Photon shell game in three-resonator circuit quantum electrodynamics

Journal name:
Nature Physics
Year published:
Published online

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.

At a glance


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

    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=1.9fF. fa6.29GHz, fb6.82GHz and fc6.34GHz 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 |gright fence or excited state |eright fence. d, Two-dimensional swap spectroscopy for Q1. The probability Pe to find the qubit in |eright fence 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 g1a17.580.01MHz, g1b20.650.02MHz, g2b20.430.01MHz and g2c17.960.01MHz.

  2. Photon shell game.
    Figure 2: Photon shell game.

    a, Block diagram of the sequence used to coherently transfer a single-photon Fock state |1right fence 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 |1right fence 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 |eright fence 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).

  3. Quantum-mechanical realization of the /`Towers of Hanoi/' and shell game phase coherence.
    Figure 3: Quantum-mechanical realization of the ‘Towers of Hanoi’ and shell game phase coherence.

    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 |1right fence 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 |ψXright fence (top row) and |ψYright fence (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).

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

    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.

  5. Two-resonator Rabi swaps.
    Figure 5: Two-resonator Rabi swaps.

    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.


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Author information


  1. Department of Physics, University of California, Santa Barbara, California 93106-9530, USA

    • Matteo Mariantoni,
    • H. Wang,
    • Radoslaw C. Bialczak,
    • M. Lenander,
    • Erik Lucero,
    • M. Neeley,
    • A. D. O’Connell,
    • D. Sank,
    • M. Weides,
    • J. Wenner,
    • T. Yamamoto,
    • Y. Yin,
    • J. Zhao,
    • John M. Martinis &
    • A. N. Cleland
  2. California NanoSystems Institute, University of California, Santa Barbara, California 93106-9530, USA

    • Matteo Mariantoni
  3. Green Innovation Research Laboratories, NEC Corporation, Tsukuba, Ibaraki 305-8501, Japan

    • T. Yamamoto


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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