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The strong-coupling regime of cavity quantum electrodynamics for a single two-level quantum system is reached when the coupling rate g at which a single photon is absorbed or coherently emitted into the cavity mode by the quantum system is larger than both the field decay rate κ out of the cavity and the dipole decay rate γ of the quantum system1,2. Under these conditions, the quantum processes of absorption and stimulated emission dominate the dynamics and can lead to a coherent exchange of a photon in the cavity and an excitation of the quantum system. This regime has been successfully accessed with neutral atoms2,4,5, and recently with quantum dots6 and Cooper pairs7. Although single ions have also been applied in CQED-related experiments3,21,22, the single-particle strong-coupling regime has not yet been reached. For an ensemble of N quantum systems interacting with a photon in a specific cavity mode, a collective coupling rate gN≡g N1/2 is obtained. When gN exceeds both κ and γ, the so-called collective strong-coupling regime is reached, which has so far only been realized using cold gases of neutral atoms in the form of atomic beams23, magneto-optical traps24 or Bose–Einstein condensates8,9.

In contrast to neutral atomic gases, ion Coulomb crystals have a series of attractive solid-state properties of importance for collective strong-coupling CQED studies and for the development of quantum-information devices. In Coulomb crystals, the ions have a uniform density and are positioned at distances of ∼10 μm (refs 16, 25). Consequently, there are no close-distance interactions between the particles that would lead to either temporal or spatial inhomogeneous spectral broadening. Ion Coulomb crystals can, furthermore, be stably trapped for hours, which allows for repeatable experiments with a coupling strength being constant to within a few per cent. Moreover, the crystal structure allows for various cavity modes to interact with the same crystal without interference and with identical coupling strengths. Finally, the low density of Coulomb crystals has the advantage that all of the interacting ions can very efficiently be optically pumped into a specific desired state.

To reach the collective strong-coupling regime, here we use a linear radiofrequency ion trap that incorporates an 11.8-mm-long optical cavity with its axis coinciding with the central trap axis26 (z axis), as shown in Fig. 1a. With a moderately high finesse F=3,000±200 at the 866 nm resonant wavelength of the 3d2D3/2–4p2P1/2 transition in 40Ca+ addressed in the experiments, the cavity has a decay rate of κ=2π×2.15 MHz. The natural dipole decay rate of an excitation on the 3d2D3/2–4p2P1/2 transition is γ=2π×11.15 MHz. With the 5 mm focal length of both curved cavity mirrors, this leads to a single-photon coupling rate of g=2π×0.53 MHz for an ion positioned at an anti-node of the standing-wave field in the centre of the TEM00 cavity mode. With these parameters, the collective strong-coupling regime is entered when the effective number of ions interacting with the photons in a cavity mode exceeds ∼500 (see the Methods section).

Figure 1: Schematic diagram and ion Coulomb crystal images.
figure 1

a, Schematic diagram of the experimental set-up with the linear radiofrequency ion trap incorporating an optical cavity along its radiofrequency-field-free axis (z axis). The incoupling mirror of the cavity is mounted on a piezoelectric transducer (PZT) allowing for tuning of the cavity around the atomic resonance. LC: laser cooling beam, RP: repumping beam, OP: optical pumping beam, PP: probe pulse, RB: reference beam, CM: cavity mirrors, PZT: piezoelectric transducer, CCD: CCD camera, APD: avalanche photodiode. b, Energy levels of 40Ca+ including the relevant transitions for the three main parts of the experimental sequence. First, the ions are Doppler laser cooled by driving the 4s2S1/2–4p2P1/2 transition using 397 nm laser beams and repumping on the 3d2D3/2–4p2P1/2 transition by light at 866 nm. Next, the ions are optically pumped into the mj=+3/2 Zeeman substate of the 3d2D3/2 level using the optical pumping beam in combination with the laser cooling beams. Finally, the coupling of the Coulomb crystals to the cavity field is measured by injecting a weak probe pulse of 866 nm light into the cavity and detecting the reflection signal by an avalanche photodiode. c,d, Projection images of a 1-mm-long ion Coulomb crystal recorded with the CCD camera by collecting 397 nm fluorescence light emitted during laser cooling: the whole crystal (c) and ions within the cavity mode volume (d) (see the text for details).

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The trap can be loaded controllably in situ with a few to tens of thousands of 40Ca+ ions through isotope-selective two-photon ionization26. The trapped ions are Doppler laser cooled into a Coulomb crystal16,25 by driving the 4s2S1/2–4p2P1/2 transition using two counter-propagating 397 nm light beams with left- and right-handed circular polarizations and by repumping on the 3d2D3/2–4p2P1/2 transition using an 866 nm light beam, propagating along the y axis and linearly polarized along x (Fig. 1a, b). Using a charge-coupled device (CCD) camera, projection images of the crystals can be recorded by collecting 397 nm fluorescence light emitted by the ions during the cooling process. From these images, the effective number of ions N in the crystal interacting with the TEM00 mode of the cavity can be extracted (see the Methods section). Figure 1c shows a projection image of a Coulomb crystal containing 6,400±200 ions of which 536±18 ions effectively interact with the cavity mode. For a visual impression of the overlap of the ions in the crystal and the TEM00 mode of the cavity, the repumping beam can be resonantly injected into the cavity. A resulting image is shown in Fig. 1d; in this case, the ions outside the cavity mode appear dark owing to the lack of repumping from the metastable 3d2D3/2-level. After the Coulomb crystallization and in the presence of a magnetic bias field along the z axis B=(Bx=0.0 G, By=0.0 G, Bz=2.6 G), the ions are optically pumped with high efficiency (∼97%) into the mJ=+3/2 Zeeman substate of the long-lived metastable 3d2D3/2-level of 40Ca+ (lifetime: ∼1s) using a beam making an angle of 45∘ with respect to the z axis and having a proper elliptical polarization (Fig. 1a,b).

To investigate the collective coupling of the ions in the Coulomb crystal to the TEM00 mode of the cavity, a 1.4-μs-long and 99%-mode-matched probe pulse is injected. The field strength of this pulse corresponds to an average intra-cavity photon number of about one or less at any time. The frequency of the probe light can be tuned around the 3d2D3/2–4p2P1/2 transition, and it is left-handed circularly polarized to obtain the strongest coupling with the optically pumped ions (Fig. 1b). A weak off-resonant continuous-wave laser beam at 894 nm is also injected into the cavity as a reference of the cavity fluctuations and drifts. Photons in the probe pulse and the reference beam are detected in reflection and transmission, respectively, by avalanche photodiodes with an overall efficiency of 16% after spectral and spatial filtering (Fig. 1a).

An accurate way to quantify the strength of the collective coupling is, for a series of detunings of the probe frequency ωp with respect to the 3d2D3/2–4p2P1/2 transition frequency ωa, to measure the width and resonance phase shift of the reflected probe signal when scanning the cavity resonance frequency ωc around ωp. Figure 2a, b shows results of such measurements for a crystal similar to the one shown in Fig. 1. According to the two-level atom theory2, the half-width of the probe signal κ′ (Fig. 2a) should have the following Lorentzian dependence,

where Δ=ωp−ωais the probe detuning from the atomic resonance and γ′ is the effective dipole decay rate. The corresponding cavity resonance phase shift δc′ (Fig. 2b) should vary dispersively according to

A fit to the data of Fig. 2a with gN and γ′ as free parameters results in a collective coupling rate gN=2π×(12.2±0.2) MHz and a dipole decay rate γ′=2π×(11.9±0.4) MHz. The value of gN is in very good agreement with the calculated collective coupling rate of 2π×(12.1±0.3) MHz using the effective number of ions N=536±18 deduced from the CCD image and the optical pumping efficiency into the 2D3/2 mJ=+3/2 Zeeman substate of 97%. The slightly higher value of γ′ as compared with the natural dipole decay rate γ=2π×11.15 MHz is consistent with a Doppler broadening corresponding to a temperature of the crystal of 20–30 mK. Micromotion of the ions induced by the trapping radiofrequency field25,27 could potentially contribute to the observed broadening. However, both simulations of Coulomb crystals in linear ion traps27 and observed identical coupling rates for the TEM00 and TEM01,10 modes do not suggest measurable effects. A complementary method for quantifying the coupling strength is based on locking the cavity resonance to the atomic resonance (ωc=ωa) and scanning the probe field frequency ωp around the common cavity–atom resonance. This leads to the observation of the so-called single-photon Rabi splitting arising from the resonantly coupled crystal–cavity system23. From the measured Rabi splitting in the data presented in Fig. 2c, a collective coupling rate of 2π×(12.2±0.2) MHz is deduced, consistent with the previous results presented in Fig. 2a,b.

Figure 2: Collective coupling measurements.
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For all of the data presented, a crystal similar to that of Fig. 1c with 520±18 ions effectively interacting with the fundamental TEM00 cavity mode was used. a, Measured half-widths of the probe signal, κ′, when scanning the cavity resonance frequency, ωc, for a series of detunings of the probe from atomic resonance, Δ. The inset shows a sample set of 100 averaged scans of the cavity resonance frequency at a given frequency of the probe pulse (Δ=10 MHz). The widths 2κ′ obtained from a Lorentzian fit of five of these data sets are averaged to give one point in the absorption curve (orange solid circle). b, The corresponding shift of the cavity resonance, δ′c, as a function of Δ. The solid lines are fits to the theoretical absorption and dispersion curves (see the text for details). In a and b, the cavity resonance frequency is scanned over 1.3 GHz around the atomic resonance, ωa, at a rate of 30 Hz. During the frequency scan, the cooling, optical pumping and probing sequence is repeated at a rate of 50 kHz, with a 1.4 μs probe detection time. c, Measured probe reflection spectra when locking the cavity resonance to the atomic resonance (ωc=ωa) and scanning the probe field frequency ωp around the common cavity–atom resonance. The red data points (squares) are obtained with an empty cavity, whereas the blue data points (circles) are obtained with an ion Coulomb crystal present. The solid lines are theoretical fits23. For each detuning of the probe from the common cavity and atomic resonance (ωc=ωa), one data point is the average of 20,000 sequences. The error bars are ±1 s.d.

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The effective number of ions N in the cavity mode volume can be controlled with high accuracy by varying either the crystal shape or density through changes of the trapping potentials. An example of the observed collective coupling effect dependence on N is presented in Fig. 3, where the cooperativity parameter C=gN2/2κ γ′ is plotted as a function of N. In accordance with the expression for gN, the cooperativity scales linearly with N. For the largest number of ions in the cavity mode, N∼1,600, which corresponds to a Coulomb crystal length of 3 mm and an ion density of 6×108 cm−3, one obtains a cooperativity of 8, or an equivalent effective optical depth of 16. This shows that the collective strong-coupling regime can be entered with large ion Coulomb crystals, and the cooperativity supersedes previously obtained values with ions by at least an order of magnitude3,21,22. Furthermore, the measured stable coupling strength for a single crystal over two hours presented in the inset of Fig. 3 highlights the solid-state nature of Coulomb crystals.

Figure 3: Collective coupling dependence on the number of ions.
figure 3

Cooperativity, C, versus the effective number of ions, N. The blue data points (circles) represent results obtained when applying left-handed polarized probe light, resonantly interacting with the ions, whereas the red data points (squares) are obtained with a right-handed circularly polarized probe where no coupling to the ions is expected to be observed. The solid lines are linear fits. The dashed horizontal line indicates the region above which collective strong coupling is achieved (gN>κ,γ). The inset shows the coupling rate for a single crystal over 2 h. A constant coupling fit to the data (solid blue line) results in C=2.04±0.03. All vertical error bars: ±1 s.d., horizontal error bars: see the Methods section.

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To evaluate the prospect of using ion Coulomb crystals for quantum storage, the temporal stability of collective Zeeman substate coherences in the 3d2D3/2-level is measured. In these experiments, the optical pumping light beam is propagating along the y axis and it is polarized linearly along the z axis and the bias magnetic field has components both along the x axis and the z axis B=(Bx=0.15 G, By=0.0 G, Bz=0.15 G). After the optical pumping preparation, this bias field configuration gives rise to an induced coherent Larmor precession between the Zeeman substates of the 3d2D3/2-level with respect to the quantization axis defined by the propagation direction of the probe light (z axis). This coherent dynamics is probed by varying the delay between the end of the optical pumping period and the injection of the probe pulse. The decay of the resulting coherent oscillations in the cooperativity presented in Fig. 4 leads to a Zeeman substate coherence time τE=1.7−0.8+100 ms or τG=0.5−0.2+0.6 ms, assuming an exponential or Gaussian decay, respectively (see the Methods section). This value, obtained without any advanced shielding against stray magnetic fields, is already at the level of the best reported coherence times for single ions in equivalent magnetic-field-sensitive states28.

Figure 4: Collective Zeeman substate coherence time measurements.
figure 4

a, The blue points represent the relative change in the cooperativity, , as a function of the delay time τ between optical pumping and probing, in the presence of a magnetic bias field having components both along the x axis and the z axis B=(Bx=0.15 G, By=0.0 G, Bz=0.15 G). is the mean cooperativity averaged over one oscillation period of the Larmor precession between the four Zeeman substates of the 3d2D3/2- level. A fit to the results (solid line) corresponds to a coherence time of τE=1.7−0.8+100 ms or τG=0.5−0.2+0.6 ms (see the Methods section). b, For comparison, the cooperativity values (normalized to the mean value of all points, Cm) for a bias field oriented along the z axis (Bz=0.15 G) are presented as well (red points). In all measurements, the effective number of ions interacting with the cavity field was N=570±20. Error bars are ±1 s.d.

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The effective optical depth of 16 is comparable to those typically used in light storage experiments with neutral atoms18, and it is large enough for quantum state storage and retrieval with potential fidelities >90% (ref. 29). By conducting similar CQED experiments using two-component Coulomb crystals30, it should be feasible in the near future to increase the optical depth by at least a factor of two, partly by confining more ions within the TEM00 mode volume, and partly by forcing the long-range ordered ions in the central crystal component30 to ‘line-up’ with the anti-nodes of the standing-wave cavity field. Possibilities for multi-mode light storage can also be investigated. A further interesting aspect of CQED experiments with Coulomb crystals yet to be explored is the extent to which these observed collective effects may be used to reveal information about the crystals, such as structural phase transitions. Finally, CQED experiments with Coulomb crystals might provide insight into light-induced mechanical effects on more traditional solids such as micro-mechanical oscillators20.

Methods

Determination of the effective number of ions.

For an ensemble of Ntot ions, one can define the effective number of ions, N, interacting with a TEM mode of the cavity through the expression for the collective coupling strength gN≡g N1/2 with . Here, ri is the position of the ith ion and ψ(r) is the standing-wave mode function. For the considered TEM00 mode, one finds

where w0 is the waist, k is the wave number, w(z)=w0(1+z2/z02)1/2, R(z)=z+z02/z and z0=k w02/2. As, under the current experimental conditions, the positions of the individual ions and the phase of the cavity field are not correlated, N can be well approximated by , where VCoul is the volume of the Coulomb crystal and ρ0 is the uniform density of ions in the Coulomb crystal31. For the trap used here26 ρ0=(6.03±0.08)×103×Urf2[V2] cm−3, and in the experiments ρ0 was varied between 1×108 cm−3 and 6×108 cm−3. The Coulomb crystal volume is determined from the projection fluorescence images and the known rotational symmetry of the crystals. With the typically low-aspect ratio crystals used in the experiments, the total relative uncertainty in N is 2–5%.

Coherence time measurements.

Given the magnetic field configuration and the optical pumping preparation, the collective coupling of the crystal with the circularly polarized probe after a time τ is dependent on the population in the mJ=+3/2 and mJ=+1/2 Zeeman substates of the 3d2D3/2-level. The cooperativity C(τ) is consequently expected to vary as a+bcos(ωLτ)+ccos(2ωLτ), where the Larmor frequency is given by ωL=(ωx2+ωz2)1/2, with ωx and ωz being the Larmor frequencies induced by the Bx and Bz magnetic field components, respectively, and the parameters a,b,c are functions of Bx, Bz and the optical pumping efficiency. The coherence time is evaluated by fitting the measured data to the above expression with all of the oscillating terms multiplied by a decaying function being either exponential, expected if the decoherence processes give rise to a homogeneous broadening of the energy levels, or Gaussian, obtained for inhomogeneous broadening decoherence processes. Owing to current technical limitations related to the data acquisition at long probe pulse delays, it has not been possible to discriminate between the two types of process.