Delayed coherent quantum feedback - QCC 2015...20 January 2015 a) Vr Vin 320 um320 um b) L Vr Vin...

Delayed coherent quantum feedback

Quantum Cybernetics & Control 2015

Arne L. Grimsmo Université de Sherbrooke

!20 January 2015

a)

Vr

Vin

320 um320 um

b)L

Vr

Vin

Figure 1: An artificial atom in front of a mirror. a) A micrograph of the atom-mirrorsystem, a superconducting transmon embedded at a distance L from the end of a 1Dtransmission line. (zoom in) The transmon. The atom size is small compared to thewavelength of the microwave field. We characterize the system by sending in a coherentprobe field, Vin, at Êp ¥ 5GHz and measuring the reflected field, Vr. Measurements aredone at T = 50mK, where thermal excitations of the field are negligible. b) Cartoon ofthe atom-mirror system. The blue and red curves show the mode structure of the voltagealong the transmission line at the atom frequency for L = ⁄/2 and L = 3⁄/4, respectively.By tuning ⁄ of the two-level atom via an external magnetic flux, Õ, the coupling betweenthe field and the atom can be turned o↵ when the atom sits at a node of the resonant EMfield (blue). The atom is maximally coupled at the antinode (red).

13

(I.-C. Hoi et al., arXiv:1410.8840)

Outline

• Quick recap on open quantum systems

• Delayed Coherent quantum feedback: What is the difficulty?

• Primer on “tensor network” (TN) diagrams

• Re-deriving the Lindblad master equation with TN diagrams

• A solution to the feedback problem

• Example: Resonance fluorescence of a two-level atom into a coherent feedback loop a)

Vr

Vin

320 um320 um

b)L

Vr

Vin


13

First: Open Quantum Systems Recap

Input/output picture (Heisenberg picture):

Example:

Cascaded systems

(van Loo et al., Science 342, 2013)

Cascaded systems

Example:

Delayed Coherent Quantum Feedback

Simplest example: Self feedback (no controller)

Simplest example: Self feedback (no controller)

a)

Vr

Vin

320 um320 um

b)L

Vr

Vin


13

Quantum delayed feedback: What is the difficulty?

• Correlations between system and field are crucial: can hold an enormous amount of information.

• Even after discretization & truncation one faces the quantum many-body problem (exponentially large Hilbert space)

• Same problem faced in condensed matter and quantum information. Can we learn something from recent developments in these field?

Tensor Network Representations for Open Quantum Systems

3

Later, when dealing with system-bath dynamics, it will be useful to have a notation that distinguishes betweensystem and bath states. We here introduce a set of specialized conventions for this purpose. Consider a system-bathtensor product, S ⌦B. System states are denoted as in Eq. (1):

⇢S = ⇢S 2 L(HS). (15)

We distinguish this from states of the bath, B, by drawing the latter with the leg going downwards:

�B = �B 2 L(HB), (16)

A superoperator, ASB , that acts on the composite system-bath space we then draw

ASB = ASBS S

B

B

, (17)

where the labels indicate the system associated to the legs. This convention is practical for dealing with superoperatorsacting on product states, ⇢S ⌦ �B , which we can now draw

ASB⇢S ⌦ �B = ASB ⇢S

�B

. (18)

We need to compose such superoperators in two di↵erent ways. Superoperators, say ASB and BSB0 , that act on twodi↵erent bath systems, B and B0:

AS,BBS,B0S S

B

B

B0

B0

. (19)

And, superoperators that act on the same bath system, B, but two di↵erent systems S and S0:

AS,B

BS0,B

SS

S0S0

B

B

. (20)

Finally, we consider how the generalized trace defined in Eq. (10) acts on the level of superoperators. Consider asuperoperator, A, acting on two isomorphic systems A and A0:

A :L(HA)⌦ L(HA0) ! L(HA)⌦ L(HA0)

= AA

A0

A

A0.

(21)

We then have that

TR(A,A0)A : L(HA) ! L(HA0)

= A

=X

ij

hiA|A⇣(·)⌦ |iA0ihjA0 |

⌘|jAi,

(22)

where {|iAi} and {|iA0i}, are orthonormal bases for the two respective systems.

3


⇢S = ⇢S 2 L(HS). (15)


�B = �B 2 L(HB), (16)


ASB = ASBS S

B

B

, (17)



�B

. (18)


AS,BBS,B0S S

B

B

B0

B0

. (19)


AS,B

BS0,B

SS

S0S0

B

B

. (20)



= AA

A0

A

A0.

(21)

We then have that


= A

=X

ij


⌘|jAi,

(22)


3


⇢S = ⇢S 2 L(HS). (15)


�B = �B 2 L(HB), (16)


ASB = ASBS S

B

B

, (17)



�B

. (18)


AS,BBS,B0S S

B

B

B0

B0

. (19)


AS,B

BS0,B

SS

S0S0

B

B

. (20)



= AA

A0

A

A0.

(21)

We then have that


= A

=X

ij


⌘|jAi,

(22)


3


⇢S = ⇢S 2 L(HS). (15)


�B = �B 2 L(HB), (16)


ASB = ASBS S

B

B

, (17)



�B

. (18)


AS,BBS,B0S S

B

B

B0

B0

. (19)


AS,B

BS0,B

SS

S0S0

B

B

. (20)



= AA

A0

A

A0.

(21)

We then have that


= A

=X

ij


⌘|jAi,

(22)


1

AS⇢S = AS ⇢S (1)

AB�B = AB

�B

(2)

1

AS⇢S = AS ⇢S (1)

AB�B = AB

�B

(2)

A tensor network primer

1

AS⇢S = AS ⇢S (1)

AB�B = AB

�B

(2)

tr⇢S = ⇢S . (3)

1

AS⇢S = AS ⇢S (1)

AB�B = AB

�B

(2)

tr⇢S = ⇢S . (3)

ASS0 = ASS0S

S0

S

S0. (4)

TR(S,S0)ASS0 = ASS0

=X

ij

hiS |A⇣(·)⌦ |iS0ihjS0 |

⌘|jSi,

(5)

1

AS⇢S = AS ⇢S (1)

AB�B = AB

�B

(2)

tr⇢S = ⇢S . (3)

ASS0 = ASS0S

S0

S

S0. (4)

TR(S,S0)ASS0 = ASS0

=X

ij


⌘|jSi,

(5)

A tensor network primer

A “conveyor belt” of harmonic oscillators

A “conveyor belt” of harmonic oscillators

5

FIG. 1: A “conveyor belt” of harmonic oscillators, labeled by n, moves past the system as shown in the figure. The n’thoscillator interacts with the system for a short time �t.

where in the last equality, we have introduced the system operator L =pa, and the Fourier transformed field

b(t) =1p2⇡

Zd!e�i!tb(!), (30)

which satisfies [b(t), b†(t0)] = �(t� t0). Formally, the solution for the system state at a time, t, can be written

⇢S(t) = trBW(t)⇢S(0), (31)

where the evolution superoperator, W(t), is given through

W(t)⇢ = T exp

✓�i

Z t

0dt0H(t0)

◆⇢ T exp

✓i

Z t

0dt0H(t0)

◆, (32)

where H(t) = HS + V (t) is the interaction picture Hamiltonian.The next step is to go to a “discretized time” picture. This will allow us to expose the structure of the evolution

operator in terms of a “tensor network”, using the diagrammatic notation from the previous section. We do this byfirst introducing ladder operators

bn =1p�t

Z tn

tn��tb(t0)dt0, (33)

which satisfy the usual commutation relation [bn, b†m] = �nm. Here, tn = n�t, for n = 0, . . . , N , where t = N�t.Using the Suzuki-Trotter expansion, the evolution operator can then be written

W(t) = limN!1

UN . . .U2U1. (34)

Here, and in the following, N ! 1 refers to the continuum limit, �t ! 0, such that N�t = t is held fixed. Theunitary superoperator Un is defined through

Un⇢ =exp⇣�i�tHS +

p�t(Lb†n � L†bn

⌘⇢ exp

⇣i�tHS �

p�t(Lb†n � L†bn

⌘, (35)

We now work with finite N , and eventually take the continuum, N ! 1, limit. Eq. (34) and Eq. (35) then suggestthe following picture: The system interacts with a collection of harmonic oscillators, labelled by n, where the n’thoscillator interacts with the system at time tn, for a short time �t, before leaving the system. After this it neverinteracts with the system again. We will refer to this as the “conveyor belt picture,” and it is illustrated pictoriallyin Fig. 1.

The superoperator Un, we denote diagramatically as

Un = UnS S

n

n

, (36)

where the labels indicate the system associated to the legs: S for the system and n for the n’th oscillator. The resultof acting with this superoperator on a product state ⇢S ⌦ �n is then denoted

Un⇢S ⌦ �n = Un ⇢S

�n

. (37)

6

We are now in a position to represent diagrammatically the system-bath state at time tN , assuming an initial state⇢S ⌦ �1 ⌦ �2 ⌦ · · ·⌦ �N :

⇢SB(tN ) =

�N �2 �1

UN · · · U2 U1 ⇢S . (38)

As noted in the previous section, the shapes appearing in the diagram above, Eq. (38), can be thought of as tensors.A diagram consisting of many tensors, connected by joining their legs, like in Eq. (38), is therefore often called a“tensor network” [5]. In fact, the tensor network representation of the state in Eq. (38) is analogous to a matrixproduct state (MPS) [6], but with superoperators and density operators in place of operators and kets. MPS arefamous for their role as a variational ansatz in the density matrix renormalization group method [6], used to find thelow-energy spectrum of many-body Hamiltonians. In light of this similarity, we call a state on the form of Eq. (38)a superoperator product state (SPS) [12]. If taking the continuous limit (more on this below), N ! 1, the resultingstate we can similarly call a continuous SPS (cSPS), in analogy with the recently introduces cMPS [7]. MPS andcMPS are subclasses of SPS and cSPS, respectively. This connection has been exploited to use open quantum systemsto generate cMPS states, in the context of quantum simulation, both in theory [8, 9] and in experiment [citation toEichler?].

In the present context, however, we are chiefly interested in the reduced system dynamics, ⇢S(t) = trB⇢SB(t). Thatis, we want to trace over the bath degrees of freedom in Eq. (38). This gives a reduced system state (cf. Eq. (14)):

⇢S(tN ) =

�N �2 �1

UN · · · U2 U1 ⇢S . (39)

We would now like to retrieve the continuum, N ! 1 limit, to find a more compact form of Eq. (31), for ⇢S(t). Wefirst note that Eq. (39) can be built from a single tensor:

En =

�n

Un = trn�Un

� · �⌦ �nU†n

(40)

= exp

✓�i�tH[HS ]� i

p�tGn[L] +�tFn[L,L]� �t

2Cn[L,L]

◆, (41)

where we have defined superoperators

H[X] (·) =[X, ·] (42)

Gn[X] (·) =hb†niX (·)� hbniX† (·) + H.c (43)

Fn[X,Y ] (·) =hbnb†niX (·)Y † + hb†nbniX† (·)Y � h(b†n)2iX (·)Y � hb2niX† (·)Y † (44)

Cn[X,Y ] (·) =hbnb†niX†Y (·) + hb†nbniXY † (·)� h(b†n)2iXY (·)� hb2niX†Y † (·) + H.c. (45)

In Eq. (40) higher order terms in �t have been neglected, and the expectation values are taken with respect to�n. Note that the Hermitian conjugate of A (·)B refers to B† (·)A†. In the continuum limit, N ! 1, we define atime-propagator, E(t), which can be written as a product integral [4]:

E(t) = limN!1

NY

n=1

En (46)

=T exp

✓�itH[HS ]� i

Z t

0dG(s)[L] +

Z t

0

✓dF(s)[L,L]� 1

2dC(s)[L,L]

◆◆. (47)

Assume:

1

AS⇢S = AS ⇢S (1)

AB�B = AB

�B

(2)

tr⇢S = ⇢S (3)

tr�n = �n (4)

ASS0 = ASS0S

S0

S

S0. (5)

TR(S,S0)ASS0 = ASS0

=X

ij


⌘|jSi,

(6)

Recall:

6


⇢SB(tN ) =

�N �2 �1

UN · · · U2 U1 ⇢S . (38)



⇢S(tN ) =

�N �2 �1

UN · · · U2 U1 ⇢S . (39)


En =

�n

Un = trn�Un

� · �⌦ �nU†n

(40)

= exp



2Cn[L,L]

◆, (41)


H[X] (·) =[X, ·] (42)





E(t) = limN!1

NY

n=1

En (46)

=T exp

✓�itH[HS ]� i

Z t

0dG(s)[L] +

Z t

0

✓dF(s)[L,L]� 1

2dC(s)[L,L]

◆◆. (47)

6


⇢SB(tN ) =

�N �2 �1

UN · · · U2 U1 ⇢S . (38)



⇢S(tN ) =

�N �2 �1

UN · · · U2 U1 ⇢S . (39)


En =

�n

Un = trn�Un

� · �⌦ �nU†n

(40)

= exp



2Cn[L,L]

◆, (41)


H[X] (·) =[X, ·] (42)





E(t) = limN!1

NY

n=1

En (46)

=T exp

✓�itH[HS ]� i

Z t

0dG(s)[L] +

Z t

0

✓dF(s)[L,L]� 1

2dC(s)[L,L]

◆◆. (47)

Assume:

In the continuum limit:

6


⇢SB(tN ) =

�N �2 �1

UN · · · U2 U1 ⇢S . (38)



⇢S(tN ) =

�N �2 �1

UN · · · U2 U1 ⇢S . (39)


En =

�n

Un = trn�Un

� · �⌦ �nU†n

(40)

= exp



2Cn[L,L]

◆, (41)


H[X] (·) =[X, ·] (42)





E(t) = limN!1

NY

n=1

En (46)

=T exp

✓�itH[HS ]� i

Z t

0dG(s)[L] +

Z t

0

✓dF(s)[L,L]� 1

2dC(s)[L,L]

◆◆. (47)

A Tensor Network Solution to the Feedback Problem

Un Vn−M

2

⇢SL(tM+1) = UM+1 V1 UM V0 · · · U1 V1�M ⇢S (12)

UM V0 · · · U1 V1�M ⇢S

UM+1 V1

(13)

E(k)n = E(k)

n...

...=

V(1)nU (1)

n

V(2)nU (2)

n

...

V(k)nU (k)

n

(14)

E(k)n = E(k)

n...

...

S1

S2

Sk

S1

S2

Sk

(15)

E⌧ (t)S1

S2

S3

S1

S2

S3

⇢S(t) = E⌧ (t)⇢S1

(16)

⇢S(t2M+1) =

UM V0 · · · U1 V1�M ⇢S

U2M VM · · · UM+1 V1

U2M+1 VM+1

(17)

Un Vn−M

Using the same approach and taking the continuum limit:

Time:

Recall:

1

AS⇢S = AS ⇢S (1)

AB�B = AB

�B

(2)

tr⇢S = ⇢S (3)

tr�n = �n (4)

ASS0 = ASS0S

S0

S

S0. (5)

TR(S,S0)ASS0 = ASS0

=X

ij


⌘|jSi,

(6)

Un (7)

Vn�M (8)

(9)

⇢SB(tM ) = UM V0 · · · U1 V1�M ⇢S (10)

⇢SL(tM ) = UM V0 · · · U1 V1�M ⇢S (11)

Example for :

2

⇢SL(tM+1) = UM+1 V1 UM V0 · · · U1 V1�M ⇢S (12)

UM V0 · · · U1 V1�M ⇢S

UM+1 V1

(13)

E(k)n = E(k)

n...

...=

V(1)nU (1)

n

V(2)nU (2)

n

...

V(k)nU (k)

n

(14)

E(k)n = E(k)

n...

...

S1

S2

Sk

S1

S2

Sk

(15)

E⌧ (t)S1

S2

S3

S1

S2

S3

⇢S(t) = E⌧ (t)⇢S1

(16)

2

⇢SL(tM+1) = UM+1 V1 UM V0 · · · U1 V1�M ⇢S (12)

UM V0 · · · U1 V1�M ⇢S

UM+1 V1

(13)

E(k)n = E(k)

n...

...=

V(1)nU (1)

n

V(2)nU (2)

n

...

V(k)nU (k)

n

(14)

E(k)n = E(k)

n...

...

S1

S2

Sk

S1

S2

Sk

(15)

E⌧ (t)S1

S2

S3

S1

S2

S3

⇢S(t) = E⌧ (t)⇢S1

(16)

a)

Vr

Vin

320 um320 um

b)L

Vr

Vin


13

(I.-C. Hoi et al., arXiv:1410.8840)

Example: Resonance fluorescence of two-level atom into a coherent feedback loop

Example: Resonance fluorescence of a two-level atom into a coherent feedback loop

Dashed: no feedback

0.0 0.5 1.0 1.5 2.0 2.5 3.0

�t

0.0

0.2

0.4

0.6

0.8

1.0

h�+�

�i

0.0 0.5 1.0 1.5 2.0 2.5 3.0

�t

0.0

0.2

0.4

0.6

0.8

1.0

h�+�

�i

Thank you for your attention

Acknowledgements: Thanks to Howard Carmichael, Simon Whalen and Alexandre Blais for many discussions on this topic

Delayed coherent quantum feedback - QCC 2015...20 January 2015 a) Vr Vin 320 um320 um b) L Vr Vin...

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