Ultimate squeezing through coherent quantum feedback: A fair comparison with measurement-based schemes
aa r X i v : . [ qu a n t - ph ] O c t Ultimate squeezing through coherent quantum feedback:A fair comparison with measurement-based schemes
Alfred Harwood and Alessio Serafini Department of Physics & Astronomy, University College London,Gower Street, WC1E 6BT, London, United Kingdom (Dated: October 17, 2019)We develop a general framework to describe interferometric coherent feedback loops and provethat, under any such scheme, the steady-state squeezing of a bosonic mode subject to a rotating wavecoupling with a white noise environment and to any quadratic Hamiltonian must abide by a noise-dependent bound that reduces to the 3dB limit at zero temperature. Such a finding is contrasted,at fixed dynamical parameters, with the performance of homodyne continuous monitoring of theoutput modes. The latter allows one to beat coherent feedback and the 3dB limit under certaindynamical conditions, which will be determined exactly.
Introduction –
Feedback is one of the main avenues to ex-ert and refine control on physical systems. In quantummechanics, feedback control may be applied in two, radi-cally different, fashions: as measurement-based feedback[1], where measurements are used to purify the systemand to inform operations on it, or as coherent feedback[2], where only deterministic manipulations of subsys-tems coupled to the system of interest (part of the latter’senvironment) are performed. While in measurement-based feedback the quantum information is turned intoclassical information by the act of measuring, in coherentfeedback the information stays quantum at all stages ofthe control loop.It might be argued that, since the manipulations in-volved are deterministic, coherent feedback loops shouldnot be considered as feedback control at all, but ratheras a class of open-loop control strategies where only cer-tain auxiliary degrees of freedom are accessible. However,quantum optics allows us to disregard such a terminolog-ical dispute (although the discussion of this issue in [3]is worth mentioning), through the adoption of the input-output formalism, which is tailored to describe the inter-action of a countable set of localised modes (e.g., a setof cavity modes) with a neighbouring field’s continuum(the electromagnetic field outside a cavity). We shallthen, as customary in the context of quantum optics, de-fine a coherent feedback loop as one where a set of outputmodes, interacting with a system at an input-output in-terface, may be manipulated through quantum CP-mapsand then fed back into a system as input modes at an-other input-output interface. This approach is similar tothat used in the established field of ‘cascaded quantumsystems’, where the output of one system is used as theinput of another [4–6], and of the related “all optical”feedback [7], the difference being that here the output isfed back into the system whence it came.Notice that the input-output paradigm finds successfuland broad application to a number of quantum set-upswhere a high degree of coherent control is achievable,ranging from purely optical set-ups to optomechanics [8], nanoelectromechanics, atomic ensembles, and cav-ity QED waveguides [9], to mention but a few. Giventhe impressive recent advances in the realisation of quan-tum technologies, connectivities (especially via fibres andwaveguides) are quickly nearing a point where quantumcontrol loops will be feasible and pivotal in harnessingquantum resources, such as quantum coherence and en-tanglement. Indeed, coherent control loops have beendemonstrated in optical [10] and solid-state [11] systems,whilst measurement-based feedback has been by now ap-plied to a variety of systems, with the aim of performingquantum operations, of enhancing cooling routines [12]or of entangling quantum systems [13]. It is thereforeparamount to understand the ultimate limits of feedbackstrategies as well as which class of loops, coherent ormeasurement-based, is advantageous to perform a giventask or optimise a given figure of merit.After the seminal study [7] – which, at variance withthe present enquiry, does not deal with a squeezingHamiltonian acting on the system, but rather with moregeneral forms of coupling between system cavity and thefeedback loop, such a theoretical comparison has beenaddressed only for specific tasks in finite-dimensional sce-narios [3, 14], or for the realisation of protocols involvingout-of-loop degrees of freedom [15] (i.e., concerning therelationship between input and output degrees of free-dom). Treatment [15], in particular, adopts a frame-work that is wholly analogous to ours, and establishesa few remarkable impossibilities for measurement-basedfeedback. In our study, the input-output formalism,compounded with general linear operations, will yield aframework for a fair comparison between the two classesof control schemes, which may be contrasted at givenconnectivities and other technical and environmental pa-rameters (such as detection efficiencies and temperature).Note that the set of operations encompassed by co-herent and measurement-based feedback differ, since thestochastic dynamics originating from measurements can-not be reduced to deterministic operations, which isthe essence of the so called “measurement problem”of quantum mechanics. Nevertheless, coherent feed-back has been proven superior in a number of tasksand contexts, so that our ability to demonstrate situa-tions where measurement-based schemes do in principleprove superior is all the more striking and consequential.Measurement-based strategies, it turns out, prove partic-ularly effective in stabilising in-loop figures of merit (i.e.,quantities pertaining to the localised modes).Specifically, in this paper we shall consider a singlebosonic mode coupled to a white noise continuum atfinite temperature through the input-output formalism.We shall assume the mode to be subject to a squeezingHamiltonian and, as a significant case study, shall adoptthe optimisation of steady-state squeezing as our figure ofmerit. First, as proof of principle, we shall present a sim-ple coherent feedback loop using a single feedback modesubject to losses, show that it can enhance the achiev-able squeezing, and contrast it with what is achievablethrough (feasible) homodyne measurements of the out-put field. Then, we consider the most general possiblecoherent feedback setup, letting an arbitrary number ofoutput modes at one interface undergo the most generaldeterministic Gaussian CP-map not involving any sourceof squeezing (i.e., the most general open, passive opti-cal transformation, corresponding in practice to leakage,beam splitters and phase shifters), before being fed backinto the system. This will prove that the simple schemewe considered is indeed optimal, and that our comparisonis therefore conclusive.
Continuous variable systems –
A system of n bosonicmodes can be described as a vector of operators ˆ r =(ˆ x , ˆ p ... ˆ x n , ˆ p n ) T . These obey canonical commutation re-lations (CCR) [ˆ x i , ˆ p j ] = iδ ij ˆ where we have set ~ = 1.The CCR for multiple modes can be described usingthe symmetrised version of the commutator [ˆ r , ˆ r T ] =ˆ r ˆ r T − (ˆ r ˆ r T ) T = i Ω n where Ω n is a 2 n × n matrixknown as the symplectic form: Ω n = L nj =1 Ω , withΩ = (cid:18) − (cid:19) . In the rest of this paper, we will omit thesubscript from Ω, letting the context specify the appro-priate dimension. For a quantum state ˆ ρ , the expectationvalue of the observable ˆ x is given by h ˆ x i = Tr[ˆ ρ ˆ x ]. Usingvector notation, this can be generalised to give the firstand second statistical moments of a state: ¯ r = Tr[ˆ ρ ˆ r ] σ = Tr[ { (ˆ r − ¯ r ) , (ˆ r − ¯ r ) T } ˆ ρ ] . (1)The above definition leads to a real, symmetric covari-ance matrix σ .The steady-states we will focus on are Gaussian states,which may be defined as the ground and thermal statesof quadratic Hamiltonians. Such states are fully char-acterised by first and second statistical moments, as de-fined above. Unitary operations which map Gaussianstates into Gaussian states are those generated by aquadratic Hamiltonians. The effect of such operations on the vector of operators is a symplectic transforma-tion ˆ r → S ˆ r where S is a 2 n × n real matrix whichsatisfies S Ω S T = Ω. The corresponding effect on thecovariance matrix of the system is the transformation σ −→ S σ S T . In this study, we will make use of so-called‘passive’ transformations, which do not add any energy tothe system and therefore do not perform any squeezing.Passive transformations must satisfy the extra constraintthat S is orthogonal, ie. SS T = . The Input-Output Formalism –
The input-output for-malism is a method for dealing with the evolution ofsystems coupled to a noisy environment, consisting ofa continuum of modes (e.g., the free electromagneticfield). The interaction of the system with such an en-vironment can be modelled as a series of instantaneousinteractions with different modes at different times [16].The incoming mode which interacts with the system attime t is known as the input mode and is labelled ˆ x in ( t ),The mode scattered at time t is labelled ˆ x out ( t ) and isknown as the output mode. The input modes satisfythe continuous CCR: [ˆ r in ( t ) , ˆ r T in ( t ′ )] = i Ω δ ( t − t ′ ) whereˆ r in ( t ) = (ˆ x in, , ˆ p in, ... ˆ x in,m , ˆ p in,m ) T . The coupling of thesystem to the input fields is given by a Hamiltonian ˆ H C :ˆ H C = 12 ˆ r T SB H C ˆ r SB = 12 ˆ r T SB (cid:18) CC T (cid:19) ˆ r SB , (2)where ˆ r SB = (ˆ x , ˆ p , ... ˆ x n , ˆ p n , ˆ x in, , ˆ p in, ... ˆ x in,m , ˆ p in,m ) T is the total vector of the n system modes and m bathmodes. The square matrix H C is the coupling Hamil-tonian matrix and C is a 2 n × m matrix known as thecoupling matrix. The Heisenberg evolution of the systemoperators is given by a stochastic differential equationknown as the quantum Langevin equation [17]:dˆ r ( t ) = A ˆ r ( t ) d t + Ω C ˆ r in ( t ) d t (3)The matrix A is known as the drift matrix of the systemand is given by A = Ω n H S + Ω C Ω C T . The symmet-ric square matrix H S specifies the system Hamiltonianˆ H S = ˆ r T H S ˆ r . The vector ˆ r in ( t ) is a stochastic pro-cess known as a quantum Wiener process which, in anal-ogy with the classical Wiener process, obeys the relations[ˆ r in ( t ) , ˆ r T in ( t )](d t ) = i Ω d t and h{ ˆ r in ( t ) , ˆ r T in ( t ) }i (d t ) = σ in d t where σ in is the covariance matrix of the inputmodes. This relationship implies delta correlations be-tween bath modes interacting at different times (the well-known “white noise” condition) and hence the Marko-vianity of the free dynamics, which we are thus assuming.Eqs. (1) and (3) can be combined to obtain an equationfor the evolution of the system covariance matrix:˙ σ = A σ + σ A T + D , (4)where D = Ω C σ in C T Ω T is known as the diffusion ma-trix. The condition required for Eqs. (4,3) to admit asteady-state (stable) solution is that the matrix A mustbe ‘Hurwitz’, meaning that all of the real parts of itseigenvalues are negative. If this condition is satisfied,then the steady-state solution reads σ ∞ = Z ∞ e At De A T t d t . (5) Squeezing with no control –
Squeezing is the process ofreducing the variance of one quadrature and correspond-ingly increasing the variance of its conjugate. Our figureof merit for this study is σ , the element of the covari-ance matrix corresponding to twice the variance of theˆ x -quadrature. The smaller the value of σ , the moresqueezed the system is. We consider a single bosoniccavity mode, subject to the Hamiltonian ˆ H = ˆ H S + ˆ H C ,where ˆ H S = − χ { ˆ x, ˆ p } / χ > x quadrature. This corresponds to aHamiltonian matrix H S = − χ σ x , where σ x is the Pauli x -matrix. Losses in the cavity are modeled by couplingthe cavity mode to an external fields through the Hamil-tonian ˆ H C that allows for the exchange of excitations:ˆ H C = √ γ (ˆ p ˆ x in − ˆ x ˆ p in ) , (6)This corresponds to a coupling matrix C = √ γ Ω T where γ is the strength of the coupling. When this is the form ofsystem-environment coupling, the so-called input-outputboundary condition relates the system modes to the inputand output as follows [16]:ˆ r out ( t ) = √ γ ˆ r ( t ) − ˆ r in ( t ) . (7)For this example and for the remainder of this paper, wewill consider the case when all input fields are in Gibbsthermal states of the free Hamiltonian, so that σ in = ¯ N ,where ¯ N = 2 N + 1 and N ≥ σ = ¯ N γ/ ( χ + γ ). The condition for stabilityis that | χ | < γ , which means that, if the loss rate orsqueezing parameter can be tuned, and the input fieldsare taken to be vacua (so ¯ N = 1) the maximum steady-state squeezing that can be achieved is σ = . This isknown in the literature as the 3dB limit, as 10 log (2) ≈ .
01 (this is the noise, in decibels, associated with thesmallest eigenvalue of σ in units of vacuum noise).We note that steady-state squeezing could be improvedupon if the input state were squeezed, but in this studywe will only consider naturally occurring, non-squeezedreservoirs (as opposed to squeezed ones, which have onlybeen envisaged through very demanding engineering),also in view of comparing different feedback strategiesunder the assumption that the only squeezing source isconstituted by the system Hamiltonian with a certainstrength χ . Homodyne Monitoring –
In order to provide the readerwith a comparison between measurement-based and co- herent feedback, let us also consider the continuous mon-itoring of the output ˆ x -quadrature through a homodynedetector with efficiency ζ , which yields a relevant elementof the covariance matrix given by [18]: σ m = a + √ a + b ζ , (8)for a = [2 ¯ N ζ − (1 + ( ¯ N − ζ )(1 + χγ )] and b = 4 ¯ Nζ (1 − ζ ).This monitoring maximises the steady-state squeezingamong all general-dyne detections at zero temperature,i.e., for ¯ N = 1 [19], but is beneficial at finite temperaturetoo (for ¯ N >
Simple Coherent Feedback –
As a preliminary piece of in-quiry, let us report the treatment [17], and examine theperformance of the simplest possible coherent feedbackloop by feeding the output of one interface into the inputof the other after undergoing losses. To do this we willconsider a system mode coupled to two input fields, eachthrough a Hamiltonian of the form given in (6). To avoidambiguity, we will use a subscript e to refer to environ-mental white noise modes, and the the subscript in whenreferring to the input interacting with the system throughthe Hamiltonian given in (6). Adding coherent feedbackinvolves setting ˆ r in, = ˆ r e, and ˆ r in, ( t ) = Φ(ˆ r out, ( t )),where Φ is the CP-map corresponding to losses. Theselosses can be modelled as mixing at a beam splitter withan environmental mode ˆ r e, . This means that coherentfeedback can be achieved by setting :ˆ r in, = √ η ˆ r out, + p − η ˆ r e, = √ η ( √ γ ˆ r − ˆ r e, ) + p − η ˆ r e, , (9)where η is the loss rate and we have used the input out-put relation (7). It is important to note that we are as-suming instantaneous feedback, with no delays betweenthe mode put out at interface 1 and fed back at inter-face 2, which will preserve the Markovianity of the dy-namics. Making this substitution into Eq. (6) results inthe system ˆ r being effectively coupled to the environ-ment ˆ r e,tot = (ˆ r T e, , ˆ r T e, ) T through the coupling matrix C = √ γ (1 − √ η Ω T , √ − η Ω T ). Such a system requires γ (1 − √ η ) > χ in order to be stable. The steady-state squeezing achieved in these conditions is σ = ¯ Nγ (1 −√ η ) χ + γ (1 −√ η ) , which is minimised by letting √ η → − χ γ ,resulting in a squeezing of σ → ¯ N . Thus, at zero tem-perature (i.e., for ¯ N = 1), coherent feedback allows the3dB limit to be approached (but not beaten) for any choice of parameters satisfying < χ γ <
1. Notice that, regard-less of χ , no stable squeezing is achievable if ¯ N ≥ ζ of the detector satisfies ζ ≥ γ − χ )2 ( γ − χ ) + ¯ N (2 χ − γ ) (10)and the denominator of the RHS of (10) is positive. For χ < γ/
2, either the denominator is negative or the boundabove is larger than 1, which proves that homodyne mon-itoring does not beat coherent feedback at such weakinteraction strengths. For χ ≥ γ/
2, there is always adetection efficiency threshold above which the coherentfeedback loop we considered is outperformed by homo-dyne monitoring; this threshold, quite interestingly, de-creases with increasing noise (although the absolute per-formance of monitoring at given χ still deteriorates as thenoise increases). As the upper limit for stability χ = γ is approached, the efficiency threshold falls to zero, sothat detection with any efficiency will be better than ourcoherent loop in this limit. The ultimate performanceof homodyne monitoring is obtained at ζ = 1, where σ m = ¯ N (1 − χ/γ ): hence, monitoring can in princi-ple achieve stable squeezing ( σ m <
1) for all values of¯ N , although only for χ > γ (1 − / ¯ N ). Notice that, inprinciple, arbitrarily high squeezing may be stabilised atall noises (temperatures), whereas the coherent feedbackloop we studied is bounded by the value ¯ N /
2. In orderto achieve a conclusive comparison between coherent andmeasurement-based loops, we need to extend our treat-ment beyond a specific coherent feedback loop to includeany possible interferometric scheme without additionalsources of squeezing.
General Passive Coherent Feedback –
Let us thereforeconsider the most general possible coherent feedback pro-tocol which does not include any extra source of squeez-ing. A single system mode is coupled to l + m inputmodes through a coupling Hamiltonian of the same formas (6). We will give the label a to the modes interact-ing at the first l input-output interfaces, and the label b to the modes interacting at the remaining m interfaces.The first l input modes are environmental white noise,meaning we can write ˆ r in,a = ˆ r e,a = (ˆ r T e, ... ˆ r T e,l ) T . Thecorresponding output modes then undergo the most gen-eral Gaussian CP-map which does not include any form ofsqueezing. This is achieved by applying a passive trans-formation on the output modes, along with n ancillarywhite noise modes before tracing out the ancillas. Afterthe transformation, the resulting m modes are then fedinto the remaining m input interfaces of the system. We shall assume that the additional ancillary modes are alsoaffected by the same thermal noise as the environment,so that it is still σ in = ¯ N [21]. Let us stress that, byconsidering the most general passive symplectic transfor-mation, the formalism below will allow for an elegant andcompact description of the most general interferometricscheme mediating the coherent feedback loop.The passive transformation on the output and ancillamodes can be represented as an orthogonal, symplectic,2( l + n )-dimensional, square matrix Z :ˆ r in,b ⊕ ˆ r anc,f = Z (ˆ r out,a ⊕ ˆ r anc,i ) , (11)where ˆ r in,b = (ˆ r T in, ( l +1) . . . ˆ r T in, ( l + m ) ) T and ˆ r out,a =(ˆ r T out, . . . ˆ r T out,l ) T . The initial and final states of the an-cilla modes are given by ˆ r anc,i and ˆ r anc,f respectivelywhere ˆ r anc,i = (ˆ r T e, ( l + m +1) ... ˆ r T e, ( l + m + n ) ) T The orthogonal symplectic matrix Z can be decom-posed into block matrices Z = ( E FG H ). This representa-tion of Z allows us to write ˆ r in,b = E ˆ r out,a + F ˆ r anc,i .It is shown in [18] that the overall effect of this coher-ent feedback protocol is to couple the system mode tothe white-noise environment, now given by ˆ r e,a ⊕ ˆ r anc,i ,through the coupling matrix: C cf = (cid:0) C l − C m E | C m F (cid:1) , (12)where C j indicates a 2 × j dimensional matrix of the form √ γ (Ω T . . . Ω T ). It is also shown that adding coherentfeedback modifies the system Hamiltonian matrix H S byaddition of a matrix: H cf = H S + C m E Γ l + Γ T l E T C T m , (13)where Γ l is a 2 l × l = √ γ ( . . . ) T . Optimal Coherent Feedback for Squeezing –
Workingwithin this general framework of coherent feedback, wewill now find the optimal steady-state squeezing achiev-able. In particular, we will show that no coherent feed-back protocol can improve upon the 3dB squeezing limit,for any choice of quadratic system Hamiltonian. This re-sult will be outlined here, with some details left to [18].We are after the smallest eigenvalue of σ ∞ , as given byEq. (5), with D and A which are modified by the coherentfeedback loop as per Eqs. (12,13) (recall that A and D arein turn functions of H S , C and σ in ), which is equivalentto the smallest value of v † σ ∞ v for a normalised vector v . It may be shown that [18], for any coherent feedbackloop, the diffusion matrix is proportional to the identityand therefore has only a single eigenvalue given by δ =¯ N γ ( l + m − ǫ ) where ǫ = P jk E jk is the sum over the11-elements of each 2 × E jk of E .This simplifies our task greatly, since it is now apparentthat the smallest value of v † σ ∞ v is obtained by setting v = λ where λ is the normalised eigenvector of A cor-responding to the eigenvalue λ with most negative realpart: indeed, this choice minimises the positive integrand v † e A T t e At v at all times t . Whence the bound v † σ ∞ v ≥ δ Z ∞ e ( λ ∗ + λ ) t d t = − δλ ∗ + λ . (14)How negative the eigenvalue of A can be made is limitedby the stability criterion, since making one eigenvaluemore negative makes the other less negative. To ensurestability, the most negative eigenvalue of A must satisfy λ > γ (2 ǫ − l − m ) [18]. In deriving this bound, thesystem Hamiltonian was assumed to be quadratic, butotherwise completely general. This takes into accountthe modifications made to the system Hamiltonian dueto the coherent feedback, as well as any deliberate tuning.Inequality (14) along with the expressions for δ and λ yield v T σ ∞ v > ¯ N . (15)This result shows that there exists no combination ofquadratic Hamiltonians and coherent feedback protocolwhich can beat the 3dB limit. Therefore, our compar-ison with the measurement-based strategy extends be-yond the specific example we made above. Summarising,we have found that, in regard to the squeezing Hamilto-nian − χ { ˆ x, ˆ p } , coherent feedback loops are superior for χ < γ/ χ ≥ χ/ and efficienciessatisfying (10). Our comparison is definitive at zero tem-perature, for ¯ N = 1, in the sense that both measurement-based and coherent feedback were fully optimised for vac-uum input noise (homodyning is then optimal), and thatat optical frequencies one has ( ¯ N − ≈ − .It is also worth noting that our result hinges on thephase-insensitive nature of the input-output coupling (6),which implies a diffusion matrix D proportional to theidentity, and would not apply, for instance, to a quan-tum Brownian motion master equation. In this regard,our finding may be considered as an extension of the well-known 3dB squeezing limit that affects phase-insensitiveamplifiers [22], which we showed to bound in-loop, stable squeezing too under any interferometric, coherent feed-back scheme and any system quadratic Hamiltonian. No-tice that stability is another essential ingredient in estab-lishing the bound, as unstable coherent feedback loopswould be able to achieve higher squeezing (but are typi-cally not desirable in practice). Conclusions and Summary –
We have developed a gen-eral framework for passive coherent feedback in theGaussian regime and shown that no protocol withinthis framework can beat the 3dB squeezing limit atsteady state. In contrast, homodyne monitoring of out-put fields can stabilise arbitrarily high squeezing at lowenough noise and provided that detection efficiency ishigh enough. The general treatment developed here provides thegroundwork for further inquiries on passive coherent feed-back, which may be extended to the optimisation of en-tanglement in multimode systems, to more general noisemodels, as well as applied to the cooling of concrete sys-tems, such as quantum optomechanics [23].We acknowledge discussions with M. Brunelli, whomade us aware of additional literature, and M. Genoni,who flagged inaccuracies in the manuscript. [1] H. M. Wiseman and G. J. Milburn,
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HOMODYNE MONITORING AT FINITE TEMPERATURE
Continuous, general-dyne monitoring of the output field turns the diffusive equation (4) into the following Riccatiequation (see, e.g., [17] for a complete treatment of the theory):˙ σ = ˜ A σ + σ ˜ A T + ˜ D − σ BB T σ , (16)for ˜ A = A + GB T , ˜ D = D − GG T , (17) B = C Ω( σ in + σ m ) − / G = Ω C σ in ( σ in + σ m ) − / , (18)where the covariance matrix σ m parametrises the choice of measurement. We will consider the homodyne detectionof the output field of a single-input single-output system as described in the section titled ‘ Squeezing with no control ’.Homodyne detection of the ˆ x quadrature with efficiency ζ is obtained by setting σ m = lim z → z +1 − ζζ z +1 − ζζ ! , (19)which leads to a diagonal quadratic equation for the monitored steady state covariance matrix, whose diagonalelements σ m and σ m must satisfy (the off-diagonal elements vanish) γζζ ( ¯ N −
1) + 1 σ m + (cid:18) γ + χ − γζ ¯ Nζ ( ¯ N −
1) + 1 (cid:19) σ m + γζ ¯ N ζ ( ¯ N −
1) + 1 − γ ¯ N =0 , (20)( γ − χ ) σ m − γ ¯ N =0 , (21)with physical solutions σ m = 2 ¯ Nζ − (1 + ( ¯ N − ζ )(1 + χγ ) + q [2 ¯ Nζ − (1 + ( ¯ N − ζ )(1 + χγ )] + 4 ¯ Nζ (1 − ζ )2 ζ , (22) σ m = ¯ N − χγ . (23)The other solution for σ m must be discarded since it is negative or zero, and would thus violate the strict positivity of σ m , stemming from the uncertainty principle. The solution for σ m shows that, even under monitoring, the condition | χ | < γ is necessary for stability. EFFECTIVE COUPLING MATRIX FOR GENERAL PASSIVE COHERENT FEEDBACK
The Hamiltonian which couples system and input modes can be written as:ˆ H C = 12 ˆ r T tot H C ˆ r tot , (24)where ˆ r tot = (ˆ r T , ˆ r T in,a , ˆ r T in,b ) T . There are l input modes at a and m input modes at b , so ˆ r tot is a (2 + 2 l + 2 m )-dimensional vector. The coupling Hamiltonian matrix takes the form: H C = C l C m C T l C T m , (25)where C j indicates a 2 × j matrix of the form √ γ (Ω T . . . Ω T ). This allows for an exchange of excitations betweensystem and input field. When no coherent feedback is present, both ˆ r in,a and ˆ r in,b are white noise environmentalmodes. When coherent feedback is included, the input modes at a are still white noise. We will call these modes ˆ r e,a to indicate this. The output modes at a undergo a passive Gaussian CP-map and then are used to replace ˆ r in,b .The passive Gaussian CP-map is achieved by performing a passive symplectic operation on the joint state ˆ r out,a ⊕ ˆ r anc,i where ˆ r anc,i is a 2 n -dimensional vector representing the initial state of n environmental white noise modes. Theresulting mode to be input at interface b , along with the final state of the ancilla mode can be written (ˆ r in,b ⊕ ˆ r anc,f ) = Z (ˆ r out,a ⊕ ˆ r anc,i ). The 2( l + n )-dimensional square matrix Z is symplectic, which ensures that the linear operationis physical, and orthogonal, which ensures that the operation is passive (i.e., that it does not involve any squeezing).We can write Z in terms of block matrices: Z = (cid:18) E FG H (cid:19) . (26)Once the ancilla modes have been traced out, the effect of the CP-map can be written as ˆ r out,a → E ˆ r out,a + F ˆ r anc,i which allows us to write (note that E and F are, respectively, 2 m × l and 2 m × n matrices):ˆ r in,b = (cid:0) E F (cid:1) (cid:18) ˆ r out,a ˆ r anc,i (cid:19) . (27)We now write the matrix form of the multimode input-output boundary condition in order to write ˆ r out,a in termsof ˆ r in,a = ˆ r e,a : (cid:18) ˆ r out,a ˆ r anc,i (cid:19) = (cid:18) Γ l − l
00 0 n (cid:19) ˆ r ˆ r e,a ˆ r anc,i with Γ l = √ γ ... . (28)Combining the above equations, we obtain: ˆ r ˆ r in,a ˆ r in,b = E F l −
00 0 ˆ r ˆ r e,a ˆ r anc,i = L ˆ r ˆ r e,a ˆ r anc,i . (29)The effect of adding coherent feedback is therefore to couple the system to a white noise environment given by(ˆ r T e,a , ˆ r T anc,i ) T through a coupling Hamiltonian characterised by the matrix H cfC = L T H C L . This matrix is: H cfC = C m E Γ l + Γ T l E T C T m C l − C m E C m FC T l − E T C T m F T C T m , (30)which couples the system to the environment through the Hamiltonian operatorˆ H cfC = 12 (ˆ r T , ˆ r T e,a , ˆ r T anc,i ) H cfC (ˆ r , ˆ r e,a , ˆ r anc,i ) . (31)Notice that this results in a matrix equal to C m E Γ l + Γ T l E T C T m being added to the system Hamiltonian matrix andchanges the effective coupling matrix to: C cf = ( C l − C m E | C m F ) . (32) PROPERTIES OF THE ORTHOGONAL SYMPLECTIC MATRIX
We have considered an orthogonal symplectic matrix of the form Z = (cid:18) E FG H (cid:19) , (33)which transformed a vector of operators as per ˆ r Z ˆ r . Here, E is a (2 m × l ) matrix and F is a (2 m × n ). Thecondition of orthogonality means that ZZ T = , which gives us the following conditions on the submatrices: ZZ T = (cid:18) E FG H (cid:19) (cid:18) E T G T F T H T (cid:19) = (cid:18) EE T + F F T EG T + F H T GE T + HF T GG T + HH T (cid:19) = (cid:18) (cid:19) (34)In particular, we shall make use of the relation EE T + F F T = . The condition of symplecticity means that Z Ω Z T = Ω. Recall that we are using the convention that the dimension of Ω is specified by the context. In terms ofthe submatrices, this means that: Z Ω Z T = (cid:18) E FG H (cid:19) (cid:18)
Ω 00 Ω (cid:19) (cid:18) E T G T F T H T (cid:19) = (cid:18) E Ω E T + F Ω F T E Ω G T + F Ω H T G Ω E T + H Ω F T G Ω G T + H Ω H T (cid:19) = (cid:18) Ω 00 Ω (cid:19) . (35)From this we obtain the condition E Ω E T + F Ω F T = Ω, which will be key later.The vector of operators ˆ r was ordered so that ˆ r = (ˆ x , ˆ p ... ˆ x n , ˆ p n ) T . We can also consider an orthogonal symplecticmatrix S acting on a vector of differently ordered operators: ˆ s −→ S ˆ s where ˆ s = (ˆ x ... ˆ x n , ˆ p ... ˆ p n ). In this case, thetransformation matrix takes the form [17]: S = (cid:18) X Y − Y X (cid:19) with XY T − Y X T = 0 n and XX T + Y Y T = n . (36)When we use the ordering of variables ˆ s = (ˆ x . . . ˆ x n , ˆ p . . . ˆ p n ) T , the symplectic condition is SJS T = J , where J isthe symplectic form J = (cid:18) n n − n n (cid:19) . (37)Transforming between the two representations means that we can write each 2 × Z as: Z = Z . . . Z n ... . . . ... Z n . . . Z nn Z jk = (cid:18) x jk y jk − y jk x jk (cid:19) , (38)where x jk and y jk are the elements of matrices X and Y respectively. This fact will be used later. EIGENVALUES OF THE DRIFT MATRIX
The drift matrix A can be expressed in terms of the Hamiltonian and coupling matrices H S and C as A = Ω H S + 12 Ω C Ω C T . (39)Note that C Ω C T is an 2 × T = − Ω. Therefore, for a single mode, Ω C Ω C T isproportional to the identity. We shall set Ω C Ω C T = β . Also, since H S is a symmetric matrix, T r [Ω H S ] = 0,meaning that the eigenvalues of Ω H S can be written as ± h and the eigenvalues of A can be written λ = β ± h .We will now find the value of β in the coherent feedback framework, where Eq. (32) determines the coupling matrix.We use the notation Γ k from earlier to indicate a 2 k × k = √ γ ( . . . ) T . Thissatisfies Ω C k = Γ T k and Ω C T k = − Γ k . We can writeΩ C cf Ω C T cf = (Ω C l − Ω C m E )(Ω C T l − Ω E T C T m ) + Ω C m F Ω F T C T m = − Γ T l Γ l − Γ T l Ω E T C T m + Γ T m E Γ l + Γ T m E Ω E T C T m + Γ T m F Ω F T C T m . (40)We now use the symplectic property E Ω E T + F Ω F T = Ω, derived in the previous section, to writeΩ C cf Ω C T cf = − Γ T l Γ l − Γ T l Ω E T C T m + Γ T m E Γ l + Γ T m Ω C T m (41)(recall that the notation Ω refers to symplectic forms of different dimension, as appropriate for matrix multiplicationsto be consistent). Noting now that Γ T k Γ k = kγ , one hasΩ C cf Ω C T cf = − ( l + m ) γ − Γ T l Ω E T C T m + Γ T m E Γ l , (42)which can be written in terms of the 2 × E : E = E . . . E l ... . . . ... E m . . . E ml E jk = (cid:18) e jk e jk e jk e jk (cid:19) . (43)The matrix Γ T m E Γ l can be written as γ P i,j E jk , while Γ T l Ω E T C T m can be calculated in the same way:Γ T l Ω E T C T m = √ γ Γ T l Ω P mj =1 E T j Ω... P mj =1 E T jl Ω = √ γ Γ T l P mj =1 Ω E T j Ω... P mj =1 Ω E T jl Ω = γ X j,k Ω E T jk Ω= γ X j,k (cid:18) − (cid:19) (cid:18) e jk e jk e jk e jk (cid:19) (cid:18) − (cid:19) = γ X j,k (cid:18) − e jk e jk e jk − e jk (cid:19) . (44)Now, we use Eq. (38) to write e jk = e jk . This allows us to put together the above results to obtain β = γ (2 ǫ − l − m )where ǫ = P j,k e jk = P j,k e jk [since e jk = e jk as in Eq. (submatrices)].Recall that the two eigenvalues of A are λ = β ± h . In order for the system to be stable, we must have Re [ λ ] < β is negative. It also means that the most negative eigenvalue of A cannotbe lower than 2 β , since this would mean that the other eigenvalue would violate the stability criterion. We havetherefore obtained the bound λ > γ (2 ǫ − l − m ) on the most negative eigenvalue of A . EIGENVALUES OF THE DIFFUSION MATRIX
The diffusion matrix D takes the form D = Ω C σ in C T Ω T . Notice that since Ω is a unitary matrix, the eigenvaluesof D are the same as the eigenvalues of C σ in C T . The input state is taken to be a vacuum or thermal state withuniform noise, so σ in = ¯ N with ¯ N ≥
1. This means that we can find the eigenvalues of D for coherent feedback byfinding the eigenvalues of the matrix C cf C T cf and multiplying them by ¯ N : C cf C T cf = C l C T l − C m EC T l − C l E T C T m + C m EE T C T m + C m F F T C T m . (45)Using the orthogonality condition EE T + F F T = and the fact that C k C T k = kγ , we obtain: C cf C T cf = γ ( l + m ) − C m EC T l − C l E T C T m . (46)Writing in terms of the 2 × E gives: C cf C T cf = γ ( l + m ) + γ X j,k (cid:18) − e jk e jk + e jk e jk + e jk − e jk (cid:19) = γ ( l + m ) − γǫ , (47)where we have used e jk = − e jk and e jk = e jk , as per Eq. (38), and ǫ = P jk e jk = P jk e jk . Therefore, the diffusionmatrix under coherent feedback is proportional to the identity with eigenvalue δ = ¯ N γ ( l + m − ǫǫ