Quantum information processing with superconducting qubits in a microwave field

We investigate the quantum dynamics of a Cooper-pair box with a superconducting loop in the presence of a nonclassical microwave field. We demonstrate the existence of Rabi oscillations for both single- and multi-photon processes and, moreover, we propose a new quantum computing scheme (including one-bit and conditional two-bit gates) based on Josephson qubits coupled through microwaves.


A. Background
Quantum computing deals with the processing of information according to the laws of quantum mechanics. Within the last few years, it has attracted considerable attention because quantum computers are expected to be capable of performing certain tasks which no classical computers can do in practical time scales. Early proposals for quantum computers were mainly based on quantum optical systems, such as those utilizing lasercooled trapped ions, 1,2 photon or atoms in quantum elctrodynamical (QED) cavities, 3,4 and nuclear magnetic resonance. 5 These systems are well isolated from their environment and satisfy the low-decoherence criterion for implementing quantum computing. Moreover, due to quantum error correction algorithms, 5 now decoherence 6 is not regarded as an insurmountable barrier to quantum computing. Because scalability of quantum computer architectures to many qubits is of central importance for realizing quantum computers of practical use, considerable efforts have recently been devoted to solid state qubits. Proposed solid state architectures include those using electron spins in quantum dots, 7,8,9 electrons on Helium, 10 and Josephson-junction (JJ) charge (see, e.g., Refs. 11,12,13 and 15) and JJ flux (see, e.g., Refs. 14 and 15) devices. These qubit systems have the advantage of relatively long coherent times and are expected to be scalable to large-scale networks using modern microfabrication techniques.
The Josephson charge qubit is achieved in a Cooperpair box, 11 which is a small superconducting island weakly coupled to a bulk superconductor, while the Josephson flux qubit is based on two different flux states in a small superconducting-quantum-interferencedevice (SQUID) loop. 14,15 Cooper-pair tunneling and energy-level splitting associated with the superpositions of charge states were experimentally demonstrated in a Cooper-pair box, 16,17 and recently the eigenenergies and the related properties of the superpositions of different flux states were observed in SQUID loops by spectroscopic measurements. 18 In particular, Nakamura et al. 19 demonstrated the quantum coherent oscillations of a Josephson charge qubit prepared in a superposition of two charge states. In addition, Vion et al. 20 extended coherent oscillations to the charge-flux regime and Chiorescu et al. 21 studied the quantum dynamics of the flux qubit. Moreover, two charge qubits were capacitively coupled by Pashkin et al. 22 and coherent oscillations were also observed in this coupled-qubit system. Furthermore, other superconducting devices (e.g., Refs. 23 and 24) have also exhibited coherent oscillations. In addition, several other types of studies (see, for instance, Refs. 25 and 26) have been made on superconducting qubits.

B. This work
In this paper, we show that the coupled system of a Cooper-pair box and a cavity photon mode undergoes Rabi oscillations and propose a new quantum computing scheme based on Josephson charge qubits. 27 The microwave-controlled approach proposed in our paper has the significant advantage that any two qubits (not necessarily neighbors) can be effectively coupled through photons in the cavity. In addition to the advantages of a superconducting device exhibiting quantum coherent effects in a macroscopic scale as well as the controllable feature of the Josephson charge qubit by both gate voltage and external flux, the motivation for this scheme is fourfold: (i) the experimental measurements 16 showed that the energy difference between the two eigenstates in a Cooper-pair box lies in the microwave region and the eigenstates can be effectively interacted by the microwave field; (ii) a single photon can be readily prepared in a high-Q QED cavity using the Rabi precession in the microwave domain. 28 Moreover, using a QED cavity, Ref. 29 produced a reliable source of photon number states on demand. In addition, the cavity in Ref. 29 was tuned to ∼ 21 GHz, which is close to the 20 GHz microwave frequency used in a very recent experiment 30 on the Josephson charge qubit. Furthermore, the Q value of the cavity is 4 × 10 10 (giving a very large photon lifetime of 0.3 sec); (iii) our quantum computer proposal should be scalable to 10 6 to 10 8 charge qubits in a microwave cavity, since the dimension of a Cooper-pair box is ∼ 10µm to 1µm; (iv) the QED cavity has the advantage that any two qubits (not necessarily neighbors) can be effectively coupled through photons in the cavity.
Also, we study multi-photon processes in the Josephson charge qubit since, in contrast to the usual Jaynes-Cummings model (see, e.g., Chap. 10 in Ref. 31), the Hamiltonian includes higher-order interactions between the two-level system and the nonclassical microwave field. As shown by the very recent experiment on Rabi oscillations in a Cooper-pair box, 30 these higher-order interactions may be important in the Josephson charge-qubit system.
Note that the driving microwave field is typically generated using an electrical voltage acting on the charge qubit via a gate capacitor. Here, the microwave field is applied as a magnetic flux piercing the SQUID loop of the qubit in order to perform the unitary transformations needed for quantum computing.
The dynamics of a Josephson charge qubit coupled to a quantum resonator was studied in Ref. 32. In contrast to our study here, the model in Ref. 32 involves: (a) only one qubit, (b) only the Rabi oscillation with a single excitation quantum of the resonator (as opposed to one or more photons), and (c) no quantum computing scheme.

A. Cooper-pair box with a SQUID loop
We study the Cooper-pair box with a SQUID loop. 11,15,19 . In this structure, the superconducting island with Cooper-pair charge Q = 2ne is coupled to a segment of a superconducting ring via two Josephson junctions (each with capacitance C J and Josephson coupling energy E J0 ). Also, a voltage V g is coupled to the superconducting island through a gate capacitor C g ; the gate voltage V g is externally controlled and used to induce offset charges on the island. A schematic illustration of this single-qubit structure is given in Fig. 1

(a). The Hamiltonian of the system is
where is the single-particle charging energy of the island and are canonically conjugate variables. The gauge-invariant phase drops ϕ 1 and ϕ 2 across the junctions are related to the total flux Φ through the SQUID loop by the constraint where Φ 0 = h/2e is the flux quantum. This structure is characterized by two energy scales, i.e., the charging energy E c and the coupling energy E J0 of the Josephson junction. In the charging regime E c ≫ E J0 and at low temperatures k B T ≪ E c , the charge states |n and |n+1 become dominant as the controllable gate voltage is adjusted to V g ∼ (2n + 1)e/C g . Here, the superconducting gap is assumed to be larger than E c , so that quasiparticle tunneling is greatly reduced in the system.
Here we ignore self-inductance effects on the singlequbit structure. 33 Now Φ reduces to the classical variable Φ e , where Φ e is the flux generated by the applied static magnetic field. In the spin-1 2 representation with charge states | ↑ = |n and | ↓ = |n + 1 , the reduced two-state Hamiltonian is given by 11,15 where This single-qubit amiltonian has two eigenvalues with and eigenstates Using these eigenstates as new basis, the Hamiltonian takes the diagonal form where ρ z = |e e| − |g g|.
Here we employ {|e , |g } to represent the qubit.

B. Interaction of the charge qubit with a microwave field
When a nonclassical microwave field is applied, the total flux Φ is a quantum variable where Φ f is the microwave-field-induced flux through the SQUID loop [see Fig. 1(b)]. Here we assume that a singlequbit structure is embedded in a QED microwave cavity with only a single photon mode λ. Generally, the vector potential of the nonclassical microwave field is written as where a † (a) is the creation (annihilation) operator of the cavity mode. Thus, the flux Φ f is given by where the contour integration is over the SQUID loop.
Here, θ is the phase of the mode function u λ (r) and its value depends on the chosen microwave field (see, e.g., Chap. 2 in Ref. 31). For instance, if a planar cavity is used and the SQUID loop of the charge qubit is perpendicular to the cavity mirrors, one has θ = 0. We shift the gate voltage V g (and/or vary Φ e ) to bring the single-qubit system into resonance with k photons: Expanding the functions cos(πΦ f /Φ 0 ) and sin(πΦ f /Φ 0 ) into series of operators and employing the standard rotating wave approximation, we derive the total Hamiltonian of the system in this situation (with the photon Hamiltonian included) with where φ = π|Φ λ |/Φ 0 , and with m = 1, 2, 3, . . ., and where g (k) (a † a) is the k-photon-mediated coupling between the charge qubit and the microwave field. This Hamiltonian (18) is a generalization of the Jaynes-Cummings model to a solid state system. Here multiphoton processes 34 are involved for k > 1, in contrast with the usual Jaynes-Cummings model for an atomic two-level system interacting with a single photon mode, where only one photon is exchanged between the twolevel system and the external field. 31

III. RABI OSCILLATIONS IN MULTI-PHOTON PROCESS
The eigenvalues of the total Hamiltonian (18) are and the corresponding eigenstates, namely, the dressed states are given by |+, l = e −ikθ cos η |e, l + sin η |g, l + k , |−, l = − sin η |e, l + e ikθ cos η |g, l + k , where Ω l,k = 2g (k) (l + k) [(l + 1)(l + 2) · · · (l + k)] 1/2 /h (25) is the Rabi frequency, and Here, k is the number of photons emitted or absorbed by the charge qubit when the qubit transits between the excited state |e and the ground state |g , and l is the number of photons in the cavity when the qubit state is |e .
When the system is initially at the state |e, l , after a period of time t, the probabilities for the system to be at states |g, l + k and |e, l are | g, l + k|ψ(t) | 2 = Ω 2 l,k δ 2 l,k + Ω 2 l,k and | e, l|ψ(t) | 2 = 1 − | g, l + k|ψ(t) | 2 .
Thus, the probabilities are oscillating with frequency This is the Rabi oscillation with k photons involved in the state transition; when k = 1, it reduces to the usual single-photon Rabi oscillation. Very recently, Nakamura et al. 30 investigated the temporal behavior of a Cooper-pair box driven by a strong microwave field and observed the Rabi oscillations with multi-photon exchanges between the two-level system and the microwave field. Different to the case studied here, the microwave field was employed there to drive the gate voltage to oscillate. Here, in order to implement quantum computing, we consider the Cooper-pair box with a SQUID loop and use the microwave field to change the flux through the loop.

A. Analogies between Rabi oscillations and the AC Josephson effect
Rabi oscillations have been observed a long time ago in atomic physics. It is a relatively new development to observe Rabi oscillations in a condensed matter system. Since the Josephson effect can be used for this purpose, it is instructive to point out analogies and differences between Rabi oscillations and the Josephson effect.
(i) Both Rabi oscillations and the AC Josephson effect involve interactions of the photons with electrons (for Rabi oscillations) or a junction (for AC Josephson effect); (ii) the radiation must be tuned creating two-level transitions; (iii) the junction behaves like an atom undergoing transitions between the quantum states of each side of the junction as it absorbs and emits radiation.
However, the Rabi oscillation is a strong-coupling effect 31 and produces long-lived coherent superpositions.

IV. QUANTUM COMPUTING
Let us consider more than one single charge qubit in the QED cavity, and the cavity initially prepared at the zero-photon state |0 . We first show the implementation of a controlled-phase-shift operation. Here a single photon process, k = 1, is used to implement quantum computing.
Furthermore, the gate voltage for a control qubit, say A, is adjusted to have the qubit on resonance with the cavity mode (E =hω λ ) for a period of time (where single photon is involved in the state transition), while all other qubits are kept off-resonant. The interaction Hamiltonian (in the interaction picture with H 0 = 1 2 Eρ z ) is given by and the evolution of qubit A is described by This unitary operation does not affect state |g A |0 , but transforms |g A |1 and |e A |0 as where α = g (1) (1)/h. To obtain the controlled-phaseshift gate, we need the unitary operation with θ = 0 and interaction time t 1 = π/2α, which gives This operation swaps the qubit state and the state of the QED cavity. A similar swapping transformation was previously used for the quantum computing with lasercooled trapped ions. 1 (ii) While all qubits are kept off-resonant with the cavity mode and the flux Φ e is originally set to Φ e = 1 2 Φ 0 for each qubit, we change Φ e to zero for only the target qubit, say B. In this case, the evolution of the target qubit B is described in the interaction picture by where the Hamiltonian is This Hamiltonian can be used to produce conditional phase shifts in terms of the photon state of the QED cavity. 3 Applying this unitary operation to qubit B for a period of time where β = πf (0)/2|f (1) − f (0)|.
(iii) Qubit A is again brought into resonance for t 3 = π/2α with θ = 0, as in step (i). Afterwards, a controlled two-bit gate is derived as a controlled-phase-shift gate combined with two one-bit phase gates. In order to obtain the controlled-phase-shift gate U AB , which transforms |g A |g B , |g A |e B , |e A |g B , and |e A |e B as one needs to further apply successively the unitary operation given in step (ii) to the control and target qubits with interaction times t 4 = 3πh/4|f (0)| and t 5 = (2π − |β|)h/|f (0)|, respectively.
In analogy with atomic two-level systems, 1,3 one can use an appropriate classical microwave field 36 to produce one-bit rotations for the Josephson charge qubits. When the classical microwave field is on resonance with the target qubit B, the interaction Hamiltonian becomes withh where the value of the phase ν depends on the chosen microwave field (see, e.g., Chap. 2 in Ref. 31) and Φ f is the flux through the SQUID loop produced by the classical microwave field. For the interaction time t 6 = π/2Ω, the unitary operation transforms |g B and |e B as In terms of this one-bit rotation, the controlled-phaseshift gate U AB can be converted to the controlled-NOT gate, 1 A sequence of such gates supplemented by one-bit rotations can serve as a universal element for quantum computing. 37

V. DISCUSSION AND CONCLUSION
For microwaves of wavelength λ ∼ 1 cm, the volume of a planar cavity is ∼ 1cm 3 . For SQUID loop dimension ∼ 10µm to 1µm, then 10 3 to 10 4 charge qubits may be constructed along the cavity direction. Furthermore, for a 2D array of qubits, 10 6 to 10 8 charge qubits could be placed within the cavity. 38 This number of qubits is large enough for a quantum computer. For practical quantum information processing, one needs to improve the experimental setup to have a QED cavity with a high enough Q value so as to implement more quantum operations within the long photon lifetime of the cavity. Alternatively, one can also increase the number of permitted quantum operations within the given photon lifetime of the cavity by strengthening the coupling between the charge qubit and the microwave field. Because the typical interaction energy between the charge qubit and the microwave field is propositional to Φ λ , the qubit-photon coupling can be strenghthened by increasing the area enclosed by the SQUID loop and the field intensity (e.g. by putting a high-µ material inside the SQUID loop).
In the conditional gates discussed above, the two charge qubits are coupled through photons in the QED cavity. Our approach is scalable, but similar to the coupling scheme using an LC-oscillator mode, 11,15 only a pair of charge qubits at a time can be coupled. In order to implement parallel operations on different pairs of qubits, one can make use of a multi-mode QED cavity or more than one cavity, where different cavity modes couple different pairs of qubits simultaneously. Moreover, our approach might have potential applications in quantum communications using both the qubit-photon coupling (to convert quantum information between charge qubits and photons) and the photons, acting as flying qubits, to transfer quantum information between remotely separated charge-qubit systems.
In conclusion, we have studied the dynamics of the Cooper-pair box with a SQUID loop in the presence of a nonclassical microwave field. Rabi oscillations in the multi-photon process are demonstrated, which involve multiple photons in the transition between the twolevel system and the microwave field. Also, we propose a scheme for quantum computing, which is realized by Josephson charge qubits coupled through photons in the QED cavity.