Superconducting qubits - Riken · In superconducting qubits, there is no obvious analog for such...

1

Superconducting qubitsFranco Nori

Physics Dept., The University of Michigan, Ann Arbor, USA

Frontier Research System, RIKEN, Japan

Yu-xi Liu, L.F. Wei, S. Ashhab, J.R. JohanssonGroup members:

Collaborators:

J.Q. You, C.P. Sun, J.S. Tsai, M. Grajcar, A.M. Zagoskin,Y. Nakamura, Y. Pashkin

Funding 2002 -- 2005: NSA, ARDA, AFOSR, NSFFunding 2006 --: NSA, LPS, ARO, NSF, JST

2

Quick overview (just three slides!)

of several types ofsuperconducting qubits

The following figures are taken from the short review in:

You and Nori, Physics Today (November 2005)

(complimentary color reprints available after this talk)

3

Charge qubit

4

Flux qubit

6

Now we will focus on a few results from our group

No time to cover other results

7

ContentsFlux qubits as micromasers and artificial atoms

Cavity QED on a chip

Coupling qubits

Controllable coupling between qubits:

via variable frequency magnetic fields

via data buses

Scalable circuits

Testing Bell inequalities. Generating GHZ states.

Quantum tomography

Conclusions

8

ContentsFlux qubits as micromasers and artificial atomsCavity QED on a chip

Coupling qubits



via data buses

Scalable circuits


Quantum tomography

Conclusions

9

Qubit = Two-level quantum system

Chiorescu et al, Science 299, 1869 (2003) You and Nori, Phys. Today 58 (11), 42 (2005)

Reduced magnetic flux: f = Φe / Φ0. Here: Φe = external DC bias flux

10

Flux qubit (here we consider the three lowest energy levels)

ΦeI

EJ2 EJ2

αEJ2

( ) ( )

2 2

0 ( , , )2 2

2 1 cos cos 1 cos 2 2

p mp m

p m

J p m J m

P PH U fM M

U E E f

ϕ ϕ

ϕ ϕ α ϕ π

= + +

= − + − +⎡ ⎤⎣ ⎦0

ef Φ=

Φ

Phases and momenta (conjugate variables) are (see, e.g., Orlando et al, PRB (1999))

1 2 1 2 ( )/2; ( )/2; / ( , )p m k kP i k p mϕ φ φ ϕ φ φ ϕ= + = − = − ∂ ∂ =h

Effective masses

( )20 / 2 2 ; 2 (1 2 ) with capacitance of the junctionp m pM C M M Cπ α= Φ = +

11

Flux qubit: Symmetry and parity

Φe+ Φa(t)I

EJ2 EJ2

αEJ2m pParity of U( , ) Uϕ ϕ ≡

( ) ( )2 1 cos cos 1 cos 2 2J p m J mU E E fϕ ϕ α ϕ π= − + − +⎡ ⎤⎣ ⎦

( ) ( )

2

0

(0)

2

J

0

(

2( ) sin 2 2 cos

, , )2 2

am i

mm

j

p mp

p

EV

P PH UM

t

f

t

M

fα π ω

ϕ ϕ

π ϕΦ= − +

Φ

= + +

0

0

(|)

| m

H HH

V tm E m

= +

=

Transition elements are

( )(0)

J

0

2 |sin 2 2 |aij m

Et i f jαπ π ϕΦ= − +

Φ

Liu, You, Wei, Sun, Nori, PRL 95, 087001 (2005)

( 0 )( ) cos( )a a ijt tωΦ = Φ

Time-dependent magnetic flux

( ) ( )2 1 cos cos 1 cos 2J p m J mU E Eϕ ϕ α ϕ= − + +⎡ ⎤⎣ ⎦

1/2 ( , ) even function of and m p m pf U ϕ ϕ ϕ ϕ= ⇒

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Flux qubit: Symmetry and parity

In standard atoms, electric-dipole-induced selection rulesfor transitions satisfy :

1 and 0, 1l m∆ = ± ∆ = ±

In superconducting qubits, there is no obvious analog for such selection rules.

Here, we consider an analog based on the symmetry of the potential U(ϕm, ϕp)and the interaction between:

-) superconducting qubits (usual atoms) and the

-) magnetic flux (electric field).

Liu, You, Wei, Sun, Nori, PRL (2005)

13

Allowed three-level transitions in natural atoms

V - type Ξ - type or ladder

Λ - type No ∆ – typebecause of the electric-dipole selection rule.

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Some differences between artificial and natural atoms

In natural atoms, it is not possible to obtain cyclic transitions by only using the electric-dipole interaction, due to its well-defined symmetry.

However, these transitions can be naturally obtained in the flux qubit circuit, due to the broken symmetry of the potential of the flux qubit, when the bias flux deviates from the optimal point.

The magnetic-field-induced transitions in the flux qubitare similar to atomic electric-dipole-induced transitions.


15

Different transitions in three-level systems

V - type Ξ- type or ladder( f = 1/2 )

Λ - type∆ – typecan be obtained using flux qubits(f away from 1/2)


16

Flux qubit: micromaser

You, Liu, Sun, Nori, quant-ph / 0512145

We propose a tunable on-chip micromaser using a superconducting quantum circuit (SQC).

By taking advantage of externally controllable state transitions, a state population inversion can be achieved and preserved for the two working levels of the SQC and, when needed, the SQC can generate a single photon.

17

Adiabatic control and population transfer

Φe + Φa (t)

EJ2 EJ2

α EJ2

( ) ( ) ( )

( ) ( )( )

2

0

2

int0

exp H.c.

exp H.c.

a mn mnm n

mn mnm n

t t i t

H t i t m n

ω> =

> =

Φ = Φ − +⎡ ⎤⎣ ⎦

= Ω ∆ +

∑

∑

Flux qubit:

Liu, You, Wei, Sun, Nori, PRL 95, 087001 (2005)

The applied magnetic fluxes and interaction Hamiltonian are:

18

ContentsFlux qubit as an artificial atom (micromaser, etc)

Cavity QED on a chipCoupling qubits



via data buses

Scalable circuits


Quantum tomography

Conclusions

19

Cavity QED: Charge-qubit inside a cavity

H = Ec (n – CgVg / 2e )2 – EJ(Φe) cosφ,

φ = average phase drop across the JJ

Ec = 2e2/(Cg+2CJ0) = island charging energy;

EJ(Φe) = 2 EJ0 cos(πΦe/Φ0).You and Nori, PRB 68, 064509 (2003)

Here, we assume that the qubit is embedded in a microwave cavitywith only a single photon mode λ providing a quantized flux

Φf = Φλ a + Φ*λ a† = |Φλ| (e-iθ a + eiθ a†),

with Φλ given by the contour integration of uλdl over the SQUID loop.

Hamiltonian: H = ½ E ρz + ħωλ(a†a + ½) + HIk,

HIk = ρz f(a†a) + [e-ikθ |e><g| ak g(k)(a†a) + H.c.]

This is flux-driven. The E-driven version is in: You, Tsai, Nori, PRB (2003)

20

Circuit QEDCharge-qubit coupled to a transmission line

( ) [ ]† , . .2 ze gH a a a H cn gσ ω σω +Φ= + + +h

h hYale group

ω(Φe,ng) can be changed by the gate voltage ng and the magnetic flux Φe .

21


Based on the interaction between the radiation field and a superconductor, we propose ways to engineer quantum states using a charge qubit inside a microcavity.

This device can act as a deterministic single photon source as well as generate any photon states and an arbitrary superposition of photon states for the cavity field.

The controllable interaction between the cavity field and the qubit can be realized by the tunable gate voltage and classical magnetic field applied to the SQUID.

Liu, Wei, Nori, EPL 67, 941 (2004); PRA 71, 063820 (2005); PRA 72, 033818 (2005)

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Interaction with

microcavity

Before

After

JJ qubit photon generator Micromaser

JJ qubit in its ground state then excited via

JJ qubit interactswith field via

Excited JJ qubit decaysand emits photons

Atom is thermallyexcited in oven

Flying atoms interactwith the cavity field

Excited atom leaves the cavity, decays to its ground state providing photons in the cavity.

Liu, Wei, Nori, EPL (2004); PRA (2005); PRA (2005)

24

Interaction between the JJ qubit and the cavity field

Liu, Wei, Nori, EPL 67, 941 (2004);PRA 71, 063820 (2005); PRA 72, 033818 (2005)

details

25

The qubit is in its ground state

Qubit is in a superposed

statecarrier

red sidebandexcitation

The qubit returnsto its ground state and a superpositionof the vacuum and single-photon statesis created.

+


26

ContentsFlux qubits


Coupling qubitsControllable coupling between qubits:


via data buses

Scalable circuits


Quantum tomography

Conclusions

27

Qubits can be coupledeither directly

or indirectly, using a data bus

(i.e., an “intermediary”)

28

Let us now very quickly (fasten your seat belts!) see a few experimental

examples of qubits coupled directly

29

NEC-RIKEN

Capacitively coupled charge qubits

Entanglement; conditional logic gates

30

A. Izmalkov et al., PRL 93, 037003 (2004)

Inductively coupled flux qubits

Entangled flux qubit states

31J.B. Majer et al., PRL 94, 090501 (2005). Delft group


32J. Clarke’s group, Phys. Rev. B 72, 060506 (2005)


33

Berkley et al., Science (2003)

Capacitively coupled phase qubits

McDermott et al., Science (2005)

Entangled phase qubit states

34

Let us now consider qubits coupled indirectly

Scalable circuits: using an LC data bus

LC-circuit-mediated interaction between qubitsLevel quantization of a superconducting LC circuit has been observed.

Delft, Nature, 2004 NTT, PRL 96, 127006 (2006)35

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Switchable coupling: data bus

A switchable coupling between the qubit and a data buscould also be realized by changing the magnetic fluxes through the qubit loops.

Liu, Wei, Nori, EPL 67, 941 (2004) Wei, Liu, Nori, PRB 71, 134506 (2005)

Single-mode cavity field Current-biased junction

( )0cos Xπ Φ

ΦThe bus-qubit coupling is proportional to

Scalable circuitsCouple qubits directly via a common inductance

You, Tsai, and Nori, Phys. Rev. Lett. 89, 197902 (2002)

Switching on/off the SQUIDs connected to the Cooper-pair boxes, can couple any selected charge qubits by the common inductance (not using LC oscillating modes).

37

38

ContentsFlux qubits


Coupling qubits

Controllable coupling between qubits:via variable frequency magnetic fields (no data bus)via data buses

Scalable circuits


Quantum tomography

Conclusions

39

( )22

( ) ( ) ( )l2 2 cos cos cos 2 f +2

2 2pl l l lml

ql Jl Jl Jl m P Jl mml pl

PPH E E E EM M

α ϕ ϕ α π ϕ= + + + − −

M

1 2 IH M I I=

EJ1 EJ1

αEJ1

Φ(1)e

I1

EJ2 EJ2

αEJ2

Φ(2)e

I2

Coupling qubits directlydirectly and (first) without VFMF (Variable Frequency Magnetic Flux)

H0 = Hq1 + Hq2 + HI = Total Hamiltonian

l=1,2

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Hamiltonian in qubit basis

(1) (2) (1) (2)0 1 2 +H.c.

2 z zH gω σ ω σ σ σ+ −⎡ ⎤ ⎡ ⎤= + +⎣ ⎦ ⎣ ⎦h

2( ) ( ) 202 / 2l l

l e lI tω ⎡ ⎤= Φ − Φ +⎣ ⎦

1 2

2 2(1) (2)

0 1 2

2 22 2z z

g

g gH

ω ω

ω σ ω σ

∆ = − >>

⎡ ⎤ ⎡ ⎤≈ + + −⎢ ⎥ ⎢ ⎥

∆ ∆⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

h h

Qubit frequency ωl is determined by the loop current I(l) and the tunneling coefficient tl

( )1 2

(1 ) ( 2 )0 1 2

/ 0

2 2z z

g

H

ω ω

ω σ ω σ

− ≈

≈ +h h

Decoupled Hamiltonian

You and Nori, Phys. Today 58 (11), 42 (2005)

41

Now: let’s consider a Variable-Frequency-Magnetic-Flux (VFMF)

42

Controllable couplings via VFMFsWe propose an experimentally realizable method to control the coupling between two flux qubits (PRL 96, 067003 (2006) ).

The dc bias fluxes are always fixed for the two inductively-coupled qubits. The detuning ∆ = | ω2 – ω1| of these two qubits can be initially chosen to be sufficiently large, so that their initial interbit coupling is almost negligible.

When a time-dependent, or variable-frequency, magnetic flux (VFMF) is applied, a frequency of the VFMF can be chosen to compensate the initial detuning and couple the qubits.

This proposed method avoids fast changes of either qubitfrequencies or the amplitudes of the bias magnetic fluxes through the qubit loops

43

Controllable couplings via VFMFs

Applying a Variable-Frequency Magnetic Flux (VFMF)

( )0H H H t= +B(t)

Φ(1)e Φ(2)

e

EJ1

EJ1

αEJ1

EJ2

EJ2

αEJ2

(1) (2)( ) M I t I

(1)I

(2)I

(1) (1)( ) ( )I t I I t≈ +%

Liu, Wei, Tsai, and Nori, Phys. Rev. Lett. 96, 067003 (2006)

44

Controllable couplings via VFMFsFrequency or mode matching conditions

1ω

2ω

ω

1 2ω ωω+ =

1ω

2ω

ω

1 2ω ωω− =

( ) ( )

(1) (2)int 1 1 2

(1) (2)2 2 1

exp H.c.

exp H.

c

H i t

i t

σ σ ωω

σ ωω

ω

σ ω

+ +

+ −

= Ω − − − +⎡ ⎤⎣ ⎦

+ Ω − + − +⎡ ⎤⎣ ⎦

If ω1 – ω2 = ω, then the exp[…] of the second term equals one, while the first term oscillates fast (canceling out). Thus, the second term dominates and the qubits are coupled with coupling constant Ω2

If ω1 + ω2 = ω, then the exp[…] of the first term equals one, while the second term oscillates fast (canceling out). Thus, the first term dominates and the qubits are coupled with coupling constant Ω1

Thus, the coupling between qubits can be controlled by the frequency of the variable-frequency magnetic flux (VFMF) matching either the detuning (or sum) of the frequencies of the two qubits.

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Logic gates

ω1 − ω2 = ω

H1 = Ω2 σ+(1) σ−

(2) + H.c.

( )1 2 1 21| | , | ,2

e g g eψ ± = +

ω1 + ω2 = ω

H2 = Ω1 σ+(1) σ+

(2) + H.c.

( )1 2 1 21| | , | ,2

g g e eψ ± = +

Mode matching conditions

12 | |t πΩ= h

22 | |t πΩ= h


Quantum tomography can be implemented via an ISWAP gate, even if only one qubit measurement can be performed at a time.

46

Experimentally realizable circuits forVFMF controlled couplings

Grajcar, Liu, Nori, Zagoskin, cond-mat/0605484DC version in Jena exper., cond-mat/0605588

We propose a coupling scheme, where two flux qubits with different eigenfrequencies share Josephsonjunctions with a coupler loop devoid of its own quantum dynamics.

Switchable two-qubit coupling can be realized by tuning the frequency of the AC magnetic flux through the coupler to a combination frequency of two of the qubits.

The coupling allows any or all of the qubits to be simultaneously at the degeneracy point and their mutual interactions can change sign.

details

47

Switchable coupling proposals(without using data buses)

Feature Proposal

Weak fields

Optimal point

No additional circuitry

Rigetti et al. (Yale) No Yes Yes

Liu et al. (RIKEN-Michigan) OK No Yes

Bertet et al. (Delft)Niskanen et al. (RIKEN-NEC)Grajnar et al. (RIKEN-Michigan)

OK Yes No

Ashhab et al. (RIKEN-Michigan) OK Yes Yes

Depending on the experimental parameters, our proposals might be useful options in certain situations. details

48

Now: let’s consider a Variable-Frequency-Magnetic-Flux (VFMF)

and alsoa data bus

49

ContentsFlux qubits


Coupling qubits

Controllable coupling between qubits:via variable frequency magnetic fieldsvia data buses

Scalable circuitsTesting Bell inequalities. Generating GHZ states.

Quantum tomography

Conclusions

50

Scalable circuits: data bus + VFMF

We propose a scalable circuit with superconducting qubits (SCQs) which is essentially the same as the successful one now being used for trapped ions.

The SCQs act as "trapped ions" and are coupled to a "vibrating" mode provided by a superconducting LC circuit, acting as a data bus (DB).

Each SC qubit can be separately addressed by applying a time-dependent magnetic flux (TDMF).

Single-qubit rotations and qubit-bus couplings and decouplings are controlled by the frequencies of the TDMFs. Thus, qubit-qubit interactions, mediated by the bus, can be selectively performed.

Liu, Wei, Tsai, and Nori, cond-mat/0509236

Scalable circuits: using an LC data bus

LC-circuit-mediated interaction between qubitsLevel quantization of a superconducting LC circuit has been observed.

Delft, Nature, 2004 NTT, PRL 96, 127006 (2006)51

52

Scalable circuits: LC versus JJ-loop data-bus

Controllable interaction between the data bus and a flux qubitInductive coupling via M

EJ EJ

αEJ

ΦeΦe(t)

L C0

M

1 2qubit busH H H MI I= + +

EJ2 EJ2

αEJ2

Φe I2

M

I1EJ1

Φe

Φe (t)Data bus

Data bus

The circuit with an LC data bus models the Delft circuit in Nature (2004), which does not work at the optimal point for a TDMF to control the coupling between the qubit and the data bus.

This TDMF introduces a non-linear coupling between the qubit, the LC circuit, and the TDMF.

Replacing the LC circuit by the JJ loop as a data-bus, with a TDMF,then the qubit can work at the optimal point

Liu, Wei, Tsai, Nori, cond-mat/0509236

details

53

A data bus using TDMF to couple several qubits

IEJ

Φe+

Φ(t)

EJ1 EJ1

αEJ1

Φe I1

EJ2 EJ2

αEJ2

Φe I2I1

Φdc(1)

+Φ(1)(t)

Φdc(2)

+Φ(2)(t)I2

C

L

(1)(2)

I

A data bus could couple several tens of qubits.

The TDMF introduces a nonlinear coupling between the qubit, the LC circuit, and the TDMF.

Liu, Wei, Tsai, Nori, cond-mat/0509236 details

54

Comparison between SC qubits and trapped ions

Quantized bosonic mode

Trapped ions Superconducting circuits

Vibration mode

Lasers

LC circuit

Magnetic fluxes

Qubits

Classical fields

55

ContentsFlux qubits


Coupling qubits



via data buses

Scalable circuits


Quantum tomography

Conclusions

56

Testing Bell’s inequality

We propose how to use coupled Josephson qubits to test Bell’s inequality

Wei, Liu, Nori, Phys. Rev. B (2005)

57

Generating GHZ states

We propose an efficient approach to produce and control the quantum entanglement of threemacroscopic coupled superconducting qubits.

Wei, Liu, Nori, Phys. Rev. Lett. (June 2006)

We show that their Greenberger-Horne-Zeilinger (GHZ)entangled states can be deterministically generated byappropriate conditional operations.

58

ContentsFlux qubits


Coupling qubits



via data buses

Scalable circuits


Quantum tomographyConclusions

59

Quantum tomography

We propose a method for the tomographic reconstruction of qubit states for a general class of solid state systems in which the Hamiltonians are represented by spin operators, e.g., with Heisenberg-, XXZ-, or XY- type exchange interactions.

We analyze the implementation of the projective operator measurements, or spin measurements, on qubit states. All the qubit states for the spin Hamiltonians can be reconstructed by using experimental data.

This general method has been applied to study how to reconstruct any superconducting charge qubit state.

Liu, Wei, Nori, Europhysics Letters 67, 874 (2004); Phys. Rev. B 72, 014547 (2005)

60

EJ0

CJ

Vg

Cg

XΦ

Superconducting charge qubit

Quantum tomography for superconducting charge qubits

Liu, Wei, Nori, Phys. Rev. B 72, 014547 (2005)

61

Quantum tomography

Quantum tomography

detailsLiu , Wei, Nori, Phys. Rev. B72, 014547 (2005)

62

63

ContentsFlux qubits


Coupling qubits



via data buses

Scalable circuits


Quantum tomography

Conclusions

Summary

• Studied SC charge, flux, charge-flux, and phase qubits

• Studied many analogies between atomic physics and SC

qubits including ion traps, on-chip micromasers, cyclic

transitions, generating photon states using SC qubits

• We proposed and studied circuit QED. It has been

verified experimentally, years after our prediction

• Proposed several methods of controllable couplings

between different qubits

• Studied how to dynamically decouple qubits with always-

on interactions

• Introduced solid state quantum tomography64

66

End of theQuick overview

of several types ofsuperconducting qubits

For a short pedagogical overview, please see:

You and Nori, Physics Today (November 2005)

67

Switchable qubit coupling proposals

E.g., by changing the magnetic fluxes through the qubit loops.

You, Tsai, Nori, PRL (2002) Y. Makhlin et al., RMP (2001)

( )(1) (2)(1) 2)

0 0

(

, cos cosee e

eχ π π⎛ ⎞ ⎛ ⎞

Φ Φ ∝ ⎜ ⎟ ⎜ ⎟Φ ΦΦ

⎝ ⎠ ⎝ ⎠

ΦCoupling:

68

J.B. Majer et al., PRL94, 090501 (2005)

How to couple flux qubits

We made several proposals on how to couple qubits.No auxiliary circuit is used in several of these proposals to mediate the qubit coupling. This type of proposal could be applied to experiments such as:

A. Izmalkov et al., PRL 93, 037003 (2004)

69

Controllable couplings via VFMFsCoupling constants with VFMFWhen |g|/(ω1−ω2) <<1, the Hamiltonian becomes

( ) ( )(1) (2) 1

(1) (2) (1) (2)1 22 exp H.c. exp .

2H c

2.z zH i t i tω ωω σ ω σ σ σ σ σ+ + + −⎡ ⎤ ⎡ ⎤+ Ω − + + Ω= + − +⎣ ⎦ ⎣ ⎦

h h

1 and pa rity2lf =

(2)1 2 2

(1)1 1 1

(2)2 2 2

(1)1 1 1

| |

| cos(2 2 ) |

| |

| cos(2 2 ) |

P

P

e I g

e f g

g I e

e f g

ϕ π

ϕ π

Ω ∝ ×

+

Ω ∝ ×

+

( 2 )3( 2 ) ( 2 )

21

3

( 2 )12

sin

1 1with

ici

i i

i Ji

II CC

C C

ϕ=

=

=

=

∑

∑Liu et al., PRL 95, 087001 (2005)

f1=1/2 even parity

f2=1/2 odd parity

|gl> and |el> have different parities when fl=1/2

70


The couplings in these two circuits work similarly

Optimal point

Not optimal point

optimal point

Optimal point

Not at the Optimal point backGrajcar, Liu, Nori, Zagoskin,

cond-mat/0605484

Scalable circuitsrf SQUID mediated qubit interaction

Liu et al, unpublishedFriedman et al., Nature (2000)

Radius of rf SQUID ~100 µm; Radius of the qubit with three junctions~1—10 µm. Nearest neighbor interaction. 71

72

Cirac and Zoller, PRL74, 4091 (1995)

SC qubits as analogs of trapped ions

C0LΙ

Φe(2)

Φ(2)(t)I(2)

Μ(2)

I(1)Φe

(1)Φ(1)(t)

Μ(1)

Φe(3)

Φ(3)(t)I(3)

Μ(3)

ΦΦ(t)

q

q

q q

q

q

q

q

Liu, Wei, Tsai, Nori, cond-mat/0509236

back

73

Cavity QED: Controllable quantum operations

t1 t2 t3 t4 t5

1

0.5

0

t1 t2 t3 t4 t5

time

time

1

0.5

0

74

|0, g>

|0, e>

|1, e>|2, e>

|1, g>|2, g>

…

…

Three-types of excitations

|n+1,g> |n,e>|n,g> |n,e> |n,g> |n+1,e>

Carrier process:ωq = ωc

Red sideband excitation:ωc = ωq - ω

Blue sideband excitation:ωc = ωq + ω

75


Initially, the qubit is in its ground state

Superposition state

Excited stateThere is no interaction between the qubit and the cavity field at this stage.

Now turnng = 1/2

76

νh

where

Initially , the qubit is in its excited state

ng = 1

Red sideband excitation is provided by turning on the magnetic field such that .

Finally, the qubit is in its ground state and one photon is emitted.


back

77

( ) ( ) ( )

( )( )

( )( ) ( ) ( )

*1 1

† † *2 2

† *

1 exp exp2

1 2

exp exp

q z C C

C C

H i t i t

a a a a

a a i t i t

ω σ σ σ ω ω

ω σ σ

σ σ ω ω

+ −

− +

+ −

= − Ω +Ω − +⎡ ⎤⎣ ⎦

⎛ ⎞+ + − + Ω +Ω⎜ ⎟⎝ ⎠

− + Ω +Ω − +⎡ ⎤⎣ ⎦

h

( ) ( ) ( )

( )( ) ( ) ( )

*1 1

† † *

1 exp exp2

1 exp exp2

q z C C

C C

H i t i t

a a a a i t i t

ωσ σ σ ω ω

ω σ σ ω ω

+ −

+ −

= − Ω +Ω − +⎡ ⎤⎣ ⎦

⎛ ⎞+ + − + Ω +Ω − +⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠h

Mode match and rotating wave approximation

Blue , c qω ω ω= +

( ) ( )

†

† *

1 12 2

exp exp

q z

C C

H a a

a i t a i t

ω σ ω

σ ω σ ω+ −

⎛ ⎞= + +⎜ ⎟⎝ ⎠

+Ω − + Ω

h

Controllable interaction between data bus and a flux qubit

, Carrierq cω ω=

( ) ( )( )*1 1

1 exp exp2 q z C CH i t i tω σ σ ω ω σ+ −= − Ω − +Ω

|| |-| :detuning Large 2q Ω>>ωω

, ed Rc qω ω ω= −

( ) ( )

†

* †

1 12 2

exp exp

q z

C C

H a a

a i t a i t

ω σ ω

σ ω σ ω+ −

⎛ ⎞= + +⎜ ⎟⎝ ⎠

+ Ω − + Ω

h

( )e x p H .c .CH a i t i tσ ω+= Ω ∆ − +

ωω −=∆ q

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Dynamical decouplingMain idea:

Let us assume that the coupling between qubits is not very strong (coupling energy < qubit energy)

Then the interaction between qubits can be effectively incorporated into the single qubit term (as a perturbation term)

Then single-qubit rotations can be approximatelyobtained, even though the qubit-qubit interaction is fixed.

Wei, Liu, Nori, Phys. Rev. B 72, 104516 (2005)

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Testing Bell’s inequality

Wei, Liu, Nori, Phys. Rev. B (2005)

1) Propose an effective dynamical decoupling approach to overcome the “fixed-interaction” difficulty for effectively implementing elemental logical gates for quantum computation.

2) The proposed single-qubit operations and local measurements should allow testing Bell’s inequality with a pair of capacitivelycoupled Josephson qubits.

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Generating GHZ states

2) We show that their Greenberger-Horne-Zeilinger (GHZ) entangled states can be deterministically generated by appropriate conditional operations.

3) The possibility of using the prepared GHZ correlations to test the macroscopic conflict between the noncommutativityof quantum mechanics and the commutativity of classical physics is also discussed.

Wei, Liu, Nori, Phys. Rev. Lett. (June 2006)

We propose an efficient approach to produce and control the quantum entanglement of threemacroscopic coupled superconducting qubits.

1)

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Quantum tomographyQuantum states

z

y

x

Liu, Wei, and Nori, Europhys. Lett. 67, 874 (2004)

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Quantum tomography

z

y

x

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Quantum tomography

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