Observation and stabilization of photonic Fock states in a hot radio-frequency resonator

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Science  08 Mar 2019:
Vol. 363, Issue 6431, pp. 1072-1075
DOI: 10.1126/science.aaw3101

Going quantum with radio waves

It becomes increasingly difficult to detect long-wavelength single photons because of thermal fluctuations in the background. This can pose problems for single-photon detection for fields such as astronomy and nuclear magnetic resonance imaging. Gely et al. used a superconducting qubit, initially developed for circuit quantum electrodynamics (cQED) and quantum information processing for microwaves, to directly observe the quantization of radio-frequency electromagnetic fields stored in a photonic microresonator. They were then able to manipulate the quantum state of the radio-frequency field, forming one- and two-photon Fock states within the microresonator, and analyze how the system interacts dynamically with its environment. The cQED approach could be used for fundamental studies in quantum thermodynamics and also find practical application in imaging.

Science, this issue p. 1072


Detecting weak radio-frequency electromagnetic fields plays a crucial role in a wide range of fields, from radio astronomy to nuclear magnetic resonance imaging. In quantum optics, the ultimate limit of a weak field is a single photon. Detecting and manipulating single photons at megahertz frequencies presents a challenge because, even at cryogenic temperatures, thermal fluctuations are appreciable. Using a gigahertz superconducting qubit, we observed the quantization of a megahertz radio-frequency resonator, cooled it to the ground state, and stabilized Fock states. Releasing the resonator from our control, we observed its rethermalization with nanosecond resolution. Extending circuit quantum electrodynamics to the megahertz regime, we have enabled the exploration of thermodynamics at the quantum scale and allowed interfacing quantum circuits with megahertz systems such as spin systems or macroscopic mechanical oscillators.

Detecting and manipulating single photons becomes more difficult at lower frequencies because of thermal fluctuations. A hot environment randomly creates and annihilates photons, causing decoherence in addition to creating statistical mixtures of states from which quantum-state preparation is challenging. This can be mitigated by using a colder system as a heat sink to extract the entropy created by the environment. Such a scheme, known as reservoir engineering, was first developed in trapped ions (1), in which hot degrees of freedom are cooled through the atomic transitions of ions.

Using superconducting electronics, circuit quantum electrodynamics (cQED) has made extensive use of reservoir engineering to cool but also manipulate electromagnetic fields at the quantum level. With the prospect of building a quantum computer, or to demonstrate fundamental phenomena, experiments have shown the cooling or reset of qubits to their ground state (24), also in the megahertz regime (5), quantum state stabilization (68), and quantum error correction (9). Using superconducting circuits, reservoir engineering is commonplace in electromechanical systems (10, 11), but with weak nonlinearity, such schemes have only limited quantum control (12, 13) compared with that of typical cQED systems. Despite the many applications of quantum-state engineering in cQED, obtaining control over the quantum state of a hot resonator, where the environment temperature is a dominant energy scale, remains a largely unexplored and challenging task.

We directly observed the quantization of radio-frequency electromagnetic fields in a thermally excited megahertz photonic resonator and manipulated its quantum state using reservoir engineering. Specifically, we cooled the 173-MHz resonator to 90% ground-state occupation and stabilized one- and two-photon Fock states. Releasing the resonator from our control, we observed its rethermalization with photon-number resolution.

We used the paradigm of cQED, in which a resonator can be read out and controlled by dispersively coupling it to a superconducting qubit. Achieving this with a gigahertz qubit and megahertz photons is challenging because in a conventional cQED architecture, the coupling would be far too weak (14). To overcome this, we present a circuit that enables a very strong interaction, resulting in a cross-Kerr coupling larger than the qubit and resonator dissipation rates despite an order-of-magnitude difference in their resonance frequencies.

The circuit (Fig. 1A) comprises a Josephson junction [inductance LJ = 41 nanohenrys (nH)] connected in series to a capacitor (capacitance CL = 11 pF) and a spiral inductor (inductance L = 28 nH). At low frequencies, the parasitic capacitance of the spiral inductor is negligible, and the equivalent circuit (Fig. 1B) has a first transition frequency ωL = 2π × 173 MHz. At gigahertz frequencies, CL behaves as a short, and the capacitance of the spiral inductor CH = 40 fF becomes relevant instead. The resulting parallel connection of LJ, L, and CH (Fig. 1C) has a first transition frequency ωH = 2π × 5.91 GHz. The two modes share the Josephson junction. The junction has an inductance that varies with the current fluctuations traversing it, and consequently, the resonance frequency of the high-frequency (HF) mode shifts as a function of the number of excitations in the low-frequency (LF) mode and vice versa. This cross-Kerr interaction is quantified by the shift per photon χ=2AHAL, where the anharmonicities of the LF and HF modes AL = h × 495 kHz and AH = h × 192 MHz, where h is the Planck constant, are respectively given by (15) AL=e22CL(LJL+LJ)3,AH=e22CH(LL+LJ)(1)The system is described by the Hamiltonian (15)H^=ωHa^a^+ωLb^b^AH2a^a^a^a^AL2b^b^b^b^χa^a^b^b^(2)where a^ is the annihilation operator for photons in the HF mode, b^ is the annihilation operator for photons in the LF mode, and ℏ is Planck’s constant h divided by 2p. The second line describes the anharmonicity or Kerr nonlinearity of each mode. The last term describes the cross-Kerr interaction. By combining it with the first term as (ωHχb^b^)a^a^, the dependence of the HF mode resonance on the number of photons in the LF mode becomes apparent.

Fig. 1 Cross-Kerr coupling between a transmon qubit and a radio-frequency resonator.

(A) False-colored optical micrograph of the device overlaid with the equivalent lumped-element circuit. (B and C) Effective circuit at low and high frequencies. At low frequencies, the femtofarad capacitances of the circuit are equivalent to open circuits, and the device is equivalent to a series JJ-inductor-capacitor combination, whereas at high frequencies, the picofarad capacitances of the circuit are equivalent to short circuits, and the device is equivalent to a parallel JJ-inductor-capacitor combination. The circuit has thus two modes, a 173-MHz resonator and a 5.9-GHz qubit. (D) Microwave response |S11|. Through cross-Kerr coupling, quantum fluctuations of a photon number state |n=0,1,... in the resonator shift the qubit transition frequency. Peak heights are proportional to the occupation of state |n, and we extract a thermal occupation nth = 1.6 in the resonator corresponding to a temperature of 17 mK.

The cross-Kerr interaction manifests as photon-number splitting (16) in the measured microwave reflection S11 (Fig. 1D). Distinct peaks correspond to the first transition frequency of the HF mode |g,n|e,n, with frequencies ωHnχ/ℏ, where χ/h = 21 MHz. We label the eigenstates of the system|j,n, with j = g, e, f, ... and n = 0, 1, 2, ... corresponding to excitations of the HF and LF modes, respectively. The amplitude of peak n is proportional toPnκextn(3)where Pn is the occupation of photon-number level |n in the LF mode and κextn is the ratio of external coupling κext/2π = 1.6 × 106 s–1 to the total linewidth κn of peak n. From the Bose-Einstein distribution of peak heights Pn, we extracted the average photon occupation nth = 1.6 corresponding to a mode temperature of 17 mK.

The resolution of individual photon peaks is due to the condition κnχ/. The peak linewidths increase with n following κn=κ[1+4nth(H)]+2γ[n+(1+2n)nth], where κ/2π = 3.7 × 106 s–1 is the dissipation rate of the HF mode, nth(H)0.09 is its thermal occupation (fig. S10), and γ/2π = 23 × 103 s–1 is the dissipation rate of the LF mode (obtained through time-domain measurements). The condition κnAH/ makes the HF mode an inductively shunted transmon qubit (17), making it possible to selectively drive the |g,n|e,n and |e,n|f,n transitions. Despite its low dissipation rate γ, the LF mode has a linewidth of a few megahertz (measured with two-tone spectroscopy) (fig. S15), which originates in thermal processes such as |g,n|e,n occurring at rates κnth(H) larger than γ (15). The LF mode linewidth is then an order of magnitude larger than AL, making it essentially a harmonic oscillator that we will refer to as the resonator.

The junction nonlinearity enables transfer of population between states by coherently pumping the circuit at a frequency ωp. The cosine potential of the junction imposes four-wave mixing selection rules, only allowing interactions that involve four photons. One such interaction isH^int=gn+1|f,ng,n+1|+H.c.(4)which is activated when driving at the energy difference between the two coupled states ωp = 2ωH – ωL – (2nχ + AH)/ℏ, where H.c. means Hermitian conjugate. This process, enabled by a pump photon, annihilates a photon in the resonator and creates two in the transmon. The number of photons involved in the interaction is four, making it an allowed four-wave mixing process. The induced coupling rate is g=AH34AL14ξp, where |ξp|2 is the amplitude of the coherent pump tone measured in number of photons (15).

We used this pump tone in combination with the large difference in mode relaxation rates to cool the megahertz resonator to its ground state (Fig. 2A). The pump drives transitions between |g,1 and |f,0 at a rate g. The population of |g,1, transferred to |f,0, subsequently decays at a rate 2κ to the ground state |g,0. Cooling occurs when the thermalization rate of the resonator nthγ is slower than the rate Cγ at which excitations are transferred from |g,1 to |g,0, where C = 2g2/κγ is the cooperativity [proportional to cooling-pump power (15)].

Fig. 2 Ground-state cooling of the radio-frequency resonator.

(A) Energy ladder of the coupled transmon qubit and resonator. Meandering arrows indicate relaxation and thermal processes. The resonator is cooled by driving a transition (black arrow) that transfers excitations from the resonator to the qubit, where they are quickly dissipated. (B) Photon-number spectroscopy of the resonator for different cooperativities C (proportional to cooling-pump power). C = 0.01, 6, 47, and 300 from top to bottom. Ground-state occupations P0 are extracted from Lorentzian fits (black curves). (C) Vertical lines indicate the datasets of (B). A simulation (curve) predicts the measured (dots) high-C decrease of P0 through the off-resonant driving of other sideband transitions.

For different cooling pump strengths, we measured S11 (Fig. 2B). The pump frequency was adapted at each power because the AC-Stark effect increasingly shifted the qubit frequency as a function of power (fig. S9). The data are fitted to a sum of complex Lorentzians, with amplitudes given by Eq. 3 and linewidths κn, from which Pn is extracted. Thermal effects lead to the ratio Pn+1/Pn = nth/(1 + nth) between neighboring photon number states for n ≥ 1, and the cooling pump changes the ratio of occupation of the first two statesP1P0nth1+nth+C(5)Hence, the ground-state occupation increases with cooperativity, and we attain a maximum P0 = 0.82. At higher cooperativity, P0 diminishes because of the off-resonant driving of other four-wave mixing processes such as |f,n+1g,n|+h.c., which tend to raise the photon number of the resonator. This effect was simulated by using an adaptive rotating-wave approximation (Fig. 2C and fig. S6) (18).

Neighboring four-wave mixing processes are measured by sweeping the pump frequency while monitoring the spectrum (Fig. 3A). When cooling with a single pump, they eventually limit performance but can be resonantly driven to our advantage. By driving multiple cooling interactions |g,n|f,n1, less total pump power is required to reach a given ground-state occupation, minimizing off-resonant driving. By maximizing the ground-state peak amplitude as a function of the power and frequency of four cooling tones, we achieved P0 = 0.90 (Fig. 3B).

Fig. 3 Enhanced cooling and Fock-state stabilization by using multiple tones.

(A) |S11| as a function of pump and probe frequency. Vertical lines correspond to the photon number–splitted qubit frequencies. Horizontal and diagonal features appear at pump frequencies that enable the transfer of population between Fock states of the resonator. Arrows indicate three example transitions: (1) the cooling transition of Fig. 2; (2) the transition |g,2|f,1 transferring |2 to |1; and (3) the transition |g,1|f,2, which raises |1 to |2. (B) By simultaneously driving four cooling transitions (|g,n+1|f,n), cooling is enhanced to P0 = 0.9. (C) Using these transitions in conjunction with raising transitions |g,n|f,n+1, we stabilized Fock states 1 and 2 with 59 and 35% fidelity, respectively. We fit a sum of complex Lorentzians to the spectrum, showing only the relevant Lorentzian (black curve) whose amplitude provides Pn. Off-resonant driving results in population transfer to higher energy states visible as features in the lower frequencies of the spectrum.

By combining cooling |g,n|f,n1 and raising |g,n|f,n+1 tones (Fig. 3C, inset), we demonstrate stabilization of higher Fock states, non-Gaussian states commonly considered to be nonclassical phenomena (19). The optimum frequencies for the raising and cooling tones adjacent to the stabilized state were detuned by a few megahertz from the transition frequency (Fig. 3C, inset, dashed lines), otherwise one pump tone would populate the |f level, diminishing the effectiveness of the other.

Last, we investigated dynamics in a photon-resolved manner (Fig. 4). While probing S11 at a given frequency, we switched the cooling or single-photon stabilization pumps on and off for intervals of 50 μs. We performed this for a sequence of probe frequencies, resulting in S11 as a function of both frequency and time [full spectrum is provided in (15)]. The spectrum is fitted at each time to extract Pn as a function of time. After reaching the steady state, the pumps were turned off, and we observed the thermalization process that follows the semiclassical master equationP˙n=nγ(nth+1)Pn+nγnthPn1(n+1)Pnγnth+(n+1)Pn+1γ(nth+1)(6)Our cQED architecture enables the readout and manipulation of a radio-frequency resonator at the quantum level. Using the fast readout methods of cQED, single-shot readout or the tracking of quantum trajectories could enable even finer resolution of thermodynamic effects at the quantum scale. Coupling many of these devices together could enable the exploration of many-body effects in Bose-Hubbard systems with dynamically tunable temperatures (20, 21). This circuit architecture could also be used to interface circuit quantum electrodynamics with different physical systems in the megahertz frequency range, such as spin systems (22) or macroscopic mechanical oscillators (10). Last, this circuit could enable sensing of radio-frequency radiation with quantum resolution, a critical frequency range for a number of applications, from nuclear magnetic resonance imaging to radio astronomy.

Fig. 4 Fock state–resolved thermalization dynamics of the resonator.

(A and B) At ton, pumps are turned on, and the resonator evolves into (A) the ground state or (B) a single-photon state. At toff, control is released, and we observe photon number–resolved thermalization of the resonator. The extracted Fock-state occupation (dots) is fitted to Eq. 6 (black curve).

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S15

Tables S1 and S2

References (2330)

References and Notes

  1. Materials and methods are available as supplementary materials.
Acknowledgments: We acknowledge Ya. M. Blanter, S. M. Girvin, and J. D. P. Machado for useful discussions. Funding: This work was supported by the European Research Council under the European Union’s H2020 program under grant agreements 681476-QOM3D and 785219-GrapheneCore2, by the Dutch Foundation for Scientific Research (NWO) through the Casimir Research School, and by the Army Research Office through grant W911NF-15-1-0421. Author contributions: M.F.G. and R.V. developed the theoretical description of the experiment. M.F.G. designed the device. M.F.G. fabricated the device with help from J.D. and M.K.; M.F.G., M.K., C.D., J.D., and M.D.J. participated in the measurements. M.F.G. and C.D. analyzed the data. B.B. provided the software and input for the adaptive rotating-wave-approximation simulation. M.F.G. wrote the manuscript with input from M.K., C.D., and G.A.S. All coauthors reviewed the manuscript and provided feedback. G.A.S. supervised the project. Competing interests: The authors declare that they have no competing interests. Data and materials availability: Raw data as well as all measurement, data-analysis, and simulation code used in the generation of main and supplementary figures are available in Zenodo with the identifier 10.5281/zenodo.2551258.
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