Abstract
The generation and control of quantum states of light constitute fundamental tasks in cavity quantum electrodynamics^{1,2,3,4,5,6,7,8,9,10} (QED). The superconducting realization of cavity QED, circuit QED (refs 11, 12, 13, 14), enables onchip microwave photonics, where superconducting qubits^{15,16,17,18} control and measure individual photon states^{19,20,21,22,23,24,25,26}. A longstanding issue in cavity QED is the coherent transfer of photons between two or more resonators. Here, we use circuit QED to implement a threeresonator architecture on a single chip, where the resonators are interconnected by two superconducting phase qubits. We use this circuit to shuffle one and twophoton Fock states between the three resonators, and demonstrate qubitmediated vacuum Rabi swaps between two resonators. By shuffling superposition states we are also able to demonstrate the highfidelity phase coherence of the transfer. Our results illustrate the potential for using multiresonator circuits as photon quantum registers and for creating multipartite entanglement between delocalized bosonic modes^{27}.
Main
The combination of highfinesse electromagnetic cavities with atoms or qubits enables fundamental studies of the interaction between light and matter. The cavity provides a protected environment for storing and tailoring individual photonic excitations^{1,2,3,14}. Both stationary^{6,21,23} and propagating^{7,9} nonclassical fields can be synthesized using such systems, enabling quantum memory and quantum messaging^{8}. A critical challenge however is the extension from single to more versatile multicavity architectures^{27,28}, allowing manipulation of spatially separated bosonic modes. Although the entanglement of different modes of a single cavity^{4} and of freespace modes^{10} has been shown in atomic systems, and a coupled low and highqualityfactor resonator studied in circuit QED (refs 25, 26), coherent dynamics between two or more highqualityfactor cavities has yet to be demonstrated. Here we describe a tripleresonator system, where three highqualityfactor microwave resonators are coupled to two superconducting phase qubits (see Fig. 1). The qubits serve as quantum transducers^{29} that create and transfer photonic states between the resonators. Both Fock states and linear superpositions of Fock states are transferred, thus demonstrating a fully phase coherent process. The quantum transduction is carried out by means of purely resonant qubit–resonator interactions, rather than dispersive coupling^{27}, enabling rapid transfers between resonators with significantly different frequencies. As an important example, we demonstrate singlephoton Rabi swaps between two resonators detuned by ≃12,000 resonator linewidths.
Figure 1a,b shows the main experimental elements, which comprise three coplanar waveguide resonators, R_{a},R_{b} and R_{c}, two phase qubits, Q_{1} and Q_{2}, and two superconducting quantum interference devices used for qubit state readout. Each qubit is coupled to a control line that is used to adjust the qubit operating frequency f_{1,2} and couple microwave pulses for controlling and measuring the qubit state. During operation, the device is attached to the mixing chamber of a dilution refrigerator at ≃25 mK.
The circuit layout (see Fig. 1b) can be decomposed into two unit cells, R_{a}–Q_{1}–R_{b} (green area) and R_{b}–Q_{2}–R_{c} (blue area). The shared resonator R_{b} connects the two cells and protects the two qubits from unwanted crosstalk. The resonator frequencies f_{a},f_{b} and f_{c} are measured with qubit spectroscopy (not shown). The vacuum Rabi couplings between each qubit and its corresponding resonators, g_{1a} and g_{1b} for Q_{1} and g_{2b} and g_{2c} for Q_{2}, are determined by their respective coupling capacitors (see Fig. 1b). The coupling strengths are measured using twodimensional swap spectroscopy (see Fig. 1c–e). We note that the swap spectroscopy provides an excellent tool for revealing the presence of spurious twolevel systems as well as frequencies with short qubit relaxation times. In all of the experiments, the qubits are initialized in the ground state g〉 and are typically tuned to the idle point, where the qubit Q_{1} (Q_{2}) g〉↔e〉 transition frequency f_{1} (f_{2}) is set inbetween, and well away from, the resonator transition frequencies f_{a} and f_{b} (f_{b} and f_{c}).
When Q_{1} (Q_{2}) is at the idle point, the qubit–resonator detuning is sufficiently large that the qubit–resonator interactions are effectively switched off. A particular qubit–resonator Q_{p}–R_{q} (p=1,2 and q=a,b,c) interaction is switched on by shifting the qubit transition frequency f_{p} to equal the resonator frequency f_{q}, thus setting the detuning to zero and enabling quantum energy transfers. The timedependent control of the qubit transition frequency consequently enables highly complex quantum control of the resonators^{23}.
Figure 2a shows a diagram of the pulse sequence used to implement the singlephoton equivalent of the ‘shell game’, in which a pea is hidden under one of three shells and the contestant must guess where the pea is after the shells have been shuffled. The three resonators play the role of the shells and a singlephoton Fock state 1〉 that of the pea. The system is initialized in the ground state, with the qubits at their idle points, so that all interactions are effectively switched off. Qubit Q_{1} is used to pump a single photon into resonator R_{a} (see Fig. 2a(I)–(III)). The photon state can then be transferred to either of the other two resonators, using the qubits in a similar fashion to mediate the singleexcitation transfer. A transfer from R_{a} to R_{b} is shown in steps (IV)–(VI), and a second transfer from R_{b} to R_{c} carried out using Q_{2} in steps (VII)–(IX). The final location of the photon can be determined by employing the qubits as photon detectors, through the vacuum Rabi oscillations that occur when a qubit is brought in resonance with a resonator storing a photon (steps (X)–(XI)).
In the data shown in Fig. 2b(i)–(v), which represents different versions of the game, a photon was stored in one of the three resonators, shuffled between the resonators, and all three resonators then measured. In the shell game of Fig. 2b(i), no photon was placed in any resonator, whereas for example in game (iv), a single photon was transferred from R_{a} to R_{c} through R_{b}, and then detected using qubit Q_{2}; measurements of the other resonators R_{a} and R_{b} show no oscillations, that is, no photonic excitation.
We also explored a variant of the shell game, transferring a twophoton Fock state 2〉 from R_{a} to R_{b} to R_{c}. The twophoton Fock state (see ref. 21) is first generated in R_{a}, as shown by the measurements in Fig. 3a(i),(ii). Figure 3a also shows the measurements after this state is transferred from R_{a} to R_{b} and then to R_{c}. Each transfer takes two steps, starting with for example the state Q_{1}R_{a}R_{b}〉=g20〉. The qubit is brought into resonance with R_{a}, and one photon is Rabiswapped to the qubit, at a rate faster than the usual onephoton rate^{21}, leaving the system in the state e10〉. The qubit is then tuned into resonance with resonator R_{b} for a onephoton Rabi swap, resulting in the state g11〉 (see Fig. 3a(iii)). The qubit is subsequently placed back in resonance with R_{a} for a onephoton swap, yielding e01〉, and brought into resonance with R_{b} to transfer the second photon, ending with the state g02〉 (see Fig. 3a(iv)). To finally transfer the photons to resonator R_{c}, the process is repeated using qubit Q_{2}, which completes the full transfer of Fock state 2〉 (see Fig. 3a(v)). This process resembles the wellknown game ‘The towers of Hanoi’, where a set of discs with different diameters has to be moved between three posts (the three resonators) while maintaining the larger discs (Fock state 1〉, with the longer swapping time) always at the bottom of each post, and the smaller discs (Fock state 2〉, with shorter swapping time) on top.
Figure 3b,c shows the density matrices and corresponding Wigner functions^{24} for the coherent transfer of the phasesensitive state ψ_{X}〉=0〉+1〉 (top panels) and ψ_{Y}〉=0〉+e^{iπ/2}1〉 (bottom panels) between R_{a} and R_{b}. From left to right, each state is first prepared in R_{a}, shuffled to R_{b}, thus leaving R_{a} in the vacuum state, and finally shuffled back to R_{a}. For a phase coherent transfer, the offdiagonal terms of the first and third density matrices on the top row (red arrows) should be orthogonal to those in the bottom row. We find a transfer orthogonality ≃0.96 and a total transfer fidelity ≃0.92 and ≃0.88 for ψ_{X}〉 and ψ_{Y}〉, respectively (see Supplementary Information).
Another fundamental question for resonatorbased quantum computing is whether quantum states can be stored in a resonator and later extracted and stored elsewhere. We demonstrate this functionality in Fig. 4, where a single photon is stored in each of the three resonators for a variable time τ_{st}/3 before being transferred to the next resonator. Qubit measurements of the resonator containing the photon (for example resonator R_{c} in Fig. 4a) exhibit clear oscillations for total storage times τ_{st} exceeding 3 μs. Figure 4b is the same experiment repeated with a twophoton Fock state; clear oscillations are visible for total storage times τ_{st} exceeding 1.5 μs. These experiments demonstrate the realization of a programmable quantum information register.
We further show tworesonator vacuum Rabi swaps that can lead to quantum state entanglement between the two resonators R_{a} and R_{b}. The protocol is diagrammed in Fig. 5a. A onephoton Fock state is first stored in R_{a}, placing the system in the state Q_{1}R_{a}R_{b}〉=g10〉 (see Fig. 5a(I)–(III)). The qubit is then used to carry out a partial transfer of the photon to R_{b}, by placing Q_{1} in resonance with R_{a} and varying the transfer time τ (see Fig. 5a(IV)). This leaves the system in the entangled state αg10〉+βe00〉, with amplitudes^{30} α=cos(πg_{1a}τ) and β=−i sin(πg_{1a}τ). The qubit frequency f_{1} is then tuned from f_{a} to f_{b}, and left there for a time equal to the Q_{1}–R_{b} swap time, thus mapping the qubit state onto the resonator and resulting in the tworesonator entangled state
We then use both qubits to simultaneously measure the two resonators, Q_{1} measuring R_{a} for a measurement time Δτ_{1} and Q_{2} measuring R_{b} for a time Δτ_{2}. Figure 5b(c) shows the resulting oscillations in P_{1e} (P_{2e}) for Q_{1} (Q_{2}) (colour bar scale), as a function of the measurement time Δτ_{1,2} (horizontal axis) and Q_{1}–R_{a} transfer time τ (vertical axis). The data exhibit a single Rabi oscillation along the horizontal axis, which would repeat if the measurement time were increased, and also shows clear swaps as a function of the transfer time τ, as expected from the functional dependence of α and β. If the measurement times Δτ_{1,2} are chosen to equal a full qubit–resonator swap time (dashed white lines in Fig. 5b,c), the system will be in the state Q_{1}R_{a}R_{b}Q_{2}〉=αe00g〉+βg00e〉. An ideal measurement of Q_{1} and Q_{2} would then yield probabilities P_{1e}=α^{2}=cos^{2}(πg_{1a}τ) and P_{2e}=β^{2}=sin^{2}(πg_{1a}τ) for Q_{1} and Q_{2}, respectively, as a function of the transfer time τ. In Fig. 5d we show this functional dependence, with a clear 180° phase difference between the two probabilities and the summed probability P_{1e}+P_{2e} close to unity, as expected. The probabilities decrease with τ, owing to the finite energy relaxation time of Q_{1} and R_{a}. The decay time of R_{b} does not contribute noticeably (see Supplementary Information). Fitting yields an effective tworesonator decay time approximately equal to the harmonic mean of the energy relaxation times T_{1}^{rel}≃340 ns and T_{a}^{rel}≃3.9 μs of Q_{1} and R_{a}, respectively, T_{ab}^{rel}≈(1/2T_{1}^{rel}+1/2T_{a}^{rel})^{−1}≃626 ns. We note that the phase qubit enables a photon transfer between R_{a} and R_{b} even though the two resonators are separated in frequency by ≃12,000 resonator linewidths.
This last experiment demonstrates a true quantum version of the ‘shell game’, where the ‘pea’ (the photon Fock state) is simultaneously hidden under two shells, and the contestant’s selection of a shell constitutes a truly probabilistic measurement. More generally, we have experimentally demonstrated an architecture with three resonators and two qubits that exhibits excellent quantum control over single, double and superpositions of microwave photon Fock states. From a fundamental perspective, these results demonstrate the potential of multiresonator circuit QED (refs 27, 28), both for scientific study and for quantum information.
Methods
Sample fabrication. The resonators are made from a 150nmthick rhenium film grown in a molecularbeamepitaxy system on a polished sapphire substrate. All the other wiring layers are made from sputtered aluminium with Al/AlO_{x}/Al Josephson tunnel junctions. All of the microstructures on the different layers are patterned by means of optical lithography and etched by means of inductively coupled plasma etching. Amorphous silicon is used as a dielectric insulator for the qubit shunting capacitors and crossovers. Our sample fabrication clearly shows the flexibility offered by multilayer processing.
The complete device is wire bonded by aluminium wirebonds to an aluminium sample holder, which is bolted to the mixing chamber of a dilution refrigerator operating at ≃25 mK. A detailed description of the fabrication techniques, electronics and qubit calibration procedures can be found elsewhere^{23}.
Threeresonator circuit QED Hamiltonian. The Hamiltonian for the circuit of Fig. 1b can be written as the sum of the Hamiltonians of each unit cell, and . Neglecting the driving and dissipative terms for simplicity, the Hamiltonian of the first circuit unit cell can be expressed in the interaction picture with respect to Q_{1} and R_{a} and R_{b} as the combination of two Jaynes–Cummings interactions:
where are the rising and lowering operators for Q_{1}, and are the bosonic annihilation and creation operators for R_{a} and R_{b}, respectively, and Δ_{1a}≡f_{1}−f_{a} and Δ_{1b}≡f_{1}−f_{b} are the qubit–resonator detunings. The Hamiltonian for the second circuit unit cell has an analogous expression.
To effectively switch off a particular qubit–resonator Q_{p}–R_{q} interaction, the condition Δ_{p q}≫g_{p q} must be fulfilled. This is the case when Q_{1} (Q_{2}) is at the idle point. On the contrary, when Δ_{p q}0, a resonant Jaynes–Cummings interaction takes place enabling state preparation and transfer in and between the resonators as well as photon detection.
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Acknowledgements
This work was supported by IARPA under ARO award W911NF0810336 and under ARO award W911NF0910375. M.M. acknowledges support from an Elings Prize Postdoctoral Fellowship. Devices were made at the UC Santa Barbara Nanofabrication Facility, a part of the NSFfunded National Nanotechnology Infrastructure Network.
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M.M. carried out the experiments with the help of H.W. M.M. analysed the data and carried out the numerical simulations. M.M. and H.W. fabricated the sample. M.N. provided software infrastructure. J.M.M. and E.L. designed the custom electronics. E.L. took the sample picture. All authors contributed to the fabrication process, qubit design or experimental setup, and discussed the data analysis. M.M., J.M.M. and A.N.C. conceived the experiment and cowrote the paper.
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Mariantoni, M., Wang, H., Bialczak, R. et al. Photon shell game in threeresonator circuit quantum electrodynamics. Nature Phys 7, 287–293 (2011). https://doi.org/10.1038/nphys1885
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DOI: https://doi.org/10.1038/nphys1885
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