The generation and control of quantum states of light constitute fundamental tasks in cavity quantum electrodynamics
(QED). The superconducting realization of cavity QED, circuit QED (refs 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 11, 12, 13, 14), enables on-chip microwave photonics, where superconducting qubits control and measure individual photon states 15, 16, 17, 18 . 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 modes 19, 20, 21, 22, 23, 24, 25, 26 . 27
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At a glance
: Experimental architecture and two-dimensional swap spectroscopy.
a, Photograph of a sample mounted on an aluminium holder, showing three coplanar waveguide resonators (R a,R b and R c, with meander design) capacitively coupled to two superconducting phase qubits (Q 1 and Q 2). b, Block diagram showing the main elements, which comprise two circuit unit cells R a–Q 1–R b (light green area) and R b–Q 2–R c (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 C 1a= C 1b= C 2b= C 2c=1.9 fF. f a ≃6.29 GHz, f b ≃6.82 GHz and f c ≃6.34 GHz are the measured resonator frequencies, and f 1 and f 2 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 Q 1 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 Δ τ, 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), Q 1 is excited by a π-pulse (red), then, (III), brought into resonance with for example R a (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 Δ τ. (IV), A measurement pulse projects the qubit onto its ground state |g or excited state |e . d, Two-dimensional swap spectroscopy for Q 1. The probability P e to find the qubit in |e is plotted versus z-pulse amplitude and resonator measurement time Δ τ. The typical chevron pattern generated by a qubit–resonator swap (arrows) is evident for both the Q 1–R a (dashed black box) and Q 1-R b 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 Q 2. From these measurements, we find the coupling strengths g 1a ≃17.58 ∓0.01 MHz, g 1b ≃20.65 ∓0.02 MHz, g 2b ≃20.43 ∓0.01 MHz and g 2c ≃17.96 ∓0.01 MHz.
: Photon shell game.
a, Block diagram of the sequence used to coherently transfer a single-photon Fock state |1 from R a to R c through R b. After initializing the system in the ground state, (I), Q 1 is excited by a π-pulse and, (II), z-pulsed into resonance with R a for a full Rabi swap (Rabi π-swap) at the end of which, (III), R a is populated by the one-photon Fock state |1 and Q 1 is in its ground state, at the idle point. In (IV), Q 1 is z-pulsed into resonance with R a for a Rabi π-swap and then, (V), z-pulsed into resonance with R b for another Rabi π-swap at the end of which, (VI), R b is populated by the one-photon Fock state and both Q 1 and Q 2 are in the ground state at the idle point. (VII), Q 2 is z-pulsed into resonance with R b for a Rabi π-swap and, (VIII), z-pulsed into resonance with R c for another Rabi π-swap at the end of which, (IX), R c is in the one-photon Fock state and Q 2 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 R c is detected by its interaction with Q 2. b, Measurement outcomes for different photon shell games. Each plot shows the probability P e to measure a qubit in the excited state |e as a function of the qubit–resonator measurement time Δ τ. The blue circles are data; the magenta lines are a least-squares fit to an exponentially damped squared sine. The data for Q 1 are in the first two columns, for Q 2 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 R a contains one photon (
), with the other two resonators in the vacuum state. In row (iii) the photon has been placed in R b (
), and in row (iv) the photon is in resonator R c (
). In row (v), we have taken the photon from R c and placed it back in R b (
), 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).
: Quantum-mechanical realization of the ‘Towers of Hanoi’ and shell game phase coherence.
a, Format as in Fig. 2b, showing the probability P e of measuring Q 1 or Q 2 in the excited state as a function of interaction time Δ τ 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 R a with both R b and R c in the vacuum state, and, (ii), a two-photon Fock state in R a (fidelity
) with the other two resonators in the vacuum state. In (iii), one photon has been transferred from R a to R b, so one-photon oscillations are seen when Q 1 measures R a or R b, and when Q 2 measures R b. In (iv), the second photon has been transferred to R b (
), yielding the
increase when either Q 1 or Q 2 measures R b. In (v), both photons have been transferred to R c (
). Note that even with this complex protocol, both R a and R b exhibit negligible oscillations (
). See Supplementary Information for further analysis. b, From left to right: Density matrix
associated with resonator R a (resonator Hilbert space truncated to lowest four bosonic states) for | ψ X (top row) and | ψ Y (bottom row) in R a, then shuffled to R b and finally back in R a.
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( ) associated with each
in b, where is the complex resonator amplitude in square root of photon number units (colour scale bar on the far right). α
: Combined quantum state transfer and storage for one- and two-photon Fock states.
a, Probability P e to find Q 2 in the excited state (colour bar scale, right side) versus measurement time Δ τ (horizontal axis), and total storage time τ st (vertical axis). A one-photon Fock state is created and stored in resonator R a for a time τ st/3, transferred to R b and stored for the same time, then transferred to R c, 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.
: Two-resonator Rabi swaps.
a, Block diagram of the preparation and measurement protocol. (I), Q 1 is excited by a π-pulse and, (II), brought into resonance with R a for a Rabi π-swap, at the end of which, (III), R a is populated by a one-photon Fock state. In (IV), Q 1 is brought into resonance with R a for a variable transfer time τ at the end of which, (V), R a is left partially populated, and, (V) and (VI), the remaining energy is fully transferred to R b by means of a Rabi π-swap with Q 1. The probabilities for having the photon in R a or R b can be varied continuously by changing the transfer time τ (see main text). These probabilities are simultaneously measured for R a with Q 1 and for R b with Q 2(VII). b, Qubit probability P 1e (colour scale bar) as a function of the Q 1–R a measurement time Δ τ 1 (horizontal axis) and the transfer time τ (vertical axis). c, The same as in b, but for Q 2’s probability P 2e as a function of the Q 2–R b measurement time Δ τ 2. 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 Q 1-R a 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 Q 1’s probability P 1e (dark blue circles) (Q 2’s probability P 2e; light green circles), with the summed probability P= P 1e+ P 2e (magenta circles) showing the expected slow decay (see main text). The solid lines are fits to data.