2.3.1 Charge qubits
A superconducting Josephson junction qubit in which the charging energy is much large than the Josephson coupling, EC EJ, is called a charge qubit. In this regime, a convenient basis is formed by the charge states, and the phase terms can be consid-ered as perturbation. This is why this kind of qubits are called charge qubits. The necessary of one-qubit and two-qubit gates can be performed by controlling applied gate voltages and magnetic fields. Different designs will be presented that not only in complexity, but also in flexibility of manipulations.
In this subsection, the simplest charge qubit, cooper-pair box, Fig. 2.3, is pre-sented in details. This example illustrates how charge qubits provide two energy states, which satisfy the requirements of qubits.
In charge regime, at first we expand all operators in the basis of the charge states {|ni}. The Hamiltonian of a cooper-pair box, Eq. (2.13), is
H = Eˆ C(ˆn − ng)2− EJcos ˆδ .
Then by using the properties of orthonomal and complete set, hn | ˆn | n0i = δn,n0 and I =P
n|nihn|, the first term is rewritten as X
n
EC(n − ng)2|nihn| (2.22)
and by using the commutator relation,
The commutator relation Eq. (2.23) is similar to the commutator relation of number operator ˆa+ˆa and the creation operator ˆa+, [ˆa+a, ˆˆ a+] = ˆa+. So, eiˆδ and e−iˆδ can be presented in charge basis,
eiˆδ=X
n
|n + 1ihn| , e−iˆδ =X
n
|nihn + 1| , (2.24)
and the second term of Eq. (2.13) is 1
2EJX
n
(|nihn + 1| + |n + 1ihn|) . (2.25)
By combining Eq. (2.22) and Eq. (2.25), in this basis the Hamiltonian reads H =ˆ X
The energy spectrum of Eq. (2.26) is shown in Fig. 2.6a.
Under suitable conditions, when charge number on a gate capacitor ng controlled by gate voltage Vg equals half integers, the lowest two energy states are well-isolated from other states, shown in Fig. 2.6b. Because of that, near ng = 1/2, the Hamilto-nian can be reduced to
H = −ˆ 1
n
g-2 -1 0 1 2 3 0.5
n
g(a) (b)
0.5
Figure 2.6 (a) The energy spectrum of a charge qubit versus gate voltage.
(b) The lowest two energy levels near Vg = 0.5, the part of (a) circumscribed by dashed lines.
So, under suitable conditions charge qubits provide physical realizations of qubits with two charge states differing by one cooper-pair charge on a small island. For quantum computation, it is required to have the ability to rotate a state on the Bloch sphere to any position at will, and consequently σz and σx rotation are necessary. In a cooper-pair box, pure σx rotation is acquirable, as ng = 1/2, but pure σz rotation is not, since EJ is fixed. In previous section, an important concept is mentioned. A two-junction loop can substitute for the single Josephson junction, creating a SQUID-controlled qubit, Fig. 2.7. Thus, the effective Josephson energy EJ is tunable and pure σz rotations can be performed.
2.3.2 Advanced charge qubits
Operated in EJ/EC 1 regime, basic charge qubits have good anharmonicity to form two-level systems but their energy bands shown in Fig. 2.6 have slopes, making them very sensitive to low-frequency charge noise. The magnitudes of charge dispersion and
V g I
Figure 2.7 The single Cooper pair transistor. A superconducting loop with two Josephson junctions replaces the single junction in a SCB for a tunable EJ.
anharmonicity are both determined by the ratio EJ/EC. The low value of the ratio of EJ/EC brings not only good manipulations of qubits but also serious decoherence.
Many researchers keep trying to find solutions for this problem. A famous example is the transmon [17], Fig. 2.8. The fundamental idea of the transmon is to shunt the Josephson junction of a small Cooper-pair box with a large external capacitor to increase the charging energy EC and to increase the gate capacitor to the same size. This make the charge dispersion reduces exponentially in EJ/EC, while the anharmonicity only decreases algebraically with a slow power law in EJ/EC.
2.3.3 Flux qubits
In the previous section, we describe the quantum dynamics of low-capacitance Joseph-son devices where the charging energy dominates over the JosephJoseph-son energy, EC EJ, and the relevant quantum degree of freedom is the charge on superconducting
C
BC
rL
rV
gC
inC
gC
JE
JΦ
island
Figure 2.8 The equivalent circuit of a transmon.
island. We now talk about another quantum regime, the phase regime, EJ EC, in which the flux states are the better basis. This kind of qubits are called flux qubits.
A rf-SQUID is the simplest example of a flux qubit. The Hamiltonian, Eq. (2.17), is
H = Eˆ Cˆn2− EJcos ˆδ + EL(ˆδ − ˆδe)2
2 ,
and in the phase regime, the potential energy is given by
U (δ) = −EJcos δ + EL(δ − δe)2
2 . (2.29)
The potential energy is cosine function added a second power function. δe in a flux qubit play as the same role as ng do in a charge qubit. The lowest area can be approximated to a double-well. When δe equals π or odd π, a symmetric double-well potential energy appears. It is similar to that of ng equal 1/2 in a charge qubit.
Because of the tunneling through center barrier, the lowest two energy level split with a gap ∆, which depends on the height of the barrier. When δe doesn’t equal π
or odd π, the potential energy becomes unsymmetric, the probability of the lowest energy pair is not half in each well. This situation is like when ng is near 1/2, in a charge qubit, the probability is not the same in |0i and |1i. The Hamiltonian of a flux qubit can be truncated to the lowest two energy states in a simple form of
H = −ˆ 1
2(σz+ ∆σx) , (2.30)
where ∆ depends on EJ and is given by
= 4π s
6 EJ EL − 1
EJ
Φe Φ0 − 1
2
. (2.31)
In this form, the pure operator X-rotation can be performed by setting Φ/Φe = 1/2, but the pure Z-rotation can not. In order to solve this problem, we can replace the single junction with a two-junciton loop that introduces an additional external flux Φ˜e as another control variable. Therefore, the effective Josephson energy becomes tunable.
2.3.4 Advanced flux qubits
The main idea in a SQUID is to create a double-well potential, requiring large enough inductance. This implies that the qubit contains a large qubit loop, making itself influenced by magnetic fluctuations of environment seriously. One way to overcoming this difficulty is using a three-junction device pointed out by Mooij et al. [18]. In a three-junction-loop qubit, as shown in Fig. 2.9, ELis not the only element to creating a double-well potential. The loop, therefore, can be much small than a rf-SQUID and the qubit is relatively free from charge and magnetic environment fluctuations.
⊕ Φ
Figure 2.9 The three-junction SQUID.