Beam splitting fidelity

In the case our initial state is a Gaussian state, the problem simplifies greatly. We introduce the Wigner distribution W( ¯ξ), with ¯ξ the coordinates in the phase space, that relates a given density matrix ρ to a quasi-probability distribution in the phase space,

W( ¯ξ) =

Z dζ¯ 4π²Tr

ρexp(iζ¯^T ·J·ξ)

exp(iζ¯^T ·J ·ξ).¯ (6.34) The Wigner distribution of a Gaussian state has the form

W( ¯ξ) = 1

π² exp −( ¯ξ−ξ¯₀)^TV⁻¹( ¯ξ−ξ¯₀)

, (6.35)

with V the real symmetric variance matrix defined by V_ij =V_ji = 1

2h{ξˆ_i,ξˆ_j}i= Z

dξ¯ξ¯_iξ¯_jW( ¯ξ), (6.36) and {A, B} = AB+BA the anticommutator. Under the unitary transformation U(S) associated with the symplectic transformation S, the density matrix changes as ρ⁰ = U(S)ρU⁻¹(S), and the variance matrixV of the Wigner distribution changes asV⁰ =SV S^T [9–11]. The symplectic matrix S admits a unique polar decomposition that satisfies [11]

S =S(X, Y)P (6.37) transformation U(S(X, Y)) conserves the photon number and for this reason it is called

”passive”, whereas the transformationU(P) does not conserve the number of photons and it is called active. When starting from a non-squeezed state only theU(P) transformation can produce squeezing. For this reason we can study all the departures from a perfect beam splitting transformation as arising uniquely from the presence of an active term P.

6.5 Beam splitting fidelity

We consider the initial state ρi = |ψiihψi| and as a final state ρf = |ψfihψf|, with |ψii and |ψ_fi given by Eqs. (6.22,6.23). The fidelity of the gate can be written as F(t) =

|hψ_f|U(t)|ψ_ii|² = Tr[ρ_fρ(t)], withρ(t) = U(t)ρ_iU^†(t). The state ρ(t) can be written as

6.5 Beam splitting fidelity 121

By cycling the operators inside the trace sign, the fidelity then becomes

F(t) = Tr[ ˜ρ_f(−t)ρ₀(t)], (6.42) with the backward-in-time evolved non-normalized state ˜ρf(−t) =a1(−t)ρfa^†₁(−t) and the forward-in-time evolved vacuum ρ₀(t) = U(t)|0ih0|U^†(t). By writing

a1(−t) = X

i=1,2

(S_(c)(t)⁻¹)1,iai+ (S_(c)(t)⁻¹)1,i+2a^†_i, (6.43) a^†₁(−t) = X

i=1,2

(S_(c)(t)⁻¹)_3,ia_i+ (S_(c)(t)⁻¹)_3,i+2a^†_i, (6.44)

the state ˜ρ_f(−t) can be written as ˜ρ_f(−t) =|ψ(−t)ih˜ ψ(−t)|, with˜

|ψ(−t)i˜ =c₀₀(t)|0i+c₁₁(t)|1,1i+c₂₀(t)|2,0i+c₀₂(t)|0,2i, (6.45) with the coefficient c_i(t) linear combinations of the entries of the matrix S_(c)⁻¹(t),

c₀₀(t) = ([S_(c)⁻¹(t)]₁₁+e^−iϕ[S_(c)⁻¹(t)]₁₂)/√

2, (6.46)

c₁₁(t) = ([S_(c)⁻¹(t)]₁₄+e^−iϕ[S_(c)⁻¹(t)]₁₃)/√

2, (6.47)

c₀₂(t) = e^−iϕ[S_(c)⁻¹(t)]₁₄, (6.48)

c₂₀(t) = [S_(c)⁻¹(t)]₁₃. (6.49)

In the Fock basis ˜ρ_f(−t) is a 4×4 matrix. We can choose to normalize the state ˜ρ_f(−t) and write

F(t) = Tr[ ˜ρ_f(−t)]Tr[ρ_f(−t)ρ₀(t)], (6.50) with

ρ_f(−t) = ρ˜f(−t)

Tr[ ˜ρ_f(−t)]. (6.51)

The time-evolved vacuumρ₀(t) is initially a pure Gaussian state and its time evolution is easily calculated its Wigner distribution. By expressing the Wigner distribution ofρ₀(t) in the Fock basis, we have an operative way to calculate the fidelity of the beam splitter,

F(t) = Tr[ ˜ρ_f(−t)] X

{n},{m}

ρ_f(−t)_{n},{m}ρ₀(t)_{m},{n}, (6.52)

with {n}=n1, n2 and ρ{n},{m} =hn1n2|ρ|mˆ 1m2i.

We now express the Wigner function defined in Eq. (6.34) as a function of the complex amplitudes of the fields, ¯α= (α₁, α₂, α^∗₁, α₂^∗)^T. The Wigner function of the squeezed vacuum ρ₀(t) is given by

W_0,t( ¯α) = 4

π²e^{−¯α†V(c)−1(t) ¯α} (6.53)

122 Chapter 6. Photon beam splitting with superconducting resonators

0 2 4 6 8 10

time (ns)

0 0.2 0.4 0.6 0.8 1

Fidelity

Exact RWA

Figure 6.2: Fidelity of the beam splitting gate. Parameters: ω₁/2π = 7 GHz, ω₂/2π = 7.2 GHz, λ = 1, δf /2π = 0.1 GHz, and ϕ =π/2. The black solid curve represents the result of the exact calculation provided by Eq. (6.42), whereas the blue dashed curve shows the result of the rotating wave approximation as given by Eq. (6.24).

with the matrixV_(c)(t) =S_(c)(t)V_(c)(0)S_(c)(t)^†, the matrixV_(c)⁻¹(0) =I^4×4, withV_(c) = Σ^†VΣ.

The matrix elements of a density operator in the Fock space can be obtained from the Wigner function by

ρ{n},{m} = Z dα¯

π²χ_W( ¯α)D_n1,m1(α₁)D_n2,m2(α₂) (6.54) where χ_W(~α, ~α^∗) is the characteristic function, that is defined by χ_W( ¯α) ≡ Tr[ρD( ¯α)], with D( ¯α) = exp(−α^†·Σ_Z·a) the two-mode displacement operator and the matrix Σ_Z = diag(I,−I). The characteristic function is related to the Wigner function by a Fourier transfom,

χ^W( ¯α) = Z

dβ¯ exp ¯β^†·Σ_Z·α¯

W( ¯β) (6.55)

The coefficients D_n,m(α) that appear in Eq. (6.54) are defined by D_n,m(α)≡ hn|D(α)|mi=

rm!

n!α^n−me^−|α|2/2L^(n−m)_m (|α|²), (6.56) with L^(n−m)m (r) the Laguerre polynomials and D(α) = exp(αa^† −α^∗a) the single-mode

6.6 Conclusion 123

displacement operator. The characteristic function of the squeezed vacuum Eq. (6.53) is χ^W_0,t( ¯α) = exp

−1

4α¯^†·Σ_ZV_(c)(t)Σ_Z·α¯

. (6.57)

In Fig. 6.2 we plot the fidelity of the beam splitting gate obtained by exact integration of the equation of motion and compare it to the result of the rotating wave approximation, Eq. (6.24). We chose the two resonant frequencies to beω₁/2π= 7 GHz ω₂/2π = 7.2 GHz, respectively. The interaction potential has been chosen to couple the two modes in a symmetrical way, that is λ = 1. The amplitude of the driving is δf /2π = 0.1 GHz.

The dashed blue curve represents the result of the RWA and clear Rabi oscillations at a frequency Ω_R = 2δf appear. The exact solution is given by the black solid curve. The beam splitting is obtained when the fidelity reaches one. We see that there is only partial agreement between the exact solution and the rotating wave approximation. In particular the crude RWA provides a good approximation for the Rabi period, and the error in fidelity of the beam splitting gate is on order of 5%. A better result can be obtained by taking into consideration one- and two-photon processes, that result in Bloch-Siegert oscillations superimposed to the crude RWA envelope.

6.6 Conclusion

In this work we have provided an exact derivation of the fidelity of a beam splitting gate in a system of two superconducting line resonators, coupled by an externally driven SQUID. We have used an effective equivalent circuit for the two coupled resonators and studied its quantum dynamics. In the linear response of the SQUID dynamics, the two resonators turn out to be quadratically coupled and in the rotating wave approximation we find analytically that the fidelity of the beam splitting reaches unity. We have then studied the accuracy of the rotating wave approximation by providing an exact derivation of the fidelity of the beam splitting gate and by making comparison with rotating wave result. We find that the exact solution beam splitting fidelity can reach more than 95% in a time τ =π/4δf.

References

[1] Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, Measurement of conditional phase shifts for quantum logic, Phys. Rev. Lett. 75, 4710 (1995).

[2] E. Knill, R. Laflame and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature 409, 46 (2001).

[3] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.- S. Huang, J. Majer, S. Kumar, S.

M. Girvin, R. J. Schoelkopf, Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics, Nature 431, 162-167 (2004).

124 Chapter 6. Photon beam splitting with superconducting resonators

[4] M. Hofheinz, H. Wang, M. Ansmann, Radoslaw C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, J. M. Martinis and A. N. Cleland, Synthesizing arbitrary quantum states in a superconducting resonator, Nature459, 546-549 (2009).

[5] S. Kumar and D. P. DiVincenzo, Exploiting Kerr cross non-linearity in circuit quantum electrodynamics for non-demolition measurements, arXiv: 0906.2979 (2009).

[6] G. Burkard, R. H. Koch, D. P. DiVincenzo, Multilevel quantum description of deco-herence in superconducting qubits, Phys. Rev. B 69, 064503 (2004).

[7] F. Bloch and A. Siegert, Magnetic resonance for nonrotating fields, Pfys. Rev.57, 522 (1940).

[8] M. S. Shahriar, P. Pradhan, J. Morzinski, Driver-phase-correlated fluctuations in the rotation of a strongly driven qubit, Phys. Rev. A 69, 032308 (2004).

[9] R. Simon, N. Mukunda, B. Dutta, Quantum-noise matrix for multimode systems:

U(n) invariance, squeezing, and normal forms, Phys. Rev. A 49, 1567 (1994).

[10] Arvind, B. Dutta, N. Mukunda, R. Simon, Two-mode quantum systems: Invariant classification of squeezing transformations and squeezed states, Phys. Rev. A52, 1609 (1995).

[11] Arvind, B. Dutta, N. Mukunda, R. Simon, The real symplectic groups in quantum mechanics and optics, quant-ph/9509002 (1995).

A. Matrices C , C _V , M ₀ , and N

The definitions of the derived matrices C, C_V, M₀ and N that enters the Hamiltonian are given in [6, 27] for the general case. Here we apply the theory and derive the matrices for the particular case of the circuit of Fig. 2.2. The derived capacitance matrices are

C ≡ C_J +

The inductance matrices that enter the potential are M₀ = 1

The three-dimensional problem is mapped into a two-dimensional one in Sec. 2.3 with the matrix the inductance linearized matrix L⁻¹_lin;L,R is given by

L⁻¹_lin;L,R =M₀+L⁻¹_J cosφ_L,R;i. (A.7)

Because of the symmetry of the potential, we drop the subscripts R and L. Applying the matrix P we obtainL⁻¹_lin,P =P^TL⁻¹_linP,

126 Chapter A. Matrices C,C_V ,M₀, and N

In order to simplify the calculation we assume the two capacitance C₁ and C₂ to be equal, C₁ =C₂ ≡C and defineγ =C/C_J. The projected capacitance matrix C_P =P^TCP is then found to be

C_P =C_J

1 +γ+α −α

−α 1 +γ+α

. (A.9)

In this case, the orthogonal matrices that diagonalize the capacitance matrix C_P the lin-earized inductance matrix L⁻¹_lin,P are identical, C_P = O^TC_dO and L⁻¹_lin,P = O^TΛO. The frequency matrix Ω= diag(ω⊥, ωk) is given by

Ω² =ω_LC²

1

4α²(1+γ)² 0 0 _4α2(1+2α+γ)^{1−4α2 2}

!

, (A.10)

whereω_LC² = 1/L_JC_J. The matrix Mis then diagonalized by the same orthogonal matrix O and, in the basis where it is diagonal, can be written as

M= r E_J

8EC



 q1+γ

2α 0

0

q(4α²−1)(1+2α+γ) 2α



. (A.11)

B. The functions F ₁ and F _φ

We give here an explicit formula for the intra-cell and inter-cell overlaps s₁ and s₂ as functions of α, E_J/E_C and C/C_J, the matricesC andMdefined in Appendix A. The gap|∆|and the relative phase between the states |0i and |1i are given by

1

C. Perturbation theory in ∆

In order to determine the evolution governed by the Hamiltonian Eq. (3.11) we single out the term H₀ diagonal in the{|s, ni} basis, with |si the eigenstates of σ_Z and |ni the oscillator Fock states,

H=H₀+f(a+a^†) + ∆

2σ_X, (C.1)

with H₀ =σ_Z/2 + ∆ω_Za^†a. We then work in the interaction picture with respect toH₀. The Heisenberg equation for the density operator reads ˙ρ_I =−i[H_I,ρ_I], with

H_I = H⁽⁰⁾_I +V_I, (C.2)

H⁽⁰⁾_I = f(ae^−i∆ωZt+a^†e^i∆ωZt), (C.3) V_I = ∆

2

e^iΩˆntσ₊+e^−iΩˆntσ−

, (C.4)

where we define ˆΩ_n=+ 2ga^†a, andσ± = (σ_X ±iσ_Y)/2. We will call U_I(t) the evolution operator generated by H_I.

The evolution operator is given by U(t) = exp(−iω_dta^†a− iH₀t)U_I(t). For the mea-surement procedure so far defined we are interested in the evolution operator in the frame rotation at the bare harmonic oscillator frequency. Therefore

U_R(t) = exp(−itσ_Z/2−iH_intt)U_I(t). (C.5) For the case ∆ = 0 the model is exactly solvable and U_I⁽⁰⁾(t) can be computed via a generalization of the Baker-Hausdorff formula [35],

U_I⁽⁰⁾(t) =D(γ_Z(t)), (C.6)

with the qubit-dependet amplitude γ_Z(t) = −ifRt

0 dse^i∆ωZs. The operator D(α) = exp(a^†α −aα^∗) is a displacement operator [36], and it generates a coherent state when applied to the vacuum |αi ≡ D(α)|0i =e^−|α|2/2P

n(αⁿ/√

n!)|ni. In the frame rotating at the bare harmonic oscillator frequency, the state of the oscillator is a coherent state whose amplitude depends on the qubit state. A general initial state

ρ_tot(0) = X

ij=0,1

ρ_ij|iihj| ⊗ |ˆ0ihˆ0|, (C.7) where |ˆ0i is the harmonic oscillator vacuum state, evolves to

ρ_R(t) = X

ij=0,1

ρij|iihj| ⊗ |αi(t)ihαj(t)|, (C.8)

129

130 Chapter C. Perturbation theory in ∆ where we define the qubit operators α_Z(t)≡γ_Z(t)e^−igtσZ, and the object

|α_Z(t)i ≡D(α_Z)|ˆ0i, (C.9)

that gives a qubit-dependent coherent state of the harmonic oscillator, once the expectation value on a qubit state is taken, |α_i(t)i=hi|α_Z(t)|ii, fori= 0,1.

For non-zero ∆, a formally exact solution can be written as U_I(t) = U_I⁽⁰⁾(t)T exp

−i ∆ Z t

0

dt⁰V_I(t⁰)

, (C.10)

with V_I(t) =U_I^(0)†(t)V_I(t)U_I⁽⁰⁾(t) and T the time order operator. For a time scalet1/∆

we expand the evolution operator in powers of ∆t1 , U_I(t)≈ U_I⁽⁰⁾(t)

I−i∆t Z 1

0

dsV_I(s t)−(∆t)² Z 1

0

ds Z s

0

ds⁰V_I(s t)V_I(s⁰ t)

. (C.11) The interaction picture potential can be written as

V_I(t) = 1 2

D(t)σ⁺+D^†(t)σ⁻

, (C.12)

with the oscillator operators D(t) defined as

D(t) = D^†(γ₀(t))e^iΩntD(γ₁(t)) (C.13)

= exp (it−iIm[α₀(t)α₁(t)^∗])D(−δα(t)e^igt)e^2igta†a. (C.14) Hereδα(t) = α0(t)−α1(t) is the difference between the amplitudes of the coherent states associated with the two possible qubit states.

D. First and Second order contribution to the two-measurement POVM

In this Appendix we provide the full expression for the quantitiesC⁽¹⁾,F⁽²⁾ andC⁽²⁾that enter in the expression of the two-measurement POVM. For time t≈ 1/, we expand the evolution operator in ∆tand collect the contributions that arise at second power in (∆/).

By making use of the expression Eq. (C.12) for the perturbation in the interaction picture we can compute the qubit components of the second order contribution to the continuous POVM. We define

O_s(t⁰, t⁰⁰) = exp is(t⁰ −t⁰⁰)−isψ(t⁰²−t⁰⁰²)

× hδα(t⁰)e^isgt0|δα(t⁰⁰)e^isgt00i, (D.1) and hα|βi is the overlap between coherent states, and

ξ_s⁽²⁾(t⁰, t⁰⁰) = δx⁽¹⁾_s (t⁰)^∗+δx⁽¹⁾_s (t⁰⁰), (D.2) ζ_s⁽²⁾(t⁰, t⁰⁰) = −δx⁽¹⁾_s (t⁰) +δx⁽¹⁾_s (t⁰⁰). (D.3) The first term ξs⁽²⁾(t⁰, t⁰⁰) represents the complex displacement of the oscillator position due to the perturbation acting one time at t⁰⁰ < t (forward in time), and one time at

−t⁰ >−t (backward in time). The second term ζs⁽²⁾(t⁰, t⁰⁰) represents the displacement of the oscillator due to the perturbation acting two times at t⁰⁰ < t⁰ < t. Between the two perturbations the system evolves freely for the time t⁰⁰−t⁰ and accumulates a phase that depends on the difference of the effective qubit-dependent frequencies. In the short time approximation t ≈ 1/ such a phase can be neglected. Integrating the position degree of freedom over the subsetsη(s⁰), we obtain

F⁽²⁾(s⁰, t)ss = −s⁰s² 4

Z t 0

dt⁰ Z t⁰

0

dt⁰⁰Re{Os(t⁰, t⁰⁰)

×

"

erf δx(t) +ξ_s⁽²⁾_¯ (t⁰, t⁰⁰) σ

!

+ erf δx(t) +ζs⁽²⁾(t⁰, t⁰⁰) σ

!#) ,

(D.4) where ¯s = −s. This expression has meaning only in the short time approximation. By setting t ≈1/, the correction F⁽²⁾(t) at second order to the discrete POVM is evaluated

131

132 Chapter D. First and Second order contribution to the two-measurement POVM

to be

F⁽²⁾(t) = ² 4

Z t 0

dτ₁ Z t⁰

0

dt⁰⁰cos (t⁰−t⁰⁰)−ψ t⁰²−t⁰⁰²

e^{−12|A|2(t0−t00)2}

× (erf(|A|(t+t⁰+t⁰⁰)) + erf(|A|(t−t⁰+t⁰⁰))). (D.5) In an analogue way we calculate the elements of the first and second order contributions to the double measurement operatorC. The off-diagonal matrix element of the first order contribution C⁽¹⁾ is

C⁽¹⁾(t⁰;t) = 1

2(Γ(t)−1)F⁽¹⁾(t⁰−t) + i∆

4 erf

δx(t⁰−t) σ

Z t 0

dt⁰⁰e^it00Γ(t⁰⁰), (D.6) and the full expression of the diagonal matrix element of the second order contribution C⁽²⁾ is

C⁽²⁾(t) = ² 8

Z t 0

dt⁰ Z t

0

dt⁰⁰e^{i((t0−t00)−ψ(t02−t002))}e^−|A|

2

2 (t⁰−t⁰⁰)²erf(|A|(t+t⁰+t⁰⁰))

− Im

F⁽¹⁾(t) Z t

0

dt⁰e^−it0Γ(t)Γ(t⁰)^∗e^{−12|At0|2+|A|2t0t}erf

δx⁽¹⁾₋ (t⁰)/σ

. (D.7)

E. Eigenvalue problem

Defining

u± = 1

2 tan(γ) 1−sinc()± s

(1 + sinc())²−4sinc() cos²(γ)

!

(E.1) The matrix U assumes the simple form

U =









1 0 0 0

0 1 0 0

0 0 u₊ u−

0 0 1 1









, (E.2)

with sinc() = sin(π)/π, that allows for a simple solution of the eigenvalue problem in terms of a Jordan decomposition, Q=U⁻¹diag(1,sin(π)/π, λ−, λ₊)U, with

λ± = 1

2cos(φ)(1 + sinc())

± 1 2

q

cos²(φ)(1 + sinc())²−sinc²(). (E.3)

133

F. Double ring transmission and reflection amplitudes

The transmission amplitude for electrons coming from the left leadL and going to the right lead R can be calculated as

t=τ_B(I−Γ)⁻¹S_p⁰τ_A, (F.1) with Γ=S_p⁰ρ¯_AS¯_p⁰ρ_B, and S_p⁰ =

Sp 0 0 1

. We define the following transmission matrices in node A and B that take into account the lower arm of the large ring,

t⁰_A = transmission amplitudes τ_A and τ_B are given by the matrices

τA = t⁰_A I−P¯rLP¯r⁰_A−1

PtL, (F.6)

τ_B = t_L I−P¯r⁰_BP¯r_R−1

Pt⁰_B, (F.7)

with dimension respectively 3×1 and 1×3. The effective reflection amplitudes ¯ρ_A and ρ_B are given by the matrices

¯

G. Electron-hole switch

Let us consider in details the mechanism suggested to inject and collect electron from the MZ. The system, depicted in Fig. 4.9 a), is composed by a cavity, formed by a circular edge state, that is coupled to a linear edge channel by a QPC_V of transmission ˜tand reflection ˜r.

It has been experimentally demonstrated [20, 21] that such a device, if periodically driven by a time dependent potential V(t), produces a periodic current composed by an electron in one half-period and a hole in the other half-period, shown in Fig. 4.9 b). We wish to separate the electron and the hole by transmitting the electron through a barrier towards contact 3, and reflecting the hole into the contact 4. A time-dependent QPC_U driven by an external potential U(t) behaves like a beam splitter that mixes the incoming channels, from the contact 1 and 2, into the outgoing channels 3 and 4. If properly driven, it works as a switch that separates the electrons and holes generated by the cavity into different edge channels. Following [22, 23] we describe the effect of the time-dependent potential QPC_U by a scattering matrix

S_U(t) =

S₃₁(t) S₃₂(t) S₄₁(t) S₄₂(t)

. (G.1)

In the symmetric case one has S₃₁(t) = S₄₂(t) and S₃₂(t) = S₄₁(t). From the unitarity of S_U(t) follows that

1 = X

j

|S_jk(t)|², (G.2)

0 = S₃₂^∗ (t)S₃₁(t) +S₄₂^∗ (t)S₄₁(t). (G.3) The dynamics of the cavity can be described by a time-dependent scattering amplitude S_c(t, E), that describes the amplitude to be reflected at QPC_V towards QPC_U, and satisfies

|S_c(t, E)|² = 1. In the adiabatic regime, keeping all the reservoirs at the same chemical potential µ, the zero-temperature current in the contacts 3 and 4 can be written as

I_j(t) =|S_j1(t)|²I_c(t) + e 2πi

X

k=1,2

S_jk(t)∂

∂tS_jk^∗ (t), (G.4) withj = 3,4. HereI_c(t) is the current produced by the cavity, that can be written as[22, 23]

I_c(t) = e

2πiS_c(t, µ)∂

∂tS_c^∗(t, µ), (G.5)

with µ the chemical potential of all the leads. The current I_c(t) is plotted in Fig. 4.9 b) for a harmonic driving V(t) = V₀cos(Ωt), for the choice Ω/2π= 1 GHz and |˜t|² = 0.1. We

137

138 Chapter G. Electron-hole switch

choose theQP C_U such that we can writeS₃₁(t) =p

T(t) andS₄₁(t) =ip

1−T(t). It then follows that I₃(t) =T(t)I_c(t) and I₄(t) = (1−T(t))I_c(t), withT(t) related to the applied external potential U(t). By choosing a proper modulation of QPC, it is then possible to separate the electrons from the holes.

H. Inductance Matrices

We can write M=F_CLL˜⁻¹_L LL¯ ⁻¹_LLF_CL, and given that L_LK = 0, L_K =K, with FCL =

1 0 0 1

, (H.1)

and FKL = (1,1), we obtain

L_LL =L+K˜ (H.2)

with the definition

K˜ =F^T_LKL_KF_LK =K

1 1 1 1

. (H.3)

We then have M= (L+K)˜ ⁻¹. M≈L⁻¹−L⁻¹KL˜ ⁻¹.

139

Curriculum Vitae

January 2010 Dissertation in Theoretical Physics about

Quantum Control and Quantum Measurement in Solid State Qubits, under the supervision of Prof. Dr. Guido Burkard.

2005 - 2009 Graduate student under the supervision of Prof. Guido Burkard.

05.2008 - 12.2009 University of Konstanz, Germany.

05.2008 - 05.2009 Scientific visit in Scuola Normale Superiore, Pisa, Italy.

05.2007 - 05.2008 RWTH Aachen, Germany.

09.2005 - 05.2007 University of Basel, Switzerland.

March 2005 Laurea in Theoretical Physics. Title of the thesis:

Entanglement and phase transitions in spin-1/2 chains, under the supervision of Prof. Giuseppe Morandi.

1999 - 2005 Study Physics at the University of Bologna, Italy.

1999 Diploma di Maturita’, Liceo Scientifico Niccolo’ Copernico, Bologna, Italy.

25.09.1980 Born in Parma, Italy.

141

Publications

• E. Strambini, L. Chirolli, V. Giovannetti, F. Taddei, R. Fazio, V. Piazza, F. Beltram, Coherent detection of electron dephasing

arXiv:0909.2197 (2009).

• Luca Chirolli, Guido Burkard,

Quantum non-demolition measurements of a qubit coupled to a harmonic oscillator Phys. Rev. B 80, 184509 (2009).

• Luca Chirolli, Guido Burkard, Decoherence in Solid State Qubits Advances in Physics57, 225 (2008).

• Matthias Braun, Luca Chirolli, Guido Burkard,

Signature of chirality in scanning-probe imaging of charge flow in graphene Phys. Rev. B 77, 115433 (2008).

• Luca Chirolli, Guido Burkard,

Full control of qubit rotations in a voltage-biased superconducting flux qubit Phys. Rev. B 74, 174510 (2006).

143

Im Dokument Quantum control and quantum measurement in solid state qubits (Seite 134-157)