Raíces del postgraffit

A significant modification of the traditional charge qubit was proposed in Ref. [63] and measured in Ref. [108]. This turned out to be a significant change in course for cQED systems. This transmon qubit boasts several advantages over the original charge qubit. The first and most important difference is an exponentially reduced sensitivity to charge noise with only an algebraic reduction in anharmonicity. This means the qubit will lose its sensitivity to charge fluctuations much faster than its anharmonicity. Another significant difference compared with the charge qubit is the lack of external dc biasing circuitry. Additionally, the typically large capacitors used to fabricate transmons, which push the charging energies EC lower, allow for large capacitive

coupling to transmission lines and therefore easy integration into a cQED architecture. The transmon has become a popular choice for cQED style experiments due to its ease of fabrication and measurement compared to other qubit types. More importantly, the transmon also has excellent coherence properties due to its insensitivity to charge noise.

V

Figure 2-9: Transmon diagram. Sketch of a transmon inside a CPW resonator. The blue strips represent ground relative to the center strip of the resonator. The teal and green blocks form the capacitor making up the transmon with x’s marking the locations of the Josephson junctions. Adapted from [63].

The Hamiltonian of the Cooper pair box is given by [63] ˆ

H = 4EC(ˆn − ng)2− EJcos ˆφ, (2.30)

where ng is the offset charge produced by an applied gate voltage or environmentally

induced charge offset. It is significant to note that the Hamiltonian for the transmon is structurally similar to that of the Cooper-pair box charge qubit. The only difference is the ratio of the energy scales EJ/EC. The ’transmon regime’ is defined by EJ/EC ≫

1. This means the first term in equation (2.30) becomes small compared to the term involving the Josephson coupling energy EJ and the role of the offset charge ng is

reduced (see Figure 2-10). Recall that these energy scales are set by the total qubit capacitance CΣ and the sum of junction critical currents Ic. In addition, the EJ/EC

ratio can be set and fixed at the time of fabrication by using a single Josephson junction or can be tuned in situ by using two junctions to form a SQUID loop. Traditionally a split junction device is used to allow some tunability and certain types of logic gates, one of which we will discuss further in Chapter 3. In this case, the Josephson energy EJ can be tuned in the same manner as in the charge qubit

The eigenenergies of Eq. (2.30) take the form

Em(ng) = ECa2[ng+k(m,ng)](−EJ/EC), (2.31) [63] where m is the relative energy level of the qubit; ng is again the gate offset

charge; aν(q) are the Mathieu characteristic values and k(m, ng) is a sorting function

to arrange the eigenvalues properly for ng > 1. Given these eigenenergies and a

definition of the charge dispersion,

Em(ng) ∼ Em(ng = 1/4) −

ϵm

2 cos(2πng), ϵm ≡ Em(ng = 1/2) − Em(ng = 0),

(2.32)

[63] defined as the energy difference between a gate voltage at integer and half-integer values of ng (see Figure 2-10), the asymptotics of Eq. (2.32) can be found using

approximate WKB methods [32]: ϵm ∝ " EJ 2EC #m2+ 3 4 e−√8EJ/EC (2.33)

From this it is clear from Eq. (2.32) and Eq. (2.33) that the charge dispersion, de- fined as the variation in qubit energy splitting with gate charge nq (Equation (2.32)

and Equation (2.33)), in the transmon shrinks exponentially with EJ/EC ratio for

all energy states m. At the same time, the relative anharmonicity defined by the energy difference between successive energy level spacings, shrinks by a weak power law as αr ≃ −(8EJ/EC)−1/2 [63]. This scaling can be derived from the first order

perturbation theory which is outlined below in Eq. (2.37) [63].

For a split-junction transmon with two junctions, the junction term becomes a sum of two terms HJ = −EJ1cosφ1− EJ2cos φ2. The Josephson energy can now be

Figure 2-10: Transmon charge dispersion. The reduction in charge dispersion for four different values of EJ/EC plotted vs. gate charge. √8EJEC denotes the

tuned with external magnetic flux Φ EJ(Φ) = EJΣcos " πΦ Φ0 # , (2.34) [63] and EΣ

J is the sum of the two Josephson energies. In Eq. (2.34) it is assumed

that EJ1 ≈ EJ2. This condition defines the symmetric transmon case. For arbitrary

values of the two Josephson energies, equation (2.34) is modified

EJ → EJΣcos " πΦ Φ0 #* 1 + d2_tan2" πΦ Φ0 # , (2.35) [63_{] where d ≃} EJ 1−EJ 2

EJ 1+EJ 2. We refer to a transmon with intentional asymmetry as an asymmetric transmon, and the properties of such a device will be important inChapter 3 and Chapter 5.

The transmon can also be understood in terms of perturbation theory, viewing the anharmonicity as a perturbation of an otherwise harmonic oscillator [63]. In this case, the cosine term in (2.30) can be expanded to third order. Following the derivation in Koch et al. [63] and grouping harmonic and anharmonic terms, the Hamiltonian becomes

H =!8ECEJ(ˆb†ˆb + 1/2) − EJ −

EC

12(ˆb + ˆb

†₎4 _(2.36)

where ˆb and ˆb† _{are the standard creation and annihilation operators for a harmonic}

oscillator. The energy eigenstates of Eq. (4.6) are

Em ≃ −EJ+!8ECEJ " m + 1 2 # − E₁₂C(6m2+ 6m + 3), (2.37) [63] for energy level m. Some very useful approximations for transmon qubits are for the transmon anharmonicity α ≈ −EC. This is the energy difference between

the transition for ground to excited state vs. the transition from excited to next- excited state. The negative sign indicates the spacing between the energy levels of the transmon get smaller with increasing energy. This will be important inChapter 4

to model the cross-resonance effect including higher energy levels. Also, the plasma oscillation frequency, ωp = √8EJEC/¯h is a good approximation for the ground to

excited state transition in a transmon qubit.