as researchers improve their ability to engineer quantum electromagnetic ( em ) devices , we should start to look beyond individual components and find experimental and theoretical techniques that exploit coherent networks of quantum devices .for instance , it is important to consider whether useful functions are best achieved at the _ network - level _ of interacting , modular , and increasingly generic devices .this approach is routine in classical electronics .for example , an unreliable operational amplifier ( `` op - amp '' ) may be combined with a reliable impedance network in a feedback configuration to realize a reliable amplifier .this strategy is usually more efficient than developing high quality amplifier chips for each new use .if quantum engineering is to consider this approach , it would greatly benefit from some systemization of device and interconnection laws , in very rough analogy to kirchoff s laws systemization of practical electronics .because classical network theory has so many established , practical techniques , we might expect that hybrid approaches towards this end may handle network complexity better than approaches entirely native to physics .feedback engineering addresses two types of problems : feedback can enhance a system s robustness and/or tune the dynamics of an otherwise untunable system . as a general rule ,modern quantum em devices feature increasingly monolithic and insular designs , as robustness is generally prized over tunability in single - device physics . in the field of electromechanics ( also known as `` optomechanics '' ) for example , devices in which mechanical simple harmonic oscillators are coupled to itinerant em fields have become increasingly powerful , e.g. .mechanical oscillators are more isolated from thermal baths , interactions with distinct em modes are better controlled and quantum effects are starting to be observed .but as a consequence , these devices are often module - like and unadjustable : important characteristics like em center frequency and linewidth are often fixed by construction , and often a single em input / output ( i / o ) port admits access to all internal mechanisms . in principle , tunability may be won back without compromising device integrity by employing a tunable , coherent controller that exchanges continuous coherent signals with this port . just as a reliable , negative feedback network can ameliorate the unreliable gain of an op - amp , perhaps combining a tunable coherent feedback network with an insulated electromechanical circuit may bring out the best of both , and at low cost .  herewe demonstrate a coherent feedback network of em devices , a superconducting electromechanical device and a superconducting microwave controller , that provides us with a type of dynamic flexibility previously unavailable in electromechanics .namely , while previous electromechanical systems make some compromise between better mechanical control and better mechanical measurement capabilities , we demonstrate that this network may be dynamically modulated between both extremes .while our network is too complex for traditional electromechanical models , it requires only three components , all of which are accessible to superconducting circuit labs ( and feature regularly in our own work ) , and is efficiently and intuitively modeled as a linear feedback system . in other words , this network is easy to build ( given familiar technologies , specifically , devices `` generic '' in our lab ) and its mechanisms easy to intuit .but it is difficult to model quantitatively , meriting methods that naturally handle complex networks .we do not characterize performance in the quantum regime , but these devices feature regularly in such work and these capabilities are novel regardless .moreover , because the network is linear and driven by gaussian fields , the network characterization done in the classical regime should apply in the quantum one .thus , we use the term `` coherent '' in the sense that an interferometer or resonator is coherent , the primary prerequisite for quantum dynamics .because a systematic network theory perspective is relatively novel in electromechanics , this article features a pedagogical modeling section that outlines the modeling method and its advantages ( rigorous derivations are well - covered in the literature , e.g. ) .also , because it employs theoretical approaches developed by an electrical engineering community , familiar concepts in electromechanics are described in more detail than usual .the outline of the paper is as follows , in section [ sec : network ] we introduce the network construction and its physical intuition .section [ sec : results ] presents our measurements of this network , its essential operation and agreement with the network model .section [ sec : model ] describes the procedure for constructing the network model and section [ sec : con ] concludes .many superconducting microwave realizations of electromechanical systems consist of a high quality microwave lc resonator ( hereon `` resonator '' ) , in which the capacitance is modulated by a spring - loaded , mechanical simple harmonic oscillator ( hereon `` mo '' ) .motion in this mo adjusts the capacitance of the resonator , and thus acts on microwave power in the resonator . andthe voltage potential across the capacitor creates a coulomb force that acts on the mo , and thus microwave power in the resonator acts on the position of the mo . via an inductive transformer ( or some other `` input coupler '' ) , this high - quality resonator couples to a transmission line , which serves as the i / o port through which the electromechanical device is probed , driven and controlled .these microwave systems are accurately described at a quantum level with a model that is formally equivalent to the models used to describe electromechanical systems of all scales and realizations .the choice of an input coupler in an electromechanical system typically represents a compromise . for example , microwave signals inside a resonator leave the resonator more slowly if it couples to a transmission line weakly . as a consequence , microwave signals in resonators with weak couplersboth affect and are affected by the mo more strongly : the weak coupling causes electromechanical effects to integrate for a longer time .but too much integration can also be undesirable .for example , weak couplers frustrate high - bandwidth detection of mechanical motion by low - pass filtering mechanical information on em signals leaving the resonator and encouraging loss of the same through parasitic channels . the usual distinction between these two regimesis whether or not the resonator linewidth is narrower than the mo s center frequency .more strongly ( weakly ) coupled systems with ( ) are known as `` unresolved sideband '' ( `` resolved sideband '' ) systems .experimentalists typically use resolved sideband systems to control and prepare specific states in mos .unresolved sideband systems are most useful when high - bandwidth readout of the mo s motion is the priority .sometimes , intermediate regimes are ideal ( ) . in all electromechanical systems to our knowledge , though , the resonator s coupling strength is fixed at the time of construction .how might one construct a more flexible system that could be tuned dynamically between the resolved and unresolved sideband regimes ?our solution was inspired by considering what usually goes `` outside '' emc devices .many superconducting microwave electromechanical experiments conducted by us and others are partitioned into an `` upstream '' electromechanical circuit ( emc ) and a `` downstream '' cryogenic low- or near quantum limited - noise temperature amplifier , for high - fidelity readout of microwave probes .our group typically uses near quantum - limited josephson parametric amplifiers ( jpas ) for readout .jpas are composed of a 20 db directional coupler followed by a single - port `` tunable kerr circuit '' ( tkc ) , a nonlinear microwave resonator whose center frequency is drive power - dependent and tunable with an applied magnetic flux .the essential novelty in this work is that rather than measure the signal that comes out of the jpa , a portion of the signal emitted by the emc and passed to the jpa is fed back into the emc coherently .this feedback gives microwave signals more opportunities to interact with the om , which is the essential quality of a weakly - coupled / resolved - sideband system .moreover , because the jpa is dynamically tunable , we can choose whether or not this feedback occurs , and access both coupling regimes dynamically . and because the critical components , the emc and jpa , are `` generic '' in our lab , we can pursue this scheme using existing devices .a ) an experimental schematic of the coherent feedback network .a microwave tee connects an electromechanical circuit ( emc ) to a josephson parametric amplifier ( jpa ) here broken down into its directional coupler ( dc ) and tunable kerr circuit ( tkc ) subcomponents and provides an overall input - output port .the in port is typically driven by a strong electromechanical coupling tone , but this tone is typically cancelled at the dc , preventing its strong carrier from driving the tkc nonlinear .microwave signals are detected at the out port .b ) conceptual schematic of coherent interconnections .ideally , the jpa acts as an over - coupled and wide - band linear resonator with a tunable center frequency . ]housed in a dilution refrigerator , our network is the interconnection of a single - port emc , a jpa , and a commercial microwave tee ( fig . [the emc and jpa are connected to one tee port each , while the remaining tee port serves as the overall i / o network port .the network output is amplified by a cryogenic hemt for analysis .while the emc and tkc were originally designed for different experiments and were mounted in separate sample boxes , and interconnections were made using - length coplanar waveguides and low - loss , semi - rigid coaxial cables , an analogous network could be fabricated on a single si substrate .although the specific electromechanical circuit ( emc ) used here is somewhat incidental , it is a new design of this type , built from a 4.672 ghz center frequency and over - coupled mhz lumped - element resonator , employing a mechanical membrane oscillator with an effective mass of ng , center frequency of .6 khz and intrinsic linewidth of .81 hz , and in which each photon exerts a force on the mo ( in coherent amplitude units ) of hz . in this report , we do not pump our jpa ; effectively , it is operated as a gain-1 amplifier , or more precisely , as a linear resonator whose center frequency can be tuned by applying either magnetic flux or a moderate amount of microwave drive power .we do , however , typically employ a cancellation tone that prevents the carrier of any strong , electromechanical coupling tone driving the network from reaching the tkc and driving it nonlinear .the network s operation is depicted conceptually in fig .microwave signals leaving the emc and carrying information about the motion of the mo are split three ways at the tee : most of the amplitude is split evenly between the network output and the jpa input , and a small amount is reflected .modeling the jpa as a linear resonator with a tunable center frequency , the jpa reflects its signal portion with a tunable phase shift .similarly split by the tee , a portion of the jpa - reflected signal interferes with the emc - to - output signal , enhancing or diminishing the rate of mechanical information leaving the network .this same interference also enhances or diminishes the effective microwave linewidth of the network , as seen by the microwave input port .finally , the portion of the jpa - reflected signal fed back to the emc can exert a force on the mo , amplifying or counteracting its motion .all of these effects are controlled by a single , continuous parameter the jpa center frequency and both effects improve as the jpa bandwidth exceeds the bandwidth of the emc and the emc bandwidth exceeds the mo s center frequency .there are many ways to view this network .for instance , this network may be interpreted as analogous to a traditional emc , except with a user - controlled knob that modifies the coupling between the transmission line and resonator ( the resonator center frequency may also be affected ) . the specific network in fig .[ fig1]b can be viewed a form of microwave stub tuning , although the jpa bandwidth generalizes the usual model of a `` stub . ''this network may also be interpreted as a feedback control system : rather than make a room - temperature measurement of the jpa output , this signal is fed back directly to the emc without leaving the cryostat . in this work ,the first interpretation aids physical intuition , but is not predictive .the second has a physical intuition and is quantitatively predictive , but is limited to the specific wiring configuration used here .the third , however , is also physically - intuitive and predictive , but may be applied quite generally .although it is not our focus here , it is worth clarifying that coherent feedback approaches are distinct from measurement - based feedback approaches .for example , in a measurement - based version of our system , coherent signals produced by the emc and jpa would be measured by an incoherent controller ( e.g. a computer ) , which would then synthesize new signals that act back on the mo . while a measurement - based approach can alter the response function of a mo , the system s effective em linewidth is unchanged . here, the jpa s transformation of the microwave signal is ideally unitary ( i.e. adds no noise ) .this network is thus fully coherent and thus fundamentally different from any measurement - based control system .for example , it has been noted that coherent feedback in the quantum regime can outperform even ideal measurement - based feedback when one considers the cost of the controller s action a measurement - based controller must act to cancel both environmental disturbances and quantum projection noise . at a more practical level ,coherent control systems tend to be physically compact .we note that techniques borrowed from network systems theory are vital for tractable modeling of many coherent networks .  many of the demonstrations in the next section are analogous to the widely used electromechanical technique of sideband cooling .the canonical sideband cooling system consists of an emc , or some other electromechanical device , driven at a frequency below the resonator s center frequency .mechanical cooling via this method is usually explained in analogy to early ion trapping experiments .a small portion of the low - entropy em drive is inelastically scattered by a mo inside the resonator , with each scattered photon having gained or lost a phonon s worth of energy .if the center frequency of the resonator is higher than the drive frequency , then more frequency up - converted photons are scattered than down - converted photons .consequently , the mo gives up energy on average to the em field per scattering event , reducing its motion .this phenomenon is usually visible through `` blue '' ( up - converted ) and `` red '' ( down - converted ) sidebands to the `` coupling tone '' ( drive carrier ) emitted by the resonator .a cooling mo is indicated by more blue than red sideband power , less total sideband power , and sideband power spectral densities broader than the intrinsic mo linewidth . to connect our coherent feedback network to sideband cooling, we point out that there are equivalent , coherent feedback interpretations of these phenomena .these interpretations involve mechanically - modulated em signals interfering with the coupling tone , say , at the input coupler .dependent on this interference , em power is continuously added to or removed from the resonator in such a way to counteract mechanical motion .the network s response to microwave probe tones is depicted in fig .[ fig2 ] . by monitoring the network s phase response( i.e. the phase with which microwave tones are reflected by the network ) we see that the microwave response may be finely tuned using a static , magnetic flux bias applied to the jpa .viewing the response over a 2 ghz frequency range in fig .[ fig2]a , the jpa center frequency strongly varies with applied flux .the other visible resonances , the narrow - band emc and very broad - band resonances arising from standing waves resonances between the subcomponents , are only affected as the jpa tunes through them .focusing on the 10 mhz band around the intersection of the jpa and emc resonances (  flux coil bias , fig . [ fig2]b ) , we see that as the flux increases and the jpa - like resonance approaches , the emc - like resonance first broadens slightly , then abruptly narrows and shifts higher , then becomes under - coupled , and finally re - broadens and returns to its original center .these effects indicate that the two device modes are coupling coherently and hybridizing .microwave - related parameters may be estimated given these 2 ghz and 10mhz probe ranges . using a linear control systems model built from interconnected device models ( employing matlab s control systems toolbox , see section [ sec : model ] ) , we infer an emc coupling linewidth of mhz and internal loss rate mhz ; a jpa coupling linewidth of mhz and internal loss rate of mhz ; and tee - to - emc round trip phase shift of rad and tee - to - jpa round trip phase phase shift of rad .these parameters are consistent with parameter estimates from the devices in isolation .although difficult to distinguish losses internal to these over - coupled devices from interconnection losses , a network model dominated by losses internal to the emc and jpa is more consistent with observations than a network model dominated by interconnection losses .similarly , the tee is assumed to be ideal and properly terminated , reflecting 1/9 of the incident power with a phase shift and splitting the remaining power between the other two ports . with these parameters ,the linear control systems model accurately reproduces the network s microwave response as the flux varies , fig .[ fig2]c .  only the emc - like resonance ( i.e. the resonance in the vicinity of 4.672 ghz ) has significant coupling to the mo .how this resonance responds to applied flux is more easily seen by taking a few , constant - flux line cuts through fig .[ fig2]d depicts three such line cuts . with the jpa far - detuned ( 50 bias current ), the microwave phase response indicates an overcoupled , mhz linewidth network resonance at 4.672 ghz . note that even with the jpa far detuned , the network causes the effective linewidth of the emc - like resonance to differ from , the linewidth of the bare emc . at different flux biasings ,this resonance frequency and linewidth vary together , each by values of order the emc s coupling linewidth .the network s mechanical response .a ) employing , the power spectral density of the blue , thermally - excited mechanical sideband is shown for three flux settings ( same flux settings as in fig .[ fig2]d ) , keeping the coupling tone fixed at 4.672 ghz .b ) points represent the observed linewidth of blue thermal sidebands as the jpa is tuned , employing a coupling tone at a fixed frequency and power such that .the line is a theoretical prediction with no free parameters .c ) from the same experiment , the observed changes in the mo center frequency and theoretical prediction are shown .d ) cooling efficiency ( ce ) , as inferred from ring - down measurements and measured red and blue sideband powers . left , ce for different coupling tone frequencies when jpa is far - detuned .right , ce for a fixed coupling tone and different jpa tunings .lines are theoretical predictions .numbered call outs in right plot correspond to jpa tunings used in fig .[ fig4]a . ]  in fig .[ fig3 ] , we see that the dynamics of the mo also varies with applied flux . far - detuning the jpa once more ( back to 50 flux coil bias ) and driving the network with a strong coupling tone at the 4.672 ghz network resonance ( and protecting the tkc with a cancellation tone through the directional coupler , fig. [ fig1]a ) , the thermal motion of the mo is visible via red and blue inelastically - scattered network output signals , i.e. sidebands .the power spectral densities of these sidebands ( fig .[ fig3]a ) indicate a mechanical center frequency of khz and intrinsic damping rate of hz .next , keeping the the coupling tone fixed at the same power and frequency , but varying the flux bias , the mechanical motion inferred from the sidebands varies .in particular , as the network s resonance moves higher and its linewidth narrows in fig .[ fig2]d , in fig .[ fig3]a the apparent mechanical motion is damped more heavily ( sideband linewidth becomes ) , is cooled , and its center frequency moves ( ) .these effects are analogous to sideband cooling in traditional electromechanical systems . in our system , these effects may be interpreted as sideband cooling , or as the jpa acting as a coherent controller , exerting different control laws for different amounts of applied flux .it is important to note that , the number of photons in the emc induced by the coupling tone , does not change as the tkc ( the tunable component of the jpa ) is tuned . while the network s response to probes and mechanical sidebands varies with the tkc state , the coupling tone is canceled at the directional coupler before the tkc ( fig .[ fig1]a ) and is thus unaffected by the tkc s center frequency .more precisely , the mean power in the coupling tone is dissipated in the terminated port in the directional coupler through interference with the cancellation tone , while information - carrying fluctuations about the mean power are not .these fluctuations are fed back to the tee and emc after reflecting off the tkc , while the mean coupling power experiences an entirely different network without tkc feedback . except for inducing ,this mean coupling power is unimportant and it carries no information .the assumption of a constant is made throughout , and would have resulted in very inaccurate predictions in figs .[ fig3]b & c ( discussed next ) if incorrect .while fig .[ fig3]a establishes the qualitative mechanical response , the rest of the figure considers it quantitatively .driving the network with a coupling tone fixed at a frequency just 13 khz below the ( 4.672 ghz ) network resonance with the jpa far - detuned and with a fixed power such that is expected , figs .[ fig3]b & c depict the linewidth and center frequency of the mechanical sidebands as a function of flux biasing .starting from a linewidth of with the jpa far - detuned , reaches a maximum of hz , with a simultaneous frequency shift of hz , when the flux bias is 56.4 .this flux bias corresponds to a jpa center frequency 3.4 mhz higher than the coupling tone , and an undercoupled network response with a khz internal loss rate . if internal device losses were lower , this effective internal loss rate would be lower , and the maximum on fig .[ fig3]b would have been higher .because the near - resonant coupling tone has no significant effect on the mechanical state when the jpa is far - detuned , it is natural to interpret the jpa as `` controlling '' the mo as it is tuned through the emc . using a control - systems model ( here employing standard - issue mathematica toolkits ) and no free parameters , predictions for the expected mechanical linewidth and frequency shift as a function of flux biasare also included in figs .[ fig3 ] b & c.  one metric for quantifying the efficiency with which the network controls the mo is the probability that a mechanically scattered photon dissipated by the network is a frequency up - converted photon minus the probability it is a down - converted photon .this `` cooling efficiency '' ( ce ) is independent of , but is coupling - frequency dependent and is positive when the network cools the mo ( and is negative when it amplifies thermal motion ) .the maximum ce obtainable over all coupling frequencies is , where is the total , emc - like resonance linewidth . in fig .[ fig3]d , left plot , we plot ce as a function of coupling tone frequency and with the jpa far - detuned .these are measured by first inducing coherent oscillations in the mo well above their thermal occupation through amplitude modulation of the coupling tone at frequency , and then stopping the modulation and measuring the red and blue sidebands emitted by the network as the mechanical state re - equilibrates .data is averaged over 25 trials .the power emitted by both sidebands , and , decays exponentially ( `` rings down '' ) in time at rate , while ce= is constant .the extremum of ce are , implying a mhz - linewidth network coupled to the significantly slower khz mo , an unresolved - sideband network .underlying this data is the expected ce versus coupling frequency , using independent network and mo calibrations . driving the network as in figs .[ fig3 ] b & c and tuning the jpa , but now making ring down measurements , ce peaks at 0.98 at 56.4 .a ce of 0.98 is only possible for networks with at most khz , a resolved - sideband network and a 10-fold reduction in consistent with the thermal data estimate above .such a reduction was apparent from microwave measurements in fig .[ fig2 ] , but ce represents an effective `` '' measurement from the mechanical bath `` port '' to microwave output .ce also reaches a minimum of -0.04 at 57.8 , indicating that the network resonance can also dip slightly below its far - detuned jpa resonance .an independently calibrated control systems prediction using \right|^2-\left|\xi_{\{\mu_o^\ast\text{th}_i\}}[j\omega]\right|^2}{\left|\xi_{\{\mu_o\text{th}_i\}}[j\omega]\right|^2+\left|\xi_{\{\mu_o^\ast\text{th}_i\}}[j\omega]\right|^2}\ ] ] ( see eq . ,section [ sec : model ] ) underlies the data .dynamic jpa - control .a ) red and blue sideband powers ringing down from a very high coherent amplitude . from 0 - 25ms ( 25 - 40 ms ) the network is in state 1 ( state 2 ) , see figs . [ fig3]d & [ fig4]b .inset , detail of network transition , depicted with 25 khz bandwidth .b ) microwave probe measurements for the network states employed in a , with arrows indicating coupling tone and sidebands .c ) switched re - thermalization of a cooled mo . at 100 ms ,jpa is tuned from ( cooling ) state to ( bath - thermalizing ) state . at 600 ms ,jpa is switched to the cooling state for 6 ms only .plot represents the ensemble distribution of blue sideband amplitudes in time for 300 trials .d ) sideband amplitude variance for 6 ms= and 3 ms= cooling - interval experiment ensembles . ]  finally , we demonstrate that the state of the jpa , and thus the dynamics it effects , may be switched far faster than the mo can equilibrate . while above we controlled the jpa state using static flux biasing , below we apply both static flux biasing and an additional microwave tone at a frequency 10mhz below the emc - like resonance . the amplitude of this new , `` control '' tone is switched dynamically and is not cancelled at the directional coupler . at its strongest ,the control tone is too weak and off - resonant with the emc to affect the mo , but is sufficiently strong and resonant with the wider - band jpa to strongly shift its center frequency .the rate with which the jpa may be switched in this manner should lie between the jpa linewidth ( 50 mhz ) and the network s em linewidth as a whole ( khz ) , orders of magnitude faster than the mechanical response rate ( 0.81 hz intrinsic linewidth ) . by performing similar ring down measurements starting from an even larger mechanical coherent amplitude ( inducing khz pk - pk resonance shift in the emc ) ,this switched control may be observed with high visibility and bandwidth . in fig .[ fig4]a , the power measured in the blue and red mechanical sidebands is depicted in time where , from 0 to 25 ms , the network is in a resolved - sideband and blue detuned state ( state 1 in fig .[ fig4]b , with ce called out in fig .[ fig3]d ) , while from 25 to 40 ms , the network is in an unresolved - sideband and slightly red detuned state ( state 2 ) . in the first segment , almost all the mechanically - scattered power is in the blue sideband , and the total power emitted decays in time , indicating mo cooling ( the large amplitude motion employed here induces a variation in the emc center frequency of order the state 1 linewidth . as a consequence , the ring down power is more linear than exponential in time ) . in the second segment ,the two powers are roughly equal , with the red sidband slightly greater , and the total power hardly decays .the inset depicts the transition between states 1 & 2 , through which the blue sideband stays constant , but the red sideband power jumps by an order of magnitude .this is only possible because state 2 has a much larger linewidth than 1 ( fig .[ fig4]b ) .the transition occurs at least as fast as the 25 khz detection bandwidth , much faster than the mechanical decay rate in state 1 ( hz ) or 2 ( hz ) .switched control over incoherent thermal motion may also be demonstrated , at the expense of reducing the detection bandwidth to 100 hz ( so that 10 min averaging times yield reasonable signal to noise ratios ) . in fig .[ fig4]c & d , the coupling tone and jpa are tuned such that the mechanics are cooled for several seconds with hz ( the most aggressive , resolved - sideband cooling configuration in fig .[ fig3]b ) . at time 100ms , the jpa is rapidly far - detuned from the emc , so that the network is on - resonance with the coupling tone and is in an unresolved sideband limit . for 500 ms ,the mechanical state then thermalizes with its bath at rate . after 500 ms ,the jpa is briefly tuned to the cooling state for 6 ms , and then back to the thermalization state for a final 400 ms .[ fig4]c depicts the 300-trial ensemble distribution of continuously monitored blue sideband amplitudes . before 100 ms ,the distribution is dominated by amplifier noise , but broadens by a factor of 3 during the thermalization periods .the sideband amplitude is proportional to mechanical displacement ( by a factor that differs for the cooling and thermalization states ) , and the broadening of the distribution is indicative of a warming mo ( to phonon occupation , or 26 mk , or ( 63 hz) variance in the emc resonance , although precise knowledge of the thermal occupation is incidental to our purposes ) .the position variance of the hot hz mo is expected to drop by in 6 ms , by in 3 ms , and the variance of the cold , -linewidth mo should rise to of the bath equilibrium value in 200 ms .[ fig4]d depicts the ensemble variance of 6 & 3 ms cooling - interval experiments and is consistent with these expectations .each subcomponent ( the emc , jpa and tee ) is separably described by an i / o dynamical model , a familiar concept in quantum optics and superconducting microwave systems . moreover , despite fundamental nonlinearities , the device dynamics essential to the network s operation are accurately described by common linear approximations , as discussed more below .it is well known that a network of linear , coherent i / o devices coupled to itinerant gaussian fields may be modeled as a classical control system of interconnected , linear state space models .the residual `` quantumness '' of these systems is captured by nonclassical `` noise '' driving the network inputs , but such quantum - level accuracy is not required in this work as the dynamics considered are classical . moreover ,when vacuum fluctuations in the microwave fields are dynamically significant , each instance of microwave dissipation inside each device or interconnection between devices must be modeled using an additional i / o port pair ( and a beamsplitter component interrupting lossy interconnections ) , but such accuracy is not required here . in sum ,the physics " of our model is captured by well - known i / o device models and cascaded interconnections , but our manipulation of these device models adopts techniques from network theory . to many readers ,the least familiar aspect of the model is likely to be these manipulations , which are expected ( and partially tested ) to be the same in classical and quantum linear systems .more precisely , the tee is modeled as a 6-input and 6-output ( 6-i / o ) itinerant field scattering device ( three physical ports , each of which is both a field input and output , each field described by two quadrature degrees of freedom ) with no internal degrees of freedom .the jpa is modeled as a 2-i / o , over - coupled linear resonator device ( whose center frequency is a tunable parameter ) with two internal degrees of freedom , representing the two quadratures of the mode inside the tkc . andthe emc is a 4-i / o device , with two i / o ports connected to a microwave field ( effectively at zero temperature ) and two i / o ports connected to a thermal mechanical bath .the emc also contains four internal degrees of freedom , representing the quadratures of coupled microwave and mechanical modes .consequently , a linear network model representing the schematic depicted in fig .[ fig1]b consists of ten coupled linear equations of motion , with six internal degrees of freedom , and two microwave and two mechanical bath i / o port pairs . despite this complexity , standard control systems software toolkits make the construction of network dynamical models intuitive and efficient linear system theories may be applied to their analysis .we now describe the model construction in more detail .while the electromechanical interaction is fundamentally nonlinear , the dynamics we consider may be modeled by linearizing the coupling of the microwave and mechanical modes about the large microwave coherent state induced in the resonator by the coupling tone , an approximation that is nearly ubiquitous in electromechanics . in a frame in which the em degrees of freedom are rotating with the coupling tone s carrier frequency ,the hamiltonian that describes the internal dynamics of the emc in the linearized approximation is ( ) where is the annihilation operator for the mechanical mode and the annihilation operator for fluctuations in the microwave mode about the large coherent state ( the creation operators ) .additionally , is the center frequency of the mo , is the frequency detuning between the resonator center frequency ( ) and the carrier of the coupling tone ( ) , and , where is the number of intra - resonator photons induced by the coupling tone and is the fundamental coupling rate between photons and phonons .  in the usuali / o formulation for an emc , the heisenberg equations of motion may be written as   & = & \left[\begin{array}{cccc }  j\omega-\frac{\gamma_0}{2 } & jg & 0 & jg\\ jg & j\delta - \frac{\kappa_l+\kappa_c}{2 } & jg & 0\\ 0 & -jg & -j\omega-\frac{\gamma_0}{2 } & -jg\\ -jg & 0 & -jg & -j\delta - \frac{\kappa_l+\kappa_c}{2}\end{array}\right]\left[\begin{array}{c}a_1\\ a_2\\ a_1^\dag\\ a_2^\dag\end{array}\right]+\left[\begin{array}{cccc}-j\sqrt{\gamma_0}&0&0&0\\ 0&-j\sqrt{\kappa_c}&0&0\\0&0&j\sqrt{\gamma_0}&0\\0&0&0&j\sqrt{\kappa_c}\end{array}\right]\left[\begin{array}{c}b_{th , in}\\ b_{e , in}\\ b_{th , in}^\dag\\ b_{e , in}^\dag\end{array}\right]\nonumber\\ \left[\begin{array}{c}b_{th , out}\\ b_{e , out}\\ b_{th , out}^\dag\\ b_{e , out}^\dag\end{array}\right]&=&\left[\begin{array}{cccc}j\sqrt{\gamma_0}&0&0&0\\ 0&j\sqrt{\kappa_c}&0&0\\0&0&-j\sqrt{\gamma_0}&0\\0&0&0&-j\sqrt{\kappa_c}\end{array}\right ] \left[\begin{array}{c}a_1\\ a_2\\ a_1^\dag\\ a_2^\dag\end{array}\right ] + \left[\begin{array}{cccc}-1&0&0&0\\ 0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{array}\right]\left[\begin{array}{c}b_{th , in}\\ b_{e , in}\\ b_{th , in}^\dag\\ b_{e , in}^\dag\end{array}\right]\end{aligned}\ ] ]  where , and ( and ) are standard mathematical objects in i / o theory that are functions of time and may be roughly considered annihilation operators on the infinitesimal segment of free field incident on ( leaving from ) the device at time .specifically , and are associated with the fields that drive the system through the thermal mechanical port and the emc s microwave port , respectively . as with , is associated with the microwave field fluctuations about the coupling tone amplitude .moreover , is the intrinsic mechanical decay rate , ( ) is the emc coupling ( internal ) decay rate , and is the imaginary number with the electrical engineering sign convention.=\int_{-\infty}^{\infty}f(t)e^{i\omega t}dt ] where is the imaginary number . ]are unwieldy , but they also contain relatively few free parameters .linear , coherent i / o systems feature such restrictions in general , which may be traced back to fundamentally unitary dynamics ( e.g. i / o theory in the quantum stochastic differential equation formulation ) ) and are the primary distinction between coherent and more general linear i / o systems . rather than specify the fully reduced form of eqs . , we simply note that they may be written as where ^t ] , and similarly for .the matrices define the device dynamics .[ eq : emc2 ] are directly relatable to a common representation of a classical linear system known as a state space model . emulating this representation even more fully , we can represent eqs .[ eq : emc2 ] using the mathematical object , which is defined by , , and and notated \ ] ] ( each matrix and its location in the above array implies eqs . [ eq : emc2 ] ) .we can similarly generate state space model representations for the jpa and tee devices ; call them and , respectively .control system schematic of the network depicted in fig .[ fig1]b . the tee , emc and jpa components are represented as multi - input multi - output ( mimo ) linear state space models that exchange signals continuously .the interconnections are labeled here such that , e.g. stands for the two free field channels labeled and , which are explained in the text .transmission line delays may be modeled here as static phase shifts and .standard software toolkits are available that reduce the network on the left to a single mimo state space model .this approach is compatible with both quantum and classical investigations . ]we now come to the problem of constructing the network model . as discussed in refs .  andelsewhere , the formal relation between linear quantum i / o device models and classical state space models means that procedures for constructing dynamical models of linear quantum networks are analogous to those for classical linear networks . through these methods, we may derive a new state space model that describes the network as a whole . in the remainder of this section, it will be ambiguous whether we are constructing a quantum or coherent classical model .it is only when we neglect the effects of quantum fluctuation in the free fields when we compare our model to experiment that the approach becomes a classical approximation .the network model we need is depicted schematically in fig .[ figfb ] , as inspired by fig .[ fig1]b , where the emc and jpa each exchange input and output signals with two ports of the tee .the `` '' and `` '' blocks represent the phase shifts accumulated just from transmission line delays around the tee - and - emc and tee - and - jpa network loops , respectively .we thus define our network by defining the state space model objects , , and , and labeling the channels associated with each component s input and output fields according to the schematic fig . [ figfb ] . for instance, the label `` '' ( the label `` '' ) in fig .[ figfb ] is applied to both the channel through which the output field ( ) `` flows '' and the appropriate channel through which one of the inputs is driven .the label `` '' is applied to the channel through which the ( heretofore unmentioned ) field drives a input associated with the network s microwave input , etc .  while algorithmic , constructing using a linear systems approach is extremely tedious . however , software toolboxes are available in , for example , matlab and mathematica that completely automate the procedure .this basic fact is under - recognized by the physics community and deserves explicit emphasis : after defining the three state space models and labeling the component input and output channels according to fig .[ figfb ] , the entire state space model may be obtained in matlab using the single command n = connect(t , m , j ) .( other software that can deal with nonlinear quantum i / o networks has also been developed ) using such methods , these control systems toolkits determine an effective network state space model defined by matrices ( dimensions ) , ( ) , ( ) , and ( ) , ,\ ] ] which are too unwieldy to write out in general here .it is useful , though , to consider the network s equation of motion for just , the annihilation operator for the microwave fluctuations in the emc .( the network modifies the dynamics of the mo only inasmuch as and are modified ) for the emc in isolation , and taking for clarity , the equation of motion for this mode is ( eq . ) where . in the network , taking and in the limit that the jpa has no internal loss and couples much more strongly to transmission lines than the emc does , we find that where and where is the total phase shift acquired by a narrow - band signal that makes a round trip from the tee to jpa and back ( i.e. is determined by the length of that interconnection and the center frequency of the jpa ) .  from the correspondence between eqs .& , we see that the emc resonator response still acts like a single - mode resonator in this limit , except one that is now driven by the inputs `` '' and has a new detuning from the coupling tone ( ) , a new decay rate ( ) , and a new phase shift between inputs and outputs ( controlled by ) that are all controlled by , which is in turn controlled by the jpa center frequency .new effective system parameters , such as new effective decay rates and detunings , is a primary reason for constructing coherent feedback networks in general . in this case , these modifications quantitatively express the physical mechanisms qualitatively described in section [ sec : network ] .for example , the effective coupling rate comes from the interference between signals emitted by that are passed to the network output directly and those that reflect off before being passed to the same output port .the effective detuning , , represents the interference between fields inside and those that exit , but are fed back into by the network . in general , and especially when intra - jpa loss is not negligible , these same qualitative effects still hold , although they lose their quantitative accuracy , in which case the full network model is needed .if losses were negligible and the jpa bandwidth effectively infinite , the effective microwave parameters given in eq .( [ eq : primes ] ) could be substituted immediately into a traditional analysis of an electromechanical circuit , whose physical interpretations are discussed in section [ sec : network ] . eq .( [ eq : a2eff ] ) could also be extended to incorporate non - idealities with yet more complicated expressions , but this is beside the point . the modularity of the devices that make up the network permits us to use modular network techniques to construct a dynamical model and make predictions using general and efficient algorithms .analytic expressions of the full equations of motion are readily available , but they complicate matters unnecessarily here , and they are efficiently reproduced through the construction outlined above . furthermore , we argue that such an approach is often efficient , useful , and appropriate as electromechanics and quantum engineering in general begins to move beyond the physics of individual devices .  in section [ sec : results ] , we compare the network model s steady state output predictions to data . this is most naturally done through a laplace transform of the field and internal variables such that , e.g. = \int_0^\infty e^{-st}b_{\mu , in}dt\ ] ] where ( as above ) is a function of time . in the laplace domain, it is appropriate to write the network equations of motion in terms of susceptibilities and transfer functions such that &=&(si - a_n)^{-1}b_n\breve{b}^\prime_{in}[s]\equiv\chi[s]\breve{b}^\prime_{in}[s]\nonumber\\ \breve{b}^\prime_{out}[s]&=&\left(c_n(si - a_n)^{-1}b_n+d_n\right)\breve{b}^\prime_{in}[s]\equiv\xi[s]\breve{b}^\prime_{in}[s]\nonumber\\\end{aligned}\ ] ] where ^t ] , and is a identity matrix . while manually calculating the multi - input multi - output ( mimo ) transfer function  ] is applied several times in section [ sec : results ] to make predictions about the signals emitted by the network .for instance , the network s phase response to microwave probes of varying frequency is simulated in fig .[ fig2]c by calculating the phase angle of the matrix element of  ] , as runs along the imaginary axis .similarly , the absolute value squared of the matrix element of  ] , represents the gain with which -energy excitations in the thermal bath induce -energy excitations in the microwave network output ( i.e. induce blue sideband power in the network output ) , through effective -type interactions ( used in eq . ) .conversely , \right|^2 ] , the matrix element of may be used to predict the rate with which mo excitations decay out the network s blue sideband , potentially enhancing the total mo decay rate , and similar expressions may be used to predict red sideband decay and losses dissipated in the network internally ( used in fig . [ fig3]b ) .thus , the functions  ] are extremely convenient for modeling steady state network dynamics . andwhile the matrix of expressions represented by  ] is easily manipulated algebraically and computationally .we have demonstrated a small coherent feedback network of modular superconducting microwave devices that provides a type of dynamical flexibility previously unavailable to electromechanics .the network has at least three natural interpretations : a dynamically - tunable input coupler , a tunable microwave stub , or a gain-1 amplifier that feeds back coherent signals . despite the simplicity of the components and construction( coaxial cabling between pre - existing and familiar superconducting devices ) , the network is too complex to be modeled using traditional electromechanical techniques ( e.g. , such approaches assume a single resonator mode , while our network features coupled resonators ) .however , it is efficiently and intuitively modeled using coherent network techniques , an unconventional approach that will find increasing utility as electromechanics ( or quantum engineering in general ) begins to move beyond the physics of individual devices .although we only demonstrate this network s operation in the classical regime , on the basis of the well - known connections between classical and quantum dynamics in linear coherent systems , we expect that the essential mechanisms of the network should work analogously in the presence of unambiguously quantum fields and states .we have not yet probed the quantum regime because of our use of a hemt ( non - quantum limited ) amplifier for readout and the dynamical richness to be explored first in the classical regime .there are several worthwhile directions for future investigations with this same network .first , the tkc may be pumped by a third microwave tone to add a parametric gain - and - squeezing element ( or a more general kerr nonlinearity ) into the feedback and read - out dynamics .recent theoretical work suggests that quantum coherent feedback to an electromechanical system from a parametric gain controller can outperform any type of ideal measurement - based feedback or passive coherent controller .second , the ability to dynamically and continuously modulate the network s coupling to its i / o port could be leveraged to shape the waveform of signals read into and out of the emc .this capability could facilitate high - fidelity coherent state transfer between the mo and arbitrary coherent devices either `` up- '' or `` down - stream '' from the network . and finally , the model could be considered from the perspective of well - developed theories of classical optimal and robust control .such investigations are likely to yield recommendations for a more precisely - controlled network construction . in particular , different applications are helped and hindered by different round trip phase shifts over the emc and tkc network branches .while these phases were uncontrolled by the use of bulky cable interconnections in this work , a more integrated network could be constructed with much better precision .j. gough and m. r. james , _ the series product and its application to quantum feedforward and feedback networks _ , ieee trans . auto .control .* 54 * , 2530 ( 2009 ) ; _ quantum feedback networks : hamiltonian formulation _ , commun . in math .phys . * 287 * , 1109 ( 2009 ) .k. j. strm and r. m. murray , _ feedback systems : an introduction for scientists and engineers _ ( princeton university press , princeton , new jersey , 2008 ) . h. m. wiseman and g. j. milburn , _ quantum measurement and control _( cambridge university press , cambridge , england , 2009 ) .t. j. kippenberg and k. j. vahala , _ cavity optomechanics : back - action and the mesoscale _ , science * 321 * , 1172 ( 2008 ) .a. d. oconnell _ et al ._ , _ quantum ground state and single - phonon control of a mechanical resonator _ , nature ( london ) * 464 * , 697 ( 2010 ). j. d. teufel , t. donner , d. li , j. w. harlow , m. s. allman , k. cicak , a. j. sirois , j. d. whittaker , k. w. lehnert , and r. w. simmonds , _ sideband cooling micromechanical motion to the quantum ground state _ , nature ( london ) * 475 * 359 , ( 2011 ) .j. chan , t. p. mayer alegre , a. h. safavi - naeini , j. t. hill , a. krause , s. grblacher , m. aspelmeyer , and o. painter , _ laser cooling of a nanomechanical oscillator into its quantum ground state _, nature ( london ) * 478 * , 89 ( 2011 ) .a. h. safavi - naeini , j. chan , j. t. hill , t. p. mayer alegre , a. krause , and o. painter , _ observation of quantum motion of a nanomechanical resonator _ ,lett . * 108 * , 033602 ( 2012 ) .t. p. purdy , r. w. peterson and c. a. regal , _ observation of radiation pressure shot noise on a macroscopic object _ ,science * 339 * , 801 ( 2013 ) .t. a. palomaki , j. w. harlow , j. d. teufel , r. w. simmonds , and k. w. lehnert , _ coherent state transfer between itinerant microwave fields and a mechanical oscillator _ , nature ( london ) * 495 * , 210 ( 2013 ) .j. d. teufel , j. w. harlow , c. a. regal and k. w. lehnert , _ dynamical backaction of microwave fields on a nanomechanical oscillator _ ,* 101 * , 197203 ( 2008 ) .m. d. lahaye , j. suh , p. m. echternach , k. c. schwab , and m. l. roukes , _ nanomechanical measurements of a superconducting qubit _ , nature ( london ) * 459 * , 960 ( 2009 ) .j. d. teufel , t. donner , m. a. castellanos - beltran , j. w. harlow , and k. w. lehnert , _ nanomechanical motion measured with imprecision below that at the standard quantum limit _ , nature nanotechnol .* 4 * , 820 ( 2009 ) . j. b. hertzberg , t. rocheleau , t. ndukum , m. savva , a. a. clerk , and k. c. schwab , _ back - action - evading measurements of nanomechanical motion_ , nature phys .* 6 * , 213 ( 2010 ) .f. massel , s. u. cho , j .-pirkkalainen , p. j. hakonen , t. t. heikkil , and m. a. sillanp , _ multimode circuit optomechanics near the quantum limit _ , nature comm .* 3 * , 987 ( 2012 ) . c. a. regal and k. w. lehnert , _ from cavity electromechanics to cavity optomechanics _ , journal of physics : conference series * 264 * , 012025 ( 2011 ) .r. w. andrews __ , _ in preparation ._  m. a. castellanos - beltran , k. d. irwin , g. c. hilton , l. r. vale , and k. w. lehnert , _ amplification and squeezing of quantum noise with a tunable josephson metamaterial _ , nature phys . * 4 * , 929 ( 2008 ) .j. kerckhoff and k. w. lehnert , _ superconducting microwave multivibrator produced by coherent feedback _ , phys* 109 * , 153602 ( 2012 ) .  i. wilson - rae , n. nooshi , w. zwerger , and t. j. kippenberg , _ theory of ground state cooling of a mechanical oscillator using dynamical backaction _ , phys .lett . * 99 * , 093901 ( 2007 ) .f. marquardt , j. p. chen , a. a. clerk , and s. m. girvin , _ quantum theory of cavity - assisted sideband cooling of mechanical motion _lett . * 99 * , 093902 ( 2007 ) .j. e. gough , m. r. james , and h. i. nurdin , _ squeezing components in linear quantum feedback networks _ , phys .rev . a * 81 * , 023804 ( 2010 ) .m. r. james , h. i. nurdin , and i. r. petersen , _h control of linear quantum stochastic systems _ , ieee trans .53 , 1787 ( 2008 ) .h. i. nurdin , m. r. james , and i. r. petersen , _ coherent quantum lqg control _, automatica * 45 * , 1837 ( 2009 ) .r. hamerly and h. mabuchi , _ advantages of coherent feedback for cooling quantum oscillators _ , phys .* 109 * , 173602 ( 2012 ) .h. mabuchi , _ coherent - feedback quantum control with a dynamic compensator _ , phys .rev . a * 78 * , 032323 ( 2008 ) .s. iida , m. yukawa , h. yonezawa , n. yamamoto , and a. furusawa , _ experimental demonstration of coherent feedback control on optical field squeezing _ , ieee trans .* 57 * , 2045 ( 2012 ) .m. tsang and c. m. caves , _ coherent quantum - noise cancellation for optomechanical sensors _ , phys .105 * , 123601 ( 2010 ) .t. botter , d. w. c. brooks , n. brahms , s. schreppler , and d. m. stamper - kurn , _ linear amplifier model for optomechanical systems _ , phys . rev .a * 85 * , 013812 ( 2012 ) .h. m. wiseman and g. j. milburn , _ all - optical versus electro - optical quantum - limited feedback _ , phys .a * 49 * , 4110 ( 1994 ) .j. kerckhoff , h. i. nurdin , d. s. pavlichin , and h. mabuchi , _ designing quantum memories with embedded control : photonic circuits for autonomous quantum error correction _ , phys .lett . * 105 * , 040502 ( 2010 ) .  k. r. brown , j. britton , r. j. epstein , j. chiaverini , d. leibfried , and d. j. wineland , _ passive cooling of a micromechanical oscillator with a resonant electric circuit _lett * 99 * , 137205 ( 2007 ) .d. kleckner and d. bouwmeester , _ sub - kelvin optical cooling of a micromechanical resonator _ , nature ( london ) * 444 * , 75 ( 2006 ) .e. gavartin , p. verlot , t. j. kippenberg , _ a hybrid on - chip optomechanical transducer for ultrasensitive force measurements _ , nature nanotechnol .* 7 * , 509 ( 2012 ) .y. yin _ et al ._ , _ catch and release of microwave photon states _lett . * 110 * , 107001 ( 2013 ) .m. d. reed , b. .r .johnson , a. a. houck , l. dicarlo , j. m. chow , d. i. schuster , l. frunzio , r. j. schoelkopf , _ fast reset and suppressing spontaneous emission of a superconducting qubit _ , appl .. lett . * 96 * , 203110 ( 2010 ) .m. d. reed , l. dicarlo , s. e. nigg , l. sun , l. frunzio , s. m. girvin , and r. j. schoelkopf , _ realization of three - qubit quantum error correction with superconducting circuits _ , nature ( london ) * 482 * , 382 ( 2012 ) .z. leghtas , g. kirchmair , b. vlastakis , r. schoelkopf , m. devoret , and m. mirrahimi , _ hardware - efficient autonomous quantum error correction _ , arxiv:1207.0679 .v. b. braginskii , a. b. manukin , and m. y. tikhonov , _ investigation of dissipative ponderomotive effects of electromagnetic radiation _ , sov .jetp * 31 * , 829 ( 1970 ) .s. gigan , h. r. bhm , m. paternostro , f. blaser , g. langer , j. b. hertzberg , k. c. schwab , d. buerle , m. aspelmeyer , and a. zeilinger , _ self - cooling of a micromirror by radiation pressure _ , nature ( london ) * 444 * , 67 ( 2006 ) . o. arcizet , p .- f .cohadon , t. briant , m. pinard , and a. heidmann , _ radiation - pressure cooling and optomechanical instability of a micromirror _ , nature ( london ) * 444 * , 71 ( 2006 ) .r. l. hudson and k. r. parthasarathy , _ quantum ito s formula and stochastic evolutions _ ,* 93 * , 301 ( 1984 ) . c. w. gardiner and m. j. collett ,_ input and output in damped quantum systems : quantum stochastic differential equations and the master equation _a * 31 * , 3761 ( 1985 ) . c. w. gardiner and p. zoller , _ quantum noise _ ( springer - verlag , berlin , 2004 ) .

Abstract: we demonstrate a fully cryogenic microwave feedback network composed of modular superconducting devices connected by transmission lines and designed to control a mechanical oscillator coupled to one of the devices . the network features an electromechanical device and a tunable controller that coherently receives , processes and feeds back continuous microwave signals that modify the dynamics and readout of the mechanical state . while previous electromechanical systems represent some compromise between efficient control and efficient readout of the mechanical state , as set by the electromagnetic decay rate , the tunable controller produces a closed - loop network that can be dynamically and continuously tuned between both extremes much faster than the mechanical response time . we demonstrate that the microwave decay rate may be modulated by at least a factor of 10 at a rate greater than times the mechanical response rate . the system is easy to build and suggests that some useful functions may arise most naturally at the network - level of modular , quantum electromagnetic devices .


the volterra s model for population growth of a species within a closed system is given in as where is the birth rate coefficient , is the crowding coefficient and is the toxicity coefficient .the coefficient indicates the essential behaviour of the population evolution before its level falls to zero in the long term . is the initial population and denotes the population at time .this model includes the well - known terms of a logistic equation , and in addition it , includes an integral term that characterizes the accumulated toxicity produced since time zero .+ we apply scale time and population by introducing the non - dimensional variables to obtain the non - dimensional problem where is the scaled population of identical individuals at time and is a prescribed non - dimensional parameter .the only equilibrium solution of eq .( [ eq3 ] ) is the trivial solution and the analytical solution \tau ) \label{eq4}\ ] ] shows that for all if .  the solution of eq .( [ eq1 ] ) has been of considerable concern .although a closed form solution has been achieved in , it was formally shown that the closed form solution can not lead to any insight into the behaviour of the population evolution . some approximate and numerical solutions for volterra s population modelhave been reported .the successive approximations method was suggested for the solution of eq .( [ eq3 ] ) , but was not implemented . in this case , the solution has a smaller amplitude compared to the amplitude of for the case .  in , several numerical algorithms namely the euler method , the modified euler method , the classical fourth - order runge - kutta method and runge - kutta - fehlberg method for the solution of eq .( [ eq3 ] ) are obtained .moreover , a phase - plane analysis is implemented . in ,the numerical results are correlated to give insight on the problem and its solution without using perturbation techniques .however , the performance of of the traditional numerical techniques is well - known in that it using provides grid points only , and in addition , it requires a large amounts of calculations .  in , the series solution method and the decomposition methodare implemented independently to eq .( [ eq3 ] ) and to a related non - linear ordinary differential equation .furthermore , the padapproximations are used in the analysis to capture the essential behaviour of the populations of identical individuals and approximation of and exact value of for different were compared .small solved the volterra s population model by the singular perturbation method .this author scaled out the parameters of eq .( [ eq1 ] ) as much as possible and considered four different ways to do this .he considered two cases small and large .it is shown in that for the case , where populations are weakly sensitive to toxins , a rapid rise occurs along the logistic curve that will reach a peak and then is followed by a slow exponential decay . and , for large , where populations are strongly sensitive to toxins , the solutions are proportional to .  in adomian decomposition method and sinc - galerkin method were compared for the solution of the same integral equation in this paper .this showed that adomian decomposition method is more efficient and easier to use for the solution of volterra s population model .  in approach is based upon composite spectral functions approximations .the properties of composite spectral functions consisting of few terms of orthogonal functions utilized to reduce the solution of the volterra s model to the solution of a system of algebraic equations .  in chebyshev and hermite functions collocation approach were compared for the solution of volterra s population model growth model of a species within a closed system .they reduced the solution of this problem to the solution of a system of algebraic equations .  in applied two common collocation approaches based on radial basis functions to solve volterra s population model .  in a numerical method based on hybrid function approximationswas proposed to solve volterra s population model .these hybrid functions consist of block - pulse and lagrange - interpolating polynomials .also , in volterra s population growth model of a species within a closed system is approximated by collocation method based on two orthogonal functions , sinc and rational legendre functions .momani et al . and xu used a numerical and analytical algorithm for approximate solutions of a fractional population growth model , respectively .the first algorithm is based on adomian decomposition method ( adm ) with pad approximants and the second algorithm is based on homotopy analysis method ( ham ) .many problems in science and engineering arise in infinite and semi - infinite domains .different numerical methods have been proposed for solving problems on various domains such as fem , fdm and spectral methods and meshfree method .the use of the rbf is an one of the popular meshfree method for solving the differential equations . for many years the global radial basis functions such as gaussian, multi quadric , thin plate spline , inverse multiqudric and etc was used .these functions are globally supported and generate a system of equations with ill - condition full matrix.to convert the ill - condition matrix to a well - condition matrix , csrbfs can be used instead of global rbfs .csrbfs can convert the global scheme into a local one with banded matrices , which makes the rbf method more feasible for solving large - scale problem .the most popular family of csrbf are wendland functions .this function introduced by holger wendland in 1995 .wendland starts with the truncated power function which be strictly positive definite and radial on for , and then he walks through dimension by repeatedly applying the operator i. + * definition *  with we define it turns out that the functions are all supported on [ 0,1 ] . +* theorem 1 *  the function are strictly positive definite ( spd ) and radial on and are of the form ,\nonumber\\ 0 &    r>1 , \end{cases}\ ] ] with a univariate polynomial of degree .moreover , are unique up to a constant factor , and the polynomial degree is minimal for given space dimension and smoothness  .wendland gave recursive formulas for the functions for all .we instead list the explicit formulas of . + * theorem 2 * the function , have form ,\nonumber\\ & & \phi_{s,2}\doteq(1-r)_+^{l+2}[(l^2 + 4l+3)r^2+(3l+6)r+3],\nonumber\\ & & \phi_{s,3}\doteq(1-r)_+^{l+3}[(l^3 + 9l^2 + 23l+15)r^3+(6l^2 + 36l+45)r^2\nonumber\\ & & + ( 15l+45)r+15],\nonumber\end{aligned}\ ] ] where , and the symbol denotes equality up to a multiplicative positive constant .+ the case follows directly from the definition .application of the definition for the case yields ,\nonumber\end{aligned}\ ] ] where the compact support of reduces the improper integral to a definite integral which can be evaluated using integration by parts .the other two cases are obtained similarly by repeated application of . we showed the most of wendland functions in table [ table . 1 . ] .@*3|c| & smoothness & spd + & & + & & + & & + & & + & & + & & +  [ table . 1 . ]the one - dimensional function to be interpolated or approximated can be represented by an csrbf as where ,\nonumber\\ & & \xi = [ \xi_1,\xi_2 , \cdots , \xi_n ] ^t,\nonumber\end{aligned}\ ] ]  is the input , is the local support domain and are the set of coefficients to be determined . by using the local support domain, we mapped the domain of problem to csrbf local domain . by choosing interpolate nodes in domain : to summarize the discussion on the coefficients matrix , we define where : ^t,\nonumber\\ & & a=[\phi ^t(x_1 ) , \phi ^t(x_2 ) , \cdots , \phi ^t(x_n)]^t\nonumber\\ & & = \begin{pmatrix }  \phi_1(x_1)&\phi_2(x_1)&\cdots & \phi_n(x_1)\cr \phi_1(x_2)&\phi_2(x_2)&\cdots & \phi_n(x_2)\cr \vdots & \vdots & \ddots & \vdots \cr \phi_1(x_n)&\phi_2(x_n)&\cdots & \phi_n(x_n)\nonumber \end{pmatrix}.\end{aligned}\ ] ] note that , by solving the system ( [ system ] ) , the unknown coefficients will be achieved .   in the indirect method , the formulation of the problem starts with the decomposition of the highest order derivative under consideration into crbf .the obtained derivative expression is then integrated to yield expressions for lower order derivatives and finally for the original function itself .+ we approximate for solving the model by icsrbf : by using integral operator we have now , to obtain we define the residual functions by substituting eqs .( [ eq4])-([eq6 ] ) in eq .( [ eq3 ] ) the set of equations for obtaining the coefficients come from equalizing eq .( [ eq7 ] ) to zero at interpolate nodes from where is a last interpolate node and is a arbitrary parameter . applied the method presented in this paper to examine the mathematical structure of .table ( [ tab1 ] ) shows the maximum of for some and by using in comparison with exact solution and adm solution by wazwaz .the resulting graph of eq .( [ eq3 ] ) is shown in fig .( [ fig ] ) .  ) for and . ]  @*7|c| & & &&&& + & & &&&& + & & &&&& + & & &&&& + & & &&&& + & & &&&& +  [ tab1 ]  .minimum value of which is obtained with and for icsrbf . [ cols="^,^",options="header " , ]   [ tab2 ]  the local support domain and arbitrary parameter must be specified by the user.an important unsolved problem is to find a approach to determine the optimal size of .the accuracy of these csrbf depends on the choice of and . by the meaning of residual function in case of eq .( [ eq7 ] ) , we try to minimize by choosing good and parameters .we define as were  is - order legendre polynomial .table ( [ tab2 ] ) show the minimum of which is obtained with local support domain and arbitrary parameter .a method has been presented for solving volterra s population model which is an integro - ordinary differential equation , based on the compactly supported radial basis functions approximation . in this work ,we applied two common icsrbf methods on the volterra s population model without converting it to an ordinary differential equation .we used function .this function are proposed to provide an effective but simple way to improve the convergence of the solution by collocation method .as appeared from the figures , we have shown that , when the constant is small , this type of population is relatively insensitive to toxins , and when is large , population of this type are extremely sensitive to toxins .additionally , through the comparison with adm , we have showed that the icsrbf approach have good reliability and efficiency .

Abstract: in this paper , indirect collocation approach based on compactly supported radial basis function ( csrbf ) is applied for solving volterra s population model . the method reduces the solution of this problem to the solution of a system of algebraic equations . volterra s model is a non - linear integro - differential equation where the integral term represents the effect of toxin . to solve the problem , we use the well - known csrbf : . numerical results and residual norm ( ) show good accuracy and rate of convergence .   * keywords : * volterra s population model , compact support radial basis functions , collocation method . + * msc(2010 ) : * primary : 34g20 ; secondary : 34k28 .  corresponding author : member of research group of scientific computing fax : + 98 2122431653 . ]


as the wireless cellular networks evolve towards the 5 g generation , it is expected that the number of base stations ( bss ) per area will noticeably increase  , leading to _network densification_. the availability of multiple proximate and interconnected bss leads to the usage of cooperative transmission / reception techniques , commonly referred to as _ coordinated multi - point ( comp )_  .motivated by these recent trends , a transmission scheme for serving bidirectional traffic simultaneously via spatially separated hd - bss was investigated in  .the scheme emulates full duplex ( fd ) operation using two interconnected hd - bss , and is termed _ compflex _ : comp for in - band wireless full - duplex . in the initial work , the performance was analyzed through a simplified one - dimensional wyner - type deployment model .compflex can be seen as a generalization of fd , where a fd bs corresponds to compflex with interconnection distance zero .we show that the nonzero separation distance in compflex brings two benefits : ( i ) the distance between a bs and its associated mobile stations ( mss ) decreases ; and ( ii ) the distance between two interfering mss increases .this translates into improved transmission success probability in uplink ( ul ) and downlink ( dl ) .the use of in - band fd wireless transceivers  has recently received significant attention . however , due to the high transceiver complexity , fd is currently only feasible at the network infrastructure side  and the mss keep the hd transceiver mode .an in - band fd bs can serve one ul and one dl mss simultaneously , on the same frequency .other approaches to fd emulation by hd devices have been studied in the literature , such as having the transmissions in ul and dl ( partially ) overlap in time .that is , the ul and dl time slots , which conventionally should take place at separate time intervals , are now overlapping , an approach used by the rapid on - off division duplexing ( rodd ) in  .the authors of consider the physical ul and dl channels themselves , and have them overlap .compared to these approaches , compflex takes advantage of the spatial dimensions in a cellular network .  in this paper, we treat compflex in a two - dimensional scenario , with planar deployment of interconnected hd - bss . following the trend in the literature for modeling spatial randomness of network nodes ,we analyze the performance using the tools of stochastic geometry .stochastic geometry has been used in many papers in the literature to model the placement of network nodes , including fd capable ones as in  .the setup of compflex is shown in fig .[ fig : compflexsystemmodel ] , where one hd - bs working in the ul cooperates via a wired connection ( double solid line ) with another hd - bs that operates in the dl .the solid arrows indicate signals ; the dashed arrows interference .two interfering cells are also shown , and the boundaries between them are indicated by dotted lines . using the interconnection link ,the interference from the dl - bs to the ul - bs is perfectly canceled .note that , for the sake of clarity , not all interference from neighboring cells is shown .the rest of this paper is structured as follows . in sec .[ sec : systemmodel ] , we describe the system model , deployment assumptions and ms association , as well as the fd baseline scheme . the signal model and interference characteristicsare given in sec .[ sec : signalmodel ] . in sec .[ sec : analysis ] , we derive and prove the expression of the success probability for compflex in ul and dl , and explain the approximations used . the numerical results are provided and discussed in sec .[ sec : numericalresults ] , where the effects of the compflex scheme on signal and interference link distances is shown .the paper is concluded in sec .[ sec : conclusion ] .we consider a scenario where hd - bss serve hd - mss with bidirectional traffic .we assume rayleigh fading with unit mean power .the power of the channel between nodes and is written , and from the assumption of rayleigh fading we have ., means that is exponentially distributed with parameter . ]the distance between nodes and is written as .we use the pathloss model , where is the path loss exponent .we assume additive white gaussian noise ( awgn ) with power .full channel state information is assumed at all nodes .the default transmission power of a bs and ms is and , respectively .we assume that the bss are deployed according to a poisson point process ( ppp ) with intensity .the bs located at , defines a _ voronoi region _ , where is the euclidean distance .this region consists of those points in that are closer to the bs at than any other bs . from this definition ,the intersection of any two voronoi regions and is empty , when .this concept will be important when we consider the ms association in subsec .[ sub : userassociationandscheduling ] .we assume that the voronoi tessellation determines the rule by which the ms associates with the bs , both for dl and ul , and further that , at a specific time , only one ms randomly located in a voronoi cell is active .as stated previously , in compflex we assume that all nodes are hd .we define a _ compflex pair _ as two adjacent and connected hd - bss , one serving ul and the other dl traffic .the algorithm for pairing the bss works as follows :  given a deployment , we consider a finite observation window with dimensions km ( i.e. of size km ) , and choose a bs at random in this window .we then list all the _ unpaired _ neighbors of this bs , and choose one of those randomly .these two bss are then considered to be a compflex pair .the algorithm then proceeds to the other unpaired bss , and pairs the adjacent ones that are unpaired .the bs closest to the origin is called the _ typical _ bs .[ fig : compflexsnapshot ] shows one instance of the algorithm , where the compflex pairs are indicated by bold lines between the corresponding bss ( shown as triangles in the figure ) . in each compflex pair, one bs is assigned either ul or dl randomly with probability for each .the other bs is assigned the opposite traffic direction .given a ms , the bs serving that ms is denoted by .the algorithm terminates when it is no longer possible to pair any bss . by the rule of pairing , in any compflex pair the two bss must have adjacent voronoi regions .therefore , any bs whose voronoi region is surrounded by voronoi regions that belong to paired bss remains unpaired .we assign the unpaired bss either as ul or dl at random , with probability for each .  after the bs pairing , all bsshave been assigned either ul or dl . since bssare assigned either ul or dl at random , on average half of the bss are ul , the other half , dl .we have , where is the density of the compflex ul - bss , and the density of the compflex dl - bss .we recall that given a ppp with intensity , we can define a new process by independently selecting each point in with probability , resulting in a _ thinned _ ppp with intensity .we approximate the point processes of ul - bss and dl - bss as independent thinned ppps .this is only an approximation , as from the algorithm it follows that the selection of dl / ul is not independent , since a dl - bs must be adjacent to a ul - bs .given the voronoi regions defined by the bs deployment , the ms in a given region is scheduled to be served by the bs corresponding to that region . by the definition of a voronoi region , a ms is associated to one unique bs .more specifically , for every voronoi region one ms is attached to this bs .the position of the ms is chosen uniformly at random in the region .the traffic direction of the ms is matched to the corresponding bs , i.e. if the bs is ul , then the ms has ul traffic , similarly for dl .  a snapshot of the deployment and pairing of bss , along with the associated ul and dl mss , is shown in fig . [ fig : compflexsnapshot ] .   in the fd baseline scheme ,the bss are deployed according to a ppp with intensity . in each voronoi region , one ul ms and one dl ms is served .the location of the two mss attached to bs at location is chosen uniformly at random inside the voronoi region of the bs .the two mss are assumed to be hd devices .note that in this setting , the number of mss is the same as that of compflex , since each fd bs serves two mss .the ul signal to interference plus noise ratio ( sinr ) at bs is in the above , the numerator represents the ul signal . in the denominator ,the term is the interference from other dl bss , is the interference from other ul mss , and is the awgn . the interference can be written as : where and are the sets of interfering bss and mss respectively .the dl sinr at ms is given as in the above , the numerator represents the dl signal .the first term in the denominator , , is the aggregate interference from other dl bss to dl ms , and is the interference from the other ul mss .the interference terms equal where and are the sets of interfering bss and mss .we analyze the performance of compflex using the transmission success probability .this metric and its complement , the outage probability , are often used in works that analyze cellular networks through stochastic geometry . in the analysis, we consider a pair of typical bss and their associated mss .the typical ul - bs is denoted and the dl - bs , while the ul - ms and dl - ms are denoted and respectively .these bss and mss represent the performance of the entire network .we write the sinr at this bs as ( for ul ) .similarly , the dl sinr at a typical ms is written as .a transmission is successful if the sinr is not lower that the target threshold sinr at the receiver .recall that the bss are deployed according to a ppp with density .the distribution of the distance between a dl - bs and its associated ms is denoted . in deriving this distribution ,we assume that the bs is located at the origin , i.e. we consider a typical bs . the distanceis then denoted .similarly , the density of the distance between a ms and its ul - bs is written .  as stated from the assumptions , the location of the scheduled ms is uniform at random inside the voronoi region of the bs .for analytical tractability , we assume that the location of the ms can be any point in . under this assumption , the distance from the typical bs to its ms then has the cumulative distribution function ( cdf ) : this simplification is routinely made in the literature ( see e.g.  ) for analytical tractability .we assume that the ul distance cdf is the same as the dl .the numerical results confirm that this approximation is reasonable .   in this section ,we approximate the success probabilities in ul and dl for compflex . in the derivations, we assume that bss and mss are deployed according to independent ppps with density .note that we approximate the locations of the mss as a ppp , even though they are constrained to be inside the voronoi cell of their serving bs . also recall that the interference from the paired dl - bs to the ul - bs is cancelled , and this is reflected in the interference expressions in the proof .[ thm : successprobabilitycompflexuplink ] assuming independent ppp deployment of mss and bss , the success probability in ul in compflex is where and the laplace transforms of the interference from bss and mss are   we consider the ul sinr , and choose a typical bs .then we condition on the distance from the bs to the nearest ul - ms being .the success probability is the conditioned cdf of the sinr equals ( note that we drop the explicit notation of the conditioning for readability ) \nonumber \\   & = \exp \left ( -s\sigma^2 \right ) { \mathcal{l}}_{\psi}(s ) { \mathcal{l}}_{\varphi}(s ) \nonumber,\end{aligned}\ ] ] where in we have used that , and we set .we now derive the interference from the other dl - bss . in the derivation ,we condition on the distance to the nearest interfering dl - bs to be because this distance is independent from .the distance represents the approximation of the distance to the paired dl - bs , whose transmission is perfectly cancelled and this gives an upper bound .the density of interfering dl - bss is . then  \nonumber \\   & \stackrel{(a)}{= } { \mathbb{e}}_{i^\psi } \left [ \prod_{i\in \psi_{b(i ) } } { \mathbb{e}}_g \left [ \exp \left ( - s  p_b g_{i , b(i)}r_{i , b(i)}^{-\alpha } \right ) \right ] \right ]  \nonumber \\   & \stackrel{(b)}{= } \!\exp\! -2\pi\lambda_{c , d } \!\!\!\int_t^\infty \!\!\!\left(\ ! 1\!-\!{\mathbb{e}}_g \!\left [ \exp \!\left ( \!-\ ! s  p_b g_{i , b(i)}x^{-\alpha } \!\right ) \!\right ] \!\right ) x \mathrm{d}x \!\right )  \nonumber \\   &\stackrel{(c)}{= } \exp \left ( -2\pi\lambda_{c , d } \int_t^\infty \left ( \frac{s p_b x^{-\alpha}}{1+s p_b x^{-\alpha } } \right ) x \mathrm{d}x \right ) ,  \nonumber\end{aligned}\ ] ] where in we have used that the channels are independent , is from the probability generating functional ( pgfl ) of a ppp with density and , and in we rewrite using the moment generating function ( mgf ) of an exponential random variable . combining this with the probability density function ( pdf ) of the distance and using , we get eq . .using similar arguments , eq .  also can be derived .note however , that in eq . , the distance to the nearest interfering ul - ms follows the same distribution as the distance to the served ul - ms .  for dl ,recall that we approximate the interfering mss as a ppp , and this approximation implies that an interfering ul - ms could be inside the voronoi cell of the dl - ms .however , in compflex there is exactly one ms in each cell .therefore , the interference is overestimated .[ thm : successprobabilitycompflexdownlink ] assuming independent ppp deployment of mss and bss , the success probability in dl for compflex is where and the laplace transforms of the interference from bss and mss are   the proof follows similar steps as the one for ul , and so is omitted .note that the integration range of eq.  starts at , since there is no interference cancellation in dl , contrary to ul . from this, we obtain a lower bound on the success probability .   in the fd baseline , for dl ,( 9 ) in  gives the outage probability of a scenario similar to our fd baseline .the success probability can be directly derived from that equation .  in deriving the ul success probability, we can use a strategy similar to the one used for dl in  .then , the ul success probability in the fd baseline is where and the laplace transforms of the interference from bss and mss are  show the performance of compflex , and the comparison with the fd baseline schemes , using both numerical simulations and the analytical model given in the previous section .the simulation assumptions are shown in table [ tab : simulationparameters ] , where the densities are chosen comparable with  .  .simulation parameters . [ cols="^,<,^",options="header " , ]   [ tab : simulationparameters ]  we study the success probability in both ul and dl for compflex , in terms of varying the sinr threshold , and compare with fd .the bs and ms transmission powers are held constant according to the values in tab .[ tab : simulationparameters ] . the ul success probability for compflex and fd , both simulation and analytical ,is shown in fig .[ fig : successprobabilityulcomparison ] .here , the analytical curve approximates the simulated values quite closely .however , the success probability in ul is lower than dl , which can be partially explained by the ms power being lower than the bs power .  the resulting success probability for dl is shown in fig . [ fig : successprobabilitydlcomparison ] .in this figure , we can observe that the analytical derivations result in a lower bound on the success probability .this was to be expected , since the point processes in compflex are not truly ppp .however , as the figure shows , the ppp approximation is quite close and serves well as an indicator of the expected performance of compflex .we also observe that the success probability in compflex is about higher than fd , for most of the range of sinr thresholds .the explanation of this can be attributed to how compflex affects the distances of the signal and interference links .since one of the main features in compflex is that it brings the mss closer to the serving bss , we analyze and compare the cdfs of the various distances for signal and intra - cell interference links in compflex and fd . this comparison is shown in fig .[ fig : cdfdistances ] .the simulated cdf are shown as lines , while the analytical cdf using eq .are shown as markers ] . from this figure, we can observe two important points :  first , we compare the cdfs of the distances between an ul - ms and bs , and between a bs and dl - ms .we see that for both compflex and fd , the ul and dl distance curves overlap .this implies that the distances of ul and dl follow the same distribution .what is also interesting is that the cdfs of the distances in compflex are shifted to the _ left _ , compared to fd .this means that the lower ms to bs distances have higher probability in compflex compared to fd .second , the cdf curve of the intra - cell interference distance in compflex is shifted to the _ right _ compared to fd .this means that higher interference distances have higher probability in compflex compared to fd .taken together , these two points can explain the performance advantages of compflex over fd , which come from having a lower signal distance and a higher interference distance _in this work , we have analyzed the performance of compflex in a planar network setting , and compared it with a fd baseline scheme .we have derived the success probability for ul and dl , and validated the results via simulations .it was observed that the success probability of both ul and dl was higher for compflex than in fd , an effect which can be attributed to the effects of compflex on the node distances .thus it is beneficial to consider the usage of hd devices instead of fd , which is also useful for using already existing technology and avoiding the signal complexities of fd .one promising research direction is to consider more than two connected bss , and more general clustering criteria .also , it would be interesting to compare compflex with other interference mitigation techniques such as comp .this work was supported by innovation fund denmark , via the virtuoso project .

Abstract: a full duplex base station ( fd - bs ) can be used to serve simultaneously two half - duplex ( hd ) mobile stations ( mss ) , one working in the uplink and one in the downlink , respectively . the same functionality can be realized by having two interconnected and spatially separated half duplex base stations ( hd - bss ) , which is a scheme termed _ compflex _ ( comp for in - band wireless full duplex ) . a fd - bs can be seen as a special case of compflex with separation distance zero . in this paper we study the performance of compflex in a two - dimensional cellular scenario using stochastic geometry and compare it to the one achieved by fd - bss . by deriving the cumulative distribution functions , we show that compflex brings bss closer to the mss they are serving , while increasing the distance between a ms and interfering mss . furthermore , the results show that compflex brings benefits over fd - bs in terms of communication reliability . following the trend of wireless network densification , compflex can be regarded as a method with a great potential to effectively use the dense hd deployments .   full duplex , cellular communications , clustering , network densification , spatial model .