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dipole moment of this artificial atom is very large, often more than four orders of magni-tude greater than the typical value for an elec-tronic transition of a real atom. Because the qubit’s size and shape are adjustable, the dipole coupling can also be engineered by having the atom essentially fill the transverse dimension of the cavity, which means that the vacuum Rabi frequency (expressed as a fraction of the photon frequency) approaches a maximum value53 of a few per cent, set by the fine-struc-ture constant (see Box 2). In comparison, the best values obtained so far using real atoms in either optical or microwave cavities are much smaller, of the order of one part in 106. The very large interactions achievable in circuit QED make it easier to attain the strong cou-pling limit of cavity QED. Another advantage of circuit QED is that it avoids the difficulties of cooling and trapping the atom, as the qubit can be fabricated at precisely the desired location inside the cavity.

Several experiments with superconduct-ing qubits in the past few years have accessed the regime of strong coupling, and have reca-pitulated many classic results from quantum optics. Strong coupling with circuit QED was first achieved in 2004 (refs 23, 24), and a device like that shown in Figure 2b has been used23 to observe vacuum Rabi splitting in a solid-state, artificial system. When transmission through the cavity was measured when the qubit was tuned into resonance, two separate peaks (the vacuum Rabi splitting) could be resolved (see Fig. 3a, overleaf), corresponding to coherent superpositions of a single photon in the trans-mission line and a single excitation of the qubit. A more recent experiment54 with an optimized qubit now approaches the fine-structure limit, with a dimensionless coupling strength of about 2.5%, yielding the large splitting shown in Figure 3b. Other experiments have observed vacuum Rabi oscillations in the time domain25

and demonstrated a maser based on a single artificial atom30.

Circuit QED has also been used for quan-tum communication and coupling between qubits. A source of non-classical microwaves has been demonstrated, for example, in which single photons are produced on demand27. This experiment also showed that the quantum information contained in a superposition state of a qubit could be mapped onto the photon state, demonstrating the conversion between a standing and a flying qubit, a milestone for quantum computation. Finally, a cavity has been used to realize a solid-state quantum bus, with a quantum state being transferred from one qubit to another using a microwave photon as the intermediary. This last achieve-ment was made simultaneously in experiments with phase qubits29 and charge qubits28. Taken together, these experiments indicate that com-munication between small prototype systems of several qubits, wired together with photons and cavities, is possible. The combination of techniques and concepts from quantum optics, in conjunction with the technology for super-conducting quantum circuits, is likely to lead to continued rapid progress.

The combination of circuit QED and experi-mental advances with superconducting circuits raises many interesting questions, and next we shall discuss some possible themes and areas for future work.

New regimes of quantum opticsAs mentioned above, the relative coupling strength in circuit QED is many orders of magnitude greater than in the better-known versions of cavity QED with real atoms. This means that less-familiar, higher-order effects can have a noticeable influence. One exam-ple is the dispersive, or off-resonant, case, in which the qubit and the photon interact with-out the photon being absorbed. In the ‘strong

dispersive regime’ in circuit QED26, this inter-action, although roughly ten times smaller than the resonant case, is still larger than all sources of decoherence, a situation that has been accessed in only a few experiments with Rydberg atoms44,45. Circuit QED couplings can approach the limit where multiphoton effects, which are usually rare, play an important role. Other new phenomena include optical bist-ability of the cavity, in which the presence of a single atom makes the cavity oscillations strongly anharmonic, and causes the entangle-ment of multi-photon states. It is also possible to engineer strong photon–photon nonlineari-ties, based for example on the simultaneous interaction of two cavities with a single qubit.

What is the real limit on the strength of cou-pling? It should be possible to push coupling strengths beyond the fine-structure limit dis-cussed above for electric fields. For instance, if the current in a transmission line is passed directly through a Josephson junction53, the relative coupling can be larger than unity (g > !, where ! is the transmission frequency of the atom/cavity), so the photon and the qubit cease to be separate entities and the coupling can-not be considered as a perturbation. All these investigations could add significantly to the body of knowledge on the light–matter inter-action already gleaned from cavity QED.

What are the limits of coherence?Perhaps the greatest outstanding problem with all solid-state implementations of quantum systems is how to minimize decoherence, the inevitable loss of quantum information owing to coupling to undesired degrees of freedom, and secure enough time to allow complex manipulations. In their roughly 10 years of existence, the coherence time of supercon-ducting qubits has increased by a factor of almost 1,000 (from just nanoseconds to a few microseconds), but further improvements will

Figure 2 | Circuit QED devices. a, Schematic representation (adapted from ref. 22) of the circuit analogue of cavity quantum electrodynamics (QED), where a superconducting qubit (green) interacts with the electric fields (pink) in a transmission line (blue), consisting of a central conductor and two ground planes on either side. The cavity is defined by two gaps (the mirrors) separated by about a wavelength. The cavity and qubit are measured by sending microwave signals down the cable on one side of the cavity and collecting the transmitted microwaves on the output side.

b, Micrograph of an actual circuit QED device that achieves the strong-coupling limit. It consists of a superconducting niobium transmission line on a sapphire substrate with two qubits (green boxes) on either side. The inset shows one of the superconducting Cooper-pair box charge qubits located at the ends of the cavity where the electric fields are maximal. The qubit has two aluminium ‘islands’ connected by a small Josephson junction. Changing the state of the qubit corresponds to moving a pair of electrons from the bottom to top (shown schematically).

667

NATURE|Vol 451|7 February 2008 HORIZONS

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dipole moment of this artificial atom is very large, often more than four orders of magni-tude greater than the typical value for an elec-tronic transition of a real atom. Because the qubit’s size and shape are adjustable, the dipole coupling can also be engineered by having the atom essentially fill the transverse dimension of the cavity, which means that the vacuum Rabi frequency (expressed as a fraction of the photon frequency) approaches a maximum value53 of a few per cent, set by the fine-struc-ture constant (see Box 2). In comparison, the best values obtained so far using real atoms in either optical or microwave cavities are much smaller, of the order of one part in 106. The very large interactions achievable in circuit QED make it easier to attain the strong cou-pling limit of cavity QED. Another advantage of circuit QED is that it avoids the difficulties of cooling and trapping the atom, as the qubit can be fabricated at precisely the desired location inside the cavity.

Several experiments with superconduct-ing qubits in the past few years have accessed the regime of strong coupling, and have reca-pitulated many classic results from quantum optics. Strong coupling with circuit QED was first achieved in 2004 (refs 23, 24), and a device like that shown in Figure 2b has been used23 to observe vacuum Rabi splitting in a solid-state, artificial system. When transmission through the cavity was measured when the qubit was tuned into resonance, two separate peaks (the vacuum Rabi splitting) could be resolved (see Fig. 3a, overleaf), corresponding to coherent superpositions of a single photon in the trans-mission line and a single excitation of the qubit. A more recent experiment54 with an optimized qubit now approaches the fine-structure limit, with a dimensionless coupling strength of about 2.5%, yielding the large splitting shown in Figure 3b. Other experiments have observed vacuum Rabi oscillations in the time domain25

and demonstrated a maser based on a single artificial atom30.

Circuit QED has also been used for quan-tum communication and coupling between qubits. A source of non-classical microwaves has been demonstrated, for example, in which single photons are produced on demand27. This experiment also showed that the quantum information contained in a superposition state of a qubit could be mapped onto the photon state, demonstrating the conversion between a standing and a flying qubit, a milestone for quantum computation. Finally, a cavity has been used to realize a solid-state quantum bus, with a quantum state being transferred from one qubit to another using a microwave photon as the intermediary. This last achieve-ment was made simultaneously in experiments with phase qubits29 and charge qubits28. Taken together, these experiments indicate that com-munication between small prototype systems of several qubits, wired together with photons and cavities, is possible. The combination of techniques and concepts from quantum optics, in conjunction with the technology for super-conducting quantum circuits, is likely to lead to continued rapid progress.

The combination of circuit QED and experi-mental advances with superconducting circuits raises many interesting questions, and next we shall discuss some possible themes and areas for future work.

New regimes of quantum opticsAs mentioned above, the relative coupling strength in circuit QED is many orders of magnitude greater than in the better-known versions of cavity QED with real atoms. This means that less-familiar, higher-order effects can have a noticeable influence. One exam-ple is the dispersive, or off-resonant, case, in which the qubit and the photon interact with-out the photon being absorbed. In the ‘strong

dispersive regime’ in circuit QED26, this inter-action, although roughly ten times smaller than the resonant case, is still larger than all sources of decoherence, a situation that has been accessed in only a few experiments with Rydberg atoms44,45. Circuit QED couplings can approach the limit where multiphoton effects, which are usually rare, play an important role. Other new phenomena include optical bist-ability of the cavity, in which the presence of a single atom makes the cavity oscillations strongly anharmonic, and causes the entangle-ment of multi-photon states. It is also possible to engineer strong photon–photon nonlineari-ties, based for example on the simultaneous interaction of two cavities with a single qubit.

What is the real limit on the strength of cou-pling? It should be possible to push coupling strengths beyond the fine-structure limit dis-cussed above for electric fields. For instance, if the current in a transmission line is passed directly through a Josephson junction53, the relative coupling can be larger than unity (g > !, where ! is the transmission frequency of the atom/cavity), so the photon and the qubit cease to be separate entities and the coupling can-not be considered as a perturbation. All these investigations could add significantly to the body of knowledge on the light–matter inter-action already gleaned from cavity QED.

What are the limits of coherence?Perhaps the greatest outstanding problem with all solid-state implementations of quantum systems is how to minimize decoherence, the inevitable loss of quantum information owing to coupling to undesired degrees of freedom, and secure enough time to allow complex manipulations. In their roughly 10 years of existence, the coherence time of supercon-ducting qubits has increased by a factor of almost 1,000 (from just nanoseconds to a few microseconds), but further improvements will

Figure 2 | Circuit QED devices. a, Schematic representation (adapted from ref. 22) of the circuit analogue of cavity quantum electrodynamics (QED), where a superconducting qubit (green) interacts with the electric fields (pink) in a transmission line (blue), consisting of a central conductor and two ground planes on either side. The cavity is defined by two gaps (the mirrors) separated by about a wavelength. The cavity and qubit are measured by sending microwave signals down the cable on one side of the cavity and collecting the transmitted microwaves on the output side.

b, Micrograph of an actual circuit QED device that achieves the strong-coupling limit. It consists of a superconducting niobium transmission line on a sapphire substrate with two qubits (green boxes) on either side. The inset shows one of the superconducting Cooper-pair box charge qubits located at the ends of the cavity where the electric fields are maximal. The qubit has two aluminium ‘islands’ connected by a small Josephson junction. Changing the state of the qubit corresponds to moving a pair of electrons from the bottom to top (shown schematically).

667

NATURE|Vol 451|7 February 2008 HORIZONS

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P. R. Johnson, F. W. Strauch et al., Physical Review B 67, 020502(R) (2003).

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for this algorithm is [54]

Tmax = (Na +Nb +NaNb)π

Ω+

Na

j=1

π

2ga√j

+(Na + 1)Nb

j=1

π

2gb√j

(28)

This assumes that all of the Rabi and shift pulses are π-pulses. Of course, specific instances of the algorithm canhave times less than Tmax, as will be seen in the followingexamples.

C. Specific Examples

The simplest example is the construction of a NOONstate

|ΨNOON =1√2|0 ⊗ (|N, 0+ |0, N) . (29)

This state is particularly nice, as the solution for theinverse evolution admits the following simplification

U†b,j = R†

b,j0B†j0. (30)

The evolution times are chosen to move population from|0, 0, j → |1, 0, j−1 → |0, 0, j−1. This is done for j =N → 1, after which U †

a moves population from |0, N, 0to |0, 0, 0. Each step is a two-state rotation, and onlytrivial phases appear. We can schematically write this asU = (AR)N (BR)N , where all of the rotations have θ = πexcept for the first R, which is a π/2-pulse. The numberof steps is 4N , with a time [54]

TNOON =

2N − 1

2

π

Ω+

N

j=1

π

2ga√j

+N

j=1

π

2gb√j

(31)

An explicit list of parameters for the twelve steps neededfor N = 3 is given in Table I

A more complicated example is given by the maximallyentangled state

|Ψ = 1√N + 1

N

n=0

|n ⊗ |N − n. (32)

While this only occupies a sparse region of the Fock-statediagram, the algorithm does not have the same simplifi-cations or explicit solution as the NOON state. Figure 4shows a numerical solution for the eighteen steps of thecontrol sequence for U with N = 3.

TABLE I: Procedure for Ψ = |0, 3, 0+ |0, 0, 3

.

Step Parameters Quantum State

Ra,1 Ωtqa,1 = π/2,ωd = ω0 |0, 0, 0 − i|1, 0, 0A1 gata,1 = π/2 |0, 0, 0 − |0, 1, 0Ra,2 Ωtqa,2 = π,ωd = ω1 |0, 0, 0+ i|1, 1, 0A2 gata,2 = π/(2

√2) |0, 0, 0+ |0, 2, 0

Ra,3 Ωtqa,3 = π,ωd = ω2 |0, 0, 0 − i|1, 2, 0A3 gata,3 = π/(2

√3) |0, 0, 0 − |0, 3, 0

Rb,1 Ωtqb,1 = π,ωd = ω0 −i|1, 0, 0 − |0, 3, 0B1 gbtb,1 = π/2 −|0, 0, 1 − |0, 3, 0Rb,2 Ωtqb,2 = π,ωd = ω−1 i|1, 0, 1 − |0, 3, 0B2 gbtb,2 = π/(2

√2) |0, 0, 2 − |0, 3, 0

Rb,3 Ωtqb,3 = π,ωd = ω−2 −i|1, 0, 2 − |0, 3, 0B3 gbtb,3 = π/(2

√3) −|0, 0, 3 − |0, 3, 0

See [54].

D. Superconducting Implementation

The algorithm described above can be implementedby a tunable superconducting qubit, such as the phase[41] or transmon [42] qubit, coupled to two on-chip res-onators, as shown in Fig. 5. However, there are a numberof details that will depend on the specific experimentalimplementation. First, the sequence of operations mustbe “programmed” with a time-dependent control pulsefor the frequency ωq(t) and the microwave drive Ω(t).Such pulses are shown in Fig. 6 for the NOON statesequence with N = 3.First, when analyzing the fidelity of this control se-

quence, it is important to note that, for fixed couplingsga, gb, the natural basis to describe the various transi-tions is in fact the dressed basis, i.e. those states thatare true eigenstates of the Hamiltonian for Ω = 0 and ωq

equal to some fixed value. While these states are close tothe product states |q, na, nb, this is only an approxima-tion. It is convenient to imagine that the couplings ga, gbcan be turned on and off at the beginning and end of thealgorithm. This could be achieved by using a tunablecoupler [43, 44] or by moving the qubit frequency to theregime ωq ωa,ωb.Second, each of the number-state-dependent Rabi

transitions will in fact require optimization in both am-plitude and frequency for each desired Fock state (na, nb).The amplitude optimization is necessary because of thevariation of the matrix elements (of σx) between thedressed states, while the frequency optimization is nec-essary because the expression for ωd in Eq. (11) is onlya leading-order perturbative result for a two-level sys-tem. The results of such a partial optimization for thecontrol pulse of Fig. 6 is shown in Fig. 7. Here wehave shown the expectation values na, nb, and q(in the uncoupled basis), averaged over a window of 5nsto remove high-frequency oscillations (due to fixed cou-pling and the dressed basis). Each Rabi pulse increases

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