The quantization of a charge qubit. The role of inductance and gate capacitance

0.17 MB·English

by  Ya. S. Greenberg

The quantization of a charge qubit. The role of inductance and
gate capacitance
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0 Ya. S. Greenberg1 and W. Krech2
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1Novosibirsk State Technical University,
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20 K. Marx Ave., 630092 Novosibirsk, Russia
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8 2Friedrich-Schiller University, Jena, Germany
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] (Dated: February 6, 2008)
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Abstract
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r The Hamiltonian of a charge qubit, which consists of two Josephson junctions is found within
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u
s well known quantum mechanical procedure. The inductance of the qubit is included from the very
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a beginning. It allows a selfconsistent derivation of the current operator in a two state basis. It is
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- shown that the current operator has nonzero nondiagonal matrix elements both in the charge and
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n
the eigenstate basis. It is also shown that the interaction of the qubit with its own LC resonator
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has a noticeable influence on the qubit energies. The influence of the junctions asymmetry and
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1 the gate capacitance on the matrix elements of the current operator and on the qubit energies are
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4 calculated. The results obtained in the paper are important for the circuits where two or more
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charge qubits are coupled with the aid of inductive coil.
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0 PACS numbers: 03.67.Lx,85.25.Cp, 85.25.Dq, 85.35.Ds
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I. INTRODUCTION
Josephson-junction charge qubits are known to be candidates for scalable solid-state
quantum computing circuits [1], [2], [3]. Here we consider a superconducting charge qubit
which consists of two Josephson junctions embedded in a loop with very small inductance L,
typically in the pH range. This insures effective decoupling from the environment. However,
in the practical implementation of qubit circuitry it is important to have the loopinductance
as much as possible consistent with a proper operation of a qubit. A relative large loop
inductance facilitatesa qubit controlbiasing schemes andtheformation, controlandreadout
of two-qubit quantum gates. These considerations stimulated some investigations of the role
the loop inductance plays in the dynamic properties of charge qubits [4], [5], [6], where for
the small loop inductance the corrections to the energy levels due to finite inductance of the
loop have been found. The corrections have been obtained by perturbation expansion of the
energy over small parameter β = L/L , where L is the Josephson junction inductance.
J J
For complex Josephson circuit the construction of quantum Hamiltonian which accounts
for finite inductances of superconducting loops can be made with the aid of the graph theory
[7]. This approach has been developed in [8] for systematic derivation of the Hamiltonian of
superconducting circuits and has been applied for the calculations of the effects of the finite
loop inductance both for flux [9] and charge [10] qubits.
In principle, the account for a finite loop inductance (even if it is small) requires for the
magnetic energy to be included in quantum mechanical Hamiltonian of a qubit from the
very beginning. It allows one to obtain the effects of the interaction between two-level qubit
and its own LC circuit. In addition it allows a correct definition of the current operator in
terms of its matrix elements in a two level basis.
In this paper we investigate the effect of finite loop inductance and gate capacitance
for a asymmetric charge qubit, which consists of two Josephson junctions embedded in a
superconducting loop.
The construction of exact Lagrangian and Hamiltonian for the charge qubit is given in
Section II and Section III, respectively. The approximation for exact Hamiltonian for small
L is made in Section IV. It is shown that Hamiltonian is decomposed in three parts: qubit
part, LC-oscillator part and the qubit-LC oscillator interaction part. In this approximation
the energy levels of the charge qubit are explicitly dependent on the gate capacitance and
2
critical current asymmetry and , in addition, are shifted due to vacuum fluctuations of LC
oscillator. The current operator both in charge and in eigenstate basis is obtained in Section
V. It is shown that the asymmetry of critical currents of the Josephson junctions results in
additional terms in the operator of critical current. The corrections to the qubit energies
due to its interaction with LC circuit and their dependence on critical current asymmetry
and on gate capacitance are calculated in Section VI.
II. LAGRANGIAN FOR THE CHARGE QUBIT
We consider here a charge qubit in the arrangement, which has been first proposed in [1]
(see Fig.1). The qubit consists of two Josephson junctions in a loop with very small induc-

I
1
C
1
V
1
V C
g g
L
C
I 2
2
V
2
FIG. 1: Charge qubit with inductance coil.
tance L, typically in the pH range. This insures effective decoupling from the environment.
As a general case we assume that two junctions have different critical currents I , I and
c1 c2
capacitance C , C . The Josephson energy E = I Φ /2π, where Φ = h/2e is the flux
1 2 J c 0 0
quantum, is assumed to be much less than the Coulomb energy E = (2e)2/2C, so that the
C
charge at the gate is well defined.
The Lagrangian of this qubit is the difference between the charge energy in the junction
capacitors and the Josephson plus magnetic energy:
Φ2
L = U +E cosϕ +E cosϕ (1)
J1 1 J2 2
− 2L
3
where U is the electric energy of JJ’s and gate capacities
C V2 C V2 C V2
1 1 2 2 g
U = + + , (2)
2 2 2
Φ = V dt, where V is the voltage drop across the inductance.
L L
InRvirtue of Josephson relations
~ ~
V = ϕ˙ , i = 1,2; V = V + ϕ˙ (3)
i i g 2
2e 2e
the voltage drop across the inductance is:
dΦ d ~
X
V = V +V = (ϕ +ϕ ) Φ (4)
L 1 2 1 2 X
− dt dt 2e −
(cid:20) (cid:21)
where Φ is the external flux.
X
In terms of the phases ϕ , ϕ , ϕ Lagrangian (1) takes the form:
1 2 3
~2 C ~ 2
2 2 g
L = C ϕ˙ +C ϕ˙ + V + ϕ˙ (5)
2(2e)2 1 1 2 2 2 g 2e 2
(cid:18) (cid:19)
~2 (ϕ +(cid:0)ϕ ϕ )2 (cid:1)
1 2 X
− +E cosϕ +E cosϕ
−(2e)2 2L J1 1 J2 1
where ϕ = 2πΦ /Φ .
X X 0
Next we make the known (Likharev and Averin) redefinition of the Josephson phases:
ϕ +ϕ = ϕ; ϕ ϕ = 2δ. Lagrangian (5) takes the form:
1 2 1 2
−
~2(C +C ) ϕ˙2 ~2(C C ) C ~ ~ 2
1 2 ˙2 1 2 ˙ g ˙
L = +δ + − δϕ˙ + V + ϕ˙ δ (6)
2(2e)2 4 2(2e)2 2 g 4e − 2e
(cid:18) (cid:19) (cid:18) (cid:19)
~2 (ϕ ϕ )2 ϕ ϕ
X
− +(E +E )cos cosδ +(E E )sin sinδ
−(2e)2 2L J1 J2 2 J2 − J1 2
III. CONSTRUCTION OF HAMILTONIAN
Conjugate variables are defined in a standard way:
1∂L ~(C +C +C ) ~(C C C ) C V
1 2 g 1 2 g ˙ g g
n = = ϕ˙ + − − δ + (7)
ϕ ~∂ϕ˙ 4(2e)2 2(2e)2 4e
1∂L ~(C +C +C ) ~(C C C ) C V
1 2 g ˙ 1 2 g g g
n = = δ + − − ϕ˙ (8)
δ ~ ∂δ˙ (2e)2 2(2e)2 − 2e
4
From these equations we express phases in terms of conjugate variables:
E α n
C g
ϕ˙ = 2 n +γ(n +n ) (9)
~ ϕ − 2 δ g
h (cid:16) (cid:17) i
E α n 1
˙ C g
δ = γ n + (n +n ) (10)
~ ϕ − 2 2 δ g
(cid:20) (cid:21)
(cid:16) (cid:17)
where E = (2e)2/2C , α = C2/C (C +C ), γ = (C +C C )/C , C = C +C +C ,
C Σ Σ 1 2 g g 2 1 Σ Σ 1 2 g
−
n = C V /2e.
g g g
Now we construct Hamiltonian:
H =~n ϕ˙ +~n δ˙ L (11)
ϕ δ
−
Eliminating time derivatives of the phases from (11) with the aid of (10), (9), we obtain the
final expression for Hamiltonian of the asymmetric charge qubit:
n 2 E α n
g C 2 g
H = E α n + (n +n ) +E αγ n (n +n ) (12)
C ϕ δ g C ϕ δ g
− 2 4 − 2 −
(cid:16) (cid:17) ϕ ϕ (cid:16) (ϕ ϕ(cid:17))2 (2e)2
X 2
2E cos cosδ E ξsin sinδ +E − n
− J 2 − J 2 J 2β − 2C g
g
where E = Φ I /2π, I = (I +I )/2, ξ = (I I )/I , β = 2πLI /Φ .
J 0 C C C1 C2 C2 C1 C C 0
−
The first two equations of motion
1 ∂H 1 ∂H
˙
δ = ; ϕ˙ =
~∂n ~∂n
δ ϕ
are given by Eqs. (9) and (10). Two other equations are as follows:
1∂H 2E ϕ E ϕ
J J
n˙ = = cos sinδ + ξsin cosδ (13)
δ −~ ∂δ − ~ 2 ~ 2
1∂H E ϕ E ϕ E ϕ ϕ
J J J X
n˙ = = sin cosδ + ξcos sinδ − (14)
ϕ −~ ∂ϕ − ~ 2 2~ 2 − ~ β
Below we consider Hamiltonian (12) as quantum mechanical with commutator relations
imposed on its variables
[ϕ,n ] = i; [δ,n ] = i (15)
ϕ δ
where n = i∂/∂ϕ, n = i∂/∂δ.
ϕ δ
− −
5
IV. APPROXIMATION TO QUANTUM MECHANICAL HAMILTONIAN
Obviously, Hamiltonian (12) is 2D nonlinear oscillator. We assume L is small, so that
its frequency (LC )−1/2 >> E /~ . Therefore we can consider ϕ as fast variable with fast
Σ J
oscillations near the point ϕ , the minimum of potential U(ϕ,δ) (see (12)):
C
2
ϕ ϕ (ϕ ϕ )
X
U(ϕ,δ) = 2E cos cosδ E ξsin sinδ +E − (16)
J J J
− 2 − 2 2β
We single out of this potential the fast variable ϕ, which describes the interaction of the
qubit with its own LC circuit.
The point of minimum ϕ of U(ϕ,δ) (16) with respect to ϕ is defined from ∂U/∂φ = 0:
C
ϕ βξ ϕ
C C
ϕ = ϕ βsin cosδ + cos sinδ (17)
C X
− 2 2 2
Inwhat followswe consider δ asslowvariableandexpand U(ϕ,δ) near thepointof minimum
to the third order in ϕ (ϕ = ϕ + ϕ). In the vicinity of ϕ the potential U(ϕ,δ) can be
C C
written as:
b
EJ β ϕC βξ ϕC ⌢2
U(ϕ,δ) = U(ϕ ,δ)+ 1+ cos cosδ + sin sinδ ϕ (18)
C
2β 2 2 4 2
(cid:18) (cid:19)
EJ⌢3 ϕC ξ ϕC
ϕ sin cosδ cos sinδ
−24 2 − 2 2
(cid:18) (cid:19)
where ϕ is the operator conjugate to n .
ϕ
With the aid of (17) we write U(ϕ ,δ) to the first order in β:
C
b
ϕ ϕ
X X
U(ϕ ,δ) U(ϕ ,δ) = 2E cos cosδ E ξsin sinδ (19)
C X J J
≡ − 2 − 2
E β ϕ ξ ξ2 ϕ
J 2 X 2 2 X 2
sin cos δ sinϕ sin2δ + cos sin δ
X
− 2 2 − 4 4 2
(cid:18) (cid:19)
Therefore, we decompose Hamiltonian (12) into oscillator, qubit and interaction parts: H =
H +H +H , where
osc qb int
2 EJ⌢2
H = E αn + ϕ E α(1 γ)n n (20)
osc C ϕ 2β − C − g ϕ
E α E α
C 2 C
H = (n +n ) γn (n +n )+U(ϕ ,δ) (21)
qb δ g g δ g X
4 − 2
6
EJ⌢2 ϕX ξ ϕX
H = E αγn n + ϕ cos cosδ + sin sinδ (22)
int C ϕ δ
4 2 2 2
(cid:18) (cid:19)
EJ⌢3 ϕX ξ ϕX
ϕ sin cosδ cos sinδ
−24 2 − 2 2
(cid:18) (cid:19)
EJ⌢2 2 ϕX 2 ξ ξ2 2 ϕX 2
+β ϕ sin cos δ sinϕ sin2δ+ cos sin δ
X
8 2 − 8 8 2
(cid:18) (cid:19)
EJ⌢3 2 ξ ξ2 2
+β ϕ sinϕ cos δ cosϕ sin2δ sinϕ sin δ
X X X
96 − 2 − 4
(cid:18) (cid:19)
In the above equations we disregard the constant term which is proportional to n2.
g
The first term in (22) describes the interaction of the phase variables of the qubit, ϕ and
δ via the gate, the other terms are responsible for the interaction of the qubit with its own
LC circuit.
A. Two-level approximation
First we quantize (20) according to (n = i ∂ ; [n ,ϕ] = i):
ϕ − ∂ϕ ϕ −
1/4 1/4
1 2βE α 1 E
C + J +
ϕ = a +a ; n = i a a (23)
ϕ
√2 E √2 2βE α −
(cid:18) J (cid:19) (cid:18) C (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
where [a,a+] = 1.
In addition, we use the two level approximation in the charge basis: n = 1 (1+τ );
δ 2 Z
cosδ = τ /2; sinδ = τ /2 with Pauli operators
X Y
τ 0 = 0 ;τ 1 = 1 ; (24)
Z Z
| i −| i | i | i
τ 0 = 1 ;τ 1 = 0 ;
X X
| i | i | i | i
τ 0 = i 1 ;τ 1 = i 0 .
Y Y
| i − | i | i | i
In this approximation sin2δ=cos2δ = 0, since these operators couple charge states which
differs by two Cooper pairs. Therefore, cos2δ=sin2δ=1/2.
Now we write down Hamiltonian (20,21, 22) within two level subspace in terms of Pauli
operators τ ,τ ,τ and oscillator operators a+,a.
X Y Z
We obtain the following result:
H = W +H +H +H (25)
0 osc qb int
where
E βξ2
C
W = cosϕ (26)
0 X
32
7
1 E α γ
+ C +
H = E a a+ +i (1 γ)n a a (27)
osc 0 g
2 √2η 2 − − −
(cid:18) (cid:19)
h i
EJβ 2 ξ2 ξ2 (cid:0)+ (cid:1)2
+ η 1+ 1 cosϕ a +a
X
64 8 − − 8
(cid:18)(cid:18) (cid:19) (cid:18) (cid:19) (cid:19)
EJβ 3 ξ2 (cid:0) + (cid:1)3
+ η 1 sinϕ a +a
X
384√2 − 4
(cid:18) (cid:19)
(cid:0) (cid:1)
E α ϕ ξ ϕ
C X X
H = (1+2(1 γ)n )τ τ E cos τ E sin (28)
qb g Z X J Y J
8 − − 2 − 2 2
i ECαγ + η2 ϕX ξ ϕX + 2
H = a a τ +E τ cos +τ sin a +a (29)
int 23/2 η − Z J16 X 2 Y 2 2
(cid:18) (cid:19)
(cid:0) (cid:1) η3 ϕX ξ ϕX (cid:0) + (cid:1)3
E τ sin τ cos a +a
J X Y
− 96√2 2 − 2 2
(cid:18) (cid:19)
(cid:0) (cid:1)
where
1/2 1/4
2E E α 2βE α
C J C
E = , η =
0
β E
(cid:18) (cid:19) (cid:18) J (cid:19)
B. The energy levels of the charge qubit
Here we neglect the interaction of the qubit with its own LC circuit. It is justified if
β is sufficiently small so that the energy levels of the qubit oscillator are located much
higher than the ground level of the qubit. The approximation we make here is to average
Hamiltonian (25) over the vacuum state, a+a = 0, of the qubit oscillator. The result is as
follows:
1 1 1
H = W + Aτ + Bτ + Cτ (30)
X Y Z
2 2 2
where
β E η2 ξ2
2 J
W = E ξ 1 cosϕ (31)
C X
32 − 2 − 8
(cid:20) (cid:18) (cid:19)(cid:21)
η2 ϕ
X
A = 2E 1 cos (32)
J
− − 16 2
(cid:18) (cid:19)
η2 ϕ
X
B = E ξ 1 sin (33)
J
− − 16 2
(cid:18) (cid:19)
E α
C
C = [1+2(1 γ)n ] (34)
g
4 −
8
Hamiltonian(30)hasthecorrectionsontheorderofη2 √β whichareduetothevacuum
≈
fluctuations of the LC oscillator. Since in a charge qubit E >> E these corrections in
C J
principle might be not very small.
Hamiltonian (30) can be made diagonal in the eigenbasis with the aid of the matrix [6]:
e−iΨsin θ cos θ
S = − 2 2 (35)
 cos θ eiΨsin θ 
2 2
b  
where sinθ = ε/∆E, cosθ = C/∆E, sinΨ = B/ε, cosΨ = A/ǫ; ε = √A2 +B2, ∆E =
√ε2 +C2.
The qubit Hamiltonian in eigenstate basis, therefore, reads:
1
−1
S HS = W ∆Eσ (36)
Z
− 2
where W is given by (31) and b b
η2 2 ϕ ξ2 ϕ E α 2
∆E = 4E2 1 cos2 X + sin2 X + C [1+2n (1 γ)]2 (37)
s J − 16 2 4 2 4 g −
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
As is seen from (37) the energies of the charge qubit account for its full asymmetry and are
explicitly dependent on the inductance and on the gate capacitance.
V. CURRENT OPERATOR
From the first principles the average current in the loop is equal to the first derivative of
the eigenenergy relative to the external flux:
∂E
n
I = (38)
∂Φ
X
This expression can be rewritten in terms of exact Hamiltonian of a system:
ˆ
∂H
I = n n (39)
h | ∂Φ | i
X
From (39) we would make ansatz that the current operator is as follows:
∂Hˆ
Iˆ= (40)
∂Φ
X
However (40) is not a consequence of (39). Therefore, the ansatz (40) must be proved in
every case, since the current operator in the form of Eq. (40) has to be consistent with its
definition in terms of variables of Hamiltonian H. The prove for our case is given below.
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The current operator across every junction is a sum of a supercurrent and a current
through the capacitor:
~
Iˆ = I sinϕ + C ϕ¨ (i = 1,2) (41)
i Ci i 2e i i
We are interested in the current through the inductance coil, I in (41) (see Fig.1).
1
Direct calculation of I with the aid of (9,10,14,13) yield the result:
1
ϕ ϕ
X
I = I − (42)
1 C
− β
b
which is independent of parameters of a particular junction in the loop. From the other
hand the expression (42) can be obtained from our Hamiltonian (12) with the aid of (40).
Therefore, the equation (40) gives us the true expression for the current operator. It is
importanttonotethattheproperexpressionforthecurrentoperator(42)cannotbeobtained
without magnetic energy term in the original Lagrangian (1).
It follows from (12) and (42) that Iˆ,Hˆ = 0. Therefore, an eigenstate of H cannot
6
possess a definite current value. h i
For the charge qubit the current operator can be obtained from (30) with the aid of its
definition (40):
2π ∂W π B ξ
I = + τ + Aτ (43)
X Y
Φ ∂Φ Φ − ξ 4
0 X 0 (cid:20) (cid:21)
The transformation of (43) bin the eigenstate basis yeild the result:
−1
S IS = I +I σ +I σ +I σ (44)
0 Z Z X X Y Y
b bb
where
2π ∂W
I = (45)
0
Φ ∂Φ
0 X
1∂∆E π E2 ξ2 η2 2
I = = J 1 1 sinϕ (46)
Z X
−2 ∂Φ −Φ ∆E 4 − − 16
X 0 (cid:18) (cid:19)(cid:18) (cid:19)
π η2 ϕX ξ42 + ∆CE ξ42 −1 cos2 ϕ2X
I = E 1 sin (47)
X Φ0 J − 16 2 h cos2 ϕX(cid:16) + ξ2 s(cid:17)in2 ϕX i
(cid:18) (cid:19) 2 4 2
π ξ η2 ϕX 1+ ∆CE ξ42 −1 sin2 ϕ2X
I = E 1 cos (48)
Y −Φ0 J2 − 16 2 h cos2 ϕ(cid:16)X + ξ2 s(cid:17)in2 ϕX i
(cid:18) (cid:19) 2 4 2
10