input-output relations of Eq. (13) can be expressed for ∞ † √1 † − − each of the output ﬁelds (i = 1, . . . , n) as: b (t) = dω b ( ω)e iωt, (9b) (cid:88) (cid:2) 2π −∞ n − (cid:3) where b(t) stands for any element of a(t), ain(t), and aout,i(ω) = Zij(ω)ain,j(ω) athoeurte(fto)r.eTωheisﬁtehled soipdeerbaatonrdsfarreequinentchye i(nretelaraticvteiotno ftrhaemcea,rarinedr (cid:88)j=1 (cid:2) + Z+(ω)a† (−ω) , (16a) frequency). We also use the double-length column vectors of ij in,j the form: (cid:20) (cid:21) a† (−ω) = n Z+(−ω)∗a (ω) (cid:3) out,i ij in,j j=1 − − ∗ † − b˘(ω) = b‡ (−ω) , (10) + Zij( ω) ain,j( ω) . (16b) b ( ω) B. Squeezing spectrum where b(ω) stands for either of a(ω), a (ω), and a (ω). in out With this notation, Eq. (8) together with its Hermitian conju- gate can be transformed into one matrix equation and solved Consider the quadrature of the ith output ﬁeld in time and for a˘(ω) in the frequency domain: frequency domains: a˘(ω) = (A˘ + iωI2m)−1K˘ †S˘ a˘in(ω). (11) Xi(t, θ) = aout,i(t)e−iθ + a†out,i(t)eiθ, (17a) − † − X (ω, θ) = a (ω)e iθ + a ( ω)eiθ, (17b) i out,i out,i (cid:20) ×(cid:21) Here, I is the 2m 2m identity matrix, A˘ = ∆(V, W), 2m K˘ = ∆(K, 0), S˘ = ∆(S, 0), and we use the notation: where θ is the homodyne phase. The power spectral density A B of the quadrature’s quantum noise (commonly referred to as ∆(A, B) = ∗ ∗ . Analogously, the boundary condi- B A the squeezing spectr(cid:90)um) is [45, 46]: tion of Eq. (5) together with its Hermitian conjugate can be transformed into one matrix equation in the frequency do- ∞ main: P (cid:48)(cid:104) (cid:48) (cid:105) (ω, θ) = 1 + dω :X (ω, θ), X (ω , θ): , (18) i −∞ i i a˘out(ω) = S˘a˘in(ω) + K˘ a˘(ω). (12) where : : denotes the normal ordering of boson operators (cid:104) (cid:105) (cid:104) (cid:105) − (cid:104) (cid:105)(cid:104) (cid:105) and x, y = xy x y . Since all input ﬁelds are in the (cid:104) (cid:105) (cid:104) (cid:48) (cid:105) vacuum state, X (ω, θ) = X (ω , θ) = 0, and one obtains: In Eqs. (11) and (12), a˘(ω) is a 2m-dimensional vector, i i × a˘ (ω) and a˘ (ω) are 2n-dimensional vectors, A˘ is a 2m in out × × P N N − 2m matrix, K˘ is a 2n 2m matrix, and S˘ is a 2n 2n ma- (ω, θ) = 1 + (ω) + ( ω) i i i trix. By substituting Eq. (11) into Eq. (12), one obtains the M − M ∗ + (ω)e 2iθ + (ω) e2iθ, (19) quantum input-output relations in the matrix form: i i a˘ (ω) = Z˘(ω)a˘ (ω), (13) where (cid:90) out in ∞ N (cid:90) (cid:48)(cid:104) † − (cid:48) (cid:105) where (cid:104) (cid:105) i(ω) = −∞ dω aout,i( ω )aout,i(ω) , (20a) ∞ M (cid:48)(cid:104) (cid:48) (cid:105) − † (ω) = dω a (ω)a (ω ) . (20b) Z˘(ω) = I + K˘ (A˘ + iωI ) 1K˘ S˘ (14) i −∞ out,i out,i 2n 2m ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: V

text: K

text: iΩ

text: is

text: an

text: m

text: m

text: complex

text: matrix

text: is

text: the

text: network’s

text: transfer-matrix

text: function.

text: The

text: ma-

text: and

text: Y

text: −K†S

text: an

text: m

text: n

text: complex

text: matrix.

text: trix

text: Z˘(ω)

text: can

text: be

label: equation

text: into

text: the

text: block

text: To

text: obtain

text: the

text: transfer-matrix

text: function

text: from

text: input

text: to

text: output

text: ﬁelds,

text: we

text: seek

text: the

text: solution

text: of

text: Eq.

text: in

text: the

text: frequency

text: domain.

text: Z

text: (ω)

text: Z+(ω)

text: Using

text: the

text: Fourier

text: we

text: deﬁne:

text: Z˘(ω)

text: Z+(

text: ω)

text: Z

text: ω)

text: b(t)

text: √1

text: dω

text: b(ω)e−iωt,

text: where

text: Z−(ω)

text: and

text: Z+(ω)

text: are

text: n

text: n

text: matrices.

text: Correspond-

text: 2π

text: ingly,

text: input-output

text: relations

text: of

text: Eq.

text: can

text: be

text: expressed

text: for

text: √1

text: each

text: of

text: the

text: output

text: ﬁelds

text: (i

text: n)

text: as:

text: b

text: (t)

text: dω

text: b

text: ω)e

text: iωt,

text: 2π

text: n

text: where

text: b(t)

text: stands

text: for

text: any

text: element

text: of

text: a(t),

text: ain(t),

text: and

text: aout,i(ω)

text: Zij(ω)ain,j(ω)

text: athoeurte(fto)r.eTωheisﬁtehled

text: soipdeerbaatonrdsfarreequinentchye

text: i(nretelaraticvteiotno

text: ftrhaemcea,rarinedr

text: Z+(ω)a†

text: (−ω)

text: frequency).

text: We

text: also

text: use

text: the

text: double-length

text: column

text: vectors

text: of

text: ij

text: in,j

text: the

text: form:

text: a†

text: (−ω)

text: n

text: Z+(−ω)∗a

text: (ω)

text: out,i

text: ij

text: in,j

text: b˘(ω)

text: b‡

text: (−ω)

text: Zij(

text: ω)

text: ain,j(

text: ω)

text: b

text: ω)

text: B.

text: Squeezing

text: spectrum

text: where

text: b(ω)

text: stands

text: for

text: either

text: of

text: a(ω),

text: a

text: (ω),

text: and

text: a

text: (ω).

text: in

text: out

text: With

text: this

text: notation,

text: Eq.

text: together

text: with

text: its

text: Hermitian

text: conju-

text: gate

text: can

text: be

text: transformed

text: into

text: one

text: matrix

text: equation

text: and

text: solved

text: Consider

text: the

text: quadrature

text: of

text: the

text: ith

text: output

text: ﬁeld

text: in

text: time

text: and

text: for

text: a˘(ω)

text: in

text: the

text: frequency

text: domain:

text: frequency

text: domains:

text: a˘(ω)

text: (A˘

text: †S˘

text: a˘in(ω).

text: Xi(t,

text: θ)

text: aout,i(t)e−iθ

text: a†out,i(t)eiθ,

text: X

text: (ω,

text: θ)

text: a

text: (ω)e

text: iθ

text: a

text: ω)eiθ,

text: i

text: out,i

text: out,i

text: Here,

text: I

text: is

text: the

text: identity

text: matrix,

text: A˘

text: ∆(V,

text: W),

text: K˘

text: ∆(K,

text: S˘

text: ∆(S,

text: and

text: we

text: use

text: the

text: notation:

text: where

text: θ

text: is

text: the

text: homodyne

text: phase.

text: The

text: power

text: spectral

text: density

text: A

text: B

text: of

text: the

text: quadrature’s

text: quantum

text: noise

text: (commonly

text: referred

text: to

text: as

text: ∆(A,

text: B)

text: Analogously,

text: the

text: boundary

text: condi-

text: B

text: A

text: the

text: squeezing

text: is

text: tion

text: of

text: Eq.

text: together

text: with

text: its

text: Hermitian

text: conjugate

text: can

text: be

text: transformed

text: into

text: one

text: matrix

text: equation

text: in

text: the

text: frequency

text: do-

text: main:

text: P

text: (ω,

text: θ)

text: dω

text: :X

text: (ω,

text: θ),

text: X

text: (ω

text: θ):

text: i

text: i

text: i

text: a˘out(ω)

text: S˘a˘in(ω)

text: K˘

text: a˘(ω).

text: where

text: denotes

text: the

text: normal

text: ordering

text: of

text: boson

text: operators

text: and

text: x,

text: y

text: xy

text: x

text: y

text: Since

text: all

text: input

text: ﬁelds

text: are

text: in

text: the

text: vacuum

text: state,

text: X

text: (ω,

text: θ)

text: X

text: (ω

text: θ)

text: and

text: one

text: obtains:

text: In

text: Eqs.

text: and

text: a˘(ω)

text: is

text: a

text: vector,

text: i

text: i

text: a˘

text: (ω)

text: and

text: a˘

text: (ω)

text: are

text: vectors,

text: A˘

text: is

text: a

text: in

text: out

text: P

text: N

text: N

text: matrix,

text: K˘

text: is

text: a

text: matrix,

text: and

text: S˘

text: is

text: a

text: ma-

text: (ω,

text: θ)

text: (ω)

text: ω)

text: i

text: i

text: i

text: trix.

text: By

text: substituting

text: Eq.

text: into

text: Eq.

text: one

text: obtains

text: the

text: M

text: M

text: (ω)e

text: (ω)

text: quantum

text: input-output

text: relations

text: in

text: the

text: matrix

text: form:

text: i

text: i

text: a˘

text: (ω)

text: Z˘(ω)a˘

text: (ω),

text: where

text: out

text: in

text: N

text: where

text: i(ω)

text: dω

text: aout,i(

text: ω

text: )aout,i(ω)

text: M

text: (ω)

text: dω

text: a

text: (ω)a

text: (ω

text: Z˘(ω)

text: I

text: K˘

text: (A˘

text: iωI

text: S˘

text: i

text: out,i

text: out,i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 loss vac out7 in7 Out3 In3 OPO2 In1 Out2 vac in5 lock Out1 In2 out5 Controller theta=theta_2 Out2 alpha=xi_c vac lock kappa_1=kappa_c_1 in3 In2 out3 phi=phi_1 kappa_2=kappa_c_2 kappa_3=kappa_c_3 Delta=omega_c loss vac out6 in6 In2 Out3 In3 phi=phi_2 lock vac out2 Out2 theta=theta_1 in2 OPO1 In1 Out2 vac in1 Out1 In2 Plant theta=theta_3 alpha=xi_p Out2 kappa_1=kappa_p_1 vac loss kappa_2=kappa_p_2 in4 In2 out4 kappa_3=kappa_p_3 device=OPO_CQFN_1 Delta=omega_p output module−name=OPO_CQFN_1 out1 params=xi_p:complex;xi_c:complex;kappa_p_1:real;kappa_p_2:real;kappa_p_3:real;kappa_c_1:real;kappa_c_2:real;kappa_c_3:real;omega_p:real;omega_c:real;theta_1:real;theta_2:real;theta_3:real;phi_1:real;phi_2:real FIG. 3. A schematic depiction of the CQFC network of two coupled OPOs. TABLE II. Parameters of the CQFC network of two coupled OPOs. Parameter Type Description κ Positive Leakage rate for the left mirror of the plant OPO cavity p1 κ Positive Leakage rate for the right mirror of the plant OPO cavity p2 κ Positive Leakage rate for losses in the plant OPO cavity p3 ω Real Frequency detuning of the plant OPO cavity p ξ Complex Pump amplitude of the plant OPO p κ Positive Leakage rate for the left mirror of the controller OPO cavity c1 κ Positive Leakage rate for the right mirror of the controller OPO cavity c2 κ Positive Leakage rate for losses in the controller OPO cavity c3 ω Real Frequency detuning of the controller OPO cavity c ξ Complex Pump amplitude of the controller OPO c φ Real Phase shift of the ﬁrst phase shifter 1 φ Real Phase shift of the second phase shifter 2 θ Real Rotation angle of the ﬁrst beamsplitter 1 θ Real Rotation angle of the second beamsplitter 2 θ Real Rotation angle of the third beamsplitter 3 to denote, respectively, the transmittivity and reﬂectivity of each beamsplitter, and to denote the total phase shift for the feedback roundtrip path. Similarly to Eq. (32), we also deﬁne the scaled pump amplitudes x and x for the plant and controller OPOs, re- p c φ = φ + φ (39) spectively: 1 2 1P 1B 1tuO )−( 1tuO 1nI 1nI In1 (−) In1 (−) In1 Out1 Out1 Out1 B3 B2 P2 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: loss

text: vac

text: vac

text: lock

text: Controller

text: vac

text: lock

text: loss

text: vac

text: lock

text: vac

text: vac

text: Plant

text: vac

text: loss

text: output

text: FIG.

text: A

text: schematic

text: depiction

text: of

text: the

text: CQFC

text: network

text: of

text: two

text: coupled

text: OPOs.

text: TABLE

text: II.

text: Parameters

text: of

text: the

text: CQFC

text: network

text: of

text: two

text: coupled

text: OPOs.

text: Parameter

text: Type

text: Description

text: κ

text: Positive

text: Leakage

text: rate

text: for

text: the

text: left

text: mirror

text: of

text: the

text: plant

text: OPO

text: cavity

text: κ

text: Positive

text: Leakage

text: rate

text: for

text: the

text: right

text: mirror

text: of

text: the

text: plant

text: OPO

text: cavity

text: κ

text: Positive

text: Leakage

text: rate

text: for

text: losses

text: in

text: the

text: plant

text: OPO

text: cavity

text: ω

text: Real

text: Frequency

text: detuning

text: of

text: the

text: plant

text: OPO

text: cavity

text: p

text: ξ

text: Complex

text: Pump

text: amplitude

text: of

text: the

text: plant

text: OPO

text: p

text: κ

text: Positive

text: Leakage

text: rate

text: for

text: the

text: left

text: mirror

text: of

text: the

text: controller

text: OPO

text: cavity

text: κ

text: Positive

text: Leakage

text: rate

text: for

text: the

text: right

text: mirror

text: of

text: the

text: controller

text: OPO

text: cavity

text: κ

text: Positive

text: Leakage

text: rate

text: for

text: losses

text: in

text: the

text: controller

text: OPO

text: cavity

text: ω

text: Real

text: Frequency

text: detuning

text: of

text: the

text: controller

text: OPO

text: cavity

text: c

text: ξ

text: Complex

text: Pump

text: amplitude

text: of

text: the

text: controller

text: OPO

text: c

text: φ

text: Real

text: Phase

text: shift

text: of

text: the

text: ﬁrst

text: phase

text: shifter

text: φ

text: Real

text: Phase

text: shift

text: of

text: the

text: second

text: phase

text: shifter

text: θ

text: Real

text: Rotation

text: angle

text: of

text: the

text: ﬁrst

text: beamsplitter

text: θ

text: Real

text: Rotation

text: angle

text: of

text: the

text: second

text: beamsplitter

text: θ

text: Real

text: Rotation

text: angle

text: of

text: the

text: third

text: beamsplitter

text: to

text: denote,

text: respectively,

text: the

text: transmittivity

text: and

text: reﬂectivity

text: of

text: each

text: beamsplitter,

text: and

text: to

text: denote

text: the

text: total

text: phase

text: shift

text: for

text: the

text: feedback

text: roundtrip

text: path.

text: Similarly

text: to

text: Eq.

text: we

text: also

text: deﬁne

text: the

text: scaled

text: pump

text: amplitudes

text: x

text: and

text: x

text: for

text: the

text: plant

text: and

text: controller

text: OPOs,

text: re-

text: p

text: c

text: φ

text: φ

text: φ

text: spectively:

text: ##LTFigure##

text: ##LTLine##

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