The quantization of persistent current qubit. The role of inductance

0.15 MB·English

by  Ya. S. Greenberg

The quantization of persistent current qubit. The role of
inductance
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Ya. S. Greenberg
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Novosibirsk State Technical University,

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20 K. Marx Ave., 630092 Novosibirsk, Russia
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(Dated: February 6, 2008)
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] Abstract
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The Hamiltonian of persistent current qubit is found within well known quantum mechanical
c
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r procedure. It allows a selfconsistent derivation of the current operator in a two state basis. It is
p
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s shown that the current operator is not diagonal in a flux basis. A non diagonal element comes
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a from the finite inductance of the qubit. The results obtained in the paper are important for the
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- circuits where two or more flux qubits are coupled inductively.
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PACS numbers: 03.67.Lx,85.25.Cp, 85.25.Dq, 85.35.Ds
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I. INTRODUCTION
Josephson-junction qubits are known to be candidates for scalable solid-state quantum
computing circuits [1]. Here we consider a supercobducting flux qubit which has been first
proposed in [2] and analyzed in [3] and [4]. The qubit consists of three Josephson junctions
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 flux qubit
circuitry it is important to have the loop inductance as much as possible consistent with
a proper operation of a qubit. A relative large loop inductance facilitates a qubit control
biasing schemes and the formation, control and readout of two-qubit quantum gates. These
considerations stimulated some investigations of the role the loop inductance plays in the
operation of a flux qubit [5], [6], [7]. The main goal of these works was the calculation of
the corrections to the energy levels due to finite inductance of the loop. In the early work
[5] these corrections have been obtained by perturbation expansion of the energy over small
parameter β = L/L , where L is the Josephson junction inductance. The extension to
J J
large β’s (up to β 10) had been considered in [7]. However, it is important to realize
≈
that for finite loop inductance the interaction between two state qubit with its own LC
circuit cannot in general be neglected. If β is not small, as in [7], this interaction can have
substantial influence on the energy levels. Unfortunately, this interaction in [7] has been
completely neglected.
In principle, the account for a finite loop inductance (even if it is small) requires the
correct construction of quantum mechanical Hamiltonian of a qubit, which contains all
relevant interactions. This has been done in [6], where the effective Hamiltonian has been
obtained by a rigorous expansion procedure in powers of β. As was shown in [6], one of the
effect oftheinteractionofaflux qubit withitsownLCoscillatoristherenormalizationofthe
Josephson critical current. The inclusion of circuit inductances in a systematic derivation
of the Hamiltonian of superconducting circuits has been done in [8]. It allows the correct
calculations of the effects of the finite inductance both for flux [9] and charge [10] qubits.
In this paper we investigate another physical effect which comes from finite loop induc-
tance. Namely, we show that the finite loop inductance results in the additional term of the
current operator in the flux basis:
I = Aτ +Bτ , (1)
Z X
b 2
where τ and τ are Pauli matrices in the flux basis. The quantities A and B in (1) are
Z X
calculated in the paper the B being conditioned by the finite loop inductance: for L = 0 the
second term in (1) is absent. Though for the usual qubit design with small loop inductance
this second term is relatively small, nevertheless, it might give noticeable effects for large β’s
for the arrangements when two flux qubit are coupled either via a common inductance [7] or
inductively coupled via a term MI I in the Hamiltonian, where M is a mutual inductance
1 2
between qubit’s loops, I , I are the current operators of the respective qubits.
1 2
b b
The paper is organized as follows. In Section II we start with the exact Lagrangian of a
b b
flux qubit with finite loop inductance. In section III with the aid of well known procedure
we derive rigorously the quantum qubit Hamiltonian. The current operator is studied in
Section IV, where we show that in general it is not diagonal in the flux basis. The matrix
elements for the current operator are calculated in Section VI, where in order to obtain
analytical results we consider a flux qubit with a small loop inductance.
II. LAGRANGIAN FOR THE FLUX QUBIT
We consider here a well known design of the flux qubit with three Josephson junctions
[2], [3], [4], which is shown on Fig.1.
Two junctions have equal critical current I and (effective) capacitance C, while those
c
of the third junction are slightly smaller: αI and αC, with 0.5 < α < 1. If the Josephson
c
energy E = I Φ /2π is much larger than the Coulomb energy E = e2/2C, the Josephson
J c 0 C
phase is well defined. Near Φ = Φ /2, this system has two low-lying quantum states [3, 4].
x 0
The Lagrangian of this qubit is the difference between the charge energy in the junction
capacitors and the sum of Josephson and magnetic energy:
3 C V2 3 Φ2
L = i i + E(i)cosϕ (2)
2 J i − 2L
i=1 i=1
X X
where V is the voltage across the junction capacitance C , which is related to the phase
i i
˙
φ by the Josephson relation V = (Φ /2π)φ ; Φ is the flux trapped in the loop:
i i 0 i
3
Φ
0
Φ = ϕ Φ (3)
i X
2π −
i=1
X
3
j E C
1 J1 1
V
1
F
X
I j 2 EJ2 C2 L
V
2
C
3
j E V
3 J3 3
FIG. 1: A flux qubit, where an external magnetic flux Φ pierces the superconducting loop that
X
contains three Josephson junctions and inductance L. Two Josephson junctions are considered to
be identical, E = E = E , C =C = C, and E = αE , C = αC.
J1 J2 J 1 2 J3 J 3
Next we make the following definitions: φ = φ +φ +φ , φ +φ = 2θ, φ φ = 2χ. In
1 2 3 1 2 1 2
−
terms of these new phases Lagrangian (2) takes the form:
~2 ~2 E
L = ϕ˙2 +ϕ˙2 +α ϕ˙2+2E cosθcosχ+αE cos(ϕ 2θ) J (ϕ ϕ )2 (4)
16E 1 2 16E 3 J J − − 2β − X
C C
(cid:0) (cid:1)
where β = 2πLI /Φ .
C 0
III. CONSTRUCTION OF HAMILTONIAN
Conjugate variables are defined in a standard way:
1∂L ~
˙
n = = α ϕ˙ 2θ (5)
ϕ ~∂ϕ˙ 8E −
C
(cid:16) (cid:17)
1∂L ~ ~
˙ ˙
n = = θ α ϕ˙ 2θ (6)
θ ~ ∂θ˙ 4E − 4E −
C C
(cid:16) (cid:17)
1∂L ~
n = = χ˙ (7)
χ ~∂χ˙ 4E
C
From these equations we express phases in terms of conjugate variables:
8E 2α+1 8E
C C
ϕ˙ = n + n (8)
~ α ϕ ~ θ
4
4E 8E
˙ C C
θ = n + n (9)
~ θ ~ ϕ
4E
C
χ˙ = n (10)
~ χ
In terms of conjugate variables Lagrangian (4) takes the form:
1+2α
L = 2E n2 +2E n2 +4E n2 +8E n n (11)
C θ C χ C α ϕ C θ ϕ
E
+2E cosθcosχ+αE cos(ϕ 2θ) J (ϕ ϕ )2
J J X
− − 2β −
The Hamiltonian is constructed according to the well known rule:
H = ~n ϕ˙ +~n θ˙+~n χ˙ L (12)
ϕ θ χ
−
Finally we obtain:
2α+1
H = 4E n2 +2E n2 +2E n2 +8E n n +U(χ,θ,ϕ) (13)
C α ϕ C χ C θ C θ ϕ
where
E
U(χ,θ,ϕ) = 2E cosθcosχ αE cos(ϕ 2θ)+ J (ϕ ϕ )2 (14)
J J X
− − − 2β −
Hence, equations of motion for the phases (8), (9), (10) are simply ϕ˙ = 1 ∂H ; θ˙ =
~∂nϕ
1 ∂H; χ˙ = 1 ∂H . The equations of motion for conjugate variables are:
~∂nθ ~∂nχ
1∂H αE E
J J
n˙ = = sin(ϕ 2θ) (ϕ ϕ ) (15)
ϕ −~ ∂ϕ − ~ − − ~β − X
1∂H 2αE 2E
J J
n˙ = = sin(ϕ 2θ) sinθcosχ (16)
θ −~ ∂θ − ~ − − ~
1∂H 2E
J
n˙ = = cosθsinχ (17)
χ −~ ∂χ − ~
Below we consider Hamiltonian (13) as quantum mechanical with commutator relations
imposed on its variables
[ϕ,n ] = i; [θ,n ] = i; [χ,n ] = i (18)
ϕ θ χ
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IV. CURRENT OPERATOR
From the first principles a current in the loop is equal to the first derivative of the state
energy relative to external flux:
∂E
n
I = (19)
∂Φ
X
This expression can be rewritten in terms of exact Hamiltonian of a system:
∂Hˆ
I = n n (20)
h | ∂Φ | i
X
From (20) we would make ansatz that the current operator is as follows:
ˆ
∂H
Iˆ= (21)
∂Φ
X
However (21) is not a consequence of (20). Therefore, the ansatz (21) must be proved in
every case, since the current operator in the form of Eq. (21) has to be consistent with its
definition in terms of variables of Hamiltonian H. The prove for our case is given below.
The current operator across every junction is a sum of a supercurrent and a current
through the capacitor:
~
Iˆ = I sinϕ + Cϕ¨ (i = 1,2) (22)
i 0 i 2e i
~
Iˆ = αI sinϕ +α Cϕ¨ (23)
3 0 3 2e 3
Since the current in a loop is unique the equations (22) and (23) must give identical result.
This is indeed the case if we express phases φ (i = 1,2,3) in terms of φ, θ, χ and use the
i
equations (8), (9), (10), (15), (16), (17). For every I in (22), (23) we obtain the same
i
expression
ϕ ϕ
Iˆ= I − X (24)
0
− β
which is independent of parameters of a particular junction in the loop. From the other
hand the expression (24) can be obtained from our Hamiltonian (13) with the aid of (21).
Therefore, the equation (21) gives us the true expression for the current operator. It is
importanttonotethattheproperexpressionforthecurrentoperator(24)cannotbeobtained
without magnetic energy term in the original Lagrangian (2).
It follows from (13) and (24) that Iˆ,Hˆ = 0. Therefore, an eigenstate of H cannot
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possess a definite current value. h i
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A. Current operator in a two-state basis
Suppose a system is well described by two low lying states Ψ± with corresponding
| i
eigenenergies E±:
H Ψ± = E± Ψ± (25)
| i | i
Within this subspace Hamiltonian canb be expressed in terms of Pauli matrices σX,σY,σZ:
E+ +E− E+ E−
H = − σ (26)
Z
2 − 2
with σZ Ψ± = Ψ± . b
| i ∓| i
Now we calculate the matrix elements of the current operator (21) within this subspace.
According to (19),(20) and (21) diagonal matrix elements are:
∂E±
Ψ± I Ψ± = (27)
h | | i ∂Φ
X
In order to find nondiagonal matrix elemebnts of the current operator we use the expression
ˆ ′
∂H ∂n
′
n n = (En′ En) n (28)
h | ∂λ | i − ∂λ
(cid:28) (cid:12) (cid:29)
(cid:12)
ˆ (cid:12) ′
which is obtained by differentiating of the identity n H n = 0 with respect to parameter
(cid:12)
h | | i
λ. Hence, the nondiagonal elements of the current operator are:
∂Ψ
Ψ− Iˆ Ψ+ = Ψ+ Iˆ Ψ− = (E+ E−) Ψ− + (29)
h | | i h | | i − ∂Φ
(cid:28) (cid:12) X (cid:29)
(cid:12)
Therefore, we can express the current operator in terms of Pauli(cid:12)matrices:
(cid:12)
I = ∂ E+ +E− I ∂ E+ −E− σZ +(E+ E−) Ψ− ∂Ψ+ σX (30)
∂Φ 2 − ∂Φ 2 − ∂Φ
X (cid:18) (cid:19) X (cid:18) (cid:19) (cid:28) (cid:12) X (cid:29)
(cid:12)
wherbe I is the unity matrix. (cid:12)
(cid:12)
Below we consider two low lying states of a flux qubit
E± = E0 √ε2 +∆2 (31)
±
where E andthe tunneling rate∆areindependent of theexternal fluxΦ , andthe quantity
0 X
ε is linear function of the flux, ε = E λf , where λ is a numerical factor which depends on
J X
qubit parameters α and g = E /E , f = Φ /Φ 1/2.
J C X X 0
−
Therefore, for the flux qubit we get in eigenstate basis:
H = E ∆ σ (32)
0 ε Z
−
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∂∆ ∂Ψ
Iˆ= ε σZ +2∆ε Ψ− + σX (33)
−∂Φ ∂Φ
X (cid:28) (cid:12) X (cid:29)
(cid:12)
where ∆ = √ε2 +∆2. (cid:12)
ε (cid:12)
Transformation to the flux basis is obtained via the rotation around y axes in a two level
subspace with the aid of the matrix R = exp(iξσ /2), where cosξ = ε/∆ , sinξ = ∆/∆ :
Y ε ε
R−1σ R = τ cosξ+τ sinξ, R−1σ R = τ sinξ+τ cosξ,whereτ ,τ arePaulimatrices
Z Z X X Z X X Z
−
in a flux basis. Hence, we get for Hamiltonian (32) and current operator (33) in the flux
basis:
H = ετ ∆τ (34)
Z X
− −
∂∆ ε ∂Ψ ∂∆ ∆ ∂Ψ
Iˆ= ε +2∆ε Ψ− + τZ ε 2ε Ψ− + τX (35)
− ∂Φ ∆ ∂Φ − ∂Φ ∆ − ∂Φ
(cid:18) X ε (cid:28) (cid:12) X (cid:29)(cid:19) (cid:18) X ε (cid:28) (cid:12) X (cid:29)(cid:19)
(cid:12) (cid:12)
Therefore, the current operator(cid:12)is not diagonal neither in the flux b(cid:12)asis nor in the eigen-
(cid:12) (cid:12)
state basis.
The stationary state wave functions Ψ± can be written as the superpositions of the wave
functions in the flux basis, Ψ ,Ψ where L, R stand for the left, right well, respectively:
L R
Ψ± = a±ΨL +b±ΨR, where
∆ ε ∆
ε
a± = ;b± = ∓ ; (36)
2∆ (∆ ε) 2∆ (∆ ε)
ε ε ε ε
∓ ∓
The coefficients a±, b± are defipned in such a way, thpat τZ ΨL = ΨL , τZ ΨR = + ΨR ,
| i −| i | i | i
τ Ψ = + Ψ , τ Ψ = + Ψ . In terms of the functions Ψ ,Ψ the cross term
X L R X R L L R
| i | i | i | i
Ψ− ∂Ψ+ will read
∂ΦX
D (cid:12) E
(cid:12)
∂Ψ ∂a ∂b ∂Ψ ∂Ψ
(cid:12) + + + L R
Ψ− = a− +b− +a−a+ ΨL +b−b+ ΨR (37)
∂Φ ∂Φ ∂Φ ∂Φ ∂Φ
(cid:28) (cid:12) X (cid:29) X X (cid:28) (cid:12) X (cid:29) (cid:28) (cid:12) X (cid:29)
(cid:12) (cid:12) (cid:12)
The resu(cid:12)lts obtained up till now are exact in that w(cid:12)e did not make any (cid:12)approximation to
(cid:12) (cid:12) (cid:12)
the Hamiltonian (13). However, in order to calculate ε and cross term Ψ− ∂Ψ+ in (33),
∂ΦX
(35) we need some approximate procedure. D (cid:12) E
(cid:12)
(cid:12)
V. APPROXIMATION TO QUANTUM MECHANICAL HAMILTONIAN
In order to calculate the matrix elements of the current operator we have to find the wave
functions of two lowest levels of Hamiltonian (13). First we single out of the potential (14)
8
the fast variable ϕ, which describe the interaction of the qubit with its own LC circuit. The
point of minimum ϕ of U(χ,θ,ϕ) (14) with respect to ϕ is defined from ∂U/∂φ = 0:
C
ϕ = ϕ αβsin(ϕ 2θ) (38)
C X C
− −
In the vicinity of ϕ the potential U(χ,θ,ϕ) can be written as:
C
E αE
U (χ,θ,ϕ) U (χ,θ,ϕ )+ Jϕ2 +ϕ2 J cos(ϕ 2θ) (39)
C C
≈ 2β 2 −
where ϕ is a small operator correction to ϕ : ϕb= ϕ b+ϕ.
C C
As is known the potential U(χ,θ,ϕ ) has a degenerate point at Φ = Φ /2. Assuming
C x 0
b b
f << 1, β << 1 we obtain near this point:
X
ϕ = π +2πf αβsin2θ (40)
C X
−
From (24) and (40) we find a current operator in ”coordinate” representation:
I = I αsin2θ (41)
0
For U(χ,θ,ϕ ) we obtain near degenerbacy point
C
α2βE
U (χ,θ,ϕ ) = 2E cosθcosχ+αE cos2θ+α2πf E sin2θ J sin22θ (42)
C J J X J
− − 2
Below we follow the procedure described in [11]. At f = 0, the potential (42) has two
x
minima at χ = 0, θ = θ∗, with cosθ∗ = 1/2α (θ∗ > 0). Tunnelling lifts their degeneracy,
±
leading to energy levels E± = E0 ∆. However, at degenerate bias the current vanishes,
±
forcing one to move slightly away from this point. In order to find the levels for f 1
x
| | ≪
we expand Eq. (42) near its minima, retaining linear terms in f , β and quadratic terms in
x
r/l
χ,θ. Define θ∗ as the minima, shifted due to fx and β:
1 2α2 1 2α2
θ∗r/l = θ∗ +2πfx − β − ; (43)
± 4α2 1 ± 2α(4α2 1)
− −
that is, the upper (lower) sign refers to the right (left) well. The potential energy (42) then
reads:
Ur/l(χ,θ,ϕ )
C = Ur/l +Ar/lχ2 +Br/lθ2 (44)
E 0
J
r/l
where the operator correction θ = θ θ∗ , b b
−
1 √4α2 1 4α2 1
r/l b
U = α 2πf − β − (45)
0 − − 2α ± X 2α − 8α2
9
1 2α2 1 2α2 1
Ar/l = 2πf − +β − (46)
2α ∓ X2α√4α2 1 4α2
−
1 2α2 +1 1 5
Br/l = 2α 2πf +β + α2 2α4 (47)
X
− 2α ∓ 2α√4α2 1 −4 2 −
− (cid:18) (cid:19)
Combining (44) and (42) in (13) we obtain quadratic quantum mechanical Hamiltonian for
the flux qubit in the left and right well near the degeneracy point:
4(2α+1) E
Hr/l = E Ur/l + E nˆ2 + Jϕˆ2 + 2E nˆ2 +E Ar/lχˆ2 (48)
J 0 α C ϕ 2β C χ J
(cid:20) (cid:21)
(cid:2) E (cid:3)
+ 2E nˆ2 +E Br/lθˆ2 +8E nˆ nˆ + JCr/lϕˆ2
C θ J C θ ϕ 2
h i
where
1 2α2 1 √4α2 1 β 4α2 1
Cr/l = 1 2πf − + − + 1 2α − (49)
X
2α − ± √4α2 1 α 2α − − α
(cid:20) (cid:18) − (cid:19) (cid:18) (cid:19)(cid:21)
The first terminsquare brackets in(48) isthe HamiltonianofLC oscillator of the fluxqubit,
which is slightly modified by the last term in (48). the next two terms in square brackets are
oscillator Hamiltonians for the flux qubit variables, χ and θ, respectively. The interaction
of the θ degree of freedom with the qubit LC circuit is given by next-to-last term in (48).
Assuming the frequency (LC)−1/2 of the qubit LC circuit is much higher than the junc-
tions frequencies E /~, E /~ we neglect the interaction of the qubit variables, θ and χ with
J C
the qubit LC oscillator. This is equivalent to the averaging of Hamiltonian (48) over the
ground state of the LC Hamiltonian. Therefore, for the qubit Hamiltonian we obtain:
1 E
H = Hr/l = E Ur/l + ε + JCr/l ϕˆ2 + 2E nˆ2 +E Ar/lχˆ2 + 2E nˆ2 +E Br/lθˆ2
qb J 0 2 0 2 C χ J C θ J
(cid:10) (cid:11) (cid:10) (cid:11) (cid:2) (cid:3) h (50i)
where
1/2 1/2
8E E (2α+1) 1 8βE (2α+1)
ε = C J ; ϕ2 = C
0
β α 2 E α
(cid:18) (cid:19) (cid:18) J (cid:19)
(cid:10) (cid:11)
Next we confine ourself only to the ground state of (50) in either of the wells.
1 Cr/l ~ωr/l ~ωr/l
εr/l = ε +E Ur/l +E ϕ2 + θ + χ (51)
2 0 J 0 J 2 2 2
(cid:10) (cid:11)
where
4 2α2 1 2α2 1
~ωr/l = E 1 2πf − +β − (52)
χ J αg ∓ X2√4α2 1 4α
r (cid:18) − (cid:19)
10