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Photon shell game in three-resonator circuit quantum electrodynamics

Abstract

The generation and control of quantum states of light constitute fundamental tasks in cavity quantum electrodynamics1,2,3,4,5,6,7,8,9,10 (QED). The superconducting realization of cavity QED, circuit QED (refs 11, 12, 13, 14), enables on-chip microwave photonics, where superconducting qubits15,16,17,18 control and measure individual photon states19,20,21,22,23,24,25,26. A long-standing issue in cavity QED is the coherent transfer of photons between two or more resonators. Here, we use circuit QED to implement a three-resonator architecture on a single chip, where the resonators are interconnected by two superconducting phase qubits. We use this circuit to shuffle one- and two-photon Fock states between the three resonators, and demonstrate qubit-mediated vacuum Rabi swaps between two resonators. By shuffling superposition states we are also able to demonstrate the high-fidelity phase coherence of the transfer. Our results illustrate the potential for using multi-resonator circuits as photon quantum registers and for creating multipartite entanglement between delocalized bosonic modes27.

Main

The combination of high-finesse 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 excitations1,2,3,14. Both stationary6,21,23 and propagating7,9 non-classical fields can be synthesized using such systems, enabling quantum memory and quantum messaging8. A critical challenge however is the extension from single to more versatile multi-cavity architectures27,28, allowing manipulation of spatially separated bosonic modes. Although the entanglement of different modes of a single cavity4 and of free-space modes10 has been shown in atomic systems, and a coupled low- and high-quality-factor resonator studied in circuit QED (refs 25, 26), coherent dynamics between two or more high-quality-factor cavities has yet to be demonstrated. Here we describe a triple-resonator system, where three high-quality-factor microwave resonators are coupled to two superconducting phase qubits (see Fig. 1). The qubits serve as quantum transducers29 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 coupling27, enabling rapid transfers between resonators with significantly different frequencies. As an important example, we demonstrate single-photon Rabi swaps between two resonators detuned by 12,000 resonator linewidths.

Figure 1: Experimental architecture and two-dimensional swap spectroscopy.
figure 1

a, Photograph of a sample mounted on an aluminium holder, showing three coplanar waveguide resonators (Ra,Rb and Rc, with meander design) capacitively coupled to two superconducting phase qubits (Q1 and Q2). b, Block diagram showing the main elements, which comprise two circuit unit cells Ra–Q1–Rb (light green area) and Rb–Q2–Rc (dark blue area). The horizontal placement represents the spatial layout of the sample, whereas the vertical distribution corresponds to the frequencies of the elements. The qubit–resonator coupling capacitors are designed to be C1a=C1b=C2b=C2c=1.9 fF. fa6.29 GHz, fb6.82 GHz and fc6.34 GHz are the measured resonator frequencies, and f1 and f2 the tunable qubit transition frequencies. Control and readout wiring is also shown. c, Upper panel, pulse sequence for swap spectroscopy, with data shown in d and e. The Q1 line shows a Gaussian microwave π-pulse (red) and a qubit tuning pulse (z-pulse; hashed magenta) with variable z-pulse amplitude (zpa) and duration Δτ, followed by a triangular measurement pulse (black). Lower panel, diagrammatic representation of the pulse sequence. (I), The entire system is initialized in the ground state. (II), Q1 is excited by a π-pulse (red), then, (III), brought into resonance with for example Ra (dashed black rectangle in d) by means of a z-pulse (magenta) and allowed to interact with the resonator (or electromagnetic environment) for a time Δτ. (IV), A measurement pulse projects the qubit onto its ground state |g〉 or excited state |e〉. d, Two-dimensional swap spectroscopy for Q1. The probability Pe to find the qubit in |e〉 is plotted versus z-pulse amplitude and resonator measurement time Δτ. The typical chevron pattern generated by a qubit–resonator swap (arrows) is evident for both the Q1–Ra (dashed black box) and Q1-Rb interactions. Near the centre of the plot a qubit interaction with a spurious two-level system is seen, surrounded by two regions with short qubit relaxation time. e, The same as in d but for Q2. From these measurements, we find the coupling strengths g1a17.580.01 MHz, g1b20.650.02 MHz, g2b20.430.01 MHz and g2c17.960.01 MHz.

Figure 1a,b shows the main experimental elements, which comprise three coplanar waveguide resonators, Ra,Rb and Rc, two phase qubits, Q1 and Q2, 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 f1,2 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, Ra–Q1–Rb (green area) and Rb–Q2–Rc (blue area). The shared resonator Rb connects the two cells and protects the two qubits from unwanted crosstalk. The resonator frequencies fa,fb and fc are measured with qubit spectroscopy (not shown). The vacuum Rabi couplings between each qubit and its corresponding resonators, g1a and g1b for Q1 and g2b and g2c for Q2, are determined by their respective coupling capacitors (see Fig. 1b). The coupling strengths are measured using two-dimensional swap spectroscopy (see Fig. 1c–e). We note that the swap spectroscopy provides an excellent tool for revealing the presence of spurious two-level 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 Q1 (Q2) |g〉↔|e〉 transition frequency f1 (f2) is set in-between, and well away from, the resonator transition frequencies fa and fb (fb and fc).

When Q1 (Q2) 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 Qp–Rq (p=1,2 and q=a,b,c) interaction is switched on by shifting the qubit transition frequency fp to equal the resonator frequency fq, thus setting the detuning to zero and enabling quantum energy transfers. The time-dependent control of the qubit transition frequency consequently enables highly complex quantum control of the resonators23.

Figure 2a shows a diagram of the pulse sequence used to implement the single-photon 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 single-photon 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 Q1 is used to pump a single photon into resonator Ra (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 single-excitation transfer. A transfer from Ra to Rb is shown in steps (IV)–(VI), and a second transfer from Rb to Rc carried out using Q2 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)).

Figure 2: Photon shell game.
figure 2

a, Block diagram of the sequence used to coherently transfer a single-photon Fock state |1〉 from Ra to Rc through Rb. After initializing the system in the ground state, (I), Q1 is excited by a π-pulse and, (II), z-pulsed into resonance with Ra for a full Rabi swap (Rabi π-swap) at the end of which, (III), Ra is populated by the one-photon Fock state |1〉 and Q1 is in its ground state, at the idle point. In (IV), Q1 is z-pulsed into resonance with Ra for a Rabi π-swap and then, (V), z-pulsed into resonance with Rb for another Rabi π-swap at the end of which, (VI), Rb is populated by the one-photon Fock state and both Q1 and Q2 are in the ground state at the idle point. (VII), Q2 is z-pulsed into resonance with Rb for a Rabi π-swap and, (VIII), z-pulsed into resonance with Rc for another Rabi π-swap at the end of which, (IX), Rc is in the one-photon Fock state and Q2 in the ground state at the idle point. Measurement and qubit state readout are carried out in (X) and (XI), respectively, where the presence of the Fock state in Rc is detected by its interaction with Q2. b, Measurement outcomes for different photon shell games. Each plot shows the probability Pe to measure a qubit in the excited state |e〉 as a function of the qubit–resonator measurement time Δτ. The blue circles are data; the magenta lines are a least-squares fit to an exponentially damped squared sine. The data for Q1 are in the first two columns, for Q2 in the second two; each row corresponds to a different game. Row (i), all resonators are in the vacuum state (with fidelity ; we define as the amplitude of the fit; see Supplementary Information), that is, no stored photons. Row (ii), resonator Ra contains one photon (), with the other two resonators in the vacuum state. In row (iii) the photon has been placed in Rb (), and in row (iv) the photon is in resonator Rc (). In row (v), we have taken the photon from Rc and placed it back in Rb (), demonstrating the high degree of control and population coherence in the system. All data in the Letter are corrected for measurement errors (see Supplementary Information).

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 Ra to Rc through Rb, and then detected using qubit Q2; measurements of the other resonators Ra and Rb show no oscillations, that is, no photonic excitation.

We also explored a variant of the shell game, transferring a two-photon Fock state |2〉 from Ra to Rb to Rc. The two-photon Fock state (see ref. 21) is first generated in Ra, as shown by the measurements in Fig. 3a(i),(ii). Figure 3a also shows the measurements after this state is transferred from Ra to Rb and then to Rc. Each transfer takes two steps, starting with for example the state |Q1RaRb〉=|g20〉. The qubit is brought into resonance with Ra, and one photon is Rabi-swapped to the qubit, at a rate faster than the usual one-photon rate21, leaving the system in the state |e10〉. The qubit is then tuned into resonance with resonator Rb for a one-photon Rabi swap, resulting in the state |g11〉 (see Fig. 3a(iii)). The qubit is subsequently placed back in resonance with Ra for a one-photon swap, yielding |e01〉, and brought into resonance with Rb to transfer the second photon, ending with the state |g02〉 (see Fig. 3a(iv)). To finally transfer the photons to resonator Rc, the process is repeated using qubit Q2, which completes the full transfer of Fock state |2〉 (see Fig. 3a(v)). This process resembles the well-known 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 3: Quantum-mechanical realization of the ‘Towers of Hanoi’ and shell game phase coherence.
figure 3

a, Format as in Fig. 2b, showing the probability Pe of measuring Q1 or Q2 in the excited state as a function of interaction time Δτ with a resonator. The data are shown as blue circles, with a least-squares fit as magenta solid lines. (i), A one-photon Fock state |1〉 in Ra with both Rb and Rc in the vacuum state, and, (ii), a two-photon Fock state in Ra (fidelity ) with the other two resonators in the vacuum state. In (iii), one photon has been transferred from Ra to Rb, so one-photon oscillations are seen when Q1 measures Ra or Rb, and when Q2 measures Rb. In (iv), the second photon has been transferred to Rb (), yielding the increase when either Q1 or Q2 measures Rb. In (v), both photons have been transferred to Rc (). Note that even with this complex protocol, both Ra and Rb exhibit negligible oscillations (). See Supplementary Information for further analysis. b, From left to right: Density matrix associated with resonator Ra (resonator Hilbert space truncated to lowest four bosonic states) for |ψX〉 (top row) and |ψY〉 (bottom row) in Ra, then shuffled to Rb and finally back in Ra. is projected onto the number states , the magnitude and phase of which are represented by an arrow in the complex plane, with scale embedded in the first panel. c, Wigner function W(α) associated with each in b, where α is the complex resonator amplitude in square root of photon number units (colour scale bar on the far right).

Figure 3b,c shows the density matrices and corresponding Wigner functions24 for the coherent transfer of the phase-sensitive state |ψX〉=|0〉+|1〉 (top panels) and |ψY〉=|0〉+eiπ/2|1〉 (bottom panels) between Ra and Rb. From left to right, each state is first prepared in Ra, shuffled to Rb, thus leaving Ra in the vacuum state, and finally shuffled back to Ra. For a phase coherent transfer, the off-diagonal 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 |ψX〉 and |ψY〉, respectively (see Supplementary Information).

Another fundamental question for resonator-based 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 τst/3 before being transferred to the next resonator. Qubit measurements of the resonator containing the photon (for example resonator Rc in Fig. 4a) exhibit clear oscillations for total storage times τst exceeding 3 μs. Figure 4b is the same experiment repeated with a two-photon Fock state; clear oscillations are visible for total storage times τst exceeding 1.5 μs. These experiments demonstrate the realization of a programmable quantum information register.

Figure 4: Combined quantum state transfer and storage for one- and two-photon Fock states.
figure 4

a, Probability Pe to find Q2 in the excited state (colour bar scale, right side) versus measurement time Δτ (horizontal axis), and total storage time τst (vertical axis). A one-photon Fock state is created and stored in resonator Ra for a time τst/3, transferred to Rb and stored for the same time, then transferred to Rc, stored for the same time and then measured. b, The same as in a, but for the generation and storage of a two-photon Fock state.

We further show two-resonator vacuum Rabi swaps that can lead to quantum state entanglement between the two resonators Ra and Rb. The protocol is diagrammed in Fig. 5a. A one-photon Fock state is first stored in Ra, placing the system in the state |Q1RaRb〉=|g10〉 (see Fig. 5a(I)–(III)). The qubit is then used to carry out a partial transfer of the photon to Rb, by placing Q1 in resonance with Ra and varying the transfer time τ (see Fig. 5a(IV)). This leaves the system in the entangled state α|g10〉+β|e00〉, with amplitudes30 α=cos(πg1aτ) and β=−i sin(πg1aτ). The qubit frequency f1 is then tuned from fa to fb, and left there for a time equal to the Q1–Rb swap time, thus mapping the qubit state onto the resonator and resulting in the two-resonator entangled state

Figure 5: Two-resonator Rabi swaps.
figure 5

a, Block diagram of the preparation and measurement protocol. (I), Q1 is excited by a π-pulse and, (II), brought into resonance with Ra for a Rabi π-swap, at the end of which, (III), Ra is populated by a one-photon Fock state. In (IV), Q1 is brought into resonance with Ra for a variable transfer time τ at the end of which, (V), Ra is left partially populated, and, (V) and (VI), the remaining energy is fully transferred to Rb by means of a Rabi π-swap with Q1. The probabilities for having the photon in Ra or Rb can be varied continuously by changing the transfer time τ (see main text). These probabilities are simultaneously measured for Ra with Q1 and for Rb with Q2(VII). b, Qubit probability P1e (colour scale bar) as a function of the Q1–Ra measurement time Δτ1 (horizontal axis) and the transfer time τ (vertical axis). c, The same as in b, but for Q2’s probability P2e as a function of the Q2–Rb measurement time Δτ2. Measurement times are sufficient to show one complete Rabi oscillation between the measurement qubit and resonator, with a Rabi swap occurring at the centre of each horizontal axis (dashed white line), with multiple swaps shown as a function of the Q1-Ra transfer time τ (vertical direction). Cuts through the probabilities along the dashed white lines (full Rabi swaps) are shown in d, where the expected co-sinusoidal oscillations are observed in Q1’s probability P1e (dark blue circles) (Q2’s probability P2e; light green circles), with the summed probability P=P1e+P2e (magenta circles) showing the expected slow decay (see main text). The solid lines are fits to data.

We then use both qubits to simultaneously measure the two resonators, Q1 measuring Ra for a measurement time Δτ1 and Q2 measuring Rb for a time Δτ2. Figure 5b(c) shows the resulting oscillations in P1e (P2e) for Q1 (Q2) (colour bar scale), as a function of the measurement time Δτ1,2 (horizontal axis) and Q1–Ra 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 Δτ1,2 are chosen to equal a full qubit–resonator swap time (dashed white lines in Fig. 5b,c), the system will be in the state |Q1RaRbQ2〉=α|e00g〉+β|g00e〉. An ideal measurement of Q1 and Q2 would then yield probabilities P1e=|α|2=cos2g1aτ) and P2e=|β|2=sin2g1aτ) for Q1 and Q2, 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 P1e+P2e close to unity, as expected. The probabilities decrease with τ, owing to the finite energy relaxation time of Q1 and Ra. The decay time of Rb does not contribute noticeably (see Supplementary Information). Fitting yields an effective two-resonator decay time approximately equal to the harmonic mean of the energy relaxation times T1rel340 ns and Tarel3.9 μs of Q1 and Ra, respectively, Tabrel≈(1/2T1rel+1/2Tarel)−1626 ns. We note that the phase qubit enables a photon transfer between Ra and Rb 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 multi-resonator circuit QED (refs 27, 28), both for scientific study and for quantum information.

Methods

Sample fabrication. The resonators are made from a 150-nm-thick rhenium film grown in a molecular-beam-epitaxy system on a polished sapphire substrate. All the other wiring layers are made from sputtered aluminium with Al/AlOx/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 multi-layer processing.

The complete device is wire bonded by aluminium wire-bonds 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 elsewhere23.

Three-resonator 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 Q1 and Ra and Rb as the combination of two Jaynes–Cummings interactions:

where are the rising and lowering operators for Q1, and are the bosonic annihilation and creation operators for Ra and Rb, respectively, and Δ1af1fa and Δ1bf1fb 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 Qp–Rq interaction, the condition Δp qgp q must be fulfilled. This is the case when Q1 (Q2) is at the idle point. On the contrary, when Δp q0, 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 W911NF-08-1-0336 and under ARO award W911NF-09-1-0375. M.M. acknowledges support from an Elings Prize Postdoctoral Fellowship. Devices were made at the UC Santa Barbara Nanofabrication Facility, a part of the NSF-funded 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 set-up, and discussed the data analysis. M.M., J.M.M. and A.N.C. conceived the experiment and co-wrote the paper.

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Correspondence to Matteo Mariantoni or A. N. Cleland.

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Mariantoni, M., Wang, H., Bialczak, R. et al. Photon shell game in three-resonator circuit quantum electrodynamics. Nature Phys 7, 287–293 (2011). https://doi.org/10.1038/nphys1885

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