Quantum Dissipation and Control through Feedback Networks
John Gough
Aberystwyth University
Quantum Control Engineering:
Mathematical Principles and Applications
Isaac Newton Institute for Mathematical Sciences Cambridge, UK
21 July - 15 August, 2014
John Gough
An Introduction to Quantum Feedback Networks
Outline
Classical Feedback Networks and Control
The SLH Formalism
Quantum Feedback Networks
John Gough
An Introduction to Quantum Feedback Networks
Classical Input/Output Systems
We will investigate linear systems. Here a input xin is transformed into an output xout
via a linear relation
xout = G xin
which is depicted below as a block diagram.
Figure : The linear relation xout = G xin .
The main characteristic of the input-output equation is that it is linear:
G (c1 x1 + c2 x2 ) = c1 Gx1 + c2 Gx2 ,
for all scalars c1 and c2 .
John Gough
An Introduction to Quantum Feedback Networks
Systems run in Parallel
 
x1out
G1
 .   .
 ..  =  ..
0
xnout

···
..
.
···
John Gough
  in 
x1
0
..   .. 
.  . ,
Gn
xnin
An Introduction to Quantum Feedback Networks
Cascaded Systems
We may feed the output of G in as input to H:
xout
=
G xin
yout
=
H yin .
Figure : Cascaded systems.
We set xout = yin , then
yout = H G xin .
Figure : Cascaded systems.
John Gough
An Introduction to Quantum Feedback Networks
Summing and Take-Off Points
In order to build more sophisticated networks, we need to be able to connect the
inputs and outputs of various systems in a more flexible manner.
Figure : Summing points.
Figure : A take-off point.
John Gough
An Introduction to Quantum Feedback Networks
Feedback Loops
Figure : A simple feedback arrangement, and its equivalent system.
The output then satisfies the algebraic identity
xout = F (xin ± Hxout )
which can be rearranged to give
xout = (1 ∓ FH)−1 F xin .
The term 1 ∓ FH must be invertible and in such cases the network is said to be
well-posed.
John Gough
An Introduction to Quantum Feedback Networks
Feedback Loops
Figure : A simple feedback arrangement, and its equivalent system.
The result is that we obtain the effective transfer function
G = (1 ∓ FH)−1 F ≡ F (1 ∓ HF )−1 .
We may think of G as the “renormalized” transfer function! Formally we have the
geometric series expansion:
G
≡
=
F ∓ FHF + FHFHF ∓ FHFHFHF + · · ·
∞
∞
X
X
F
(∓HF )n =
(∓FH)n F .
n=0
John Gough
n=0
An Introduction to Quantum Feedback Networks
Fractional Linear Transformations
Let us consider a more generic model as sketched in below.
Figure : General feedback arrangement.
Here the feedback component is through the component G22 .
John Gough
An Introduction to Quantum Feedback Networks
Fractional Linear Transformations
It is convenient to split up the network into two subnetworks:
Setting x1 = xin and y1 = xout we see that the first of these subnetworks is in fact just
a linear system with the block matrix gain
y1
G11 G12
x1
=
,
y2
G21 G22
x2
while the second network implies the supplementary relation
x2
=
John Gough
T y2 .
An Introduction to Quantum Feedback Networks
Fractional Linear Transformation
We eliminate the feedback loop through a “renormalization” as before. We rearrange
the input-output relation x2 = T (z + G22 x2 ) to get x2 = R z, where
R = (1 − TG22 )−1 T .
This eliminates the feedback loop!
Figure : Feedback loop “renormalization”: R = (1 − TG22 )−1 T .
John Gough
An Introduction to Quantum Feedback Networks
Fractional Linear Transformation
This leads to the effective input-output model with transfer function
G (T ) = G11 + G12 (1 − TG22 )−1 TG21 .
The transfer function G is a fractional linear transformation of the gain T . The
network is well-posed whenever the inverse (1 − TG22 )−1 exists.
We note that the direct feedthrough is G0 = G11 , while the asymptotic gain is a Schur
complement
−1
G∞ = G11 − G12 G22
G21 ,
(whenever G22 is invertible).
John Gough
An Introduction to Quantum Feedback Networks
State-based Models
or
ẋ
y
ẋ
=
Ax + Bu
y
=
Cx + Du
=V
x
u
,
V=
A
C
B
D
.
The function T (s) = D + C (sI − A)−1 B is called the transfer function.
We define the parallel sum
V1 V2
=
A1
C1

,
B1
D1
A
1 + A2

C1
C2
John Gough
A2
C2
B2
D2

[B1 , B2 ] .
D1
0
0
D2
An Introduction to Quantum Feedback Networks
Networks
The classical plant-controller setup:
John Gough
An Introduction to Quantum Feedback Networks
Quantum Models
We have position and momentum observables satisfying the CCR [q, p] = 2i, and
introduce
1
1
b , (q + ip), b † , (q − ip)
2
2
so that
b, b † = 1,
and define the number operator
N , b † b.
John Gough
An Introduction to Quantum Feedback Networks
Quantum Models
We have position and momentum observables satisfying the CCR [q, p] = 2i, and
introduce
1
1
b , (q + ip), b † , (q − ip)
2
2
so that
b, b † = 1,
and define the number operator
N , b † b.
The Schrödinger repesentation on L2 (R, dx):
∂
φ(x).
∂x
A complete orthonormal basis is given by the Hermite polynomials
(qφ)(x) = xφ(x),
(pφ)(x) ≡ −2i
p
1
hx|ni = √ Hn (x) ρ(x),
n!
with ρ(x) =
2
√1 e −x /2 .
2π
b ∗ |ni
=
N |ni
=
(n = 0, 1, 2, · · · )
We have
√
√
n + 1|n + 1i,
b |ni =
n|n − 1i,
0,
n ≥ 1;
n = 0.
n|ni.
John Gough
An Introduction to Quantum Feedback Networks
Definition
For β ∈ C we define the exponential vector (Bargmann state) as
†
|e(β)i = e βb |0i =
John Gough
X βn
√ |ni.
n!
n
An Introduction to Quantum Feedback Networks
Definition
For β ∈ C we define the exponential vector (Bargmann state) as
†
|e(β)i = e βb |0i =
X βn
√ |ni.
n!
n
Their main properties are
|e(0)i
=
|0i,
John Gough
An Introduction to Quantum Feedback Networks
Definition
For β ∈ C we define the exponential vector (Bargmann state) as
†
|e(β)i = e βb |0i =
X βn
√ |ni.
n!
n
Their main properties are
|e(0)i
=
|0i,
b |e(β)i
=
β|e(β)i,
John Gough
An Introduction to Quantum Feedback Networks
Definition
For β ∈ C we define the exponential vector (Bargmann state) as
†
|e(β)i = e βb |0i =
X βn
√ |ni.
n!
n
Their main properties are
|e(0)i
=
|0i,
b |e(β)i
=
he(α)|e(β)i
=
β|e(β)i,
X (α∗ )n β n
∗
√
√ = eα β ,
n!
n!
n
John Gough
An Introduction to Quantum Feedback Networks
Definition
For β ∈ C we define the exponential vector (Bargmann state) as
†
|e(β)i = e βb |0i =
X βn
√ |ni.
n!
n
Their main properties are
|e(0)i
=
|0i,
b |e(β)i
=
he(α)|e(β)i
=
hx|e(β)i
=
β|e(β)i,
X (α∗ )n β n
∗
√
√ = eα β ,
n!
n!
n
X βn
p
1 2p
Hn (x) ρ(x) = e βx− 2 β
ρ(x),
n!
n
John Gough
An Introduction to Quantum Feedback Networks
Definition
For β ∈ C we define the exponential vector (Bargmann state) as
†
|e(β)i = e βb |0i =
X βn
√ |ni.
n!
n
Their main properties are
|e(0)i
=
|0i,
b |e(β)i
=
he(α)|e(β)i
=
hx|e(β)i
=
β|e(β)i,
X (α∗ )n β n
∗
√
√ = eα β ,
n!
n!
n
X βn
p
1 2p
Hn (x) ρ(x) = e βx− 2 β
ρ(x),
n!
n
∆q
=
∆p = 1
(Minimal Uncertainty States: ∆q ∆p = 1.)
John Gough
An Introduction to Quantum Feedback Networks
Definition
For β ∈ C we define the exponential vector (Bargmann state) as
†
|e(β)i = e βb |0i =
X βn
√ |ni.
n!
n
Their main properties are
|e(0)i
=
|0i,
b |e(β)i
=
he(α)|e(β)i
=
hx|e(β)i
=
β|e(β)i,
X (α∗ )n β n
∗
√
√ = eα β ,
n!
n!
n
X βn
p
1 2p
Hn (x) ρ(x) = e βx− 2 β
ρ(x),
n!
n
∆q
=
∆p = 1
(Minimal Uncertainty States: ∆q ∆p = 1.)
The situation easily generalizes to n modes:
bj , bk∗ = δjk
~ = ⊗k |e(βk )i on L2 (Rn , dx n ) ∼
and set |e(β)i
= ⊗n L2 (R, dx).
John Gough
An Introduction to Quantum Feedback Networks
The Euclidean Group on Cn
We consider three basic operations on exponential vectors
phase:
translation:
rotation:
|e(~
α)i 7→ e −iθ |e(~
α)i
~
|e(~
α)i 7→ |e(~
α + β)i
|e(~
α)i 7→ |e(R α
~ )i
~ ∈ Cn and R ∈ U(n).
where β
John Gough
An Introduction to Quantum Feedback Networks
Displacement
→
→
−† ~ −
.β−β ∗ .~
b,
~ ,eb
Introduce the Weyl unitary D(β)
~ |e(~
D(β)
α)i
†~
~
~
D(β) b D(β)
But
=
=
then
−
→
~ |2
−β ∗ .~
α− 12 |β
e
~b + β.
~
~
|e(~
α + β)i,
~2 ) D(β
~1 ) = e −iIm{β~2∗ .β~1 } D(β
~1 + β
~2 ).
D(β
John Gough
An Introduction to Quantum Feedback Networks
Displacement
→
→
−† ~ −
.β−β ∗ .~
b,
~ ,eb
Introduce the Weyl unitary D(β)
~ |e(~
D(β)
α)i
†~
~
~
D(β) b D(β)
But
=
=
then
−
→
~ |2
−β ∗ .~
α− 12 |β
e
~b + β.
~
~
|e(~
α + β)i,
~2 ) D(β
~1 ) = e −iIm{β~2∗ .β~1 } D(β
~1 + β
~2 ).
D(β
Rotation
Let R = e −iΩ ∈ U(n), so that Ω† = Ω. Set
Γ(R) , e −i
→
−† ~
b Ωb
,
then
Γ(R) |e(~
α)i
=
Γ(R)† ~b Γ(R)
=
|e(R α
~ )i,
R ~b.
And Γ(R2 ) Γ(R1 ) = Γ(R2 R1 ).
John Gough
An Introduction to Quantum Feedback Networks
The Euclidean group U(n) o Cn
The group product is
~2 ) ? (R1 , β
~1 ) = (R2 R1 , β
~2 + R2 β
~1 )
(R2 , β
i.e., we have
~1 7→ R2 (R1 α
~1 ) + β
~2
α
~ 7→ R1 α
~ +β
~ +β
John Gough
An Introduction to Quantum Feedback Networks
The Euclidean group U(n) o Cn
The group product is
~2 ) ? (R1 , β
~1 ) = (R2 R1 , β
~2 + R2 β
~1 )
(R2 , β
i.e., we have
~1 7→ R2 (R1 α
~1 ) + β
~2
α
~ 7→ R1 α
~ +β
~ +β
~ θ) by
More generally define the family of unitary operators W (R, β,
−
→
∗
~ θ) |e(~
W (R, β,
α)i , e −iθ e −β
~ |2
.R α
~ − 12 |β
~
|e(R α
~ + β)i.
Then
n
o
~ ∗ .R2 β1 )
~2 , θ2 ) W (R1 , β
~1 , θ1 ) = W (R2 R1 , β
~2 + R2 β
~1 , θ1 + θ2 + Im β
W (R2 , β
2
John Gough
An Introduction to Quantum Feedback Networks
The Euclidean group U(n) o Cn
The group product is
~2 ) ? (R1 , β
~1 ) = (R2 R1 , β
~2 + R2 β
~1 )
(R2 , β
i.e., we have
~1 7→ R2 (R1 α
~1 ) + β
~2
α
~ 7→ R1 α
~ +β
~ +β
~ θ) by
More generally define the family of unitary operators W (R, β,
−
→
∗
~ θ) |e(~
W (R, β,
α)i , e −iθ e −β
~ |2
.R α
~ − 12 |β
~
|e(R α
~ + β)i.
Then
n
o
~ ∗ .R2 β1 )
~2 , θ2 ) W (R1 , β
~1 , θ1 ) = W (R2 R1 , β
~2 + R2 β
~1 , θ1 + θ2 + Im β
W (R2 , β
2
Canonical Transformation and their Composition
~ θ)† ~b W (R, β,
~ θ) = R ~b + β
~
W (R, β,
with the cascade rule
n
o
~2 , θ2 ) C (R1 , β
~1 , θ1 ) = (R2 R1 , β
~2 + R2 β
~1 , θ1 + θ2 + Im β
~ ∗ .R2 β
~1 ).
(R2 , β
2
John Gough
An Introduction to Quantum Feedback Networks
Steps to the “SLH” Formalism
John Gough
An Introduction to Quantum Feedback Networks
Steps to the “SLH” Formalism
1
Go from Cn to Cn ⊗ L2 (R, dt), that is replace modes bk with quantum white
noises bk (t),
h
i
bj (t), bk† (s) = δjk δ(t − s).
John Gough
An Introduction to Quantum Feedback Networks
Steps to the “SLH” Formalism
1
Go from Cn to Cn ⊗ L2 (R, dt), that is replace modes bk with quantum white
noises bk (t),
h
i
bj (t), bk† (s) = δjk δ(t − s).
2
Introduce a system with Hilbert space h and replace
R ,→ S = Sjk with S † S = SS † = I ;
~ ,→ ~L = Lj ;
β
θ
,→
H = H†;
with Sjk , Lj and H operators on h.
John Gough
An Introduction to Quantum Feedback Networks
Steps to the “SLH” Formalism
1
Go from Cn to Cn ⊗ L2 (R, dt), that is replace modes bk with quantum white
noises bk (t),
h
i
bj (t), bk† (s) = δjk δ(t − s).
2
Introduce a system with Hilbert space h and replace
R ,→ S = Sjk with S † S = SS † = I ;
~ ,→ ~L = Lj ;
β
θ
,→
H = H†;
with Sjk , Lj and H operators on h.
The “series product” is
n
o
(S2 , ~L2 , H2 ) C (S1 , ~L1 , H1 ) = (S2 S1 , ~L2 + S2~L1 , H1 + H2 + Im ~L†2 S2~L1 )
John Gough
An Introduction to Quantum Feedback Networks
SLH Formalism
The SLH formalism for quantum Markov models deals with the category of models
S11
 .
S =  ..
Sn1

···
..
.
···
John Gough



S1n
L1
 . 
.. 
.  , L =  ..  , H
Snn
Ln
An Introduction to Quantum Feedback Networks
Model Matrix
These may be assimilated into the model matrix
− 12 L∗ L − iH −L∗ S
V =
L
S
P

1 P ∗
− 2 j Lj Lj − iH − j L∗j Sj1

L1
S11

= 
..
..

.
.
Ln
Sn1


V00 V01 · · · V0n
 V10 V11 · · · V1n 


=  .
..
..  .
..
 ..
.
.
. 
Vn0 Vn1 · · · Vnn
John Gough
···
···
..
.
···
−
P
j
L∗j Sjm
S1n
..
.
Snn





An Introduction to Quantum Feedback Networks
Parallel Sum
The parallel sum of models is defined by
V1
=
V2
 P

1 (k)∗ (k)
L
k=1,2 (− 2 L
L(1)
L(2)
John Gough
− iH (k) )
−L(1)∗ S (1)
S (1)
0

−L(2)∗ S (2)

0
S (2)
An Introduction to Quantum Feedback Networks
Feedback Reduction
The feedback reduction is
F(r ,s) (V, T ) αβ = Vαβ − Vαr T (1 − Vrs T )−1 Vsβ
for α 6= r and β 6= s.
John Gough
An Introduction to Quantum Feedback Networks
Proposition
As long as the network is well-posed we will have that the FLT F(r ,s) (V, T ) is again
the model matrix of an SLH system. a
a
It is a unexpected triumph over non-commutativity! It is essentially due to C.L. Siegel and was discovered in
the 1940’s. N. Young An Introduction to Hilbert Space, Cambridge Math. Textbook.
John Gough
An Introduction to Quantum Feedback Networks
Proposition
As long as the network is well-posed we will have that the FLT F(r ,s) (V, T ) is again
the model matrix of an SLH system. a
a
It is a unexpected triumph over non-commutativity! It is essentially due to C.L. Siegel and was discovered in
the 1940’s. N. Young An Introduction to Hilbert Space, Cambridge Math. Textbook.
The series product is a special case
F(2,1)
V≡
S (1)
0
0
S (2)
John Gough
(1) L
,
,H ,T = I
(2)
L
An Introduction to Quantum Feedback Networks