Linear Quantum Feedback Networks

arXiv:0807.2338v1 [quant-ph] 15 Jul 2008

Linear Quantum Feedback Networks J. Gough1 , R. Gohm1 , M. Yanagisawa2 Abstract The mathematical theory of quantum feedback networks has recently been developed [5] for general open quantum dynamical systems interacting with bosonic input fields. In this article we show, for the special case of linear dynamical systems Markovian systems with instantaneous feedback connections, that the transfer functions can be deduced and agree with the algebraic rules obtained in the nonlinear case. Using these rules, we derive the the transfer functions for linear quantum systems in series, in cascade, and in feedback arrangements mediated by beam splitter devices.

1) Institute for Mathematical and Physical Sciences, University of Wales, Aberystwyth, Ceredigion, SY23 3BZ, Wales 2) Department of Engineering, Australian National University, Canberra, ACT 0200, Australia

1

Introduction

The aim of this paper is to deduce the algebraic rules for determining the dynamical charactersitics of a prescribed network consisting of specified quantum oscillator systems connected by input-output fields [1], [2]. Physical models included cavity systems or local quantum oscillators with a quantum optical field. The resulting dynamics is linear, and the analysis is carried out using transfer function techniques [3], [4]. The rules have been recently deduced in [5] in the general setting for nonlinear quantum dynamical systems by first constructing a network Hamiltonian and transfering to the interaction picture with respect to the free flow of the fields around the network channels. However it is of interest to restrict to linear systems for two main reasons. Firstly, the derivation here for linear systems procedes by an alternative method to the general nonlinear case, and we are able to confirm the restriction of the nonlinear formula to linear systems yields the same result. Secondly, linear systems are the most widely studied models in both classical and quantum dynamical systems theory and so it is natural to develop these further. There has been recent interest in the development of coherent, or fully quantum control for linear systems [6]-[10] and this paper contributes by establishing the algebraic rules for building networks of such devices.

1

2

Linear Quantum Markov Models

The dynamical evolution of a quantum system is determined by a family of unitaries {V (t, s) : t ≥ s} satisfying the propagation law V (t3 , t2 ) V (t2 , t1 ) = V (t3 , t1 ) where t3 ≥ t2 ≥ t1 . The evolution of a state from time s to a later time t being then given by ψ (t) = V (t, s) ψ (s). In a Markov model we factor the underlying Hilbert space as h ⊗ E representing the system and its environment respectively and the unitary V (t, s) couples the system specifically with the degrees of freedom of the environment acting between times s and t. For a bosonic environment, we introduce input processes bi (t) for i = 1, · · · , n with the canonical commutation relations, [1], h i bi (t) , b†j (s) = δ ij δ (t − s) . (1) It is convenient to assemble these into the following column vectors of length n   b1 (1)   .. (2) bin (t) =  . . bn (t)

A Markov evolution can be described equivalently by the chronological-ordered and Wick-ordered expressions Z t Z t ~ V (t, s) = T exp −i Υ (τ ) dτ ≡ : exp −i ΥWick (τ ) dτ : s

s

where the stochastic Hamiltonian is (with Υ (t) =

n X

i,j=1

Eij ⊗ b†i (t) bj (t) +

n X i=1

† Eij

= Eji and K † = K)

Fi ⊗ b†i (t) +

n X j=1

Fj† ⊗ bj (t) + K ⊗ 1,

and the Wick-ordered generator is given by [13] −iΥWick (t)

=

n X

i,j=1

−

(Sij − δ ij ) ⊗ b†i (t) bj (t) +

n X j=1

L†i Sij

⊗ bj (t) −

1 2

n X i=1

n X i=1

Li ⊗ b†i (t)

L†i Li

− iH

!

⊗ 1.

The Wick-ordered coefficients are given by the Stratonovich-Ito conversion formulae, see appendix, S=

1 − 2i E , 1 + 2i E

L = −i

1 F, 1 + 2i E

H = E00 +

1 1 F †. Im F 2 1 + 2i E

(3)

Note that H isP selfadjoint, and that P S is a unitary matrix whose entries are † † operators on h: nk=1 Sik Sjk = δ ij = nk=1 Ski Skj . In fact, we may write S = E e−iJ with J = 2 arctan . 2 2

In differential form we have d V (t, s) = dt ≡

−i : ΥWick (t) V (t, s) :

n X

i,j=1

−

b†i (t) (Sij − δ ij )V (t, s) bj (t) +

n X

L†i Sij V

(t, s) bj (t) −

j=1

n 1X

2

i=1

n X

b†i (t) Li V (t, s)

i=1

L†i Li − iH

!

V (t, s) .

Note that all the creators appear on the left and all annihilators on the right. This equation can be interpreted as a quantum stochastic differential equation [1], [11], [12]. We sketch the system plus field as a two port device having an input and an output port. system

output, bout

d

d

input, bin

Figure 1: input-output component †

The output fields are defined by bout (t) = V (t, 0) bi (t) V (t, 0) and we have i the input-output relation bout (t) = i

n X

Sij (t) bj (t) + Li (t) ,

j=1

†

†

where Sij (t) = V (t, 0) Sij V (t, 0) and Li (t) = V (t, 0) Li V (t, 0). More compactly, bout (t) = S (t) bin (t) + L (t). † Let X be a fixed operator of the system and set X (t, t0 ) = V (t, t0 ) XV (t, t0 ), then we obtain the Heisenberg-Langevin equation d X (t, t0 ) = dt =

1 V (t, t0 )† [X, Υ (t)]V (t, t0 ) i †

b†i (t) V (t, t0 )

†

† Ski XSkj − δ ij X V (t, t0 ) bj (t)

† [X, Lk ] V (t, t0 ) +b†i (t) V (t, t0 ) Ski †

+V (t, t0 ) [L†i , X]Sij V (t, t0 ) bj (t) 1 † 1 † +V (t, t0 ) Lk [X, Lk ] + [L†k , X]Lk − i [X, H] V (t, t0 ) . 2 2 Note that the final term does not involve the input noises, and that the expression in braces is a Lindbladian. In the special case where S = 1, this equation reduces to the class of Heisenberg-Langevin equations introduced by Gardiner [1]. 3

2.1

Linear Models

We consider a quantum mechanical system consisting of a family of harmonic oscillators h i{aj : j h= 1, · ·i· , m} with canonical commutation relations [aj , ak ] =

0 = a†j , a†k and aj , a†k = δ jk . We collect into column vectors:  a1   a =  ...  . am 

(4)

Our interest is in the general linear open dynamical system and here we make several simplifying assumptions: 1) The Sjk are scalars. P 2) The L′j s are linear, i.e., there exist constants cjk such that Lj ≡ k cjk ak . P 3) H is quadratic, i.e., there exist constants ω jk such that H = jk a†j ω jk ak .

The complex damping is 21 L† L + iH = −a† Aa where A = − 21 C † C − iΩ with C = (cjk ) and Ω = (ω jk ). The Heisenberg-Langevin equations for a (t) = V (t, 0) aV (t, 0) and input-output relations then simplify down to Aa (t) − C † Sb(t),

a˙ (t) = out

b

(t) =

Sb (t) + Ca (t) .

(5) (6)

These linear equations are amenable to Laplace transform techniques [3],[4]. We define for Re s > 0 Z ∞ ˆ C (s) = e−st C (t) dt, (7) 0

b˙ (s) = sˆ where C is now any of our stochastic processes. Note that a a (s) − a. We find that ˆ a (s)

ôut (s) b

ˆ in (s) + (sIm − A)−1 a, = − (sIm − A)−1 C † S b ˆ in (s) + Cˆ = Sb a (s) .

The operator ˆ a (s) can be eliminated entirely to give ˆ out (s) = Ξ (s) b în (s) + ξ (s) a b

(8)

where the transfer matrix function is Ξ (s) = S − C (sIm − A)

−1

C†S

(9)

and ξ (s) = C (sIm − A)−1 . As an example, consider a single mode cavity coupling to the input field via √ L = γa, and with Hamiltonian H = ωa† a. This implies K = γ2 + iω and 4

√ C = γ. If the output picks up an additional phase S = eiφ , the corresponding transfer function is then computed to be Ξcavity (s) = eiφ

2.2

s + iω − s + iω +

γ 2 γ 2

.

(10)

The Transfer Matrix Function

The models we consider are therefore determined completely by the matrices (S, C, Ω)with S ∈Cn×n , C ∈ Cn×m and Ω ∈ Cm×m . We shall use the conA B (s) = D + C (s − A)−1 B for matrices A ∈ Cm×m , B ∈ vention C D Cm×n , C ∈ Cn×m and D ∈ Cn×n , and write the transfer matrix function as A −C † S Ξ (s) = (s) , (11) C S where A = − 21 C † C − iΩ. We note the decomposition i h A −C † −1 Ξ = In − C (sIm − A) C † S ≡ S. C In In the simplest case of a single cavity mode we have √ − γ2 − iω − γeiφ √ Ξcavity (s) = (s) . γ eiφ

Lemma 1 For each ω ∈ R, the transfer function Ξ (iω) ≡ Ξ (0+ + iω) is unitary whenever it exists. Proof. The decomposition follows immediately from (11). We have then for instance † 1 1 † † I − C C C Ξ 0+ + iω Ξ 0+ + iω = I − C 1 † 1 † ′ C C − iΩ′ 2 C C + iΩ 2 1 1 1 1 † =I −C 1 † C C + iΩ′ + C † C − iΩ′ − C † C 1 † C†, ′ ′ 2 2 2 C C + iΩ 2 C C − iΩ where Ω′ = Ω + ω. The term in braces however vanishes identically, leaving † † Ξ (0+ + iω) Ξ (0+ + iω) = I. The relation Ξ (0+ + iω) Ξ (0+ + iω) = I is similarly established. Whenever appropriate, we may determine Ξ from its (unitary) values on the imaginary axis by using the Hilbert transform Z ∞ Ξ (iω) 1 PV dω. Ξ (s) = 2πi ω + is −∞ the real and imaginary parts of A need not commute - that is, †In general, C C, Ω need to be identically zero. However, when this does occur we recover a multi-mode version of the cavity situation. 5

Lemma 2 If A is a function of C † C then Ξ (s) =

s + A˜† S, s − A˜†

where A˜ is a function of CC † and Ξ may be analytically continued into the whole complex plane. Proof. Here we must have A = − 21 C † C − iε C † C where ε is a real valued function. We set A˜ = − 21 CC † − iε CC † . From the identity Cf C † C C † = CC † f CC † for suitable analytic functions f , we have 1 † ˜ ˜ s − A Ξ (s) = s − A I − CC S = s − A˜ − CC † S s − A˜ however, −A˜ − CC † = A˜† , and this gives the result. The hermitean matrices C † C and CC † will have the same set of eigenvalues: to see this, suppose that φ is a non-zero unit eigenvector of CC † with eigenvalue γ, then ψ = γ −1/2 C † φ is a unit eigenvector of C † C with the same eigenvalue, conversely, every eigenvector ψ of C † C with non-zero eigenvalue γ gives rise to a nonzero eigenvector φ = γ −1/2 Cψ P of CC † . † Let CC have the spectral form k γ k Ek with real eigenvalues γ k and corresponding eigenprojectors Ek , then we have Ξ (s) =

X s − 1 γ k + iεk 2 Ek S, s + 21 γ k + iεk k

where εk = ε (γ k ). In particular, the rational fraction is of modulus unity for imaginary s (= iω) and we may write X iφ (ω) Ξ 0+ + iω = e k Ek S k

i(ω+εk )−γ k /2 . Note that Ξ (0+ + iω) is clearly unitary and where φk (ω) = arg i(ω+ε k )+γ k /2 the limit ω → 0 is well-defined. This limit will equal −S in the special case that K is selfadjoint (i.e., ε ≡ 0). Ξ may be analytically continued into the negative-real part of the complex plane. The poles of Ξ then form the resolvent ˜ and the zeroes being the complex conjugates. set of K,

3

Introducing Connections

The situation depicted in the figure below is one where (some of) the output channels are fed back into the system as an input. Prior to the connection between output port(s) si and input port(s) ri being made, we may model the

6

in bi component as having the total input bin = and total output bout = bin e out bi out where the bin may be multi-dimensional noises (we in fact j and bj bout e only require the multiplicities to agree for j = i, e respectively). -

ri e re e

esi ese

figure 2: A quantum system with feedback The transfer matrix function takes the general form   P P A − j Cj† Sji − j Cj† Sje . Ξ ≡  Ci Sii Sie See Ce Sei

When we make the connection, we impose the various constraints bin ri (k) (t) = bout (t − τ ) where output field labelled s (j) is to be connected to the input i si (j) field ri (k) where τ > 0 is the time delay. We assume the idealized situation of instantaneous feedback τ → 0+ . To avoid having to match up the labels of the internal channels, it is more convenient to introduce a fixed labelling and write t− = ηbin bout i (t) i where η is the adjacency matrix: 1, if (s, r) is an internal channel, η sr = 0, otherwise

The model with the connections is then a reduction of the original and the out remaining external fields are the input bin e and output be . Theorem 3 Let (η − Sii ) be invertible. The feedback system described above ˆ out în has input-output relation b = Ξred b e e + ξ red a and the reduced transfer matrix function † 1 Ared −Cred Sred Ξred ≡ , ξ red ≡ Cred , Cred Sred s − Ared where Sred Cred Ared

−1

= See + Sei (η − Sii )

Sie ,

−1

= Sei (η − Sii ) Ci + Ce , X † = A− Cj Sji (η − Sii )−1 Ci . j=i,e

7

(12)

Proof. The dynamical equations can be written as X † a˙ (t) = Aa (t) − Cj Sjk bin k (t) , bout j (t)

=

X

j,k

Sjk bin k (t) + Cj a (t) .

k=i,e out Now the constraint ηbin implies that i = bi −1

bin i (t) = (η − Sii )

(Sie bin e (t) + Ci a (t)),

and so a˙ (t) =

[A − −

X

j=i,e

X

j=i,e

Cj† Sji (η − Sii )−1 Ci ]a (t)

−1 Cj† Sje + Sji (η − Sii ) Sie bin e (t)

or ˆ a (s) = −

X † 1 1 −1 ê (s) + Cj Sje + Sji (η − Sii ) Sie b a, s − Ared s − Ared j=i,e

with Ared as above. Consequently, ˆ out b e

în ˆ in = Sei b a i + See be + Ce ˆ in ˆ + Cred ˆ a = Sred b e

în = Ξred b e + ξ red a where Ξred ξ red

= =

Sred − Cred

X

Cred

j=i,e

1 , s − Ared

1 −1 Cj† Sje + Sji (η − Sii ) Sie , s − Ared

and Sred , Cred are as in Pthe statement of the theorem. −1 † Sred . Now We now show that j=i,e Cj† [Sje + Sji (η − Sii ) Sie ] = Cred X

j=i,e

−1

Cj† [Sje + Sji (η − Sii )

Ci† [Sie + Sii (η − Sii )

Sie ] =

−1

Ci† η (η − Sii )

=

−1

Sie ] + Ce† Sred

Sie + Ce† Sred ,

† while Cred Sred = Ci† (η † − Sii† )−1 Sie† Sred + Ce† Sred . However, −1

(η † − Sii† )−1 Sie† Sred = (η † − Sii† )−1 Sei† (See + Sei (η − Sii ) 8

Sie )

and using the identities Sii† Sii + Sei† Sei = 1, Sii† Sie + Sei† See = 0, this reduces to h i −1 (η † − Sii† )−1 Sie† Sred = (η † − Sii† )−1 −Sii† Sie + (1 − Sii† Sii ) (η − Sii ) Sie h i −1 = (η † − Sii† )−1 −Sii† (η − Sii ) + (1 − Sii† Sii ) (η − Sii ) Sie =

−1

η (η − Sii )

P

Sie .

1 C † Sred , as required. s − Ared red † Cred − iΩred For consistency, we should check that we have Ared = − 1e Cred 1 † 1 † with Ωred selfadjoint. Indeed, setting A = − 2 Ci Ci − 2 Ce Ce − iΩ and substituting in for Cred and Kred we find after some algebra that o o n n Ωred = Ω + Im Ci† Sii (η − Sii )−1 Ci + Im Ce† Sei (η − Sii )−1 Ci .

Therefore Ξred = Sred −

j=i,e

Cred

The manipulation for this is trivial except for the calculation of the term of the form 12 Ci† XCi where X

= ≡ =

−1

−1

1 + 2Sii (η − Sii ) − (η † − Sii† )−1 Sei† Sei (η − Sii ) i h −1 (1 − ηSii† )−1 Sii η † − ηSii† 1 − Sii η † o n Sii η † −1 2i Im = 2i Im S (η − S ) ii ii 1 − Sii η †

where again we use the identity Sii† Sii + Sei† Sei = 1. Li Sii Sie = ,L= In terms of the parameters (S, L, H) with S = Le Sei See Ci a and H = a† Ωa, we have that the feedback system is described by the Ce a reduced parameters (Sred , Lred , Hred ) where Sred Lred Hred

= See + Sei (η − Sii )

−1

Sie

−1

= Sei (η − Sii ) Li + Le , o o n n = H + Im L†i Sii (η − Sii )−1 Li + Im L†e Sei (η − Sii )−1 Li .

(13)

The same equations have been deduced in the nonlinear case o n by different arguo n ments [5]. Note the identity Im L†i Sii (η − Sii )−1 Li = Im L†i (η − Sii )−1 Li . Remark 4 Let U be a unitary operator ona fixed Hilbert space H = H1 ⊕ U11 U12 H2 which decomposes as U = . The non-commutative M¨ obius U21 U22 2→1 transform ϕU is the superoperator defined by −1

2→1 ϕU (X) = U11 + U12 (1 − XU22 )

9

XU21

−1

defined on the domain of operators X on H2 for which the inverse (1 − XU22 ) 2→1 exists. The transform ϕU maps unitaries on H2 in its domain to unitaries in H1 [14]. −1 Remark 5 In particular, Sred is unitary as it equals ϕi→e S (ξ) where ξ = η with η being unitary. We may expand the geometric series to write

Sred = See + Sei ξSie + Sei ξSii ξSie + Sei ξSii ξSii ξSie + · · · = See +

∞ X

n

Sei ξ (Sii ξ) Sie

n=0

which shows that Sred can be built up from contributions from the various paths through the network. Likewise Lred

=

Le +

∞ X

n

Sei ξ (Sii ξ) Li ,

n=0

Hred

=

H+

∞ X

n=0

4

∞ o X n n n Im L†e Sei ξ (Sii ξ) Li . Im L†i (Sii ξ) Li + n=0

Systems in Series

As a very special case of feedback connections we consider the situation of systems in series. This is referred to as feedforward in engineering.

es2

r2 e

es1

r1 e

figure 3: Cascaded systems The individualtransfer functions before the connection e = (s1 , r2 ) is made Ai −Ci† Si are given by Ξi = with Ai = − 12 Ci† Ci − iΩi .and these may be Ci Si concatenated to give   A1 + A2 −C1† S1 −C2† S2 . Ξ =  C1 S1 0 0 S2 C2

To use the formula for the reduced transfer function following connection, we must first of all identify the internal (eliminated) and external fields: here in in out out bi b2 bi b1 in out b = = , b = ≡ , in out bin b b bout e 1 e 2

10

and

0 S1 Sii Sie ≡ , Li ≡ L1 , Le ≡ L2 , S2 0 Sei See with trivially η = 1. The reduced transfer function is then readily computed to be # " A1 + A2 − C2† S2 C1 − C2† S2 + C1† S1 . Ξseries = C2 + S2 C1 S2 S1 Likewise we deduce the relations o n (14) S = S2 S1 , L = L2 + S2 L1 , H = H1 + H2 + Im L†2 S2 L1 .

The same equations have been deduced in the nonlinear case by different arguments [6].

4.1

Feedforward: Cascades

If the two systems are truly distinct systems, that is, if they are different sets of oscillators, then we are in the situation of properly cascaded systems. In this case one would expect that the transfer function to factor as the ordinary matrix product Ξseries ≡ Ξ2 Ξ1 . We now show that this is indeed the case. Lemma 6 Let Ξj be transfer functions for mj oscillators coupled to n fields (j = 1, 2). If we consider the ampliated transfer functions for m1 +m2 oscillators coupled to n fields   A1 0 −C1† S1 ˜1 =  , 0 0 Ξ 0 (C1 , 0) S1   0 0 0 ˜2 =  , 0 A2 Ξ −C2† S2 (0, C2 ) S2 then

˜ series = Ξ2 Ξ1 . Ξ

(15)

Proof. We compute this directly,   A1 0 −C1† S1 ˜ series =   Ξ −C2† S2 C1 −A2 −C2† S2 S1 (C1 , C2 ) S2 S1 −1 s − A1 0 −C1† S1 = S2 S1 + (C1 , C2 ) C2† S2 C1 s − A2 −C2† S2 S1 ! 1 0 C1† S1 s−A1 = S2 S1 − (C1 , C2 ) 1 1 1 C2† S2 C1 s−A − s−A C2† S2 S1 s−A2 2 1 i i h h = S2 − C2 (s − A2 )−1 C2† S2 × S1 − C1 (s − A1 )−1 C1† S1 ,

giving the result.

11

5

Beam Splitters

A simple beam splitter is a device performing physical superposition of two α β input fields. It is described by a fixed unitary operator T = ∈ U (2): µ ν out in b1 α β b1 = . bout µ ν bin 2 2 This is a canonical transformation and the output fields satisfy the same canonical commutation relations as the inputs. The action of the beam splitter is depicted in the figure below. On the left we have a traditional view of the two inputs being split into two output fields. On the right we have our view of the beam splitter as being a component with two input ports and two output ports: we have sketched some internal detail to emphasize how the scattering (superimposing) of inputs how ever we shall usually just draw this as a “black box” component in the following. bout 2 6 bin 1

-

-

bout 1

6 bin 2

bout 1 bout 2

in c c b1 @ @ in @ c b2 c

Figure 4: Beam-splitter component. To emphasize that the beam splitter is an input-output device of exactly the for we have been considering up to now, let us state that its transfer matrix function is 0 0 Ξbeam splitter = ≡ T. 0 T

Our aim is to describe the effective Markov model for the feedback device sketched below where the feedback is implemented by means of a beam splitter. Here we have a component system, called the plant, in-loop and we assume that A0 −C0† S0 it is described by the transfer function Ξ0 = . C0 S0 bout 1 bin 1 -

6 -

bout 2 plant

6 bin 2

b

12

b

Figure 5: Feedback using a beam-splitter. It is more convenient to view this as the network sketched below. -

plant

cr3 s3 c

beam splitter cs1 r1 c bin 1 cs2 r2 c

bout 1

Figure 6: Network representation. Here we have the pair of internal function for the network is  A0  0 Ξunconn. =   0 C0

edges (s2 , r3 ) and (s3 , r2 ). The transfer 0 T11 T21 0

 −C0† S0  0   0 S0

0 T12 T22 0

with respect to the labels (0, s1 , s2 , s3 ) for the rows and (0, r1 , r2 , r3 ) for the in out columns. This time the external fields are bin = bout ≡ T11 bin 1 + e = b1 , be 1 in T12 b2 while the (matched) internal fields are in out in b2 b2 T21 bin in out 1 + T22 b2 bi = . , bi = ≡ bin bout S0 bin 3 3 3 + L0 That is

T22 0 Sii = 0 S0 Sei =(T12 , 0) , L0 Li = , 0

,

T21 Sie = , 0 See = T11 , 0 Li = 0, η = 1

1 0

.

Substituting into our reduction formula we obtain −1 T21 −T22 1 S = T11 + (T12 , 0) 0 1 −S0 ≡ C

= ≡

Ω

= ≡

−1

T11 + T12 (1 − S0 T22 ) S0 T21 , −1 0 −T22 1 (T12 , 0) C0 1 −S0 −1

T12 (1 − S0 T22 )

C0 , −T22 Ω0 + Im (0, L†0 ) 1 −1

Ω0 + Im C0† (1 − S0 T22 ) 13

1 −S0 C0 .

−1

0 L0

and so, when the connections are made, the transfer fmatrix function is " # −1 A0 − C0† S0 T22 C0 −C0† S0 T21 − C0† S0 (1 − S0 T22 ) T22 Ξconn. = , −1 −1 T12 (1 − S0 T22 ) C0 T11 + T12 (1 − S0 T22 ) S0 T21 Note that S = ϕT2→1 (S0 ) where ϕT2→1 (z) = T11 + T12 β z −1 − T22 T21 is the M¨ obius transformation inthe complex plane associated with T . α β If we further set T = , and x + iy = S0 ν, then µ ν 2 2 2 2 † β C C0 = 1 − |ν| C † C0 ≡ 1 − x − y C † C0 , C † C = 0 0 0 2 2 1 − S0 ν |1 − S0 ν| (1 − x) + y 2 y 1 C0† C0 . Im C0† (1 − S0 ν)−1 C0 = Im C0† C0 = 2 2 1 − x − iy (1 − x) + y In particular, if we take a single oscillator in-loop with S0 = eiφ0 , then we obtain S ≡ eiφ and the phase is determined by the M¨ obius transformation. √ √ If we further have L0 = γ 0 a, H0 = ω 0 a† a, we find that L ≡ eiδ γa and H = ωa† a where γ=

1 − x2 − y 2

(1 − x) +

y2

γ0,

ω=

y

ω0, (1 − x)2 + y 2 α β and δ is a real phase. In the specific case T = with S0 = 1, ω 0 = 0 β −α considered by Yanagisawa and Kimura [3], we have x = −α and y = 0, therefore we find 1−α γ , ω=0 γ= 1+α 0 which agrees with their findings. 2

An alternative computation of Ξ is given by the following argument. We consider the input-output relations X ôut în în ôut b Tij b b = j , 2 = Ξ0 b1 + ξ 0 a0 , i j=1,2

h i −1 in ˆ out ˆ and eliminating b ≡ (1 − T Ξ) b + T ξ a T yields 22 22 0 21 2 1 0

i h −1 −1 ˆ in ôut ξ 0 a0 . b = T11 + T12 Ξ0 (1 − T22 Ξ0 ) T21 b 1 + T21 (1 − Ξ0 T22 ) 1

That is

Ξ = T11 + T12 Ξ−1 0 − T22

−1

T21 = ϕT2→1 (Ξ0 ) .

We remark that if T12 and T21 are invertible, then we may invert the M¨ obius transformation to get Ξ−1 0 = T22 + T21 14

1 T12 . Ξ − T11

To illustrate with acavity mode in-loop, we take the beam splitter matrix α β to be T = with α2 + β 2 = 1, and the transfer function Ξ0 (s) = β −α s+iω−γ/2 s+iω+γ/2 , then we find Ξ=

6

s + iω − α + Ξ0 = 1 + αΞ0 s + iω +

1−α 1+α 1−α 1+α

γ 2 γ 2

.

The Redheffer Star Product

An important feedback arrangement is shown in the figure below. A - er1 s1 e

bin 1

6

- bout 1

er2 s2 e

bout = bin 3 2

bout = bin 2 3 ?

es3 r3 e

bout 4

es4 r4 e

bin 4

B Figure 7: Composite System We shall derivethematrices taking component be now for this system A Bto A A B B S11 S12 C1A S33 S34 C3 described , , ΩA and B by , , A A B B S21 S22 C2A S43 S44 C4B ΩB . The operators of systems A are asumed to commute with those of B. We have two internal channels to eliminate which we can do in sequence, or simulataneously. We shall do the latter. here we have A A S12 0 S11 0 , Sei = See = B 0 S43 0 S44 A A S21 0 S22 0 Sie = , Sii = B B 0 S34 0 S33 and Le =

LA 1 LB 4

,

Li =

LA 2 LB 3

15

,

η=

0 1

1 0

.

The parameters are therefore A A S11 0 S12 S⋆ = + B 0 S44 0

−1 A A 0 −S22 1 S21 0 B B B S43 1 −S33 0 S34 ! A B −1 A A B −1 A A A B S34 1 − S22 S33 S21 S12 S33 S11 + S12 S33 1 − S22 , = A B −1 A B A B −1 A B S22 S34 1 − S22 S33 S21 S44 + S43 1 − S22 S33 S43 A A −1 A A C1 S12 0 −S22 1 C2 C⋆ = + B B C4B 0 S43 1 −S33 C3B ! A B −1 B A A B −1 A A B C3 1 − S22 S33 C2 + S12 S33 C1A + S12 S33 1 − S22 , = A B −1 B B A A B −1 A B C3 S33 C2 + S43 S22 1 − S22 1 − S22 S33 C4B + S43 n B A −1 A A B A −1 B S33 C2 S22 C3 + C3B† 1 − S33 S22 Ω⋆ = ΩA + ΩB + Im C3B† 1 − S33 A† A† A B −1 A A B −1 A B C S C + C2 1 − S22 S33 +C2 1 − S22 S33 2 22 3 A† A A† A B A −1 B B B A −1 B S C C3 + C1 S12 1 − S33 S22 +C1 S12 1 − S33 S22 33 2 o A B −1 A B A B −1 A B B C2 . 1 − S22 S33 S22 C3 + C4B† S43 1 − S22 S33 +C4B† S43

7

Appendix (Stratonovich to It¯ o Conversion)

It is convenient to introduce integrated fields Z t Z t Z t Bi (t) ≡ bi (s) ds, Bi† (t) ≡ b†i (s) ds, Λij (t) ≡ b†i (s) bj (s) ds. 0

0

0

Bi†

Bi (t) and (t) are called the annihilation and creation process, respectively, for the ith field and collectivey are referred to as a quantum Wiener process. Λij (t) is called the gauge process or scattering process from the jth field to the ith field. A noncommutative version of the Ito theory of stochastic integration with respect to these processes can be built up. The quantum It¯ o table giving the product of infinitesimal increments of these process is × dBi dΛij dBj† dt

dBk 0 0 0 0

dΛkl δ ik dBl δ jk dΛil 0 0

dBl† δ il dt δ jl dBi† 0 0

dt 0 0 . 0 0

The Ito equation for the unitary process is then dV = (dG) V where dG =

n X

i,j=1

(Sij − δ ij )dΛij +

n X

Li dBi†

i=1

−

n X

n

L†i Sij Bj

j=1

−

1X † L Li − iH 2 i=1 i

The Stratonovich form is dV = −i (dE) ◦ V where dE =

n X

i,j=1

Eij dΛij +

n X

Fi dBi† +

n X j=1

i=1

16

Fj† dBj (t) + Kdt.

!

dt.

and we define the Stratonovich differential to be (dX)◦Y = (dX) Y + 21 (dX) (dY ) with the last term computed using the It¯ o table. We have the consistency condition dV = (dG) V ≡ −i (dE) V − 2i (dE) (dG) V or dG = −idE −

i (dE) (dG) , 2

and using the table we see that S−1 L 1 − L† L − iH 2

i ≡ −iE − E (S − 1) 2 i = −iF − EL 2 i = −iK − F † L 2

which can be solved to give the relations (3).

References [1] C. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 2nd ed., ser. Springer Series in Synergetics. Springer, 2000. [2] H.Wiseman, Quantum theory of continuous feedback, Phys. Rev. A, vol. 49, no. 3, pp. 2133.2150, 1994. [3] M. Yanagisawa, H. Kimura, Transfer function approach to quantum control Part I: Dynamics of quantum feedback systems, IEEE Transactions on Automatic Control, 48, No. 12, 2107-2120, December (2003) [4] M. Yanagisawa, H. Kimura, Transfer function approach to quantum control Part II: Control concepts and applications, IEEE Transactions on Automatic Control, 48, No. 12, 2121-2132, December (2003) [5] J. Gough, M.R. James, Quantum Feedback Networks: Hamiltonian Formulation, arXiv:07070048(v1) [quant-ph] [6] J. Gough, M.R. James, The series product and its application to feedforward and feedback networks, arXiv:07070048(v1) [quant-ph] [7] M. R. James, H. I. Nurdin, and I. R. Petersen, H ∞ control of linear quantum stochastic systems, 2007, to be published (accepted 24-92007) in IEEE Transactions on Automatic Control. [Online]. Available: http://arxiv.org/pdf/quant-ph/0703150(pre-print) [8] S. Lloyd, Coherent quantum control, Phys. Rev. A, 62:022108, 2000

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[9] H. Mabuchi, Coherent-feedback control with a dynamic compensator, March 2008, submitted for publication, preprint: http://arxiv.org/abs/0803.2007. [10] H. I. Nurdin, M. R. James, and I. R. Petersen, Quantum LQG control with quantum mechanical controllers, 2008, to be presented at the 17th IFAC World Congress (Seoul, South Korea, July 6-11, 2008). [Online]. Available: http://arxiv.org/pdf/0711.2551(expandedversion) [11] R. L. Hudson and K. R. Parthasarathy, Quantum Ito’s formula and stochastic evolutions, Commun. Math. Phys. 93, 301-323 (1984) [12] K. Parthasarathy, An Introduction to Quantum Stochastic Calculus. Berlin: Birkhauser, 1992. [13] J. Gough, Quantum Stratonovich calculus and the quantum Wong-Zakai theorem, J. Math. Phys., vol. 47, no. 113509, 2006. [14] N. Young, An Introduction to Hilbert Space, Cambridge Mathematical Textbooks, (1988)

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