Feedback Control of Rabi Oscillations in Circuit QED · 2020. 1. 6. · Rabi Oscillations in...

Feedback Control of Rabi Oscillations in Circuit QED

(arXiv: 1304.5919)

Wei Cui and Franco Nori

Workshop on Coherent Control of Complex Quantum Systems (C3QS)May 7-12, 2013 Okinawa, Japan

Workshop on C3QS, Okinawa, Japan 2

Motivation:To extend the feedback design method to quantum domainTo apply the result to very important quantum control problemsTo demonstrate the effectiveness of the obtained result by simulationTo draw your attention to this new growing and challenging field

Key issue:Feedback control of Rabi oscillations in circuit QED

Main method:Lyapunov stability

Perspective:Exploitation of feedback

Workshop on C3QS, Okinawa, Japan

Opportunities for Quantum Control

3

!   Extremely high-precision measurement

! Ultracold systems in condensed matter science

!  High-intensity and short-wavelength light sources

!  Ultrafast control on the motion of atoms/electrons

!  Quantum engineering on the nanoscale structures

!  Quantum computation, data security and encryption

From “Controlling the quantum world: the science of atoms, molecules and photons”, Physics 2010, National Research Council (2007)


What is quantum control?--Reverse Engineering

4

If nature does not provide the wanted solution. ----We engineer!•Quantum control aims at reaching a target with a control law.•We do not know the law a-priori.

In standard QM problems we know the way, but not the end.

In QC problems we know the end, but not the way.

What is Optimal Control? - Reverse Engineering

If nature does not provide the wanted solution? – We engineer!

• Optimal Control aims at reaching a target with a control laser.• We do not know the laser a-priori!In standard QM problems weknow the way, but not the end.

In OC we know the end, butnot the way.

vs.

Optimal Control is about searching a laser in order to reach the target.

4 of 34





vs.


4 of 34





vs.


4 of 34





vs.


4 of 34

Quantum control is about searching a law in order to reach the target.


Quantum system control model

5

1

( ) ˆ ˆ ˆ ˆ[ ( ) ( ( ))) ) ]( (=

∂ Ψ= ⊗ ⊗ ++ ⊗

∂+ Ψ∑h

r

S E S E SE ii

i E

tH ti I tI H t H t H I t

tu

Free Hamiltonian of the system

Hamiltonian for the environment

Interaction Hamiltonian between the system and the environment

controls

Semiclassical Control Hamiltonians

State of the Total System

Quantum Control

Types of Quantum Control:Open loop - control actions are predetermined, no feedback is involved.

• Open loop - control actions are predetermined, no feedback is involved.

controller quantum system

controlactions

Types of Quantum Control:

8Matt James (ANU) Quantum Feedback Control 11 / 60

Quantum Control

Closed loop - control actions depend on information gained as the systemis operating.

• Closed loop - control actions depend on information gained as the system is operating.

controller

quantum system

controlactions

information

(feedback loop)

9

Matt James (ANU) Quantum Feedback Control 13 / 60


History of quantum feedback controlTheory Researcher Year

Quantum filter theory V. P. Belavkin 1980s

Adaptive feedback controlH. Rabitz 1991

Adaptive feedback controlExperimental realization in chemical system, 1997Experimental realization in chemical system, 1997

Markovian feedbackH. M. Wiseman 1993

Markovian feedbackExperimental realization in optical system, 2002Experimental realization in optical system, 2002

Bayes feedbackA. C. Doherty 1999

Bayes feedback Experimental realizations in optical system, 2007Experimental realizations in optical system, 2007Bayes feedbackExperimental realizations in atom-cavity system,2002Experimental realizations in atom-cavity system,2002

Direct coherent feedbackS. Lloyd 2000

Direct coherent feedbackExperiment in NMR, 2000Experiment in NMR, 2000

Field-mediated coherent feedback

M. James，H. Mabuchi 2003Field-mediated coherent feedback Experimental realization in optical system 2008Experimental realization in optical system 2008

6


Quantum feedback schemes

7

Quantum feedback

Measurement-based feedback

Direct Markovian feedback

Bayes feedback

Feedback by strong measurement

Quantum coherent feedback

Direct coherent feedback

Field-mediated coherent feedback

Passive feedback schemes


Quantum feedback schemes

8


(Classical control)

Coherent feedback

(Quantum control)

Essential distinction: the control loop is quantum or not !



9

The controlled system is measured by a probe field

The output of the probe field is directly fed back to produce a control signal via linear amplification

H. M. Wiseman et al PRL 70, 548 (1993) PRA 47, 1652 (1993) PRA 49, 1350 (1994) Output feedback !

Direct Markovian feedback



10

The output is fed into an estimator to estimate the system state

A. C. Doherty et al PRA 60, 2700 (1999) PRA 62, 012105 (2000)

The estimated system state is fed into a control circuit to obtain a state-based feedback

State-based feedback !

Bayes feedback


Adaptive Homodyne Measurement of Optical Phase. PRL 89, 133602 (2002)

Stabilize and release quantum state. PRL 89, 133601 (2002)

Stabilizing nonorthogonal qubit states . PRL 104, 080503 (2010)

Preparing photon number state. Nature 477, 73 (2011)

Cooling nanomechanical resonator. Nature 444, 75 (2006)

Bayes feedback of two-atom spin states, arXiv:1206.3184v1PRL 109, 173601 (2012)

11

Experiments in optical systems


Cavity Quantum Electrodynamics (cQED)

12

2g = vacuum Rabi freq. κ = cavity decay rate γ = transverse decay rate

Strong Coupling = g > κ , γ , 1/t

t = transit time

Jaynes-Cummings Hamiltonian

† †12 ˆ ˆ2)ˆ )(

2(el Jx zr a a a a

E gH Eω σ σ σ σ− += + − +−+h h

Quantized Field Electric dipole

Interaction 2-level system


A Circuit Implementation of Cavity QED

13

L = λ ~

2.5 cm

Cooper-pair box atom10 µm 10 GHz in

out

transmission line cavity

Circuit QED proposal: A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, PRA 69, 062320 (2004)


Experiment realizations

14

Yale group, nature 431, 162 (2004) Delft group, nature 431, 159 (2004)


Comparison of cQED with Atoms and Circuits

15

Circuit QED with artificial atoms QED with atoms

Josephson junction devices Atoms

Current and voltage sources Light sources

T=30 mK T=300 K

Circuit environment Cavity

Strong JJ-environment coupling Weak atom-field coupling


Comparison of cQED with Atoms and Circuits

16

Parameter Symbol Optical cQED with Cs atoms

Microwave cQED/

Rydberg atoms

Super-conducting

circuit QED

Dipole moment d/eao Vacuum Rabi

frequency g/π" MHz kHz MHz

Cavity lifetime κ; Q ns; 3 x 107 ms; 3 x 108

ns; 104

Atom lifetime γ" ns ms µs Atom transit time ttransit µs µs Infinite Critical atom # N0=2γκ/g2 x 10-3 x 10-6 x 10-5

Critical photon # m0=γ2/2g2 x 10-4 x 10-8 x 10-6 of vacuum

Rabi oscillations nRabi=2g/(κ+γ)

A. Blais etal., PRA 69, 062320 (2004)


Dispersive Measurement

17

J-C model:

Dispersive regime:

For large detuning, g /!"1, expansion of Eq. (4) yieldsthe dispersive spectrum shown in Fig. 1(c). In this situation,the eigenstates of the one excitation manifold take the form[15]

!− ,0" # − $g/!%!↓ ,0" + !↑ ,1" , $7%

!+ ,0" # !↓ ,0" + $g/!%!↑ ,1" . $8%

The corresponding decay rates are then simply given by

#− ,0 & $g/!%2$ + % , $9%

#+ ,0 & $ + $g/!%2% . $10%

More insight into the dispersive regime is gained by mak-ing the unitary transformation

U = exp' g!

$a&+ − a†&−%( $11%and expanding to second order in g (neglecting damping forthe moment) to obtain

UHU† ) ''(r + g2! &z(a†a + '2') + g2! (&z. $12%As is clear from this expression, the atom transition is acStark/Lamb shifted by $g2 /!%$n+1/2%. Alternatively, onecan interpret the ac Stark shift as a dispersive shift of thecavity transition by &zg2 /!. In other words, the atom pullsthe cavity frequency by ±g2 /%!.

III. CIRCUIT IMPLEMENTATION OF CAVITY QED

We now consider the proposed realization of cavity QEDusing the superconducing circuits shown in Fig. 2. A 1Dtransmission line resonator consisting of a full-wave sectionof superconducting coplanar waveguide plays the role of thecavity and a superconducting qubit plays the role of theatom. A number of superconducting quantum circuits couldfunction as artificial atom, but for definiteness we focus hereon the Cooper-pair box [6,16–18].

A. Cavity: Coplanar stripline resonator

An important advantage of this approach is that the zero-point energy is distributed over a very small effective volume()10−5 cubic wavelengths) for our choice of a quasi-one-dimensional transmission line “cavity.” As shown in Appen-dix A, this leads to significant rms voltages Vrms

0 #*'(r /cLbetween the center conductor and the adjacent ground planeat the antinodal positions, where L is the resonator length andc is the capacitance per unit length of the transmission line.At a resonant frequency of 10 GHz $h* /kB#0.5 K% and fora 10 +m gap between the center conductor and the adjacentground plane, Vrms#2 +V corresponding to electric fieldsErms#0.2 V/m, some 100 times larger than achieved in the3D cavity described in Ref. [3]. Thus, this geometry mightalso be useful for coupling to Rydberg atoms [19].

In addition to the small effective volume and the fact thatthe on-chip realization of CQED shown in Fig. 2 can befabricated with existing lithographic techniques, atransmission-line resonator geometry offers other practicaladvantages over lumped LC circuits or current-biased largeJosephson junctions. The qubit can be placed within the cav-ity formed by the transmission line to strongly suppress thespontaneous emission, in contrast to a lumped LC circuit,where without additional special filtering, radiation and para-sitic resonances may be induced in the wiring [20]. Since theresonant frequency of the transmission line is determinedprimarily by a fixed geometry, its reproducibility and immu-nity to 1/ f noise should be superior to Josephson junctionplasma oscillators. Finally, transmission-line resonances incoplanar waveguides with Q#106 have already been dem-onstrated [21,22], suggesting that the internal losses can bevery low. The optimal choice of the resonator Q in this ap-proach is strongly dependent on the intrinsic decay rates ofsuperconducting qubits which, as described below, are pres-ently unknown, but can be determined with the setup pro-posed here. Here we assume the conservative case of anovercoupled resonator with a Q#104, which is preferable forthe first experiments.

B. Artificial atom: The Cooper-pair box

Our choice of “atom,” the Cooper-pair box [6,16], is amesoscopic superconducting island. As shown in Fig. 3, the

FIG. 2. (Color online). Schematic layout and equivalent lumpedcircuit representation of proposed implementation of cavity QEDusing superconducting circuits. The 1D transmission line resonatorconsists of a full-wave section of superconducting coplanar wave-guide, which may be lithographically fabricated using conventionaloptical lithography. A Cooper-pair box qubit is placed between thesuperconducting lines and is capacitively coupled to the center traceat a maximum of the voltage standing wave, yielding a strong elec-tric dipole interaction between the qubit and a single photon in thecavity. The box consists of two small $#100 nm,100 nm% Joseph-son junctions, configured in a #1 +m loop to permit tuning of theeffective Josephson energy by an external flux -ext. Input and out-put signals are coupled to the resonator, via the capacitive gaps inthe center line, from 50) transmission lines which allow measure-ments of the amplitude and phase of the cavity transmission, andthe introduction of dc and rf pulses to manipulate the qubit states.Multiple qubits (not shown) can be similarly placed at differentantinodes of the standing wave to generate entanglement and two-bit quantum gates across distances of several millimeters.

CAVITY QUANTUM ELECTRODYNAMICS FOR… PHYSICAL REVIEW A 69, 062320 (2004)

062320-3


Initial state

2

II. CIRCUIT FOR MEASUREMENT ANDFEEDBACK CONTROL

As shown in Fig.1(a), we consider a superconductingcircuit QED system with a superconducting qubit cou-pled to a microwave readout cavity and driven by twoexternal drives: (i) a read-out drive with amplitude ✏

d

(t)and frequency !

d

near the cavity resonance frequency !c

,and (ii) a Rabi drive with amplitude ✏

r

(t) and frequency!r

near the frequency of the qubit !q

, [21, 31–33]. TheHamiltonian of the entire system can be written as

H = ~!c

a†a+ ~!q2�z

+ ~g(a†�� + a�+)

+~[✏d

(t)e�i!dta† + ✏r

(t)e�i!rta† + h.c.], (1)

where a† and a are the creation and annihilation op-erators for the microwave readout cavity, �+ and ��are the raising and lowering operators of the super-conducting qubit, and g is the coupling strength be-tween the cavity and the qubit. In the dispersive regime[34, 35], |�| = |!

q

� !c

| � g, by applying the dis-persive shift U = exp[g(a�+ � a†��)/�], and movingto the rotating frames for both the qubit and cavity,Uc

= exp(�ia†a!d

t), Uq

= exp(�i�z

!r

t/2), with therotating-wave approximation, the Hamiltonian in Eq. (1)becomes

He↵ = ~�ca†a+ ~�a†a�z + ~!̃q

2�z

+ ~⌦R2

�x

+~⇥✏d

(t)a† + ✏⇤d

(t)a⇤, (2)

where

�c

= !c

� !d

, � = g2/�, ⌦R

= 2✏r

(t)g/�, (3)

and the Lamb-shifted qubit transition frequency

!̃q

= !q

� !r

+ �. (4)

If the cavity state is coherent, and the microwave cavitydecay rate is much larger than the qubit decay rate, ��1 (that allows to decouple the qubit dynamics from theresonator adiabatically), the state at time t is given by|gi⌦ |↵

g

(t)i or |ei⌦ |↵e

(t)i. Here ��↵g(e)(t)

↵are coherent

states of the cavity and, from Eq. (2), the field amplitudesare given by [37],

↵̇g

(t) = �i✏d

(t)� i(�c

� �)↵g

(t)� 2↵g

(t),

↵̇e

(t) = �i✏d

(t)� i(�c

+ �)↵e

(t)� 2↵e

(t). (5)

Thus, these coherent states ↵g(e) act as “pointer states”

[7] for the qubit. Based on homodyne detection, by ap-plying the transformation

P (t) = |eihe|D[↵e

(t)] + |gihg|D[↵g

(t)], (6)

withD[↵] = exp(↵a†�↵⇤a) as the displacement operatorof the microwave cavity, the e↵ective stochastic master

equation for the qubit degrees of freedom is

d⇢̃ = � i~!̃ac

(t)

2[�

z

, ⇢̃] dt� i⌦R2

[�x

, ⇢̃] dt+ �1D [��] ⇢̃dt

+��

+ �d

(t)

2D [�

z

] ⇢̃dt+p⌘ |�(t)|H [�

z

] ⇢̃dWt

.

(7)

Here

!̃ac

(t) = !̃q

+B(t), (8)

and

�(t) = ↵e

(t)� ↵g

(t) (9)

is the separation between the pointer states ↵g

(t) and↵e

(t), ⌘ is the measurement e�ciency, ��

is the puredephasing rate, D[A] is the damping superoperator

D[A]⇢ = A⇢A† � 12(A†A⇢+ ⇢A†A), (10)

and

H [A] ⇢̃ = A⇢̃+ ⇢̃A† � ⌦A+A†↵ ⇢̃. (11)Also,

�d

(t) = 2�Im[↵g

(t)↵⇤e

(t)] (12)

is the measurement-induced dephasing and

B(t) = 2�Re[↵g

(t)↵⇤e

(t)] (13)

is the ac Stark shift. The innovation dWt

is a Wienerprocess [7] with

E [dWt ] = 0, and E[dW2t

] = dt. (14)

Due to the qubit decay �1 and dephasing �� +�d(t), thesystem must quickly lose its quantum features.A coherent drive is turned on for 20 ns to build up

the photon population of the cavity and is then repeatedevery 100 ns [see Fig.1(b)]. The cavity pull �/2⇡ repre-sents the dispersive coupling strength between the cavityphoton number and the qubit [36, 37]. The cavity pullis designed to be �/2⇡ = 5 MHz, and the cavity decayrate is /2⇡ = 20 MHz. A homodyne detection of thereadout cavity field, with the help of the distance �(t)between the states |↵

e

(t)i and |↵g

(t)i, can then be usedto distinguish the coherent states and thus readout thestate of the qubit. By applying the P -transformation tothe in-phase quadrature amplitude

I�

=1

2

⌦ae�i� + a†ei�

↵, (15)

with � the phase of the local oscillation, the homodynemeasurement record from the microwave cavity becomes

I(t) =p⌘ |�(t)| h�

z

(t)i+ ⇠(t) = s(t) + ⇠(t), (16)

act as “pointer states”


Initial state

2



d

(t)and frequency !

d



r

(t) and frequency!r



H = ~!c

a†a+ ~!q2�z

+ ~g(a†�� + a�+)

+~[✏d


(t)e�i!rta† + h.c.], (1)


q

� !c


= exp(�ia†a!d

t), Uq

= exp(�i�z

!r


He↵ = ~�ca†a+ ~�a†a�z + ~!̃q

2�z

+ ~⌦R2

�x

+~⇥✏d

(t)a† + ✏⇤d

(t)a⇤, (2)

where

�c

= !c

� !d

, � = g2/�, ⌦R

= 2✏r

(t)g/�, (3)


!̃q

= !q

� !r

+ �. (4)


g



↵are coherent


↵̇g

(t) = �i✏d

(t)� i(�c

� �)↵g

(t)� 2↵g

(t),

↵̇e

(t) = �i✏d

(t)� i(�c

+ �)↵e

(t)� 2↵e

(t). (5)




(t)] + |gihg|D[↵g

(t)], (6)



d⇢̃ = � i~!̃ac

(t)

2[�

z

, ⇢̃] dt� i⌦R2

[�x

, ⇢̃] dt+ �1D [��] ⇢̃dt

+��

+ �d

(t)

2D [�

z

] ⇢̃dt+p⌘ |�(t)|H [�

z

] ⇢̃dWt

.

(7)

Here

!̃ac

(t) = !̃q

+B(t), (8)

and

�(t) = ↵e

(t)� ↵g

(t) (9)


(t) and↵e



D[A]⇢ = A⇢A† � 12(A†A⇢+ ⇢A†A), (10)

and

H [A] ⇢̃ = A⇢̃+ ⇢̃A† � ⌦A+A†↵ ⇢̃. (11)Also,

�d

(t) = 2�Im[↵g

(t)↵⇤e

(t)] (12)


B(t) = 2�Re[↵g

(t)↵⇤e

(t)] (13)




] = dt. (14)



e

(t)i and |↵g


I�

=1

2


↵, (15)


I(t) =p⌘ |�(t)| h�

z

(t)i+ ⇠(t) = s(t) + ⇠(t), (16)

act as “pointer states”


Measurement induced backaction

20

0 50 1000

0.1

0.2

0.3

t (ns)

Numb

er of

photo

ns

0 50 100

0

4

8

12

14

t (ns)

B(t)

−0.2 0 0.2−0.5

−0.4

−0.3

−0.2

−0.1

0

Re(α)

Im(α)

MHz

2



d

(t)and frequency !

d



r

(t) and frequency!r



H = ~!c

a†a+ ~!q2�z

+ ~g(a†�� + a�+)

+~[✏d


(t)e�i!rta† + h.c.], (1)


q

� !c


= exp(�ia†a!d

t), Uq

= exp(�i�z

!r


He↵ = ~�ca†a+ ~�a†a�z + ~!̃q

2�z

+ ~⌦R2

�x

+~⇥✏d

(t)a† + ✏⇤d

(t)a⇤, (2)

where

�c

= !c

� !d

, � = g2/�, ⌦R

= 2✏r

(t)g/�, (3)


!̃q

= !q

� !r

+ �. (4)


g



↵are coherent


↵̇g

(t) = �i✏d

(t)� i(�c

� �)↵g

(t)� 2↵g

(t),

↵̇e

(t) = �i✏d

(t)� i(�c

+ �)↵e

(t)� 2↵e

(t). (5)




(t)] + |gihg|D[↵g

(t)], (6)



d⇢̃ = � i~!̃ac

(t)

2[�

z

, ⇢̃] dt� i⌦R2

[�x

, ⇢̃] dt+ �1D [��] ⇢̃dt

+��

+ �d

(t)

2D [�

z

] ⇢̃dt+p⌘ |�(t)|H [�

z

] ⇢̃dWt

.

(7)

Here

!̃ac

(t) = !̃q

+B(t), (8)

and

�(t) = ↵e

(t)� ↵g

(t) (9)


(t) and↵e



D[A]⇢ = A⇢A† � 12(A†A⇢+ ⇢A†A), (10)

and

H [A] ⇢̃ = A⇢̃+ ⇢̃A† � ⌦A+A†↵ ⇢̃. (11)Also,

�d

(t) = 2�Im[↵g

(t)↵⇤e

(t)] (12)


B(t) = 2�Re[↵g

(t)↵⇤e

(t)] (13)




] = dt. (14)



e

(t)i and |↵g


I�

=1

2


↵, (15)


I(t) =p⌘ |�(t)| h�

z

(t)i+ ⇠(t) = s(t) + ⇠(t), (16)

ac Stark shift

2



d

(t)and frequency !

d



r

(t) and frequency!r



H = ~!c

a†a+ ~!q2�z

+ ~g(a†�� + a�+)

+~[✏d


(t)e�i!rta† + h.c.], (1)


q

� !c


= exp(�ia†a!d

t), Uq

= exp(�i�z

!r


He↵ = ~�ca†a+ ~�a†a�z + ~!̃q

2�z

+ ~⌦R2

�x

+~⇥✏d

(t)a† + ✏⇤d

(t)a⇤, (2)

where

�c

= !c

� !d

, � = g2/�, ⌦R

= 2✏r

(t)g/�, (3)


!̃q

= !q

� !r

+ �. (4)


g



↵are coherent


↵̇g

(t) = �i✏d

(t)� i(�c

� �)↵g

(t)� 2↵g

(t),

↵̇e

(t) = �i✏d

(t)� i(�c

+ �)↵e

(t)� 2↵e

(t). (5)




(t)] + |gihg|D[↵g

(t)], (6)



d⇢̃ = � i~!̃ac

(t)

2[�

z

, ⇢̃] dt� i⌦R2

[�x

, ⇢̃] dt+ �1D [��] ⇢̃dt

+��

+ �d

(t)

2D [�

z

] ⇢̃dt+p⌘ |�(t)|H [�

z

] ⇢̃dWt

.

(7)

Here

!̃ac

(t) = !̃q

+B(t), (8)

and

�(t) = ↵e

(t)� ↵g

(t) (9)


(t) and↵e



D[A]⇢ = A⇢A† � 12(A†A⇢+ ⇢A†A), (10)

and

H [A] ⇢̃ = A⇢̃+ ⇢̃A† � ⌦A+A†↵ ⇢̃. (11)Also,

�d

(t) = 2�Im[↵g

(t)↵⇤e

(t)] (12)


B(t) = 2�Re[↵g

(t)↵⇤e

(t)] (13)




] = dt. (14)



e

(t)i and |↵g


I�

=1

2


↵, (15)


I(t) =p⌘ |�(t)| h�

z

(t)i+ ⇠(t) = s(t) + ⇠(t), (16)

measurement induced dephasing


Feedback control of Rabi oscillations in circuit QED

21


4

Paramp HEMT

LO QI

IQ

RFRF

Transmonqubit

Digitizer/computer

Dilution refrigerator

Feedback circuit

Homodyne setupSignal generation

Phase error = Feedback signal

a

b

c

d

IN

OUT

Readoutcavity

Rabi reference3.0 MHz

X

X*Y

Y

AnalogMultiplier

Rabi drive

Readoutdrive LO

(x2)

10 MHz

|1>

|0>

Q

I

|1>

|0>Q

I

FIG. 1: Experimental setup. (a) shows the signal generation setup. One generator provides the Rabi drive at theac Stark shifted qubit frequency (!

01

� 2�n̄), while the output of another generator at 7.2749 GHz is split to create themeasurement signal, paramp drive and local oscillator. The relative amplitudes and phases of these three signals are controlledby variable attenuators and phase shifters (not shown). (b) shows a simplified version of the cryogenic part of the experiment;all components are at 30 mK (except for the HEMT amplifier, which is at 4 K). The combined qubit and measurementsignals enter the weakly coupled cavity port, interact with the qubit, and leave from the strongly coupled port. The outputpasses through two isolators (which protect the qubit from the strong paramp drive), is amplified, and then continues to thedemodulation setup. The coherent state at the output of the cavity for the ground and excited states is shown schematicallybefore and after parametric amplification. (c) The amplified signal is homodyne detected and the two quadratures are digitized.The amplified quadrature (Q) is split o↵ and sent to the feedback circuit (d), where it is multiplied with the Rabi referencesignal. The product is low-pass filtered and fed back to the IQ mixer in (a) to modulate the Rabi drive amplitude.

Irfan Siddiqi，et. al., Nature 490, 77 (2012)

5

Feedback ONDig

itize

r V

olta

ge

(V

)

Time (µs)

-0.04

0.00

0.04

151050 1000995990985980

-0.04

0.00

0.04

1086420

Dig

itize

r V

olta

ge

(V

)

Time (µs)

Feedback OFF

a

d

b0.020

0.010

0.0004.03.53.02.52.0

Po

we

r S

pe

ctru

m (

V2/H

z)

Frequency (MHz)

Γ/2π

FeedbackOFF

FeedbackON

Squeezed quadrature(x20)

Γ/

2π

(M

Hz)

n

c0.3

0.2

0.1

0.00.80.40.0

Γenv∼∼

FIG. 2: Rabi oscillations and feedback. (a) shows conventional ensemble-averaged Rabi oscillations measured using weakcontinuous measurement, which decay in time due to ensemble dephasing. In (b), the individual measurement traces from(a) are Fourier transformed before averaging. The averaged spectrum shows a peak at the Rabi frequency (blue trace) witha width �/2⇡ (FWHH). The grey trace shows an identically prepared spectrum for the squeezed quadrature (multiplied by 20for clarity), which contains no qubit state information. � is plotted as a function of cavity photon occupation n̄ (measurementstrength) in (c), showing the expected linear dependence. (d) shows feedback-stabilized ensemble averaged Rabi oscillations,which persist for much longer times than those without feedback seen in (a). The corresponding spectrum, shown in (b), hasa needle-like peak at the Rabi reference frequency (red trace). The slowly changing mean level in the Rabi oscillation tracesin (a) and (d) is due to the thermal transfer of population into the second excited state of the qubit. See section IV(C) ofsupplementary information for more details.

Fe

ed

ba

ck e

ffic

ien

cy (

D)

Feedback Strength (F)τtomo (ns)

η = 0.40Γ/2π = 0.154 MHz

ExperimentAnalytical TheoryNumerical Simulations

b

<σ

X,Y

,Z>

n = 0.47

-1.0

-0.5

0.0

0.5

1.0

300250200150100500

ca0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00.0750.0500.0250.000

σY

σX=0

σZ

FIG. 3: Tomography and feedback e�ciency. (a) shows quantum state tomography of the feedback-stabilized state. Weplot h�Xi , h�Y i and h�Zi for di↵erent time points ⌧tomo in one full Rabi oscillation of the qubit. The solid lines are sinusoidalfits. The magnitude of these sinusoidal oscillations is approximately equal to the feedback e�ciency D = 0.45. (b) shows avisualization of the feedback stabilized Bloch vector for D < 1. The Bloch vector during a single measurement remains roughlywithin a certain angle (shaded pie region) of the desired state; higher the e�ciency D, smaller the angle. Tomography of thestabilized state measures the average over many iterations which reduces the length of the averaged Bloch vector (red arrow).In (c), we plot D as a function of the dimensionless feedback strength F . Solid red squares are experimental data with amaximum value of D = 0.45. The dashed black line is a plot of Eq. (1) with ⌘ = 0.40 and �/2⇡ = 0.154 MHz (n̄ = 0.47,�env

/2⇡ = 0.020 MHz), while the solid black line is obtained from full numerical simulations of the Bayesian equations includingfinite loop delay (250 ns) and feedback bandwidth (10 MHz).

5

Feedback ONDig

itize

r V

olta

ge

(V

)

Time (µs)

-0.04

0.00

0.04

151050 1000995990985980

-0.04

0.00

0.04

1086420

Dig

itize

r V

olta

ge

(V

)

Time (µs)

Feedback OFF

a

d

b0.020

0.010

0.0004.03.53.02.52.0

Po

we

r S

pe

ctru

m (

V2/H

z)

Frequency (MHz)

Γ/2π

FeedbackOFF

FeedbackON

Squeezed quadrature(x20)

Γ/

2π

(M

Hz)

n

c0.3

0.2

0.1

0.00.80.40.0

Γenv∼∼

FIG. 2: Rabi oscillations and feedback. (a) shows conventional ensemble-averaged Rabi oscillations measured using weakcontinuous measurement, which decay in time due to ensemble dephasing. In (b), the individual measurement traces from(a) are Fourier transformed before averaging. The averaged spectrum shows a peak at the Rabi frequency (blue trace) witha width �/2⇡ (FWHH). The grey trace shows an identically prepared spectrum for the squeezed quadrature (multiplied by 20for clarity), which contains no qubit state information. � is plotted as a function of cavity photon occupation n̄ (measurementstrength) in (c), showing the expected linear dependence. (d) shows feedback-stabilized ensemble averaged Rabi oscillations,which persist for much longer times than those without feedback seen in (a). The corresponding spectrum, shown in (b), hasa needle-like peak at the Rabi reference frequency (red trace). The slowly changing mean level in the Rabi oscillation tracesin (a) and (d) is due to the thermal transfer of population into the second excited state of the qubit. See section IV(C) ofsupplementary information for more details.

Fe

ed

ba

ck e

ffic

ien

cy (

D)

Feedback Strength (F)τtomo (ns)

η = 0.40Γ/2π = 0.154 MHz

ExperimentAnalytical TheoryNumerical Simulations

b

<σ

X,Y

,Z>

n = 0.47

-1.0

-0.5

0.0

0.5

1.0

300250200150100500

ca0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00.0750.0500.0250.000

σY

σX=0

σZ

FIG. 3: Tomography and feedback e�ciency. (a) shows quantum state tomography of the feedback-stabilized state. Weplot h�Xi , h�Y i and h�Zi for di↵erent time points ⌧tomo in one full Rabi oscillation of the qubit. The solid lines are sinusoidalfits. The magnitude of these sinusoidal oscillations is approximately equal to the feedback e�ciency D = 0.45. (b) shows avisualization of the feedback stabilized Bloch vector for D < 1. The Bloch vector during a single measurement remains roughlywithin a certain angle (shaded pie region) of the desired state; higher the e�ciency D, smaller the angle. Tomography of thestabilized state measures the average over many iterations which reduces the length of the averaged Bloch vector (red arrow).In (c), we plot D as a function of the dimensionless feedback strength F . Solid red squares are experimental data with amaximum value of D = 0.45. The dashed black line is a plot of Eq. (1) with ⌘ = 0.40 and �/2⇡ = 0.154 MHz (n̄ = 0.47,�env

/2⇡ = 0.020 MHz), while the solid black line is obtained from full numerical simulations of the Bayesian equations includingfinite loop delay (250 ns) and feedback bandwidth (10 MHz).


Theory: Bayesian feedback

23University of California, RiversideAlexander Korotkov

"Quantum Bayes theorem“ (ideal detector assumed)

eH

I(t)

|1> |2>

2

2

11 1 22 2

1 2

01 ( )

( , ) (0) ( , ) (0) ( , )

1( , ) exp[ ( ) / 2 ],2

/ 2 , | | , /II i i

i i

I I t dt

P I P I P I

P I I I DD

D S I I I S I

τ

ττ ρ τ ρ τ

τπ

τ τ

≡

= +

= − −

= −

11 12

21 22

(0) (0)(0) (0)

ρ ρ

ρ ρ

Measurement (during time τ):

H = ε = 0 (frozen qubit)Initial state:

I_

P

I1 I2

Iactual_

2D1/2

( ) ( | )( | )

( ) ( | )k kki i

iP B P A B

P B AP B P A B

=After the measurement during time τ, the probabilities can be updated using the standard Bayes formula:

211 1

11 2 211 1 22 2

12 1222 111/2 1/2

12 22 12 22

(0) exp[ ( ) / 2 ]( )(0) exp[ ( ) / 2 ] (0) exp[ ( ) / 2 ]

( ) (0) , ( ) 1 ( )[ ( ) ( )] [ (0) (0)]

I I DI I D I I Dρ

ρ τρ ρ

ρ τ ρρ τ ρ τ

ρ τ ρ τ ρ ρ

− −=

− − + − −

= = −

Quantum Bayesformulas:

University of California, RiversideAlexander Korotkov


eH

I(t)

|1> |2>

2

2

11 1 22 2

1 2

01 ( )

( , ) (0) ( , ) (0) ( , )

1( , ) exp[ ( ) / 2 ],2

/ 2 , | | , /II i i

i i

I I t dt

P I P I P I

P I I I DD

D S I I I S I

τ

ττ ρ τ ρ τ

τπ

τ τ

≡

= +

= − −

= −

11 12

21 22

(0) (0)(0) (0)

ρ ρ

ρ ρ



I_

P

I1 I2

Iactual_

2D1/2

( ) ( | )( | )

( ) ( | )k kki i

iP B P A B

P B AP B P A B


211 1

11 2 211 1 22 2

12 1222 111/2 1/2

12 22 12 22

(0) exp[ ( ) / 2 ]( )(0) exp[ ( ) / 2 ] (0) exp[ ( ) / 2 ]

( ) (0) , ( ) 1 ( )[ ( ) ( )] [ (0) (0)]

I I DI I D I I Dρ

ρ τρ ρ

ρ τ ρρ τ ρ τ

ρ τ ρ τ ρ ρ

− −=

− − + − −

= = −


Measurement output:

Conditional probability distributions:



eH

I(t)

|1> |2>

2

2

11 1 22 2

1 2

01 ( )

( , ) (0) ( , ) (0) ( , )

1( , ) exp[ ( ) / 2 ],2

/ 2 , | | , /II i i

i i

I I t dt

P I P I P I

P I I I DD

D S I I I S I

τ

ττ ρ τ ρ τ

τπ

τ τ

≡

= +

= − −

= −

11 12

21 22

(0) (0)(0) (0)

ρ ρ

ρ ρ



I_

P

I1 I2

Iactual_

2D1/2

( ) ( | )( | )

( ) ( | )k kki i

iP B P A B

P B AP B P A B


211 1

11 2 211 1 22 2

12 1222 111/2 1/2

12 22 12 22

(0) exp[ ( ) / 2 ]( )(0) exp[ ( ) / 2 ] (0) exp[ ( ) / 2 ]

( ) (0) , ( ) 1 ( )[ ( ) ( )] [ (0) (0)]

I I DI I D I I Dρ

ρ τρ ρ

ρ τ ρρ τ ρ τ

ρ τ ρ τ ρ ρ

− −=

− − + − −

= = −


5

B. Numerical simulations

We now discuss numerical simulations of the Bayesian equations. To avoid the complications with the Stratonovichvs the Itô form for the stochastic di↵erential equations, we used a two step process to evolve equations (2) and (3).First, we set �I = 0 i.e. we suppress the measurement and evolve the the resulting ordinary di↵erential equationsusing a 4th order Runge-Kutta step. We then include the e↵ect of the measurement by performing a Bayesian2,6

update which we describe below. The measurement output Im

= ⌧�1R t+⌧t I(t

0)dt

0in a given time interval ⌧ , is drawn

from a Gaussian probability distribution with standard deviation � =pS

id

/(2⌧) and centered around I0

and I1

forthe qubit in state |0i and |1i respectively. The conditional probability distributions are given by

P (Im

| |0i) = 1p2⇡�

exp

� (Im � I0)

2

2�2

�, P (I

m

| |1i) = 1p2⇡�

exp

� (Im � I1)

2

2�2

�(19)

Given an initial qubit state ⇢(t), the measurement outcome is drawn from a combined probability distribution

P (Im

) = ⇢00

(t)P (Im

| |0i) + ⇢11

(t)P (Im

| |1i). (20)

We use a combination of a binomial and a Gaussian random number generator to create a measurement outcome Im

which is then used to update the qubit state using the following equations2

⇢

11

(t+ ⌧) =⇢

11

(t)P (Im

| |1i)P (I

m

), ⇢

00

(t+ ⌧) =⇢

00

(t)P (Im

| |0i)P (I

m

)(21)

⇢

01

(t+ ⌧) = ⇢01

(t)

p⇢

11

(t+ ⌧)⇢00

(t+ ⌧)p⇢

11

(t)⇢00

(t)(22)

This process is repeated for each time step to obtain the qubit density matrix ⇢(t) and the measurement outputI(t) as a function of time. If ⌘

det

< 1, we add ⇠add

(t) to I(t) which is generated using an appropriate Gaussianwhite noise generator. Since �

env

is now included in equation (3), the spectral density for the added noise is given byS

add

= Sout

� Sid

, where Sout

= Sid

/⌘

det

is the total output noise. In other words, the extra noise only correspondsto detector ine�ciency.

The output signal I(t) is low pass filtered with a 10 MHz cuto↵ to account for the bandwidth of the paramp. Tocreate the feedback signal, we remove any dc o↵sets and multiply this output with the reference signal sin(⌦

0

t), where⌦

0

/2⇡ = 3 MHz. We then implement feedback by modifying ⌦R ! ⌦R(t) in equations (2) and (3) using equations(10) and (11). Feedback loop delay (⌧

delay

) is included by modifying the R.H.S of equation (11) so that t ! t� ⌧delay

while feedback circuit bandwidth is included by filtering ⌦fb

(t) with a 10 MHz low-pass filter before adding it toequation (10) . With the feedback modified qubit state ⇢(t) we can compute feedback e�ciency D as described in theprevious section.

C. Thermal fluctuations and higher qubit levels

The discussion so far has assumed that the e↵ective temperature of the qubit Tqubit

⌧ ~!01

/kB where !01 is thetransition frequency between the ground and first excited state. Even though the dilution fridge temperature T = 30mK, we find significant thermal population of the first excited state corresponding to an e↵ective temperature ofapproximately 140 mK. We believe that this is due improper thermalization of the qubit sample inside the Aluminumcavity. Further, the transmon qubit has higher levels with similar transition frequencies between neighbouring levels(!

12

. !01

) and we observe few percent population in the second (and higher) excited states.

We measure these populations using strong measurements which allows us to discriminate between the first 4 levelsof the transmon with high single-shot fidelity11–13. Fig. S2 shows the single-shot histograms and one can clearlyresolve four peaks in the distribution. These correspond to populations P

0

= 0.83, P1

= 0.13, P2

= 0.03, P3+

= 0.01.Further, in the presence of Rabi driving (with or without feedback) the population in the 2nd excited state is enhancedup to 7.5 % and 2.5 % in higher levels. As mentioned in the main text, staet tomography of the stabilized state is

5




= ⌧�1R t+⌧t I(t

0)dt



id


and I1


P (Im

| |0i) = 1p2⇡�

exp

� (Im � I0)

2

2�2

�, P (I

m

| |1i) = 1p2⇡�

exp

� (Im � I1)

2

2�2

�(19)


P (Im

) = ⇢00

(t)P (Im

| |0i) + ⇢11

(t)P (Im

| |1i). (20)



⇢

11

(t+ ⌧) =⇢

11

(t)P (Im

| |1i)P (I

m

), ⇢

00

(t+ ⌧) =⇢

00

(t)P (Im

| |0i)P (I

m

)(21)

⇢

01

(t+ ⌧) = ⇢01

(t)

p⇢

11

(t+ ⌧)⇢00

(t+ ⌧)p⇢

11

(t)⇢00

(t)(22)


det

< 1, we add ⇠add


env


add

= Sout

� Sid

, where Sout

= Sid

/⌘

det



0

t), where⌦

0


delay






⌧ ~!01


12

. !01



0

= 0.83, P1

= 0.13, P2

= 0.03, P3+


5




= ⌧�1R t+⌧t I(t

0)dt



id


and I1


P (Im

| |0i) = 1p2⇡�

exp

� (Im � I0)

2

2�2

�, P (I

m

| |1i) = 1p2⇡�

exp

� (Im � I1)

2

2�2

�(19)


P (Im

) = ⇢00

(t)P (Im

| |0i) + ⇢11

(t)P (Im

| |1i). (20)



⇢

11

(t+ ⌧) =⇢

11

(t)P (Im

| |1i)P (I

m

), ⇢

00

(t+ ⌧) =⇢

00

(t)P (Im

| |0i)P (I

m

)(21)

⇢

01

(t+ ⌧) = ⇢01

(t)

p⇢

11

(t+ ⌧)⇢00

(t+ ⌧)p⇢

11

(t)⇢00

(t)(22)


det

< 1, we add ⇠add


env


add

= Sout

� Sid

, where Sout

= Sid

/⌘

det



0

t), where⌦

0


delay






⌧ ~!01


12

. !01



0

= 0.83, P1

= 0.13, P2

= 0.03, P3+


standard deviation

5




= ⌧�1R t+⌧t I(t

0)dt



id


and I1


P (Im

| |0i) = 1p2⇡�

exp

� (Im � I0)

2

2�2

�, P (I

m

| |1i) = 1p2⇡�

exp

� (Im � I1)

2

2�2

�(19)


P (Im

) = ⇢00

(t)P (Im

| |0i) + ⇢11

(t)P (Im

| |1i). (20)



⇢

11

(t+ ⌧) =⇢

11

(t)P (Im

| |1i)P (I

m

), ⇢

00

(t+ ⌧) =⇢

00

(t)P (Im

| |0i)P (I

m

)(21)

⇢

01

(t+ ⌧) = ⇢01

(t)

p⇢

11

(t+ ⌧)⇢00

(t+ ⌧)p⇢

11

(t)⇢00

(t)(22)


det

< 1, we add ⇠add


env


add

= Sout

� Sid

, where Sout

= Sid

/⌘

det



0

t), where⌦

0


delay






⌧ ~!01


12

. !01



0

= 0.83, P1

= 0.13, P2

= 0.03, P3+


Combined probability distributions:

5




= ⌧�1R t+⌧t I(t

0)dt



id


and I1


P (Im

| |0i) = 1p2⇡�

exp

� (Im � I0)

2

2�2

�, P (I

m

| |1i) = 1p2⇡�

exp

� (Im � I1)

2

2�2

�(19)


P (Im

) = ⇢00

(t)P (Im

| |0i) + ⇢11

(t)P (Im

| |1i). (20)



⇢

11

(t+ ⌧) =⇢

11

(t)P (Im

| |1i)P (I

m

), ⇢

00

(t+ ⌧) =⇢

00

(t)P (Im

| |0i)P (I

m

)(21)

⇢

01

(t+ ⌧) = ⇢01

(t)

p⇢

11

(t+ ⌧)⇢00

(t+ ⌧)p⇢

11

(t)⇢00

(t)(22)


det

< 1, we add ⇠add


env


add

= Sout

� Sid

, where Sout

= Sid

/⌘

det



0

t), where⌦

0


delay






⌧ ~!01


12

. !01



0

= 0.83, P1

= 0.13, P2

= 0.03, P3+




eH

I(t)

|1> |2>

2

2

11 1 22 2

1 2

01 ( )

( , ) (0) ( , ) (0) ( , )

1( , ) exp[ ( ) / 2 ],2

/ 2 , | | , /II i i

i i

I I t dt

P I P I P I

P I I I DD

D S I I I S I

τ

ττ ρ τ ρ τ

τπ

τ τ

≡

= +

= − −

= −

11 12

21 22

(0) (0)(0) (0)

ρ ρ

ρ ρ



I_

P

I1 I2

Iactual_

2D1/2

( ) ( | )( | )

( ) ( | )k kki i

iP B P A B

P B AP B P A B


211 1

11 2 211 1 22 2

12 1222 111/2 1/2

12 22 12 22

(0) exp[ ( ) / 2 ]( )(0) exp[ ( ) / 2 ] (0) exp[ ( ) / 2 ]

( ) (0) , ( ) 1 ( )[ ( ) ( )] [ (0) (0)]

I I DI I D I I Dρ

ρ τρ ρ

ρ τ ρρ τ ρ τ

ρ τ ρ τ ρ ρ

− −=

− − + − −

= = −


Update the qubit state as:

A. N. Korotkov, PRB 63, 115403(2001)


Circuit for measurement and feedback control

24

Feedback Control of Rabi Oscillations in Circuit QED

Wei Cui1 and Franco Nori1, 2

1Advanced Science Institute, RIKEN, Wako-shi, Saitama 351-0198, Japan

2Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

(Dated: February 12, 2013)

We consider the feedback stabilization of Rabi oscillations in a superconducting qubit which iscoupled to a microwave readout cavity. The signal is readout by homodyne detection of the in-phase quadrature amplitude of the weak measurement output. By multiplying the time-delayedRabi reference, one can extract the signal, with maximum signal-to-noise ratio, from the noise. Wefurther track and stabilize the Rabi oscillations by using Lyapunov feedback control to properlyadjust the input Rabi drives. Theoretical and simulation results illustrate the e↵ectiveness of theproposed control law.

PACS numbers: 42.50.Dv, 85.25.-j

In control theory, the signal to be controlled is com-pared to the desired reference, and the discrepancy isused to correct the control action. In contrast to classi-cal systems, where measurements do not alter the stateof the system, quantum measurements will collapse thesystem instaneously into one of its eigenstates in a proba-bilistic manner: the “measurement-induced backaction”.Based on the quantum trajectory theory, Wiseman andMilburn [1] developed a quantum conditional stochasticmaster equation (SME) to describe the dynamics result-ing from the feedback (of the measurement output at eachinstant) to the quantum system. SME has been a topicof considerable activity in recent years for it paves theway for studying real-time feedback control in quantuminformation processing [2–4].

Circuit quantum electrodynamics (i.e., circuit QED,where a superconducting qubit is coupled to a microwave-frequency resonator cavity; see, e.g., [5–8]) has beenshown to be a promising solid-state quantum computingarchitecture. Circuit QED provides several simple high-fidelity readout mechanisms, such as using large measure-ment drive powers [9], and using either quantum-limited[10] or nonlinear bifurcation amplifiers [11]. Moreover,circuit QED is an excellent test-bed for implementingquantum feedback control in either the qubits or the mi-crowave resonator [12–20]. For example, a recent work[21] has been shown that quantum feedback control canreduce dephasing and remarkably prolong the Rabi oscil-lations.

Here, we analytically derive a simple andexperimentally-feasible feedback control law for cir-cuit QED to track and stabilize Rabi oscillations. Asshown in Fig.1, we consider a superconducting circuitQED system with a superconducting qubit coupled toa microwave readout cavity and driven by two externaldrives: (i) a read-out drive with amplitude ✏

d

(t) andfrequency !

d



r

(t) and frequency!r


, [11, 21]. The

Output

Input

Amplifier

Qubit Resonator

HomodyneMeasurement

Multiplier

TimeDelay

RabiReference

+_

Rabi DriveFeedback Law

Read-outDrive

FIG. 1: (Color online) Simplified circuit diagram of measure-ment and feedback control. A superconducting qubit (yellow)is coupled to a microwave readout cavity (blue). The ampli-fied output is homodyne-detected and the quadrature signal isthen extracted from the noise by multiplying the time-delayedRabi reference (green). The discrepancy is used to design thefeedback control law to correct Rabi oscillations (red).

Hamiltonian of the whole system can be written as

H = ~!c

a†a+ ~!q2�z

+ ~g(a†�� + a�+)

+~[✏d


(t)e�i!rta† + h.c], (1)

where a† and a are the creation and annihilation oper-ators for the resonator, �+ and �� are the raising andlowering operators of the superconducting qubit, and gis the coupling strength between the resonator and thequbit. In the dispersive regime, |�| = |!

q

� !c

| � g, byapplying the dispersive shift U = eg(a�+�a

†��)/�, and

moving to the rotating frames for both the qubit andcavity, U

c

= e�ia†a!dt, U

q

= e�i�z!rt/2 [22], with therotating-wave approximation, the Hamiltonian in Eq. (1)becomes

He↵ = ~�ca†a+ ~�a†a�z + ~!̃q

2�z

+ ~⌦R2

�x

+~⇥✏d

(t)a† + ✏⇤d

(t)a⇤, (2)

where �c

= !c

�!d

, � = g2/�, ⌦R

= 2✏r

(t)g/� and theLamb-shifted qubit transition frequency !̃

q

= !q

� !r

+


System Hamiltonian

25

2



d

(t)and frequency !

d



r

(t) and frequency!r



H = ~!c

a†a+ ~!q2�z

+ ~g(a†�� + a�+)

+~[✏d


(t)e�i!rta† + h.c.], (1)


q

� !c


= exp(�ia†a!d

t), Uq

= exp(�i�z

!r


He↵ = ~�ca†a+ ~�a†a�z + ~!̃q

2�z

+ ~⌦R2

�x

+~⇥✏d

(t)a† + ✏⇤d

(t)a⇤, (2)

where

�c

= !c

� !d

, � = g2/�, ⌦R

= 2✏r

(t)g/�, (3)


!̃q

= !q

� !r

+ �. (4)


g



↵are coherent


↵̇g

(t) = �i✏d

(t)� i(�c

� �)↵g

(t)� 2↵g

(t),

↵̇e

(t) = �i✏d

(t)� i(�c

+ �)↵e

(t)� 2↵e

(t). (5)




(t)] + |gihg|D[↵g

(t)], (6)



d⇢̃ = � i~!̃ac

(t)

2[�

z

, ⇢̃] dt� i⌦R2

[�x

, ⇢̃] dt+ �1D [��] ⇢̃dt

+��

+ �d

(t)

2D [�

z

] ⇢̃dt+p⌘ |�(t)|H [�

z

] ⇢̃dWt

.

(7)

Here

!̃ac

(t) = !̃q

+B(t), (8)

and

�(t) = ↵e

(t)� ↵g

(t) (9)


(t) and↵e



D[A]⇢ = A⇢A† � 12(A†A⇢+ ⇢A†A), (10)

and

H [A] ⇢̃ = A⇢̃+ ⇢̃A† � ⌦A+A†↵ ⇢̃. (11)Also,

�d

(t) = 2�Im[↵g

(t)↵⇤e

(t)] (12)


B(t) = 2�Re[↵g

(t)↵⇤e

(t)] (13)




] = dt. (14)



e

(t)i and |↵g


I�

=1

2


↵, (15)


I(t) =p⌘ |�(t)| h�

z

(t)i+ ⇠(t) = s(t) + ⇠(t), (16)

2



d

(t)and frequency !

d



r

(t) and frequency!r



H = ~!c

a†a+ ~!q2�z

+ ~g(a†�� + a�+)

+~[✏d


(t)e�i!rta† + h.c.], (1)


q

� !c


= exp(�ia†a!d

t), Uq

= exp(�i�z

!r


He↵ = ~�ca†a+ ~�a†a�z + ~!̃q

2�z

+ ~⌦R2

�x

+~⇥✏d

(t)a† + ✏⇤d

(t)a⇤, (2)

where

�c

= !c

� !d

, � = g2/�, ⌦R

= 2✏r

(t)g/�, (3)


!̃q

= !q

� !r

+ �. (4)


g



↵are coherent


↵̇g

(t) = �i✏d

(t)� i(�c

� �)↵g

(t)� 2↵g

(t),

↵̇e

(t) = �i✏d

(t)� i(�c

+ �)↵e

(t)� 2↵e

(t). (5)




(t)] + |gihg|D[↵g

(t)], (6)



d⇢̃ = � i~!̃ac

(t)

2[�

z

, ⇢̃] dt� i⌦R2

[�x

, ⇢̃] dt+ �1D [��] ⇢̃dt

+��

+ �d

(t)

2D [�

z

] ⇢̃dt+p⌘ |�(t)|H [�

z

] ⇢̃dWt

.

(7)

Here

!̃ac

(t) = !̃q

+B(t), (8)

and

�(t) = ↵e

(t)� ↵g

(t) (9)


(t) and↵e



D[A]⇢ = A⇢A† � 12(A†A⇢+ ⇢A†A), (10)

and

H [A] ⇢̃ = A⇢̃+ ⇢̃A† � ⌦A+A†↵ ⇢̃. (11)Also,

�d

(t) = 2�Im[↵g

(t)↵⇤e

(t)] (12)


B(t) = 2�Re[↵g

(t)↵⇤e

(t)] (13)




] = dt. (14)



e

(t)i and |↵g


I�

=1

2


↵, (15)


I(t) =p⌘ |�(t)| h�

z

(t)i+ ⇠(t) = s(t) + ⇠(t), (16)

2



d

(t)and frequency !

d



r

(t) and frequency!r



H = ~!c

a†a+ ~!q2�z

+ ~g(a†�� + a�+)

+~[✏d


(t)e�i!rta† + h.c.], (1)


q

� !c


= exp(�ia†a!d

t), Uq

= exp(�i�z

!r


He↵ = ~�ca†a+ ~�a†a�z + ~!̃q

2�z

+ ~⌦R2

�x

+~⇥✏d

(t)a† + ✏⇤d

(t)a⇤, (2)

where

�c

= !c

Feedback Control of Rabi Oscillations in Circuit QED · 2020. 1. 6. · Rabi Oscillations in...

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